[Grundlehren der mathematischen Wissenschaften] Boundary Behaviour of Conformal Maps Volume 299 ||

Grundlehren der mathematischen Wissenschaften 299 ASeries ofComprehensive Studies in Mathematics

Editors

M. Artin S. S. Chern 1. Coates 1. M. Fröhlich H. Hironaka F. Hirzebruch L. Hörmander C. C. Moore 1. K. Moser M. Nagata W. Schmidt D. S. Soott Ya. G. Sinai 1. Tits M. Waldschmidt S.Watanabe

Managing Editors

M. Berger B. Eckmann S. R. S. Varadhan

eh. Pommerenke

Boundary Behaviour of Conformal Maps

With 76 Figures

Springer-Verlag Berlin Heidelberg GmbH

Christian Pommerenke Fachbereich Mathematik Technische Universität 1000 Berlin 12, FRG

Mathematics Subject Classification (1991): 30Cxx, 30D40, 30D45, 30D50, 28A78

ISBN 978-3-642-08129-3

Library of Congress Cataloging-in-Publication Data Pommerenke, Christian. Boundary behaviour of conformal maps ICh. Pommerenke. p. cm. - (Grundlehren der mathematischen Wissenschaften; 299) Includes bibliographical references and indexes. ISBN 978-3-642-08129-3 ISBN 978-3-662-02770-7 (eBook) DOI 10.1007/978-3-662-02770-7 1. Conformal mapping. 2. Boundary value problems. I. Title 11. Series. QA360.P66 1992 515'.9-dc20 92-10365 CIP

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© Springer-Verlag Berlin Heidelberg 1992

Originally published by Springer-Verlag Berlin Heidelberg New York in 1992 Softcover reprint of the hardcover 1st edition 1992

Typesetting: Data conversion by Springer-Verlag 41/3140 -5 4 3 21 0 Printed on acid-free paper

Preface

We study the boundary behaviour of a conformal map of the unit disk onto an arbitrary simply connected plane domain. A principal aim of the theory is to obtain a one-to-one correspondence between analytic properties of the function and geometrie properties of the domain.

In the classical applications of conformal mapping, the domain is bounded by a piecewise smooth curve. In many recent applications however, the domain has a very bad boundary. It may have nowhere a tangent as is the case for Julia sets. Then the conformal map has many unexpected properties, for instance almost all the boundary is mapped onto almost nothing and vice versa.

The book is meant for two groups of users.

(1) Graduate students and others who, at various levels, want to learn about conformal mapping. Most sections contain exercises to test the understanding. They tend to be fairly simple and only a few contain new material. Prerequisites are general real and complex analyis including the basic facts about conformal mapping (e.g. AhI66a).

(2) Non-experts who want to get an idea of a particular aspect of conformal mapping in order to find something useful for their work. Most chapters therefore begin with an overview that states some key results avoiding technicalities.

The book is not meant as an exhaustive survey of conformal mapping. Several important aspects had to be omitted, e.g. numerical methods (see e.g. Gai64, Gai83, Hen86, Tre86, SchLa91), probabilistic methods and multiply connected domains, also the new approach of Thurston, Rodin and Sullivan via circle packings (see e.g. RoSu87, He91).

Several principles guided the selection of material, first of all the importance for the theory and next its applicability (e.g. Chapter 3). If there is a good modern treatment of a topic then the book tries to select aspects that are of particular importance for conformal mapping or are as yet not covered in books; see e.g. Chapters 5, 7 and 9. On the other hand, many of the results in Chapters 8, 10 and 11 can be found only in research papers. It is clear that the author's bias played a major part in the choice of subjects.

There are references to further (mainly more recent) papers that contain additional material. But this is only a selection of the extensive literature.

VI Preface

Many more results and references can be found e.g. in the books of GattegnoOstrowski, Lelong-Ferrand, CaratModory and Golusin.

The figures are sketches to illustrate the concepts and proofs. They do not represent exact conformal maps.

I want to thank many mathematicians for their generous help. Above all I am grateful to Jochen Becker and Steifen Rohde who read the manuscript, eliminated a lot of errors and suggested many improvements. Jim Langley kindly advised me on language problems.

I want to thank H. Schiemanowski for typing the manuscript and its many changes and U. Graeber for drawing the figures.

My gratitude also goes to the Technical University Berlin and the Centre de Recerca Matematica in Barcelona for making it possible for me to write this book.

Berlin, December 1991 Christian Pommerenke

Contents

Chapter 1. Some Basic Facts ...................................... 1

1.1 Sets and Curves ................................................. 1 1.2 Conformal Maps ................................................. 4 1.3 The Koebe Distortion Theorem .................................. 8 1.4 Sequences of Conformal Maps ................................... 13 1.5 Some Univalence Criteria ........................................ 15

Chapter 2. Continuity and Prime Ends .......................... 18

2.1 An Overview .................................................... 18 2.2 Local Connection................................................ 19 2.3 Cut Points and Jordan Domains ................................. 23 2.4 Crosscuts and Prime Ends ....................................... 27 2.5 Limits and Cluster Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 32 2.6 Countability Theorems.......................................... 36

Chapter 3. Smoothness and Corners ............................. 41

3.1 Introduction ..................................................... 41 3.2 Smooth Jordan Curves .......................................... 43 3.3 The Continuity of Derivatives .................................... 46 3.4 Tangents and Corners ........................................... 51 3.5 Regulated Domains .............................................. 59 3.6 Starlike and Close-to-Convex Domains ........................... 65

Chapter 4. Distortion .............................................. 71

4.1 An Overview .................................................... 71 4.2 Bloch Functions ................................................. 72 4.3 The Angular Derivative .......................................... 79 4.4 Harmonie Measure .............................................. 84 4.5 Length Distortion ............................................... 87 4.6 The Hyperbolic Metrie .......................................... 90

VIII Contents

Chapter 5. Quasidisks 94

5.1 An Overview ................................................... 94 5.2 John Domains .................................................. 96 5.3 Linearly Connected Domains ................................... 103 5.4 Quasidisks and Quasisymmetric Functions ...................... 107 5.5 Quasiconformal Extension ...................................... 112 5.6 Analytic Families ofInjections .................................. 120

Chapter 6. Linear Measure ...................................... 126

6.1 An Overview ................................................... 126 6.2 Linear Measure and Measurability .............................. 128 6.3 Rectifiable Curves .............................................. 134 6.4 Plessner's Theorem and Twisting ............................... 139 6.5 Sectorial Accessibility .......................................... 144 6.6 Jordan Curves and Harmonie Measure .......................... 151

Chapter 7. Smirnov and Lavrentiev Domains .................. 155

7.1 An Overview .................................................... 155 7.2 Integrals of Bloch Functions .................................... 156 7.3 Smirnov Domains and Ahlfors-Regularity ....................... 159 7.4 Lavrentiev Domains ............................................ 163 7.5 The Muckenhoupt Conditions and BMOA ...................... 168

Chapter 8. Integral Means ....................................... 173

8.1 An Overview ................................................... 173 8.2 Univalent Functions ............................................ 174 8.3 Derivatives of Univalent Functions .............................. 176 8.4 Coefficient Problems ............................................ 183 8.5 The Growth of Bloch Functions ................................. 185 8.6 Lacunary Series ................................................ 188

Chapter 9. Curve Families and Capacity ....................... 195

9.1 An Overview ................................................... 195 9.2 The Module of a Curve Family ................................. 196 9.3 Capacity and Green's Function ................................. 203 9.4 PHuger's Theorem .............................................. 210 9.5 Applications to Conformal Mapping ............................ 215

Chapter 10. Hausdorff Measure ................................. 222

10.1 An Overview ................................. ................... 222 10.2 Hausdorff Measures ............................................ 223

Contents IX

10.3 Lower Bounds for Compression ................................. 229 10.4 Zygmund Measures and the Angular Derivative ................. 235 10.5 The Size of the Boundary ...................................... 240

Chapter 11. Local Boundary Behaviour ........................ 245

11.1 An Overview ................................................... 245 11.2 Asymptotically Conformal Curves and Bo ....................... 246 11.3 The Visser-Ostrowski Quotient ................................. 251 11.4 Module and Conformality ...................................... 256 11.5 The Ahlfors Integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

References ......................................................... 269

Author Index...................................................... 291

Subject Index ..................................................... 295

Chapter 1. Some Basic Facts

1.1 Sets and Curves

1. Throughout the book we shall use the following notation:

(1)

D(a,r) = {z E C: Iz - al < r}

][J) = {z E C : Izl < I} = D(O, 1)

'f = {z E C : I z I = I} = ß][J)

C=CU{oo}

][J)* = {z E C: Izl > I} U {oo}

(a E C, r > 0),

(unit disk) ,

(unit circle) ,

(Riemann sphere) ,

(exterior of unit circle) .

The spherical metric (or chordal metric) in C is defined by

d# (z w) = I z - w I for z, w E C , J(1 + IzI 2 )(1 + Iw1 2 )

and d#(z, 00) = 1/ Jl + Iz1 2, d#(oo, oo) = O. It is invariant under rotations of the sphere, the Möbius transformations

(2) T(Z) = (az-b)j(bz+a) , a,bEC, laI 2 +lbI 2 >O.

We shall use this spherical metric when we deal with sets in C. Thus diam# A will denote the spherical diameter of A. When we deal with sets in C we will try to avoid the awkward spherical metric and use the euclidean metric Iz-wl. Thus diam A will denote the euclidean diameter. The spherical and euclidean metrics are equivalent on bounded sets in C.

2. The set A c C is called connected if there is no partition

A = Al U A2 with Al #- 0, A2 #- 0, Al n A2 = Al n A 2 = 0.

If a connected set A intersects an arbitrary set E and also its complement C\E then it intersects its boundary ßE. The continuous image of a connected set is connected.

A closed set is connected if and only if it cannot be written as the union of two disjoint non-empty closed sets. A compact connected set with more than one point is called a continuum. It has uncountably many points.

A connected open set G is called a domain. If G is open the following three conditions are equivalent:

2 Chapter 1. Some Basic Facts

(i) G is a domain; (ii) G eannot be written as the union of two disjoint non-empty open sets;

(iii) any two points in G ean be eonneeted by a polygonal are in G.

Every open set G ean be deeomposed as

(3) G = U Gk , Gk disjoint domains, OGk C oG. k

The eountably many domains Gk are ealled the components of G.

3. A curve C is given by a parametric representation

(4) C : 'P(t) , a:::; t :::; ß with 'P eontinuous on [a, ßl.

We write C C E if 'P(t) E E for all t. It is ealled closed if 'P(a) = 'P(ß). A closed eurve ean also be parametrized as

(5) C : 'ljJ( () , (E 1I' with 'ljJ eontinuous on 1I'.

We eall C a Jordan are if (4) holds for an injeetive (= one-to-one) function 'P, and a Jordan curve if (5) holds with an injective 'ljJ. Thus a Jordan eurve is closed whereas a Jordan are has distinct endpoints. An open Jordan are has a parametrization

(6) C: 'P(t) , a < t < ß with'P continuous and injective on (a,ß).

Jordan Curve Theorem. If J is a Jordan curve in<c then <C \ J has exaetly two eomponents Go and GI, and these satisfy oGo = oGl = J.

A domain G bounded by a Jordan eurve J is ealled a Jordan domain. If Je C the bounded component of C \ J will be called the inner domain of J.

We say that two points in <C are separated by the closed set A if they He in different eomponents of iE \ A. Thus they are not separated by A if it is possible to eonnect them by a curve without meeting A. The following result is very useful for giving rigorous proofs of "obvious" facts in plane topology (New64, p.llO; Pom75, p.31).

Janiszewski's Theorem. Let A and B be closed sets in iE such that An B is eonneeted. If two points are neither separated by A nor by B, then they are not separated by A U B.

4. Let C: 'P(t), a :::; t :::; ß be a curve in Co Its length A(C) is defined by

n

(7) A(C) = sup L 1'P(tv ) - 'P(tv-l)1 1'=1

where the supremum is taken over all partitions a = to < h < ... < tn = ß and all n E N. We call C reetifiable if A( C) < 00. It is clear from (7) that

1.1 Sets and Curves 3

---1', I ,

I \ / I

/ I " , , I _' I

... ~..-" ..-

I

.... - -- ------------Fig. 1.1. Janiszewski's theorem

(8) diamC :::; A(C).

If ep is piecewise continuously differentiable then

(9) A(C) = J: lep'(t)1 dt.

In the case of a closed curve C : 'I/J«(), ( E '][' with piecewise continuously differentiable 'I/J we can write

(10) A(C) = i 1'I/J'«()lld(l·

If Cl. C2 , • •• are disjoint curves we define A(Uk Ck) = L:k A( Ck), in particular A(0) = O. The concept of length and its generalization, linear measure, will be of great importance.

Now let G be an open set in C and let C be a hal/open curve in G given by

(11) C: ep(t) , a:::: t < ß with ep continuous in [a, ß) .

Its length is

(12)

We say that C ends at b if

(13) b = lim ep(t) E G c iC t-+ß

exists. Then C = Cu {b} is a curve in G U {b}. For simplicity we shall refer to C as a curve in Gending at b.

Proposition 1.1. 1/ C is a hal/open curve in G and i/ A(C) < 00 then C ends at a point b E G.

ProoJ. Let a < T < T' < ß. Then

lep(T')-ep(T)I:::;A(Cr,)-A(Cr)-tO as T,T'-tß

by (7) and (12). It follows that the limit (13) exists. 0


1.2 Conformal Maps

1. We say that f maps the domain H C C conformally onto G C C if

(i) fis meromorphic in Hj (ii) fis injective (one-to-one), that is z, z' EH, z =1= z' :::} f(z) =1= f(z')j (iii) f(H) = G.

We say that f maps H conformally into G if (iii) is replaced by f(H) c G. A univalent function is the same as a conformal map.

Note that we define conformal maps only for (connected) open sets. Here are some elementary propertiesj see e.g. Ah166a.

(a) The inverse is also a conformal map. (b) A conformal map is a homeomorphism, Le. a continuous injective map

with continuous inverse. (c) Conformal maps are locally univalent, Le. the derivative does not vanish

and there are only simple poles. ( d) Angles between curves including their orientation are preserved by con

formal maps. (e) The conformal image of a (measurable) subset A has the area

areaf(A) = J L II'(zW dxdy.

In particular, if f(z) = ao + alZ + a2z2 + ... is a conformal map of JI} into C then

00

areaf(JI}) = 11" L nlan l2 .

n=l

Reßection on circles in C, Le. ordinary circles or straight lines, is a powerful tool for analytic continuation (AhI66a, p. 171).

Reflection Principle. Let H, G be domains and H*, G* their reftections on the circles C, B in C and suppose that H n C = 0, G n B = 0. 1f f is a conformal map of H onto G such that

f(z) ---+ B as z ---+ C, zEH

then f can be extended to a conlormal map 01 the set 01 interior points 01 HuH* u (Cn8H).

2. The domain G in Cis called simply connected if the closed set C \ Gis connected.

Riemann Mapping Theorem. Let G ~ C be a simply connected domain and let Wo E G, 0 :::; a < 211". Then there is a unique conlormal map 01 JI} onto G such that 1(0) = Wo and arg 1'(0) = a.

1.2 Conformal Maps 5

(

----~------~----B

Fig. 1.2. The reflection principle

The Riemann mapping theorem is one of the most important results of complex analysis. It implies e.g. that any simply connected domain G in C is homeomorphic to C and thus that any elosed curve in G is homotopic to a point.

If G is a simply connected domain with 00 E G that has more than one boundary point, then the transformation w = 1/(w - a), a f/. G, leads to a domain G with 0 E G ~ Co Hence there is a conformal map i of ][} onto G with i(0) = O. Thus f = a + I/i maps ][} onto G such that f(O) = 00. If 00 E G it is often more convenient to consider the exterior ][}* of the unit cirele instead of][}. Then we obtain a conformal map

(1)

of][}* onto G with g( (0) = 00. It is uniquely determined by arg b. The Riemann mapping theorem has no analogue in higher dimensions: A

theorem of Liouville (see e.g. Nev60, Geh62) states that every conformal map of a domain in !Rn with n > 2 is a Möbius transformation.

3. The Möbius transformations

() _az+b rz---, cz+d

a,b,c,d E C, ad-bc=lO

are the only conformal maps of C. They form a group with respect to composition that we denote by Möb. Möbius transformations map cireles in C onto cireles in C where a line is considered as a cirele through 00.

The conformal maps of ][} onto ][} are of the form

(2) () cz+zo r z = ,

1 + zocz Izol < 1, Icl = 1.

They form a subgroup Möb(][}) so that Möb(][}) = {r E Möb : r(][}) = ][}}. It is easy to see that

(3) (1 -lzI2 )lr'(z)1 = l-lr(zW for z E][}, rE Möb(][}).

6 Chapter 1. Sorne Basic Facts

We shall often eonsider h = fOT where f is analytie in lDl and T is given by (2). Then h(O) = f(zo) and

(4) (1 -lzI2)lh'(z)1 = (1 -IT(ZW)If'(T(Z))1 for z E lDl.

4. The non-euclidean metric (hyperbolie metrie) is defined by

(5) for Zt, Z2 E lDl

where the minimum is taken over all eurves C in lDl from Zl to Z2j it is attained for the non-euclidean segment S from Zl to Z2, that is the are of the circle through Zl and Z2 orthogonal to 1I'. A eonsequenee of (5) is the triangle inequality

and, together with (3), the invarianee property

(7)

Transforming one point to 0 we easily see that

(8) for Zl, Z2 E lDl.

It follows that

(9) = = tanh>'. I Zl - Z2 I e2A - 1 1 - ZlZ2 e2A + 1

Often the non-euelidean metric is defined with a factor 2 in (5).

5. A Stolz angle at ( E 1I' is of the form

(10) L1 = {z E lDl : I arg (1 - "( Z ) I < 0:, I Z - (I < p} (0 < 0: < i, p < 2 eos 0: ) .

The exaet shape is not relevant. The important fact is that all points of L1 have bounded non-euelidean distanee from the radius [0, (].

Now let f be a function from lDl to C. There are two important kinds of limits. We say that f has the angular limit a E C at ( E 1I' if

(11) f(z) ~ a as Z ~ (, Z E L1

for eaeh Stolz angle L1 at (; the opening angle 20: of L1 ean be any number < 1r. We say that f has the unrestricted limit a E C at ( if

(12) f(z)~a as Z~(, zElDlj

if we now define f(() = athen f beeomes continuous at ( as a function in lDlu {O.

m

--~~.-o-

\ \ \ \

1.2 Conformal Maps 7

....

Fig. 1.3. The non-euclidean segment S between Zl, Z2 E ][} and a Stolz angle Ll at (E ']['

We shall use the symbol f(() to denote the angular limit whenever it exists. Its existence does not imply that f has an unrestricted limit. For example,

(13) fo(z) = exp[-(l + z)j(l- z)] (z E][})

has the angular limit 0 at 1. But if z tends to 1 on a circle tangential to '][' then Re[(l + z)j(l- z)] and thus Ifo(z)1 are constant.

As we shall see, most of the functions we are going to encounter have angular limits at many points but unrestricted limits only in more special circumstances.

6. There are two useful differential invariants for meromorphic functions. The spherical derivative of f is defined by

(14) f#(z) = 1!,(z)lj(l + If(zW)·

It is invariant under rotations of the sphere, see (1.1.2). If f is locally univalent, then the Schwarzian derivative

(15)

is analytic. It satisfies

(16)

d f"(z) 1 (f"(Z))2 S,(z) = dz f'(z) - "2 f'(z)

and is thus Möbius-invariant, that is SO'o, = S, for 0" E Möb. Furthermore (3) and (16) imply that, if r E Möb(][}),

(17) (1-lzI2)2IS,0.,.(z)1 = (1-lr(zW)2 IS,(r(z))1 (z E][}).


1.3 The Koebe Distortion Theorem

The dass S consists of all functions

(1)

analytic and univalent in ][». Many formulas take their nicest form with this normalization 1(0) = 0, 1'(0) = 1.

The dass E consists of all functions

(2)

univalent in ][»*. If 1 E S then

belongs to E and omits o. Conversely if 9 E E and g( () f:. 0 for ( E ][»* then

(4) I(z) = l/g(Z-I) = Z - boz2 + (b~ - bl )z3 +... (Izl < 1)

belongs to S. It is not difficult to show (see e.g. Dur83, p.29) that

(5) area (C \ g(][»*)) = 7r (1 - ~ n 1bnl2) for gE E.

This implies the area theorem

00

(6) IbI12~LnlbnI2~1 for gEEj n=1

equality holds if and only if g(() == (+ bo + bl(-1 with Ibil = 1. If gE E and g(() f:. w for ( E ][»* then

(7) h(() = Jg((2) - W = (+ ~(bo - W)(-I + ...

is an odd function in E. Hence it follows from (6) that

(8) Iw-bol~2 for gEE, w\tg(][»*)

and thus from (3) that (Bie16)

(9)

Equality holds for the Koebe fu,nction

00

(10) lo(z) = (1 ~ z)2 = ; nzn.

The Bieberbach conjecture that

1.3 The Koebe Distortion Theorem 9

(11) lanl :::; n for I ES, n = 2,3, ...

was proved by de Branges (deB85) after many partial results, see e.g. Löw 23, GaSch55a, Hay58a, Ped68, PeSch72. The study of this conjecture led to the development of a great number of different and deep methods that have solved many other problems; see e.g. the books Gol57, Hay58b, Jen65, Pom75, Dur83 and also Section 8.4.

If I is analytie and univalent in lD> and if Zo E lD> then the Koebe translorm

I({:~oz) -/(zo) (1 2 !"(zo) _) 2 (12) h(z) = (1 _ Izol2) !'(zo) = z + "2 (1 - Izol ) !'(zo) - Zo z + ...

belongs to S. This fact makes it possible to transfer information at 0 to information at any point of 1D>. Thus we obtain from (9):

Proposition 1.2. 11 I maps lD> conlormally into ethen

(13) 1 (1 -lzI2) j:f:? - 2z1 :::; 4 lor z EID>.

Equality holds for the Koebe function. Integrating this inequality twiee (see e.g. Dur83, p.32) we obtain the famous Koebe distortion theorem.

Theorem 1.3. 11 I maps lD> conlormally into C and il z E lD> then

( ) 1 '( )1 1 - Izl I'()I I'()I 1 + Izl 15 I 0 (1 + Izl)3 :::; I z :::; I 0 (1 _ Izl)3 .

The distance to the boundary is an important geometrie quantity. We define

(16) df(z) = dist(f(z),81(1D>)) for z E 1D>.

Corollary 1.4. 11 I maps lD> conlormally into ethen

(17) ~(1-lzI2)1!'(z)1 :::; df(z) :::; (1-lzI2)1!,(z)1 lor z E 1D>.

Proof. Let h again be the Koebe transform. Since h ES we obtain from (14) and the minimum principle that

~ :::; liminflh(Z) I:::; Ih'(O)1 = 1 4 Izl--+1 Z


and (17) (with Zo instead of z) follows because

dj(zo) = dist(8G, I(zo)) = liminf I/«() - l(zo)l· 1'1-+1

D

We now turn to an invariant consequence of the Koebe distortion theorem in terms of the non-euclidean distance >. given by (1.2.8).

Corollary 1.5. 11 1 is a conlormal map o/][)) into C and il Zo, ZI E ][)) then

() tanh>.(zo,ZI) I/(zd - l(zo)1 [4>.()] 18 4 s (1-lzoI2)1f'(zo)1 S exp ZO,ZI,

(19) 1!,(zo)1 exp[-6>'(zo, ZI)] S I!,(zdl s 1!,(zo)1 exp[6>'(zo, zd]·

Prool. We consider the Koebe transform (12) with z = (ZI - zo)/(1 - zozd and thus ZI = (z + zo)/(1 + zoz). We obtain from (14) that

l:.l < I/(zd - l(zo)1 _ < jzl < (1 + Izl) 2 4 - (1-lzoI 2)1/'(zo)1 -lh(z)1 - (1-lzI)2 - l-lzl

and (18) follows from (1.2.8). Furthermore we see from (15) that

I f'(zd I = 11 + zozI2Ih'(z)1 S (1 + Izl)3 I'(zo) 1 - Izi

whieh implies the upper estimate (19); the lower one follows from symmetry. D

The following form is more geometrie and perhaps easier to apply.

Corollary 1.6. Let a, b, c be positive constants and let 0 < Izol = 1 - 0 < 1. 11

(20) Osl-b8slzls1-ao,largz-argzolsco

and il 1 maps ][)) conlormally into ethen

(21) I/(z) - l(zo)1 s M 10I!,(zo)l,

(22)

where the constants MI and M 2 depend only on a, b, c.

Prool (see Fig. 1.4). The circular arc from Zo to Z2 has length < co. Hence we obtain from (1.2.5) that

1.3 The Koebe Distortion Theorem 11

I , , , , , , ,. , , ,,"/ ''*(

T

Fig.1.4. The set of values described by (20) is shaded. The angle at 0 is 2c8

cti c A(ZO, Z2) < 1 _ (1 _ ti)2 = 2 _ ti < c.

Furthermore

I Z3 - Z1 I (1 - ati) - (1 - bti) b - a b - a 1 - Z1Z3 = 1 - (1 - ati)(1 - bti) = b + a - abti :::; -b- .

Hence we obtain from (1.2.6) and (1.2.8) that

1 2b- a A(ZO, z) :::; A(ZO, Z2) + A(Zl, Z3) :::; c + 2 log -a-'

Thus (21) and (22) follow from Corollary 1.5. D

We show now that the average growth of fis much lower than (1 - r)-2; see e.g. Hay58b or Dur83 for more precise results.

Theorem 1. 7. Let f map JD) conformally into C. Then

(23) f(() = lim f(r() i- 00 r-+1

exists for almost all ( E '][' and

(24)

It follows from this result (Pra27) that f belongs to the Hardy space H 2/ 5 ;

in fact fE HP holds for every p < 1/2 (see Theorem 8.2) but not always for p = 1/2 as the Koebe function (10) shows.

Proof. We may assume that fES. The function

(25) g(z) = f(Z5)1/5 = Z (f~~5)) 1/5 = ~ bnzn


is analytic and univalent in TI)). It follows from Parseval's formula that

for 0 :::; r < 1. Since n(l- r )rn - 1 :::; 1 and since 9 is univalent, this is bounded by

(26)

00 1 L nlbn l2rn - 1 = - area {g(z) : Izl :::; vr} 1 7rr

:::; ! max Ig(zW :::; (1 - r 5/ 2)-4/5 :::; (1 _ r)-4/5 r Izl:=;v'r

because of (25) and (14). Hence we conclude that

(27) ~ r1 r27r (I_r)9/ 10 19'(reitWdtdr:::; r1(I_r)-9/ 10 dr=1O. 27r Jo Jo Jo

It follows by Schwarz's inequality that

(11 19'(reit ) I dr) 2 :::; 1 1(1 _ r)-9/10 dr 1 1(1 _ r)9/10 I9'12 dr.

If we integrate we obtain from (27) that

(28) 1 r27r ( r1 )2 27r Jo Jo 19'(reit ) I dr dt:::; 100.

Hence the inner integral is finite for almost all t and lim,.--+1 g(reit ) f:. 00 exists for these t by Proposition 1.1. Hence the limit (23) exists for almost all t by (25).

Finally we see from (25) and Parseval's formula that

the last inequality follows from (26) by integrating over [0,1]. o

Exercises 1.3

1. Let 1 map lIJJ conformally onto a bounded domain. Show that (1-lzl)lf'(z)1 --+ 0 as Izl --+ 1.

2. Let 1 be a conformal map of lIJJ into C \ {cl with 1(0) = O. Show that

41czl II(z)1 ::; (1 -lzlP for zElIJJ.

1.4 Sequences of Conformal Maps 13

3. Use the Koebe transform to deduce from (14) that

1-lzl < I zj'(z) I< l+lzl for zElDl. 1 + Izl - I(z) - 1(0) - 1 -Izl

4. If 1 maps lDl conformally into C apply the preceding exercise to show that

l1og[(f(z) - I(O))/(zj'(O))) I :::; 4Iog[(1 + Izl)/(l -Izl)]

for z E lDl and deduce that e.g.

larg I(z) - I(zo) - arg j' (zo) I :::; 8.\(zo, z) + ~2 . z - Zo

5. If 1 E S show that log[z-l/(z)] belongs to the Hardy space H 2 • (Compare the proof of Proposition 3.8.)

6. Show that la~ - a31 :::; 1 if 1 E S and deduce that the Schwarzian derivative satisfies

IS,(z)1 :::; 6(1 - Iz1 2)-2

if 1 maps lDl conformally into C.

1.4 Sequences of Conformal Maps

The usual not ion of convergence of domain sequences is not adequate for our purposes where the limit has to be a domain. Caratheodory therefore introduced a different concept.

Let Wo E C be given and let Gn be domains with Wo E Gn C Co We say that

(1) Gn --+ G as n --+ 00 with respect to Wo

in the sense of kernel convergence if

(i) either G = {wo}, or G is a domain f:. C with Wo E G such that some neighbourhood of every wEG lies in Gn for large nj

(ii) for w E aG there exist Wn E aGn such that Wn --+ was n --+ 00.

lt is clear that every subsequence also converges to G. We show now that the limit is uniquely determined. Let also Gn --+ G* as

n --+ 00 and suppose that G rt. G*. Since Wo E G n G* and since G is a domain by (i), there would exist w* E GnaG* and thus, by (ii), points Wn E aGn with Wn --+ w*. This contradicts (i) because w* E G. Thus G c G* and similarly G* cG.

Kernel convergence has surprising properties and depends on the choice of the given point Wo. As an example, if

Gn = C \ (-00, -l/n] \ [l/n, +00)

then, as n --+ 00,


{{Im W > O} if Wo = i,

Gn ----; {O} if Wo = 0, {Imw < O} if Wo = -i.

The usual definition of kernel convergence is formally different; see e.g. Gol57, p.46 or Pom75, p.28. Some special cases are given in the exercises.

Theorem 1.8. Let In map lDl conlormally onto Gn with In(O) = Wo and I~ (0) > O. 11 G = {wo} let I (z) == wo, otherwise let I map lDl conlormally onto G with 1(0) = Wo and 1'(0) > o. Then, as n ----; 00,

In ----; I locally unilormly in lDl {:} Gn ----; G with respect to wo.

This is the Caratheodory kernel theorem (Car12). It is a powerful tool for constructing conformal maps with given boundary behaviour. In Corollary 2.4 we shall give a geometrie characterization of uniform convergence.

Proof. (a) Suppose first that In ----; I locally uniformly in lDl. If I is constant then G = {wo}. Otherwise G is a domain. Let wEG and let D( w, 28) c G. Then A = 1-1 (D(w, 28)) c lDl is compact so that (In) converges uniformlyon A. Hence we can find no such that Iln(z) - l(z)1 < 8 for Z E A and n 2:: no, and it follows from RoucM's theorem that D(w, 8) c In(A) c Gn for n 2:: no. Thus (i) holds.

Now let w E BG. Then there exist Zk E lDl such that II(Zk) - wl < l/k. We can find an increasing sequence (nk) such that

We define Wn as the point of BGn nearest to In(Zk) if nk ~ n < nk+1. Then

IWn - In(Zk)1 ~ (1 -IZkI2) 11~(Zk)1 < (1 -IZkI2)II'(Zk)1 + ~ ~ 4dt(Zk) + ~

by Corollary 1.4 and (2), and since dt(zk) ~ II(Zk) - wl it follows that

1 7 IWn - wl ~ 411(Zk) - wl + k + Iln(Zk) - I(Zk)1 + II(Zk) - wl < k

again by (2). It follows that W n ----; W as n ----; 00. Hence (ii) holds. (b) Conversely let Gn ----; Gas n ----; 00. Since G =I- <C it follows from (ii) that

dist(wo, BGn ) ~ M for some constant M and thus, by (1.3.14) and (1.3.17),

4Mlzi Iln(z) - wol ~ (1 -lzlF for Z E lDl.

Hence (In) is normal. Suppose now that In ----; I(n ----; 00) does not hold. Then there is a subsequence with Inv ----; f* =I- I as v ----; 00 locally uniformly in lDl. It follows from part (a) that Gnv ----; G* = f*(lDl) as v ----; 00. Since also

1.5 Some Univalence Criteria 15

Gn " ----+ G we conclude from the uniqueness of the limit that G* = G and thus that r = f because of our normalization. This is a contradiction. 0

Exercises 1.4

1. Show that an increasing sequence of domains converges to its union unless this is = C.

2. Let (Gn ) be a decreasing sequence of domains and let H be the open kernel of its intersection. If Wo E H show that Gn converges (with respect to wo) to the component of H containing Wo.

3. Let In map JI)) conformally onto the bounded domain Gn with In(O) = Wo. If Gn -t G = I(JI))), show that

J 'f 1/~(z)12dXdY-tJ'f 1!,(z)12dxdy as n-too J1z1<r J1z1<r

holds for r < 1 but not always for r = 1.

1.5 Some U nivalence Criteria

We come now to some sufficient conditions for a given analytic function to be injective. The first criterion is of a topological nature and is related to the degree of a map; see e.g. Mil65, p. 28.

Theorem 1.9. Let f be nonconstant and analytic in the domain He C and let G be the inner domain of the Jordan curve J. If

(1) f(z) ----+ J as z ----+ öH

then f(H) = G. 1I furthermore 1 assumes in H some value 01 G only once (with multiplicity 1) then 1 is injective and H is simply connected.

We mean by (1) that if (zn) is any sequence in H whose limit points lie on öH then all Hmit points of (f(zn)) He on öG = J.

Prool. (a) We show first that öf(H) c J. If w E öf(H) (possibly w = 00)

then there exist Zn E H such that 1 (Zn) ----+ wand Zn ----+ ( for some ( E H. If ( E H then f would map a neighbourhood of ( onto a neighbourhood of w = f(() contrary to w E öf(H). Hence ( E öH and it follows from (1) that wEJ.

By the Jordan curve theorem, C \ J has a component G* with 00 E G*. If f(H) r:t. G then G* would meet I(H) and also C \ f(H) because 00 rt f(H). Hence G* would meet öl(H) c J = öG* which is impossible. Thus f(H) C G and it follows from öf(H) c J that I(H) = G.


(b) Our furt her assumption implies that the set

(2) H 1 = {z EH: !,(z) =f 0, I(ZI) =f I(z) for Zl EH, Zl =f z}

is non-empty. We claim that H 1 = H. Suppose this is false. Since H is a domain there exist Zo E H n 8H1 and therefore Zn EH \ H 1 with Zn - zo(n - 00). Since Zn (j. H 1 we can find z~ =f Zn such that I(z~) = I(zn); we may assume that z~ - Zo E H. Since I(z~) - I(zo) E G it follows from (1) that Zo E H. Hence I assumes all values near I(zo) = I(zo) in a neighbourhood V of Zo and also in a neighbourhood V' of zo; if Zo = Zo then !'(zo) = 0 and these values are assumed at least twice in V. This contradicts our assumption that Zo E 8H1 • Thus H 1 = H and I is injective in H, by (2).

Hence I is a conformal map of H onto G and in particular a homeomorphism. Since G is simply connected, it follows (New64, p. 156) that H is also simply connected. This completes the proof. 0

The following boundary principle is a consequence of Corollary 2.10 below: If I is analytic in IDl and continuous and injective on l' then I is injective also in IDl.

We turn now to analytic univalence criteria.

Proposition 1.10. Let I be analytic in the convex domain H. 11 (3) Re!,(z) > 0 lor zEH

then I is univalent in H. 11 H = IDl and I is continuous in iij then I(IDl) is a Jordan domain.

Prool. Let Zl and Z2 be distinct points in H. Since H is convex the segment [ZI, z2]lies in H. Hence it follows from (3) that

(4) Re I(Z2) - I(Zl) = r1 Re[!'(zl + (Z2 - Zl)t)] dt > 0 Z2 - Zl 10

and thus that I(Zl) =f I(Z2)' If I is continuous in iij and if Zl and Z2 are distinct points on l' then (ZI, Z2) C IDl so that I(zd =f I(Z2) by (4). 0

The Becker univalence criterion (DuShSh66, Bec72, BeP084) is much deeper.

Theorem 1.11. Let I be analytic and locally univalent in IDl. 11

(1 - Iz1 2 ) Izf"(Z) I < 1 lor Z E IDl f'(z) -

then I is univalent in IDl. The constant 1 is best possible.

The proof uses Löwner chains (see e.g. Pom75, p. 172); the univalence ultimately comes from the uniqueness theorem for ordinary differential equations.

1.5 Some Univalence Criteria 17

An alternative proof (Ahl74j Scho75, p.169) uses quasiconformal mapSj the univalence follows then from topological considerations.

The Nehari univalence criterion (Neh49) uses the Schwarzian derivative defined in (1.2.15).

Theorem 1.12. Let I be meromorphic and locally univalent in][». 11

(1-lzI2)2ISj(z)1 ~ 2 lor z E][»

then I is univalent in][». The bound 2 is best possible. 111(0) E C and 1"(0) = o then I has no pole.

The proof uses either the theory of second order linear differential equations (see e.g. Hil62 or Dur83, p.261) or again quasiconformal maps. See GePo84 or Leh86, p.91 for the final statement.

The Epstein univalence criterion (Eps87) is a common generalization of the last two criteria: Let I be meromorphic and 9 analytic in ][». If both functions are locally univalent and if

then I is univalent in ][». The original proof uses differential geometry and works in a more general context, see Pom86 for a proof using Löwner chains.

Exercises 1.5

1. Let g(z) = z+bo+ ... be analytic in {I< Izl < oe} and continuous in {I:::; Izl < oe} and let J be a Jordan curve in Co If g(1l') c J and if 9 omits some point rt. J, show that 9 maps ][»* conformally onto the outer domain of J.

2. Let ~:' nlanl :::; 1. Show that j(z) = z + ~:' anZn maps][» conformally onto a Jordan domain.

Chapter 2. Continuity and Prime Ends

2.1 An Overview

Let 1 map the unit disk ]])l eonformally onto G c C = Cu { 00 }. In this ehapter, we study the problem whether it is possible to extend 1 to some or all points ( E 11' = ß]])l by defining

(1) I(() = lim I(z) E C. z-+(

If this limit exists for all ( E 11' then 1 beeomes eontinuous in iDj if 00 E ßG then eontinuity has to be understood in the spherical metrie.

The global extension problem has a eompletely topological answer. A set is ealled locally connected if nearby points ean be joined by a eonneeted subset of small diameter.

Continuity Theorem. The function 1 has a continuous extension to ]])l U 11' il and only il ßG is locally connected.

The problem whether this extension is injeetive (= one-to-one) on iD has again a topological answerj see Section 2.3. The following theorem is often used and has many important eonsequenees.

Caratheodory Theorem. The function 1 has a continuous and injective extension to ]])l U 11' il and only il ßG is a Jordan curve.

If ßG is loeally eonneeted but not a Jordan domain then parts of ßG are run through several times. Some of the possibilities are indicated in Fig. 2.1 where points with the same letters eorrespond to eaeh other. Note that e.g. the ares a2dl and al d2 are both mapped onto the segment ad.

The situation beeomes mueh more eomplieated if ßG is not loeally eonnected. CaratModory introdueed the not ion of a prime end to deseribe this situationj see Section 2.4 and 2.5.

Prime End Theorem. The points ( on 11' correspond one-to-one to the prime ends 01 G, and the limit I(() exists il and only il the prime end has only one point.

2.2 Local Connection 19

b

--a, t + t

--+ --+

, c t d -+ 4-

a ID

+- t G ,

--+

Fig.2.1

-1.; r----------------------.1.;

a

--------------------- -==.:.---=-----===------= -1 ~---------

Fig.2.2. The point a belongs to two prime ends whereas the entire segment [0,1] belongs to a single prime end

In Section 2.6 we shall see that, rat her surprisingly, certain situations can occur only a countable number of times. For instance, there are at most countably many points on 8G that have more than two preimages on 1['.

2.2 Local Connection

We shall consider bounded domains in C; everything carries over to the general case of domains in iC if the spherical metric is used instead of the euclidean metric.

The closed set A c C is called locally connected if for every t > 0 there is 8 > 0 such that, for any two points a, b E A with la - bl < 8, we can find a continuum B with

(1) a, bEB CA, diam B < t .

The continuous image of a compact locally connected set is again compact and locally connected (New64, p.89). Hence every curve is locally connected as the continuous image of a (locally connected) segment.

20 Chapter 2. Continuity and Prime Ends

The union of finitely many compact locally connected sets is locally connected (New64, p.88). This is not generally true for infinitely many sets. A counterexample is the boundary of the domain (see Figure 2.2)

(2) G = {u + iv : lul < 1,0 < v < I} \ Cl [~, ~ + 1] . n=2 n n

The points 0 and i / n cannot be connected in 8G by any connected subset of diameter < 1 so that 8G is not locally connected.

Theorem 2.1. Let I map lIJ) conlormally onto the bounded domain G. Then the lollowing lour conditions are equivalent:

(i) I has a continuous extension to iij; (ii) 8G is a curve, that is 8G = {<p«() : ( E 'Ir} with continuous <Pi

(iii) 8G is locally connected; (iv) C \ G is locally connected.

We can rephrase (ii) ::::} (i) as follows: If it is at all possible to parametrize 8G as a closed curve (perhaps with many self-intersections) then we can use the conformal parametrization

(3)

The implication (iii) ::::} (ii) is a special case of the Hahn-Mazurkiewicz theorem (New64, p.89) that every locally connected continuum is a curvej we will not use this theorem.

The proof will be based on the following estimate (Wolff's lemma) about length distortion under conformal mappingj see Chapter 9 and e.g. Ahl73 for the systematic use of this method.

Proposition 2.2. Let h map the open set He C conlormally into D(O, R). 11 cE C and C(r) = H n {Iz - cl = r} then

(4) inf A(h(C(r))) ~ ~ lor 0< p < 1 p<r<v'P log 1/ p

so that there is a decreasing sequence rn ---+ 0 with

(5) A(h(C(rn ))) ---+ 0 as n ---+ 00.

Prool. We write l(r) = A(h(C(r))) and obtain from the Schwarz inequality that

1(r)2 = (r Ih'(Z)lldZI)2 ~ r Idzl r Ih'(zWldzl JC(r) JC(r) JC(r)

2.2 Local Connection 21

It follows that

(6) l°Ol(r)2~ ::;27r ![lh'(Z) 12 dXdY =27rareah(H)

and since h(H) c D(O, R) we conclude that

1 1 1..;P dr -log - inf l(r)2::; l(r)2- ::; 27r2 R2. 2 P p<r<..;p p r

This implies (4), and (5) is an immediate consequence. o

Proolol Theorem 2.1. (i) =} (ii). If 1 is continuous in TI) we can write BQ as the curve (3).

(ii) =} (iii). This is clear because every curve is locally connected. (iii) =} (iv). Let BQ be locally connected. For c > 0 we choose 8 with

o < 8 < c as in the definition of local connectionj see (1) with A = BQ. Let now a, b E C \ Q and la - bl < 8. The case that [a, b] n BQ = 0 is trivial. In the other case let a' and b' be the first and last points where [a, b] intersects BQ. We can find a continuum B c BQ from a' to b' with diamB< c, and [a, a'] U B U [b', b] is a continuum of diameter< 3c connecting a and bin C \ Q.

Fig.2.3

(iv) =} (i). We mayassume that 1(0) = O. Then

(7) D(O, Ro) c Q c D(O, R)

with suitable Ro < R. Let 0 < c < Ro and choose 8 with 0 < 8 < c as in (1) with A = C \ Q . Next we choose p with 0 and c = z, we therefore find r with p < r < 1/2 such that

(8) A(f(C)) <8<c, C=1D>nBD(z,r).

We shall show below that


(9) If(z) - f(z')1 < 2c.

It follows that f is uniformly continuous in lIJi and therefore that f has a continuous extension to ll). 0

Suppose now that (9) is false and furthermore that 0 n 1I' -=1= 0; the other case is simpler. Then f(O) is a curve with endpoints a,b E ßG c ce \ G by Proposition 1.1. Since la - bl < (j by (8), there is a continuum B c ce \ G with a, bEB and diam B < c; see Fig. 2.3. Hence we see from (7) and (8) that

B U f(O) c D(a,c), 0 tf- D(a,c).

Consequently f(z') (or f(z)) is not separated from 0 by BUf(O). Since f(z') and 0 are neither separated by ce \ G and since the intersection

(B U f ( 0)) n (ce \ G) = B

is connected, we conclude from Janiszewski's theorem (Section 1.1) that f(z') and f(O) = 0 are not separated by the union f(O) U (ce \ G). Hence z' and 0 are not separated by 0 U 1I' which is false because Iz - z'l 0, there exists {j > 0 independent of n such that any two points an, bn E An with lan - bnl < (j can be joined by continua B n C An of diameter < c. The above proof shows:

Proposition 2.3. Let fn map lIJi conformally onto Gn such that fn(O) = O. Suppose that

(10) D(O, R o) C Gn C D(O, R) for all n.

If ce \ Gn is uniformly locally connected then the functions fn are equicontinuous in ll).

A sequence (fn) is called equicontinuous in ll) if, for every c > 0, there is p > 0 independent of n such that

I f n (z) - f n (z') I < c for z, z' E ll) , I z - z' I < p, n E N.

The ArzeUl.-Ascoli theorem states that every uniformly bounded equicontinuous sequence contains a uniformly convergent subsequence.

The Caratheodory kernel theorem (Theorem 1.8) gives a necessary and sufficient condition for locally uniform convergence. We can supplement it by the following result about uniform convergence.

Corollary 2.4. Let fn map lIJi conformally onto Gn with fn(O) = O. Suppose that (10) holds and that ce \ Gn is uniformly locally connected. If

(11) fn(z) --> f(z) as n --> 00 for each z E lIJi

then the convergence is uniform in ll).

2.3 Cut Points and Jordan Domains 23

This is an immediate consequence of Proposition 2.3. The converse also holds (Pom75, p.283): If pointwise convergence implies uniform convergence then C \ Gn is uniformly locally connected.

Exercises 2.2

1. Let j map ]j)) conformally onto a bounded domain. Use the method of Proposition 2.2 to show that almost all radii [0, () are mapped onto curves of finite length.

2. Let G be a bounded simply connected domain with locally connected boundary. Use the Poisson integral in ]j)) to solve the Diriehlet problem: If r.p is real-valued and continuous on 8G then there is a unique harmonie function u in G that is continuous in Ci and satisfies u = r.p on BG.

3. Let j(z) = ao + alZ + ... map]j)) conformally onto a bounded domain G and write

n <Xl

Sn(Z) = L avz" , 10;' = L vlavl2. v=O v=n

Show that Sn(Z) = j(TnZ)+O(e;!2) (n --+ (0) uniformly in iD where Tn = l-en/n; even O(en log I/IOn) is true.

4. If furthermore 8G is locally connected, show that the power series of j converges uniformlyon 1I'. This is Fejer's Tauberian theorem (LaGa86, p. 65); the convergence need not be absolute even for Jordan domains (Pir62).

2.3 Cut Points and Jordan Domains

Let E be a locally connected continuum. We say that a E E is a cut point of E if E \ {al is no longer connected (see e.g. Why42). For example, a (closed) Jordan curve has no cut points whereas for a Jordan are every point except the two endpoints is a cut point.

Proposition 2.5. Let f map]j)) conformally onto the bounded domain G with locally connected boundary. Let a E aG and

A = f-l({a}) = {( E 1I': f(() = a}, m = cardA:::; 00.

Then a is a cut point if and only if m > 1, and the components of aG \ {a} are f(Ik) where Ik (k = 1, ... , m) are the ares that make up 1I' \ A.

We shall see (Corollary 2.19) that the case 3 :::; m :::; 00 can occur only for countably many points a. The set A may be uncountable but has always zero measure and even zero capacity (Beu40; see Theorem 9.19).

Proof. Since f is continuous in iD by Theorem 2.1, we can write

m

(1) aG\ {al = f(1I'\A) = U f(Ik). k=l


/,/

c _///-

f(CI!

'- // '>(

/ , / ,

Fig.2.4

I \ \ / , /

,x .... / ---

The sets f(Ik) are connected as eontinuous images of eonneeted sets. It remains to show that f(Ij ) and f(h) for j =I=- k eannot be connected within ßG \ {al.

To show this (see Fig. 2.4) eonsider a cireular are C in ][)) that eonnects the endpoints of I j . Then J = f(C) is a Jordan eurve in G U {al. By the Jordan eurve theorem, C \ J has exaetly two eomponents, eaeh bounded by J. If C' is a cireular are in ][)) from I j to I k then C and C' cross exaetly onee. Henee f(C) = J and f(C') cross exaetly onee and it follows from (1) that f(Ij ) and f(Ik ) lie in different eomponents of C\ J and thus eannot be connected within ßG \ {al. 0

We now deduee the important Caratheodory theorem (Car13a).

Theorem 2.6. Let f map][)) conformally onto the bounded domain G. Then the following three conditions are equivalent:

(i) f has a continuous injective extension to iDj (ii) ßG is a Jordan curvej (iii) ßG is locally connected and has no cut points.

Proof. The implieations (i) :::} (ii) :::} (iii) are obvious. Assurne now (iii). Then fis eontinuous in iD by Theorem 2.1 and, by Proposition 2.5, every point on ßG is attained only onee on 1I'. Sinee furthermore f(][))) = G is disjoint from f(lI') = ßG it follows that (i) holds. 0

Corollary 2.7. Let G and H be Jordan domains and let the points Zl, Z2, Z3 E ßG and Wb W2, W3 E ßH have the same eyclic order. Then there is a unique conformal map h of G onto H that satisfies

(2) h(Zj) = Wj for j = 1,2,3.

2.3 Cut Points and Jordan Domains 25

Prool. By the Riemann mapping theorem there are conformal maps land 9 of lDl onto G and H. By Theorem 2.6, they are continuous and injective on 11'. We choose r E Möb(lJ))) such that r(f-l(zj)) = g-I(Wj) for j = 1,2,3. Then h = gor 0 1-1 is a conformal map of G onto H satisfying (2). If h* is another map with the same properties then r- 1 0 g-1 0 h* 0 I is a conformal selfmap of lDl keeping the three points 1-1(zj) fixed and thus the identity. Hence h* = h.

o

Corollary 2.8. A conlor:nal mcp o/lDl onto a Jordan domain can be extended to a homeomorphism 01 C onto C.

Prool. Let I map lDl conformally onto the (bounded) Jordan domain G while f* maps lDl* = {izi > I} U {oo} conformally onto the outer domain G* of the Jordan curve ßG such that f* (00) = 00. Then land f* can be extended to bijective continuous maps of iij onto G and of iij* onto G*; the inverse maps are continuous because the sets are compact . If we define <p(() = f*-I(f(()) for ( E 11' and

(3) I(r() = f*(r<p(()) for 1 < r < 00, (E 11',

then I becomes a homeomorphism of iC = iij U lDl* onto iC = G u G* . 0

A consequence is the purely topological Schoenfties theorem.

Corollary 2.9. A bijective continuous map 01 a Jordan curve onto another can be extended to a homeomorphism 01 C onto C.

Prool. Let land 9 map lDl conformally onto the inner domains G and H of the two Jordan curves and let <p be the given bijective continuous map of ßG onto ßH. We extend I and 9 according to Corollary 2.8. Then'I/J = g-1 0 <p 0 I is a bijective continuous map of 11' onto 11'. If we define

(4) 'I/J(r() = r'I/J(() for 0 ~ r < 00, z E 11'

then go 'I/J 0 1-1 is the required homeomorphic extension of <po o

Corollary 2.10. Let G be a bounded Jordan domain and let I be non-constant and analytic in lDl. 11

(5) dist(f(z), ßG) ---t 0 as Izl ---t 1

then l(lDl) = G and I has a continuous extension to iij. There exists m = 1,2, ... such that I assumes every value 01 G exactly m times in iij.

The following univalence criterion is a special case: If I is analytic in lDl and if I(z) tends to the Jordan curve ßG as Izl ---t 1 then I maps lDl conformally onto G provided that one value in G is assumed only once.


Proof. It follows from Theorem 1.9 with H = j[)) that I(j[))) = G. Let cp map j[)) conformally onto G. We see from Theorem 2.6 that cp is continuous and injective in ~. Then 9 = cp-l 0 I is an analytic map of j[)) onto j[)) and (5) implies that Ig(z)1 --+ 1 as Izl --+ 1. The reflection principle therefore implies that 9 is a rational function with g(1I') = 1I', hence a finite Blaschke product

m

(6) rr z - Zk g(z) = C _,

1- ZkZ lei = 1, Zk E j[)) (k = 1, ... , m).

k=l

Such a function assumes in ~ every value in ~ exactly m times (counting multiplicity). Hence I = cp 0 gis continuous in ~ and assumes every value in G exactly m times. 0

The next result (Rad 23) deals with sequences of Jordan domainsj compare Corollary 2.4.

Theorem 2.11. For n = 1,2, ... let

Jn : CPn((), ( E 1I' and J: cp((), (E 1I'

be Jordan curves and let In and I map j[)) conlormally onto the inner domains 01 Jn and J such that In(O) = 1(0) and I~(O) > 0, 1'(0) > 0 lor all n. 11

(7) CPn --+ cp as n --+ 00 unilormly on 1I'

then

(8) In --+ I as n --+ 00 unilormly in TI] .

Prool. It follows easily from Theorem 1.8 that we have at least pointwise convergence, i.e. that (2.2.11) holds. In view of Corollary 2.4 it thus suffices to show that the curves Jn are uniformly locally connected.

Suppose this is false. Then there are indices n v and points av , a~ E Jn "

with lav - a~1 --+ 0 (v --+ 00) that are not contained in any continuum of diameter< co in Jn". Let av = CPn,,((v) and a~ = CPn,,((~). We mayassume that (v --+ ( and (~ --+ (' as v --+ 00. It follows from (7) that av --+ cp((), a~ --+ cp((/) and thus that cp(() = cp((/). Since J is a Jordan curve, we conclude that ( = (' and thus that (v --+ (, (~ --+ (. The function CPn" maps the closed arc of 1I' between (v and (~ onto a continuum in Jn " containing av and a~. Its diameter tends to 0 as v --+ 00 by (7), and this is a contradiction. 0

Exercises 2.3

1. Let H be a Jordan domain and A a Jordan arc from a E H to b E 8H with ACH U {b}. Show that there is a unique conformal map 1 of j[)) onto H \ A that is continuous in ~ and satisfies 1{1} = a and I{±i} = b.

2.4 Crosscuts and Prime Ends 27

2. Let G be a Jordan domain and Zl, ..• , Z4 cyclically ordered points on oG. Show that there exists a unique q > 0 and a conformal map of G onto {u + iv : 0 < u< q,O < v < I} such that Zl >--+ 0, Z2 >--+ q, Z3 >--+ q + i, Z4 >--+ i.

3. Let J be a Jordan curve and A a Jordan arc that lies inside J except for its endpoints. Show that there is a homeomorphism of C onto C that maps J u A onto Tu [-1,+1].

4. Let H and G be Jordan domains in C and let f be analytic in H and continuous in H. Show that f maps H conformally onto G if and only if f maps oH bijectively onto oG.

5. Under the assumptions of Corollary 2.10, show that f is a conformal map if and only if j'(z) f= 0 for Z E ][l

2.4 Crosscuts and Prime Ends

We now consider simply connected domains in iC = Cu { oo} and use the spherical metric. All not ions that we introduce will be invariant under rotations of the sphere. If necessary we may assume that 00 E Gj otherwise we map some interior point of G to 00 by a rotation of the sphere. This has the technical advantage that fJG is now a bounded set so that we can use the more convenient euclidean metric in our proofsj the spherical and the euclidean metric are equivalent for bounded sets.

A crosscut C of G is an open Jordan arc in G such that C = C U {a, b} with a, b E fJGj we allow that a = b. We first prove two useful topological facts.

Proposition 2.12. I/ C is a cross cut 0/ the simply connected domain G c iC then G \ C has exactly two components Go and GI and these satisfy

(1) G n fJGo = G n fJGI = C.

If G is not simply connected then there are crosscuts C connecting different boundary components and G \ Cis a single domain (New64, p.120).

Proof. Let 9 map G conformally onto ][)). Then

(2) 9 Igl . h = l-lgl = l-lgl exp(zargg)

is a non-analytic homeomorphism of G onto Co If z tends to fJG then Ig(z)1 ---t 1 and thus Ih(z)l---t +00. It follows that J = h(C)U{oo} is a Jordan curve and, by the Jordan curve theorem, there are exactly two components Ho and H I

of iC \ J which satisfy fJHo = fJH1 = J. Then

(3)


are disjoint domains with G n oG; = h-l(C n oH;) = h-l(C n J) = C and Go U G l = h-l(C \ J) = G \ C. 0

Proposition 2.13. Let G be simply connected and J a Jordan curve with G n J =I- 0. If Zo E G \ J then there are countably many crosscuts Ck C J of G such that

(4)

where Go is the component of G \ J containing Zo and where Gk are dis joint domains with

(5) Ck = G n oGk c G n oGo for k ~ 1.

It follows that if z E G lies in the other component of iE \ J then z E Gk

for exactly one k. Note that Gk for k ~ 1 may contain points of Jj see Fig. 2.5.

Fig.2.5

Proof. Let Ck(k ~ 1) denote the (at most) countably many arcs of JnGnoGo. By Proposition 2.12, G \ Ck consists of two components and we denote by Gk

the component not containing zoo Then (5) holds by (1) and it follows from the Jordan curve theorem that Go lies in the same component of iE \ Jas zoo

It is clear that the right-hand side of (4) lies in G. Conversely let z E G but z ~ Gk U Ck for all k ~ 1. There is a curve in G from Zo to z, and since diam# Ck ---t 0 (k ---t 00) it meets only finitely many crosscuts Ck. Hence there exists m such that Zo and z are not separated by BUoG where B = Cm +! U .... Also Zo and z are not separated by Cl U oG. Hence Janiszewski's theorem shows that Zo and z are not separated by the union (B U Cl) U oG because the intersection (B n Cd n oG = oG is connected. Continuing we see that Zo


and z are not separated by (B U Cl U· .. U Cm) U oG and thus not by J U oG. It follows that z E Go. This proves (4). 0

We show now that the preimages in the "good" domain ][J) of curves in the possibly "bad" domain Gare again curves. It follows that the preimage of a crosscut of Gis a crosscut of][J). Note that the converse does not hold because the image may oscillate and thus not end at a definite point of oG.

Proposition 2.14. If f maps][J) conformally onto G then the preimage of a curve in Gending at a point of oG is a curve in][J) ending at a point ofT, and curves with distinct endpoints on oG have preimages with distinct endpoints on T.

Thus we are given a curve r : w(t), 0 < t ::::: 1 in G with w(t) ----t w E oG as t ----t O. Then z(t) = r 1 (w(t)) has a limit ( ETas t ----t O.

ProoJ. Applying Proposition 2.2 with h = f- 1 , H = G and c = w, we find rn ----t 0 such that

By Proposition 2.13 there is a cross cut Cn of G with Cn C Bn that separates w(1) from w(t) (t > 0 small)j see Fig. 2.6. If Un denotes the component of G \ Cn that does not contain w(1), then f-1(Un ) is a domain in ][J) not containing z(1) that is bounded by f-1(Cn ) and part of T. It follows from (6) that A(f-1(Cn )) ----t 0 and thus that diamr 1(Un ) ----t 0 as n ----t 00. We conelude that ( = limt_O z(t) exists and ( E T because f is a homeomorphism.

To prove the second assertion we may assurne that 00 E G and thus that oe is bounded. Let rand r* be two curves in e and suppose that f- 1(r) and f- 1(r*) end at the same point ( E T. We now apply Proposition 2.2 with h = f restricted to a neighbourhood of ( (in ][J)) where f remains bounded. We find arcs Qn = ][J) n oD((,rn ) with A(f(Qn)) ----t 0 as n ----t 00. Thus there are points on f- 1(r) and f- 1(r*) arbitrarily elose to ( such that the corresponding points on rand r* have arbitrarily small distance. It follows that the endpoints of rand r* have to coincide. 0

We say that (Cn ) is a null-chain of G if

(i) Cn is a crosscut of G (n = 0,1, ... )j (ii) Cn n Cn +! = 0 (n = 0,1, ... )j

(iii) Cn separates Co and Cn+! (n = 1,2, ... )j (iv) diam# Cn ----t 0 as n ----t 00.

If oG is bounded we can replace the spherical diameter diam # by the euclidean diameter. By Proposition 2.12 and by (i) there is exactly one component Vn

of e \ Cn that does not contain Co. Thus we can write (iii) as

(7) Vn +! C Vn for n = 1,2, ....


wl11

Fig.2.6

Fig. 2.7. Two null-chains; the second null-chain cuts off a whole boundary segment

Fig. 2.8. None of these four sequences is a null-chain because they violate the corresponding condition

Let now (C~) be another null-chain of G. We say that (C~) is equivaZent to (Cn ) if, for every sufficiently large m, there exists n such that

(8) C:" separates Cn from Co, Cm separates C~ from Cb.

If V~ is the component of G \ C~ not containing cb, this can be expressed as

(9)

It is easy to see that this defines an equivalence relation between null-chains. The equivalence classes are called the prime ends of G. Let P( G) denote the set of all prime ends of G. We now prove the main result (Car13b), the prime end theorem.

Theorem 2.15. Let f map][]) conformally onto G. Then there is a bijective mapping

(10) [: 1f -+ P(G)

such that, if ( E 1f and if ( Cn ) is a null-chain representing the prime end [( (), then (f-l(Cn )) is a null-chain that separates 0 from ( for Zarge n.


The last statement means more precisely that

for suitable On, O~ satisfying 0 < On < O~ --+ 0 (n --+ 00). In particular we see that the prime ends of 11)) correspond to the points of 'lI'.

Proof. (a) We may assume again that 00 E G so that ßG is bounded. Let ( E 'lI' be given and apply Proposition 2.2 with h = f restricted to a neighbourhood of ( in 11)). We obtain a sequence of arcs Qn = 11)) n ßD(, rn) such that

(12) rn > rn+l --+ 0, A(f( Qn)) --+ 0 as n --+ 00.

Our null-chain (Cn ) will be constructed by slightly modifying the sets f(Qn). This could be avoided by using Theorem 1. 7 or modifying instead condition (ii)j see e.g. Pom75, p.272.

Let An = {z Eil)): rn - (rn - rn+t}/3 < Iz - (I < rn + (rn-l - rn )/3}. Suppose that crosscuts Cv of G with disjoint endpoints and with f-l(Cv ) C A v have already been constructed for v < n. Since f(Qn) has finite length, it ends at two points an, bn E ßG that may coincide (see Proposition 1.1). We now apply Proposition 2.2 with h = f- 1 and c = an to obtain a circular crosscut of G that intersects f(Qn). We can choose the radius such that the endpoints of this crosscut do not coincide with the endpoints of any Cv for v < n. Furthermore we can choose our crosscut such that its preimage in 11))

lies in An. We do the same for c = bn choosing in particular a different radius if bn = an. Then Cn is obtained from f(Qn) by replacing the ends near an and bn by parts of the crosscuts just defined (see Fig. 2.9). This completes our construction for v = n.

..... , \ \ I I

\ / '.... / ---,..-- ........ ,

/ \ \

ean )

'" ./

I I

Fig.2.9. Construction of Cn (indicated by arrows) for the case that Cn - 1 ends at the point bn


Thus conditions (i) and (ii) are satisfied, and since f-l(Gn) C An we see that (iii) is also satisfied. Finally (iv) follows from (12). Thus f-l(Gn) is a null-chain separating 0 from (. Let now /(() be the prime end of G represented by (Gn ). Thus the mapping (10) is weIl defined.

(b) Let ( and (' be distinct points of 1l'. If (Gn ) and (G~) are the nullchains constructed in (a) it is clear that f- 1(G:n) (m 2: mo) does not separate f-l(Gn) from f-l(GO) for large n. Since f is a homeomorphism we conclude that (8) does not hold so that (Gn ) and (G~) are not equivalent, i.e. that /(() =I 1(('). Hence (10) is injective.

(c) Let now a prime end of G represented by G~ be given. Then Bn = f-l(G~) is an open Jordan arc. By Proposition 2.14, it follows from (i) and (ii) that Bn is a crosscut of lD> and that B n n B n+1 = 0. Let Un denote the component of lD> \ Bn not containing O. Then Un+! C Un for large n and there is thus a point ( with

(13) ( E 1l' n X n 8(1D> \ X) where X = n Une iij. n

Suppose now that there is another point (' E X and consider again the crosscuts Gm constructed in part (a). If m is sufficiently large then f- 1 (Gm) C D((, I( - ('1/2). Hence Bn intersects f-l(Gm) for large n so that G~ = f(Bn) intersects Gm and also Gm+!. It would follow that these two sets have a distance :::; diamG~ -+ 0 (n -+ 00) which is false because Gm n Gm+1 = 0.

Thus there is a unique point (satisfying (13). It follows that diamBn -+ 0 as n -+ 00. Since B n n B n+! = 0 and ( E U n+! we see that each Bn has a positive distance from (. Hence (11) holds with suitable 0 < On < O~ -+ o. It is now easy to see that (G~) and (Gn ) are equivalent. Hence (10) is surjective.

o

We have taken the classical definition of a prime end due to Caratheodory. There are several alternative (equivalent) definitions of this concept, see e.g. Maz36, CoPi64, Oht67, Ah173.

2.5 Limits and Cluster Sets

Let G be a simply connected domain in C and let p be a prime end of G represented by the nuIl-chain (Gn ). We denote again by Vn the component of G \ Gn not containing Go. It follows from (2.4.7) that V n is a decreasing sequence of non-empty compact connected sets in C. The impression of p defined by

00

(1) I(P) = n V n n=1

is therefore a non-empty compact connected set and thus either a single point or a continuumj if I(P) is a single point we call the prime end degenerate. It

2.5 Limits and Cluster Sets 33

follows from (2.4.9) that equivalent null-chains lead to the same intersection so that our definition is independent of the choice of the null-chain representing the prime end p.

We say that w E C is a principal point of p if this prime end can be represented by a null-chain (Cn ) with

Cn c D(w,c) for c > 0, n > no(c) .

Let II (p) denote the set of all principal points of p.

Fig.2.10. Three prime ends. I(pI) = II(pI) is a single point; I(P2) is a segment, II(P2) is a point; I(P3) is the right-hand border, II(P3) is a smaller segment

Let! be any function in [J) with values in C. We define the (unrestricted) cluster set C(f, () of ! at ( E 1l' as the set of all w E C for which there are sequences (zn) with

(2) Zn E [J), Zn --+ (, !(Zn) --+ w as n --+ 00 .

It is easy to see that

(3) C(f, () = n clos{f(z) : Z E [J), Iz - (I < r} . r > O

If ! is continuous in [J) it follows that C(f, () is compact and connected and thus either a continuum or a single point.

Let now E c [J) and ( E 1l' n E. The cluster set CE(f, () of! at ( along E consists of all w E C such that, for some sequence (zn) ,

(4) Zn E E, Zn --+ ( , !(Zn) --+ w as n --+ 00 .

We shall be interested in the cases that E is a Stolz angle Ll at ( or that E is a curve r in [J) ending at (. It is easy to see that CLl(f,() and Cr(!, () are connected compact sets if ! is continuous in [J). See the books Nos60 and CoL066 for the theory of cluster sets.


The next theorem (CarI3b, Linl5) relates the geometrically defined prime ends with the analytically defined cluster sets. Let J( () again denote the unique prime end of G that corresponds to ( according to Theorem 2.15.

Theorem 2.16. Let f map IIJ) conformally onto G c C. If ( E 'f then

(5) J(J(()) = C(f,(),

(6) II(j(()) = nCr(f,() = C[O,o(f,() = C4(f,(), r

where r runs through all curves in IIJ) ending at ( and where Ll is any Stolz angle at (.

Proof. (a) Let the null-chain (Cn) represent the prime end J((). It follows from (2.4.11) that

so that (5) is a consequence of (1) and (3) . (b) We may assurne that 00 E G. Let w be a principal point of j(() and

r a curve in IIJ) ending at (. If (Cn ) is a null-chain representing J(() that converges to w then there exists Zn E f-l(Cn) n r for large n and (4) holds with E = r. Hence w E Cr(f' ().

It is clear that the intersection of all cluster sets Cr(f, () lies in C[O,o(f, () C C4(f,().

Let finally w E C4(f, (). Then there are Zn E Ll with f(zn) -+ wand Pn = IZn - (I -+ 0; we may ass urne that Pn+l < Pn/4. It follows from (2.2.6) that 12pn l(r)2 L -dr<oo,

n Pn r l(r) = A(f(IIJ) n 8D((,r))).

Hence there exist rn with Pn < rn < 2pn such that

We put z~ = (1 - rn )(. Since Zn lies in a Stolz angle Ll we obtain from Corollary 1.6 and Corollary 1.4 that

We now make the same modification of f(Qn) as in part (a) of the proof of the prime end theorem to construct a null-chain (Cn ) with f (z~) E Cn . Since f(zn) -+ w as n -+ 00 it follows from (7) and (8) that Cn converges to w so that w E II(J(()). 0

We defined in (1.2.11) that f has the angular limit f(() at ( E 'f if

(9) f(z)-+f(() as z-+(, zELl

2.5 Limits and Cluster Sets 35

for each Stolz angle .1. If this is even an unrestricted limit, Le. if (9) holds with ~ instead of .1, then 1 is continuous in ~U {Cl (in the spherieal metric). Since the cluster set reduces to a point if and only if the corresponding limit exists, we obtain from Theorem 2.16 the following geometrie characterization of limits:

Corollary 2.17. 111 maps ~ conlormally onto G c ethen

(i) 1 has the angular limit a at ( {:} 1 has the radial limit a at ( {:} 1 has the limit a along some curve ending at ( {:} II (j( ()) = {a};

(ii) 1 has the unrestricted limit a at ( {:} 1(j(()) = {al.

Thus 1 is continuous at ( if and only if the prime end j( () is degenerate. Furthermore we see from Theorem 1. 7 that 1 has a finite angular limit almost everywhere.

It is not possible (GaPo67) to replace the Stolz angle .1 in (9) by any fixed domain tangential to 1'. Much more is however true almost everywhere (NaRuSh82, Two86): For almost all eiiJ E 1', the conformal map I(z) tends to a finite limit as

(10)

for every c > o. This allows strongly tangential (but not unrestrieted) approach.

Exercises 2.5

1. Let E =1= 0 be aperfeet set (i.e. eompact and without isolated points) on l' that eontains no are. Show that

G = ~ \ {r( : 1/2 :::; r < 1, ( E E}

has uneountably many prime ends whose impression is a eontinuum.

2. Let G = ~ \ {(I - e-t)eit : 0 :::; t < oe}. Show that G has a prime end p with I(P) = II(p) = ']['.

3. Let H be any simply eonneeted domai,n. Construet a simply eonneeted subdomain G with a prime end p such that I(p) = II(p) = ßH. Deduee that I(p) and II(p) need not be loeally eonneeted.

4. Show that II(p) is a eontinuum or a single point.

5. Show that f has a radial limit at ( if and only if the prime end [( () is accessible, i.e. if there is a Jordan are that lies in G exeept for one endpoint and that interseets all but finitely many crosscuts of some null-ehain of [( ().

6. Let f map ~ eonformally onto a bounded domain. Use Exercise 2.2.3 to show that the power series of f eonverges at ( E l' if and only if f has a radial limit. (See (5.2.23) for the absolute eonvergenee and Hay70 for the ease of unbounded domains.)


2.6 Count ability Theorems

Most of the following results have a strong topological flavour. A coloured triod is the union of three Jordan arcs B, R, Y that have all one endpoint, the junction point, in common and are otherwise disjoint. These arcs are labeled blue, red and yellow while the junction point is black. We start with a colour version (G. Jensen) of the Moore triod theorem (Moo28).

Proposition 2.18. Let M be a system 01 coloured triods in<C with distinct junction points such that sets 01 different colours do not meet. Then M is countable.

Prool. We consider the system M* of all "coloured circles" with rational radii and centers that are divided by three points with rational central angles into three arcs which we colour blue, red and yellowj circles with different colour schemes are considered to be different. To each triod in M, we assign a circle in M* such that the junction points lies inside and each arc of the triod meets the circle with the first point of intersection being on a circular arc of the same colour. Since M* is countable it suffices to show that different coloured circles are assigned to different coloured triods.

Suppose that the same coloured circle C* with arcs B*, R* and y* is assigned to the triods T and T'. We replace the arcs of T and T' by the subarcs from the junction points to the circle (see Fig. 2.11). Now RU B U Y divides the inner domain of C* into three subdomainsj this follows from the Jordan curve theorem. The junction point of T' lies in one of these three domains, say the one bounded by B, Rand parts of B* and R*. But Y ' goes to y* and therefore has to meet BuR which contradicts our assumption. 0

Fig.2.11

An immediate consequence is: Every system of disjoint (uncoloured) triods in the plane is countable. We deduce now that, if the conformal map 1 is continuous in iij then

2.6 Countability Theorems 37

(1) card{( E '][' : J(() = a} ~ 2

except possibly for countably many a. More generally:

Corollary 2.19. Let J be a homeomorphism oJ If} into C. Then there are at most countably many points a E C such that

(2) J(r(j) --+ a as p --+ 1 - (j = 1,2,3)

JOT three distinct points (1, (2, (3 on ']['.

ProoJ. Since J is a homeomorphism the sets Aj = {J(r(j) : 1/2 ~ r < I} are disjoint, and Al U A2 U A3 U {al is a triod by (2). Hence Proposition 2.18 shows that there can be only countably many distinct points a. 0

The Bagemihl ambiguous point theorem (Bag55) deals with completely arbitrary functions. We say that ( E '][' is an ambiguous point of J if there are two ares r1 and r2 ending at ( such that

(3)

For example, if J(z) = exp[-(1 + z)/(1 - z)] then 1 is an ambiguous point because different circles r touching '][' at 1 have disjoint cluster sets, namely different circles concentric to O.

Corollary 2.20. Any function from If} to C has at most countably many ambiguous points.

ProoJ. Let 1i denote the system of all open sets in C that are the union of finitely many disks in C with rational center and rational radius. Let ( E '][' be an ambiguous point of J. Since Cr1 (1,() and Cr2 (1,() are disjoint compact sets we can find disjoint H j E 1i such that Cr ; (I, () c H j for j = 1,2. Since H j is open we can replace rj by a subare ending at ( such that

(4)

Now 1i x 1t is countable and we may therefore restrict ourselves to those ( for which (4) holds with fixed Hl,H2 E 1t and suitable rl,r2'

We now define a coloured triod with junction point (. The blue arc is rl, the red arc is r 2 and the yellow arc is ((,2(]. It follows from (4) and H 1 n H 2 = 0 that blue ares do not intersect red ares, and trivially neither meets a yellow arc. Hence, by Proposition 2.18, there are only countably many such triods and thus only countably many points ( satisfying (4). 0

We come now to the Collingwood symmetry theorem (CoI60). The lefthand cluster set C+ (I, () at ( E '][' consists of all w E C for which there are (zn) with


(5) znE][}, argzn>arg(, zn--+(, I(zn)--+w as n--+oo.

The right-hand cluster set C- (f, () is defined similarly with arg Zn < arg ( in condition (5). It is dear that C±(f, () c C(f, ().

Proposition 2.21. 11 1 is any function from ][} to ethen

(6) C+(f,() = C-(f,() = C(f,()

exeept lor at most eountably many ( E ']['.

Proof. Let Dk and Die be countably many disks in C with Dk c Die such that each point of C lies in infinitely many Dk and that the spherieal diameter satisfies diam# Die --+ 0 (k --+ 00). Omitting the symbol 1 we define

(7) Ak = {( E '][': C(() n Dk # 0, c+(() n Die = 0}, k = 1,2, ....

We daim that if ( E Ak then there is an open arc Bk(() of,][, with ( as its right endpoint such that

(8)

Suppose this is false for some k. Then there are points (n E '][' with (n --+ ( and argen > arg ( such that Wn E C((n) nDk for some Wn. By (2.5.2), we can choose Zn so dose to (n that arg Zn > arg ( and I(zn) E Dk; we may assume that I(zn) --+ W E Dk as n --+ 00. Then W E C+(() n Die contrary to (7).

Let now (,(' E Ak and arg ( < arge'. Then (' f/. Bk(() by (7) and (8) so that Bk(() n Bk((') = 0. It follows that each Ak is countable and therefore also its union A.

Finally let ( E '][' \ A and W E C(() . We have W E Dk " for suitable kv --+ 00. Since ( f/. A k" we see from (7) that C+(() n D=" # 0 and since W E Di.." diam# Di.., --+ 0, we condude that W E C+((). Hence C+(() = C(() except on the countable set A; we handle C-(() in a similar way. 0

We say that p = f(() is a symmetrie prime end if (6) holds. In Fig. 2.10, for example, PI and P2 are symmetrie whereas P3 is not symmetrie; neither is the "comb" in Fig. 2.2. It seems that our idea of prime ends is strongly inßuenced by these asymmetrie examples whereas all but countably many prime ends are symmetrie by Proposition 2.21.

Proposition 2.22. Let 1 map][} eonlormally onto G. Let ( and (' be distinct points on '][' where 1 has radial limits wand w'. Suppose that the prime end p = !(() is symmetrie and that w' E I(p). Then I(p) is the smallest eompaet eonneeted subset 01 8G eontaining wand w'. In partieular, il w = w' then I(p) = {w}.

Together with the Collingwood symmetry theorem this shows that, except possibly for countably many exceptions, the radial limits are either injective or there is an unrestrieted limit.

2.6 Countability Theorems 39

/ /

/

/

/ /

/

oe( u· \ , , , , , , ,

Fig.2.12

Proof (see Fig. 2.12) . If w = w' let E = {w}, otherwise let E be any eontinuum with w, w' E E c ßG. The eomponent H of C \ E eontaining f(O) is simply eonneeted. The images C and C' of [0, () and [0, (') end at wand w' in E. Henee H \ (C u C') has exaetly two eomponents H± by Proposition 2.12. Choosing the notation appropriately we have f(U±) C H± and thus, by (5),

I(p) = C(() = C+(() nC-(() c H+ nH- c EUCuC'.

Sinee I(p) c ßG we eonclude that I(p) C E . o

A set E C '[' is of first category if it is the eountable union of nowhere dense sets, Le. if

00

E = U En , En contains no proper are . n= l

This is a topological (as opposed to measure-theoretie) generalization of eountability (Oxt80) . Non-empty open sets are not of first category by the Baire eategory theorem.

Proposition 2.23. If f is continuous in ][} then

cu, () = C[O,OU, ()

for ( E '[' except for a set of first category.

This is the Collingwood maximality theorem (CoI57) ; see e.g. CoLo66, p. 76 for the proof. The prime ends are classified aeeording to the following table:

II(p) = I(p) II(p) # I(p)

II (p) singleton First kind Second kind

II (p) not singleton Third kind Fourth kind


Let E j = {( E 1l': n() is ofthe jth kind} for j = 1, ... ,4. It follows from Proposition 2.23 and Theorem 2.16 that E 2 and E4 are of first category. The sets E 3 and E 4 have zero measure hy Theorem 1. 7.

There is a domain with EI = 0 (Car13h, CoL066, p.184) so that CU, () is a continuum for each ( E 1l'. In this case E 2 has measure 211" hut is of first category whereas E3 has measure 0 hut is not of first category. Frankl (see CoPi59) has constructed a domain such that diamI(p) ?: 1 and I(p)nI(p') = 0 for all distinct prime ends p, p' .

There are many furt her results ahout prime ends; see e.g. Pir58, Pir60, Ham86, Näk90 and the hooks CoL066, Oht70. See e.g. McM67, McM69a for furt her countahility theorems.

Exercises 2.6

1. Show that a conformal map has no ambiguous points.

2. In Exercise 2.5.1, show that there are infinitely many non-symmetrie prime ends.

3. Draw a symmetrie prime end of the fourth kind.

4. Let 9 be a function in ][)) such that (( - z)g(z) has an angular limit a(() for each ( E 1l'. Consider j(z) = (1 - JzJ)g(z) to show that a(() = 0 except possibly for countably many (.

5. Show that, with at most countably many exceptions, all graph-theoretieal eomponents of a planar graph are Jordan curves or Jordan ares (possibly without endpoints). A planar graph consists of Jordan ares ("edges") that can intersect only at their endpoints ("nodes"), and two nodes are in the same component if they can be connected by finitely many edges.

Chapter 3. Smoothness and Corners

3.1 Introduction

We now study the behaviour of the derivative f' for the case that the image domain G = I(IT») has a reasonably smooth boundary C; the general case will be studied in later chapters.

The most c1assical case is that C is an analytic curve, i.e. there is a parametrization C : <p((), ( E 1[' where 1 and perhaps undefined otherwise.

Proposition 3.1. Let I map IT» conlormally onto the inner domain 01 the Jordan curve C c <C. Then C is an analytic curve il and only il I is analytic and univalent in {Izl < r} lor some r > 1.

Prool (see Fig. 3.1). First let C be an analytic curve. Then there is r > 1 such that h = 1 as Izl ----> 1, the reflection principle (Section 1.2) shows that h can be extended to an analytic univalent function in {l/r< Izl < r} that satisfies 11 p < Ih(z)1 < p. We conc1ude that 1= <p 0 his analytic and univalent in {1 Ir< I z I < r} and thus in {I z I < r}. The converse is tri via!. D

'f

Fig.3.1

In the next two sections we restrict ourselves to Jordan curves. A theorem of Lindelöf states that

42 Chapter 3. Smoothness and Corners

C smooth {::} arg I' is continuous in jij .

This is a very satisfactory result because it gives a one-to-one correspondence between a geometrie and an analytie condition. The fact that Im log f' = arg I' is continuous in l!)) U 11' does not imply that Re log I' = log I I' I is continuous there. No geometrie characterization for the continuity of log I' in l!)) U 11' is known. We shall show

C Dini-smooth ::::} I' and log I' are continuous in jij.

Corresponding results hold for higher derivatives. For example, the KelloggWarschawski theorem states that

C E Coo {::} all derivatives are continuous in jij

where COO is the dass of all smooth curves with some infinitely differentiable parametrization.

Many domains that occur in applications have corners. If there is a corner of opening 7rCt at I(() then I(z) - I(() behaves like (z - ()<> as z --+ (. The detailed behaviour is more complicated than might be expected (Section 3.4).

The Schwarz-Christoffellormula states that

n

(1) I'(z) = 1'(0) II (1 - ZkZ)<>k- 1 for z E l!))

k=l

if 1 maps l!)) conformally onto a polygonal domain with inner angles 7rCtk(k = 1, ... , n) at the corners Wk = I(Zk). Unfortunately it is not easy to determine the points Zk E 11' if Wk is givenj see e.g. Neh52, KoSt59 or LaSc67. Domains bounded by finitely many circular ares can be handled via the Schwarzian derivative Sfj see Neh52, p.201.

We shall generalize the Schwarz-Christoffel formula to the wide dass of regulated domains: The boundary of a regulated domain has everywhere forward and backward halftangents that vary continuously except for countably many jumps. All domains with piecewise smooth boundary are regulated, furthermore all convex domains. This is related to the starlike and dose-to-convex domains.

We shall not consider the important numerical methods for conformal mappingj see e.g. Gai64, Gai83, Hen86, Tre86.

The main tool is the Poisson integral. If 9 is analytie in l!)) and if v = Im 9 has a continuous extension to jij then the Schwarz integmllormula states that

(2) i 12 ... eit + z . g(z) = Reg(O) + -2 -·t-v(ed)dt 7r 0 e' - z

for z E l!)).

Taking the imaginary part leads to the Poisson integml lormula

(3) 1 12 ... 1 Izl2 1 Img(z)=- I.; 12v(eit )dt= p(z, ()v(()ld(1

27r 0 e' - z 'f

3.2 Srnooth Jordan Curves 43

where p(z, () is the Poisson kernel

(4) 1 1 - Izl2 1 ( + z

p(z, () = 211" I( _ zl2 = 211" Re ( _ z > 0 (Izl < 1(1 = 1) .

Conversely if g is given by (2) then (Ahl66a, p.168)

(5) v continuous at (E 11' => Img(z) --+ v(() as z --+ (, z E][}.

It is easy to prove geometrically that

(6) leit - reiß I ~ max Ct ~ 191,1 - r)

for 0 :::; r :::; 1 and It -191 :::; 11".

Exercises 3.1

We assurne that I rnaps ][} eonforrnally onto the inner domain of the Jordan eurve Ccc. 1. Suppose that the open are A of 11' is rnapped onto an analytic are on C. Show

that I is analytic in sorne neighbourhood of A.

2. Let I be analytic in sorne neighbourhood of ( E 11' and let f' «() = O. Show that C eonsists near fee) of two analytic ares that form an inward pointing (zero-angle) eusp.

3. If I is analytic near z = eit and f'(z) -I- 0, show that the eurvature K(Z) of C at I (z) satisfies

K(Z) = 1!'(z)I-1 Re[l + zl"(z)/ !,(z)].

%tK(z) = -1!,(z)I-1 Irn[z2S/(z)].

3.2 Smooth Jordan Curves

Let C be a Jordan curve in C. We say that C is smooth if there is a parametrization C : w(r), 0 :::; r :::; 211" such that w'(r) is continuous and ::J 0; we have chosen the parameter range [0,211"] for convenience. We extend w(r) to -00 < r < +00 as a 211"-periodic function.

The curve C is smooth if and only if it has a continuously varying tangent, i.e. if there is a continuous function ß such that, for all t,

(1) arg[w(r) - w(t)]--+ ß(t) as r --+ t+, --+ ß(t) + 11" as r --+ t - .

We call ß(r) the tangent direction angle of C at w(t). Now let f map ][} conformally onto the inner domain of C. Since the

characterization of smoothness in terms of tangents does not depend on the parametrization, we may choose the conformal pammetrization


(2) C: w(t) = f(e it ) , 0:::; t:::; 271".

We first give an analytic characterization of smoothness (LinI6).

Theorem 3.2. Let f map J[]) conformally onto the inner domain ofthe Jordan curve C. Then C is smooth if and only if arg f' has a continuous extension to iD. If C is smooth then

(3) argf'(eit ) = ß(t) - t - ~ (t E lR) 2

for the conformal parametrization and

(4) i 1211" eit + z ( 71") log I'(z) = log 11'(0)1 + -2 -·t- ß(t) - t - -2 dt (z E J[])).

71" 0 e' - z

Proof. (a) First let C be smooth. The functions

(5) f(ei/nz) - f(z)

9n(Z) = log (ei/n _ l)z (n = 1,2, ... )

are analytic in J[]) and continuous in iD by Theorem 2.6. Hence we see from (3.1.2) that

(6)

It follows from (5) and (1) that

Im9n(eit) = arg[f(eit+i/n ) - f(e it )]- t - arg(ei / n -1)

~ß(t)-t-7I"/2

as n ~ 00. The convergence is uniform in t because ß(t) is continuousj see e.g. Proposition 3.12 below. Hence (4) follows from (6) for n ~ 00 because 9n(Z) ~ log f'(z) for each z E J[]) and we conclude from (3.1.5) that

(7) argl'(z)~ß(t)-t-7I"/2 as z~eit, zEJ[]).

(b) Conversely, let v = arg f' be continuous in iD and let first (, z E J[]) and ( i= z. We write

q(z) == f(z) - f(() e-i1J() = t 11'(( + (z _ ()s)lei1J(+(z-()8)-i1J() ds. z - ( 10

If 0 < e < 71"/2 and if 8 > 0 is chosen so small that Iv(z) - v(()1 < e for Iz - (I < 8, it follows that

I Imql :::; sine 1111'1 ds, Req ~ cOSe 1111'1 ds

3.2 Smooth Jordan Curves 45

for I z - (I < () and therefore I arg q( z) I ::::: c. Hence, by continuity,

(8) f(z) - f()

arg z-( =v()+argq(z)----tv()

and it follows that C has at f() a tangent of direction angle v() + arg (+1f /2 which varies continuously. Thus C is smooth. D

The smoothness of C does not imply that I' has a continuous extension to lTh. To see this let h be analytic in j[]), Im h continuous in lTh and

IReh(z)1 < 1, IImh(z)1 < 1f/2 for z E j[]).

If log I' = h then e- 1 < 11'1 = eReh < e and I arg 1'1 = I Imhl < 1f/2. Hence f maps j[]) conformally onto a Jordan domain G by Proposition 1.10, and BG is smooth by Theorem 3.2. We can choose h such that Re h is not continuous in lTh. This is for instance the case if h maps j[]) conformally onto the domain of Fig. 2.2 as Corollary 2.17 (ii) shows. Thus I' = eh has no continuous extension to lTh.

The next result (War30) provides a substitute that is sufficient for our purposes.

Proposition 3.3. Let C be smooth. Then C is rectifiable and

(9)

for tl ::::: t2 ::::: h + 21f where M is a constant.

Proof. By Theorem 3.2, the function arg I'(eit ) is continuous and 21f-periodic. By the Weierstraß approximation theorem (Zyg68, I, p. 90), it can therefore be approximated by a trigonometrie polynomial. Hence we can find a polynomial p such that I arg I'(z) - Imp(z)1 < 1f /6 holds for z E 1I' and thus also for z E j[]).

Let MI denote the maximum of Re p on lTh. We write z = reit with 0 < r < 1 and obtain

where M 2 is constant by the residue theorem. Let n E N and h = 190 < 191 < ... < 19n = t2. Applying the Schwarz inequality twice we see that


If we use the above estimate and let r ---+ 1 we deduce that n

L If(ei11~) - f(ew~-l)1 S M3(t2 - td1/ 2

v=l

which implies (9). o

Exercises 3.2

1. Show that the tangent direction angle changes by 271" if a smooth Jordan curve is run through once in the positive sense.

2. Let j map ][)) onto the inner domain G of a smooth Jordan curve. Show that j can be extended to an angle-preserving homeomorphism of jij onto G.

3. Show that j(z) = 2z + (1 - z) log(1 - z) maps ][)) onto the inner domain of a smooth Jordan curve but that f'(x) ---+ +00 as x ---+ 1-.

3.3 The Continuity of the Derivatives

We need a few facts about the modulus of continuity. Let the function 'P be uniformly continuous on the connected set A c Co Its modulus of continuity is defined by

for 8 S o. This is an increasing continuous function with w(O) = o. If A is convex it is easy to see that

(2) w(n8) S nw(8) for 8:::: 0, n = 1,2, ....

If 'P is analytic in ][)) and continuous in jij then the moduli of continuity in jij and '][' are essentially the same because (RuShTa75)

(3) w( 8, 'P, '][') S w( 8, 'P, jij) S 3w( 8, 'P, '][') for 8 S 'Ir /2 j

see e.g. Tamr73 and Hin89 for further results in this direction. The function 'P is called Dini-continuous if

(4)

the limit 'Ir could be replaced by any positive constant. For Dini-continuous 'P and 0 < 8 < 'Ir, we define

(5) w*(8) :=w*(8,'P,A) = [0 w(t) dt+8 r w~) dt. Jo t Jo t

The following estimates are closely connected with conjugate functions (see e.g. Gar81, p.106).

3.3 The Continuity of the Derivatives 47

Proposition 3.4. Let cp be 27r-periodic and Dini-continuous in R Then

(6) i 1 211" eit + z g(z) = -2 -'t-cp(t) dt (z E lDJ) 7r 0 e' - z

has a continuous extension to ll). Furthermore

(7) 1 '()I 2 w(l - r) 2 111" w(t) d 2 w*(l - r) 9 z < - + 7r -2- t < 7r -'------'-- 7r 1 - r l-r t - 1 - r

for Izl ::; r < 1, and if Zl, Z2 E ll) then

The function cp is Hölder-continuous with exponent 0(0< 0 ::; 1) in A if

(9)

i.e. if w( 8) ::; M 80.. Then cp is Dini-continuous and w* (8) = 0(80.) if 0 < 0 < 1 but only w*(8) = 0(8 log 1/8) if 0 = 1. Thus Proposition 3.4 implies that 9 is Hölder-continuous in ll) if 0 < 0 < 1.

Praof. It follows from (6) that

(10) i 1211" eit g'(z) = - ('t )2CP(t)dt for z E lDJ.

7r 0 e' - z

Since the integral vanishes for constant cp, the substitution t = {} + r shows that

for 0 ::; r < 1 and thus, by (1),

, i'19 2111" w(r) Ig(re )1::; - I' 12dr.

7r 0 e''T - r

If we consider the integrals over [0,1 - r] and [1 - r,7r] separately and use (3.1.6), we obtain the first inequality (7), and the second one then follows from (see (2))

~w(8) ::; w (~)::; r6 w(t) dt::; r6 w(t) dt. 3 3 16/3 t 10 t

The estimates hold also for Izl < r by the maximum principle. Let Zj = rj(j, rj < 1, (j E '][' (j = 1,2) with IZ1-Z21 ::; 8 and put r = 1-8.

Integrating the second estimate (7) over [Zl, Z2] we obtain

Suppose now that rj > r for some j. From the first estimate (7) we see that


Exchanging the order of integration we obtain that the last term is

= 211' fO w(t) dt + 211'8 r w~) dt Jo t Jo t

so that Ig(r(j) - g(rj(j)1 :S 7w*(8) by (5). Furthermore Ig(r(d - g(r(2)1 :S 211'w* (8) by (11). It follows that (8) holds in all cases. 0

We say that the curve C is Dini-smooth if it has a parametrization C : w( T), 0 :S T :S 211' such that w' (T) is Dini-continuous and i- O. Let Mb M2, ... denote suitable positive constants.

Theorem 3.5. Let 1 map ]]J) conlormally onto the inner domain 01 the Dinismooth Jordan curve C. Then I' has a continuous extension to ll) and

(12) I(() - I(z) ~ I'(z) --I- 0 I I' I']]J) ~ -r Jor '> -t Z, ,>, z E , (-z

See Ke112, War32, War61. Gur assumption is that

(14)

where (4) holds and w* is defined by (5). We can write the assertion (13) as w(8, 1', ll)) :S M l w*(8, w', IR).

Proof. Let 0 < t2 - tl :S 8 :S 11' and define Tj (j = 1,2) by wh) = I( eitj ).

Since IW'(T)I ?: 11M2 we obtain from Proposition 3.3 that

(15) T2 - Tl :S M2 r2 Iw'(T)1 dT :S M3 (t2 - tl )1/2 :S M38l/2. J 7'1

Let ß(t) again denote the tangent direction angle at I(e it ). It follows from (3.1.6) that

Irleiß1-r2eiß21?:~Iß2-ßll for rl?:r, r2?:r, Ißl-ß21:S1I'· 11'

Since IW'(T)I :S 11M2 and since w is increasing we therefore get from (14) and (15) that

Iß(tl) - ß(t2)1 :S 1I'M2w(T2 - Td :S M4w(M3Vb) . Since t :S M5w(t) by (2), we conclude that the modulus of continuity of ,(t) = ß(t) - t -11'/2 satisfies w(8,,) :S M6w(J8) and therefore

3.3 The Continuity of the Derivatives 49

r w(8,')') d8:::; M 6 r w(v'8) d8 = 2M6 (.fiT w(x) dx < 00

10 8 10 8 10 x

so that ')' is Dini-continuous. Hence log f' has a continuous extension to iij by (3.2.4) and Proposition 3.4 with cp = ')'.

Hence f' is continuous and =I- 0 in iij and thus

(16) f(() - f(z) = 11 !,(z + s(( - z)) ds for z, ( E iij (- z 0

which implies (12). Since we know now that f' is bounded we see as in (15) that r2 -rl :::; M 7(t2 -tl) and therefore that w(8, ')') :::; M gw(8). Hence I log f'(zt}logf'(z2)1 :::; M 9w*(8) by Proposition 3.4 which implies (13). 0

We now turn to higher derivatives restricting ourselves for simplicity to Hölder conditions.

The Jordan curve C is of dass C" (n = 1,2, ... ) if it has a parametrization C : w(r), 0 :::; r :::; 211' that is n times continuously differentiable and satisfies w'(r) =I- O. It is of dass C",OI. (0<: 0::::; 1) iffurthermore

(17)

The following Kellogg- Warschawski theorem (War32) shows that, for 0 < 0:

< 1, we can always take the conformal parametrizationj this is not always true ifo:=1.

Theorem 3.6. Let f map II} conformally onto the inner domain of the Jordan curve C of class C",OI. where n = 1,2,... and 0 < 0: < 1. Then f(n) has a continuous extension to iij and

Proof. We prove by induction on n = 1,2, ... that the assertions of the theorem hold and that furthermore, with ')'(t) = arg f'(eit ),

(19) ( d) n 'l-n 1271' it + z-d log !,(z) = _z - ;"-'=,),(n)(t) dt (z E II}), Z 211' 0 e' - z

(20) ( d )n-l (d )n-l 1!,(eit)l- l dt [')'(t) + t +~] = Iw'(r)l- l dr argw'(r)

where t and r are related by f(eit ) = w(r). Let first n = 1. We know from Theorem 3.5 that log!, is continuous in iij

and that (18) is true. It follows from (3.2.4) by differentiation that

d i 1271' 2zeit

Z-d log!'(z)=-2 ('t )2')'(t)dt, z 11' 0 e' - z


and we obtain (19) for n = 1 integrating by parts and using that 'Y is 211"periodic. Finally (20) for n = 1 is equivalent to (3.2.3).

Suppose nOw that our claim holds for n and that C is of class cn+1,a. Then Iw' (T) I i= 0 and arg w' (T) have nth Hölder-continuous derivatives (with exponent a) so that the right-hand side of (20) has a Hölder-continuous derivative with respect to T. Since

(21)

is Hölder-continuous, the left-hand side of (20) has a Hölder-continuous derivative with respect to t and has the form Ij'I-(n-l)'Y(n-l) + ... where the remaining terms are already known to have a Hölder-continuous derivative. Hence 'Y(n) exists and is Hölder continuous.

Applying Proposition 3.4 with cP = 'Y(n) we see from (19) that (zjJnlog!'(z) is Hölder-continuous in jij. Since j(v) is Hölder-continuous for v ::; n, it follows that j(n+l) is Hölder-continuous. Finally, if we differentiate (19) for n and then integrate by parts we obtain (19) for n + 1; if we differentiate (20) for n and use (21) we obtain (20) for n + 1. 0

It follows at Once from Theorem 3.6 that all derivatives j(n) are continuous in jij if the Jordan curve C is of class C=, i.e. of class cn for all n. The COnverse is trivial because we can choose the conformal parametrization.

Exercises 3.3

1. If cp is uniformly continuous on the convex set A, show that

w(81 + 82) ::; W(81) + w(82 ).

2. Let w(8) = O((log 1/8)-C) with e > 1. Show that w*(8) = O((log 1/8)-C+I).

3. Let 9 be analytic in ]]J) and let 0 < a < 1. Show that 9 is Hölder-continuous in jij with exponent a if and only if g'(z) = 0((1 - Izl)"'-l) as Izl -+ 1- (HardyLittlewood, see e.g. Dur70, p.74).

4. Let G be the inner domain of the Dini-smooth Jordan curve C and let 1/J be a real-valued continuous function on C with Je 1/Jldzl = O. Show that there is a harmonie function u in G with grad u continuous in G whose normal derivative at z E C is 1/J(z). If G = ]]J) show that

112,.. °t °t u(z) =const-- . 1/J(e' )logle' -zldt. 11" 0

5. Let the Jordan curve C be of dass Cl,,,, with 0< a < 1. Show that the conformal map f can be extended to a (non-analytic) homeomorphism of C onto C with non-vanishing Hölder-continuous gradient; compare Corollary 2.9.

6. Let C be of dass C2 ,,,, with 0 < a < 1. Show that, uniformly in z, ( E jij,

!'() 1 1 j"(z) '" f() - f(z) = (- z +"2 f'(z) + O(I( - zl) as Iz - (I -+ O.

3.4 Tangents and Corners 51

3.4 Tangents and Corners

Let now G be any simply connected domain in C with locally connected boundary. By Theorem 2.1 this means that the conformal map f of][)) onto G is continuous in iD. We shall study the local behaviour at a point ( = eUJ E 'lI'.

We say that {JG has a corner of opening 7ra(O ~ a ~ 27r) at f(() # 00 if

(1) arg[f( eit ) _ f( ei..?)] ---+ { ß as t ---+ ß+ , ß + 7ra as t ---+ ß - .

It would be more precise to say "at the prime end corresponding to '" instead of "at f(()" because there may exist other (' E 'lI' with f((/) = f(() where there may be a corner of opening 7ra' or none at all, see Fig. 3.2.

Fig.3.2

If a = 1 then we have a tangent of direction angle ß; compare (3.2.1). If a = 0 we have an outward-pointing cusp; if a = 2 we have· an inward-pointing cusp.

As an example, the function

f(z) = (~~;)a =e-i1ra(;~~)a (0<a~2)

maps ][)) onto the sector G = {I arg wl < 7ra/2}, and {JG has a corner of opening 7ra with ß = -7ra/2 at O. If a = 2 then G = C \ (-00,0] and there is an inward-pointing cusp at O. We always use the branch of the logarithm determined by

(2) ß + 7r/2 < Imlog(z - ei ..?) = arg(z - ei ..?) < ß + 37r/2.

Theorem 3.7. Let f map ][)) conformally onto G where {JG is locally connected. Let ( = ei ..? E 1I' and f(() # 00. Then {JG has a corner of opening 7ra(O ~ a ~ 2) at f(() if and only if

(3) arg f~:) ~ 6~() ---+ ß - a( ß + 7r /2) as z ---+ (, z E ][)).

To prove this theorem (LinI5) we need the following representation formula.


Proposition 3.8. 1f f maps lDl conformally onto G c <C and if a i=- 00, a (j. G then

(4) i 1271" eit + z . log(f(z) - a) = log If(O) - al + - -Ot- arg(f(ett ) - a) dt.

271" 0 et - z

Proof. We may assurne that a = O. We have

1 1271" 112

71" -2 I log f(reitW dt = const +- (log If(reit )l)2 dt 71" 0 71" 0

for 0 :s: r < 1. Since x 2 :s: eX + e- X we have

(2)2 11

12/5

Slog Ifl :s: If1 2/ 5 + 7

and since both fand 1/ f are analytic and univalent in 1Dl, it follows from Theorem 1. 7 that the integral is bounded. Hence 9 = log f belongs to the Hardy space H 2 , and (4) follows from the Schwarz integral formula (3.1.2) which is valid for gEHl and thus for 9 E H 2 • 0

Proof of Theorem 3.7. Let oG have a corner at f((). Taking the imaginary part in (4) we easily see that

(5) f(z) - f(() _ ~ 1271" 1 -lzl 2 f(e it ) - f(() d arg ( ) - I °t 12 arg (0 ) t . z - ( '" 271" 0 et - z ett - ( '"

Since arg( eit - eil?) --; {) + 71" /2 as t --; {)+, --; {) + 371" /2 as t --; {)-, we conclude from (1) that

(6) f(eit ) - f(() (71") arg (e it _ ()", --; ß - a {} + "2 as t--;{)±.

Hence (3) follows from (5) by (3.1.5). Conversely, if (3) holds it follows by continuity that (6) is true which at once implies (1). 0

We say that oG has a Dini-smooth corner at f(() if there are closed arcs A ± C 1I' ending at ( E 1I' and lying on opposite sides of ( that are mapped onto Dini-smooth Jordan arcs C+ and C- forming the angle 7I"a at f((). Then Theorem 3.7 can be strengthened (War32).

Theorem 3.9. 1f oG has a Dini-smooth corner of opening 7I"a (0 < a :s: 2) at f ( () i=- 00 then the junctions

f(z) - f(() (z - ()", and

J'(z) (z - ()",-l

are continuous and i=- 0,00 in IT» n D((, p) for some p > O.


Proof. We shall reduce the present situation to the case of a Dini-smooth Jordan curve by localizing and then straightening out the angle. We may assurne that f(() = O.

If PI > 0 is sufficiently small we can find circles that touch 8D(0, pd and C±. Let G' be the subdomain of D(O, pd n G indicated in Fig. 3.3. We next apply the transformation w 1--+ w1jcx to G' and obtain a domain H that is bounded by a Jordan curve with a tangent at O.

----

, " "

/ //

\ \ \ \ I

/~ .... ( +'-. , " \

\ \ \ \ I~

/ /

/ ./

, I

..... ....------Fig.3.3. Localization (shaded) and straightening

We now show that 8H is Dini-smooth. Let w±(1'), 0 ::; l' ::; b1 be parametrizations of the arcs C± n 8G' such that w± (0) = 0 and w± ( 1') i=- 0 is Dini-eontinuous. The eorresponding are er on 8H ean then be parametrized as

(7) v(t) = w(ctcx )l jcx, 0::; t ::; b2

where c = l/lw'(O)I; we have dropped the subscripts ±. It follows that

(8) v'(t) = c1jCXW'(ctCX)u(ctCX), u(1') == (1'- 1w(1'))-1+1 jcx

with v'(O) = [cw'(o)j1jcx and thus Iv'(O)1 = 1. If 0< 1'1 < 1'2 ::; b1 then

where w is the modulus of continuity (3.3.1). Since 11'-1"-'(1')1 is bounded from below (because w'(O) i=- 0) it easily follows, by (8), that w(8, u) ::; M 1w(8, w') and thus w(8, v') ::; M 2w(c8CX , w'). Hence

(9) l b2 8-1w(8,v±)d8::; M 2a-11b1 C 1w(t,w')dt < 00.


Now Iv±(O)1 = 1 and argv~(O) = argv~(O) because BH has a tangent at O. It follows that v~ (0) = v~ (0). Hence (9) shows that the curve are Cl U ct : v±(ltl), 0 :::; ±t :::; b2 is Dini-smooth and therefore also BH.

Let h map lIJ) conformally onto H such that h(l) = O. Since G' c G there is a conformal map cp of lIJ) into lIJ) such that cp(l) = ( and

(10) h(s) = j(cp(s))l/a for sE lIJ);

see Fig. 3.3. We see from Theorem 3.5 that h' (s) is continuous and =J. 0, 00

for s E TI}. Since cp maps an are of l' with 1 E l' onto Ai U At c 1', the refiection principle shows that cp maps some neighbourhood of 1 conformally onto a neighbourhood of (. Thus our assertions follow because, with z = cp(s),

j(z) (h(S) s _l)a f'(z) ah'(s) (h(S) s -1 )a-1 (z-()a= s-lcp(s)-( '(z_()a-l= cp'(s) s-lcp(s)-(

and h(s)j(s - 1) tends to a limit =J. 0 as s --> 1 by (3.3.12). D

We now define a corner at 00 by reducing it to the finite case. We say that BG has a corner of opening 7W (0 :::; a :::; 2) at 00 if W f-+ 1 j w leads to a corner of opening 7ra at o. By Theorem 3.7 this holds if and only if arg[(z - ()a j(z)] -->, + a7r + m') as z --> (, z E jj) for some, and thus

(11)

where ( = e i 1'J, see (2). This means that the domain approximately contains an infinite sector of opening a7r and midline inclination angle ,. We call the corner at 00 Dini-smooth if the corresponding corner at 0 is Dini-smooth.

It follows at once from Theorem 3.9 that the functions (z - ()a j(z) and (z - ()a+1 f'(z) are continuous and =J. 0,00 in TI} n D(p, () if BG has a Dinismooth corner of opening 7ra > 0 at 00.

Fig. 3.4. Corners at 00 of openings 7rQ and 7r with , = 0


Proposition 3.10. Let 8G have a Dini-smooth corner 01 opening 7r0 with 0< 0 < 2 at J(() = 00 where (= eiiJ . Suppose that

(12)

110 < 0 < 1 then

(13) a

I(z) = (z _ ()a + Wo + 0(1) as z = r(, r -+ 1

with some constant a =I- 0, arga = 'Y + 07r + oß. 110= 1 then (14)

a . b+ - b- (1 ) I(z) = --,. +ie'l' log(z-()+o log--z-~ 7r 1-r

as z = r(, r -+ 1.

1/1 < 0 < 2 then (13) holds with an additional term a' /(z - ()a-l where a' E C.

The assumption (12) says that the curves forming the angle 7r0 at 00 are asymptotic to the lines W = eil'±i7ra/2(s 1= ib±), s E IR.. If 0 =I- 1 they intersect at .

(15) Wo = ei-r(b- ei7ra/ 2 + b+ e-i7ra/ 2)/ sin(7ro)

and this is the constant in (13). For example, the function [(z + l)/(z - l)]a maps JD) onto {-37r0/2 < arg W < -7r0/2} according to (2); we have 'Y = -7r0 and b+ = b- = O.

If 0 = 1 then the two asymptotes are parallel. For example, the function

(16) l+z . l+z 7r J(z) = -- -zlog-- +-1-z 1-z 2

maps JD) conformally onto the domain of Fig. 3.5; we have

J(e it ) = { i(cott/2 -log(cott/2)) + 7r for 0 < t < 7r, -i(1 cot t/21 + log I cot t/21) for - 7r < t < o.

1.; -1 o ------------

Fig.3.5

Prool. (a) Let first 0 < 0 < 1. Then Wo exists and we may assume that 'Y = 0, b+ = b- = 0 and ( = 1. Then Wo = 0 and (12) shows that


f( it) . 7rQ ( 1) 7rQ (I I"') arg e = ±T + 0 If(eit)1 = ±T + 0 t as t -> O±

because the corner is Dini-smooth. It follows that

(17) arg[(eit - 1)'" f(e it )] = 7rQ + cp(t), cp(t) = o(ltl"') as t -> 0 ± .

Given c > 0 we can therefore find 8 such that Icp(t)1 < cltl'" for Itl :S 8. Assuming for simplicity that Wo = 0 rf. G we have as in Proposition 3.8

(18) i 171" eit + z g(z) == log[(z - 1)'" f(z)] = a1 + - -'t-cp(t) dt 27r -71" e' - z

for z E ]]J) and thus

, li 171" eitcp(t) I 1171" Icp(t)1 Ig (z)1 = ;: -71" (eit _ z)2 dt :S;: -71" leit _ Zl2 dt.

We see from (3.1.6) that, for 0 < r < 1 - 8,

2c l 1-r (1 - r)'" 2c jO 7r2t'" 171" t'" Ig'(r)l:S - ( )2 dt+ - -2-dt+M1 2'dt

7r 0 1 - r 7r 1-r tot

= (2c + 2c7r ) (1- r)"'-l + M2 (c). 7r 1- Q

Since 0 < Q < 1 it follows by integration that

Ig(l) - g(r)1 :S cM3 (1 - r)'" + M2 (c)(1 - r).

Hence g(r) = g(l)+o((l-r)"') as r -> 1-, and (17), (18) show that Img(l) = 7rQ. Thus (13) follows by exponentiation.

(b) Now let Q = 1. We may assume that l' = 0 and ( = 1. The function

(19) h(z) = i7r- 1(b+ - b-) log(z - 1) + 3b+ /2 - b- /2

satisfies Re h( eit ) -> b± as t -> O±. Since also Re f( eit ) -> b± by (12) and since If(eit )I-1 = O(ltl) as t -> O±, we conclude from (12) that

cp(t) == arg[(eit - l)(f(eit ) - h(eit ))]_ 7r 7r 7r 't 1 = 4=2' ± 2' + o(lf(e' )1- ) = o(ltl).

As in part (a), we deduce that, as r -> 1-,

log[(r - l)(f(r) - h(r))] = ao + 0 ((1 - r) log _1_) 1-r

from which (14) follows by (19). (c) Finally let 1 < Q < 2. We define 9 again by (18) and show now that

g"(r) = 0((1 - r)"'-2) as r -> 1-. It follows that g(r) = g(l) + (r - l)g'(l) + 0((1- r)"') which leads to (13) with an additional term g'(l)eg (1)(r -1)1-"'.

D


We return to the case of a corner at a finite point by w f---+ l/w. The asymptotes now become circles of curvature.

Theorem 3.11. Let ßG have a Dini-smooth corner of opening 7I"a(O < a < 2) at f (() i- 00 and suppose that the ares C± forming the corner have curvature K:± at f((). If 0< a < 1 then, as z = rC r -t 1-,

(20) f(z) = f(() + b1(z - ()<> + b2(z - ()2<> + 0((1 - r)2<»

where arg b1 = ß - 7I"a/2 - aß and

(21)

If a = 1 then

(22) f(z) = f(() + b1(z - () + b2(z - ()21og(z - () + 0 ((1 _ r)21og _1_) l-r

where b2 = e-ißbr(K:+ - K:-)/(271"). If 1 < a < 2 then (20) holds with an additional term b'(z - ()<>+!.

Fig.3.6. Corner and circles of curvature with /'i;+ > 0, /'i;- < 0

For example, if 0 < a < 1 and a = ei7r<>/2 then

(23) f(z) = 1/ (a (z + 1)<> + 1) = 0; (z -1)<> _ 0;2 (z _1)2<> + ... Z - 1 2<> 22<>

maps j[)) onto the domain bounded by [0, 1] and a circular are; we have K:+ = 0, K:- = 2sin7l"a and ß = 0 at f(l) = O. If a = 1 we see that there is always a logarithmic term except if K:+ = K:-. The case a = 0 of an outward pointing cusp will be considered in Chapter 11.

Proof. We may assume that f(() = O. Now w f---+ l/w transforms the circles of curvature of C± at 0 into the lines Im[e-iJ'fi7r<>/2w] = =fK:± /2 where, = -ß-7I"a/2. The existence of the curvature is equivalent to (12) with b± = K:± /2. Hence we can apply Proposition 3.10 to 1/ f. If 0 < a < 1 we conclude from (13) that


and (20) with (21) follows from (15). The cases 1 :::; 0: < 2 are similar. 0

In most applications we have the case of an analytic corner, Le. a corner of opening 1m formed by two arcs C± : w±(t), 0 :::; t :::; 8 where w± is analytic in Itl < 8 and w±(O) = !((), w±(O) =1= o.

Lewy and Lehman (Lehm57) have given an asymptotic expansion: If 0: > 0 is irrational then, as z ~ (, z E iij,

00 00

(24) !(z) '" L L akj(z - ()k+a j , aOl =1= 0; k=Oj=l

if 0: = p/q with relatively prime positive p, q then

00 q [k/p] (25) !(z) '" L L L akjm(Z - ()k+pj/q(log(z - ())m, aOlO =1= O.

k=Oj=l m=O

We now give an example to show that the logarithmic terms may appear. Let q = 1,2, .... It follows from (16) that

(26) ( l+Z l+z 1I")-1/q

!(z) = 1 _ z - i log 1 - z + 2"

= b1(z - l)l/q + b2 (z - 1)1+1/q log(z - 1) + ... maps j[» conformally onto the domain

{I argwl < 1I"/(2q)} \ c, C: t(i + 1I"tq)-1/q, 0:::; t :::; 1.

Thus 0 is a corner of opening 11"/ q formed by a line and the analytic curve C. These asymptotic expansions have been generalized to corners with weaker

conditions (Wig65). See e.g. PaWaHo86 for a discussion ofthe numerical computation of conformal maps at corners. Many explicit examples can be found in KoSt59.

Exercises 3.4

1. Let 8G have a corner of opening 011" > 0 at f(() "I- 00 formed by two Höldersmooth curves with exponent -y(O < -y < 1). Show that, with a "I- 0,

f(z) = f(() + a(z - ()'" + O(lz - (1"'+"'''1) as z --+ ( , z E iij.

2. Let f(z) = z + b(z - 1)21og(z - 1) where b > 0 is sufficiently small. Show that f maps j[» conformally onto a smooth Jordan domain and determine jt±.

3. Let f map j[» conformally onto {u + iv : u > 0, v > 1/u} such that f(1) = 00.

Use "squaring" to show that f(x) = a(1 - X)-1/2 + 0((1- X)1/2) as x --+ 1-.

3.5 Regulated Domains 59

4. Show that a Jordan curve can have only countably many corners of angle #- 1r.

(Apply the Bagemihl ambiguous point theorem to f(e- t () = arg[w(eit () - w(()] for t > 0, ( E 11'.)

3.5 Regulated Domains

The typical domain in the applications of conformal mapping is bounded by finitely many smooth arcs that may form corners or may go to infinity; parts of the boundary may be run through twice. We shall introduce (Ost35) a dass of domains, the regulated domains, that is wide enough to allow all these possibilities but also narrow enough to have reasonable properties.

We call ß a regulated junction on the interval [a, b] if the one-sided limits

(1) ß(t-) = lim ß(r) , ß(t+) = lim ß(r) 'T-+t- 'T-+t+

exist for a :S t :S b. The function ß is regulated if and only if (Die69, p.145) it can be uniformly approximated by step functions, i.e. if for every e > 0 there exist a = to < tl < ... < tn = band constants ')'1, ... ,')'n such that

(2) Iß(t) - ,),,,1< e for t,,-1 < t < t", ZI = 1, ... ,no

We have ß(t+) = ß(t-) except possibly for countably many t. We assume again that G is a simply connected domain in IC with locally

connected boundary C and that f maps ]IJ) conformally onto G, so that fis continuous in jjj by Theorem 2.1. We shall use the conformal parametrization

(3) C = 8G : w(t) = f(eit ) , O:S t :S 211";

this curve may have many multiple points and may pass through 00.

We call G a regulated domain if each point on C is attained only finitely often by f and if

(4) { lim arg[w(r) - w(t)] for w(t) =I- 00,

ß(t) = 'T-+t+ lim argw(r) + 11" for w(t) = 00

'T-+t+

exists for all t and defines a regulated function. The limit ß(t) is the direction angle of the forward tangent of C at w(t), more precisely of the forward halftangent. Thus a domain is regulated ifit is bounded by a (possibly non-Jordan) curve C with regulated forward tangent.

It follows from Proposition 3.13 below that ß(t+) = ß(t) and that ß(t-) is the direction of the backward tangent at w(t). Since ß is continuous except for countably many jumps it follows that C has a tangent except for at most countably many corners or cusps. If w(t) =I- 00 is a corner we determine the argument by

(5) ß(t+) - ß(t-) = 11"(1 - a) , 1I"a opening angle


in accordance with (3.4.1). If w(t) = 00 we define

(6) ß(t+) - ß(t-) = 7r(1 + 0:), 7r0: sector angle;

see (4) and (3.4.11). If C is piecewise smooth then G is regulated and ß is continuous except for

the finitely many values that correspond to the corners. As another example consider the function (3.4.16), see Fig. 3.5. Then ß(t) = 37r /2 for 0 :::; t < 7r /2 and for 7r :::; t < 27r while ß(t) = 7r/2 for 7r/2 :::; t < 7r.

-'JI' +'JI' • +- f(1 ) +-

G -+

0

--+- 0 1 'JI' T

~ 2

t I -2

-+

0

Fig.3.7. A regulated domain with the values of ß(t) written to the corresponding parts of the boundarYj in this example ß(t) is constant on each linej the apparent jump at f(l) is caused by the periodicity

Suppose now that we have another parametrization C : w* (t*), a :::; t* :::; b of the boundary. Then w* = wo 'P where 'P is a strictly increasing continuous map from [a, b] to [0,27r] and

arg[w*(T*) - w*(t*)] = arg[w(T) - w(t)] -+ ß(t)

as T = 'P(T*) -+ t = 'P(t*), T* > t* if w(t) I:- 00. Hence wo 'P is again a regulated function and vice versa. Our definition is therefore independent of the choiee of boundary parametrization. This is important because in general the conformal map f is not explicitly known.

We first develop some geometrie properties.

Proposition 3.12. Let G be regulated. 1f 0< c < 7r and

(7) Iß(t) - ,I < c, w(t) I:- 00 for t E 1

where 1 is an interval on IR, then

(8) larg[w(T) -w(t)]-,I < c for T,t E 1, t < T.

Proof. Suppose that (8) is false and consider (see Fig. 3.8) the sector A = {z : I arg(z - w(t)) - ,I < c}. Since w( T') E A for small T' > t but w( T) (j. A there is a last point w(t') E GA with t < t' < T. Since


I arg[w(T') - w(t')]-,I ~ c for t' < T' < T

we condude from (4) that I argß(t') -,1 ~ ein contradiction to (7). 0

Fig.3.8

Proposition 3.13. Let G be regulated. Then ß(H) = ß(t) for all t, and if w(t) =f. 00 then

(9) ß(t-) = lim arg[w(t) - W(T)]. r--+t-

Proof. We assume first that w(t) =f. 00. Since ß(t-) exists by the definition of a regulated function, there is 8 > 0 such that Iß( T) - ß( t- ) I < dor t - 8 < T < t. Hence it follows from Proposition 3.12 by continuity that

I arg[w(t) - W(T)]- ß(t-)I ::; c for t - 8< T < t.

This implies (9), and ß(H) = ß(t) for w(t) =f. 00 is proved similarly. Assume now that w(t) = 00. We can choose T' with T' > t such that

Iß(T) - ß(H)I < c for t < T::; T'. Hence

I arg[w(T') - W(T)]- ß(H)I < c for t < T < T'

by Proposition 3.12 and thus

I argw(T) + 7r - ß(H)I < jarg W(T~(T~(T') j + c < 2c

if T is sufficiently dose to t. It follows that arg W ( T) + 7r -+ ß ( H) as T -+ H and therefore ß(t) = ß(H) by (4). The proof of (9) is similar. 0

Theorem 3.14. The domain G c C is regulated if and only if, for every c > 0, there are jinitely many points 0 = to < h < ... < tn = 27r such that

(10) w(t) =f. 00, I arg[w(T) - w(t)]-,vl < c (tv-l < t < T < tv)

for v = 1, ... ,n with real constants IV'


Condition (10) says that Cv : w(t), tv-l < t < t v are rat her Hat open Jordan arcs in Co These arc may be unbounded and are obtained by rotating the graph of a suitable real Lipschitz function with small constant.

Fig.3.9

Proof. First let G be a regulated domain. Since ß is a regulated function, given c > 0 we can find t v and IV such that (2) holds. Thus (10) holds by Proposition 3.12.

Assume now conversely that, for every c > 0, condition (10) holds with suitable t v and IV. Let s be givenj we restrict ourselves to the case wes) =I 00. There exists v such that tv-l S s < t v. By continuity we have

(11) I arg[w(r) - w(t)]- lvi Sc for s S t < r < t v .

Hence the limit points of arg[w(r) - wes)] as r --+ s+ lie in some interval of length 2c for every c > 0 and it follows that the limit ß(s) exists and satisfies Iß(s) - lvi S c. We also see from (11) that Iß(t) - lvi S c for s S t < t v. Hence ß(t) --+ ß(s) as t --+ s+.

Furthermore we have tv-l < S S t v for some v and an analogue of (11) is valid for tv-l < t < r S s. This can be used to show that ß(s-) exists. Hence ß is regulated. 0

We now turn to a representation formula.

Theorem 3.15. Let f map lI} conformally onto a regulated domain. Then, for z E lI},

(12) i 1271" eit + z ( 7r) logl'(z) = log 11'(0)1 + -2 -·t- ß(t) - t - -2 dt 7r 0 e' - z

where ß(t) is the direction angle of the forward tangent at f(e it ), and furthermore

(13) , ß(H) - ß(t-) ( it 7r) 7r arg f (z) + arg(z - e ) - t - - --+ ß(t) - t - -7r 2 2

as z --+ eit , z E lI} for each t.

It follows that ß(t + 27r) = ß(t) + 27r, furthermore that

(14) f '( it) ß(H) + ß(t-) t 7r arg re --+ - --2 2

as r--+1-.


We mention without proof that the converse is also true: If a conformal map has an integral representation (12) where ß is regulated, then the image domain is regulated.

Proof. By Theorem 3.14 there is a constant M' such that I arg(w - w)1 < M' for w, w E C. It follows from Proposition 3.8 that this also holds for wEG. The function

(15) v(z,()=arg[(f(z)-f(())/(z-()] for ZE][J), (EIT), z#(

is defined except for finitely many ( and satisfies Iv(z, ()I < M = M' + 271" for z E ][J) and ( E 11'. Since v(z, () is harmonie in ( E ][J) (for fixed z E ][J)) the maximum principle shows that Iv(z, ()I ::::: M holds for z, ( E ][J). Hence the analytie functions 9n defined as in (3.2.5) satisfy

(16) IIm9n(z)I=lv(eijnz,z)I:::::M for ZE][J), n=1,2, ....

We apply the Schwarz integral formula (3.1.2) to 9n(rz). If we let r ----; 1- we obtain from (16) and Lebesgue's bounded convergence theorem that

for z E ][J). We now let n ----; 00. Since 9n(0) = log 1'(0) and Im9n(eit) ----; ß(t) - t - 71"/2 by (4), we obtain that (12) holds for each z E][J).

To prove (13) we may assurne that t = O. We use that

r27r arg(z - 1) - i = Ja p(z, eit ) [arg(eit -1) - iJ dt

where p is the Poisson kernel (3.1.4). Taking the imaginary part in (12) we see that, with ß(O) - ß(O-) = 7I"(J',

arg j'(z) + (J' (arg(z - 1) - i) = 127r p(z, eit ) [ß(t) - t - ~ + (J' (arg(eit - 1) - ~)] dt.

The function in the square bracket has the same limit ß(O) - 71"/2 as t ----; 0+ and as t ----; 0-. Hence (13) is a consequence of (3.1.5). 0

Now we introduce a stronger condition (Paa31; Dur83, p. 269). The regulated domain G is of bounded boundary rotation if ß(t) has bounded variation, i.e. if

(17) r27r n

Jn Idß(t)1 = sup L Iß(t ll ) - ß(tll-1)1 < 00 a (t.,) 11=1


for all partitions 0 = to < tl < ... < t n = 211'. This definition does not depend on the parametrization of C = 8G because the value of (17) is not changed if we replace ß by ß 0 <p with continuous increasing <po

Corollary 3.16. If f maps JI)) conformally onto a domain G of bounded boundary rotation then

(18) 1 12~ . log j'(z) = log 1'(0) - - log(1 - e-dz)dß(t) (z E JI))). 11' 0

This is a Stieltjes integral. The branch of the logarithm is determined such that 'arg(1- e-itz)' < 11'/2 for z E JI)).

Proof. Since (( + z)/(( - z) = 1 + 2(z/(1 - (z) for '(I = 1 we see from (12) that

i 12~ e-itz ( 11') logj'(z)=c+- 1 't ß(t)-t--2 dt

11' 0 - e-' z with a complex constant c. Since ß(t) - t - 11'/2 is 211'-periodic an integration by parts gives

1 12~ . log!,(z) = c - - log(1- e-dz)d(ß(t) - t) 11' 0

and (18) follows because J log(1 - e-it z) dt = 0; the constant c is determined by putting z = O. 0

Every function of bounded variation can be written as

(19) ß = ßjump + ßsing + ßabs

where ßjump is constant except for countably many jumps, ßsing is continuous and ß~ing = 0 for almost all t, and ßabs is absolutely continuous, Le. the indefinite integral of its derivative. We shall now consider the case that there are only finitely many jumps and that ßsing = O.

Corollary 3.17. If ß is absolutely continuous except for the jumps 1I'0'k at tk (k = 1, ... ,n) then

Proof. In the decomposition (19) we have dßabs(t) = ß'(t)dt except at the jumps tk. Since

we obtain (20) from (18) byexponentiation. o

3.6 Starlike and Close-to-Convex Domains 65

This is the generalized Schwarz-ChristoJJel lormula. The classical form (3.1.1) holds for the special case that the domain is bounded by line segments and thus ß'(t) = 0 except at the corners. The difficulty in applying (20) is that ß(t) is the forward tangent direction at I(eit ) and therefore refers to the conformal parametrization which is not explicitly known if the boundary is given geometrically.

If C has a finite corner of opening 7rO!k at 1 (eitk ) then O'k = 1 - O!k by (5) j if eitk corresponds to an infinite sector of opening 7rO!k then O'k = 1 + O!k by (6). The difficulty remains that the tk are not explicitly known.

Exercises 3.5

1. Let G be a bounded regulated domain. Show that 8G has finite length and that its arc length parametrization is differentiable with at most countably many exceptions.

2. If I'(z) = (1 - z6)1/2(1 - Z2)-5/2 determine I(lD»).

3. Let G be a bounded regulated domain without outward-pointing cusps. Show that I is Hölder-continuous in lD>.

4. Let I map lD> conformally onto a domain containing 00 such that 1(0) = 00.

Generalize the concept "regulated" to this case and show that

2 I i 12" eit + z log[z I (z)] = const +- -.t-(ß(t) + t) dt.

211" 0 e' - z

5. Construct a Jordan curve that has forward and backward tangents everywhere but is not regulated.

3.6 Starlike and Close-to-Convex Domains

The domain G c C is called convex if w, w' E G =} [w, w') c G, and the analytic function 1 is convex in lD> if it maps lD> conformally onto a convex domain. Every convex domain is regulated and furthermore of bounded boundary rotation. The direction angle ß( t) of the forward (half-) tangent is increasing and satisfies ß(27r) - ß(O) = 27r.

It follows from Corollary 3.16 that, if 1 is convex, then

(1) I"(z) 1 1211" 2e-it z 1 12

11" eit + z 1 + z 1'( ) = 1 + -2 1 °t dß(t) = -2 -Ot-dß(t) z 7r 0 - e-' z 7r 0 e' - z

for z E lD>. Since dß(t) 2:: 0 we conclude from (3.1.4) that

(2) [ I';(Z)] Re 1+z l ,(z) >0 for zElD>.

Conversely, if I'(z) =1= 0 and (2) holds then f is convex (Pom75, p.44j Dur83, p.42) and (1) holds, furthermore (ShS69, Suf70)


(3) Re( 2z!,{z) _z+(»o ~ r ID> J{z) - J{() z _ (- or z, .. E .

A deep property (RuShS73;Dur83, p.247) is that the Hadamard convolution

00

(4) J * g(z) = L anbnzn (z E ID» n=O

of two convex (univalent) functions J(z) = I: anzn and g(z) = I: bnzn is again convex. See e.g. Rus75, Rus78, ShS78 for furt her developments.

Fig.3.10. A starlike domain

The domain G is called starlike with respect to 0 if wEG::::} [0, w] c G, and the analytic function 9 is starlike in ID> if it maps ID> conformally onto a starlike domain such that g(O) = O. This holds (Pom75, p.42; Dur83, p.41) if and only if g'{O) =I 0 and

(5) Re[zg'(z)/g(z)] > 0 for z E ID>.

Hence we see from (2) that

(6) J(z) convex {:} g(z) = z!,(z) starlike.

Theorem 3.18. IJ 9 is starlike then the limits

(7)

exist Jor all t and

(8) ( 112". . ) g(z) = zg'(O)exp -; 0 log(1- e-dz)dß(t) (z E ID».

Conversely, iJ ß(t) is increasing and ß(211") - ß(O) = 211" then (8) represents a starlike fu,nction.

The radial limit g( eit ) is an angular limit by Corollary 2.17. For the present case of starlike functions the approach may be rat her tangential (Two86).


Proof. We apply Corollary 3.16 to the convex function f defined as in (6). We replace ß(t) by 71,/2+ (ß(t+ )+ß(t- ))/2 which does not change the value of the integral. Then (8) follows from (3.5.18), and (3.5.14) gives the first limit (7). The second limit then also exists because Ig(reit)1 is increasing in rj indeed (5) shows that

(9) ~ log Ig(reit)1 = Re[eitg'(reit)/g(reit)] > 0 for 0< r < 1. ar

Conversely if gis given by (8) then, as in (1),

(10) g'(z) _ 1 1211" eit + z

z-(-) - - -'t-dß(t) . 9 z 27l' 0 e' - z

Since dß(t) ~ 0 it follows from (3.1.4) that (5) holds so that 9 starlike. 0

As an example, let Zk E T, ak > 0 (k = 1, ... , n) and al + ... + an = 2. Then

m

(11) g(z) = z rr (1 - ZkZ)-O'k (z E]j)))

k=l

is starlike and g(]j))) is the plane slit along n rays {sei1h : Pk :::; s < oo} (k = 1, ... , n) where (h - (h-l = 7l'ak. This is a consequence of the following result (ShS70).

Proposition 3.19. Let 9 be starlike and 0:::; {} :::; 27l'. If ß({}+) - ß({}-) = 7l'a > 0 then

(12) Ig(re iiJ ) 1 ~ rlg'(O)12-2+0'(1- r)-O' for 0:::; r < 1,

and g(]j))) contains the infinite sector {ß( {}-) < arg w < ß( {}+)} which cannot be replaced by a larger infinite sector.

Proof. We can write ß = ßo + ßl, where ßo is constant except for a jump of height 7l'a at {} and where ßl is increasing. Hence we see from (8) that

Ig(rei'l9) 1 = rlg'(O)I(I- r)-O' exp (-~ 1211" log 11- e-it+i'l9rldßl(t))

which implies (12) because log 1 ... 1:::; log 2 and ßl(27l') - ßl(O) = 27l' -7l'a. Now suppose that the above sector does not lie in g(]j))). Then there exists

w E C with ß({}-) < argw < ß({}+) such that [O,w) c g(]j))) but w E ag(]j))). Hence we see from Corollary 2.17 that w = g(eit ) for some t which is =I=- {} by (12). But ß(t) = argg(eit ) by (7), and this is:::; ß({}-) for t < {} and ~ ß({}+) for t > {} which is a contradiction.

On the other hand there are points t; ---+ {}± (n ---+ 00) such that f( eit;) =I=-

00. Since arg f( eit;) ---+ ß( {}±) it follows that g(]j))) contains no larger sector {a- < argw < a+} with either a- < ß({}-) or a+ > ß({}+). 0


The height ?Ta of the largest jump of ß(t) determines the growth of g. If M(r) = max{lg(z)1 : Izl = r} then (Pom62, Pom63, ShS70)

(13) log M(r)

log 1/(1 - r) -t a,

M'(r) (l-r)---ta

M(r) as r-t1-.

It follows from Proposition 3.19 that ß is eontinuous, Le. a = 0, if and only if g(lI))) eontains no infinite sector. In Theorem 6.27 we shall give a geometrie eharacterization for the absolute eontinuity of ß.

The function 1 is ealled close-to-convex (Kap52) in II)) if it is analytic and if there exists a eonvex (univalent) function h such that

(14) Re[j'(z)/h'(z)] > 0 for z E II));

by (6) this is equivalent to the existenee of a starlike function 9 such that Re[z!,(z)/g(z)] > O. Henee (5) shows that every starlike function is close-toeonvex.

As an example, eonsider the eonvex function h(z) = log[(l + z)/(l - z)] that maps II)) onto a horizontal strip. Then (14) is equivalent to

(15) Re[(l- z2)j'(z)] > 0 for z E II)).

Proposition 3.20. Every close-to-convex function 1 is univalent in II)), and il (14) holds with the convex function h then

(16)

Proof. The function cp = 10 h-1 is analytic in the eonvex domain H = h(II))). Henee cp is univalent in H by Proposition 1.10 and it follows that 1 = cp 0 his univalent in II)). The inequality (16) follows from (1.5.4) applied to cp. 0

Let 1 be close-to-eonvex in II)) and let (14) be satisfied where h is eonvex. We denote by ß(t) the tangent direetion angle of the eonvex eurve 8h(II))); see (3.5.4). We need the eoneept of a prime end and its impression defined by (2.5.1). Let 1(t) denote the impression of the prime end [(eit); see (2.4.10).

Theorem 3.21. Let G = I(II))) and 0:::; t < 2?T. 111(t) contains finite points then there exists b(t) E 1(t) such that

lies in C \ G and contains 1(t). The radial limit I(eit ) E iC exists unless ß(H) - ß(t-) =?T.


If the increasing function ß is continuous at t then S(t) is the halßine b(t) - iseiß(t), 0 ::::; s < 00; if ß has a jump at t then S(t) is an infinite sector.

S(t\ )

Fig. 3.11. A close-to-convex domain with two non-degenerate prime ends. The function ß has a jump at h but not at t2

Prooj. Let b(t) be a principal point of [(eit ). By definition there are crosscuts Qn of [j) separating 0 from e it such that diamQn --+ 0 and j(Qn) --+ b(t) as n --+ 00 . We assume that ß(H) > ß(t-); the other case is simpler. If ß(t-) ::::; 0 ::::; ß(H) = ß(t) then we can find Zn E Qn with

arg(zn - eit ) --+ 1I"(ß(t) - O)/(ß(t) - ß(t-)) + t + 11"/2 as n --+ 00

and thus argznh'(zn) --+ 0 - 11"/2 by Theorem 3.15. Furthermore it follows from (16) and (3) that

I j(z) - J(zn) I 11" arg h(z) - h(zn) <"2'

for Izi < IZnl < 1 and thus 0 < arg[j(z) - j(zn)]- argznh'(zn) < 211". Since j(zn) E j(Qn) and therefore j(zn) --+ b(t) as n --+ 00, we conclude that, for all Z E ]]J),

0::::; arg(f(z) -b(t)) -0+11"/2::::; 211" for ß(t-)::::; 0 ::::;ß(H).

Equality cannot hold by the maximum principle and it follows from (17) that S(t)c<C\G.

Let eit; be the endpoints of Qn. Since S(t~)nG = 0 we see that j(]]J)\ Qn) does not meet S(t;t)US(t;;)Uj(Qn)' Since ß(t~) --+ ß(t±) and f(Qn) --+ b(t), it follows from (2.5.1) that I(t) C S(t) .

Finally if j(eit ) has a principal point b*(t) =I- b(t) then b*(t) E I(t) C S(t) and thus S*(t) C S(t) by (17), hence S*(t) = S(t) by symmetry. This is possible only if S(t) is a halfplane, i.e. if ß(H) - ß(t-) = 11". Hence the radial limit exists except possibly in this case; see Corollary 2.17. 0


It follows from Theorem 3.21 that at every point of aG there is a halfline that lies in C \ G, and two such halflines do not properly cross each other. It is easy to fi11 in any rest of C \ G by non-intersecting halflines.

This gives one part of the fo11owing geometrie characterization (Lew58, Lew60): A univalent function is close-to-convex if and only if the complement of the image domain is the union of halflines that are dis joint except possibly for their initial points.

For further results on close-to-convex functions, see e.g. Tho67, Scho75, Mak87 and Section 5.3.

Exercises 3.6

1. If fis convex show that Re[zj'(z)/f(z») > ~ for z E]I)).

2. Show that z(l - 2az2 + z4)-1/2 (a > 0) is starlike in ]I)) and determine the image domain.

3. Let 9 be starIike. Show that (Keo59)

121< Ig'(reit)1 dt = 0 (M(r) log 1 ~ r) as r--+l-.

4. Let a close-to-convex function map ]I)) onto the inner domain of the smooth Jordan curve C. Show that the tangent direction angle of C never turns back by more than'lr (Kap52).

5. Let f satisfy (15). Show that fis convex in the direction of the imaginary axis (Rob36), i.e. every verticalline intersects f(lD» in an interval or not at al1. (The converse holds if f (x) is real for real x.)

Chapter 4. Distortion

4.1 An Overview

Let 1 map ][} conformally onto G c C. We shall study the behaviour of I' for general domains G. The tangent angle is related to arg I' whereas 11'1 describes how sets are compressed or expanded. The results of this chapter will be often used later on.

The functions 9 = log f' and 9 = log(f - a) (a ~ G) are Bloch functions, Le. satisfy

IIgl18 = sup (1-lzI2)1g'(z)1 < 00. Izl<1

Bloch functions satisfy a non-trivial maximum (and minimum) principlej see Theorem 4.2. This is actually true for the wider dass of normal junctions (LeVi57, see e.g. Pom75), Le. meromorphic functions 9 in ][} with

(1) _ 2 # _ (1 - Iz12) 19'(z)1

sup (1 Izl)g (z) - sup 1 I ()12 < 00. Izl<1 Izl<1 + 9 z

Lehto-Virtanen Theorem. For normal junctions, every asymptotic value is an angular limit, i.e. il g(z) ---? a as z ---? ( E 'll' along some curve then g(z) ---? a as z ---? ( in every Stolz angle. In particular, radial and angular limits are the same.

While a conformal map has an angular limit I(() at almost all ( E 'll' the angular derivative

(2) !'(() = lim I(z) - I(() , Li Stolz angle at ( E 'll' z-+(,zELl Z - (

may exist almost nowhere. The angular derivative is particularly useful if it is finite. Then it is equal to the angular limit of 1'. If it is moreover nonzero then 1 is angle-preserving at (.

Now let cp be analytic in ][} and cp(O) = 0, cp(][}) c ][}. Then cp is expanding on the parts of 'll' that are mapped to 'll':

Julia-Wolff-Lemma. 11 ( E 'll' and cp(() E 'll' then cp'(() exists and satisfies 1 :s Icp'(()1 :s 00.

72 Chapter 4. Distortion

Löwner's Lemma. 11 Ac 1I' and <p(A) c 1I' then A(A) :-s: A(<p(A)).

These results lead to monotonicity properties of the angular derivative (Theorem 4.14) and the harmonie measure (Corollary 4.16).

The harmonie measure w(z, A) of a set A c 1I' at z E ]]J) is the total noneuclidean angle under whieh A is seen from z normalized such that w(z, 1I') = 1; see Fig. 4.7. While we use it mainly in order to have an elegant conformally invariant formulation, the true power of harmonie measure appears only for multiply connected domains.

A useful fact about conformal maps 1 is (Corollary 4.18) that every point z E ]]J) can be connected to the arc 1 c 1I' by a curve S such that

(3) A(f(S)):-S: K(w) dist(f(z), äG) :-s: K(w)(1-lzI2)lf'(z)l,

where the constant K(w) depends only on the harmonie measure w(z, 1) and Adenotes again the length.

Gehring-Hayman Theorem. 11 C is a curve in ]]J) and il S is the noneuclidean segment connecting the endpoints 01 C then

A(f(S)) :-s: KA(f(C)) , K absolute constant.

Thus the hyperbolic (non-euclidean) geodesie between two points of G is not much longer in the euclidean sense than any curve in G between these two points. We shall discuss these geodesics in Section 4.6 and compare the (analytical) hyperbolic metric with the (geometrieal) quasihyperbolic metric.

4.2 Bloch Functions

The function g is called aBloch junction if it is analytie in ]]J) and if

(1) Ilglls = sup(1-lz I2)lg'(z)1 < 00. zElIli

This defines a seminorm, and the Bloch functions form a complex Banach space 13 with the norm Ig(O)1 + Ilglls, see AnCIPo74.

It follows from (1.2.4) that

for 7 E MÖb(]]J)). Hence Ilglls is conformally invariant, Le.

(3) Iig 0 711s = Ilglls for 7 E Möb(]]J)) .

Furthermore (1) shows that

4.2 Bloch Functions 73

for z E JI)) and thus

(4)

where A is the non-euclidean distance (1.2.8). Using the identity (3) with r(z) = (z + zd/(l + ZlZ) we can write (4) in the invariant form

Every bounded analytic function belongs to B. Indeed if Ig(z)1 :::; 1 for z E JI)) then (1 -lzI2)lg'(z)1 :::; 1-lg(z)12 :::; 1. An example of an unbounded Bloch function is g(z) = 10g[(1 + z)/(l- z)]. It satisfies IIgl18 = 2 and equality holds in (4) for z > o.

All Bloch functions are normal, i. e. satisfy (4.1.1). Furthermore all analytic normal functions belong to the MacLane dass A, see MacL63; Hay89 , p.785.

There is a dose connection between Bloch functions and univalent functions, in particular their derivatives.

Proposition 4.1. 11 1 maps JI)) conlormally into ethen

(6) Illog(f - a) 118 :::; 4 lor a rf- I(JI))) ,

(7) 11 log 1'118 :::; 6.

Conversely if Iig 11 B :S 1 then 9 = log f' for some conformal map f·

Proof. It follows from Proposition 1.2 that, for z E JI)),

(1 - Iz12) / d: log I'(z)/ = (1 - Iz12) / j:gj / :::; 4 + 21z1 < 6

and since I/(z) - al ;::: df(z) we see from Corollary 1.4 that

2 I d I (1-lz I2)II'(z)1 (1 - Izl ) dz log(f(z) - a) = I/(z) _ al :::; 4.

In the opposite direction, define 1 by 1(0) = 0 and 9 = log 1'. If IIgl18 :::; 1 then

(1-lz I2) Iz~:g] I:::; (1-lz I2)lg'(z)l:::; 1 for z E JI))

so that 1 is univalent by Theorem 1.11. 0

Theorem 4.2. Let C be a circle or line with CnlI' =1= 0 and let G be a domain with


(8) C c j[)) \ C, A = oC \ C C j[)).

If gE Band

(9) "( c = -2 -. -IlgilB (0:::; "( < 7r)

SIll"(

where"( is the angle between C and,][, towards C, then

(i) Ig(z)l:::; a:::; c for z E A => 2c

Ig(z)1 :::; log(c/a) + 1 for z E C,

(ii) Ig(z)1 2:: a 2:: ec for z E A => Ig(z)l2:: e-1a for z E C,

(iii) dist(g(z),g(A)):::; ec for z E C.

/

1"=T/2

Fig.4.1. The situation of Theorem 4.2; the domain G lies on one side of the circle C

The important point is that we have to know abound for Ig(z)1 onlyon A = oC \ C and not on C. Thus (i) is a non-linear maximum principle for Bloch functions while (ii) is a minimum principle. Both are consequences of the Lehto-Virtanen maximum principle (LeVi57; Pom75, p.263) for normal functions; see AnCIPo74 for (iii).

We must have some restriction on a in (ii) because Ig(z)1 2:: e-1a excludes any zeros in C. The proof will show that, in (9), we may replace Ilgll by sup{(1-lzI2)1g'(z)1 : z E C}.

If "( = 0 then C touches '][' and c = (1/2) Ilgll; the case "( = 7r is not allowed. Since 1 :::; 7r"(-1(7r - "()-l sin"(:::; 4/7r we see from (9) that

7r 4 (10) c :::; 2( ) IlgilB :::; -c. 7r-"( 7r

Proof. (a) We first study the case that "( > 0 and OC C j[)). In order to establish (i) and (ii) we consider the increasing function

(11) <p(x) = x log:: for e-1a:::; x < +00 a

which satisfies <p(e-1a) = -e-1a, <p(a) = 0 and <p(+oo) = +00. Hence, for each a > 0, there exists b > a such that <p(b) = c, and if c :::; e-1a then there exists b* with e-1a :::; b* < a such that <p(b*) = -co


We may assurne that ±i E C and that G lies to the right of Cj otherwise we replace 9 by go T with suitable T E Möb([J)) which does not change the angles, the bounds and the seminorm, by (3). For 0 < " < , let G' denote the open subset of G cut off by the circle C' through ±i that forms the angle " with Tj see Fig. 4.2.

(i) Let Ig(z)1 S a S c for z E A = 8G \ C. We claim that Ig(z)1 s b for z E G. Suppose this is false. Since a < band since 9 is continuous in G c [J) there exists " < , such that

(12) Ig(z)1 Sb = Ig(xo)1 (z E G') with Xo E C' n 8G' ,

and we mayassume that Xo E IRj otherwise we consider 9 0 (j with suitable (j E Möb([J)). We have (xo, Xo + 8) c G' for some 8 > 0 because otherwise Ig(xo)1 S a < b. Let now u(z) = arg(i + z) - arg(1 + iz) for z E [J). Then (13)

u(z) > 0 (z E [J)), u(z) = " (z E C'), 7r

u(x) = "2 - 2arctanx (x E IR).

Thus v = cuj(b"(') -log Igl satisfies v(z) 2': -loga for z E 8G' \ C' c 8G \ C and furthermore, by (12),

c v(z)2':b-Iogb=-loga for zE8G'nC'

with equality for z = Xo. Since v is harmonie except for positive logarithmic poles, we conclude from the minimum principle that v (z) 2': - log a for z E G'. Hence

v(x) 2': -loga = v(xo) for Xo < x < Xo +8

and therefore, by (1), (12) and (13),

0< v'(xo) = _ 2cj(b"(') _ Re g'(xo) < _ 2cj(b"(') + Ilgll . - 1 + x6 g(xo) - 1 + x6 (1 - x6)b

Hence we see from (9) and (13) that

~ = c < ,'llgll 1 + x6 = ,'llgll 2 sin , - 2 1 - x6 2 sin,

which contradicts "(' < ,. Hence we have shown that Ig(z)1 Sb for z E G. Since c = cp(b) = blog(bja)

by (11), we have

- = - log - 2': - log - + log log - + 1 = - log - + 1 11 b 1[ b b] 1[ c ] b c a 2c a a 2c a

and this proves (i)j the right-hand side is > 0 because a S c. (ii) Now let Ig(z)1 2': a 2': ec for z E 8G \ C. We claim that Ig(z)1 2': b* for

z E G. Since b* < a there would otherwise exist " < , such that

Ig(z)1 2': b* = Ig(xo)1 (z E G') with Xo E C' n 8G'


where we may again assume that Xo E R Since <p(b*) = -c the function v = cu/(b*'Y')+log Igl satisfies v(z) 2:: loga both for z E BG'\C' and for z E C', hence for z E G' by the minimum principle. We conclude that v' (xo) 2:: 0 which leads to a contradiction as above. Hence we have proved that Ig(z)1 2:: b* 2:: e-1a for z E G.

(iii) Let z E G and suppose that dist(g(z),g(A)) 2:: ec. Then Ig(()-g(z)1 2:: ec for ( E A and thus Ig(() - g(z)1 2:: c > 0 for ( E G by (ii) which contradicts z E G.

-i

Fig.4.2

/'----- ....... ....... // .... -----................ "~n

~-r-___ " \ \c\

\ \ \ \ \ I J J I I

I I / /

/ I '" / /

.... '" i'--_"-

Fig.4.3

(b) We now turn to the general casej see Fig. 4.3. Let Cn be circles with Cn n Gi 0 and Cn n C = 0 such that Cn -t C(n -t (0) and let Gn be the open subsets of G cut off by Cn. Then BGn C JD) and 'Yn > 0 so that we can apply part (a) to Gn , and our assertions follow for n -t 00 because 'Yn -t 'Y'o

Theorem 4.3. Let r be a Jordan arc in JD) ending at ( E 'll'. If gE Band if

(14)

then 9 has the angular limit b at (.

This result (LeVi57) can be expressed as folIows. For Bloch functions (and more generally for normal functions), every asymptotic value is an angular limitj we say that 9 has the asymptotic value b at ( if (14) holds for some curve r. The proof will show that we have only to assume that r is a half open Jordan arc and that ( E r \ r C 'll'.

Proof. Let L1 be a Stolz angle at (. We first consider the case that bi 00. Let 37r/4 < 'Y < 7r and define c by (9). Let r* be a subarc of rending at ( such that

(15) Ig(z) - bl < c < c for z E r* .


Fig.4.4

We construct disks D± as in Fig. 4.4 such that C± = aD± intersects 'll' at ( under the angle 'Y and r* \ (D+ U D-) "# 0. If 8 > 0 and 'Ir - 'Y are sufficiently small then .1* = {z E .1 : Iz - (I < 8} C D+ n D-.

Let V± be the component of D± \ r* that contains points of 'll' and let G~ = D± \ (V± u r*). Since r* connects {Cl and ][)) n (C+ u C-) without meeting V+ U V-, it follows that V+ and V- are disjoint. Hence

(16) .1* C D+ n D- C G+ U G- U r*.

We see from (15) and Theorem 4.2 (i) that

Ig(z) - bl < 2c/[log(c/c) + 1]

for z E G± and thus for z E .1* by (16). Since the right-hand side tends -+ 0 as c -+ 0 we conclude that g has the angular limit b at (. The case b = 00 is handled the same way using now (ii) instead of (i). D

In a similar way, we obtain (AnCIPo74) from Theorem 4.2 (iii):

Proposition 4.4. Let r be a Jordan arc in ][)) ending at ( E 'll' and let g E B. For 0 < 0: < 'Ir / 2 there exists K = K ( 0:) such that

(17) dist(g(z) , ger)) :::; KllgllB lor zELl,

where .1 = {I arg(1 - (z)1 < 0:, Iz - (I < p} with suitable p > O.

In particular it follows for 9 E B that

(18) Igl bounded on r => Igl bounded on .1;

(19) Reg -+ ±oo on r => Reg has the angular limit ± 00 at (.

Corollary 4.5. Let h be analytic and nonzero in ][)). 1/ log hEB and if h has the asymptotic value b E C at ( E 'll', then h has the angular limit b at (.

Proof. This follows at once from Theorem 4.3 if b "# 0,00. If b = 0 then Re log h has the asymptotic value -00 at ( and thus the angular limit -00 by (19). It follows that h has the angular limit 0 at (. The case b = 00 is handled similarly. D


Another way to prove this would be to show that h is normal and then appeal to the normal TIlllction version of Theorem 4.3. The next result implies that every Bloch function is radially bounded on a dense subset of ']['; see StSt81. In fact much more is true as Makarov has proved; see (10.4.12).

Proposition 4.6. Let I be an arc of'][' and let 9 E B. Then there exists ( E I such that

(20) Ig(r() - g(O)1 :::; 221IgIIB/A(I) for 0:::; r < 1.

Proof (see Fig. 4.5). Let iJ = A(I)/2. We may assume that g(O) = 0, IlgilB = 1 and that I is the arc of '][' between e-it9 and eit9 • The circle C through eit9 ,

e-it9 and 0 forms the inner angle 'Y = 7r - iJ with '][' at e±it9. Let G be the component of

{Z inside C : Ig(z)1 < a}, a = (e7r)/(2iJ)

with 0 E 8G. We claim that I n 8G i- 0. Since a ~ ec by (10), it would otherwise follow from Theorem 4.2 (ii) that 0 = Ig(O)1 ~ e-1a > O. Thus there exist Zn E G with Zn --+ ( EIn 8G as n --+ 00. Since 0 and Zn can be connected by a curve Pn C G it follows from Theorem 4.2 (iii) with 'Y = 7r /2 (hence c = 7r /4 by (9)) that

e7r e7r 11 Ig(rzn)1 :::; 2iJ + 4 < -:0 for 0:::; r :::; 1

and (20) follows for n --+ 00. 0

OT--_-;

Fig.4.5

The name "Bloch function" derives from the close connection to the Bloch constant problem. Let dg(z) denote the radius of the largest unramified disk around g(z) on the Riemann image surface g(][}). Then (Ah138; Gol57, p.320)

(21) dg(z):::; (1-lzI2)1g'(z)1 :::; 2(3M - dg(z)) Jd;t) :::; 4~ va for z E][}

if M = sup{dg(() : ( E][}} < 00; see also Bon90.

4.3 The Angular Derivative 79

The space Bo consists of all Bloch functions 9 such that

(22)

For example all functions that are analytic in Il} and continuous in Il}U1l' belong to Bo. See e.g. AnC1Po74, ArFiPe85, Bis90 and Section 11.2 for a discussion of the "little Bloch space" Bo.

Exercises 4.2

1. Let fES. Show that g(z) = log(f(z)/z) is aBloch function.

2. Let f map Il} conformally into C. Show that f E B if and only if feIl}) contains no arbitrarily large disks.

3. Let 9 E B and g(O) = 0, IIgl18 = 1. Show that 11' n 8H =1= 0 where H is the component of {z EIl}: Ig(z)1 < r} (r > e/2) that contains 0 (StSt81, GnHPo86).

4. Let 9 be a nonconstant Bloch function and g(z) ~ b as Izl ~ 1, zEr where r is a half open Jordan arc in Il} with r \ r c 1I'. Show that r = rU {Cl for some ( E 11'.

5. Let f map Il} conformally into C and suppose that I arg 1'1 is bounded on some curve ending at ( E 11'. Show that I arg I' (r() I is bounded for 0 < r < 1.

6. Let 9 be analytic and let (1 - IzI)IIm[zg'(z)ll be bounded in Il}. Show that

(1-lzl)21g"(z)1 is bounded and deduce that 9 E B.

4.3 The Angular Derivative

Let I be analytic in Il}. We say that I has the angular derivative f'(() at ( E 11' if the angular limit I (() =f. 00 exists and if

(1) I(z) - I(() -+ !'(() as z -+ (, zELl z-(

for each Stolz angle Ll at (. We allow that f' (() = 00 though this case has properties different from those of the finite case. For the wide class of John domains we shall show (Theorem 5.5) that if f'(() =f. 00 exists then (1) holds with Il} instead of Ll.

Proposition 4.7. The analytic function I has a finite angular derivative f'(() il and only il f' has the finite angular limit f'(() at (.

The proof uses the following auxiliary result that we formulate with a weaker hypothesis in view of a later application. We omit the proof which either uses the Schwarz integral formula (3.1.2) or normal families as in Pom75, p.306.


Proposition 4.8. Let h be analytic in JD). Iflm h(z) has a finite angular limit at ( E 11' then (z - ()h'(z) has the angular limit 0 at (.

Proof of Proposition 4.7. Let (1) hold. Proposition 4.8 applied to h(z) (f(z) - f(())/(z - () shows that

!,(z) - f(z) - f(() = (z - ()h'(z) ~ 0 as z ~ (, zELl z-(

so that f'(z) ~ f'(() again by (1). Conversely suppose that f' has the finite angular limit f'((). It follows

that

f(z) = f(O) + l z !,(s)ds

has a finite angular limit f (() and that

(2) f(z) - f(() = r1 !,(t( + (1 - t)z) dt ~ !'(() as z ~ (, zELl. 0 z - ( Jo

We now turn to univalent funetions. The next result follows at onee from Proposition 4.1 and Corollary 4.5.

Proposition 4.9. Let f map JD) conformally into C and let f have an angular limit f( () -j. 00 at ( E 11'. If

f(z) - f(() -----'----'-----:-~ ~ a as z ~ (, zEr

z-( (3)

for some curve r in JD) ending at (, or if

(4) !,(z) ~ a -j. 00 as z ~ ( , zEr,

then f' (() = a exists.

We say that f is conformal at ( E 11' if the angular derivative f' (() exists and is -j. 0,00. We say that f is isogonal at ( if there is a finite angular limit f(() and if

(5) f(z) - f(()

arg z _ ( ~ 'Y as z~(, zELl

for some finite 'Y and every Stolz angle Ll. It is clear that

(6) f eonformal at ( =} f isogonal at ( .

We have seen in Seetion 3.2 that the eonverse does not hold. Ostrowski (Ost36, Fer42) has given a geometrie eharacterization of isogonalitYi see Theorem 11.6. We shall study eonditions for eonformality in Seetion 11.4.


If ofCIf)) is locally connected then, by Theorem 3.7 with 0: = 1,

(7) oG has a tangent at f(() * f isogonal at (.

The converse holds for John domains (Theorem 5.5) and thus for quasidisks (see Section 5.4).

The following property justifies the name isogonal (= angle-preserving).

Proposition 4.10. If the conformal map f is isogonal at ( E '][' then smooth curves in a Stolz angle at ( are mapped onto smooth curves, and the angles between curves are preserved.

Praof. Let r j : Zj(T), 0 :::; T :::; 1 for j = 1,2 be two smooth curves in .1 with zl(l) = z2(1) = ( and consider the image curves f(rj ) : Wj(T) = f(Zj(T)), o :::; T :::; 1. It follows from (5) that

f(() - f(Z·(T)) arg[wj(l) - Wj(T)] = arg (_ Zj(~) + arg[( - Zj(T)]

--+ "( + arg zj (1) as T --+ 1 - .

Hence f(rj ) has a tangent of direction angle "( + argzj(l) at f(() and it follows from (8) below that the direction angle of f (rj ) depends continuously on T E [0,1] so that f(rj ) is smooth. Finally the angle between r l and r 2 at (is argz~(l) - argz~(l) which is also the angle between f(rt} and f(r2). 0

Proposition 4.11. The conformal map f is isogonal at ( E '][' if and only if fand arg f' have finite angular limits at (. In this case

(8) (Z - ()f'(z) --+ 1 f(z) - f(()

as Z --+ ( in every Stolz angle.

The Visser-Ostrowski condition (8) will be studied in Section 11.3; see Vis33 and WaGa55.

Praof. If fis isogonal at ( it follows from (5) and from Proposition 4.8 applied to

(9) h(z) = log f(z) - f(() (z E If)) z-(

that (8) holds, and using again (5) we conclude that arg f'(z) --+ "( in every Stolz angle.

Conversely suppose that

f(z) --+ f(() #- 00, argf'(z) --+ "( as z --+ (, zELl

for every Stolz angle .1. Since


I(z) - I(() e-i-y = {l l/'(t( + (1 _ t)z)lei(argf'-'Y) dt z - ( 10

and since arg I' (t( + (1 - t) z) --+ 'Y uniformly in t as z --+ (, zELl it follows that

tan [arg I(z; = t() -'Y] = 1111'1 sin(arg I' - 'Y) dt /1 11!'1 cos(arg I' - 'Y) dt

tends --+ o. Hence (5) holds. o

Proposition 4.12. There is a conformal map onto a Jordan domain (even a quasidisk) that is nowhere isogonal and thus nowhere conformal on 11'.

It follows from (6) that this conformal map has nowhere on 11' a finite nonzero angular derivative. It does however have infinite and zero angular derivatives at very many points (see Theorem 10.13). Furthermore (7) shows that the bounding Jordan curve has nowhere a tangent even though it may be a quasicircle (see Section 5.4). Most Julia sets (see e.g. Bla84, Lju86, Bea91) have this property. We postpone the proof to the end of Section 8.6.

The Julia- W olfJ lemma deals with selfmaps of JI)) that need not be univalent.

Proposition 4.13. Let <p be analytic in JI)) with an angular limit <p(() at ( E 11'. 1f

<p(JI))) C JI)), <p( () E 11'

then the angular derivative <p' (() exists and

(10) 0< (<p'(() = sup 1-lz12 l<p(() - <p(z)1 2 < +00. <p(() zEll} I( - zl2 1-1<p(z)12 -

Proof. We may assume that ( = <p(() = 1; otherwise we consider <p((z)/<p((). The function h = (1 + <p)/(1 - <p) is analytic in JI)) and satisfies

(11) 1 - 1<p(z)12

Reh(z) = 11- <p(z)1 2 > 0 for z E JI)).

Let Zn E JI)) and Tn(S) = (s + zn)/(1 + Zns). Then

(12) I h(Tn(S)) - iImh(zn) I < 1 + Isl < 4 for sE JI)) Reh(zn) - 1-lsl - 1 -lsl2

because the left-hand function is analytic and of positive real part for s E JI)) and has the value 1 at s = O.

We define

(13) . 11 - zl2

c = mf I 12 Reh(z) zEll} 1- z


and choose (zn) such that this infimum is approached as n ----+ 00. Let 0 < x < 1. It follows from (12) with s = r;l(x) that

1 () . ()I 411 - zn x I2 ( )

h x - dmh Zn ::; (1-lznI2){1 _ x 2 ) Reh Zn .

Since h( x) ----+ 00 as x ----+ 1- we conclude that

. 1 - X 11 - znl2 (14) hmsup -1 -lh(x)l::; 1 1 12 Reh(zn), thus::; c

x-+1- + X - Zn

by our choice of (Zn). Since [(1 - x)/(1 + X)] Reh(x) 2 c by (13), it follows from (14) that

1 - X 1 + ip(x) = 1 - X h(x) ----+ c as X ----+ 1 _ . 1 + X 1 - ip(x) 1 + X

Hence (1 - ip(x))/(1 - X) ----+ l/c because ip(x) ----+ 1. It is easy to see that 10g[(I- ip(z))/(I- z)] is aBloch function (with norm

::; 4). Hence we see from Theorem 4.3 that (1- ip(z))/(I- z) has the angular limit l/c at 1 so that ip'(I) = l/c, and (10) now follows from (11) and (13).0

We now prove a comparison theorem that goes back to Caratheodory and Lelong-Ferrandj see Fig. 4.6.

Theorem 4.14. For j = 1,2 let /j map][} conlormally onto Gj where GI C G2 • Let r be a Jordan are in GI ending at w E BG1 nBG2 and let r j = I j- 1(r) end at (j E ']['. 11 the angular derivative IH(d i- 00 exists then IH(2) i- 00

exists and 1/~((2)1 = cl/{((l)1 with 0::; C < 00.

It follows from Proposition 2.14 that r j is a Jordan are ending on '['. If GI and G2 are Jordan domains then the assumptions involving r can be replaced simply by h((d = 12((2) =w.

Fig.4.6

Proof. The function ip = 1:;1 0 h is analytic in ][} and satisfies ip(][}) C ][} and ip(r1 ) = r 2 • Hence ip has the asymptotic value (2 along r 1 by Proposition 2.14


and therefore the angular limit (2 at (1. Henee cp'((1) exists by the Julia-Wolff lemma (Proposition 4.13) and thus

!z((2) - !z(cp(r(1)) (1 - r(1 11((1) - l1(r(1) ff((d ~~--~~~-~~--~--~ ~----

(2 - cp(r(1) cp((d - cp(r(1) (1 - r(1 cp'((d as r ~ 1-. This means that (12((2) - !z(Z))/((2 - z) has the asymptotic value ff((1)/cp'((1) along the eurve cp(r(1), 0 :::; r :::; 1 ending at (2 and it follows from Proposition 4.1 and Corollary 4.5 that f~((2) exists and that If~((2)1 = elff((dl with 0 :::; e = 1/lcp'((dl < +00. D

As an applieation, let J1, J and J2 be Jordan eurves such that J1 lies inside J and J lies inside J2 exeept for the point W E J1 nJnJz. Let 11, fand!z map J[» onto the corresponding inner domains such that 11(() = f(() = !z(() = w. We suppose that the eomparison eurves J1 and Jz are Dini-smoothj see Section 3.3. Then it follows from Theorem 3.5 that ff(() and fHO exist and are finite and nonzero. Henee we eonclude from Theorem 4.14 that f'(() =I 0,00 exist so that f is eonformal at (. More general eomparison curves ean be obtained from the results in Section 11.4.

Exercises 4.3

Let f map J[» conformally onto G.

1. Let G be bounded by the (non-Jordan) curve C and let (, (' be distinct points on 1l' where f is isogonal. If fee) = f((') = w show that C has a tangent at w.

2. Let Zo E [» and let fee) be the boundary point nearest to f(zo). Show that I'(() exists and is finite but may be zero.

3. If G is bounded and convex show that I' (() # 0 exists for all ( E 1l'.

4. Find (n E 1l' and 0 < r n < 1 such that I' (() exists and is infinite for all ( with fee) E 1l' if

n

5. Let r.p be analytic in J[» and r.p(J[») c J[». Show that r.p has at most one attractive

fixed point in~, i.e. a point Zo E ~ such that r.p(zo) = Zo and Ir.p'(zo)I < 1.

6. Prove the converse of Proposition 4.10.

4.4 Harmonie Measure

Let A be a measurable set on 1l'. The harmonie measure of A at z E J[» is defined by

(1) _ 1 r 1-lz12

w(z, A) = w(z, A, J[») = 21f JA I( _ zl2ld(1 ,

i.e. by the Poisson integral of the eharaeteristie function of A. It is clear that this is a harmonie function satisfying 0 :::; w(z, A) :::; 1 for z E J[». Furthermore (e.g. Dur70, p. 5), as r ~ 1-,

(2)

4.4 Harmonie Measure 85

{ I for alm ost all ( E A , w(r(, A) -7 0

for alm ost all ( E '][' \ A .

If T E Möb(J[))), i.e. if T is a Möbius transformation of J[)) onto J[)), then

(3) 1 - IT(ZW I 1 - IzI 2

IT(() - T(z)i2IT (()I = I( - zI 2 •

Hence (1) shows that

(4) W(T(Z), T(A)) = w(z, A) for TE Möb(J[))).

Furthermore

(5) w(O, A) = ~ r Id(1 = A(A) 21l" JA 21l"

is the normalized total angle under which the set A is seen from O. Hence it follows from (4) and the conformal invariance of angles that 21l"w(z, A) is the total angle under which A can be seen from Z along non-euclidean lines; see Fig.4.7.

Fig. 4.7. The sum of the angles is 27rw(z, A)

Proposition 4.15. Let 'P be analytic in J[)) and let A, B c '][' be measurable. If the angular limit 'P( () exists for all ( E A and if

(6) 'P(J[))) c J[)), 'P(A) c B c ']['

then

(7) W('P(z),B) 2: w(z,A) for z E J[)).

If 'P(O) = 0 then (5) and (7) show that

(8) A(B) 2: A(A)

if (6) is satisfied. This is Löwner's lemma. The problem of the measurability of the image set will be discussed in Section 6.2.


Prool. Let V be any open subset ofT with Be V. Then w(z, V) is eontinuous on l!)) U V. The function u(z) = w('P(z), V) is harmonie in l!)) and satisfies o ~ u ~ 1. If ( E Athen 'P(() E B c V so that u has the radial limit 1 at (. Henee

1 12,.. 1 Izl 2 w('P(z), V) = u(z) = 211' 0 I( _ Zl2 u(()ld(1 :2: w(z, A)

by (1), and (7) follows from (1) beeause B is measurable. D

Now let G be a simply eonneeted domain with loeally eonnected boundary and let 1 map l!)) eonformally onto G. Then 1 is eontinuous in TI) by Theorem 2.1. If A is a Borel set of 8G then 1-1(A) is a Borel set on T and therefore measurable. We define the harmonie measure of A at z E G relative to G by

(9) w(z, A, G) = w(f-1(Z), 1-1(A), l!))).

It follows from (4) that this definition does not depend on the ehoiee of the function 1 mapping l!)) onto G.

CoroUary 4.16. Let GI, G2 be Jordan domains. 11

(10)

where A is a. Borel set then

(11) w(z, A, Gd ~ w(z, A, G2) lor z E GI .

Proof. Let fi(j = 1,2) mapl!)) eonformally onto Gj • We see from (9) and (7) with 'P = 12-1 011 and thus 'P 0 11 1 = 1:;1 that

y;'(z, A, GI) = w(f11(z), 11 1(A)) ~ w(f:;l(z), l:;l(A))

beeause I j- 1 (A) C Tj this is again a Borel set and thus measurable. Henee (11) follows lrom (9). D

This is Carleman's prineiple 01 domain extension for harmonie measure (Nev70, p.68). For reasonable sets A (say union of ares) it follows easily from the mruumum principle for harmonie functions applied to w(z, A, GI) -w(z, A, G2 ). This proof has the advantage that it earries over to multiply eon-nected domains. .

Harmonie measure ean be generalized (see e.g. Nev70, Ah173) to domains of any eonnedivity where it is of great importaneej eonformal mapping fails here and its generalization, the universal eovering map of l!)) onto G, is not easy to handlle.

Exercises 4.4

1. If A(Z, z') denotes the non-euclidean distance then w(z', A) ~ e2>.(z,z'lw(z, A).

2. Let the functions 1j;± be continuous in [a, b] and positive except for 1j;±(a) = 1j;±(b) = O. Let G be the domain between the graphs C± : ±1j;±({), a ~ { ~ b. If 1j;- ~ 1j;+ show that w(x, C-, G) 2: w(x, C+, G) for a < x < bj see Küh67.

4.5 Length Distortion 87

3. Let 9 be analytic in the Jordan domain G and continuous in G. If Ig(z)1 :<:::: a for z E A, Ig(z)1 :<:::: 1 for z E 8G \ A where A C 8G is a Borel set, show that

Ig(z)1 :<:::: a",(z,A,G) for z E G.

This is the two-constants theorem, see Nev70, p.41.

4.5 Length Distortion

Let Kl, K2 , ••• be positive absolute constants and let w(z, A) again denote the harmonie measure of A eTat z E lD> relative to 1D>.

Proposition 4.17. Let I map lD> conlormally into C. If Zo E lD> and ill is an arc of T, then there is ( E 1 such that

(1) Ilogj'(z)-logj'(zo)I<Kdw(zo'!) lor zES

where S is the non-euclidean segment from Zo to (.

Proof. We consider the Bloch function

g(z) = logj'(r(z)), r(z) = t + Zo (z E 1D». +zoz

We see from (4.2.3) and (4.2.7) that IIgl18 ::<:::: 6. Hence, by Proposition 4.6, there exists (* E r- 1(1) such that

Ilogj'(r(r(*)) -logj'(zo)1 ::<:::: 132/A(r-1(/)) for 0::<:::: r < 1

which implies (1) with ( = r((*) E I. 0

Let d j (z) again denote the distance of I (z) to the boundary. It follows from Corollary 1.4 that, for z E 1D>,

Corollary 4.18. If Zo E ]])l and if 1 is an arc ofT then

(3)

where S is the non-euclidean segment from Zo to a suitable point on I.

Proof. We may assume that Zo = 0; otherwise we consider I 0 r with suitable rE Möb(lD» and use djo.,.(O) = dj(zo) and (4.4.4). Now we see from (1) that

1j'(r()1 < 1j'(O)lexp[Kdw(O,/)] (0::<:::: r < 1)

for some ( E I, and (3) follows from (2) because

A(f(S)) = 11 1j'(r()1 dr, S = [0, (]. o


Later we shall prove the much stronger result (see Corollary 9.20) that A(I(S)) is not much larger than df(zo) except for the non-euclidean segments connecting Zo to a set of small capacity and thus of small measure. The next estimate goes in the opposite direction.

Proposition 4.19. Let 1 map][J) conlormally into C. 11 Zo E][J) and I is an arc on 11' then

(4) . diam/(I)

df(zo) S dlSt(l(ZO), 1(1)) S K 3 ( 1)5/2 W zo,

where 1(1) is the set 01 angular limits on I whenever they exist.

Prool. We may assume that Zo = 0 and 1(0) = O. Choose a E 1(1). Since 1/(1 - a) is analytic and univalent in ][J) it follows from Theorem 1.7 (for r --+ 1-) that

because lai ~ df(O) ~ 11'(0)1/4. Since a E 1(1) it follows that

dist(O, 1(1)) A(I)5/2 < 4K diam/(I) - 4·

Theorem 4.20. Let 1 map ][J) conlormally into C. 11 Zl, Z2 E iij then

(5)

(6)

A(I(S)) S K 5A(I(C)),

diam/(S) S K 6 diam/(C),

o

where S is the non-euclidean segment from Zl to Z2 and C is any Jordan arc in ][J) from Zl to Z2.

This is the Gehring-Hayman theorem (GeHa62, Pom64, Jae68), see Fig. 4.8 (i). The proof is based on the following lemma, see Fig. 4.8 (ii).

Lemma 4.21. Let -1 S 6 < 6 S 1 and assume that

(7)

For j = 1,2, let T j be the non-euclidean line through ~j orthogonal to lR if I~jl < 1, and Tj = {~j} otherwise. Then

(8) diam/(S) S K 7 diam/(C)

where S = (~17 6) and C is any Jordan arc from Tl to T2.

4.5 Length Distortion 89

(i)

ID

Z, /

/ I (ii)

Fig.4.8

Proof. Let P = diamf(C) and let Zj E Tj n C. We obtain from (1.3.18) that

The quotient is 2: K;;l by (7). Henee we have

(9)

for suitable a E öf(JI))). Now let 6 ::; x ::; 6. Applying Corollary 4.18 to 9 = 1/(f - a) we find (±

on the two ares of 11' between Tl and T2 such that, for Z on the non-euelidean segments S± from x to (±,

Ig(z)1 ::; Ig(x)1 + Ig(z) - g(x)1 ::; Ig(x)1 + KlOdg(x) ::; Klllg(x)1

beeause 0 f/. g(JI))). Sinee C eonneets Tl and T2 this inequality holds for some z E C and we eonclude from (9) that

If(x) - al ::; Klllf(z) - al ::; Kll(lf(z) - f(Zl)1 + If(zd - al) ::; Kll(p+ K 9P)

whieh implies (8). o

Proof of Theorem 4.20. We may assume that Zj = Xj is real and that Xl < X2.

We furthermore assume that >'(Xl' X2) 2: 1; the ease >'(Xl, X2) < 1 ean easily be handled by the Koebe distortion theorem. Then (6) follows at onee from Lemma 4.21.

In order to prove (5) we divide (Xl, X2) into a finite or infinite number of segments Sn = (en, en+d with 1 ::; >.(en, en+d ::; 2. If Tn denotes the noneuelidean segment through en orthogonal to lR then there exist ares Cn of C that eonneet Tn and Tn+l and are disjoint exeept for their endpoints, Henee, by (8),

diamf(Sn) ::; K 7 diamf(Cn ) ::; K 7A(f(Cn)).

We see from Corollary 1.5 that, for X E Sn,


(~n+1-~n)lf'(X)I:::; eXP[6A(~n,~n+1)ll~~+~ e~n (1-~~) 1!'(~n)1 n n+l

~ K12If(~n+l) - f(~n)1 ~ K 12 diamf(Sn).

It follows that

r'2 l~n+l j", 1!,(x)1 dx = L 1!,(x)1 dx Xl n ~n

n

The diameter estimate in Theorem 4.20 and related results can be generalized to quasiconformal maps in any dimension (Zin86, HeNä92).

Exercises 4.5

We assume that j maps [J) conformally onto G c C. Let Kl,K2, ... denote suitable absolute constants.

1. Let I be an arc of 1I' and let r = 1- A(I)j1l" > O. Show that there is a crosscut Q of [J) that separates 0 from I such that A(f(Q)) ::; Kldl(rz) for z E I.

2. Let C be a crosscut of G and let B be the image of a non-euclidean segment. If both endpoints of B lie in one component of G \ C, show that dist(C, w) ::; K2 diamC holds for all points w of Bin the other component of G \ C (Ost36).

3. Let ( E 1I' and suppose that the corresponding prime end p is rectifiably accessible, Le. the Jordan arc of Exercise 2.5.5 has finite length. Show that the curve {f(r() : 0 ::; r ::; I} has finite length.

4. Let C be any curve in [J) from (1 E 1I' to (2 E 1I'. Show that

dl(O)I(l - (212 ::; K 3 diamj(C).

(Consider the point nearest to 0 on the non-euclidean line from (1 to (2.)

4.6 The Hyperbolic Metric

Let f map [J) onto G c C. The hyperbolic metric (or Poincare metric) of G is defined by

(1) AG(WbW2) = AJ[)(ZbZ2) for Wj = f(zj), Zj E [J) (j = 1,2)

where AJ[) is the non-euclidean metric in [J). By (1.2.7) this definition is independent of the choice of the function f mapping [J) onto G because all others have the form f 0 'T with 'T E Möb([J).

The definition can be generalized to multiply eonneeted domains with at least three boundary points. The eonformal map is then replaeed by the universal eovering map of [J) onto G; see e.g. Ah173 and Hem88. The hyperbolie metric has several important monotonicity properties, see Hem79, Wei86, Min87, Hem88 and Hay89, p.698.

4.6 The Hyperbolic Metric 91

It follows from (1.2.5) that

(2) Aa(WI, W2) = minAa(C), Aa(C) == r Idzl c Jj -l(C) l-lzl 2

where the minimum is taken over all curves C in G from Wl to W2. It is attained for the hyperbolic geodesie from Wl to W2, that is the image of the non-euclidean segment from Zl to Z2. It follows from Theorem 4.20 and from the definition that

(3) A(S) ::; KA(C) , Aa(S)::; Aa(C)

where S is the hyperbolic geodesic and C is any curve in G from Wl to W2.

Thus the hyperbolic geodesic is almost as short as possible in the euclidean sense; see Fig. 4.9. For non-convex domains there is not always a shortest curve in G from Wl to W2.

There is another geometrie property (J0r56, see also Küh69, Min87), see Fig.4.9.

Proposition 4.22. Let D c G be a disk and S a hyperbolic geodesie of G. If 8 D is tangential to S then D n S = 0.

1G

Fig.4.9

Proof. We mayassume S is the image of areal interval in IDJ and that 8D and S touch at f(O), furthermore that f(O) = 0 and 1'(0) = 1. Then

1 1 h(z) = f(z) - ~ - z (Izl < 1)

is analytie in IDJ. Since 8D is tangential to Sand thus to ffi. at 0 we see that {l/w: W E D} = {w* : Imw* > a} (or = {Imw* < a}). Since D n 8G = 0 it follows that

1 limsupImh(z) = limsupIm f( ) ::; a.

Izl--+l Izl--+l z The maximum principle therefore implies that Im h( z) < a for z E IDJ, in partieular that


1 Im f(x) = Imh(x) :::; a for -1< x < 1

which is equivalent to D n S = 0. o

The hyperbolic metrie is not a geometrie quantity in contrast to the quasihyperbolic metric

(4) ,* ( ) . 1 Idwl "'G Wl, W2 = mln d· ( J'lG)

C eist w,u

where the minimum is taken over all curves C c G from Wl to W2. It follows from (4.5.2) that

(5) 1 < If'(z)1 < 4 1 - Izl 2 - dist(J(z), ßG) - 1 - Izl 2

(z E][)))

and thus, by (2) and (4),

(6)

See e.g. GeOs79 and GeHaMa89 for further results. If G is multiply connected then AG can be much larger than AG; see e.g. BeaPo78, Pom84.

We use the quasihyperbolic metrie to give a geometrie characterization (BePo82) of Hölder domains, i.e. of domains G = f(][))) such that fis Hölder continuous in jj) for some exponent 0: with 0 < 0: :::; 1; see (3.3.9). This holds if and only if (Dur70, p.74)

(7) If'(z)1 :::; M l (1-lzl)"-1 (z E][)))

for some constant MI whieh is equivalent to

1 + Izl 1 0: log 1-lzl :::; M 2 + log (1-lz I2)1f'(z)l·

Hence it follows from (5) that Gis a Hölder domain if and only if

(8) 20:AG(J(0), w) :::; M 3 + log 1/ dist[w, ßGJ (w E G)

and thus, by (6), if and only if

(9) AG(J(O), w) = 0(1/ dist[w, ßGJ) as w --+ ßG;

the exponent 0: is not completely determined by (9). See e.g. NäPa83, NäPa86, SmSt87 for furt her results about Hölder do

mains. If f(z) = L::'=o anzn then (SmSt91; see also Sta89, SmSt90, JoMa91)

00

(10) L n1+6 lan l2 < 00 for some 8 > O. n=l

Hence it follows by the Schwarz inequality that the power series converges absolutelyon 11'. For general bounded domains, the estimate (10) holds only

4.6 The Hyperbolic Metric 93

for 6 = 0 and the power series need not converge absolutely even in the case of Jordan domains; see e.g. Pir62.

Exercises 4.6

1. Let D C G be a disk and S a hyperbolic geodesie of G. Show that D n S is connected (J0r56).

2. Let G be as in Fig. 4.10 and w± = -1±i/2. Show that the hyperbolic geodesie S intersects the positive real axis at a point u with 1/ K :::; u :::; K. Give numerical upper and lower bounds for AG ( w + , W - ).

3. Show that Hölder domains have no outward pointing (zero-angle) cusps.

-1+i

.------ - -----...s \

~!'""-------=--+.----o /u .. -----------.."",

-i

Fig.4.10

Chapter 5. Quasidisks

5.1 An Overview

A quasicircle in C is a (not necessarily rectifiable) Jordan curve J such that

diam J(a, b) ::; Mla - bl for a, bE J

where J(a, b) is the smaller are of J between a and b. The inner domain is called a quasidisk. This important concept was introduced by Ahlfors and appears in many different contexts (see e.g. Geh87).

There are two one-sided versions of quasidisks, the John domain and the linearly connected domain (see Fig. 5.1). We will discuss these types of domains in detail because they appear quite often. The corresponding conformal maps satisfy upper and lower Hölder conditions respectively and behave "tamely" in many respects. For example, finite angular derivatives are automatically unrestricted derivatives for John domains. A quasidisk is a linearly connected John domain.

(ii)

Fig.5.1. A John domain (i) and a linearly connected domain (ii). The definitions essentially say that the shaded domains must not be much longer than la - bl

Quasicircle Theorem. Let J be a Jordan curve in C and let f map ]j})

conformally onto the inner domain of J. Then the following conditions are equivalent:

(a) J is a quasicircle; (b) f is quasisymmetrie on 'IT'; (c) f has a quasiconformal extension to C; (d) there is a quasiconformal map of C onto C that maps 'IT' onto J.

5.1 An Overview 95

We shall now explain the concepts used here and describe the plan of proof.

A quasisymmetrie map of 1l' into C is a homeomorphism h such that, for ZI, Z2, Z3 E 1l',

(Ahlfors); we shall actually use Väisälä's formally stronger (but equivalent) definition. The implication (a) =? (b) follows from results about John domains and linearly connected domains.

There are several excellent books about quasieonformal maps, e.g. Ah166b, LeVi73 and Leh87. We will therefore say little about the general theory and concentrate on the relation to conformal mapping.

A quasiconformal map of C onto C is a homeomorphism h such that small circles are approximately mapped onto small ellipses of bounded axes ratio. The analytic formulation of this condition is that h( x + iy) is absolutely continuous in x for almost all y and in y for almost all x and that the partial derivatives are locally square integrable and satisfy the Beltrami differential equation

(2) Bh __ lI.(z)Bh c 1 11 Ir< ,.., lor a most a z E \L-, Oz Bz

where ft is a complex measurable function with

(3) Ift(z)1 ::; K < 1 for z E C.

We say then that h is K-quasiconformal. A geometrie definition can be given in terms of modules of curve families.

Our proof of (b) =? (c) uses the recent Douady-Earle extension method. This is more difficult than the usual Beurling-Ahlfors method but has the great advantage of giving a Möbius-invariant extension.

The implication (c) =? (d) is trivial. In order to show (d) =? (a) we use a deep theorem of Bojarski, Ahlfors and Bers without proof. It says that every quasiconformal map can be embedded into a family of quasiconformal maps that depends analytieally on a complex parameter >. E ]jJ) and reduces to the identity for >. = O. More generally we shall prove a result of Maue, Sad and Sullivan that such analytic families of injective maps always lead to quasicircles.

Where<J.s there are no conformal maps in jRn(n > 2) except for Möbius transformations, there are many interesting quasiconformal maps in jRn, see e.g. Väi71.

There is an extensive literat ure on conformal maps of]jJ) with quasieonformal extension to C, see e.g. Leh87. Their Schwarzian derivatives Sf form the universal Teichmüller space with the norm sup(1-lzI2)2ISf(z)l. Its closure is strictly less than the corresponding space for all conformal maps; see Geh78, AsGe86, Thu86, Ast88a, Kru89, Ham90.

96 Chapter 5. Quasidisks

5.2 J ohn Domains

Let MI, M 2 , ... denote suitable positive constants. The bounded simply connected domain G is called a John domain (MaSa79) if, for every rectilinear cross cut [a, b] of G,

(1)

holds for one of the two components H of G \ [a, b]; see Proposition 2.12 and Fig. 5.1 (i). It is clear that 8G is locally connected. If 8G is a piecewise smooth Jordan curve then G is a John domain if and only if there are no outward-pointing cusps.

We now show that the rectilinear cross cuts of the definition can be replaced by any crosscuts.

Proposition 5.1. Let G be a John domain. I/ C is a crosscut 0/ G then

(2)

holds fOT one 0/ the components H 0/ G \ C .

Proof. Let M 2 = 2MI + 2, furthermore 8 = diam C and S = [a, b] where a and bare the endpoints of the crosscut C. If diam G ~ M 28 then (2) is trivial. In the other case we can find Zo E G with Izo - al > (MI + 1)8. Let H be the component of G \ C that does not contain Zo and let Go be the component of G \ S that contains Zo; see Fig. 5.2.

We claim that H n Go c D(a,8). Otherwise there is zEH n Go with Iz - al > 8. Since also Izo - al > 8 it follows that S U C does not separate Zo and z. Since Su8G does not separate Zo and z and since (SuC)n(Su8G) = S is connected, it follows from Janiszewski's theorem (Section 1.1) that the union S U C u 8G does not separate Zo and z which is false because zEH.

Every component Gk(k = 1,2, ... ) of G\ Go is, by Proposition 2.13, also a component of G\ Sk for some segment Sk eS. The other component contains Zo and therefore has a diameter> M18. Hence (1) shows that diamGk ~ MI 8 so that H n Gk C D(a, (MI + 1)8). It follows that H C D(a, (MI + 1)8) which implies (2). 0

• Zo

Fig. 5.2 The shaded domain is H

5.2 John Domains 97

We now give an analytic characterization of John domains (Pom64, JeKe82a). We define (see Fig. 5.3)

(3) B(reit ) = {pei..? : r :::::: p:::::: 1,111 - tl :::::: 1[(1 - r)} (0:::::: r < 1),

(4) 1(reit ) = 1l' n BB(reit ) = {ei..? : 111 - tl :::::: 1[(1 - r)}.

The factor 1[ has been chosen so that B(O) = ~ and 1(0) = 1l'. Let again df(z) = dist(J(z), BG).

Theorem 5.2. Let f map [JI conformally onto G such that

(5) df(O) 2': cdiamG.

Then the following conditions are equivalent:

(i) G is a John domain; (ii) diamf(B(z)) :::::: M 3df(z) for z E [JI;

(iii) there exists 0: with 0 < 0: :::::: 1 such that

If'(()1 :::::: M 4 If'(z)1 C = I::) <>-1 for (E [JI n B(z), z E [JI,

(iv) there exists ß > 0 such that, for all arcs A c 1 c 1l',

A(A) :::::: ßA(I) 1

=? diamf(A):::::: 2" diamf(I)·

The function f is continuous in ~ by Theorem 2.1. The constants M3 , M4 ,

0:, ß depend only on each other, on MI and on Cj we shall use the assumption (5) only in the proof of (i) =? (ii). It is possible to replace (iv) by

(iv') for every c: > 0 there exists 8 > 0 such that

A(A) :::::: 8A(1) =? diamf(A):::::: c:diamf(I)

for all arcs A c 1 c 1l'.

r Fig.5.3. The shaded domains is B(z)

Proof. Let K 1 , K 2 , .•. denote suitable absolute constants. We write z = reit and (= pei..?


(i) * (ii) (see Fig. 5.3). If r S 1/2 then

diamf(B(z» S diamG S c-1df(O) S K 1c-1df(z)

by (5) and Corollary 1.6. Let now 1/2 < r < 1. Then the two arcs J± of 11' with (371"/2)(1 - r) S 1'19 - tl s 271"(1- r) do not intersect So = [_eit,O). By Corollary 4.18 there are non-euclidean segments S± from z to J± such that

(6)

Our choice of J± implies that S+ U S- separates B(z) from So. Hence f(B(z» c Hand f(So) C H' where Hand H' are the two components of G \ f(S+ U S-). We see therefore from (2) and (6) that

diamf(B(z» S diamH S 2K2M2df(z)

or that df(O) S diamf(So) S diamH' S 2K2M2df (z).

In the second case it follows from (5) that

diamf(B(z» S diamG S c-1df(O) S c-12K2M 2df(z).

Hence (ii) holds in both cases. (ii) * (iii). We define

(7) <p(r) = 11(1- x)I!,(x)1 2 dx for Os r < 1

and L1(r) {z: r S x < 1, ° S y S 1 - x} where z = x + iy. Since 1f'(x)1 s K3 1f'(z)1 for z E L1(r) by Corollary 1.6 and since f is univalent, we see that

<p(r) sKi 11 11-'" 1!'(zW dydx = Ki areaf(L1(r».

Since L1(r) C B(r) we thus obtain from (ii) that

(8)

where 0: = 1/(8K4 Mi}. Since <p'(x) = -(1 - x)If'(x)12 by (7), we conclude that, for r S P < 1,

log <p(p) = r <p' (x) dx < _ r ~ dx = 20: log 1 - P . <p(r) Jr <p(x) - Jr I-x l-r

Using that 1f'(p)1 s K5 1f'(x)1 for p S x S (1 + p)/2 we deduce from (7) that

-(1- p)21!,(p)12 < Ki<p(p) S Ki<p(r) - p 1 (1 )2a 4 l-r

and thus, by (8),

5.2 John Domains 99

(9) 1!,(p)1 :s: (2/a)1/2 K 51!,(r)1 (1- p)a-l for r:S: p < 1. 1-r

We now apply (9) to f(eitz). If ( E B(z) then, again by Corollary 1.6,

whieh is our eondition (iii). (iii) =} (iv). Let ( E B(z). Integrating (iii) on the eireular are from reit to

reiiJ and then on [reiiJ , pe iiJ ] we obtain .

If(z) - f(()1 :s: M4 1!,(z)1 (It -191 + (1 - r)-a+1 I P (1 - X)"'-l dX) .

Sinee It - 191 :s: 7r(1 - r) it follows that

(10) diamf(B(z)) :s: M5(1- r)I!,(z)1 for z E lIJl.

Let A eIbe ares on 1I' and ehoose (, z such that A = 1((), 1= l(z). Then ( E B(z) and therefore, by (10) and (iii),

diamf(A) :s: M5(1- p)I!'(()1 :s: M6 (~ =:) a (1- r)I!,(z)1

:s: K7 M6 C =:) a diam f(I);

the last estimate follows from Proposition 4.19. This implies (iv') and thus (iv) beeause (1- p)/(l- r) = A(A)/A(I).

(iv) =} (i). Let [Wl, W2] with Wj = f(zj), Zj E 1I' be a crosscut of G and let I be the (shorter) are of 1I' between Zl and Z2, furthermore So the non-euelidean line from Zl to Z2 and Zo the point of So closest to O. We set M7 = exp(27rKs/ß) where K s is the eonstant in Corollary 4.18 and write

(11) {( EI: If(() - f(zo)1 > M7d,(zo)} = UA k

k

with open ares A k on I. It follows from Corollary 4.18 that 27r K s/ ß :s: Ks/w(zo, Ak ). Sinee I( - zol :s: y'2(1 - Izo!} for ( E I we see from (4.4.1) that

A(Ak ) < 47r(1 - Izol)w(zo, Ak ) _ (A) ß A(1) - 2(1 -Izol) - 27rw ZO, k:S: .

Henee (11) and (iv) show that

diam f(1) :s: 2M7d,(zo) + s~p diam f(A k ) :s: 2M7d,(zo) + ~ diam f(l)

and therefore that diam f (I) :s: 4M7d, (zo). Finally

d,(zo) :s: A(f(So)) :s: Kglwl - w21


by Theorem 4.20. If His the component of G \ [Wl, W2] bounded by f(l) and [Wl, W2] it follows that

diamH = diamf(I) ::; 4KgM71wl - w21.

Hence G is a John domain. This completes the proof of Theorem 5.2. D

Corollary 5.3. Let f map][Jl conformally onto a John domain. 1f z E D then

(12) If((l) - f((2)1 ::; Msdf(z) (1~1~1~11) a for (1, (2 E B(z).

Proof. Let r = Izl and 8 = 1(1 - (21. If 1 - 28 < r then (12) follows at once from (ii). Suppose therefore that P = 1 - 28 2': rand that (j = pjei{)j with PI ::; P2· If PI < P = 1 - 28 then If'(() I ::; KlOlf'((l)1 for I( - (11 ::; 8 by Corollary 1.6, and (12) follows easily from (iii) by integration over [(1, (2].

Let now P ::; PI ::; P2. Since (1 - r)If'(z)1 ::; 4df(z) by Corollary 1.4, it follows from (iii) that

If(p.ei{)j)-f(pei{)j)l< Mgdf(z) {Pj(l-x)a- l dx J - (l-r)a lp

< Mgdf(z) (1- p)a = Mg d (z) (~)a - (l-r)a a a f 1-r

for j = 1,2 and furthermore that

and these estimates together imply (12). D

Let again M1, M2 ,... denote positive constants. We prove a result (SmSt90) about the average growth; compare (4.6.10).

Theorem 5.4. 1f 00

(13)

maps ][Jl conformally onto a J ohn domain then

00

(14) L n l+Olan l2 < öo for some 8 > O. n=l

Proof. For 0 ::; r < 1 we consider

(15)

5.2 John Domains 101

Since 1 is univalent it follows from Theorem 5.2 (ii) that

If we integrate this inequality over 0 ::; t ::; 211" and substitute T = 79 + t we see from (15) that, for 1/2 ::; r < 1,

11" 11 cp(p) dp ::; M1(1 - r)cp(r).

Writing 8 = 1I"/(2Mt} we therefore have

! [(1 - r)-2611 cp(p) dP] ::; 0 for 1/2::; r < 1.

Since cp is increasing we conclude that

(1 - r)-26+1cp(r) ::; (1 - r)- 2611 cp(p) dp::; 226 t cp(p) dp r Jl/2

for 1/2 ::; r < 1 and thus, by (15),

We now return to the angular derivative and the isogonality defined in Section 4.3. We show that for John domains the angular approach can be replaced by unrestricted approach in ][»j compare War35b.

Theorem 5.5. Let 1 map ][» conlormally onto a John domain G and let ( E 1I'. (i) 11 f'(() =I- 00 exists then

(16) I(z) - I(() -t f'(() as z -t (, z E iij. z-(

(ii) 111 is isogonal at ( then there is'Y such that

(17) I(z) - I(() as z -t r, z E iij arg z _ ( -t 'Y ..

and 8G has a tangent at I(().

Prool (see Fig. 5.4.). (i) It follows from (4.3.1) that there are Pn with 0 < Pn < 1 such that

(18) I I(z*) - t() -!'(()I < .!. for z* E Lln , n = 1,2, ... , z* - n


where L1n is the Stolz angle {larg(l - (z)1 < 7r/2 - l/n, Iz - (I< Pn}. Let now z E ]]J) and Iz - (I < Pn. By (18) it is sufficient to consider the case z 'f- L1n · We choose z* E 8L1n with Iz* - (I = Iz - (I and write

j(z) - j(() _ !'(() = (j(z*l- j(() _ !'(()) z* - ( + z* - z !'(() z-( z -( z-( z-(

j(z*) - f(z) z-(

Since z, z* E B(r() it follows from Corollary 5.3 that, with r = 1 - Iz - (I,

I j(z*) - j(z) I ::; M4 df(r() I z* - z 1° ::; M41!'(r()I~

z - ( 1 - r 1 - r n°

because Iz* - zl ::; Iz - (I/n, hence from (18) that

I j(z) - j(() - !'(()I ::; (1 + I!'(()I).!. + M41!'(r()I~ z - ( n n°

and (16) follows because 1f'(r()1 is bounded (Proposition 4.7). (ii) Let now j be isogonal at (. By (4.3.5) and (4.3.8), there are Pn > 0

such that

(19) larg j(z*) - j(() - ,I < .!., z* - ( n I j(z*) - j(() I 1

(z* - ()f'(z*) >:2 for z* E L1n

where L1n is defined as in (i). Given z E ]]J) with Iz - (I < Pn and z 'f- L1n we choose z* E 8L1n with Iz* - (I = Iz - (I and write

j(z) - j(() = j(z*) - f(() z* - ( (1- f(z*) - f(Z)) . z-( z*-( z-( f(z*)-j(()

It follows from Corollary 5.3 and (19) that

I f(z*) - j(z) I 2M4 df (z*) I z - z* 1° M 1 -lz*1 M5

j(z*) - j(() ::; Iz* - (l1f'(z*)1 1-lz*1 ::; 5 1z* - (I ::; ~. Since 1 arg[(z* - ()/(z - ()]I < l/n we conclude from (19) that

larg j(z) - j(() _ ,I < ~ + M6 •

z-( n n

Hence (17) holds, and it easily follows from (17) that 8G has a tangent at j((); see Section 3.4. D

It follows from Corollary 5.3 that

(20) Ij(() - j(r()1 ::; M7df (r() for 0::; r < 1, (E 1l.

This means that the "curvilinear sector"

(21) {w E C : Iw - f(r()1 < Mi1Ij(() - f(r()I, 0::; r < I}

5.3 Linearly Connected Domains 103

Fig. 5.4. The Stolz angle .1n is shaded

of "vertex" f( () lies in the image domain G. If G contains however a rectilinear sector of fixed size at f (() for each ( E 1I', then ßG is rectifiable (and even f' E HP for some p > 1; FiLe87). There exist John domains such that G contains a sector at f(() for almost no ( E 1I' (Section 6.5). See e.g. GeHaMa89 and NäVä91 for furt her results on John domains.

It also follows from Corollary 5.3 that f satisfies a Hölder condition. Hence every John domain is a Hölder domain (see Section 4.6) but not conversely (BePo82).

Exercises 5.2

1. Show that every bounded convex domain is a John domain.

2. Let GI and G 2 be John domains. Show that GI U G2 is a John domain if it is simply connected.

3. Let f map ][Jl conformally onto a John domain. Show that

11 1!,(p()1 dp = O(df(r()) = 0((1 - r)") as r ---+ 1-

uniformly in ( E 1I' for some Q > O.

4. Show that condition (ii) of Theorem 5.2 can be replaced by

diamf(I(z)) ::::: Mdf(z) for z E []).

5. If the John domain is bounded by a Jordan curve without tangents, show that there is nowhere a finite nonzero angular derivative.

5.3 Linearly Connected Domains

We turn now to a concept that is dual to the concept of a John domain. We say that the simply connected domain G c C is linearly connected if any two points Wt,W2 E G can be connected by a curve A c G such that (Fig. 5.1 (ii))


by Mb M 2 , • .• we denote suitable positive constants. Linearly connected domains have been studied e.g. by NäPa83 and Zin85. The inner domain of a piecewise smooth Jordan curve is linearly connected if and only if it has no inward-pointing cusps (of inner angle 27r). For John domains, we had to exclude outward-pointing cusps (of inner angle 0).

Proposition 5.6. Let f map I!)) conformally onto a linearly connected domain G. Then f is continuous in TI) with values in C and

(2)

where S is the non-euclidean segment !rom Zl to Z2.

Hence every finite value can be assumed only once. (The value 00 can be assumed finitely often, see Pom64, Satz 3.9.) In particular a bounded linearly connected domain is a Jordan domain.

Proof. Let ( E '][' and let p be the prime end corresponding to (. Suppose that its impression is not a single point, see Section 2.5. Let Wo be a principal point of p (possibly = 00) and w #- 00 some other point in the impression. We choose p with 0 < 4p < Iw - wol. By Theorem 2.16 there exists Zl E I!))

such that If(zd -wl < p/M1 • Since Wo is a principal point, there is a crosscut Q of I!)) with dist (f (Q), w) > 3p that separates Zl from ( and therefore also from some Z2 EI!)) with If(Z2) - wl < p/M1 ; see Fig. 5.5. By (1) we can find a curve A c G from f(Zl) to f(Z2) such that diamA ::::: M1If(Zl) - f(Z2)1 < 2p. Hence A c D (w,3p) so that f(Q) nA = 0 which is false because f-l(A) has to intersect Q. Hence the impression of p reduces to a point so that f has an unrestricted limit at ( by Corollary 2.17. It follows that f has a continuous extension to TI). The estimate (2) with M 2 = K 6 M 1 is therefore an immediate consequence of (1) and Theorem 4.20 with C = f-l(A). 0

_-r---_~!

lT z, ....... _

Fig.5.5

The next result is a dual to Theorem 5.2 (iii) and Corollary 5.3 on John domains. Let I(z) again be defined by (5.2.4).

Theorem 5.7. Let f map I!)) conformally onto a linearly connected domain. Then there exists ß < 2 such that

5.3 Linearly Connected Domains 105

(3) ( )ß-l

11'(p()1 ~ ~1f'(r()1 ! =: lor (E1I', O~r~p<l,

The last inequality (for z = 0) implies the lower Hölder condition

(5) 11(d - 1(2)1 ~ Mill(l - (21ß for (1, (2 E 1I'

which is equivalent to the Hölder condition

for the inverse function. See Gai72 and NäPa83 for explicit bounds in terms of the constant Ml in (1).

Proof. (a) For fixed b we consider the family.VJt of all functions I(z) = z + a2z2 + '" that are analytic and univalent in JI) and satisfy

(7)

where S is the non-euclidean segment from Zl to Z2. It is dear that the family VJt is compact with respect to locally uniform convergence in JI). Hence

(8)

is assumed. If la21 = 2 then I(z) = z(1- az)-2 with lai = 1 so that I(JI)) is a slit plane and (7) does not hold. It follows that ß < 2.

The Koebe transform (1.3.12) of I E VJt also satisfies (7) and thus belongs to VJt. Hence we see from (8) and (1.3.12) that

(9) - (1 - Iz12) -- - Z < ß 11 f"(z) 1

2 j'(z)- for z E JI), I E VJt

and we condude that

(10) 1 ~IOg[(1-r2)f'(r()ll = If"(r() _ 2r( I ~~. 8r I'(r() 1 - r2 1 - r2

If we integrate and then exponentiate we obtain

(1- p2)1f'(p()1 > (1- P 1+r)ß for 0~r<p<1 (1- r2)11'(r()1 - 1 + p 1- r

which implies (3) for I E VJt. Furthermore Corollary 1.4 shows that

(11) for 0~r~p<1, (E1I'.


(b) Let now I(lD») be linearly connected. To show (3) and (4) we may assume that 1(0) = 0 and 1'(0) = 1. Then I E rot with b = M 2 by (2), and (3) follows from part (a). Let p( be the point on the non-euclidean segment from (1 to (2 that is nearest to O. Let first Izl = r < p < 1. We see from (2) that

M 211((t) - 1((2)1 ~ diaml(S) ~ df(P()

and (4) follows from (11) because 1- P ~ 1(1 - (21/2 and df(r() ~ Mildf(Z) by Corollary 1.6. The case p < r follows easily from Corollary 1.6 and (2). 0

Remark. A family J of analytic functions

(12) I(z) = z + a2z2 + ... , I'(z) i=- 0 (z E j])))

is called linearly invariant (Pom64) if

(13) I E J, 7 E Möb(j]))) ::::} 1 0 7 - 1(7(0)) 7'(0)1'(7(0)) E J.

The order of J is defined by SUp{la21 : I E J}. The family S has order 2. In part (a) of the above proof, we have shown first that rot is linearly

invariant of order ß < 2 and next that (3) holds (with a different constant) for any linearly invariant family of order ß < 00.

Let 0 ~ 0: ~ 1. The function I is called o:-close-to-convex if it is analytic and univalent in j])) and if

(14) I arg[l'(z)/h'(z)JI ~ o:7r/2 (z E j])))

for some convex univalent function h. The case 0: = 1 has been considered in Section 3.6.

Proposition 5.8. Let 0 ~ 0: < 1. 11 1 is o:-close-to-convex then I(j]))) is linearly connected.

Prool. The function cp = 10 h-l is analytic in the convex domain H = h(j]))) and satisfies largcp'(v)1 ~ o:rr/2 for v E H by (14). If Zj E j])) (j = 1,2) and Vj = h(zj) then

cp(V2) - cp(Vl) = rl CP'(VI + t(V2 _ vt}) dt = rl eiargrp'lcp'l dt ~-~ h h

and thus, because cp(Vj) = I(zj),

Re I(Z2) - I(zt} ~ cos o:rr t Icp'l dt ~ cos(o:rr/2) A(C) V2 - VI 2 10 lVI - v21

where C = 10 h- l ([vl,V2]) is a curve in I(j]))) from I(zt) to I(Z2)' Hence (1) holds with MI = l/cos(o:rr/2) < 00. 0

5.4 Quasidisks and Quasisymmetric Functions 107

Exercises 5.3

1. Let J be a Jordan curve in C. Show that the inner domain of J is a John domain if and only if the outer domain is linearly connected (allowing now domains that contain 00).

2. Let 0 :S 0 :S 1. Show that the o-elose-to-convex functions form a linearly invariant family of order 1 + 0 (Rea56).

3. Let ~ be a linearly invariant family of order ß. Show that (Pom64, Satz 2.4)

2 (2) 2 2 2ß - 2 :S sup sup 1 -Izl IS,(z)1 < 2ß + Kß 'E'8 zEll)

where S, is the Schwarzian derivative and K is an absolute constant. (For the lower bound, consider a function where la21 is elose to ß.)

5.4 Quasidisks and Quasisymmetrie Functions

Let J be a Jordan eurve in C and let MI, M 2 , ••• denote suitable positive eonstants that depend only on J and the function f.

We say that J is a quasicircle (or quasiconformal curve) if

(1) diamJ(a,b):SM1Ia-bl for a,bEJ

where J (a, b) is the are (of smaller diameter) of J between a and b. The inner domain G of a quasicircle J is ealled a quasidisk. Note that J need not be reetifiable. Domains where (1) holds with diameter replaeed by length are ealled Lavrentiev domains and will be studied in Seetion 7.4.

A pieeewise smooth Jordan eurve is a quasicircle if and only if it has no cusps (of angle 0 or 271"). As a further example consider the domain

G={u+iv:0<u<1, u"'sin(7I"/u)<v<1} (0)0).

If a > 2 then J = oG is pieeewise smooth and has no eusps. If a = 2 then J is still a quasicircle. If however 0 < a < 2 then an = 1/(2n) E J, bn = 1/(2n + 1) E J and

diamJ(an,bn) 2n(2n+1) -.,.----'--:-'"-;--"'- > -+ 00 as n -+ 00 lan - bnl (2n + 1)'"

so that J is not a quasidisk. For z E JI)) we define again df(z) = dist(f(z),oG) and the are I(z)

{eit : It - arg zl S; 71"(1 - Izl)}.

Theorem 5.9. Let f map JI)) conformally onto G. Then the following conditions are equivalent:

(i) G is a quasidisk; (ii) G is a linearly connected John domain;


(iii) 1 is continuous in ll}, and iII(z)(1/2 S Izl < 1) has the endpoints eit!, eit2

(tl S t2 S h + 71") then

If G is a quasidisk then J = oG is a Jordan eurve so that 1 is eontinuous and injeetive in ll}. By Theorem 5.5, a finite angular derivative is an unrestricted derivative, and 1 is isogonal at ( E 11' if and only if J has a tangent at I((). Beeause of (ii) we ean use all the estimates obtained in Seetions 5.2 and 5.3. We shall establish furt her estimates in Seetion 5.6.

Proof. (i)::::} (ii). Let (i) be satisfied. If [a, b] is a erosseut of G then J(a, b) U [a, b] bounds a eomponent of G \ [a, b] that has diameter S Mlla - bl. Henee G is a John domain, see (5.2.1).

In order to show that G is linearly eonnected (see (5.3.1)), eonsider Wl, W2 E G; see Fig. 5.6. We may assume that [WI, W2] ct. G; otherwise we ehoose A = [WI,W2]. Let I((j) ((j E 11') be the boundary point on [WI,W2] nearest to Wj. By (1) one of the ares I of 11' from (1 to (2 satisfies diam/(I) S Mll/((I) - 1((2)1. If r is elose enough to 1 then l(rI) is a eurve in G of diameter 2Ml l/((I) - 1((2)1 < 2Ml lwl - w21 which ean be eonnected within G by eurves oflength < IWI - w21 to Wj. Together this gives a eurve A c G from Wl to W2 with diamA S (2Ml + 2)lwl - w21·

(ii) ::::} (iii). Sinee G is a John domain we see from Theorem 5.2 that (see (5.2.3) )

(2) diam{f(eit ) : h S t S t2} S diam/(B(z)) S M 2df(z).

Sinee G is linearly eonneeted it follows from Theorem 5.7 with (j = eit; that

(3)

This gives (iii) with M 3 = M 2 M 4 •

(iii)::::} (i). To see this we ehoose I(eit !) = a and l(eit2 ) = b. o

Fig.5.6. Condition (iii) and the proof of (i) ::::} (ii)

5.4 Quasidisks and Quasisymmetric Functions 109

The map h of 1l' into <C is called quasisymmetrie if it is injective (one-toone) and if there exists a strictly increasing continuous function >.(x) (0 ~ x < +00) with >'(0) = 0 such that

(4)

see BeuAh56, TuVä80, Väi81, Väi84. Reversing the roles of Zl and Z3 we deduce that

(5)

Since >.(x) -+ 0 as x -+ 0 it follows from (4) that h is continuous. Another consequence is (5.1.1) which conversely implies (TuVä80) that h is quasisymmetric.

Proposition 5.10. If h is a quasisymmetrie map of 1l' into <C then J = h(1l') is a quasicircle.

Proof. We write Wj = h(zj), Zj E 1l' (j = 1, ... ,4). If Wl and W3 are given let Z2 lie on the (smaller) arc of 1l' between Zl and Z3' We choose Z4 on the opposite side of Z3 such that IZ4 - z31 = IZ3 - zll. Then IZ2 - z31 ~ IZ3 - z41 and thus, by (4),

! W2 - W3! = ! W4 - W3!! W2 - W3! ~ >'(1)2. Wl - W3 w3 - wl w3 - w4

D

We have thus proved (b) ::::} (a) of Section 5.1, and we now prove (a) ::::} (b).

Theorem 5.11. If f maps lDl conformally onto a quasidisk then f restricted to 1l' is quasisymmetrie.

We prove (4) with

(6) for 0 ~ x ~ 1,

for 1 < x < 00,

where Q > 0 and ß < 2 are the constants of Theorem 5.2 (iii) and Theorem 5.7. It is easy to modify the right-hand side such that it becomes continuous.

Proof. Let Wj = f(zj), Zj E 1l' for j = 1,2,3. We choose Zo E lDl such that I(zo) is the arc of 1l' between Zl and Z3 that contains Z2.

We first consider the case that IZI-Z21 ~ IZ2-Z31. It follows from Corollary 5.3 that

(7) IWI-W21~M7df(ZO)(1;1IZ211)a ~M8df(ZO)!Zl-Z2!a - Zo Z2 - Z3


because 27r(1 -Izol) ;::: IZ2 - z31. Since IW2 - w31 ;::: Mg-Idj(zo) by (5.3.4), we conclude that

by (5.3.4) while IWI - w21 :::; Mndj(zo) by the first inequality in (7). Hence

if 1 ZI - Z21 > l. Z2 - Z3 -

o

We need some further results about general quasisymmetrie maps. The first result will be used for the proof of (b) ::::} (c).

Proposition 5.12. If hand h* are quasisymmetrie maps of'IT' onto the same Jordan curve then <p = h*-I 0 h is a quasisymmetrie map of'IT' onto 'IT'.

Proof. Since h = h* o<p it follows from (5) and (4) that, with z; = <p(Zj),

l/A* (I:~ = :~ I) ~ 1 ~:~:g = ~:~:i~ 1 = 1 ~i;~~ = ~~;:~ 1 :::; A (I;~ = ;: I)

so that (4) for <p holds with A replaced by l/A*-I(l/A). o

Corollary 5.13. If fand g map]]J) and]]J)* conformally onto the inner and outer domains of a quasicircle, then <p = g-I 0 f is a quasisymmetrie map of 'IT' onto 'IT'.

Proof. Let Wo lie in the inner domain. It is easy to see that the function h(z) = l/(g(z-l) - wo) maps ]]J) conformally onto the inner domain of a quasicircle. Hence its restriction to 'IT' is quasisymmetrie by Theorem 5.11, and since M 1./ :::; Ih(z)1 :::; M I2 for Z E 'IT', it follows that g is quasisymmetrie on 'IT'. Hence g-I 0 fis quasisymmetrie by Theorem 5.11 and Proposition 5.12.0

Now we give a formulation (Väi84) in terms of crossratiosj these are invariant under all Möbius transformations.

Proposition 5.14. If his a quasisymmetrie map of'IT' into ethen, for Zj E 'IT' (j=1, ... ,4),

5.4 Quasidisks and Quasisymmetrie Functions 111

(8)

where the function >. * depends only on >..

Proof. We write

(9) b = IZI - z311 z2 - z41 . IZI - z411z2 - z31

By symmetry we may assurne that

(10)

hence that a2 :::; b. The quasisymmetry and (9) show that

If a :::0: 1 it follows that b* :::; >.( Jb)>'(b); if a :::0: b it follows that b* < >.(Jb)>'(l). This leaves the case that a< 1 and a < b. Then

IZ2 - z41 :::; IZ2 - z31 + IZ3 - zll + IZI - z41 :::; (2 + a)lzl - z41 < 31z1 - z41

by (9) and (10), and since furthermore

we conclude from (9) and the quasisymmetry that

o

Exercises 5.4

1. Write w = u + iv and consider the domain

G = {O < u < 1, 0< v < I} \ 0 {21n :::; u:::; ~:, 0 :::; v :::; 2~} n=l

where 1 < an < 2. When is Ga John domain, linearly connected or a quasidisk?

2. The snowfiake curve is defined by the following infinite construction: Start with an equilateral triangle and replace the middle third of each side by the other two sides of an outward-pointing equilateral triangle. In the resulting polygon, replace the middle third of each of the 12 sides by the other two sides of an equilateral triangle, and so on. Show that the snowflake curve is a non-rectifiable quasicircle.


3. Let f map !DJ eonformally onto a quasidisk. Show that there are eonstants 0 < CI: < ß < 2 and e > 0, M < 00 such that

for Zl, Z2 E '][' and for (1, (2 on the smaller are of '][' between Zl and Z2.

4. If G is a quasidisk show that A(C) :::; Mlwl - w21 where C is the hyperbolie geodesie from Wl to W2 in G. (Use e.g. Theorem 5.2 (iii) and Proposition 5.6.)

5. Let h be aquasisymmetrie map of'][' onto J and 'P of'][' onto ']['. If 9 is univalent in a domain G :::::> J, show that go h 0 'P is quasisymmetric.

6. Let hn be a sequenee of quasisymmetric maps of '][' into a fixed disk D with a fixed function >. in (4). Show that (hn ) is equicontinuous and deduee that some subsequenee eonverges uniformly to a quasisymmetric map or a eonstant.

5.5 Quasiconformal Extension

1. We first consider the extension of quasisymmetrie maps of '][' to quasiconformal maps of !DJ U ']['.

Theorem 5.15. The sense-preserving homeomorphism <P of'][' onto'][' can be extended to a homeomorphism 0 of lTh onto lTh that is real-analytic in !DJ and has the following properties:

(i) If a, T E Möb(!DJ) then the extension of a 0 <P 0 T is given by a 0 00 T.

(ii) If<p is quasisymmetrie then 0 is quasiconformal in !DJ.

A complex function is real-analytic if its real and imaginary parts are realanalytic functions of x and y. We shall show that

(1) ~/~ < Ii < 1 Ifr frl 8-z 8z - for z E !DJ

where the constant Ii depends only on the function >. in the definition (5.4.4) of quasisymmetry.

The proof of this important result (BeuAh56, DoEa86) will be postponed to later in this section. The Beurling-Ahlfors extension does not satisfy (i) and is not difficult to construct (see e.g. LeVi73, p.83); it would suffice to prove (b) :::} (c) in Section 5.1.

We shall construct the Douady-Earle extension. It is uniquely determined by the conformal naturalness property (i) together with

(2) i <p(()ld(1 = 0 :::} 0(0) = o.

We now deduce (BeuAh56, Tuk81 , DoEa86):

5.5 Quasiconformal Extension 113

Theorem 5.16. Every q3!asisJ/mmet"ic map h 011I' into C can be extended to a quasiconlormal map h 01 C onto C such that the extension 01

(3) (10 ho 7 , (1 E Mäb, 7 E Mäb([]))

is given by (1 0 h 0 7.

Our extension h is a homeomorphism of C onto C that is real-analytic in C \ 1I' and satisfies

(4) - - <K<l I {)k/{)k I {)z {)z -

for Izl -=1= 1.

We will not need the general theory of quasiconformal maps at this stage; we remark only that a homeomorphism k of C onto C satisfying (4) is quasiconformal because 1I' is a removable set for quasiconformality.

Proof. Since h is quasisymmetric it follows from Proposition 5.10 that J = h(1I') is a quasicircle which we may assume to be positively oriented. Let land 9 map []) and [])* onto the inner and out er domain of J such that g( 00) = 00. The restrictions of land 9 to 1I' are quasisymmetric by Theorem 5.11. Hence

(5)

are quasisymmetric (Proposition 5.12). Let ({5 denote the Douady-Earle extension of cp to [» (see Theorem 5.15).

Similarly there is an extension ;j of '1fJ to [»*. We define

(6) - {/(({5(Z)) for Izl ::=; 1, h(z) = _

g('1fJ(z)) for Izl ::::: 1;

if Izl = 1 then both definitions agree and k(z) = h(z) by (5). It follows that k is a homeomorphism of C onto C. This function is real-analytic in C \ 1I' by Theorem 5.15. If Izl < 1 then

{)k = 1,(-){)({5 {)z cp {)z'

{)h = I' (-) {)({5 {)z cp {)z '

so that (4) follows from (1); the case Izl > 1 is handled in a similar manner. Let (1 E Mäb and 7 E Mäb([])) , thus also 7 E Mäb([])*). Consider h* =

(10 h 0 7 and the conformal maps !* = (10 I 0 7 and g* = (10 9 0 7. The functions corresponding to (5) are cp* = 7-1 0CP07 and '1fJ* = 7-1 O'1fJ07. Hence it follows from the conformal naturalness property (i) that

-* {!*0({5*(z)=(1010({507(Z) forlzl::=;l, h (z) = - -

g*O'1fJ*(z) = (10g0'1fJ07(Z) forlzl:::::1.

Hence k * = (1 0 k 0 7 by (6). D


Theorem 5.17. Every conlormal map 1 01 lDl onto a quasidisk can be extended to a quasiconlormal map 1 01 C onto C. The extension 010' 0 1 0 r with 0' E Möb and r E Möb(lDl) is given by 0' 0 10 r.

This result (Ahl63, DoEa86) establishes (a) => (c) and thus (b) => (c) in Section 5.1 because (a) {:} (b) by Section 5.4.

Proof. We choose h in (5) as the restriction of 1 to '][' which is quasisymmetric by Theorem 5.11. Then <p is the identity and it follows from (i) and (2) that ip is also the identity, and (6) therefore defines the quasiconformal extension of 1 to C. 0

2. Let now <p be a sense-preserving homeomorphism of '][' onto ']['. The Douady-Earle extension of<p is based on the function

(7) 1 w-<p(() F(z,w) == F<p(z,w) = (()p(z, ()ld(1 (z,w E lDl)

l' 1 - w<p

where p(z, () is the Poisson kernel (3.1.4). Since J1'P(z, ()ld(1 = 1 we can rewrite (7) as

(8) F(z, w) = w _ 1 - Iwl 2 [ <p(() 1 - Iz I2 Id(l. 211" 11' 1 - w<p(() I( - zl2

Let 0', r E Möb(lDl). We have

(9) O'(w)-O'(w) () w-w ---'----===-'-'-q w = 1 - O'(w)O'(w) 1 - ww

Iq(w)1 = 1

where q is independent of w. Hence it follows from (7) with the substitution ( = r(s) that

F(r(z), w) = [ w - <p 0 r((\p(r(z), r(s))lr'(s)lldsl 11' 1 - w<p 0 r s

= q(w) [ o'(w) - 0' 0 <p 0 r(s) p(z, s)ldsl 11' 1 - O'(w)o' 0 <p 0 r(s)

and thus, by (7),

(10) F(r(z), O'- 1(w)) = q(w)Fuo<po7"(z, w), Iq(w)1 = 1.

This property will make it possible to transfer information from (0,0) to any point of lDl x lDl.

We now study (DoEa86) the local behaviour of F at (0,0). The Fourier coefficients of <p are

(11) an = - (<p(()ld(1 (n E Z). I1-n

211" 11'

We ass urne that

(12)


ao = ~ r <p(()ld(1 = 0. 27r J][

It follows from (8) that the partial derivatives satisfy

(13)

(14)

(15)

(16)

Fz(O,O) = -~ r (<p(()ld(1 = -a-l, 27r J][

Fw(O,O) = 1,

We deduee from Lemma 5.18 below that

(17)

We see from (16) that

00

(18) n=2

where we have used Parseval's formula and 1<p(()1 = 1. Henee it follows from (14), (15) and (16) that the Jacobian satisfies

(19) _ 2 2 2 82 1::::: Jw(O, O) = IFw(O, 0)1 -lFw(O,O)1 = 1-lbl ::::: 4'

Sinee F(O,O) = ° by (12), it follows from the implieit function theorem that F(z,w) = ° has a unique solution w = 0(z) near ° with 0(0) = 0. The partial derivatives are

(20) 0z = (FzFw-FJ;iw)jJw , 0z= (FzFw-FzFw)jJw

so that, by (13)-(16),

(21)

Henee we deduee from (17) and (19) that

(22)

so that 0 is a loeal diffeomorphism near 0.


Lemma 5.18. Write cp(eit ) = eiu(t) where u is real and continuous with u(t + 27r) = u(t) + 27r and suppose that

(23) u(t+s)-u(t)2:a

with 0 < a ::=; 7r /3. Then

(24)

(25)

for 7r 27r - <s<-3 - - 3 ' tE lR

Since cp is a sense preserving homeomorphism of 1r onto 1r it is clear that (23) holds for some a > 0 and (17) thus follows from (24).

Proof. We see from (11) that

1 1211" 1211" 8 = -2 (ei(t-r-u(t)+u(r)) - ei(-t+r-u(t)+u(r») dTdt. 47r a a

Writing T = t + sand taking the real part we obtain

(26)

1 r211" 12

11" {j = 27r2 Ja a sinssin[u(t + s) - u(t)] dtds

= 2~2111" sin s (111" v(t, s) dt) ds

where, because u(t + 27r) = u(t) + 27r,

v(t, s) = sin[u(t + s) - u(t)] + sin[u(t + 27r) - u(t + s + 7r)]

+ sin[u(t + s + 7r) - u(t + 7r)] + sin[u(t + 7r) - u(t + s)].

The terms in the square brackets sum up to 27r. Each is nonnegative for o ::=; s ::=; 7r and 2: a for 7r /3 ::=; s ::=; 27r /3 by (23). Hence Lemma 5.19 below shows that

{ 0 for 0 < s < 7r ,

v(t,s) > - -- 4sin3 a for7r/3::=;s::=;27r/3

and (24) follows from (26) . Furthermore, by (21) and (18),

(lall2 -la_112) (1 -lbI2) > 83

lal - a_ l bl 2 - 16

which implies (25). D


Lemma 5.19. 1f Xj ~ 0: > 0 (j = 1,2,3,4) and Xl + X2 + X3 + X4 = 211" then

(27) . . . . {4Sin3 0: if 0::::; 11"/3,

0' == smXI + smX2 + smX3 + smX4 ~ . o otherwzse.

Proof. We consider only the case 0: :::; 11"/3 and may assume that 0: :::; Xl :::; X2 :::; X3 :::; X4. We consider first the case that X4 :::; 11". Then all terms in (27) are non negative, furthermore 2XI + 2X3 :::; 211" and thus X3 :::; 11" - 0: and sin Xj ~ sin 0: for j = 1,2,3. It follows that

0' ~ 3sino: ~ 4sin30:.

Consider next the case that X4 > 11". Then X4 = 211" - Y with 0 :::; y < 11". Hence y = Xl + X2 + X3 and thus 0: :::; Xl :::; X2 :::; 11"/2, 0: :::; X3 :::; 11" - 20:. It follows that

therefore

siny = COS(XI + x2)sinx3 + COSX3 sin(xi + X2)

:::; COS 20: sin X3 + (cos Xl sin X2 + COS X2 sin Xl) cos 0:

:::; (sinX3 + sinX2 + sinXI) cos2 0: - sin3 0:

0' = sinXI + sinX2 + sinX3 - siny

~ (sinXI + sinX2 + sinX3) sin2 0: + sin3 0: ~ 4sin3 0:. 0

Lemma 5.20. For each z E ][)) there exists a unique W E ][)) with Fcp(z, w) = O.

Proof. Let Z E ][)) be fixed. We see from (8) that

(28) 1 r 1 - Iwl 2 1 - IzI2

IF(z,w) - wl :::; 211" Ir Icp() - wl I( _ ZI2Id(l.

Let now Wo E '][' and W n E][)) such that W n --+ Wo as n --+ 00. For ( (j. cp-I(WO) the integrand tends --+ 0, and since it is bounded by 4/(1 - Izl), it follows from Lebesgue's bounded convergence theorem that IF(z,wn ) - wnl --+ 0 as n --+ 00. Hence F( z, w) becomes continuous in ii» if we define F( z, w) = w for w E']['.

We conclude from (10) and (19) that w f-+ F(z, w) has a positive Jacobian for all w with F( z, w) = O. Hence our assertion is a consequence of the following special case of the Poincare-Hopf theorem (Mil65, p.35):

Consider a vector field that is continuous in ii» and continuously differentiable in][)). If it points outward at all points of,][, and if its Jacobian is positive at all its zeros, then it has exactly one zero in ][)). 0

3. Let again cp be a sense-preserving homeomorphism of 11' onto 11'. We define

(29) _() {CP(z) for z E 11'. cp z =

w where Fcp(z,w) = 0 for z E][));


there is a unique W E lDl with Fcp(z, w) = 0 by Lemma 5.20. We call 0 the Douady-Earle extension (or barycentric extension) of r.p (DoEa86). It is thus uniquely determined by

(30) Fcp(z, 0(z)) == ~ ( 0(z~ - r.p(() 1 - Izl: Id(1 = O. 27r Ir 1 - r.p(z)r.p(() I( - zi

The complex harmonie extension

r.p*(z) = ~ r.p(()p(z, Old(1 (z E lDl), r.p*(z) = r.p(z) (z E 'll')

is continuous in lD. We remark that r.p* is a homeomorphism of lD onto lD; we shall not use this result (Ch045). See e.g. CIShS84 for the theory of injective complex harmonie functions.

Proo! o! Theorem 5.15. (a) We deduce from (8) and (30) that, for (0 E 'll',

and thus

10(z) - r.p*(z)1 ~ ~ Ir.p(() - r.p((o)lp(z, Old(l·

If now z ...... (0, z E lDl then the right-hand side tends ...... 0 because r.p is continuous. Since r.p* is continuous in lD it follows that 0 is continuous in lD. The conformal naturalness condition (i) is a consequence of (30) and (10). We denote the extension 0 also by r.p.

Given Zo E lDl we choose 0', T E Möb(lDl) such that T(O) = Zo, 0'(0) = r.p(zo) and define h = 0'-1 0 r.p 0 T. Then h(O) = 0 and

(31) hz(O) = r.pz(ZO)T'(O)/O"(O) ,

hz(O) = r.pz(ZO)T'(O)/O"(O)

and thus 1r.pz(zo)1 2 -1r.pz(zoW > 0 by (22). Since r.p(lDl) c lDl and r.p('ll') = 'll' it follows that r.p is an unbranched proper covering and thus a homeomorphism of lD onto lD.

(b) Now let r.p be quasisymmetric on 'll'. We see from (31) that

(32)

We write h = eiu . Suppose that (23) does not hold so that u(t + s) - u(t) < Q: for some t and s with 7r/3 ~ s ~ 27r/3.Then Ih(ei(t+sl) - h(eit)1 < 2sin~ < Q:. If

Z - eit+117ri/9 3 - ,

then


IZI - z311 z2 - z41 < 2 = MI.

IZI - z411z2 - z31 - sin(7r/18)

We apply Proposition 5.14 to the quasisymmetrie function r.p. Sinee (5.4.8) is Möbius invariant and h = (J"~I 0 r.p 0 T we therefore see that

where A * depends only on A. We eonclude that

Henee one of the two ares of 1I' either between ZI and Z3 or between Z2 and Z4 is mapped onto an are of 1I' of length < M 3a l / 2 . Adding the are between ZI

and Z4 we get, in all four possible eases, an are I c 1I' such that

(33) A(I) ;:::: 1~7r, A(h(I)) < a + M 3a l / 2 < M 4 a l / 2 .

Sinee Ih(Z4)1 = 1 we eonclude that

The first integral is zero by (30) beeause h(O) = O. It follows that (23) holds if a < M5~1. Note that M 5 depends only on A* and thus A and not on Zoo

Consequently (32) and (25) show that

Ir.pz(zo)/r.pz(zo) 1 :::; Ii < 1 for Zo E lDl

for some eonstant Ii. Thus r.p is quasieonformal in lDl. D

For a later applieation we need a furt her property; again we use the Möbius invariant eondition (5.4.8).

Proposition 5.21. Let r.p be a sense-preserving quasisymmetrie map oi 1I' onto 1I' and suppose that

(34) Wv = eW+27riv/3, r.p(w) - w lor v-I 2 3 v - v J' -".

Then the Douady-Earle extension 0 oi r.p satisfies

(35) 10(0)1:::; ro < 1

where the constant ro depends only on A.

Proof. We may ass urne that 0(0) = r with 0 :::; r < 1. It follows from (30) that

(36) 27r = { (1 _ r - r.p(() ) Id(1 = { (1 - r)(r.p(() + 1) Id(l. Ifl 1 - rr.p( () Ifl 1 - rr.p( ()


We consider the interval

(37) 1= {( E 11': l<p(() - rl :::; 6(1- r)}

and obtain from (36) that

271":::; [2Id(1 + [ 2~1 - r~ Id(1 :::; 2A(I) + 2371" J1 Ilfv 6 1- r

and thus A(I) 2: 271"/3. Hence W v EI for some ZI, say ZI = 3, and there exists Zl E I with IZ1 - w31 = 1. We obtain from (34) and from Proposition 5.14 with Z2 = Wb Z3 = W2 and Z4 = W3 that

(38)

the second factors are = 1. Since l<p(zt} - w31 :::; l<p(zt} - rl + 1<p(W3) - rl :::; 12(1 - r) by (37) it follows from (38) that

1< IW2 - w31 :::; 1<P(Zl) - w21 + l<p(zt} - w31 :::; 12(1 - r)(1 + A*(2))

and thus <p(O) = r < 1 - [12(1 + A*(2))]-1. o

Exercises 5.5.

1. Let f map !Dl conformally onto a quasidisk in C and let T be the quasiconformal extension of Theorem 5.17. Show th~t the extension of the K~be transform of f is formally the Koebe transform of f. If f is odd, show that f is odd.

2. Let cpn be quasisymmetric maps of 11' onto 11' with the same function A in (5.4.4). If cpn -> cp as n -> 00 uniformlyon 11', show that CPn -> cp uniformly in !Dl.

3. Let cp be a sense-preserving homeomorphism of1l' onto 11' such that Icp(()-cp((')I :S MI( - (l Prove that

cp(r()-cp(()=O((1-r)IOg1~r) as r->1-.

See Ear89 for more precise results.

5.6 Analytic Families of Injections

We now prove a surprising result (MaSaSu83) of a different nature.

Theorem 5.22. Suppose that

(i) 1>.. : !Dl -> C is injective for each A E !Dl, (ii) 1>. (z) is analytic in A E !Dl for each fixed z E !Dl,

(iii) fo(z) = z for z E !Dl.

5.6 Analytic Families of Injections 121

Then each 1>.(>.. E ][))) can be extended to a homeomorphism ofITJ onto a closed quasidisk in C.

Thus 1>.., >.. E ][)), is an analytic family of injections of][)) into C that reduce to the identity for >.. = O. Note that the continuity of 1>.. is a conclusion, not an assumption.

Even more is true though we do not need this (BeRo86): Under the above assumptions, 1>. is the restriction to ][)) of a I>"I-quasiconformal map of C. Furthermore (SuThu86, BeRo86) the extension can be made such that it is analytic in {I>..I < 1/3}.

The proof of these additional facts is quite deep whereas Theorem 5.22 is an easy consequence of Schottky's theorem: If h is analytic and =I 0,1 in ][)) then

(1) Ih(>")1 ::; 4>(lh(O)I, (1 + 1>"1)/(1 - 1>"1)) for 1>"1 < 1

where 4>(x, t) (0 ::; x < +00) is a (universal) strictly increasing continuous function with 4>(0, t) = O. Very good bounds (Hem80, Hem88) are

(2) ( ) { exp[t1l'2/log(16/x)] for 0 < x::; 1 4> x, t < (e1l'x)t for 1 < x < 00

which we will however not need.

Proof of Theorem 5.22. Let Zl, Z2, Z3 be distinct points in][)). The function

is analytic and =I 0,1 with h(O) = (Zl - Z2)/(Z3 - Z2). Hence we obtain from (1) that

(3)

First let>.. E ][)) be fixed. If also Z2, Z3 are fixed we deduce from (3) that 1>.. is bounded in ]jJ). If we let Zl -t Z2 and use that 4>(0, t) = 0 we see that 1>.. is uniformly continuous in ]jJ) and thus has a continuous extension to ITJ. Hence (3) holds for Zb Z2, Z3 E ITJ. It follows that 1>.. is injective in ITJ and thus a homeomorphism. Furthermore (3) shows that 1>. is quasisymmetric on 1l'; see (5.4.4). Therefore Proposition 5.10 shows that 1>.(1l') is a quasicircle and thus fA (ITJ) a closed quasidisk in C. 0

The first application is to the Becker univalence criterion (Bec72):

Corollary 5.23. Let f be analytic and locally univalent in]jJ). If

(4)

then f maps ]jJ) conformally onto a quasidisk.


Proof. We may assume that 1(0) = 0 and 1'(0) = 1. For ), E ]j)) let 1>.. be defined by

(5) ),

log I~ = -log!, , f>..(0) = o. '"

Then (1 -lzI2)lzlnz)j IHz)1 ~ 1)'1 < 1 by (4) so that 1>, is univalent in]j)) by Theorem 1.11. It is clear that f>.. is analytic in ), and (5) implies lo(z) = z. Hence our assertion follows from Theorem 5.22 because I". = I. D

We turn now to the Nehari univalence criterion (AhWe62) on the Schwarzian derivative SI defined in (1.2.15).

Corollary 5.24. Let I be meromorphic and locally univalent in ]j)) and let

(6)

11 1(0) E C and 1"(0) = 0 then I maps ]j)) conlormally onto a (bounded) quasidisk.

If 1(0) = 00 or 1"(0) =I 0 we can find a Möbius transformation a such that 9 = a 0 I satisfies g(O) = g"(O) = O. Since Sg = SI we conclude from the corollary that I(]j))) is a quasidisk in C, Le. the image of a bounded quasidisk under a Möbius transformation.

Proof. We may additionally assume that 1(0) = 0 and 1'(0) = 1. Let UA and VA be the solutions of the differential equation

(7) ),

w" + 2",SI(Z)W = 0 (z E]j)))

that satisfy the initial conditions uA(O) = 0, u~(O) = 1 and vA(O) = 1, v~(O) = O. Since also u~(O) = 0 by (7) we see that

(8)

Since w" = 0 for), = 0 we have uo(z) == z, vo(z) == 1 and thus lo(z) == z. Using (7) it is easy to see that Sb = ~SI' Thus SI .. = SI and since I(z) = z+O(z3) we conclude from (8) that I". = I. If ), E ]j)) then

by (6). Hence IA is univalent in]j)) by the Nehari univalence criterion (Theorem 1.12) and has no pole because InO) = O. Finally U A and VA depend analytically on), and therefore also IA' Hence we conclude from Theorem 5.22 that f>..(]j))) is a (bounded) quasidisk, hence also I(]j))) = I".(]j))). D


Related univalence criteria hold for any quasidisk instead of ]]J) (and only for quasidisks)j see e.g. Ahl63, MaSa78 and Geh87.

We now need a deep embedding theorem for general quasiconformal maps (AhBer60j Leh87, p. 70):

Let h be a quasiconformal map of e onto e that keeps 0, 1 and 00 fixed. For A E ]]J) let h>. be the quasiconformal map that satisfies

8h>./8h>. A (9) &z 8z = ;,J-L(z) for almost all Z E C, IJ-L(z) I ~ K < 1

(see (5.1.2) and (5.1.3)) and keeps 0,1,00 fixed. Then h", = hand h>.(z) depends analytically on A E ]]J) for each z E ]]J).

Using this result we can now prove (d) =* (a) and thus complete the proof of the quasicirde theorem stated in Section 5.1.

Theorem 5.25. If his a quasiconformal map ofe onto e that keeps 00 fixed, then h(ll') is a quasicircle.

Proof. We may assume that h(O) = 0 and h(l) = 1. We consider the quasiconformal maps h>. (A E ]]J)) defined above. It follows from (9) and the normalization that ho(z) = z. Hence h(ll') = h",(ll') is a quasicirde by Theorem 5.22. 0

Another consequence of the embedding theorem is the Lehto majorant principle (see e.g. Leh87, p.77) that we formulate here for the dass S of normalized analytic univalent functions.

Theorem 5.26. Let cI! : S --+ C be an analytic functional that vanishes for the identity and satisfies

(10) 1cI!(f) I ::::; M for all fES.

If fES and if f has a K-quasiconformal extension to e keeping 00 fixed then

Here we call cI! an analytic functional if cI!(f>.) is an analytic function whenever f>. depends analytically on A. If f>. E S then If>.(z)1 ::::; (1 - Izl)-2 for z E ]]J) by the Koebe distortion theorem. An application of Vitali's theorem therefore shows that fik)(z) is analytic in A for k E N, z E ]]J). It follows that

any rational combination of fik ) (Zj) with Zj E ]]J) is an analytic functional provided that it is finite for all fES.

Proof. We apply the embedding theorem to the extension h of f / f(I). It follows from (9) that 8h>.(z)/&z = 0 for almost all z E ]]J) so that h>. is conformal in]]J) (LeVi57, p.183). The functions

f>.(z) = h>.(z)/h~(O) (z E]]J)), A E]]J)


belong therefore to Sand depend analytically on A; as above it can be shown that h~(O) is analytic. Hence <p(A) = €P(!>.) is analytic in ][Jl and satisfies <p(0) = 0 and 1<p(A)1 ::; M by (10). Hence 1<p(A)1 ::; IAIM by Schwarz's lemma and thus 1€P(f) I = 1<p(K)1 ::; KM because I", = I. 0

Applications. We assume that 1 E S has a K-quasiconformal extension to iC that keeps 00 fixed. Then the Schwarzian derivative satisfies (Küh69b)

(11)

this follows by considering the analytic functional Sf(z) (z fixed) and using that the left-hand side of (11) is bounded by 6 for 1 E S. Furthermore

(12) I f'(z) I l+lzl

log z 1 ( z ) ::; K log 1 _ I z I for z E ][Jl

because log[zf'(z)j I(z)] is an analytic functional vanishing for the identity which is bounded by log[(l + Izl)j(l-lzl)] for an 1 E S (Gru32; Pom75, p. 66; Dur83, p.126). If z, ( E ][Jl then (Küh71)

I (z - ()2 f'(z)f'() I 11 - z(1 2 (13) log (f(z) _ 1())2 ::; Klog (1 -l zI2) (1-1(12) ;

the corresponding estimate for the fun dass S is a consequence of the Golusin inequality (see e.g. Pom75, p.64; Dur83, p.126). See e.g. Sch075 for further estimates.

Exercises 5.6.

1. Show that Theorem 5.22 remains true if (iii) is replaced by the assumption that 10 maps ][Jl conformally onto a quasidisk.

2. Let 1 be analytic and j'(z) E A for z E ][Jl where A is a compact convex set with 0 rt A. Show that 1 maps ][Jl conformally onto a quasidisk. (Let 'lj;()..) map ][Jl conformally onto a flat rectangle containing 0 and 1 with 'lj;(0) = 0 and apply Proposition 1.10 to (1- 'lj;()..))z + 'lj;()..)/(z).)

3. Let bn()") (n = 0,1, ... ) be analytic in ][Jl and bn(O) = O. If

00

g>..(z) = z + I>n()..)z-n (Izl > 1) n=O

is univalent for ).. E ][Jl show that g>.. (1I') is a quasicircle and (Küh71) that Enlbn ()..)1 2 ::; 1)..1 2 •

4. In the Lehto majorant principle, replace €P(id) = 0 by €P(id) = a and show that now (Leh76)

I€PU)1 :::: (lai + KM)/(l + laIK/M).


5. Let ( E 11' and 0 ::; r < p < 1. If f maps j[J) conformally onto a John domain, show that

~ ::; If(p() - f(r()I(p - r)-II!'(r()1- 1 ::; MI

for some constant MI. If f has a II':-quasiconformal extension to iC keeping 00

fixed, deduce that (compare Theorems 5.2 and 5.7)

~ (1- P)1< < If'(p() I <M3 (1-r)1< M2 1 - r - f'(r() - 1 - P

Chapter 6. Linear Measure

6.1 An Overview

Linear measure A is a generalization of length and was studied in detail by Besicovitch (Bes38). We follow the excellent presentation in Fa185. The linear measure is an important special case of a Hausdorff measure to be discussed in Section 10.2; see Rog70. We shall write E ~ E' if the plane sets E and E' differ only by a set of zero linear measure.

We assume that J maps lIJl conformally onto G c <C. Then the angular limit J ( () exists and is finite for alm ost all ( E 1r.

Riesz-Privalov Theorem. 1J G is bounded by a rectifiable Jordan curve then f' ,belongs to the Hardy space H 1 and, Jor E C T,

(1) A(E) = 0 {=} A(f(E)) = O.

We shall study domains with rectifiable boundary in more detail in Chapter 7. In the present chapter we are more concerned with domains that have a "bad" boundary.

The Riesz-Privalov (or F. and M. Riesz) theorem together with Fatou's theorem leads to a fundamental result in the general theory of boundary behaviour.

Plessner's Theorem. 1J 9 is meromorphic in lIJl then, Jor almost all ( E T, either 9 has a finite angular limit at (, or g(..1) is dense in iC Jor every Stolz angle ..1 at (.

Thus the behaviour is, at almost all points, either very good or very bad. A consequence is:

Privalov Uniqueness Theorem. 1J a meromorphic function in lIJl has the angular limit 0 on a set oJ positive measure on T, then it vanishes identically.

We defined that J is conJormal at ( E T if

(2) !'(() = lim J(z) - J() = lim !,(z) -10,00 z->(,zELl z - ( z->(,zELl

6.1 An Overview 127

for every Stolz angle ..1 at (. We say that I is twisting at ( if I(z) winds around the boundary point I() infinitely often in both direetions as z ---+ (,

( E r for every eurve rending at (. This is the very opposite of eonformal.

McMillan Twist Theorem. At almost all points 01 '][', the map I is either conlormal or twisting.

Let Seet(f) denote the set of all ( E '][' sueh that I() is seetorially accessible from Gj this means that G eontains an open triangle of vertex I(c). Then Seet(f) ,g,{ ( E '][' : I is eonformal at (} by the twist theorem. The next result generalizes the absolute eontinuity property (1).

McMillan Sector Theorem. 11 E c Sect(f) then

A(E) = 0 {::} A(f(E)) = O.

The image of Sect(f) has a-finite linear measure, i.e. is the union of eountably many sets of finite linear measure.

Makarov Compression Theorem. There is a partition

'][' = Sect(f) U Ao U Al with A(f(Ao)) = 0, A(AI) = O.

If almost no point of BG is sectorially aeeessible then it follows that almost all of '][' is eompressed by I into a set of zero linear measure. This is an extreme ease of the "erowding effect" in numerical eonformal mapping. Another result of Makarov (Theorem 10.6) will show that a set of positive measure on '][' eannot be eompressed mueh further.

These results ean be expressed as relations between the linear measure A and the harmonie measure w of sets on BG. The Makarov eompression theorem shows that w is eoneentrated on a set of a-finite linear measure which proves a eonjeeture of 0xendal (0xe81).

The ease that Gis bounded by a Jordan eurve J is of particular interest. Then we ean also eonsider the harmonie measure w* with respeet to the outer domain of J. If T is the set of points on J where J has a tangent then there is a partition (BiCaGaJo89)

(3) J=TUBUB* with w(B)=O, w*(B*) =0.

Finally we mention a theorem about tangent direetions of arbitrary plane sets E. Its proof is similar to that of Proposition 6.23j see Sak64, Chapt. IX for details. For w E E, let C (w) denote the union of all halfiines L : w + eiiJ s, o :::; s < +00 sueh that there exist W n E E, W n =I- w with arg(wn - w) ---+ iJ, W n ---+ W as n ---+ 00.

Kolmogoroff-Vercenko Theorem. For every set E C C there is a canonical partition E = Eo U EI U E2 U E3 with

128 Chapter 6. Linear Measure

(0) A(Eo) = 0; (i) C(w) is a (complete) straight line for w E EI;

(ii) C(w) is a halfplane for w E E2 ;

(iii) C(w) = C for w E E3 •

Furthermore EI U E2 has a-finite linear measure. 1f B C EI U E2 and if ßC( w) is orthogonal to IR. for all wEB then the projection of B onto IR. has zero measure.

As an example, consider

E=[O,ljU U [O,ei/nj. nEZ,n;ioO

Then Eo = {O} U {e i / n : n E Z, n =f. O}, EI = U(O, ei / n), E2

E3 = (0,1).

6.2 Linear Measure and Measurability

{I} and

1. We begin with some concepts from general measure theory (Fa185). An outer measure J-t in C assigns to every set E c C a number J-t( E) 2 0 which may be infinite, such that J-t(0) = 0,

(1)

(2)

The outer measure J-t is called metric if

The set E is called J-t-measurable if

(4) J-t(X) = J-t(X n E) + J-t(X \ E) for all sets Xc C;

note that :S always holds by (2). Every set E with J-t(E) = 0 is J-t-measurable because J-t(X n E) + J-t(X \ E) :S J-t(E) + J-t(X) by (1).

The a-algebra of Borel sets is, by definition, the smallest a-algebra containing the open sets, Le. the intersection of all such a-algebras.

We now state a basic fact (Fa185, Th. 1.2, Th. 1.5): Let J-t be an outer measure. The collection Wl of all J-t-measurable sets is a a-algebra and the restriction of J-t to Wl is a (possibly infinite) measure, Le.

(5) J-t (91 En) = ; J-t(En) for disjoint En E Wl.

If J-t is metric then every Borel set belongs to Wl.

6.2 Linear Measure and Measurability 129

2. Next we introduce the concept of linear measure. We say that (Bk) is an c-cover of E if

(6) E c U Bk , diam Bk :::; c for all k; k

the countably many sets Bk are arbitrary but may be assumed to be convex. The outer linear measure A(E) of E c C is defined by

(7) A(E) = lim inf LdiamBk e-+O+ (Bk) k

where the infimum is taken over all c-covers of E. This infimum increases as c decreases so that the limit exists but may be infinite.

Proposition 6.1. The outer linear measure A is a metric outer measure. The collection ml of all A-measurable sets is a a-algebra that contains all Borel sets, and A is a measure on ml.

This is the linear measure. We shall say linearly measurable instead of A-measurable.

Proof. In view of the fact stated above it suffices to show that A is a metric outer measure.

It is trivial that A(0) = 0 and that (1) holds. To prove (2) let (Bnk)k be c-covers of En- Their totality is an c-cover of E = Un En . Hence

inf L diam Bk :::; L inf L diam Bnk (Bk) k n (Bnk) k

and (2) follows for c -+ O. Hence A is an outer measure. Now let p = dist(EI, E 2 ) > 0 and let (Bk) be an c-cover of E = EI U E 2 •

If c < p then no Bk intersects both EI and E2 so that we have a disjoint partition of (Bk) into c-covers (Bjk) of Ej (j = 1,2). It follows that

inf L diam Bk ~ inf L diam B 1k + inf L diam B 2k k k k

and therefore that A(E) ~ A(Ed + A(E2 ). Together with (2) this implies that (3) holds. Hence A is metric. D

For sets on IR the definitions of outer linear measure and outer Lebesgue measures coincide. Hence the linear measure on IR is identical with the Lebesgue measure. The same is true for sets on the unit circle ']['.

Every countable set in C has linear measure 0 by (2) because a point has zero measure. Every set E with A(E) = 0 is linearly measurable.

Every non-empty open set in C has infinite linear measure because it contains uncountably many disjoint line segments each of which has positive measure. Furthermore linear measure is not a-finite, i.e. it is not possible to write C as the union of countably many sets of finite linear measure.


We write E ~ E' if the sets E and E' in C differ only by a set of measure zero, i.e.

(8) E ~ E' {:} A(E \ E') = A(E' \ E) = O.

This is clearly an equivalenee relation. It is obvious that A is linear in the sense that

A(h(E)) = laIA(E) for h(z) = az + b.

If h is a contraction, i.e. a map from E into C with

(9)

then diamh(B) :::; diamB and therefore

(10) A(h(E)) :::; A(E).

In particular it follows that A(P) :::; A(E) if Pis the projeetion of E onto any line in Co

Proposition 6.2. Let J be a Jordan arc. Then A(J) is equal to the length of J. 1f A(J) < 00 and if J : 'IjJ(s), 0 :::; s :::; A(J) is the arc length parametrization of J then

(11) A('IjJ(A)) = A(A) for measurable Ac [0, A(J)].

Thus our present notation agrees with the notation introdueed in Seetion 1.1. In spite ofthis result, "irregular sets" offinite linear measure have nothing to do with lengthj see e.g. Bes38 and Fa185.

Proof. Let b:::; +00 be the length of J and let J = J1 U·· ·UJn be any partition into subares J" from W,,-l to W" (1/ = 1, ... , n). The projeetion PlI of J" onto the line through W,,-l and w" satisfies [W"-bW,,] C PlI and A(P,,) :::; A(J,,) so that

n n n

L Iw" - w"-ll :::; LA(P,,):::; LA(J,,) = A(J) ,,=1 ,,=1 ,,=1

beeause the sets J" are linearly measurable and disjoint exeept for the endpoints. Henee b :::; A(J) by (1.1.7). In particular b = 00 implies A(J) = 00.

Now let b< 00. The are length parametrization satisfies

1'IjJ(S2) - 'IjJ(sdl :::; S2 - SI for 0:::; SI :::; S2 :::; b

and is thus a eontraetion of 1= [0, b] onto J. Henee (10) shows that A(J) :::; A(I) = band thus A(J) = b. If Ac I then A('IjJ(A)) :::; A(A) and A(J\'IjJ(A)) :::; A(1 \ A) by (10). Henee the subadditivity (2) shows that

A(J) :::; A('IjJ(A)) + A(J \ 'IjJ(A)) :::; A(I) = b = A(J)


if A is measurablej see (4) with X = I. Hence equality holds in all these inequalities, in particular (11) holds. 0

Proposition 6.3. (i) If E is any set in C then there are open sets Hn ::l E such that

(12)

(ii) If E C C is linearly measurable and A(E) < 00 then there are closed sets An C E such that

(13)

See Fal85, p.8 for the proof of these results (Bes38). Part (i) shows that A is a regular outer measure. Intersections of countably many open sets are called Go-sets. Part (ii) says that every linearly measurable set E of finite measure can be written as

(14) E = B U Eo , B Borel set, A(Eo) = 0 j

more specifically B is an Pu-set, Le. the union of countably many closed sets.

Proposition 6.4. Let A be any set on 11' with A(A) > O. Then there exists ( E A such that

(15) A(A n I)/ A(I) --t 1 as A(I) --t 0,

where I are the ares of 11' with ( EI.

Here A is the outer measure. If (15) holds we say that ( is a point of density of A. See e.g HeSt69, p.274 for the proof. The concept of density plays an important role also for sets in Cj see e.g. Fal85.

3. Finally we consider the problem of measurability. Most questions dealing with preimages can be handled by the following result together with Proposition 6.1 and Proposition 6.3.

Proposition 6.5. Let f be a continuous map of]j]) into C. Then the 1!.0ints ( E 11' where the radial limit J«() exists form a Borel set, and if B c C is a Borel set then

(16) f-I(B) = {( E 11': J«() exists, f«() E B}

is also a Borel set.


Proof. Let d# denote the spherical distance on C and let Tn = 1 - l/n. For m ~ n and k = 1,2, ... , the set

is open because f is continuous. Hence

00 00 00

(17) A= nun Vmnk k=l m=l n=m

is a Borel set, and it is easy to see that f(() exists if and only if ( E A. Now let B be an open set and (Bk) an exhaustion of B by open sets with

Bk c B. Then

V~nk = {( E A : f(T() E Bk for Tm ~ T ~ Tn}

is a Borel set by the above result and because f is continuous. Hence

00 00 00

A'= nun V~nk k=l m=l n=m

is a Borel set and ( E A' if and only if ( E A and f(() E B. Consider finally the dass 9Jt of all sets B c C such that f-l(B) is a Borel

set. Then 9Jt is a a-algebra and we have shown above that 9Jt contains all open sets. Hence 9Jt contains the a-algebra of Borel sets. 0

The situation is more complicated with respect to images.

Proposition 6.6. Let f be continuous in IDJ and let the radial limit f(() exist fOT all ( in the measurable set A c 1I' . Then there are closed sets An C A such that f(An) is closed fOT n = 1,2, . .. and

00

(18) A = Eu U An, A(E) = O. n=l

Proof. By Egorov's theorem there are dosed sets An with An C An+! C A such that A(A \ An) < l/n and

f((1 - l/k)() --+ f(() as k --+ 00 uniformly for ( E An.

Thus f(An) is dosed because f is continuous in IDJ and (18) holds. 0

The continuous image of a Borel set need not be a Borel set. A Souslin set (or analytic set) has the form

(19)


where the sets A(nI, ... , nk) are compact and where the union is taken over NN, that is over the uncountably many sequences (nk)kEN of natural numbers. Every Borel set is a Souslin set, moreover the continuous image of a Borel set is a Souslin set (Kur52, p. 360, p. 390). Every Souslin set is linearly measurable (Rog70, p.48).

If fis continuous in Jl)l and if Ac 11' is a Borel or Souslin set then (see e.g. McM66, Theorem 7 (i))

(20) f(A) = {f(() : ( E A, f(() exists}

is a Souslin set and therefore linearly measurable, possibly with A(f(A)) = 00.

Conversely (BerNi92), given any nowhere dense perfect set A c 11' and any Souslin set E c C, there is a function f analytic in Jl)l such that the set of radial limits on A is equal to E.

The image of a measurable set under a continuous mapping need not be measurable (assuming the axiom of choice):

Proposition 6.7. Let f be continuous in Jl)l and let B c 11' be measurable. 1f the radial limit f(() exists for ( E Band if

(21) E c B, A(E) = 0 A(f(E)) = 0

then

(22) A c B , A measurable => f (A) linearly measurable.

1f f(B) c 11' and if (21) is false then (22) is false.

Condition (21) is a weak version of absolute continuity and will playa great role in the following sections.

Proof. By Proposition 6.6 we can write A = EU A* where A(E) = 0 and f(A*) is a Borel set. Hence f(A) is linearly measurable by (21).

Assume now that f(B) c 11' and that E C B has zero measure but f(E) has positive outer measure. Since every set on IR with positive outer Lebesgue measure contains a non-measurable set (Oxt80, p.24), we can find AcE such that f(A) is non-measurablej the set A is measurable because it has outer measure O. 0

Exercises 6.2

1. Let E be a continuurn. Use projection to show that A(E) ~ diamE.

2. Let J and J' be Jordan curves such that J lies in the inner domain of J'. If J is convex show that A(J) < A(J').

3. Let f map Jl)l conformally onto G and let F() denote the impression of the prime end of G corresponding to ( E 11'. Show that the sets A = {( E 11' : F() is one point} and U'EA F() are linearly measurable.


4. Let A'(E) = infLk diamBk where the infimum is taken over an covers of E (of any diameter). Show that A' is a non-metric outer measure. Show that A'(E) = diam E if E is a continuum and deduce that compact sets are in general not A' -measurable.

5. Let 9 be an increasing function on the interval I and let Ao, Al, A oo be the subsets where 9 has a zero, finite nonzero or infinite derivative respectively. Use the Kolmogoroff-Vercenko theorem (Section 6.1) to show that

A(Ao) + A(AI ) = A(!), A(Aoo ) = 0,

A(g(Ao}} = 0, A(g(AI }} + A(g(Aoo }} = A(g(I}}.

6.3 Rectifiable Curves

Let J be a Jordan curve in C and G its inner domain. If I maps]l)) conformally onto G then I has a continuous extension to iij and I maps 1[' one-to-one onto Jj see Theorem 2.6. Let Adenote the linear measurej it is the same as the length for rectifiable curves by Proposition 6.2.

The following theorem (Rie16, Pri19) is the basis of many proofs in the following sections.

Theorem 6.8. Let I map]l)) conlormally onto the inner domain 01 the Jordan curve J. Then

(1) A(J) < 00 {::} !' E H 1 •

11 A( J) < 00 and il A c 1[' is measurable then

(2) A(f(A)) = i 1!'(()lld(1

and I(A) is linearly measurable, furthermore

(3) A(E) = 0 {::} A(f(E)) = 0

lor E c 1[' and

(4) lim I(z) - I(() =f. 0,00 exists = !'(() lor almost all (E 1['. z---+(,zED z - (

The Hardy space H 1 consists of all functions 9 analytic in ]I)) with

(5) sup [lg(r()lld(1 < 00 j O<r<l lJr

see e.g. Dur70. If gEHl then g( () exists and is =f. 0 for almost all ( E 1[' and is integrable, furthermore

(6) ~ Ig(r() - g(()lld(1 -t 0, ~ Ig(r()lld(1 -t ~ Ig(()lld(1 as r -t 1.

6.3 Rectifiable Curves 135

Proof. (a) First let A(J) < 00. Since

n un(z) = L If(ze21riv/n) - f(ze 21ri(v-l)/n) 1

v=l

is subharmonic in ]jJ) and continuous in ~, it follows from the maximum principIe and from the definition (1.1.7) of the length that un(z) ::; A(J) for z E ]jJ).

We conclude that

(7) 121r r 1f'(reit)1 dt = lim un(r) ::; A(J) o n-+oo

for 0 ::; r < 1. Hence f' E H 1 .

(b) Conversely let f' E H 1 • Let A be an arc of 1!' and let to < h < ... < tn

where eito and eitn are the endpoints of A. Then

n n

"If(eitv ) - f(eitv-1)1 = lim "If(reitv ) - f(reitv-1)1 ~ r---+l~ v=l v=l

::; lim f 1f'(r()lld(l. r---+lJA Hence the definition oflength and (6) show that

(8) A(f(A)) ::; L 1f'(()lld(l·

We next show that this inequality holds for all measurable A c 1!'. If the arcs Ik cover Athen, by (6.2.1) and (6.2.2),

A(f(A)) ::; L A(f(Ik)) ::; L f 1f'lld(1 < f 1f'lld(1 + € k k hk ~

if the cover is suitably chosen. This implies (8). We see from the subadditivity and from (8) that, for measurable A,

A(J) = A(f(1!')) ::; A(f(A)) + A(f(1!' \ A))

::; f 1f'lld(1 + f 1f'lld(1 = f 1f'lld(1 ::; A(J) JA Jr\A Jr because of (7). It follows that A(J) < 00 and furthermore that equality holds in (8), Le. that (2) holds.

If A(E) = 0 then A(f(E)) = 0 by (2). Conversely suppose that A(f(E)) = O. Since f is a homeomorphism of 1!' onto J it follows from Proposition 6.3 (i) that there is a Borel set B :J E such that A(f(B)) = O. We conclude from (2) that f'(() = 0 for almost all ( E B so that A(E) ::; A(B) = 0 because f' E H 1 . Thus (3) holds, and f(A) is measurable by Proposition 6.7.


Since f' E H 1 it easily follows from (6) that

f(e it ) - f(l) = i l t eiT 1'(eiT ) dr

so that f(e it ) is absolutely continuous and has a finite nonzero derivative (as a function oft) almost everyVihere. Hence (4) is a consequence ofthe following proposition. 0

Proposition 6.9. Let f be analytic and bounded in 1Dl. 1f ( E 'll' and if

(9) f(z) - W tr' ::""":""-'--...,-- ---+ a E ""' as z ---+ ( z-(

restricted to z E 'll', then (9) holds fOT all z E 1Dl.

Proof. We write g(z) = (f(z) - w)j(z - () and g(() = a. For n = 1,2, ... the functions (z - ()l/ng(z) (z E 1Dl) belong to H 1 because (z - ()l/n-l does and f is bounded. They can therefore be represented as Poisson integrals

(z - ()l/ng(z) = hP(z,z')(z' - ()l/ng(z')ldz'l for z E 1Dl.

Since gis, by assumption, bounded on 'll', we obtain for fixed z E lDl and n ---+ 00

that g(z) = h p(z, z')g(z')ldz'l for z E 1Dl.

Since 9 restricted to 'll' is continuous at ( by (9), it follows (AhI66a, p.168) that g(z) ---+ g(() = a as z ---+ (, z E 1Dl. 0

The assumption of Theorem 6.8 that G = f(lDl) is a Jordan domain can be relaxed. Applying Theorem 6.8 to suitable Jordan domains obtained by covering ßG by finitely many small disks it is not difficult to show (Pom75, p.320) that

(10) A(ßG) < 00. {:} l' E H 1 •

The following theorem (Ale89) is deeperj we shall not give the proof.

Theorem 6.10. Let B be a continuum in C with A(B) < 00 and let Gk be the components of C \ B. 1f fk maps lDl conformally onto Gk then f~ E H 1

(with modifications fOT the unbounded component) and

(11) 2A(B) = L r If~(z)lldzl. k Ir

In particular, if f maps lDl conformally onto G and if A(ßG) < 00 then

(12) A(ßG) ~ h 11'(z)lldzl ~ 2A(ßG);

the left-hand estimate can be established as in the proof of Theorem 6.8.

6.3 Rectifiable Curves 137

We now prove an estimate (Lav36) that sharpens the implication A(E) > o '* A(f(E)) > O.

Proposition 6.11. Let f map ][)) conformally onto the inner domain of a Jordan curve of finite length L. 1f E c 11' is measurable then

(13) ( 211" L) A(f(E» ~ Lexp - A(E) log 11"11'(0)1 .

Proof. We may assume that L = 1. It follows from the inequality

(14) exp (A(~) L log <p(t) dt) :::; A(~) L <p(t) dt

between the geometrie and the arithmetie mean (HaLiPo67, p.137) that, with A = A(E) and 0 < r < 1,

211" log 11'(0)1 = { log II'(rz)IIdzl + ( log II'(rz)II dzl JE J1f\E

:::; AIog(~ L II'(rz)lldz l)

+ (211" - A) log (211" ~ A i\E II'(rz)lldzl) .

If we let r - 1 and then use (2) we obtain from L = 1 that

211" log 11' (0)1 :::; A log A(f(E)) - A log A - (211" - A) log(211" - A)

and (13) follows because the last two terms together are bounded by - 211" log 11". o

Finally we consider sequences of conformal maps (War30).

Theorem 6.12. Let fn(n = 1,2, ... ) and f map ][)) conformally onto the inner domains of the rectifiable Jordan curves Jn and J. 1f

(15) fn - f as n - 00 locally uniformly in][)),

(16)

then

(17)

In Theorem 2.11 we proved that (15) holds (even uniformly) if there are parametrizations Jn : <Pn((), ( E 11' and J: <p((), ( E '][' such that <Pn - 0, 1'(0) > O.


Proof. Let n = 1,2, .... By Theorem 6.8 the analytic functions

00 00

k=O k=O

belong to the Hardy space H 2 and satisfy, by Parseval's formula,

(19) A(Jn ) = ~ Ib 12 27f ~ nk ,

k=O

Since I/~ - 1'1 ::; (219nI2+21912)1/219n -91 it follows from the Schwarz inequality that

(20) 00

= 47f (A(Jn ) + A(J)) L Ibnk - bk l 2

k=O

by (18) and Parseval's formula. Given c: > 0 we choose m so large that L::::m Ibk l2 < c: and therefore

For large n, the first term on the right-hand side is < c: by (19) and (16) while the second term is < c by (15). Using again (15) we conclude that

00 m-I 00 00

k=O k=O k=m k=m

< c: + 6c: + 2c: = 9c:

for large n. Hence (17) follows from (20) and (16). D

Exercises 6.3

1. Show that a rectifiable Jordan curve has a tangent except possibly on a set of zero linear measure.

2. Let I map j[J) conformally onto a Jordan domain. Show that I is absolutely continuous on l' if and only if I has bounded variation on 1'.

3. Show that every real function <.p of bounded variation can be written as <.p = 7jJ 0 X where 7jJ is absolutely continuous and X is continuous and strictly increasing. (Map j[J) onto a domain bounded in part by the graph of <.p.)

4. Prove the converse of Theorem 6.12 assuming that In(O) -+ 1(0), Le. show that (17) implies (15) and (16), even In -+ I as n -+ 00 uniformly in jjj.

6.4 Plessner's Theorem and Twisting 139

6.4 Plessner's Theorem and Twisting

1. One of the central results (Ple27) on the boundary behaviour of general meromorphic functions is Plessner's theorem.

Theorem 6.13. 1f 9 is meromorphic in ]]Jl then, for almost all ( E '][', either

(i) 9 has a finite angular limit at (, or (ii) g(..:1) is dense in iC for every Stolz angle Ll at (.

Hence {I(I = I} can be partitioned into a nullset, a set where g(z) tends to a finite limit g( () as z ----> ( in every Stolz angle however broad it may be, and a set where g(z) comes elose to every c E iC as z ----> ( in every Stolz angle however thin it may be. In fact even more is correct (CoL066, p.147): The Stolz angle in (ii) can be replaced by any triangle in ]]Jl with vertex (. We say that ( is a Plessner point if (ii) holds.

If 9 is bounded and analytic in ]]Jl then (ii) is impossible so that 9 has an angular limit almost everywhere. This is Fatou 's theorem; we will however need it in our proof below. Another key ingredient will be the Lusin-Privalov construction of a suitable subdomain of ]]Jl with rectifiable boundary.

Proof. We consider all disks D with rational center c E Q x iQ and rational radius {j > 0, furthermore all rational numbers a and r with ° < a < 7r /6 and 1/2 < r < 1. Let

be an enumeration of these countably many objects. Let En be the set of all ( E '][' where 9 has no finite angular limit and where also

(1) {g(z) : I arg(l - (z)1 < an, rn ::::: Izl < I} n Dn = 0.

The union of these sets consists of the points where neither (i) nor (ii) hold. Hence it suffices to show that A(En ) = ° for all n.

We now drop the index n and consider the starlike Jordan domain

(2) H = D(r,O) U U {larg(l- (z)1 < a, r::::: Izl < I} ; (EE

see Fig. 6.1. It is easy to see that

(3) A(8H) ::::: 27r/ sina < 00.

Let 'P map ]]Jl conformally onto H. If Zl, ... , Zm are the zeros of 9 - c in D(O, r) then

(4) h(z) = (z - Zl)··· (z - zm)/(g(z) - c) ;t ° is bounded in H, by (1) and (2); note that cis the center of D. Hence ho 'P is bounded in ]]Jl and therefore has a nonzero angular limit almost everywhere on '][' by Fatou's theorem and the Riesz uniqueness theorem (CoL066, p. 22).


Now let

(5) A = {( E 1[': cp conformal at (, cp(() E E}.

If ( E Athen h has no nonzero angular limit at cp(() by (4) and the definition of E. Since cp is conformal and thus isogonal at ( it follows from Proposition 4.10 that ho cp has no nonzero angular limit at (; the domain H = cp(lDl) is tangential to 1[' at cp((). Hence A(A) = 0 by the last paragraph. Since cp is conformal almost everywhere on 1[' by (3) and Theorem 6.8, we conclude from (5) and (6.3.3) that A(E) = o. D

Fig.6.1. The starlike domain H = <p(lDl) obtained by the Lusin-Privalov construction

An easy consequence (Pri19, Pri56) is the very general Privalov uniqueness theorem.

Corollary 6.14. Let g(z) =t 0 be meromorphic in lDl. Then 9 has the angular limit 0 almost nowhere on 1['.

Proof. The function I/gis meromorphic in lDl. If 9 has the angular limit 0 at ( E 1[' then 1/ 9 has the angular limit 00 so that neither (i) nor (ii) hold. The set of these ( has therefore measure zero by Plessner's theorem. D

In the case of Bloch functions the Stolz angle in (ii) can be replaced by the radius.

Corollary 6.15. If 9 E T3 then, for almost all ( E 1[', either 9 has a finite angular limit, or the radial cluster set C[O,()(g, () is C.

Proof. Suppose that C[O,() (g, () =I- C. Then there exist c E C, 8 > 0 and ro < 1 such that Ig(r() - cl > 8 for ro < r < 1. Since 9 E T3 we can, by (4.2.5), choose the Stolz angle ..1 so thin that Ig(z) - g(lzl()1 < 8/2 for zELl. It follows that Ig(z) - cl > {j /2 for zELl, ro < Izl < 1 so that (ii) is not satisfied. Hence our assertion follows from Plessner's theorem. D


Applying this result to the Bloch function 9 = log!, (see Proposition 4.1) we obtain:

Corollary 6.16. 1f f maps]])) conformally into ethen, for almost all ( E 'lI', either f has a finite nonzero angular derivative !'((), or

(6) liminfl!,(r()1 = 0, limsupl!,(r()1 = +00. r~l r~l

In the general case it is not possible to replace the Stolz angle in (ii) by the radius. There exist (non-normal) functions 9 analytic in ]])) such that

(7) Ig(r()1 --+ +00 as r --+ 1 for almost all ( E 'lI'.

An example comes from iteration theory (Fat20, p.272): If 0 < lai< 1 then there is an analytic function 9 such that

(8) 9 (zt::z) = ag(z) for z E]]))

and this function satisfies (7). It is clear by Plessner's theorem that almost none of the radial limits can be an angular limit.

Another surprising example (CoLo66, p.166) states that for every set E c 'lI' of first category there is a non-constant analytic function 9 in ]])) such that g(r() --+ 0 (r --+ 1) for ( E E. There are sets of first category with measure 27r; see Section 2.6. Thus it is not possible to replace angular by radial limit in the Privalov uniqueness theorem.

Fig. 6.2. An example of twisting:Local pictorial representation of the boundary at each stage blowing up by some factor

2. Now let f map]])) conformally onto G c C. We say that fis twisting at ( E 'lI' if the angular limit f(() =I 00 exists and if

(9) liminf arg[J(z) - f(()] = -00, limsup arg[f(z) - f(()] = +00 z-C~Er z-C~Er

for every curve r c ]])) ending at (. Thus G twists around the boundary point f(() infinitely often in both directions.

Proposition 6.17. Let f map]])) conformally into C and let f(() =I 00 exist at ( E 'lI'. Then f is twisting at ( if and only if (9) holds with r replaced by a Stolz angle .1.


Proof. Let f not be twisting at ( so that (9) does not hold for so me curve r. Since log(f - f(()) is aBloch function by Proposition 4.1, it follows from Proposition 4.4 that arg(f(z) - f(()) is bounded below or above in .1. The converse is trivial because [0, () C .1. D

Theorem 6.18. Every conformal map of]]J) into C is either conformal or twisting at almost all points of 1r.

This is the McMillan twist theorem (McM69b). The points of1f where the behaviour is neither good nor exceedingly bad thus form a set of zero me.asure.

For the proof we need the following stronger version of Corollary 4.18; see Theorem 9.19 below or Pom75, p. 311, p.314.

Lemma 6.19. Let f map]]J) conformally into C and let 0 < 8 < 1. If z E ]]J)

and I is an are of 1f with w(z, I) 2: a > 0 then there exists a Borel set Bel with w(z, B) > (1 - 8)w(z, I) such that

If(() - f(z)1 :::; A(f(S)) < K(8,a)df(() for (E B

where S is the non-euclidean segment from z to ( and where K(8, a) depends only on 8 and a.

Proof of Theorem 6.18. We consider the Stolz angle

(10) .1(() = {z E]]J): larg(l- (z)1 < 7f/4, Iz - (I< 1/2} (( E 1f)

and define A = {( E 1f : f (() =/= 00 exists}. The functio~

(11) cp(z,()=argf(z)-f(() (z=/=(), =argf'(z) (z=() z-(

is continuous in ]]J) x ]]J). Let E denote the Borel set of ( E A such that both

(12) argf'(r() is unbounded above and below for 0 < r < 1,

(13) arg[f(z) - f(()] is bounded above or below in .1(().

Suppose that A(E) > O. It follows from (13) that there exist M < 00 and E' c E with A(E' ) > 0 such that, say,

(14) cp(z,() < M for z E .1((), (E E'.

The set E' has a point of density by Proposition 6.4 which we may assurne to be ( = 1. By (11) and (12) there exist rn -+ 1- such that

(15)


Fig.6.3

Let K b ... denote suitable absolute constants. Let In be the are of 'lI' indicated in Fig. 6.3. By Lemma 6.19, there exists (n E In n E' such that

r 1!,(z)lldzl:::; K1df(rn) 1sn

where Sn is the non-euclidean segment from rn to (n' If S~ is the subsegment (say from Zn to (n) that has non-euclidean distance ~ 1 from rn, then If(z) - f(rn)1 ~ K21df(rn) for z E S~ by (1.3.18). Hence

r r I !,(z) I 1st Idarg[f(z) - f(rn)]1 :::; 1st f(z) _ f(rn) Idzl:::; K 1K 2 n n

and it easily follows from (11) and Exercise 1.3.4 that

ICP((n, rn) - cp(rn, rn)1 :::; ICP((m rn) - cp(zn, rn)1 + Icp(zn, Tn) - cp(rn, rn)1 :::; K 3 •

Since rn E Ll((n) for large n, we therefore deduce from (14) that cp(rn, rn) < K 3 + M which contradicts (15). Hence we have shown that A(E) = O.

Now let ( E A \ E so that (12) or (13) are false. If (12) does not hold then the radial cluster set G[O,() (log f', () is different from iC so that, by Proposition 4.1 and Corollary 6.15, the Bloch function log f' has a finite angular limit at ( except possibly for a set of zero measure. On the other hand, if (13) does not hold then f is twisting at ( by Proposition 6.17. This proves Theorem 6.18 because A(A) = 211'. 0

Corollary 6.20. Let f map 11)) conformally into 11)) and let A c 'lI' be measurable. If f(() exists for ( E A and if f(A) c 'lI' then f is conformal at almost all points of A.

This is an immediate consequence (McM69b) of Theorem 6.18: If ( E A then f(() E 'lI' but f(lI))) eil)) so that f cannot be twisting at (.

Corollary 6.21 A conformal map of 11)) into C is conformal at almost all points where it is isogonal.


The map f was defined (Section 4.3) to be isogonal at ( E 'll' if f(() i= 00

exists and if arg(f(z) - f(()) - arg(z - () has a finite angular limit. Then f cannot be twisting at ( because (9) does not hold for r = [0, (). Hence the assertion follows from Theorem 6.18.

Note that conformal always implies isogonal. Hence it follows from Proposition 4.12 that there exists a conformal map onto a quasidisk that is twisting almost everywhere.

Exercises 6.4

1. Let 9 be non-constant and meromorphic in lI» and suppose that all angular limits g(() lie in the same countable set. Show that 9 has angular limits almost nowhere.

2. Consider a meromorphic function in lI» that has angular limits almost nowhere. Use the Collingwood maximality theorem to show that all points of 'll' \ E are Plessner points where the exceptional set E is of first category.

In the following problems let f map lI» conformally onto G c Co

3. Show that f cannot be twisting everywhere.

4. Show that G contains a spiral

f(() + r,(t)eit , 0::; t < +00 with r,(t) --> 0 (t --> 00)

for almost no ( E 'll'.

5. Let G lie in the inner domain of the rectifiable Jordan curve J. Show that f is conformal at almost all points ( E 'll' for which f(() E J.

6. Let J be a rectifiable Jordan curve. If A(J n BG) = 0 show that A(r1 (J n BG)) = 0 provided that G lies in the inner domain of J. (It is a deep result that the final assumption is redundant; see BiJo90.)

6.5 Sectorial Accessibility

1. Let f map lI» conformally onto G. We define Sect(f) as the set of all ( E 'll' such that f (() is sectorially accessible, Le. the angular limit f (() i= 00 exists and f maps a domain V c lI» with ( E V onto an open triangle ofvertex f(()j see Fig. 6.4.

m

Fig.6.4

6.5 Sectorial Accessibility 145

Proposition 6.22. All points where f is conformal belong to Sect(J), and f is conformal at almost all points of Sect(J).

This follows at once from Proposition 4.10 and from Theorem 6.18 because f is not twisting at any point of Sect(J).

We now use a variant of the Lusin-Privalov construction (Sak64, Chapt. IX; McM69b).

Proposition 6.23. There exist countably many domains Gn C G and closed sets An C '][' n af-l(Gn ) such that aGn is a rectifiable Jordan curue, f(An ) is closed and

(1) Sect(J) = UAn .

n

It follows that Sect(J) is an F".-set, furthermore that

(2) f(Sect(J)) = U f(An ) ,

n

where f(An ), the set of angular limits, is a closed set of finite linear measure. Hence

(3) f(Sect(J)) is an F".-set of a-finite linear measure.

Praof. We consider all rational numbers a E (0,1l") and alllines L : at + b, -00 < t < 00 with a, b E Q> x iQ>, a i- O. These countably many pairs can be enumerated as (Ln, an). Let A~ denote the set of all ( E Sect(J) with f(() C D(O,n) for which there is an open isoseeles triangle T C Gofbase line Ln, height 2: I/n and vertex angle an at f((); see Fig. 6.5. It follows from the definition that

00

(4) Sect(J) = U A~ . n=l

We now show that B~ = f(A~) is closed. Let Wk E B~ and Wk --+ Wo as k --+ 00. The corresponding triangles Tk converge to a triangle To of vertex Wo because the heights are 2: I/n. We have To C G because To is open, and Vo = f-l(To) has a boundary point (0 E '][' with f((o) = wo; see Exercise 2.5.5. Since (0 E A~ by the definition of A~ it follows that Wo E B~.

Let G~ be the union of all the open triangles corresponding to any ( E A~. This is an open set bounded by part of Ln together with a rotated Lipschitz graph because all triangles are isosceles and have the same vertex angle. The open set G~ has only finitely many components because the vertices lie in D(O, n) and the heights are 2: I/n. Each component is bounded by a rectifiable Jordan curve. These components partition B~ into finitely many closed sets.

We now renumber these components and corresponding sets as Gn and Bn . Since any two points wb W2 E Bn can be connected by a curve of diameter


< Mnlwl - w21 for some constant Mn we see from Exercise 4.5.4 that An = f-l(Bn ) n Sect(f) is also closed. Finally (1) follows from (4). D

~ G~

Fig.6.5

We deduce a generalization (McM69b) of (6.3.3), the McMillan sector theorem.

Theorem 6.24. Let f map JI)) conformally into Co If E C Sect(f) then

(5) A(E) = 0 ~ A(f(E)) = o.

Prooj. (a) Let E C Sect(f). We suppose first that A(E) = 0 and determine domains Gn and closed sets An according to Proposition 6.23. Let hn map JI))

conformally onto H n = f-l(Gn ) C JI)); see Fig. 6.6. Then gn = f 0 hn maps JI))

conformally onto Gn . We define

E n = An nE, Zn = h;;;l(En ) == {( : hn(() exists, hn(() E E n }

so that E = U En by (1). Since En = hn(Zn) C 1I' and hn(JI))) C JI)) it follows from Löwner's lemma (Proposition 4.15) and A(En ) = 0 that A(Zn) = o. Since aGn is rectifiable we conclude from Theorem 6.8 that

A(f(En )) = A(gn(Zn)) = 0 for all n

which implies A(f(E)) = O. (b) Conversely let A(f(E)) = O. Using the notation ofpart (a) we define

(6) E~ = {( E En : f is conformal at (}, Z~ = h;;;l(E~).

Let ( E E~. Since Gn contains a triangle ofvertex f(() and since f is conformal and thus isogonal at (, it follows from Proposition 4.10 that H n contains a triangle at (. Hence

(7) Z~ C Sect(hn ) .

Furthermore A(gn(Z~)) :::; A(gn(Zn)) = A(f(En )) :::; A(f(E)) = 0 and thus A(Z~) = 0 by Theorem 6.8. Hence we conclude from (7) and part (a) that A(E~) = A(hn(Z~)) = 0 and thus from (6) and Proposition 6.22 that A(En ) = 0 for all n. Since E = U En it follows that A(E) = O. D


Fig.6.6

2. Now we come to the Makarov compression theorem (Mak84, Mak85).

Theorem 6.25. If f maps]]J) conformally onto Q c ethen

(8) 1I' = Sect(f) U Ao U Al ,

where A(f(Ao)) = 0 and A(AI ) = O.

Together with Theorem 6.24 this shows that

(9) E c Sect(f) U Ao , A(E) = 0 =} A(f(E» = 0,

(10) E C Sect(f) U Al, A(f(E» = 0 =} A(E) = O.

Furthermore it follows from (3) that

(11) f(Sect(f) U Ao) has a-finite linear measure;

see JoWo88 for a generalization to multiply connected domains. If the total boundary oQ does not have a-finite linear measure we conelude

from (8) and (11) that f(A I ) is much larger than oQ \ f(At) in spite of the fact that A(At) = O. Another consequence of (5) and (8) is:

Corollary 6.26. If the points on oQ that are vertex of an open triangle in Q form a set of zero linear measure then there is a set Ao C 1I' such that

(12) A(Ao) = 21i", A(f(Ao)) = o.

This surprising fact (Mak84) says that, for very bad boundaries, the conformal map f compresses almost the entire unit drele into a set of zero linear measure. The first example for this was given in McMPi73.


In the opposite direction, even more is possible (Pir72): Let A be any perfect set on 11' and B any compact totally disconnected set in <C. Then there exists a conformal map 1 onto a Jordan domain a such that

Be I(A) c Ba. Arecent result (BiJo90) states that if J is a rectifiable Jordan curve and

1 any conformal map of ][} into ethen

(13) E c 11' , I(E) c J, A(E) > 0 =? A(f(E)) > O.

Thus the set I(Ao) of Corollary 6.26 cannot lie on any rectifiable curve.

3. We introduce (see Fig. 6.7) the dyadic points

(14) Znv = (1 - 2~ ) exp i7r(2;n - 1) (v = 1, ... , 2n ; n = 1,2, ... )

and the corresponding dyadic ares

(15) 1 - I( ) _ { it . 27r(v - 1) 27rV} nv - Znv - e. 2n ::; t::; 2n •

Dyadie arcs have the technical advantage that any two can intersect at inner points only if one is contained in the other. Every point of ][} is within a bounded non-euclidean distance of some dyadie point and the harmonie measure satisfies W(Znv, I nv ) > 1/2.

Fig.6.7. Some dyadic points; the heavily drawn four ares eorrespond to the outer four points (lengths are distorted)

Proolol Theorem 6.25. Let KI, K 2 , ••• denote suitable absolute constants. It follows from Proposition 6.22 and Corollary 6.16 that

(16) 11' ,g, Sect(f) U A , A = {( E 11': liminfl!,(r()1 = O} . r-+l

Let k 1,2, .... For ( E A we can find r < 1 such that 1!,(r()1 < 2-k •

We choose n such that 1 - 2-n ::; r < 1 - 2-n- 1 and v such that ( lies in the dyadie interval I nv• Let Znv be the corresponding dyadie point. It follows from (1.3.22) that I!,(znv) I < K 12-k • For fixed k we consider all dyadie arcs obtained this way and we delete those that are contained in a larger one. This gives us countably many dyadie points that we now write as (kj and corresponding dyadie arcs Jkj that cover A and are disjoint except for their endpoints. Note that 1!'((kj)1 < K 12-k •


By Lemma 6.19 we can choose K 2 so large that, if

(17)

then I(Bkj ) c D kj with suitable Borel sets Bkj C Jkj such that

(18)

We define the Borel sets

00

(19)

It follows that, for all m, the set I(Ao) is covered by the disks Dkj with k 2: m where, by (17),

indeed l-I(kjl < A(Jkj ) and the ares Jkj C 1f' (k fixed) are essentially disjoint so that the inner sum is < 211". It follows that A(f(Ao)) = O.

Let ( E A \ Zm and k 2: m. Then ( E Jkj for some j = j(). Since Jkj n (A \ Zm) C Jkj \ Bkj by (19) and since B kj C Jkj, we see that

because of (18). Since ( E Jkj and A(Jkj) --+ 0 as k --+ 00 we conclude that A \ ZTn has no point of density so that A(A \ ZTn) = 0 by Proposition 6.4. Since Zm C A we therefore see from (19) and (16) that

Ao ~ A ~ 1f' \ Sect(f) . o

4. Now let 1 map ][)) conformally onto a domain G that is starlike with respect to 0 such that 1(0) = O. Then, by Theorem 3.18,

(20) ( 1 [21< . ) I(z) = z!,(O)exp -:;;: Jo 10g(l- e-·tz)dß(t) for z E]j))

where the increasing function ß is given by

(21) ß(t) = lim arg I(reit ) = arg I(eit ). r-+l

Theorem 6.27. Let 1 be starlike in]j)). Then 1 is conlormal and satisfies

(22)


Jor almost all eit . Furthermore ß is absolutely eontinuous iJ and only iJ, Jor almost all {}, there exist w E 8G with arg w = {} sueh that G eontains some triangle oJ vertex w symmetrie to [0, w].

I ',,- I

X "-... ,,.. I'J-G '?_r ___ _

Fig. 6.8. The condition of Theorem 6.27 holds for {j but not for {j'

PraoJ. Let Adenote the set of all t such that J is conformal at eit and (22) holds. The starlike function J is nowhere twisting and is thus conformal almost everywhere on 11' by Theorem 6.18; this also follows from the fact that Re[zf'(z)/ J(z)] > o. Taking the logarithmic derivative in (20) we see that

(23) [ f'(z)] 1 1271" 1 - Izl 2

Re z J(z) = 271" 0 leit _ zI2dß(t) for z E]]J).

This Poisson-Stieltjes integral tends to ß' (t) for z = reit, r ----; 1- if ß has a finite derivative (Dur70, p. 4), hence almost everywhere by Lebesgue's theorem. It follows that A(A) = 271".

Let B denote the set of t such that, with ( = eit , the domain G contains a triangle of vertex J(() symmetrie to [0, J(()]. If t E Athen G contains a sector of vertex J(() and of any angle< 71" but not of angle> 71". Its midline and [0, J(()] form the angle .A = arg[(f'(()/ J(()]; see Fig. 6.8. It follows by (22) that AnB = {t E A: ß'(t) > O}.

We write ß = a + , where a is absolutely continuous and , is increasing with ,'(t) = 0 for almost all tEl = [0,271"]. We can interpret a,ß" also as measures. Since ,'(t) ~ ß'(t) < 00 for tE A it follows (Sak64, Chapt. IX) that ,(A) = 0 and thus ß(AnB) = a(AnB). Since a is absolutely continuous and a'(t) ~ ß'(t) = 0 for t E A \ B, we see that a(A n B) = a(A). Since A ~ I we conclude that ß(A n B) = a(A) = a(I). Hence it follows from (21) that the set of f) satisfying the last condition of our theorem has measure a(I); note that A(B \ A) = 0 by Theorem 6.18 so that J(B \ A) has linear measure 0 by Theorem 6.24 and therefore makes no contribution to the angular measure. Finally ß is absolutely continuous if and only if a(I) = ß(I) = 271". D

Exercises 6.5

1. Let f map ]]J) conformally into C. Show that, for E C Sect(f),

E measurable {? f(E) is linearly measurable.

Construct a function fand a measurable set E C 11' \ Sect(f) such that f(E) is not linearly measurable.

6.6 Jordan Curves and Harmonie Measure 151

2. Let B c C be compact. Deduce from Proposition 6.23 that the points of B that are vertex of an open triangle in C \ B form a set of O"-finite linear measure. (This also follows from the Kolmogoroff-Vercenko theorem in Section 6.1.)

3. Let J : w(t), 0 :S t :S 271" be a Jordan curve in C such that the right-hand derivative w~ (t) exists and is finite for almost all t. Show that the derivative w'(t) exists for almost all t. (Apply the Kolmogoroff-Vercenko theorem to the graphs of Re wand Im w.)

4. Show that the set Ao in the Makarov compression theorem can be chosen such that f is twisting at every point of Ao.

5. Let f be starlike with f(lIJ)) = {pe i19 : 0 :S p < r( 19)}. Show that ß is singular (Le. ß'(t) = 0 for almost all t) if and only if the last condition of Theorem 6.27 is satisfied for almost no 19.

6. Under the same assumption, suppose that r(19) :S 1 for all 19 and that E = {19 : r( 19) < I} is countable. If E is closed show that ß is absolutely continuous. If E is dense show that ß may be singular.

6.6 Jordan Curves and Harmonie Measure

We assume throughout this section that J is a Jordan eurve in C with inner domain G and outer domain G*. Let I map IIJ) eonformally onto G while f* maps IIJ)* = {Izl > I} U {oo} eonformally onto G*; see Fig. 6.9. Then land f* have eontinuous injeetive extensions to 11' by Theorem 2.6. We eonsider the harmonie measures

(1)

and define

w(E) == w(f(O), E, G) = A(f-1(E))j(27l") ,

w*(E) == w(f*(oo), E, G*) = A(f*-1(E))j(27l") ,

(2) S = {w E J: there is an open triangle of vertex w in G} = I(Seet(f)) ,

similarly S* with G* instead of G, and

(3) T = {w E J : J has a tangent at w} .

It is clear that T c S n S* . The MeMillan sector theorem states that

(4) A(E) = 0 ~ w(E) = 0, A(E*) = 0 ~ w*(E*) = 0

for E c Sand E* c S*. It follows that, for E c T,

(5) A(E) = 0 ~ w(E) = 0 ~ w*(E) = o.

Let again ,g, mean "equal up to a set of zero linear measure". We see from (4) and Proposition 6.22 that I is eonformal at 1-1 (w) for almost all wES and that f* is eonformal at f* -1 (w) for almost all w E S*. It follows that Sn S* ,g,T.


Fig.6.9. Here Wl E T, W2 E S \ S·, Wa E S· \ S, W4 ft S U S·

The Makarov compression theorem now reads

(6) J = S U Bo U BI = S* U B~ U B~

where A(Bo) = 0, w(B1) = 0 and A(B~) = 0, w*(Bi) = O. Thus the harmonie measure w is concentrated on the set S U Bo of a-finite linear measure (see (6.5.3)) in the sense that w(SUBo) = 1. In partieular (Mak84), if A(SUS*) = 0 then the harmonie measures wand w* are concentrated on a set of zero linear measure.

Proposition 6.28. 11 J is a quasicircle then

(7) T,g, S,g, S*, 1-1(T),g, Sect(J) .

Prool. Let A and A * be the sets on 'll' where 1 and rare conformal. Since A,g, Sect(J) by Proposition 6.22 we see from Theorem 6.24 that

Tc S,g,/(A) c 'll'j

the last inclusion is a consequence of Theorem 5.5 because a quasidisk is a John domain. Thus T,g, S and similarly T,g, S*. The second relation (7) now follows from (4). 0

Proposition 6.29. There is a starlike Jordan curve J such that

(8) T = 0, Sect(J) ,g, Sect(J*),g, 'll'.

Prool. Let r(t?) (0 ~ t? ~ 211') be a continuous positive function with r(O) = r(211') that does not have any finite or infinite derivative at any pointj see e.g. Fab07 for the existence of such a function. Then

J : r( t?)eUJ , 0 ~ t? ~ 211'

is a starlike Jordan curve with T = 0, and Sect(J),g, 'll' holds because 1 is conformal almost everywhere by Theorem 6.27, similarly Sect(J*),g, 'll'. 0

6.6 Jordan Curves and Harmonie Measure 153

Thus (7) may fail for non-quasicircles. The next result (BiCaGaJ089) holds for a11 Jordan curves.

Theorem 6.30. There is a partition

(9) J=TuBuB* with w(B)=O, w*(B*)=O.

Together with (5) this shows that wand w* are mutually singular if and only if A(T) = O. It follows (BrWe63) that the continuous functions in C that are analytic in C \ J form a Dirichlet algebra if and only if A(T) = O. In the opposite direction (Bis88), there are quasicircles of very big size (HausdorfI dimension> 1) such that I/M ::; w*(E)/w(E) ::; M for E c J. See e.g. BiJ092 for furt her results.

The sense-preserving homeomorphism ("conformal welding")

(10) cp = r-1 0 f

of,[, onto '[' was studied in Sections 2.3 and 5.4. It follows from (9) and (5) that cp is singular if and only if A(T) = O. The curve of Proposition 6.29 shows this may happen even if fand j* are both conformal almost everywhere.

Proof. We modify the proof of Theorem 6.25. Let Mb ... denote constants. It follows from Corollary 9.18 (to be proved later) that

Since Bkj c f-1(Dkj ) and cp(Bkj ) C j*-l(Dkj), we conclude from (6.5.17) that

and thus from (6.5.18) that

Hence, for k = 1,2, ... ,

j j

because 1 - I(kj I < A( Jkj) where the arcs Jkj C '[' are essentially disjoint. Therefore, by (6.5.19),

A(cp(Ao)) ::; A(cp(Zm)) ::; L L A(cp(Bkj )) < 2M42-2m k2:m j

so that A(cp(Ao)) = O.


Using partitions as in Theorem 6.25 we define

B = [j*(Aü) U !(Ad U (S n SO)] \ T,

B* = [!(Ao) U j*(Ai) u (S n SO)] \ T.

Since A(cp(Ao)) = 0 and since A(Ai) = 0 by Theorem 6.25, we see that

A(j*-I(B*)) :s; A(cp(Ao)) + A(Ai) + A(j*-I([S n S*] \ T)) = 0

because (S n S*) \ T has linear measure zero and thus also its preimage, by (4). Hence w*(B*) = 0 and similarly w(B) = O. Since

B U B* U T = !(Ao) U !(AI ) U j*(Aü) U j*(Ai) u (S n SO) = J

by (6.5.8), our assertion therefore follows if we replace B by B \ (B n B*). 0

Exercises 6.6

1. If J is Dini-smooth show that 1/M :S w*(E)/w(E) :S M for E C J where M is a constant.

2. Show that cp = r- 1 0 fis absolutely continuous if and only if w*(T) = 1.

3. If J is a quasicircle show that cp is absolutely continuous if and only if f is conformal almost everywhere.

4. Suppose that A(T) = 0 and extend f to j[))* by defining f(r() = r(rcp(()) for

r ~ 1 and ( E 'll'. Show that f is a homeomorphism of iC onto iC that satisfies 18f /8zl = 18f /ml for almost all z E j[))* ("almost nowhere quasiconformal").

5. Construct a Jordan curve J of a-finite linear measure and a closed set E C J such that A(E) = 0 and w(E) > O.

Chapter 7. Smirnov and Lavrentiev Domains

7.1 An Overview

We now consider domains with rectifiable boundaries in more detail. Let f map IlJ) conformally onto the inner domain G of the Jordan curve J. We have shown in Section 6.3 that J is rectifiable if and only if f' belongs to the Hardy space H I .

We call G a Smirnov domain if

(1) log 1!,(z)1 = 2~ 127r ~(~I::: log 1!'(()lld(1 for z E IlJ).

In the language of Hardy spaces this means that f' is an "outer function". This is not always true, and the Smirnov condition has not been completely understood from the geometrical point of view. See e.g. Dur70, p. 173 for its significance for approximation theory.

Zinsmeister has shown (Theorem 7.6) that Gis a Smirnov domain if J is Ahlfors-regular, i.e. if

(2) A({ZEJ:lz-wl<r})::;M1r for wEC, O<r<oo

for some constant MI' The condition of Ahlfors-regularity appears in many different contexts, for instance in G. David's work on the Cauchy integral operator.

We call G a Lavrentiev domain if

(3) A(J(a, b)) ::; M2 1a - bl for a, bE J,

where J(a, b) is the shorter are of J between a and b. This is the quasicircle condition (5.4.1) with diameter replaced by length, and Lavrentiev domains are the Ahlfors-regular quasidisks (Proposition 7.7). Tukia has proved (Theorem 7.9) that Lavrentiev curves can be characterized as the bilipschitz images of circles.

A different characterization was given by Jerison and Kenig (Theorem 7.11): A domain is a Lavrentiev domain if and only ifit is a Smirnov quasidisk such that I f' I satisfies the Coifman-Fefferman (Aoo ) condition. This is closely connected to the Muckenhoupt (Ap ) and (Bq) conditions, also to the space BMOA of analytic functions of bounded mean oscillation, the dual space of H I by a famous result of Fefferman. We shall only quote some results of this

156 Chapter 7. Smirnov and Lavrentiev Domains

rich theory (see e.g. Gar81) which has elose connections to stochastic processes (see e.g. Durr84).

Let again 1(reit ) = {eiß: 1'!9 - tl ~ 1T(1- rH. The condition

(4)

is related to the (Bq) condition and will appear repeatedly in this chapter: a) If (4) holds for some q > 0 then G is a Smirnov domain (Proposition

7.5) and even log f' E BMOA (Proposition 7.13). b) If J is Ahlfors-regular then (4) is satisfied with q < 1/3 (Theorem 7.6). c) The quasidisk G is a Lavrentiev domain if and only if (4) is true for

q = 1 (Theorem 7.8). Then (4) automatically holds for some q > 1 (Corollary 7.12) by a result of Gehring.

7.2 Integrals of Bloch Functions

It will be convenient first to consider integrals

(1) h(z) = l z (g(() - g(O))C I d( (z E lDJ)

of Bloch functions. These will also be needed in Section 10.4. Let K I , K 2 , • ••

denote suitable absolute constants and MI, M 2 , • .• other positive constants.

Proposition 7.1. 1f 9 E ß then h is continuous in iD and, for Izl ~ 1, O<ltl~1T,

(2) Ih(eitz~t-h(Z) -g((l_ I!I)z) +g(O)1 ~KIllgIIB.

Prooj. It follows from (4.2.4) that h'(z) = 0((1 -lzl)-I/2) as Izl --t 1. Hence his (Hölder-) continuous in iD; see Dur70, p. 74. In order to prove (2) we may assurne that g(O) = 0, 1/2 < Izl < 1 and 0 < Itl ~ 1T/2; see (4.2.4). Now let o < t ~ 1T/2 and ITI ~ t. We choose a such that t = 1T(1 - e-lT ). Integration by parts shows that

1:~: g'(() log ei~ z d( = h(eiT z) - h(e- lT z)(a + iT)g(e- lT z).

It follows from (4.2.1) and from

that the integrand is bounded by K 2 11g11B. Hence

7.2 Integrals of Bloch Functions 157

(3)

and we obtain (2) if we choose 7 = ±t and 7 = 0 and then subtract. 0

The next result (Zyg45) characterizes the integrals h of Bloch functions defined in (1).

Theorem 7.2. Let 9 be analytic in]j)). Then gE B if and only if h is continuous in ~ and

(4) max Ih(eitz) + h(e-itz) - 2h(z)1 :::; MIt for t > O. Izl=1

We say that a continuous function h : '[' -+ C belongs to the Zygmund class A. if (4) is satisfied. We easily obtain from Theorem 7.2 that

(5) h analytic in ]j)), h E A. <=? h' E B.

Proof. First let 9 E B. If we use (2) with t and -t and then subtract we obtain that

Ih(zeit ) + h(ze-it ) - 2h(z)1 :::; 2K1tllgll B ·

Conversely let (4) hold. Since h is analytic in ]j)) and continuous in ]j)) U '['

the Poisson integral formula shows that, for 0 < r < 1,

h(reis) _ h(eiiJ ) = ~ [21< 1- r2 [h(eiT ) _ h(eiiJ)] d7. 271" Jo 1- 2rcos(s - 7) + r2

Differentiating twice with respect to s and then substituting s = {}, 7 = {} ± t, we obtain from (l)that

iiJ,( iiJ) 1 - r211< (1 + r2) cos t - 2r - 2r sin2 t e 9 re =--71" 0 (1-2rcost+r2)3

x [h(ei(iJ+t») + h(ei(iJ-t») - 2h(eiiJ )] dt.

Using (3.1.6) it can be shown that the inner quotient is bounded by K 4 /[t(1 - r )(1 - 2r cos t + r2)]. Hence it follows from (4) that

(1 - r2)lg'(reiiJ )1 :::; 2K4M1 r 1 - r2 2 dt = 2K4M1

71" Jo 1 - 2r cos t + r

so that 9 E B. o

Remark. If we only assume (4) with h replaced by Im h, then we obtain that (1- r2)lIm[eiiJg'(reiiJ)] I :::; K sM 1 • This is enough to conclude that 9 E Bj see Exercise 4.2.6.

The non-negative finite measure p, on '[' is called a Zygmund measure if

(6)


where 1 and l' are adjacent ares of equal length. Hence the corresponding increasing function p,*(t) = p,({eiT : O:S: T:S: t}) is continuous and (6) can be written as

(7) 1p,*('!9 + t) + p,*('!9 - t) - 2p,* ('!9)I :s: M2t for t > O.

These measures can be used to characterize Bloch functions of positive real part (DuShSh66):

Proposition 7.3. The function 9 belongs to 8 and satisfies Re g( z) > 0 (z E ]]J)) ij and only ij it can be represented as

(8) . 1 (+z g(z) = zb + -( - dp,(() 1l' -z

(z E ]]J))

where p, is a Zygmund measure and b E IR.

Proof. Every analytic function 9 of positive real part can be represented in the form (8) where p, is a non-negative finite measure, and vice versa (Dur70, p.3). We may therefore assume that Reg(z) > 0 and g(O) = l.

Let j(z) = zexph(z) where h is defined by (1). Then

z!,(z)/ j(z) = 1 + zh'(z) = g(z) (z E ]]J))

has positive real part so that j is univalent and starlike in ]]J); see (3.6.5). It follows from (3.6.7) and (3.6.10) that, for so me constant a,

a+27rp,*(t) = lim argj(reit)=t+Imh(eit ). r-+l-

Thus (7) is equivalent to (4) with h replaced by Im h. Hence our assertion follows from Theorem 7.2 and the remark after its proof. D

Exercises 7.2

1. Let g(z) = 2:: bnzn be aBloch function. Show that

2. Deduce (DuShSh66) that 2:::'=1 n2 1bn l2 = O(m2 ) as m --+ 00.

3. Let 8 0 be defined as in (4.2.22). Show that (Zyg45)

gE 8 0 {o} max Ih(eitz) + h(e-itz) - 2h(z)1 = o(t) (t --+ 0). 1%1=1

7.3 Smirnov Domains and Ahlfors-Regularity 159

7.3 Smirnov Domains and Ahlfors-Regularity

Let f map lIJ) conformally onto G c C and let the linear measure A( oG) be finite. We do not assurne that Gis a Jordan domain. Then I' E H 1 by (6.3.10) so that f has a finite nonzero angular derivative I'«() for almost all ( E ']['.

Since I' has no zeros there is a representation (Dur70, p. 24)

(1) . 11(+z 11(+z log!'(z)=tb+-2 -I' -logl!'«()lld(I--2 -I' -dJ.L«() 11" 'll''' -z 11" 'll'''-z

for z E lIJ), where b = arg 1'(0) and where J.L is a non-negative singular measure, Le. the corresponding increasing function has derivative zero almost everywhere. It follows from (1) that

(2) 10gl!,(z)1 = ~p(z,()logl!'«()lld(l- ~P(z,()dJ.L«()

for z E lIJ) where p is again the Poisson kerne!. We say that f satisfies the Smimov condition if

(3) 10gl!,(z)l= ~p(z,()logl!'«()lId(1 for ZElIJ),

Le. if the singular measure J.L is zero. Then (Gar81, p.15)

(4) ~ Ilog 1!,(pz)1 - log 1!,(z)llldzl -+ 0 as p -+ 1 - .

We call G a Smimov domain if A(oG) < 00 and if (3) is satisfied. The Smirnov condition holds if log I' E H1 (hut not vice versa, see Dur72).

If log I' E H1 then (3) is satisfied even with I I' I replaced by 1'. We now construct an example (DuShSh66) to show that the Smirnov

condition is not always satisfied (KeLa37). There exists (Pir66) a strictly increasing continuous function J.L* that is

singular and satisfies

(5) sup 1J.L*(t + T) + J.L*(t - T) - 2J.L*(t)1 = O(T) as T -+ 0 O:S; t:S; 2 ...

("Zygmund smooth"). The corresponding measure J.L on '][' is thus a Zygmund measure so that (7.2.8) defines aBloch function with Reg(z) > 0 for z E lIJ).

We define

(6) f(z) = 1% exp( -ag«()) d( for z E lIJ)

where a = (2I1gI18)-1. Then

(1 -lzI 2 )1!"(z)/ !,(z)1 = a(1 -lzI2 )lg'(z)1 :::; 1/2 for z E lIJ)


so that f maps [J) conformally onto a quasidisk C by Corollary 5.23. Since M*'(t) = 0 for almost all t it follows (Dur70, p.4) that Reg(reit ) ----> 0 as r ----> 1- and thus, by (6),

(7) 1f'(()1 = 1 for almost all z E '][', 1f'(z)1 < 1 for z E [J);

the last fact follows from Reg(z) > o. Hence !' E H 1 and thus A(8C) < 00.

But the Smirnov condition (3) is not satisfied because of (7).

Let K 1 , K 2 , ••. denote suitable absolute constants and M1 , M 2 , ... constants that may depend on fand other parameters. We write again

(8) l(z) = {eit : It - arg zl :::; 7r(1 - Izl)} for z E [J).

Proposition 7.4. If f maps [J) conformally onto a Smirnov domain then, for z E [J),

(9) ! A(:(z)) l(z) log 1f'(()lld(1 -log 1f'(Z)I! :::; K 1 .

Proof. We may assurne that !,(O) = 1. We consider the Bloch function g = log!, and define h by (7.2.1); note that g(O) = O. Let Izl = r < 1. It follows from Proposition 7.1 that, with 0< P < 1 and t = 7r(1 - r),

!2~t (h (~zeit) - h (~ze -it )) - g(PZ)! :::; K 211 g 11 B :::; 6K 2 ;

see Proposition 4.1. Taking the real part we obtain from (7.2.1) that

! ~ r log 1f'(p()lld(l-log 1f'(PZ)I! :::; 6K2 • 2t JI(z)

It follows from (4) that, as P ----> 1-, the integral converges to the corresponding integral with P = 1. Hence (9) holds because 2t = A(I(z)). D

No geometrie characterization of the Smirnov condition is known. We shall give two sufficient conditions.

Proposition 7.5. Let f map [J) conformally onto C with A(8C) < 00. Suppose that q > 0 and

Then C is a Smirnov domain. If 0 < q < 1 then furthermore

(11) i p(z, ()If'((Wld(1 :::; M 2 1f'(zW for z E [J);

if C is in addition linearly connected then (10) implies (11) for 0< q :::; 1.

7.3 Smirnov Domains and Ahlfors-Regularity 161

The univalent function j(z) = (1 - z)2 satisfies (10) for all q > 0 but not (11) for q = 1 so that the assumption about linear connectedness cannot be deleted.

Proof. We first prove (11) where we may assurne that z = r > O. Let q :::; 1 if Gis linearly connected and q < 1 otherwise. We define L l = 0 and

m= [IOg_l_], Tn =l-(l-r)en , In=I(Tn ) (n=O,l, ... ,m) 1-T

so that TO = T and 0 :::; Tm < 1 - e- l , furthermore Tm+l = 0 and Im+! = 1(0) = 1I'. It follows from (10) that

Since 0 :::; Tn :::; T we see from (5.3.3) that

where ß < 2 if G is linearly connected and ß = 2 in the general case; see Theorem 5.7 and the remark following its proof. Since 1 - Tn 2: en - l (l - T)

and (ß - l)q < 1 we conclude that

1 p(T, ()If'(()lqld(1 :::; K 4M l lf'(TW f e-n(l-(ß-l)q) T n=O

and the sum converges. Now let q > o. It follows from Hölder's inequality that if (10) holds for

q > 1/2 then it also hold:, for q = 1/2. Hence (11) holds for some q > o. The inequality (6.3.14) between the geometrie and arithmetic mean therefore shows that

Hence we see from (2) that

r p(z,()dJ-l(():::; ~logM2 for z E][)) Ir q

and since J-l is a singular measure it follows (Gar81, p. 76) that J-l is zero. Hence j satisfies the Smirnov condition. 0


Let again MI, M2, ... denote suitable eonstants. We say that the eurve C c C is Ahlfors-regular if

(12) A(CnD(w,r))~Mlr for WEC, O<r<oo.

We ean write (12) as A(T(C) n][j)) ~ MI where T(Z) = az + b (z E C). It is more diffieult to show (Meyer, see Dav84) that C is Ahlfors-regular if and only if

(13) A(T(C)) ~ M 2 diamT(C) for all TE Möb.

If Eis bounded then (12) implies A(E) < 00 but not viee versa. For example, if

E = {x + ix3/ 2 sin(l/x) : 0 < x ~ 1}

then A(E n D(O, r)) > vr/M3. We eall G an Ahlfors-regular domain if oG is (loeally eonneeted and)

Ahlfors-regular. This is the weakest known geometrie eondition for Smirnov domains (Zin84).

Theorem 7.6. Let f map ][j) conformally onto an Ahlfors-regular domain. Then (10) holds for q < 1/3 so that G is a Smirnov domain.

Praof. We rest riet ourselves to prove (10) for q = 1/4. Hölder's inequality shows that, with z E ][j) and I = I(z),

(14) r 1f'((W/4Id(1 < (r 1f'(()lld(1 )1/4 ( r If(() _ f(Z)12/5Id(I)3/4. JI - J1 If(() - f(z)16/5 JI We split the seeond integral into the eontributions from

{( EI: en8 ~ If(() - f(z)1 < en+l8} (n = 0,1, ... )

where 8 = df(z) = dist(j(z), oG). The Ahlfors-regularity eondition (12) shows that

(15) r 1f'(()lld(1 00 M 1en +18 M 4

J1 If(() - f(z)1 6/ 5 ~ ~ (en8)6/5 = 81/ 5 .

Let tp(s) = (s + z)/(l + zs) for s E ][j). Then Itp'(s)1 = 11- ztp(s)12 /(1-lzI 2 ) ~ K 5(1-lzI 2) for tp(s) EI and thus, by Theorem 1.7,

r If(() - f(z)1 2/5Id(1 ~ K 5(1-lzI 2) r If(tp(s)) - f(tp(0))1 2/5Idsl J 1 Jcp-l{I)

~ K 6(1-lzI 2)7/51f'(z)1 2/5 .

Therefore we see from (14) and (15) that

l lf'((W/4 Id(1 ~ M58-1/20(1_lzI2)21/201f'(z)16/20 ~ M1A(I)If'(zW/4

beeause 8 2': (1 - IzI 2 )1f'(z)I/4 by Corollary 1.4. o

7.4 Lavrentiev Domains 163

A problem that has been extensively studied is the following: For what curves L in C is it true that

(16)

for all conformal maps f of ]]J) into C? Partial answers were given in HayWu81, FeHa87, FeZi87 and FeHeMa89,

see also Bae89. The complete solution (BiJo90) is that (16) holds if and only if L is Ahlfors-regular.

Ahlfors-regularity also plays a central role in the study of singular integral operators such as the Cauchy integral operator (Dav84)j see also e.g. Jo90.

Exercises 7.3

1. Show that the definition of a Smirnov domain G does not depend on the choice of the function f mapping ]]J) conformally onto G.

2. Let cp map]]J) conformally onto a Smirnov domain in]]J). If f maps ]]J) conformally onto the inner domain of a Dini-smooth curve, show that f 0 cp(]]J)) is a Smirnov domain.

3. Show that the inner domain of a rectifiable starlike curve is a Smirnov domain, also every bounded regulated domain (Section 3.5).

4. Show that the spiral w(t) = e-t+it" , 0 ~ t < 00, is Ahlfors-regular if a = 1 but rectifiable and nonregular if a > 1.

5. Let f be a conformal map such that !' E H 1 and 1/!, E HP for some p > O. Show that log!, E H 2 so that f(]]J)) is a Smirnov domain.

6. Let f map ]]J) conformally onto an Ahlfors-regular domain. Show (Dav84, Zin85) that

[ 1!,(z)!'«)1 ~~~(1 -Izl) JE' If(z) _ !«)12Id(1 < 00.

7.4 Lavrentiev Domains

Let Mb M2,' .. again denote positive constants. A Lavrentiev curve is a rectifiable Jordan curve J such that

(1) A(J(a,b))::;M1Ia-bl for a,bEJ

where J(a, b) is the shorter arc of J between a and bj see Fig. 7.1. Lavrentiev curves are also called "quasismooth" or "chord-arc-curves". The inner domain of a Lavrentiev curve is called a Lavrentiev domain.

For example, if cp( x) (0 ::; x ::; 1) is positive and satisfies a Lipschitz condition then {x + iy : 0 < x < 1, 0 < Y < cp(x)} is a Lavrentiev domain.

Proposition 7.7. A domain is a Lavrentiev domain if and only if it is an Ahlfors-regular quasidisk.


Proof. If Gis an Ahlfors-regular quasidisk then (1) follows at once from (5.4.1) and (7.3.12). Conversely let G be a Lavrentiev domain. It is trivial that G is a quasidisk.

To show that J is Ahlfors-regular, let w E C and first assume that r :s ro = diam J j(2MI + 3). Then there exists c E J with le - wl > (MI + l)r. If J n D(w,r) = 0 there is nothing to prove. Otherwise there exist a,b E J n aD( w, r) such that the are J' of J between a and b containing e lies outside D(w,r); see Fig. 7.1. Since A(J') 2:: 2(le-wl-r) > 2MIr 2:: MIla-bl it follows for the other arc J" of J between a and b that

A(J n D(w, r)) :s A(J") :s Mila - bl :s 2MIr.

Now let r > ro. Then A(J n D(w, r)) :s A(J) < [A(J)jro]r. o

c

Fig. 7.1. A Lavrentiev domain and the proof of its Ahlfors-regularity

We now give an analytic characterization (Zin85). Let l(z) be defined as in (7.3.8).

Theorem 7.8. Let f map JI)) conformally onto the inner domain G of a rectifiable Jordan curve. Then G is a Lavrentiev domain if and only if G is linearly conneeted and

(2) A(I~Z)) l(z) 1!'«()lld(1 :s M 2If'(z)1 for z E JI)).

The condition that Gis linearly connected (Section 5.3) cannot be deleted as again the example f(z) = (l-z)2 shows. It can however be proved (Pom82, Zin85) that Gis an Ahlfors-regular John domain (Section 5.2) if (2) holds.

Proof. Let G be a Lavrentiev domain. Then G is linearly connected by Theorem 5.9 (ii). It follows from Theorem 5.2 (ii) that, with I = l(z),

diamf(l) :s M 3df(z) :s M 3 (1 -lzI 2)1!,(z)1

so that f(l) lies in a disk of this radius. Hence it follows from the Ahlforsregularity condition (7.3.12) that


Conversely let G be linearly connected and let (2) hold (Fig. 7.2). Given a, bE J we choose z E IDJ such that J(a, b) = f(1) where 1= 1(z). Then

A(J(a,b)) = l lf'(()lld(1 ::; 27r(1-lzl)M2 1f'(z)1

::; K 1M 2df (z) ::; K 2M 2d f (z*)

by Proposition 1.6. If S is the non-euclidean segment connecting the endpoints of I then df(z*) ::; diamf(S) so that (1) follows from Proposition 5.6. 0

Remark. The opposite inequality

(3) 1f'(z)1 ::; A(~:)) l(z) 1f'(()lld(1 (z E IDJ)

holds for all conformal maps because A(1(z))If'(z)1 ::; K 3 diamf(1(z)) by Proposition 4.19.

~ f(z*)'/

L \ \ , ' ......... _--_.,/' fIS)

Fig.7.2

We say that h is a bilipschitz map of A into C if

A sense-preserving bilipschitz map of C onto C is quasiconformal but not vice versa. For example, a conformal map onto a quasidisk has a quasiconformal extension to C (Section 5.1) but no bilipschitz extension if it has unbounded derivative.

We give a characterization (Tuk80, Tuk81 , JeKe82b) of Lavrentiev domains that is analogous to the characterization of quasicircles as quasiconformal images of circles.

Theorem 7.9. The curve J is a Lavrentiev curve if and only if there is a bilipschitz map h of C onto C such that J = h(1I').

Proof. Let h be a bilipschitz map with J = h(1I') and let h((l), h((2) E J be given. We divide the shorter arc of 1I' between (1 and (2 into subarcs by points Zk (k = 1, ... ,n) where Zo = (1 and Zn = (2. It follows from (4) that


n n

L Ih(Zk) - h(Zk-dl ::; M 5 L IZk - Zk-ll ::; 1I"M5 1(1 - (21 k=1 k=1

and thus, again by (4),

Conversely let J be a Lavrentiev curve. We may assume that A( J) = 27L

The arc length parametrization J : h(eis ), 0::; s ::; 211" satisfies

for ISI - s21 ::; 11" and furthermore, by (1),

leisl - eiS2 1 ::; ISI - s21 ::; M 1 Ih(eisl ) - h(eis2 )1.

Hence h is a bilipschitz map from 1I' into tC which can be extended to a bilipschitz map of tC onto tC by the following result (Tuk80, Tuk81): 0

Theorem 7.10. Every bilipschitz map h 011I' into tC can be extended to a bilipschitz map 01 tC onto tC.

Prool (see Fig. 7.3). The function h is quasisymmetric on 1I'; indeed (5.4.4) is satisfied with A(X) ::; M~x (x > 0) by (4). Hence J = h(1I') is a quasicircle (Proposition 5.10) which we may assum~ to be positively oriented.

As in the proof of Theorem 5.16, let fand 9 map j]]J and j]]J* conformally onto the inner and out er domain of J respectively; we assurne that g( 00) = 00. Then cp = 1-1 0 hand 'Ij; = g-1 0 h are quasisymmetric by Proposition 5.12. Let (j5 and ;j; denote the Douady-Earle extensions of cp and 'Ij;; for simplicity we shall write cp, 'Ij; instead of (j5 and ;j;. We extend h to a homeomorphism of tC onto tC by defining

(5) h(z) = I(cp(z)) for Izl::; 1, = g('Ij;(z)) for Izl ~ 1

as in (5.5.6). Let z E ]]J) be given and write w = e27ri / 3 . We choose 7 E Möb(]]J)) such

that 7(0) = Z, 7(W2 ) = z/Izl and define Zy = 7(W Y ) for v = 1,2,3. Finally we choose a E Möb(]]J)) such that

(6)

Then cp* = a- 1 0 cp 0 7 gives a sense-preserving quasisymmetric map of 1I' onto 1I' satisfying cp*(wY ) = wY • Using the Möbius-invariant condition (5.4.8) of Proposition 5.14, we conclude from Proposition 5.21 that Icp*(O)1 ::; ro < 1 where ro does not depend on z.

It follows from (5) that ho 7 = 10 cp 0 7 = 10 a 0 cp* in]]J) and therefore, with (= a(cp*(O)) = cp(z) and hz = 8h/8z, that


(7)

By our choice of T, all three distances IZ1 -z21, IZ2-Z31, IZ3 -zll are comparable to l-Izl, and since a E Möb(lDl) and 1<p*(0)1 ::::; TO, at least one ofthe distances 1(1 - (21, 1(2 - (31, 1(3 - (11 is comparable to 1 - 1(1. Hence we may assume that

(8)

Since /((v) = h(zv) by (5) and (6), we can write

(1-1(12)1!'(()1 _ (1-1(12)1!'(()1 Ih(zd - h(Z2)1 IZ1 - z21 1 -lzl2 1/((1) - /((2)1 IZ1 - z21 1 -lzl2 .

All factors on the right-hand side are bounded from above and below by positive constants: The first factor because J is a quasicircle and in view of (8) (see Theorems 5.2(ii) and 5.7), the second factor because h is bilipschitz on 'lI', and the third factor because of (8). Ifwe now write X(z) = (<p*(z) - <p*(0))/ (1- <p*(O)<p*(z)) then X(O) = 0 and we conclude from (7) and the analogous identity for Ihz(z)1 that

Msl<p;(O)1 (9) Ihz(z)l::::; 1 -1<p*(0)12 = MsIXz(O)1 ::::; Mg, Ihz(z)l::::; MsIXz(O)1 ::::; Mg,

(10) Ihz(z)12 -lhz(zW ~ (IXz(OW -IXz(0)12)/MlO ~ I/Mn;

the last estimates follow from (5.5.21) and (5.5.22) as in part (b) of the proof of Theorem 5.15 using that X = "( 0 <p 0 T with "(, TE Möb(IDl).

It follows from (9) and (10) that hand its inverse function h-1 have bounded partial derivatives for z E lDl and similarly for z E 1Dl*. The righthand inequality (4) follows at once by integration over [ZI, Z2] because h is continuous on 'lI'.

To prove the left-hand inequality (4) we consider W1,W2 E C. If W1 and W2 lie in the closure of the same component of C \ J then we integrate over the hyperbolic geodesic from W1 to W2. In the other case, we choose Wo E J n [WI, W2] and integrate over the hyperbolic geodesics from W1 to Wo and from Wo to W2. Since J is a quasicircle, these geodesics have lengths comparable to the distance of their endpoints (Exercise 5.4.4). Hence we obtain

Thus (4) holds so that his a bilipschitz map. D


h > -z, lT

~ i~

(,}=1 w3

'f* ~

Fig.7.3. Note that h = f 0 cp and 0" 0 cp. = cp 0 T

Exercises 7.4

Let f map ]jJ) conformally onto G c C and let MI, ... denote suitable constants.

1. Let G be a Lavrentiev domain. Show that

j 1!'(()lld(1 ::; MII!,(O)I for some arc I c 11' with A(I) = 7r

where MI depends only on G and not on f. 2. Let G be linearly connected. Use Proposition 7.5 to show that G is a Lavrentiev

domain if and only if

1!,(z)1 ::; ~ p(z,()I!'(()lld(1 ::; M2 1!,(z)1 for z E]jJ);

the left-hand inequality is always true.

3. Show that the bilipschitz image of a Lavrentiev domain is again a Lavrentiev domain.

4. Deduce Proposition 7.7 from Theorem 7.9.

5. Let G be a quasidisk. Show that f can be extended to a bilipschitz map of C onto C if and only if 11M3 < 1!,(z)1 < M3 (z E ]jJ)). (This is in contrast to the case of quasiconformal extension.)

7.5 The Muckenhoupt Conditions and BMOA

1. We introduee several rather teehnical hut important eonditions on "weights", see ReRy75 or Gar81, Seet. VI, 6. Let <p he a nonnegative measurahle function defined on 11'. We say that <p satisfies the Coifman-Fefferman (Aoo ) condition (CoFe74; ReRy75, p.52) if one of the following three equivalent eonditions holds:

7.5 The Muekenhoupt Conditions and BMOA 169

(i) For all ares I c 'lI',

(ii) There exists Q: > 0 such that

for all ares I c 'lI' and measurable sets E c I. (iii) There exists ß > 0 such that, for ares I c 'lI' and measurable E c I,

L epld(l/ j epld(1 < ß =} A(E)/ A(I) < 1/2.

Here Mb M 2 , • .• again denote suitable positive constants. Condition (i) is a reversed geometric-arithmetie mean inequality, while (ii) is a strong form of condition (iv') for John domains (Seetion 5.2). We now prove another eharaeterization of Lavrentiev domains (Lav36, JeKe82b).

Theorem 7.11. Let f map j[J) conformally onto G. Then G is a Lavrentiev domain if and only if

(1) G is linearly connected, (2) G is a Smirnov domain, (3) 11'1 satisfies the (Aoo ) condition.

It is clear that every Lavrentiev domain satisfies (1) and (2); see Proposition 7.7 and Theorem 7.6. In spite of this fact it is not possible to delete either eondition. The function defined in (7.3.6) maps j[J) conformally onto a non-Smirnov quasidisk with II'(()I = 1 for almost all ( E 'lI' so that 11'1 satisfies the (Aoo ) condition (i).

A domain is an Ahlfors-regular John domain if and only if (2) and (3) are satisfied (Pom82). This holds in particular (FiLe86) if G is a Jordan domain such that every w E oG is vertex of an open triangle in Gwhich is eongruent to some fixed triangle.

Proof. First let G be a Lavrentiev domain. It follows from Theorem 7.8 and Proposition 7.5 that G is a Smirnov domain. If I is an are on 'lI' we ehoose z E j[J) such that 1= I(z) and obtain from (7.4.2) and Proposition 7.4 that

A~I) jlf'lld(1 ::; M 3 1f'(z)1 ::; M3 exp ( K 1 + A~I) jlog 1f'lld(l) .

Henee (i) holds with ep = Ifl Conversely let eonditions (1), (2) and (3) be satisfied. It follows from (i)

and Proposition 7.4 that


A~I) i 1/'lld(1 ::; Mi exp (A~I) i log 1f'lld(l)

::; Mi exp(Ki + log If'(z)!) = MieK11f'(z)1

so that Gis a Lavrentiev domain by Theorem 7.8. o

There are further conditions related to reversed Hölder inequalities. If 1 1 then <p 2: 0 satisfies the (Bq) condition if

(5) (A~I) 1 <P«()qld(l) i/q ::; :(;) 1 <p«()ld(l·

It is remarkable that (Muc72j Gar81, p.262)

(6) 1,

(7) q.

The following conditions are equivalent (CoFe74j ReRy75, p.50) to (i), (ii) and (iii):

(iv) 1.

These facts can be used to sharpen Theorem 7.8.

Corollary 7.12. 11 I maps lDJ conlormally onto a Lavrentiev domain then there are 6 > 0 and q > 1 such that

(8) A(I~Z)) l(z) 1f'«(Wld(1 ::; M6 1f'(zW lor z E lDJ,

1 r Id(1 M 7 (9) A(I(z)) JI(z) 1f'«)1<5 ::; 1/'(z)1<5 lor z E lDJ.

The first assertion follows at once from Theorems 7".8 and 7.11 because (v) is equivalent to (i). The second assertion follows from (iv) and (7.4.3).

It is not possible to choose the exponent 6 independent of I. For every 60 > 0 there exists (Zin85) a conformal map onto a Lavrentiev domain such that (9) becomes false for 6 = 60• The corresponding fact holds for q > 1 (JoZi82).

7.5 The Muckenhoupt Conditions and BMOA 171

The Helson-Szegö theorem (Gar81, p.149, p.255) implies for Smirnov domains that 11'1 satisfies the (A2 ) condition if and only if

(10) with sup I Reg(z)1 < 00, 7r

sup I Imh(z)1 < -. zElIli 2 zElIli

2. Let BMO(1l') denote the space of all integrable functions 9 on 1l' such that

(11) s~p At!) 1Ig() - allld(1 < 00, aI = At!) 1 g()ld(1

where the supremum is taken over all arcs I c 1l'. These are the functions of bounded mean oscillation. It is a basic fact (CoFe74; Gar81, pp. 235, 258) that

(12) 9 E BMO(1l') {::} g* E BMO(1l') {::} eqg satisfies (Aoo ) for some q > 0,

where 9 is real-valued and where g* is the conjugate function of g. We say that 9 E BMOA if 9 E H 1 and if the boundary values of 9 on 1l'

belong to BMO(1l').

Proposition 7.13. Let f map ]jJ) conformally into C and let I' E HP for some p > O. Then log I' E BMOA if and only if there exists q > 0 such that

(13) A(I~Z)) l(z) 11'(Wld(1 ::::: Msll'(zW for z E]jJ).

In particular, it follows from Theorem 7.6 that log I' E BMOA if f(]jJ)) is Ahlfors-regular (Zin84).

Proof. Let fq be defined by f~(z) = I'(z)q for z E ]jJ). If 0< q < 1/6 then, by Proposition 4.1,

(1-lzI2)lf~/(z)/f~(z)1 = q(I-lzI2)1f"(z)/f'(z)1 ::::: 6q < 1

for z E ]jJ) so that fq maps]jJ) conformally onto a quasidisk by Corollary 5.23. Iflogl' E BMOA then If~1 = exp(q Re log 1') satisfies (Aoo ) ifq E (0,1/6)

is chosen small enough; see (12). Since log f~ E H 1 the Smirnov condition is satisfied. Hence fq(]jJ)) is a Lavrentiev domain by Theorem 7.11, and (13) follows from Theorem 7.8.

Conversely suppose that (13) holds for some q > 0; we may assurne that q < p and q < 1/6. Then fq(]jJ)) is a Lavrentiev domain by Theorem 7.8, and Theorem 7.11 shows that If~1 satisfies (Aoo ). By Theorem 7.6, logf~ can be represented by its Poisson integral and it follows (Dur70, p.34) that log f~ belongs to H 1 . Hence we see from (12) that log f~ and thus log f' belong to BMOA. D

A function gin H 2 belongs to BMOA if and only if (see e.g. Bae80)

(14) Ilgll* = Ig(O)1 + sup ( [ Ig() - g(z)1 2p(z, Old(l) 1/2 < 00, zElIli J1f


and BMOA is a Banach space with this norm. The last integral can be rewritten as

(15) 2~ h Igz((Wld(1 with gz(() == 9 C(: ;() -g(z)

which is 2:: Ig~(O)12 = (1-lzI2)21g'(z)12. It follows that

(16) BMOA c B.

The subspace VMOA ("vanishing mean oscillation", Sar75) consists of all 9 such that

(17) h Ig(() - g(zWp(z, ()ld(1 ~ 0 as Izl ~ 1.

It is easy to see that VMOA C Bo. There is the deep duality (Fef 71)

(18) VMOA * ~ H 1 , H h ~ BMOA

where ~ denotes topological isomorphism. If f is a conformal map of ][)l onto the inner domain of a Jordan curve J

then log f' E VMOA if and only if

(19) A(J(a,b))/la-bl~1 as la-bl~O, a,bEJ;

compare (7.4.1). Such curves are called asymptotically smooth (Pom78), and are , in particular, asymptotically conformal (Section 11.2).

See e.g. Zin84, Sem88, Zin89, AsZi91 and BiJo92 for further results connecting BMOA and conformal mapping.

Exercises 7.5

Let j map ][)l conformally onto G.

1. If G is a Lavrentiev domain show that I' E Hq for some q > 1. Show (Zin85) furthermore that, for each q > 1, there exists a Lavrentiev domain such that I' ~ Hq.

2. If j satisfies (8), show that (compare (ii))

A(f(E))jA(f(I)) ~ Mg[A(E)jA(I)]l-l/q

for all arcs I C 11' and measurable E C I.

3. Let G be a Lavrentiev domain and h a conformal map of G onto ][)l. Show that there exists p > 1 such that

(A(~) i Ih'(W)IVldWI) I/V ~ :~) i Ih'(w)lldwl

for all arcs C c BG.

4. If arg I' is bounded in ][)l show that log I' E BMOA.

5. If I arg[zl'(z)j j(z)]1 ~ 7rClj2 for z E ][)l with Cl < 1, show that Gis a Lavrentiev domain and that 11'1 satisfies (A2 ).

Chapter 8. Integral Means

8.1 An Overview

We first consider the classical problem how the integral means

r27r 10 I/(reitW dt (0 < r < 1)

of the conformal map I grow as r --+ 1-. N ext we consider the integral means of the derivative which are more important for our purposes. Let P E IR and let ßf(P) be the smallest number such that

(1) as r --+ 1-

for every c: > o. We shall show that

(2)

(3)

supßf(P) = 3p -1 for P? 2/5, f

0.117p2 < supßf(P) < 3.001p2 for smallpj f

these are results of Feng-MacGregor, Clunie, Makarov and others. An interesting open problem is the Brennan conjecture that

(4)

It is only known that ßf( -2) < 1.601. The proofs use differential inequalities. The case P = 1 is connected with the growth of the coefficients. Not even

the exact order of growth is known for bounded univalent functions or for functions in E. We shall give abrief survey of results.

In Section 8.5 we consider integral means of Bloch functions 9 and deduce:

Makarov Law of the Iterated Logarithm. 11 9 E B then

(5) lim sup Ig(r()1 :S IIgl18 lor almost all ( E ']['.

r--+l Vlog l:r log log log l:r

174 Chapter 8. Integral Means

(6)

In the last section we consider lacunary series. The typical case is

00

g(z) = Lbkzqk, q=2,3, .... k=l

If (bk) is bounded then 9 E B. The theory of Bloch functions and lacunary series hasmany connections to probability theory. We shalllist some of the remarkable properties of lacunary series.

The lower bound in (3) comes from a function such that log f' has the form (6), and lacunary series show that (5) is essentially best possible.

In Chapter 10, we shall apply the results of this section to the problem how Hausdorff measure behaves under conformal mapping.

8.2 Univalent Functions

Let 9 be analytic in !I} and p E lR. We write z = reit, where 0 ~ r < 1. A calculation shows that

If we integrate over 0 ~ t ~ 211" the second integral vanishes by periodicity and we obtain Hardy's identity

(2)

Proposition 8.1. Let f map!l} conformally into C and let

(3) M(r) = max If(z)1 for 0 ~ r < 1. Izl=r

If f(O) = 0 and p > 0 then

(4) [21< r Jo If(reitW dt ~ 211"p Jo M(p)pp-l dp

and, for 1/2 ~ r < 1,

(5) [21< If(reit)IP-21f'(reit)12 dt ~ K(p) M(r)P ~ K(p)If'(O)IP Jo l-r (l-r)2P+1

where K(p) depends only on p.

Proof. Integrating Hardy's identity (2) (with 9 = J) we see that

r dd [21< If(reitW dt = p2 J' [ If(z)IP- 21f'(zW dxdy. r Jo J1z 1<5.r

8.2 Univalent Functions 175

We substitute w = I(z). Sinee I is univalent we see from (3) that the last expression is

/1 lM(r) ~ p2 Iwlp-2 dudv = 27rp2 sp-l ds = 27rpM(r)P ,

Iwl::oM(r) 0

and (4) follows by a further integration beeause M(O) = 1/(0)1 = o. If z = peit , 1- 3(1- r)/2 ~ p ~ r, then

by Corollary 1.6. We integrate over p and see as in the first part that

p2 1 ; r 121< I/(reitW-21I'(reit)12 dt ~ K o(p)87rpM(r)P

whieh implies (5) beeause M(r) ~ 11'(0)1(1 - r)-2 by Theorem 1.3. D

Theorem 8.2. Let I map ][)) conlormally into C and suppose that

(6) I(z) = O((I-lzl)-a) as Izl ~ 1

where 0 ~ a ~ 2. IIp > l/a then

(7) 121< I/(reit ) IP dt = 0 ((1 _ :)ap-l ) as r~l;

il 0 < p < I/athen I belongs to the Hardy space HP.

This result (Pra27) follows at onee from (4). See Hay58b for generalizations to multivalent funetions. Sinee (6) with a = 2 holds for all eonformal maps, we eonclude that (eompare Theorem 1. 7)

(8) 121< I/(reitW dt = 0 ((1 _ :)2P-l ) as r ~ 1

for p > 1/2 while I E HP for p < 1/2. We remark that the integral is maximized in S by the Koebe function (Bae74; Dur83, p. 215; Hay89, p.674).

If Pa denotes the Legendre function then (MaObSo66, p.184)

1 [21< dt 1 ( 1 + r2 ) (9) 27r Jo 11 - reitlq = (1- r 2)q/2 P- q/2 1- r 2 .

As r ~ 1 - 0 this is (MaObSo66, p. 157, formula 10)

rv 2-q7r- 1/ 2 r C ; q) r (1 - V-I if 0 ~ q < 1,

rv 2-17r-1/ 2 r (q; 1) r G) (1 - r)-q+1 if q> 1.

(10)


The conformal map (1 - z)-a: (z E [J»), 0 < Q < 2 therefore shows that Theorem 8.2 is best possible.

Exercises 8.2

1. Use Hardy's identity to show that f'r Ig(r(Wld(1 is increasing in 0 < r < 1 if p~ o.

2. Let f map [J) conformally onto a domain symmetrie with respect to o. Show that

as r--+1-.

3. Let f map [J) conformally into ce. Use Hilbert's inequality to show that

4. Use (3.1.6) to prove directly that, for q > 1,

8.3 Derivatives of Univalent Functions

Let f be a conformal map of [J) into ce. For p E lR we define

(1) ß(p) = ßf(P) = limsup (log r 1!'(r()lPld(l) flog 1 ~ . r->l- }'r r

Thus ß(p) is the smallest number such that, for every € > 0,

(2) as r-+l-.

Proposition 8.3. The junction ßf is continuous and convex in lR and satisfies

(3) { ß(p) + 3q if q > 0, ß(p + q):::; ß(p) + Iql if q < O.

Proof. The Hölder inequality shows that

for p, q E lR and 0 :::; >. :::; 1. Hence we see from (1) that

8.3 Derivatives of Univalent Functions 177

(4) ßf(>IP + (1 - A)q) :::; Aßf(P) + (1- A)ßf(q) (0:::; A :::; 1)

so that ßf is convex and therefore continuous. Inequality (3) follows from (1) and the fact (Theorem 1.3) that

[f'~0)[ (I-[z[) :::; [!,(z)[ :::; 2[!,(0)[(I-[zl)-3 for z E 1Dl. 0

As an example, consider the conformal map

I,,(Z)=(~~;)", I (1 + z),,-l I,,(z) = 20: (1 _ z)"+1 '

where 1 < 0: :::; 2. We easily see from (8.2.10) that

{ (o:+I)p-I ifI/(o:+I) <p<+oo,

(5) ßf,,(P) = (0: -I)[p[-I if - 00 2/5 and for close-to-convex functions (e.g. starlike functions, see Section 3.6).

Theorem 8.4. Let 1 map lDl conlormally into Co Then

(6) ßf(P) :::; 3p - 1 lor 2/5:::; p < +00.

11 1 is close-to-convex then

(7) { [p[-I

ßf(P) :::; 0 3p-I

lor - 00 < P < -1 ,

for - 1 :::; p :::; 1/3,

lor 1/3 < P < 00 .

Proof. We may assurne that 1 E S. First let 2/5 < P < 2 and choose (2 - p)/4 < x< p. Hölder's inequality shows that

Estimating the second integral by (8.2.5) and the third integral by (8.2.8), we see that the right-hand side is bounded by a constant times

[(1 - r )-2(2-2X/P)-1 t/2 [(1 _ r )-2.2x/(2-p)+l] I-p/2 = (1 _ r) -3p+1 .

Hence (6) follows from (1) for 2/5 0


for z E ]]) and it follows from Hölder's inequality that

Since g is univalent the second integral is 0(1) if 0 :::; P < 1/3 and 0((1- r)-3p+1) ifp > 1/3; see (8.2.8).

Since q has positive real part we can furthermore write

q(z) = q(O) + q(O)<p(z) with 1<p(z)l:::; 1, <p(0) = 0, 1 - <p(z)

so that, by Littlewood's subordination theorem (Dur70, p. 10),

(10) r Iq(r()1 3P ld(l:::; r I q(O) + q(O)( 1

3P Id(1 (0:::; r < 1). Ir Ir 1-(

By (8.2.9) the last integral is 0(1) if 0 :::; p < 1/3 and 0((1 - r)-3p+1) if p > 1/3. Hence (7) follows from (9) for p 2: 0; the case p = 1/3 follows from the continuity of ß j.

Finally if p < 0 we write

'( )-1 Z 1 f z = g(z) q(z) for z E ]]).

The first factor is bounded because g is univalent and g(O) = O. The second factor has positive real part. Hence (7) follows as in (10) but with Ipl instead ~~ 0

The next result (CIP06 7, Pom85) covers the other ranges of p in the general case but is not sharp.

Theorem 8.5. 1f f maps]]) conformally into ce then

(11) 1 ( 1)1/2 ß j (p) :::; p - "2 + 4p2 - P + 4

for p E lR, furthermore

(12) ßj( -1) < 0.601.

The right-hand side of (11) is < 3p2 for p < 0 and < 3p2 + 7p3 for p > O. A miracle happens for p = -1 which allows us to give a much better estimate; this estimate is in fact true even if f has a pole in ]]). In the opposite direction we have (Mak86, Roh89):

Theorem 8.6. There exists a conformal map f of]]) onto a quasidisk such that ßj( -1) > 0.109 and

(13) ßj(p) 2: 0.117p2 for smallipi·


In particular it follows that (6) need not hold for P = 1/3. We shall prove this theorem in Section 8.6 using lacunary series.

The Brennan conjecture (Bre78) states that

(14) ßf(P) ::::; Ipl - 1 for - 00 < P ::::; -2;

this would be sharp by (5) with Q: = 2. In view of (3) the Brennan conjecture is equivalent to ßf( -2) ::::; 1. It is only known that

(15) ßf(P) < Ipl- 0.399 for - 00 < P::::; -1

which follows from (12) and (3). In Fig. 8.1, the upper curve gives the best known bounds for ßf(P) and comes from Theorems 8.4 and 8.5 together with convexity. The lower curve comes from the example in Theorem 8.6 and from (5) with Q: = 2.

1,0

Fig.8.1. The values of sUPf ßf(P) that are possible according to present knowledge lie between the two curves

If ßf(P) < 1 then, with ßf(P) < Q: < 1,

11 127r II'(reitW r dtdr = 11

0((1- r)-a) dr < 00.

Hence it follows from (15) and (6) that

(16) l' E LP(]jJ)) for - 1.399 ::::; P < 2/3.

The Brennan conjecture would imply that l' E LP(]jJ)) for -2 < P < 2/3. For the proof of Theorem 8.5 we need a result about differential inequalities

(see e.g. BeBe65, p.139; Her72).


Proposition 8.7. Let n ::::: 2 and let ak (r) (k = 0, ... , n - 1) be continuous in ro ::; r < 1. Assume that

(17) ak(r):::::O for k=0, ... ,n-2.

Let u and v be n times differentiable in [ro, 1) and

n-l n-l (18) u(n) < L aku(k) , v(n) = L akv(k) .

k=O k=O

1f u(k)(ro) < v(k)(ro) for k = 0, ... , n - 1 then

(19) u(r) < v(r) for ro::; r < 1.

Thus u satisfies a differential inequality while v satisfies the corresponding differential equation. Note that an-l may have any sign. In the case n = 2 the assumptions reduce to

(20) u" < alu' + aou , v" = al v' + aov , ao::::: 0

and u(ro) < v(ro), u'(ro) < v'(ro).

Proof. Suppose that (19) is false and write rp = u - v. Since rp(ro) < 0 there exists rn E (ro, l).such that

rp(rn) = 0, rp(r) < 0 for ro < r < rn .

Since also rp'(ro) < 0, the function rp has a first local minimum at rn-l in (ro, rn) , so that

rp'(rn-d = 0, rp"(rn-d::::: 0, rp'(r)::; 0 for ro < r < rn-I·

Continuing this argument we find ro < rl < ... < rn-I< rn < 1 such that, for k = 0, ... ,n - 1,

rp(k) (rn-k) = 0, rp(k+l)(rn_k)::::: 0, rp(k)(r)::; 0 for ro < r < rn-k.

Hence rp(k)(rl) ::; 0 (k = 0, ... ,n - 2), rp(n-l)(rd = 0 and rp(n)(rd ::::: O. Therefore (17) and (18) show that

n-2

rp(n)(rl) < L ak(rl)rp(k) (rl) ::; 0

k=O

which contradicts rp(n)(rd ::::: O. o

Proof of Theorem 8.5. (a) We write

(21) u(r) = i 1!,(r()JPld(1 for 0< r < 1.


Differentiating under the integral sign we obtain

(22) u'(r) = p h If'lP Re[(!,,(r()/ f'(r()lld(l,

and Hardy's identity (8.2.2) shows that

Since u'(r) 2:: 0 by Exercise 8.2;1, we conclude from (21), (22) and (23) that

"( ) 4rp ,( ) 4r2p2 () 2 [I 'IP 1 !,,(r() 2r 121 I u r - 1 _ r2 u r + (1 _ r2)2 u r 5. p Ir f (f'(r() - 1 _ r2 d(

and this is bounded by 16p2(1 - r2)-2u(r) because of Proposition 1.2. Hence

"( ) 4rp ,() (16 - 4r2)p2 () 2p + c: '() 3p2 + c: () u r 5. -1 2 U r + ( 2)2 ur< --u r + ( )2 U r -r l-r l-r l-r

if ro ( c:) 5. r < 1. The corresponding differential equation

( ) "() 2p + c: '() 3p2 + c: () ( ) 24 v r = 1 _ r v r + (1 _ r)2 v r ro 5. r < 1

has the solution c(1 - r)-.x where A = A(C:) is the positive root of

(25) A(A + 1) = (2p + C:)A + 3p2 + c:.

Since (20) is satisfied we conclude from Proposition 8.7 that

h 1J'(r(Wld(1 = u(r) < (1 ~~].x(e) for ro(c:) 5. r < 1

if c(c:) is chosen large enough. Hence (11) follows from (2) because A(C:) tends to the expression on the right-hand side of (11) as c: --+ o.

(b) N ow consider

00

(26) g(z) = J'(z)-1/2 = L bnzn . n=O

Parseval's formula shows that

(27) ( ) _ 1 [ld(1 1 [I 2 ~ 2 2n U r = 271" Ir 1f'(r()1 = 271" Ir g(r()1 Id(1 = ~ Ibnl r

for 0 5. r < 1, hence


00

n=2

Computation shows that g" = -Sf9/2 where Sf denotes the Schwarzian derivative. Since ISf(r()1 ::::; 6(1 - r 2 )-2 by Exercise 1.3.6 we see from (27) that

u(4)(r) ::::; ~ f ISf(r(Wlg(r(Wld(1 27r Ir;

144 9.01 ::::; (1_r2 )4 u(r) < (1_r)4 u(r) (ro::::;r< 1).

The corresponding differential equation

(4) ( ) _ 9.01 () (ro::::; r < 1) v r - (1- r)4 v r

has the solution c(l- r )-A where >. is the positive root of >'(>' + 1)(>' + 2)(>' +3) = 9.01; calculation shows that 0.600 < >. < 0.601. It follows from Proposition 8.7 that

1 f C 27r Ir; 1!,(r()1-1Id(1 = u(r) < (1 _ r)A for ro::::; r < 1

if cis chosen sufficiently large, and we conclude that ßf(-l)::::; >. < 0.601. 0

Exercises 8.3

1. Show that b = lim",~+oo x-1ß(x) exists and that ß(p+q) ::; ß(p) +bq for p, q > O.

2. Let f map ]]J) conformally into ce. Show that

() { 0.6491pl - 0.048 for - 1 ::; p ::; -0.2,

ßf p < 0.545p - 0.018 for 0.1 ::; p ::; 0.4.

3. Show that the Brennan conjecture is equivalent to the following assertion: If g maps G c C conformally onto ]]J) then g' E U (G) for 4/3 < q < 4.

4. Suppose that (1 - IzI2~lzf"(z)/f'(z)1 ::; ,.., for z E ]]J); compare Corollary 5.23. Show that ßf(P) ::; ,..,2p /4 for pE IR.

5. Let f lie in a linearly invariant family of order a where 1 ::; a < 00; see (5.3.13). Show that

1 J 1 2 2 ßf(P) ::; P - 2 + a 2p2 - p + 4: "" (a - 1)p (p ---+ 0).

8.4 Coefficient Problems 183

8.4 Coefficient Problems

We now consider the power series expansion

(Xl

(1) J(z) = L anzn for Izl < 1. n=O

If n = 2,3, ... and rn = 1 - 1/n then

(2) nlanl = 27rr~ 1 11h J'(rneit)e-i(n-l)t dtl ::; 2: 127r

II'(rneit)1 dt.

This allows us to apply the results of Section 8.3 to estimate lan I.

Theorem 8.8. Let f map j[)) conformally into C and suppose that

(3) f(z) = O((1-lzl)-a) as Izl -t 1.

1f 1/2 < Cl ::; 2 then, as r -t 1 and n -t 00,

1f f is bounded, i.e. if Cl = 0, then

(5) 127r 11' (reit) I dt = 0 ((1 _ ~)0.491)' an = O( n°.491-1) .

The first part (LiPa32) is sharp as the functions (1 - z)-a show. For functions of large growth, the size of the coefficients is mainly determined by the growth of M(r). These results also hold for multivalent functions (see e.g. Hay58b, p.46).

For bounded functions however, the coefficient size is mainly determined by the complexity of the boundary; see also Corollary 10.19. The result for Cl = 0 (ClPo67, Pom85) is not sharp.

The situation is less clear for the range 0 < Cl ::; 1/2. The estimate (4) still holds (Bae86) at least for Cl 2': 0.497. See e.g. Pom75, p.131, p. 146 for furt her results.

Proof. We may assurne that f(O) = O. First let 1/2 < Cl < 2 and choose 1/Cl < q < 2. We see from the Schwarz inequality that

by (3), Proposition 8.1 and Theorem 8.2. This implies (4); the second estimate follows from (2).


Now let f be bounded. We see from (8.2.5) for p = 2 that

(6) [21< 1!,(reitW dt = 0 (_1_) as r --+ 1. Jo 1- r

Hence ßf(2) ::; 1 using the notation (8.3.1), and it follows from Theorem 8.5 by the convexity of ßf that, for 0 < P < 1,

ßf(l) ::; -ßf(P) + -ßf(2) ::; - - + 4p2 - P + - . 1 1 - p 1 ( 1 ( 1) 1/2) 2-p 2-p 2-p 2 4

If we choose p = 0.07 we condude that ßf(l) < 0.491 which implies (5). We remark that ßf(2 - 6) ::; 1 - 6 + 0(62 ) as 6 --+ 0; see JoMa92. 0

It might appear that one loses much by applying the triangle inequality in (2). But this is not the case because (CaJ091)

. log(nlanl) (7) 'Y == supßf(l) = suphmsup-:-"'-'----"-

f f n---+oo logn

where the supremum is taken over all bounded univalent functions. A direct construction (Pom75, p.133; Dur83, p.238) shows that 'Y > 0.17. Numerical calculations using (7) and Julia sets show (CoJ091) that 'Y > 0.23 so that, together with (5),

(8) 0.23 < 'Y < 0.491 .

Carleson and Jones have conjectured that 'Y = 1/4. Closely related is the coefficient problem for the dass E of the univalent

functions 00

(9) g(() = ( + L bnCn for 1(1 > 1. n=O

If 'Y is defined by (7) then (CaJ091)

. log(nlbnl) 'Y = sup hm sup ---=:-'--'--'-'-~

gEE n---+oo log n (10)

so that 'Y is the smallest number such that, for all 9 E E,

(11) bn = O(n1'+e-l) (n --+ 00) for each € > o. The method of integral means will not give sharp coefficient bounds. It is

often important to get sharp estimates for the coefficients and their combinations because these estimates can be transferred to bounds for other functionals via the Koebe transform (1.3.12).

The dass S consists of all analytic univalent functions f(z) = z+a2z2+ ... in 1Dl. It is now known (deB85) that

(12) lanl ::; n for fES, n = 1,2, ....

8.5 The Growth of Bloch Functions 185

This Bieberbach conjecture had been open for a long timej see (1.3.11) and e.g. Dur83. The bound la21 :$ 2 leads to the Koebe distortion theorem as we have seen in Section 1.3.

We state some further sharp bounds without proof. Let fES. If 0:$ {) :$ 271" and 0 < A :$ 4 then (Jen60, p.173)

(13) Re[(a3 - a~)e2i~ + Aa2ei~] :$ 1 + ~A2 + ~A210g ~ ,

furthermore (FeSz33j Dur83, p.104)

(14) la3 - c);a~1 :$ 1 + 2exp[-2a/(l- a)] for 0:$ a < 1.

If we write log[f(z)/z] = Eenzn then (deB85)

(15)

which implies (12) by the Lebedev-Milin inequalities (Dur83, p.143). Now let g E E and 0 :$ {) :$ 271". Then (Jen60, p.184)

(16) for 2

--<A<2 3 - - ,

in particular Ib2 1 :$ 2/3 (Schi38). Furthermore (Jen60, p.191j Pom75, p.121)

(17) [ ( 1 2) 2i~ i~] 1 3 2 1 2 4 Re b3 + - bl e + Abl e > - - - - A - - A log-2 - 2 16 8 A

for 0 < A :$ 4, in particular Ib3 1 :$ 1/2 + e-6 (GaSch55b).

Exercises 8.4

1. Let J E Sand write [z-l J(z)]'" = L:: cnzn • Show that Cn = O(n2>'-1) as n -+ 00

if A > 1/4. (See HayHu86 for more precise resultsj the Koebe function is not extremal for 0.25 < A < 0.4998.)

2. Suppose that J(z) = alZ + a2z2 + ... maps lD> conformally into 1D>. Show that la21 :::; 2lall(1 -lall) by considering J(z)/(1 - eiß J(Z))2 (Picl7j see e.g. Tam78 or Pom75, p. 98 for further results.)

8.5 The Growth of Bloch Functions

We now consider the dass B of Bloch functions, Le. of functions g analytic in lD> such that

(1) IlgilB = sup(l -lzI2) 19'(z)1 < 00 j zElIli

see Section 4.2. There are non-trivial bounds of their integral means (Mak85):


Theorem 8.9. If 9 E Band g(O) = 0 then

(2) 21 r /g(r()/2n/d(/ :::; nl//g//;n (lOg ~)n nIr l-r

for 0 < r < 1 and n = 0, 1, ....

Proof. The case n = 0 is trivial. Suppose that (2) holds for some n. Hardy's identity (8.2.2) shows that

Writing ..\(r) = log[lj(l- r2 )] we see from (2) and (1) that this is

:::; 4(n + 1)2r nl//g//;n ..\(rt . (1 - r2)-2//g //;

:::; (n + l)l//g//2n+2~ [r~..\(rt+1] 8 dr dr

Hence we obtain by integration that

~ (~ r /g(r()/2(n+1)/d(/):::; (n+l)l//g//2(n+1)~ (..\(rt+1) dr 2n Ir 8 dr

and (2) for n + 1 follows by another integration because both sides vanish for r =0. 0

We deduce the Makarov law of the iterated logarithm (Mak85).

Theorem 8.10. If gE B then, for almost all ( E 'll',

(3) limsup /g(r()/ :::; //g//8' r---+1 Jlog _1_ log log log _1_ 1-r 1-r

There exists 9 E B such that this limes superior is > 0.685//g//8 for almost all (E 'll'.

Here is a probabilistic interpretation: Consider the probability measure (2n)-1 A on 'll'. Then

(4)

defines a complex stochastic process with expectation g(O). Its variance satisfies

(5) U(t)2 :::; I/g//;t (0:::; t < 00)

by (2) with n = 1. We can write (3) as

8.5 The Growth of Bloch Functions 187

(6) . IZtl hmsup < 1. hoo IlgllBvtloglogt-

This is therefore a one-sided form of the law of the iterated logarithmj see e.g. Bin86 for a survey. The last assertion of Theorem 8.10 shows that (3) and thus (6) are not far from being best possible. See Baii86 and Ly090 for an approach using Brownian motion and Mak89d using martingales.

Proof. We may assurne that IlgilB = 1. Consider the maximal function

(7) g*(s, () = max Ig(r()1 for e::; s < 00, (E 11'. O::;r::;l-e- S

The Hardy-Littlewood maximal theorem (Dur70, p. 12) applied to the analytic function g2n and Theorem 8.9 show that

where K is an absolute constant. We multiply by s-n1jJn(s) where

Integrating we obtain from Fubini's theorem that

Hence there are sets An C 11' (n = 1,2, ... ) with A(An ) > 27l' - K/n2 such that

(8) 100 g*(s, ()2n s-n1jJn(S) ds ::; n! n3 for (E An.

Since -t8 [s-n(logs)-l-l/nj ::; 3ns-n1jJn(s) we therefore conelude that, for ( E An and e ::; a < 00,

(9) g*(a,()2na -n(loga)-1-1/n::; 3n 100 g*(s,()2n s-n1jJn(s)ds::; 3n!n4 .

The set A = U~l n:=k An satisfies A(A) = 27l' because A(An) :2: 27l' -K/n2. Now let ( E A. Then ( E An for n :2: k and suitable k = k((). If r < 1 is sufficiently elose to 1 then

n= [logloga]:2:k where a=logl/(l-r).

Since log log a :2: n and log a < en +1 we therefore see from (7) and (9) that

Ig(r()IZ < g*(a,()2 < (3n!n4 )1/n e(n+l)2/n2 alogloga - alogloga - nn


which tends -+ 1 as n -+ 00 by Stirling's formula. This proves (3). We postpone the proof of the second assertion to the end of the next section. D

Corollary 8.11. 11 I maps][} conlormally into ethen

(10) . !log l'(r()1 hm sup :::; 6 lor almost all ( E 1[' ,

r-+l Jlog -1-logloglog _1_ l-r l-r

in particular I' (r() = O( (1 - r) -E) as r -+ 1 lor e > 0 and almost all (.

This result (Mak85) follows at once from Theorem 8.10 and the fact that 11 log I'IIB :::; 6 by Proposition 4.1.

Exercises 8.5

1. Let g(z) = L:: bnzn . Show that (bn ) is bounded if 9 E Band that bn -> 0 if gE Bo• (See Fern84 for deeper results.)

2. If 9 E B, g(O) = 0 and 0 < o:lIgllB < 1, show that

r [2 2 1] 211" J'f exp 0: Ig(rC)1 flog 1- r2 IdCI::; 1- 0:211gll; (0::; r < 1).

3. Let 9 E B, g(O) = O. Use (2) to prove that, with some constant K,

A({CE1[': Ig(r()1 2:: x})::;Kxe-.,2 for O<r<l, x2::1. IIgllBvlog 1/(1 - r2 )

This is related to the central limit theorem.

4. If! maps][} conformally into C and if 0: < 1, show that I!(()-!(rC)1 = O((l-r)"') as r -> 1 for almost all C E 1['.

8.6 Lacunary Series

1. We say that the function 9 has Hadamard gaps if

00

(1) g(z) = bo + ~:~>kZnk , (k=1,2, ... ) k=1

for some real q > 1.

Proposition 8.12. 11 9 has Hadamard gaps then

9 E B {:} sup Ibk 1 < 00 ,

9 E Bo {:} bk -+ 0 as k -+ 00 .

8.6 Lacunary Series 189

Proof. The implications =} hold for all Bloch functionsj see Exercise 8.5.1. Now let Ibkl :s: M and Izl = r < 1. It follows from (1) that

rlg'(z)1 < ~ ~n rnk = M ~ ("" n )rm l-r -1-r~ k ~ ~ k ,

k=l m=l nk$m

furthermore that

(2) for k:S: j , for k::::: j.

We conclude that

rlg'(z)1 < M ~ qm r m = Mq r l-r - ~q-l q-l (l-r)2

so that (1- r 2)lg'(z)1 :s: 2Mq/(q -1). Hence 9 E B. If bk --t 0 then we choose N so large that Ibkl < c for k ::::: N and write g(z) = p(z) + 2:%"=N bkznk where p is a polynomial. As above we see that

limsup(1 - r 2 )19'(z)1 :s: lim(1 - r 2)lp'(z)1 + 2cq/(q - 1) r-l r_l

for every c > O. Hence the limes superior is = 0 and thus 9 E Bo. D

Proposition 8.13. Let the lacunary series 9 be given by (1). 1frj = 1-I/nj for j = 1,2, ... then

(3) Ibo + t bkCk - g(r()1 :s: K(q) sup Ibkl (( E 11')

for rj :s: r :s: rj+! where K(q) depends only on q > 1.

Proof. We may assurne that Ibkl :s: 1. It follows from (1) that

Ibo + t, b,("' ~ .(r() I '" E(i ~ r"') + ,~:n .. Since 1 - rnk :s: nk(1 - rj) = nk/nj and rnk :s: exp( -nk/nj+I), we see from (2) that the right-hand side is

j 00 00

:s: L qk- j + L exp( _qk-j-l) < q ~ 1 + L exp( -ql/). D k=l k=j+l 1/=0

Lacunary series are useful for constructing "pathological" conformal maps (see e.g. GnHP086).

Proposition 8.14. We define 00

(4) n=-oo


Jor q = 2,3, .... IJ

(5) 00

log j'(z) = L bkzqk , k=O

then J maps lD conJormally onto a quasidisk.

q Upper büund für Bq q

2 2.8913 12 3 1.8291 13 4 1.4614 14 5 1.2754 15 6 1.1642 20 7 1.0907 30 8 1.0386 40 9 0.9998 50

10 0.9699 80 11 0.9460 00

Upper büund für Bq

0.9266 0.9104 0.8968 0.8851 0.8455 0.8074 0.7890 0.7782 0.7621 0.7358

Proof. The sum in (4) is continuous in x and has the period 1. Hence the maximum Bq exists. Let 0 < Izl < 1. We write

Izl = exp( _q-j+X) with j E Z, 0:::; x < 1.

If Ibkl :::; M < I/Bq then, by (5),

(I-lzI2 ) Iz~:gj I:::; 2 log I~I ~ Ibkillzlqk

00

:::; 2M L qk-j+x exp( _qk-j+x)

k=O

which is < MBq < I by (4). Hence J maps lD conformally onto a quasidisk by the Becker univalence criterion (Corollary 5.23). The upper bounds are obtained by splitting [0,1] into 128 subintervals and using the fact that te- t

increases for 0 < t :::; 1 and then decreases. D

It follows from Proposition 8.14 that

(6) J(z) = r exp (b f (qk) d( = z + ... , q = 2,3, ... Jo k=l

maps lD conformally onto a quasidisk if Ibl < I/Bq. The modified Bessel functions are defined by

(7) (n=O,I, ... );


see e.g. MaObS066, p.66, p. 70. They appear in the Fourier series

00

(8) exp(x cos'!9) = Io(x) + 2 L In(x) cosn'!9. n=l

We give a lower estimate (Roh89) of the integral means of 1'.

Proposition 8.15. Let f be the conformal map defined by (6) with q odd and 0 0 and a = 10gIo(bp)/logq.

Proof. First we prove by induction on m = 0,1, ... that

(10) r27r m

io rr cos(nkt)dt 2': 0 o k=l

for n1, ... ,nm E Z.

The cases m = 0 and m = 1 are trivial. Suppose that m 2': 2 and that (10) holds for m -1. Using cosacosß = [cos(a + ß) + cos(a - ß)]/2 we can write our integral as

I127r (m-2 ) 2 rr cosnkt [cos(nm + n m-1)t + cos(nm - nm-dt] dt o k=l

which is the sum of two integrals with m-I factors and thus 2': O. Furthermore, if n1 + ... + nm is odd then the integral in (10) changes sign if we replace t by t + 'Ir and is thus = O.

If sm(t) = 2:;:'=1 cos(qkt) then

127r 127r m exp[bpsm(t)] dt = rr exp[bpcOS(qkt )] dt,

o 0 k=l

and by (8) this is

We multiply out and integrate term-by-term. We see from (7) that In! (bp) ... In", (bp) 2': 0 if n1 + .. ·+nm is even so that the result is 2': 27rlo(bp)m. Hence we obtain from Proposition 8.13 that, for 1 - q-m ::; r ::; 1 _ q-m-1,

~ 1f'(r()lPld(1 2': Cl loh exp[bpsm(t)] dt 2': 2'1rC1IO(bp)m

and (9) follows because (m + 1) logq 2': log[1/(1 - r)]. D


Proof of Theorem 8.6. We choose q = 15 and b = 1.129 < 1/BI5 • Then

ß(-l)::::: logIo(1.129)/log15 > 0.109.

Furthermore log 10 (x) ""' x2 /4 as x -; 0 and thus ß(p) ::::: 0.117p2 for small Ipl by Proposition 8.15. D

2. We now list without proof some of the surprising properties of lacunary series. We assurne throughout that 9 is given by (1) so that 9 has Hadamard gaps.

If 9 has a finite asymptotic value (see (4.2.14)) at some point of 1I' then (HaLi26, Binm69)

(11) bk -; 0 as k -; 00

and thus 9 E Bo. Conversely every function in Bo (lacunary or not) has finite radial limits on a set of Hausdorff dimension 1; see (11.2.7).

If however 9 has finite radial limits on a set of positive measure then (Zyg68, I, p. 203)

(12)

Conversely if (12) holds then 9 E HP for every p < 00, even (Zyg68, I, p. 215)

(13) sup f exp (n:lg(rOI 2 ) Idel < 00 for small n: > O. r Ir

Furthermore (12) implies 9 E VMOA; see e.g. HaSa90. If gE Hoo then more is true, namely (Sid27; Zyg68, I, p. 247)

(14)

so that 9 is continuous in ll.». Furthermore (14) holds if (GnP083)

(15) [ 1!,(z)lldzl < 00 for some curve rending on 1I'.

It is remarkable that (14) already holds (Mur81) if g(][))) i- Co It is an open problem what g(][))) i- C implies under weaker gap assumptions. An interesting example is the (non-Hadamard) series

00 00

(16) ~) -1)k(2k + 1)zk(k+ I lj2 = II (1- zn)3 i- 0 k=O n=1

which comes from the theory of theta functions. If 9 r:J- B, that is if

(17)


then 9 has the asymptotic value 00 at every point of 11'; see Mur83. Note that 9 has the angular limit 00 almost nowhere on 11' by the Privalov uniqueness theorem.

There is a elose connection of lacunary series 9 to probability theory; see Kah85 for the relation to random power series. The functions g(r()(( E 11') behave like a family of weakly dependent random variables on the probability space (11', Borei, (211")-1 A).

To be more precise, let (bk) be bounded but

00

(18) b(r)2 == L Ibkl2r2nk --+ 00 as r --+ 1-k=l

so that 9 E B \ H 2. We define rt by 2b(rt)2 = t for 0 S t < 00. Then (PhSt75, p. 60) there is a probability space (.a, A, P) and a stochastic process St(O S t < 00) with

(19) peSt S x) = (211")-lA({( E 11': Reg(rt() S x})

for x E lR such that

(20) St = X t + O(t°.47) (t --+ 00) almost surely,

where X t is the standard Brownian motion (of expectation 0 and variance t). Our continuous version is equivalent to the discrete version in PhSt75 by Proposition 8.13.

There are several important consequences. We formulate them for the modulus instead of the real part. Let bo = O. The centrallimit theorem (Zyg68, II, p.264) states that, for each set E C 11' with A(E) > 0 and x E lR,

A ({( E E : Ig(r()1 S xb(r)}) --+ A~ jX e-e /2 d~ v211" -00

(21) as r --+ 1.

The law 0/ the iterated logarithm (Wei59, p.469) states that

(22) limsup Ig(r()1 = 1 for almost all (E 11'. r--+1 b( r) Jlog log b( r)

3. Using these properties we finally prove two results stated earlier.

ProO/ 0/ Proposition 4.12. Let / be defined by /(0) = 0 and

• 00

(23) log J' (z) = i- L z2k for z E ][)). k=O

It follows from Proposition 8.14 that / maps][)) conformally onto a quasidisk. Now suppose that / is isogonal at some point ei19 . Then

arg J'(rei19 ) = ~ I>2k cos(2kß) --+ 'Y as r --+ 1-k=O


for so me 'Y by Proposition 4.11. Hence 2::k cos(2k~)z2k has a finite radial limit at 1 and it follows from (11) that COS(2k~) ---+ 0 as k ---+ 00. This is impossible because cos(2k+1~) = 2 cos2(2k~) - 1. 0

Proof of Theorem 8.10 (Conclusion). We now prove the second assertion. Let q = 15 and consider the lacunary series

for z E JI)).

k=m Then (1 - IzI 2)lzg'(z)1 'S Bq < 0.886 as in the proof of Proposition 8.14. It is easy to see that this implies IlgilB < 0.886 if m is chosen large enough. With b(r) defined by (18) we have

b(r)2 = _1_ ~ r2qk = ~ ( '"' 1) r2n 1 - r2 1 - r2 ~ ~ ~

k=m n=1 qkSn,k?:m

Loo (log n ) 2n 1 Loo ( 1) 2n M 2 ---m r 2-- 1+···+- r ---

log q log q n 1 - r 2 n=1 n=1

for some constant M. It follows that b(r)2 2 log[lj(l - r2)l/logq - M and thus from (22) that, for almost all ( E 11',

limsup Ig(r()1 r-+1 Jlog _1_ log log log _1_ 1-r 1-r

Exercises 8.6

1 2 n=-::: > 0.68511g11B .

ylogq o

1. Use (6) to construct a bounded univalent function the coefficients of which are not O(nO.064-1); compare (8.4.8).

2. Construct a univalent function such that the coefficients of 1/ I' are not O(nO.064 ).

3. Let 9 E Bo have Hadamard gaps. Show that the left-hand side of (3) tends --+ 0 uniformly in 11' for Tj :s: T :s: Tj+l, j --+ 00.

4. Let 9 have Hadamard gaps. If 9 has a finite radial limit at ( E 11', show that the series converges at ( (HaLi26).

5. Use the fact that (15) implies (14) to prove Murai's theorem that g(JI))) 1= c implies (14).

6. Let the univalent function f(z) = 2::: anzn have Hadamard gaps. Show that Enlanl < 00.

7. Let f map JI)) conformally onto the inner domain of the Jordan curve J and let log I' have Hadamard gaps. If 11'1 is bounded show that I' is continuous and nonzero in IDl. If J has tangents on a set of positive linear measure show that J is rectifiable. (Use also Proposition 6.22 and Theorem 6.24).

Chapter 9. Curve Families and Capacity

9.1 An Overview

We introduce two important concepts. Let r be a family of curves in the Borel set B c C. The module of r is defined by

(1) modr = i~f J L p(z)2 dxdy

where the infimum is taken over the functions p 2: 0 with

(2) i p(z)ldzl 2: 1 for all CEr.

The quantity 1/ mod r is called the extremal length of r. The module concept has its root in the length-area method used by Grötzsch and later e.g. by Warschawski. It was introduced by Beurling and Ahlfors (AhBeu50) and has many applications in conformal mapping, in particular in connection with quadratic differentials (see e.g. Jen65, Kuz80, Str84). It plays a key role in the theory of quasiconformal maps (LeVi73), in particular in jRn (see e.g. Väi71, Vu088).

For a compact set E C <C, following Fekete, we define the logarithmic capacity by

(3) capE = lim max rr rr IZj - zkI1/n(n-l) n--->oo (Zk)

j#

where the maximum is taken over all point systems Zl, ... , Zn E E. An alternative definition is through potentials; see e.g. Lan72, HayKe76. The concept of capacity can be generalized to Borel (and even Souslin) sets.

Apart from the linear measure A, the capacity is the most important tool to describe the size of a set in conformal mapping. Exceptional sets are often of zero capacity (Car67), and we say that a property holds nearly everywhere if it holds outside a set of zero capacity. If E c '[' then capE 2: sin[A(E)/4] with equality for an arc. More generally

(4) E c <C, capE = 0 :::} A(E) = 0

so that "nearly everywhere" implies "almost everywhere". The converse does not hold; for example the classical Cantor set has positive capacity but zero

196 Chapter 9. Curve Families and Capacity

measure. See e.g. Nev70 and Section 10.2 where we study the relation to Hausdorff measure.

PHuger's Theorem. 11 rE(r) is the lamily 01 curves that connect {Izl = r} with the Borel set E c 1I' then

(5) capE = lim ~exp (_ 'Fr ()). r-->O V r mod rE r

This result can be used to establish many important connections between capacity and conformal mapping which are more precise than some of the estimates proved in the first chapters. For instance, if I is a conformal map of j]J) then (Beu40)

(6) 11 1!,(r()lld(1 < 00 for nearly all ( E 1I',

and if 1 = {eit : It - arg (I :::; 'Fr(1- r)} with (E 1I', 0:::; r < 1 then

(7) II(z) - l(r()1 :::; R(1 - r 2 )1!,(r()1 for z E 1 \ E

where capE< 11(1 - r)/v'R; see Corollary 9.20. If I maps j]J) conformally into j]J) with 1(0) = 0 and if AC 1I', I(A) c 1I' then (Theorem 9.21)

(8) capl(A) 2: 1!'(0)1-1/ 2 capA 2: capA.

This is an analogue of Löwner's lemma that A(f(A)) 2: A(A).

9.2 The Module of a Curve Family

Let B be a Borel set in <C. A curve lamily r in B consists of open, halfopen or closed arcs or curves in B. The function p in B is called admissible for r if it is non-negative and Borel measurable in Band if

(1) [p(z)ldz l 2: 1 for all CEr.

Thus the curves C in rare required to have length 2: 1 with respect to the metric p(z)ldzl; the length may be infinite.

We define the module of the curve family r by

(2) mod r = i~f J L p( z) 2 dx dy ,

where the infimum is taken over all functions p admissible for r. Note that the integral is the area of B with respect to p(z)ldzl. If Be B' then r is also a curve family in B' with the same module; we set p(z) = 0 for z E B' \ B.

9.2 The Module of a Curve Family 197

Example 1. Let r be the family of all eurves C in the reet angle

(3) G = {x + iy : 0 < x < a, 0 < Y < b}

that eonneet its two horizontal sides; see Fig. 9.1 (i). Then Po(z) = 1/b (z E G) is admissible beeause every eurve in r has length :2: b. It follows that

(4) modr::::; JipO(Z)2dXdY=~.

Now let p be any admissible funetion. Sinee the vertical segments (x, x + ib) belong to r for 0 < x < a we obtain from (1) by Sehwarz's inequality that

and thus

( b )2 b

1::::; 1 P(X+iy)dy ::::;b1p(X+iy)2dY

~::::; 1a(1bp(X+iy)2dY)dX= Jip(Z)2dXdy.

Henee we eonclude from (2) and (4) that

(5) modr = alb.

Example 2. We eonsider the annulus

(6)

Let r be the family of eurves in H that eonnect the two boundary eomponents and r' the family of eurves in H that separate them, see Fig. 9.1 (ii). Then (see e.g. Jen65, p. 18)

(7) T2

modr = 27r/ log -, Tl

, 1 T2 mod r = - log - .

27r Tl '

if Tl = 0 then modr = O.

ib a+ib

(

o a (i)

Fig.9.1. The families of Examples 9.1 and 9.2

An important property of the module is its conformal invariance. The image f(r) eonsists of all image eurves f(C) where CEr.


Proposition 9.1. Let r be a curve lamily in the domain G. 11 1 maps G conlormally into ethen

(8) mod/(r) = modr.

Proof. Let p* be admissible for I(r). We define p(z) = p*(f(z))If'(z)1 for z E G. If CEr then

(9) 1 p(z)ldzl = 1 p*(f(z))If'(z)lldzl = ( p*(w)ldwl ~ 1 c c J!(C)

so that p is admissible for r. Since I is a conformal map we have

and therefore mod r :::; mod I (r). The reverse inequality is obtained by considering 1-1 • D

The module is monotone and subadditive.

Proposition 9.2. 11 rc r' or il, lor every CEr, there exists C' E r' with C' c C, then mod r :::; mod r' _ Furthermore

(11)

In particular mod r o = 0 =? mod( r u r o) = mod r.

Prool. If p is admissible for r' then p is also admissible for r because we can find C' Er' with C' c C. Hence modr :::; modr' follows from (2). To prove (11), let Pk be admissible for r k-We set Pk(Z) = 0 where it is undefined. Then

for all k

so that pis admissible for r = U rk, and (2) shows that

modr:::; J J p2 dxdy = L J J p~dxdy k

which implies (11). The final assertion follows from the fact that

modr:::; mod(r u r o) :::; modr + modro . D

Proposition 9.3 Let n be curve lamilies in disjoint Borel sets Bk- 11 Ukrk c r then


(12) Lmodn ::; modr. k

If r is a curve family in Band if each CEr contains some Ck E r k for every k, then

(13) _1_>L_1~ modr - k modn'

The quantity 1/ mod r is ealled the extremallength of rand is often used in preference to the module; see e.g. Ah173.

I r I ( C, C2 7

J I

6 6, 62 6, 62

Fig. 9.2. The situations considered in Proposition 9.2 and Proposition 9.3

Proof. We rest riet ourselves to prove the seeond assertion and write ak 1/ modrk, a = 2: j aj. We may assume that a < 00; the case a = 00 is obtained as a limit. If Pk is admissible for r k we define

p(z) = a-1akPk(z) for z E Bk, p(z) = 0 for z E B \ U B j .

For CEr we have C ::> U Ck with Ck E r k and thus

1 pldzl 2': L a-1ak 1 Pk Idzl 2': L a-1ak = 1. C k Ck k

Thus P is admissible far r. Furthermore

fL p2 dxdy::; ~ f Lk p2 dxdy = a-2 ~a% f Lk p% dxdy

and thus, by (2),

modr::; a-2La%modrk = a-2 L ak = a-1 k k


Next we consider modules and symmetry; see Fig. 9.3.

Proposition 9.4. Let the domain G be symmetrie with respeet to lR and let G+ = {Imz > O} n G, A+ c {Imz > O} n oG, furthermore G-,A- the reflected sets in {Im z < O}. If


r = {curves ein G from A- to A+}

r± = {curves C in G± from ~ to A±}

then modr+ = modr- = 2modr.

--~r+~--r-----r-IR

Fig. 9.3. A domain G symmetrie to the real axis

Proof. It is easy to see that mod r+ = mod r-. Since every CEr satisfies C :l C+ U C- with C± E r±, we conclude from (13) that mod r+ ::::: 2 mod r.

Conversely let p be admissible for rand define p+(z) = p(z) + p(z) for z E G+. Let C+ E r+ and let C- be the reflected curve in G-. Then C+ U -+ --(C n C ) u C- E rand thus

{ p+ldzl = { p(z)ldzl + ( p(z)ldzl::::: 1 Jc+ Jc+ Jc -

so that p+ is admissible for r+. Furthermore

J i+ p+(z)2 dx dy = J i+ [p(z)2 + 2p(z)p(z) + p(z)2] dx dy

:::; 2 J i+ p(z)2 dxdy + 2 J i- p(z)2 dxdy

= 2 J [p2 dX dY

and therefore mod r+ :::; 2 mod r.

Fig.9.4

o


We now prove one of the Teichmüller module theorems (Tei38)j see Fig. 9.4 and also (18), (19) below.

Proposition 9.5. Let 0< rl < r2 and Tj = {Izl = rj} for j = 1,2. Let the Jordan curue J separate Tl and T2 and let r j denote the family of Jordan curues that separate Tj and J. 1f

(14) modrl + modr2 > (271')-llog(r2Ird - c (0< c < 1/36)

then, with an absolute constant K,

(15) 1:::; maxlzl/minlzl < 1 + KJcIOg ~. zEJ zEJ c

It follows from (7) and (12) that always

(16)

and our proposition states that approximate equality implies that J is approximately a concentric circle.

Proof. Let the functions

00 00

f(() = Lan(n, g(() = bel + Lbn(n, al > 0, b> 0 n=l n=O

map ][)) conformally onto the inner and outer domain of J. For r > r2 let r(r) denote the family of Jordan curves separating J and {Izl = r}. Since the preimage of {Izl = r} is approximately {I(I = blr} for large r, we easily obtain from (7), Proposition 9.3 and Proposition 9.1 that

271'modr2 -logr2:::; lim (271'modr(r) -logr) = -10gb T-+OO

with a similar estimate for modrl . Hence we see from (14) that

The area of the inner domain of J is 71' L:~ nlan l2 and also equal to 71'(b2 - L:~ nlbn I2). Hence

00 00

L nlan l2 + L nlbn l2 = b2 - a~ n=2 n=l

and we see from Schwarz's inequality that, for 1(1 = p = al/b,


Since P > e-27rC > 0.83 by (17) and thus 1 - p2 < e- l , it follows that

Ig(() - bel - bol ~ bT} == bKl Jdog ~, similarly I!(() - al(j ~ bry. Hence (17) implies that J lies between a circle of radius bp-l + bT} < be27rE + bT} and a circle of radius alP - bT} > be-47rE - bT} around O. 0

There are also estimates in the opposite direction due to Grötzsch and Teichmüller; see e.g. LeVi73, p. 53-61 for the proofs.

Let G be a doubly connected domain and r the family of Jordan curves that separate the two boundary components. If G c ]]J) and if G separates {O, zd from l' then

(18) 21fmodr < log(4/lzll).

If Ge C separates {O, zd from {Z2' oo} then

(19) 2 d r I 16(lzll + IZ21) 1fmo < og IZll .

Exercises 9.2

1. Use conformal mapping onto a rectangle to prove Proposition 9.4 for the case of a Jordan domain.

2. Deduce the second formula (7) from (5) and Proposition 9.4.

3. Suppose that f maps ]]J) conformally onto the open kernel of

n

U {k - 1 ::; x ::; k, 0 ::; Y ::; Yk} k=l

such that the arc {e it : 371"/4 ::; t ::; 571"/4} is mapped onto [0, iYI] while {eit

-71" / 4 ::; t ~ 7I"/4} is mapped onto [n, n + iYn]. Give two proofs that

n

4. Let the Jordan curve J be partitioned into four consecutive arcs Al, A2 , A3 , A4

and let G be the inner domain of J. Consider two curve families to prove that

dist(A I , A3 ) dist(A2 , A4 ) ::; areaG.

(See McM71 for much deeper related results.)

5. Let h be a K,-quasiconformal map from H to H* where 0 ::; K, < 1. If r is a curve family in H show that

modh(r) ::; [(1 + K,)/(1 - K,)] modr.

9.3 Capacity and Green's Function 203

9.3 Capacity and Green's Function

1. Let E be a eompaet set in C and G its outer domain, i.e. the eomponent of iC \ E with 00 E G. For n = 2,3, ... we eonsider

n n

Lln = Lln(E) = max rr rr IZk - zjl· Zl, ... ,znE E

k=l j=l

(1)

kij

The maximum is assumed for the Fekete points

(2) Zk = Znk E E (k = 1, ... , n);

this system of points is not always uniquely determined, and the maximum principle shows that Znk E BG. We ean write

(3) Lln = Ik=1~~.,n (lznk '" Z~kl)12; this is a Vandermonde determinant. We eall

n

(4) qn(Z) = rr (z - Znk) k=l

the nth Fekete polynomial and write

(5) rn = rn(E) = min Iq~(Znk)1 = min rr IZnk - znjl· k k jik

If the minimum in (5) is assumed for k = v then

Lln = rr IZnv - Znk 12 rr rr IZnk - Znj 1 ~ r;Lln- 1

kiv kij k,jiv

by (1). Henee

(6)

the seeond inequality follows from (5). We eonclude that Ll~[n(n-l)l is deereasing and eall

(7) 1

eap E = lim Lln(E) n(n-l) n->oo

the (logarithmie) capacity of E. It is also ealled the "transfinite diameter". See e.g. Lej61 for generalizations.

The next properties follow at Onee from the eorresponding properties of the quantity Lln:

(8) eap E = cap BG ;


(9)

(10)

(11)

capc,o(E) = lai capE if c,o(z) = az + bj

cap c,o(E) ::; cap E if c,o is a contraction of E .

Proposition 9.6. If E c C is compact then

(12) capE::; maxlp(z)l l / N zEE

for every polynomial p( z) = zN + ... , furthermore

(13) capE = lim 'Y;./n = lim max Iqn(z)11/n. n-+oo n-+oo zEE

Proof. Let Ip(z)1 ::; M for z E E and let m = 1,2, .... We write n = mN and consider the polynomials

(14) PI-'N+v(Z) == p(z)I-'ZV = zl-'N+v + ... for 0::; J.L < m, 0::; v < N.

Adding suitable multiples of the first j - 1 columns of the Vandermonde determinant in (3) to the jth column we see that

Lln = Ik=1~~.,n (lPl(znk) ... Pn-l(Znk)) 1

2

We now apply Hadamard's determinant inequality

(15) Ik=~~~.,n (akl ak2··· akn )12

:::; fl (t lakjl2)

If E C D(O, r) we obtain from (14) that

Lln ::; TI (t IPj(ZnkW)

m-lN-l

::; nn rr rr (r 2V M 21-') = nnrN(N-l)m MNm(m-l) .

1-'=0 v=o

Taking the n(n - l)-th root and using that n = Nm, we deduce for m --+ 00

that cap E ::; M1/N which proves (12). Furthermore (1) and (4) show that, for z E E,

n

Iqn(zW Lln = rr Iz - Znjl2 rr rr IZnk - znjl ::; Lln+1. j=l kf:.j

Hence we obtain from (6) and (12) that

(16) 1 (Ll 1 ) 2~ .!. _1 cap E ::; max Iqn (z) In::; :+ ::; 'Y;+1 ::; Ll;:~t1)

zEE ~n

and (13) follows from (7) for n --+ 00. D


2. We now introduce the Green's function with pole at 00 by the method of Fekete points (Myr33, Lej34). A probability measure p, on the compact set Eis a measure with p,(E) = 1 and p,(B) ::::: 0 for all Borel sets B C E.

Theorem 9.7. Let G be a domain with 00 E G and let E = 8G and cap E > O. Then there is a probability measure p, on E with the following properties: The junction

(17) g(z) = L log Iz - (I dp,(() -logcap E = log Izl -logcap E + 0 C~I) is sub harmonie in <C and harmonie in G \ {oo}. 1t satisfies

(18) g(z) ::::: 0 for z E G, g(z) = 0 for z E <C \ G.

1f u is any positive harmonie function in G with u(z) = log Izl + 0(1) as z ----> 00 then

(19) u(z) ::::: g(z) > 0 for z E G,

and this property uniquely determines 9 in G.

We caU p, an equilibrium measure of E and 9 the (extended) Green's funetion with pole at 00. Since it is subharmonie and thus upper semieontinuous we have

(20) g(() = limsupg(z) for (E <C. z-->(

Proof. (a) For n = 2,3, ... let Znl, ... , Znn be Fekete points of E and qn the corresponding Fekete polynomial. We introduce

(21) for z E <C,

where Vn(Znk) = -00. It foUows from Proposition 9.6 by the maximum principle that

(22)

The Lagrange interpolation formula shows that

so that, by (5), 1 ::; nlqn(z)l/bn dist(z, E)].

Hence we see from (21) that, for each z rf- E,

(23) liminfvn(z) ::::: liminf ~(logdist(z, E) -logn) = O. n---+oo n---+oo n


(b) We define probability meaSures J.Ln on E by assigning to each Fekete point Znb ... , Znn the mass l/n so that, by (4) and (21),

(24) vn(z) = r log Iz - (I dJ.Ln(() - ! log')'n for Z E C. JE n

The space of all probability measures on E is weakly compact (e.g. HayKe76, p.205), Le. there exists a subsequence (J.Lnj) and a probability measure J.L on E such that

for every function 'I/J continuous on E. We now define 9 by (17). Since log Iz - (I is subharmonie and dJ.L() ~ 0

it follows that 9 is subharmonie in Co It follows from (13), (24) and (25) that, as n = nj --t 00,

(26) Vn(z) --t g(z) for z E C \ E

and therefore from (23) that g(z) ~ 0 holds for z E C \ E and thus also for z E E = aG by (20). Hence we conclude from (22) and (26) that g(z) = 0 for z E (C \ G) \ E = C \ G. It is clear from (17) that 9 is harmonie in G \ {oo}. Since 9 is nonnegative and nonconstant in the domain G it follows that g(z) > 0 for z E G.

(c) Finally let u be positive and harmonie in G with u(z) = log Izl + 0(1) as z --t 00. We apply the maximum principle to the harmonie function vnj - u in G. Since vnj - u < vnj we conclude from (22) and (26) that 9 - u :::; 0 in G whieh proves (19).

If gis another positive harmonie function in G with g(z) = log Izl + 0(1) as z --t 00 that satisfies (19), it follows that g:::; 9 and also that 9 :::; g, hence 9 = gin G. Consequently (26) holds for all n --t 00; the case that z E C \ G is trivial by (18) and (22). D

The next result is a partial converse.

Theorem 9.8. Let u be positive and harmonie in the domain G and let

(27) u(z) = log Izl + a + 0(lzl-1 ) as z --t 00.

Then capE ~ e-a > 0 where E = aG. I/ furthermore u(z) --t 0 as z --t aG then u = 9 and capE = e-a •

Proof. As in part (c) of the foregoing proof we see that lim Vn :::; u in G and thus, by (24) and (27),

1 -log')'n = lim (log Izl- vn(z)) ~ lim (log Izl- u(z)) = -a > -00. n z-+oo z-+oo

Hence cap E ~ e-a > 0 by (5) and Proposition 9.6 so that 9 exists.


Now assume that u(z) --+ 0 as z --+ 8G. It follows from (27) and Theorem 9.7 that v = u - 9 :::: 0 in G. Since v is harmonie in G including 00 and since o ::; v(z) ::; u(z) --+ 0 as z --+ 8G, we obtain from the maximum principle that v = 0 and thus u = g. Finally a = - log cap E is a consequence of (17) and (27). 0

For simply connected domains the Green's function is closely related to conformal maps. Let again JI)l* = {Izl > I} U {oo}.

Corollary 9.9. If

(28)

maps JI)l* eonformally onto G then cap 8G = lei and

(29) g(z) = log Ih-1 (z)1 fOT z E G, g(z) = 0 fOT Z E C \ G.

This follows at once from Theorem 9.8 applied to the positive harmonie function u = log Ih-1 1. We consider two examples.

The function h() = (+ (-1 maps JI)l* conformally onto [-2,2]. Hence it follows from (10) and Corollary 9.9 that

(30) cap[a, b] = Ib - al/4 for a, bE C.

Let 0 < a ::; 11' and a = sin(a/2). Then

h() = «(a( + 1)/( + a) = a( + (1 - a2 ) + ...

maps JI)l* conformally onto the complement of a circular arc and we obtain from Corollary 9.9 that

(31) cap{eit9 : -a ::; {)::; a} = sin(a/2) (0< a ::; 11').

In partieular cap ~ = cap ']I' = 1. These examples are extremal in the following result (P6128, AhBeu50).

Corollary 9.10. If P is the orthogonal projeetion of the compact set E onto some line then

(32) A(P) ::; 4capP ::; 4capE.

If E is a eompact set on ']I' then

(33) sin[A(E)/4] ::; capE.

To prove (32) we may assume that we project onto IR. Since z r--+ Re z is a contraction of E onto P and since

cp(x) = A(P n (-00, xl) (x E P)


is a contraction of P onto a segment of length A(P), we conclude from (11) and (30) that

capE ~ capP ~ cap<p(P) = A(P)/4.

See AhBeu50 or Pom75, p. 337 for the proof of (33).

Proposition 9.11. LetE c C be compact andcapE > O. IfdiamE < band if f.L is an equilibrium measure of Ethen

(34) II.(A) < log(b/ cap E) " A r' J or compact cE,

- log(b/ capA)

in particular f.L(A) = 0 if cap A = O.

Note that capA:::; capE:::; ..1;/2:::; diarnE< bj see (7).

Proof. We may assurne that a = f.L(A) > 0 and b = 1, by (10). The function

1 f log cap E logcapE ( 1 ) (35) u(z)=~JAloglz-(ldf.L(()- a =loglzl- a +0 R is harmonie in C \ A. Since AcE and diam E < 1, we see from (17) and (18) that

liminfu(z) = liminf(~ f log -I 1 il df.L(() + g(Z)) ~ 0 z--->A z--->A a } E\A Z - ." a

and we conclude that u(z) > 0 in the outer domain H of A. Hence it follows from Theorem 9.8 applied to Hand from (35) that capA ~ exp(a-IlogcapE) whieh is equivalent to (34). 0

3. Finally we consider arbitrary sets in C. The (inner) capa city of E is defined by

(36) cap E = sup{ cap A : A compact, ACE}.

It is clear that EI C E 2 implies cap EI :::; cap E 2 .

Theorem 9.12. If E is a Borel or Souslin set then, for every € > 0, there is an open set H with

(37) E eH, cap H < cap E + € .

This important result (Cho55) states that Souslin sets are "capacitable". See e.g. HayKe76 for the proof. We deduce a subadditivity property.

Corollary 9.13.

(38)

Let E n be Borel sets and E = U~=I E n . If diam E E n such that

b b c l/log-H <1/log-E +-2 (n=I,2, ... ).

cap n cap n n (39)

Then E c H = Un H n and we may assume that diam H < b. Every compact subset A of H can be covered by finitely many disks D k such that D k C

Hnk for some nk. The compact sets An = A n Uk:nk=n Dk C Hn satisfy A = Al U ... U Am for some m. If J.L is an equilibrium measure of Athen 1 = J.L(A) S J.L(A 1 ) + ... + J.L(Am ) and therefore

b m b m b l/log-A S L 1/ log-A- < L 1/log-E +c

cap n=l cap n n=l cap n

by (34), (39) and because of An C H n. Hence (38) follows from (36). The final statement is an immediate consequence. D

We say that a property holds for nearly all ( E E if it holds for all ( E E except for a set of zero capacity. To say that a property holds nearly everywhere is much stronger than to say that it holds almost everywhere because a set of capacity zero has linear measure zero and even a-dimensional Hausdorff measure zero for every a > 0; see (10.1.3).

We now show (Fro35) that the Green's function 9 vanishes nearly everywhere on E = oG; compare also Exercise 7.

Theorem 9.14. Let G be a domain with 00 E G and capoG > O. Then

(40) g(z) -+ 0 as z -+ (,z E G for (E oG\X

where X is an Fa-set with capX = O.

Proof. It follows from (4) and Proposition 9.6 that

11 n 1 - L log IZnj - (I dJ.L() S max -log Iqn()1 -+ logcapE E n j=l CEE n

as n -+ 00 and therefore, by (17),

(41) OS inf g(z) S liminf ~ ~ g(Znj) SO. zEE n--+oo n L..J

j=l

Furthermore the sets

(42) A k = {( E E: limsup g(z) ~ -k1 } (k = 1,2 ... ) z--+C,zEG

are compact. Now suppose that cap A k > 0 for some k and let gk be the Green's function of A k. Since G lies in the outer domain of Ak where gk is


positive, it follows from (19) that gk 2: 9 in G. Since A k C ßG and gk is upper semicontinuous we deduce from (42) that, for each ( E Ak ,

gk(() 2: limsup gk(Z) 2: limsup g(z) 2: -k1 > 0 z->(,zEG z->(,zEG

in contradiction to (41) applied to gk. Thus capAk = 0 for all k, and (42) shows that g(z) --+ 0 as z --+ (, z E G for ( E E \ X where X = UAk. 0

There is a great number of further results on Green's functionj see e.g. BrCh51, duP70, Lan72, HayKe76.

Exercises 9.3

1. Use the Hadamard determinant inequality to show that exp(ia + 27rik/n) gives a system of Fekete points of 11' for each a and deduce that ..1n(1l') = nn and 'Yn(1l') = n.

2. Use the contraction property to prove (33) for the special case that E lies in one half of 11'.

3. Prove Theorem 9.12 for the special case that Eis compact.

4. Let (A k ) be an increasing sequence of compact sets. Show that capAk --+

cap (Un An) as k --+ 00.

In the following problems, let G be a domain with 00 E G and cap E > 0 where E = 8G. Let 9 be the Green's function.

5. Show that g(z) ~ log(diamE/ capE) for z E E.

6. If ACE is a continuum show that g(z) = 0 for z E A.

7. If g«() > 0 for some ( E E show that ( can be enclosed by Jordan curves in G of arbitrarily small diameter. Deduce that g«() = 0 holds for nearly all ( E E (which is stronger than (40)).

9.4 PHuger's Theorem

Now we consider sets on the unit cirde 11'. There is a dose connection to the starlike functions studied in Section 3.6. We begin with compact sets.

Proposition 9.15. Let E C 11' be compact and capE> O. Then the equilibrium measure J-L is uniquely determined. The function

(1) h(z) = z(capE)2 exp [-2 L log(l- (Z)dJ-L(()] (z E lD»

maps lD> conformally onto a starlike domain

(2) h(lD» = {sei~ : 0 ~ s < R( 19), 0 ~ 19 ~ 211"} C lD>,

where R(19) = 1 for almost all 19 and Ih(()1 = 1 for nearly all ( E E.

9.4 Pfluger's Theorem 211

Proof. The function h is univalent and starlike by Theorem 3.18 because f..L can trivially be extended to a probability measure on 'lI'. It follows from Theorem 9.7 that

(3) Ih(z)1 = Izl exp[-2g(z)] < 1 for z E ]]J).

Since h'(O) > 0 and since the Green's function g is uniquely determined by E c 'lI', the same holds for h. Hence it follows from (1) and (3.6.7) that f..L is uniquely determined. Furthermore h(() exists for ( E 'lI' and Ih(()1 = 1 for ( E E \ X where X is the union of countably many compact sets Av of capacity zero by Theorem 9.14. Hence it follows from (3.6.7) and Proposition 9.11 that

A({19 = argh((): (E ('lI'\ E) UX})

= 27r [f..L('lI' \ E) + f..L(X)] :::; 27r Lf..L(Av) = 0 v

so that R(19) = 1 for almost all 19. o

Next we consider small Fcr-sets and construct a starlike function related to the "Evans function".

Proposition 9.16. [f E c 'lI' is an Fcr-set of zero capa city then there is a starlike function h(z) = z + ... (z E ]]J)) such that

(4) Ih(z)1 --+ 00 as z --+ (, z E]]J) for each ( E E.

Proof. The Fcr-set E is the union of countably many compact sets Av of capacity zero and by Corollary 9.13 we may assurne that Av C Av+l. Let qvn denote the nth Fekete polynomial of A v defined by (9.3.4). By Proposition 9.6 we can find nv such that

(5) Iqvnv(zW/nv <exp(-4V ) for zEHv (v=O,l, ... )

where Hv is some open set containing Av. Let Zvnk be nth Fekete points of A v . Then

00 n v

(6) h(z) = z II II (1 - ZvnvkZ)-l/(nvn (z E ]]J)) v=O k=l

is starlike by Theorem 3.18 because the sum of all exponents is -2. Let ( E E. Then ( E Av for large v. Since 11 - zvnkzl :::; 2 for z E ]]J) and

IZvnkl = 1 we see from (6) and (5) that, for z E ]]J) n Hv,

Ih(z)1 ;::: 1:llqvnJz)I-1/(nv 2 V) > 1:1 exp(2V )


Finally we consider any Borel set E on 1l'. It follows from (9.3.33) and (9.3.36) that

(7) A(E) < sin A(E) < cap E . 27r - 4-


We now prove Pfiuger's theorem (Pfl55) which is very useful for estimating the size of sets (Mak87).

Theorem 9.17. Let E be a Borel set on 1I' and let rE(r) (0< r < 1) denote the family of all curves in {r < I z I < I} that connect E with {I z I = r}. Then

(8) v'r (7r) v'r -- cap E :::; exp - d r () :::; -- cap E 1 + r mo E r 1- r

for 0< r :::; 1/3 and thus

(9) cap E = lim ~ exp (- d 7r ( )) r--+O yr mo rr r

Fig. 9.5. Ptluger's theorem and its proof

Proof. (a) First we prove the lower estimate (8) assuming that Eis compact. Let h be the starlike function of Proposition 9.15. Then

(10) Ih(z)1 2: Ro == r(l + r)-2(capE)2 for Izl = r < 1

by (1) and Theorem 1.3. If R('I'J) = 1 in (2) then the preimage of [0, ei1?] under his a curve in IlJ) connecting 0 with E as we see from (3) and Theorem 9.7. If pis admissible for the family h(rE(r)) and is zero where it was undefined, it follows that

by Schwarz's inequality. Since R( 'I'J) = 1 holds for almost all 'I'J we conclude that

27r :::; log ~ J' { p(W)2 dudv o JRo<lwl<l

and thus, by (9.2.2) and Proposition 9.1,

1 1 + r 27r:::; modh(rE(r))log- = 2modrE(r) log v'r E

Ro rcap

because of (10). This gives the lower inequality (8) for the case that E is compact.

9.4 Pfluger's Theorem 213

If E is any set on '][' then we can find compact sets An C E with cap An ---+

capE by (9.3.36). Since rAn(r) C rE(r) and thus modrAJr) :::::: modrE(r) by Proposition 9.2, we obtain the lower estimate in (8) for E by letting n ---+ 00

in the estimate for An. (b) Now we prove the upper estimate (8) assuming that E is an Fa-set,

thus

E=UAn withAncompact, An CAn+1 (n=I,2, ... ). n

Let hn be the starlike functions of Proposition 9.15 for the compact sets An. The union X of all the exceptional sets is again an Fa-set of zero capacity. Let ho(z) = z + ... be the starlike function associated with X according to Proposition 9.16. It satisfies Iho(()1 ?: 1/4 for ( E '][' by Theorem 1.3.

Since An C An+1 we see from (3) and Theorem 9.7 that Ihni:::::: Ihn+11 in ]IJ). Since h~(O) > 0 it follows that hn(z) converges to a function h(z) = (capE)2 z + ... (z E ]IJ)); see Exercise 9.3.4. If (E E \ X then (E An \ X for some n and thus Ih(()1 ?: Ihn(()1 = 1.

Let 0 < 8 < 1 - log 3/ log 4. The function

f(z) = ho(z)Öh(z)l-Ö = (capE)2-2Ö z +... (z E]IJ))

is again starlike and satisfies If(()1 ?: 4- 15 for ( E E \ X and f(() = 00 for ( EX, furthermore

(11) 1

for Izl = r :::::: 3 .

Let C be a curve in rE(r) ending at ( E E. Since the angular limit f(() belongs to the cluster set of f along C by (2.5.6), we see that

liminf If(z)1 ?: If(()1 ?: 4-15 • z--->(,zEC

Together with (11) this shows that every curve in f(rE(r)) contains a curve connecting the two boundary circles of {R < Iwl < 4-Ö}. Hence we obtain from Proposition 9.2 and from (9.2.7) that

4- 15 2-15 (1 - r) modrE(r) = modf(rE(r)):::::: 27r/log - = 7r/log y'r( )1 15 R r capE -

by (11), and the upper estimate (8) follows for 8 ---+ O. Finally let E be any Borel set on '][' and let € > O. By Theorem 9.12 there

is an open set H on '][' such that E C Hand cap H < cap E + E. Since H is an Fa-set it follows that

exp (- mOd~E(r)) ::::::exp (- mOd~H(r)):::::: l~r(capE+€). 0

The following result (BiCaGaJ089, Roh91) was used in the proof of Theorem 6.30; see Fig. 9.6.


Corollary 9.18. Let hand h map llJl onto the inner and outer domain of the Jordan curve J c C and let

(12) A j = fj-I(JnD(a,R)) (a E J,R > 0)

for j = 1,2. Then

(13) A(AI)A(A2) ~ 411"2 cap Al cap A 2 ~ M R 2 for R < Rn

where the constants M and Ro depend only on hand h.

Proof. We choose Ro such that 0 < Rn ~ dist(J, h(z)) for Izl ~ 1/3 and j = 1,2. For 0 < R < Rn, let rj denote the family of all curves in {w E h(llJl) : R < Iw - al < Ro} that connect the two circles. Since h(llJl) n h(llJl) = 0 it follows from Proposition 9.2, Proposition 9.3 and (9.2.7) that

(14)

and Theorem 9.17 shows that

v: capAj ~ exp (- mOdr:j (1/3)) ~ exp (-mo~rJ because every curve in f(r A j (1/3)) contains some curve in r j . Now we multiply the inequalities for j = 1 and j = 2. Since I/al + l/a2 ~ 4/(al +a2) for ab a2 > 0 we conclude from (14) that

3 16 capA l capA2 ~ exp[-2Iog(Ro/R)] = R2/R~

and (13) follows by (7). D

The factor R 2 can often be replaced by RHO with 8 > 0 if a little more is known about the geometry of J; see Roh91.

Fig.9.6. Corollary 9.18 and its proof

9.5 Applications to Conformal Mapping 215

Exercises 9.4

1. Let E c 11' be compact and cap E > O. If Jl is an equilibrium measure of E show that

fez) = exp (LIOg(Z - () dJl (()) (Izl > 1)

maps ]IJl* conformally onto a starlike domain in {Iwl > cap E}.

2. Let E be an F,,-set in C with cap E = O. Construct a function u harmonie in C \ E with u(z) = log Izl + ... (z ---+ 00) such that u(z) ---+ -00 as z ---+ (, (E E. (This is an "Evans function" of E.)

3. Use the function of Exercise 2 to generalize (Pri19) the Privalov uniquelless theorem: Let E c C be an F,,-set of zero capacity and let 9 be meromorphie in ]IJl. If the angular limit g(() exists for eaeh ( E Ac 11' and g(() E Ethen A(A) = O.

4. Under the assumptions of Corollary 9.18, suppose that J lies in {I arg(w - a)1 < 1I"0:} near a where 0: < 1. Show that cap Al eap A 2 ::::; M' R2/[a(2-a)].

9.5 Applications to Conformal Mapping

We first generalize Theorem 1.7.

Theorem 9.19. Let f map ]IJl conformally into C. Then the angular limit f(() exists and is finite for nearly all ( E 11'. 1f R ~ 1 then

(1) If(() - f(O)1 ::::: If'(O)IR for (E 11' \ E,

(2) l ll !'(r()1 dr ::::: K 1 1!,(O)IRlog(6R) for (E 11' \ E

with an absolute constant K 1 where the exceptional set E satisfies

(3) A(E)/(27r) ::::: capE::::: l/VR.

By applying a suitable Möbius transformation we deduce (Beu40) at once

that every conformal map of]IJl into iC has angular limits nearly everywhere (which is much stronger than almost everywhere).

Proof. We may assume that f(O) = 0 and 1'(0) = 1. Let 0 < r < 1/3. The Koebe distortion theorem shows that

(4) r(l + r)-2 == r' ::::: If(()1 ::::: r" == r(l- r)-2 < 1 for Izl = r.

Let E denote the set of all ( E 11' such that

(5) f(Cd ct D(O, R) or Ilf'(z)lldzl ~ Rlog(6R) Cl


for all curves Cl from {Izl = 1/3} to ( and let rE(r) be the family of all curves from {Izl = r} to E. We define a = 10g(R/r") and

(6) 1 f'(z) 1

p(z) = al(z) if I/(z)1 ~ R, p(z) = 0 if I/(z)1 > R.

Let C E rE(r) begin at Zl with IZll = rand end at ( E E C ']['. If I(C) ct. D(O,R) then there exists a first Z2 E C with I/(Z2)1 = Rand we see from (6) and (4) that

i p(z)ldzl 211:2 !~~;) dzl = ~ IIOg ~~;:~ 12 ~ log ~ = 1.

If however I( C) c D(O, R) then we write C = Co U Cl where Cl runs from {Izl = 1/3} to E. Then the second relation in (5) holds and thus, by (6) and Exercise 1.3.3,

a1 pldzl 21 1 - Izl 1 dz 1 + J 1f'(z)lldzl C Co 1 + Izl z R

3(1 +r)2 R 2 log + log(6R) > log - > a

16r r'

by (4). Hence p is admissible for rE(r) and it follows from (6) and (4) that, with B = {z E !I) : Izl 2 r, I/(z)1 ~ R},

modrE(r) ~ J LI :;~~) 1

2 dxdy ~ !~ 1: ~s

27r 10g(R/r') 27r [log(R/r")J2 10g(R/r) + O(r)

as r ----; 0 by (4). Hence (3) follows from Theorem 9.17. Now let ( E '][' \ E. Then (5) is false for some curve Cl ending at ( and

it follows from Theorem 4.20 that (2) holds. Hence the angular limit I«() exists. By Corollary 2.17 it is equal to the limit along Cl so that I/«()I ~ R. Furthermore it follows that

(7) cap { ( E '][': 101 1!,(r()1 dr = oo} = 0

and 1 ( () =I- 00 exists outside this Borel set. o

The next result is more precise than Corollary 4.18.Let again df(z) = dist(f(z), {JG).

Corollary 9.20. Let 1 map !I) conlormally onto G C <C and let ( E '][',

o ~ r < 1, a > 0 and R 2 1. Then

(8) I/(z) - l(r()1 ~ 4Rdf (r() lor z E 1\ E,


where 1 = {eit : It - arg (I s a(l- r)} and

(9) capE S (1 + a2 )(1- r)/v'R,

and if S is the non-euclidean segment from r( to z then

(10) fs 1f'(s)lldsl S K 2df(r()Rlog(6R) for z E 1 \ E.

Proof. We write

(11) rp(s) = s + r(, g(s) = f(rp(s)) for sE]])l. 1 + r(s

Since Ig'(O)1 = (1 - r2)1f'(r()1 S 4df(r() we see from Theorem 9.19 that (8) and (10) hold for z = rp( s) and s E 11' \ E* where cap E* S 1/ VR. Since

Irp'(s)1 = 11 - r(rp~s)12 S (1 + a2)(1- r) for rp(s) E 1 1-r

it follows from (9.3.10) and (9.3.11) that E = rp(E*) satisfies (9). 0

The next result (Pom75, p.348; compare Kom42, Jen56) is an analogue of Löwner's lemma. See Ham88 and Theorem 10.11 for generalizations.

Theorem 9.21. Let f map ]])l conformally into ]])l with f(O) = O. 1f E is a Borel set on 11' such that the angular limit exists and f(() E 11' for ( E Ethen

capE capf(E) > lT7i7i\\T > capE.

- V 11'(0)1 -(12)

Proof. Let An be closed subsets of E such that cap An ~ cap E as n ~ 00.

Then f(An) is a Borel set on 11'. Since r" == max{lf(z)1 : Izl = r} rv 1f'(O)lr as r ~ 0 and since

modrf(An)(r") ;::: modf(rAJr)) = modrAn(r)

by Corollary 2.17 and by Propositions 9.2 and 9.1, we see from Theorem 9.17 that

capf(E);::: capf(An);::: 1f'(0)1-1/ 2CapAn

and (12) follows for n ~ 00. o

We formulate a special case in terms of the harmonie measure w(z, A) at z E ]])l of A c 11' with respect to ]])l defined by (4.4.1) and the non-euclidean distance A(Z, z') given by (1.2.8); see Fig. 9.7.

Corollary 9.22. Let f map ]])l conformally into ]])l \ {w}. 1f A is a Borel set in 11' and f(A) C 11' then, for z E ]])l,


(13) sin[1fw(z, A)j2] :::; (1 - exp[-4A(f(Z), W)])1/2 ;

if f(A) is an arc ofT there there is an additional factor sin[1fw(f(z) , f(A))j2] < 1 on the right-hand side.

If G = f(lDl) and E = f-l(lDlnI3G) we obtain from (13) applied to A = T\E and from w(z, E) = 1 - w(z, A) that

(14) sin [~w(z, E)] ~ exp [-2A(f(Z), w)] .

This is related to the Carleman-Milloux problem; see Nev70, p. 102.

Proof. Since the harmonie measure and the non-euclidean metrie are invariant under Möbius transformations of lDl onto lDl we may assume that z = 0 and f(O) = 0, furthermore that w > O. Since

f(z)j(l + j(Z))2 = j'(O)z +... (z E lDl)

is analytic, univalent and -I- wj(l + w)2, we see that 11'(0)1 :::; 4wj(1 + w)2. Hence we conclude from (9.4.7) and from (12) that

[1f ] A(A) 2y'W sin "2w(O,A) =sin-4- :::;capA:::; l+wcapf(A).

We have cap f(A) :::; cap T = 1 and cap f(A) = sin[1fw(O, f(A))j2] if f(A) is an arc, by (9.3.31). This implies (13) because exp[-2A(0, w)] = (l-w)j(l+w).

D

·Z

.ID

Fig. 9.7. The situation of Corollary 9.22

We now show that the estimates (12) and (13) are sharp. Let 0 < a :::; 1. Then

(15) f(z) = 4az(1 - z + J(l - z)2 + 4zar2 = az + ...

maps lDl conformally onto lDl \ [-1, -b] where b = a (1 + JI=(i) -2. If 0 < 0: :::; arcsin Va and A = {eit : Itl :::; 20:} then

f(A) = {ei19 : 1'!91 :::; 2ß}, sinß = a-1/ 2 sino:.


Since capA = sino: and capB = sinß we see that equality holds in (12) and furthermore

1 - exp[-4,X(0, -b)] = 1 - [(1 - b)/(1 + b)]2 = a

so that equality holds in (13) for z = O. The next result (Schi46, Pom68) deals with general conformal maps. It

assumes its simplest form for the dass E of univalent functions of the form g(() = (+ bo + b1(-1 + ... for 1(1 > 1.

Theorem 9.23. 1f 9 E E and A c 1I' is a Borel set such that g(() exists for (E Athen

(16) capg(A) 2: (capA)2.

In partieular it easily follows (Duf45) that, for all conformal maps f of][)) into C, (17) capA> 0 =} capf(A) > O.

Proof. We rest riet ourselves to the special case that g(A) is a continuumj see Pom75, p.344 or Mak87, p.50 for the general case. Let h(() = c( + ... with c = cap g(A) map][))* conformally onto the outer domain of g(A)j see Corollary 9.9. Then

f(z) = 1/h-1(g(1/z)) = cz + ... maps]])) conformally into ][)) and Theorem 9.21 shows that, with A* = {1/z : z E A},

capA = capA* ~ c1/ 2capf(A*) ~ C1/ 2 . o

If gE E has a K-quasieonformal extension to ethen (Küh71j see Pom75, p.346)

(18) (cap A)1+1< ~ cap g(A) ~ (cap A)l-I<

for Borel sets A c 1I'.

Theorem 9.24. Let f map][)) conformally onto G c C and let E c 1I'. 1f o < 0: ~ 1 and if H is an open subset of G such that

(19) dist (f (0) , H) 2: 0:11'(0)1, A(f(C) n H) 2: b 2: 0

for every curve C C ][)) from 0 to Ethen

(20) 15 ( 7rb2 ) A(E) ~ 27rcapE < r;:;,exp --H vo: area

This is a quantitative version of the principle that boundary sets that are difIicult to reach have small harmonie measurej compare e.g. Corollary


6.26. This fact has severe consequences ("crowding") for numerical conformal mapping; see e.g. Gai72b, PaKoH087, GaHa91. The situation is depicted in Fig.9.8.

Proof. Let r = rE (0:/3) be defined as in Theorem 9.17. The Koebe distortion theorem shows that J(z) tf. H for Izl ::; 0:/3. Hence A(f(C)nH) 2: b for CEr by (19) so that

p( w) = 1/ b for wEH, p( w) = 0 for wEG \ H ,

is admissible for J(T). Hence

modr = modJ(T) ::; J L p(W)2 dudv = b-2 areaH

and therefore, by (9.4.8),

( \1'3;;/ 4) cap E ::; exp( -7rb2 / area H)

which implies (20) by (9.4.7).

Fig. 9.8. Theorem 9.24 and its proof

Exercises 9.5

1. Let f map j[J) conformally into j[J) and let b 2: 1. Show that

111 !'(T()ldT::; b for (E 1I'\E

where cap E < exp( -cb2 ) for some constant c > O.

o

2. Let f map j[J) conformally into C and suppose that the angular limit f(() exists. Show that, for e > 0,

8- 1 cap{z E 1I': Iz - (I ::; 8, If(z) - f(()1 :::": e} ---> 0 as 8 ---> o.

3. Let ft map j[J) conformally onto G t Ce such that h(O) = 0 for tEl and suppose that G t C GT for t < T. Show that

If:(0)1- 1/ 2 capft- 1 (B) (t E 1)

is increasing if Bis a fixed compact set with B C ßGt for tEl.


4. Let A and B be Borel sets. If A u B is a continuum show that

cap(A U B) ::; capA + capB

and show that this need not be true if A U B is not connected.

5. If Ais a Borel set on'][' show that capf(A) ~ (capA)2/16 for fES.

6. For the "comb" domain defined by (2.2.2), show that the harmonie measure of [1, 1 + i/n] with respect to 3i/4 is less than exp( -7l"n2 ) for large n.

Chapter 10. Hausdorff Measure

10.1 An Overview

Let 0: > O. The o:-dimensional Hausdorff measure of a Borel set E c C is defined by

(1)

where the infimum is taken over the covers (Bk) of E with diamBk :::; c for all k. In particular Al is the linear measure A discussed in Section 6.2.

We also introduce

(2)

where the infimum is taken of all covers of E. This quantity behaves more like a capacity than a measure and satisfies (Theorem 10.3)

(3) A~(E) :::; K(capE)a.

The Hausdorff dimension is defined by

(4) dimE = inf{o: : Aa(E) = O} = inf{o: : A~(E) = O}.

Sets of non-integer dimension are often called "fractals". The dimension of a set on lR can often be determined by Theorem 10.5 (due to Hungerford in its final form).

In fact we consider the Hausdorff measure Aep with respect to any continuous increasing function r.p with r.p(0) = O. The o:-dimensional Hausdorff measure is the special case r.p( t) = ta . See e.g. Fal85 and Rog70 for a more thorough discussion.

N.G. Makarov has proved several important results about Hausdorff measures and conformal mapping. Two of these can be stated as:

Makarov Dimension Theorem. Let f map ]]J) conformally into C and let Ac ']['. IfdimA > 0 then dimf(A) > (dimA)/2 and

(5) dim A = 1 ::::} dim f(A) 2': 1.

10.2 Hausdorff Measures 223

The last implication solves an interesting problem; the best previous result, due to Carleson, was that A(A) > 0 implies dimf(A) > 1/2. In Theorem 10.6 Makarov gives a result that is much more precise than (5); it is a consequence of his law of the iterated logarithm for Bloch functions (Theorem 8.10). His Theorem 10.8 gives lower estimates for dim f(A) in terms of upper estimates for the integral means of 1/11'1 studied in Section 8.3. These theorems should be contrasted with the results of McMillan and Makarov in Section 6.5. See also Mak89d for a survey and the connection with stochastic processes.

Hamilton has proved an analogue of Löwner's lemma (for a-capacity) which implies that

(6) f(l!)) Cl!), f(A) C 11' =? dim f(A) ~ dim A;

see Makarov's Theorem 10.11 for a more precise statement.

Anderson-Pitt Theorem. If f maps l!) conformally into <C then

(7) dim{ ( E 11' : l' (() exists finitely} = 1.

This solves another problem that had been open for a long time. Makarov's Theorem 10.14 gives the best possible bound for the size of the set where the angular derivative l' (() exists. These theorems are closely connected to results about Zygmund measures and Bloch functions; see Section 10.4.

In the definition of Hausdorff measure we allowed covers by sets of widely different size. We can also consider covers by sets of fixed size. Let N(c, E) denote the minimal number of disks of diameter c that are needed to cover E. We define the (upper) Minkowski dimension by

(8) mdimE = inf{a : N(c, E) = O(c-a ) as c ----t O}

so that dim E ::; mdim E. The name "Minkowski dimension" is not standard; other names are "box dimension" or "upper entropy dimension". It was first systematically studied by Kolmogorov; see e.g. KoTi61, Haw74 and Fa190.

If fis a conformal map of l!) onto a quasidisk G then (Corollary 10.18)

(r ----t 1) .

10.2 Hausdorff Measures

1. We assume throughout this section that r.p is a positive continuous increasing function in (0, +00) and that r.p(t) ----t 0 as t ----t O. If Eis any set in <C we define

(1)

where (Bk) runs through all c-covers of E, that is,

224 Chapter 10. Hausdorff Measure

E c U Bk , diam Bk :::; c for k = 1,2, .... k

It is clear that only the behaviour of cp(t) near t = 0 matters. If a > 0 and cp(t) = t a we write Aa instead of Aep. This is the most important case. In particular we see from (6.2.7) that Al = A.

We now use the concepts introduced in Section 6.2. The following basic result is proved in the same way as Proposition 6.1.

Proposition 10.1. The quantity Aep is ametrie outer measure, and the Aepmeasurable sets form a a-algebra that contains all Borel sets. In particular

(2) Aep (U En ) = L Aep(En )

n n

if En (n = 1,2, ... ) are disjoint Borel sets.

We call Aep the Hausdorff measure with respect to cp and Aa == At<> the a-dimensional Hausdorff measure. The I-dimensional measure is thus the linear measure studied in Section 6.2.

Proposition 10.2. If cp :::; 'lj; then Aep(E) :::; A,p(E), and if cp(x)/'lj;(x) --t 0 as x --t 0 then Aep(E) = 0 whenever E has a-finite outer A,p-measure.

Proof. The first assertion follows at once from (1). In order to prove the second assertion it is sufficient to consider the case that A,p(E) < M < <Xl. Given c > 0 we choose 8 > 0 such that cp(x) < c'lj;(x)/M for 0 < x < 8 and that there is a cover (Bk) of E such that

Then

L 'lj;(diamBk ) < M, diamBk < 8 (k E N). k

L cp(diamBk ) < (c/M)M = c so that Aep(E) = O. 0 k

Thus different Hausdorff measures make a very fine distinction between the sizes of sets. Furthermore we see that Aep is not a-finite (except for cases like cp(t) = t 2 ) which reduces its value for integration theory.

The Hausdorff dimension of E C C is defined by

(3) dimE = inf{a : Aa(E) = O}.

It follows from Proposition 10.2 that

Aa (E) = <Xl for 0 < a < dirn E ,

Aa(E) = 0 for a > dimE (4)

while 0 :::; AdimE(E) :::; <Xl.


It is clear that 0 ~ dimE ~ 2 for Ecce. If E c '][' then 0 ~ dimE ~ 1, and A1(E) > 0 implies dimE = 1 but not vice versa.

As an example the snowflake curve has dimension log 4/ log 3j see e.g. Fal85 where also other examples of selfsimilar sets are discussed.

2. For E c C we also define

(5)

where (Bk) runs through all covers of E, not only through the e-covers as in the definition (1) of A<p. Hence 0 ~ A~(E) ~ A<p(E) ~ 00 and

(6) A~(E) = 0 {::} A<p(E) = O.

We write A~ instead of A~ if cp(t) = t n and a > O. The quantity A~ (E) has the advantage that it is finite for bounded sets

and it is therefore often used in complex analysis. By (6) it has the same nullsets as A<p so that, by (3),

(7) dimE = inf{a: A~(E) = O}.

Though A~ is an outer measure, it is not a metric outer measure and does not lead to a useful concept of measurej see Exercise 6.2.4. In many respects A~ behaves more like a capacity than a measure. Let M h M 2 , ••• denote suitable positive constants.

Theorem 10.3. Suppose that

(8)

for some a > O. 1f E is a Borel set in ethen

(9)

In particular we see that

so that cap E = 0 implies dirn E = O. Note that we cannot replace A~ by An because An (E) will in general be infinite. More precise results can be found in Lan72, Chapt IIIj see also Nev70 and HayKe76.

Proof. We restrict ourselves to the case that E is compactj see Lan72, p.203 for the general case. If c > cap Ethen, by Proposition 9.6, there exists n such that

(10)


where Zl, ... , Zn are nth Fekete points of E. We write

(11) "I = ce72ja , "Iv = "Ie-2vja for v = 0,1, ...

and partition C by the halfopen squares of a grid of mesh size "Iv. A square will be called "black" if it contains more than n2-v Fekete points.

Every point Z E E lies in a black square or in one of its eight neighbours for some v. This is trivial if z itself is a Fekete point. Otherwise, for every v, there would be at most 9n2-v Fekete points Zk with Iz - Zk I < "Iv and therefore

by (11), which would contradict (10). Thus E is covered by all black squares and its eight neighbours. Let Q be

one of these squares. We see from (11) that

'P(diamQ) :::; 'P(2"1v) = 'P(2ce72jae-2vja) :::; M 3e- 2v'P(c)

where we have used (8) repeatedly. Since there are n Fekete points, there are at most 2v black squares of size "Iv. Hence we obtain that

00

L 'P(diamQ) :::; M 4 L 2V 2-2v'P(c) = M 2'P(c) Q v=O

and it follows from (5) that A~(E) :::; M 2'P(c) for every c > capE which implies (8). 0

The following result (Fro35) is often useful: see e.g. HayKe76, p.223 for the proof.

Proposition 10.4. If E C C is compact and Acp(E) > 0 then there exists a measure I-l on E with 0< I-l(E) < 00 such that

(12) I-l(D(a,r)) :::; 'P(r) for a E C, 0< r:::; 1.

3. The final result (Hun88) can be used to estimate the Hausdorff measure Aa of sets on 11' from below. Let A = Al be the linear measure.

Theorem 10.5. For k = 0,1, ... let A k C [0,1] be a union of finitely many disjoint closed sets Bk,n such that A(Ao) > 0 and A k :J Ak+l. We assume that 0 < a < b < 1 and

(14)

Then

(15)


log(b/a) a = =---lo-='g (-7-:-1-'-;-/ a--:-r

Proo! (Roh89). We may ass urne that A(Ak ) ----; 0 as k ----; 00. We first show that we can replace the sets Bkn and thus A k by subsets with the additional property that equality always holds in (14). Suppose that this replacement has been made for the sets B jn with j :::; k and consider some set B == B km ; we may assurne that A(B) > O.

Let /-t 2 2 be first number such that A(Ak+JL n B) < bA(B); compare (14). Replacing A k+v n B by Ak+v+JL-2 n B for v 2 2 we may ass urne that /-t = 2. Then

A(Ak+2 n B) < bA(B) :::; A(Ak+1 n B).

Since the complement of a closed linear set is the union of countably many disjoint open intervals it is not difficult to see that there is a closed set E such that

Ak+2 nB c E c A k+1 nB, A(E) = bA(B).

We now replace the sets B k+1,n C B by Bk+1,n nE together with the extreme points of Bk+1,n so that the diameter is not changed. The union oft he new sets has linear measure bA(B). Making these replacements for all B mk we obtain a new set A k +1 such that equality holds in (14). The sets Aj for j > k + 1 are not affected because A k +2 nB c E.

We next show by induction on j = 0, ... , k that

(16)

This is trivial if j = O. Suppose that (16) holds for so me j 2 o. Then

n

where we sum over all n such that Bk-j,n C Bk-j-1,m. By (16) and (14) (with equality), this is

n

which is (16) for j + l. Let A = nk Ak be covered by the open intervals h, ... ,IN; by com

pactness we have to consider only finitely many. We choose kv such that akv+1 < A(Iv ) :::; akv • There exists m > max kv such that Am C 11 U ... U IN. We have

n

228 Chapter 10. Rausdorff Measure

Since diamBkvn :S akv by repeated application of (13), we see that the last union is contained in an interval of length 3akv . Hence it follows from (16) that

n n

:S 3(a/b)kvbm = 3aakv bm :S 3(A(Iv )/a)abm

because alb = aa . We conclude by (14) that

N

A(Ao) :S b-m A(Am ) :S 3a-a L A(Iv)a v=1

which implies Aa(A) ~ 3-1aa A(Ao) > O. o

It is often easy to give upper estimates by constructing specific covers that, together with Theorem 10.5, lead to the precise determination of the Hausdorff dimension.

Example. (See Fig. 10.1) Let p = 2,3, ... and 0 < a < l/p. Let Ao = [0,1]. The set A k consists of pk disjoint closed intervals of length ak, and A k+1 is obtained by replacing each interval by p disjoint subintervals of length ak+l.

Let A = nAk. The pk intervals of A k cover A. Hence we see from (1) that

A~(A) <_ pkaak = 1 h logp ~ w ere 0 = log(l/a)

On the other hand, it follows from Theorem 10.5 with b = ap that Aa(A) > O. Hence we see that dim A = o. For the classical Cantor set we have p = 2 and a = 1/3 and therefore dim A = log 2/ log 3. See e.g. Fa190 for further examples.

o 218 3.:.;;18 ____ -"'518 6/8

Fig.10.1. Three stages of the construction for p = 3 and a = 1/4. The dimension is log 3/ log 4 ~ 0.792

Exercises 10.2

1. Let h be a bilipschitz map of E, that is,

M-11z1 - z21 :::; Ih(Z1) - h(Z2)1 :::; Mlz1 - z21 for Z1, Z2 E E.

Show that M-'" A",(E) :::; A",(h(E)) :::; M'" A",(E).

2. Let h satisfy a Rölder condition with exponent 0 :::; 1 on E. Show that dim h(E) :::; 0-1 dimE.

10.3 Lower Bounds for Compression 229

3. Let rp(t) = 1/ log(1/t) for 0< t < 1/e. Use Corollary 9.13 to show that A<p(E) = 0 implies cap E = 0 for every compact set E. (This logarithmic measure rp is dose to being best possible, see e.g. HayKe76, p. 229.)

4. Show that dirn Uk Ek = sUPk dirn Ek and deduce that, for compact E C C, the function

lim dim [E n D(z, r)] (z E q r~O

is upper semicontinuous and has dim E as its maximum.

5. Suppose that there is a measure J-L on the compact set E such that J-L(E) > 0 and (12) is satisfied. Show that A<p(E) > O.

6. Construct uncountably many disjoint compact sets on '][' of Hausdorff dimension one.

10.3 Lower Bounds for Compression

1. We have seen (Corollary 6.26) that a set A C '][' of positive measure is often mapped onto a set of zero linear measure in C. We now show that A cannot be compressed much furt her. We tacitly assume that the angular limit f(() exists and is finite for ( E A; note that this is true except possibly for a set of zero capacity (Theorem 9.19).

Theorem 10.6. Let

(1) 'P(t)=texP (30 IOg~IOgIOgIOg~) for 0< t:::; exp(-ee).

1f f maps ][)) conformally into C then, for all Borel sets A C T,

(2) A(A) > 0 '* A<p(f(A)) > O.

This result (Mak85) is remarkably precise. It implies in particular that

(3) 0< 1, A(A) > 0 '* A",(f(A)) > O.

This solved a problem that had been open for many years (Beurling, Mat64); the best previous result (Car73) was that (3) holds for 0 < 00 with some absolute constant 00 > 1/2. See Mak89d for further results.

The theorem is best possible except for the constant (Mak85) as the function f defined by (8.6.6) and the law of the iterated logarithm for lacunary series show: There is a set A C '][' with A(A) = 27r such that A<Pl (f(A)) = 0 where 'PI is defined as in (1) but with 30 replaced by a smaller constant. See PrUrZd89 and Mak89d for examples from the theory of Julia sets.

We present an elementary proof (Roh88) of Theorem 10.6. It is based on the Makarov law of the iterated logarithm and on the following result. Let K I , K 2 , ••• denote suitable absolute constants.


Lemma 10.7. Let 0 < 8 < E, 1/2 ::::: r < 1 and A c 1I'. If diam f(A) ::::: E and

(4) If(r() - f(()1 ::::: E, (1 - r)I!,(r()1 :::: 8 for (E A,

then A can be covered by at most K 1 (E / 8)2 sets of diameter at most 1 - r.

Proof. We consider a square grid of mesh size c8 where the absolute constant c with 0 < c < 1 will be determined below. Let Qk (k = 1, ... , m) be the halfopen squares that contain points of f (r A). Since these squares are disjoint and since diam f(r A) ::::: 3E by (4), we see that

(C8)2 m = area(Ql u··· u Qm) ::::: K 2E2 .

Hence it suffices to show that Ak = {( E A : f(r() E Qk} satisfies diamAk :::::

1 - r for k = 1, ... , m. Suppose that diam Ak > 1 - r for some k. Then there are (, (' E A k with

I( - ('I > 1- r so that the non-euelidean distance satisfies A(r(, r(') :::: 1/ K 3 .

Hence, by (1.3.18) and (4),

8< (1 - r2 ) 1!,(r()1 ::::: K 4 If(r() - f(r(') I < 2K4 c8

because f(r(), f(r(') E Qk, which is a contradiction ifwe choose c = 1/(2K4 ).

o

Proof of Theorem 10.6. Let A(A) > O. By Corollary 8.11 there exists a Borel set A' c A with A(A') > 0 such that

(5) -'lj;(r) ::::: log I!,(r() I ::::: 'lj;(r) == 7 flog _I_log log log _l_ V 1-r 1-r

for ( E A' and ro ::::: r < 1 with suitable ro < 1. Hence

(6) If(() - f(r()1 ::::: 11 1!,(s()1 ds ::::: 11 eW(S) ds;

the angular limit f(() exists because the integral is finite. Now

:r [11 eW(S) ds - 2(1 - r)eW(T)] = (1 - 2(1- r)'lj;'(r))eW(T) > 0

for r elose to 1 because (1 - r)'lj;'(r) -+ 0 as r -+ 1. Since the quantity in the square bracket converges to 0 as r -+ 1 we conelude from (6) that

(7) I!(() - !(r()1 < 2(1- r)eW(T) for ro::::: r < 1, (E A' ,

increasing ro if necessary. Let (Bk) be any open cover of !(A') and let Ak = {( E A' : !(() E Bk}

for k = 1,2, .... We define Ek, rk and 8k by


where K 1 > 2 is as in Lemma 10.7. Since (4) holds by (7) and (5), we obtain from Lemma 10.7 that A k can be covered by at most K2(ck/Ök)2 sets of diameter 1 - rk so that, by (8), (1) and (5),

A(A') ::; L A(Ak) ::; K 6 L(1 - rk) exp[4'l/'(rk)] < K7 L cp(ck) j

k k k

we have used here that log l/ck '" log 1/(1 - rk) by (8). It follows that A'I'(f(A)) ~ A'I'(f(A')) ~ A(A')/K7 > O. 0

Remark. The measure A'I' defined by (1) (but with a different constant) plays also a role (Mak89c) for non-Smirnov domains: If JL is the singular measure in (7.3.1) and if Eis a Borel set on 'lI' with JL(E) > 0 then A'I'(f(E)) > 0, and this is best possible.

2. We now turn to small sets on 'lI'j see Mak87. As in (8.3.1) we define ßf( -q) as the infimum of all ß such that

Theorem 10.8. Let f map JI)) conformally into C. If A is a Borel set on 'lI' then

(10) dimf(A) > qdimA for q> O. - ßf( -q) + q + 1- dimA

The following result (Mak87) is, in part, an easy consequence.

Theorem 10.9. Let f map JI)) conformally into C. Let Ac '][' be a Borel set and a = dimA. Then

(11)

a 11 dimf(A) > 2" for 0< a ::; 12 '

a 11 dimf(A) > for 12 < a < 1.

1 + JI2(1- a)

If f is close-to-convex then

(12) dimf(A) ~ a/(2 - a) for 0< a ::; 1

and this estimate is best possible.

Taking the limit a -+ 1 in (11) we obtain that, for all conformal maps,

(13) dimA = 1 =} dimf(A) ~ 1

which is stronger than (2) because we may have A(A) = 0 but is also weaker because we do not have a function cp as precise as (1). The estimate (11) is not


sharp for 0 < a < 1. It is however not far from being sharp near a = 0 and near a = 1; see Mak87 where a wealth of further information can be found.

To prove (11) for 11/12 < a < 1 we use the estimate

ßf(-q)::; J4q2 +q+ 1/4 - q-l/2 < 3q2 (q> 0)

ofTheorem 8.5. Our assertion follows from (10) ifwe choose q = J(I- a)/3. The estimate (12) follows at once from (10) because ßf( -1) = 0 if fis closeto-convex, by Theorem 8.4. See Mak87, p. 63 for the construction of a starlike function for which (12) is sharp.

While the estimate dirn f(A) ;::: a/2 is not difficult to establish (Mat64, see Exercise 4) the proof of dirn f(A) > a/2 is much harder and uses a method of Carleson; see Mak87, p.75. If the Brennan conjecture ßf( -2) ::; 1 is true then we would obtain from (10) that

dimf(A) ;::: 2a/(4 - a) > a/2 for a > O.

The proof of Theorem 10.8 is based on the following technical result (Mak87). Let K 1 , K 2 , ... be absolute constants and df(z) = dist(f(z),oG). We consider the dyadic points

(14) z = (1- 2-n)ei7r(2v-ll/2n (v = 1, ... , 2n ; n = 1,2, ... )

and the corresponding dyadic arcs l(z); see (6.5.15).

Proposition 10.10. Let f map []) conformally onto G and let 0 < a ::; 1. If A c 1I' is a Borel set with

(15) diam f(A) ::; c: < min(a, df (0))/4,

then there are dyadic points Zl, ... ,ZN with

(16)

and a set E c 11' with A~ (E) < c:2 such that

(17)

Proof (see Fig. 10.2). Let D 1 and D 2 be concentric disks of radii c: and 2c: with D 1 n f(A) -=I- 0. It follows from Proposition 2.13 that there are countably many crosscuts Bj C f-l(G n oD2 ) of []) such that G n D 2 is covered by the sets f(Vj) where Vj is a component of []) \ Bj . Let

Ej=AnVj , Hj=f(Vj)\D1 •

We see from (15) that dist(f(O),Hj ) ;::: df(O) - 2c:;::: df (0)/2, and if Ce []) is a curve from 0 to Ej then A(f(C) n H j ) ;::: c:. Hence it follows from Theorem 9.24 that

cap E j < K 3 exp( -7rc:2 / area H j ) .


We may assume that areaHj 2:: areaHj+l. Then j areaHj ::; areaD2 = 47rC;2

and thus, by Theorem 10.3,

L A~(Ej)::; K 4 L(capEj)a::; K 5 L e-aj /4 j>N j>N j>N

< K 6 e-aN/4 < K 6 e-aN/4 < C;2 a c;

if we choose N suitably subject to (16). Hence the set E = Uj>N E j satisfies A~(E) < C;2.

Now let j = 1, ... , N. Let Ij be the are of 1I' between the endpoints of Bj

and let zj be the point on the non-euclidean line Sj connecting these endpoints that is nearest to O. Then, by the Gehring-Hayman theorem (4.5.6),

and it follows from Corollary 1.5 that there is a dyadic point Zj such that d,(zj) < Ksd,(zj) ::; K 2c; and E j C Ij C I(zj). Hence (17) holds because E=Uj>NEj. 0

Fig.1O.2

Proof of Theorem 10.8. Let

(18) a<dimA, ß>ß,(-q), ,>(l+ß-a)jq

and let MI, M 2 , ••• be suitable constants. Let m E N and (see (14))

(19) Z = {z dyadic point: Izl 2:: 1- 2-m , 1!,(z)1 < (l-lzIP}.

Since If'(lzl()1 < M11!,(z)1 for ( E I(z) by Corollary 1.6, we see from (9) and (18) that, with r n = 1 - 2-n , n 2:: m,

and therefore


00

L A(I(z))<> :::; (211")<> L (1 - rn )<> card {z E Z : Izl = rn }

(20) zEZ n=m

< M4 f T(-yq-l-ß+<»n < ~A~(A) n=m

by (18) if m is chosen sufficiently largej note that A~(A) > 0 because 0 < dimA.

Let (Bk) be any cover of f(A) with ck == diamBk sufficiently small. By Proposition 10.10 there exist dyadic points Zkj (j = 1, ... , Nk) with Nk < M5 log(1/ ck) and IZkj I ~ 1 - 2-m such that

Nk

(21) Ak == 11' n f-1(Bk) c Ek U L l(zkj) , A~(Ek) < c~ j=l

and dj(Zkj) :::; K2ck. If 1f'(Zkj)1 ~ (1 - IZkjlP then (1 - IZkjl)1+1' :::; 4K2ck and therefore

(22)

where we sum over all j with Zkj <I. Z. We see from (21) that

Ac UAk C U l(z) U U( Ek U U I(Zk j )) k zEZ k Zkjl/.Z

so that, by (20), (21) and (22),

A~(A) < ~A~(A) + L (c~ + M7c~/(1+1') log~) . k Ck

If 8:::; 2 and 8 < 0/(1 + 'Y) we conclude that

Ms L c~ > A~(A)/2 > 0, thus Ao(f(A)) > o. k

Hence dimf(A) ~ 8 and our assertion (10) follows ifwe let 'Y -+ (1+ß-o)/q, ß -+ ßj( -q) and 0 -+ dim Aj see (18). 0

The next result (Mak89b) generalizes Löwner's lemma (except for the constant).

Theorem 10.11. Suppose that <p(xt) :::; Mx<><p(t) (0< x :::; 2, t > 0) where o > O. 1f f maps!l) conformally into !I) with f(O) = 0 and if Ac 11' is a Borel set such that f(A) C 11' then

(23) A~(f(A)) ~ cA~(A)

where c is a positive constant.

10.4 Zygrnund Measures and the Angular Derivative 235

Choosing 'P(t) = tQ; we see that (Ham88)

(24) dimf(A) ;:::: dimA if Ac 1[', f(A) c 1['

for all conformal maps of l!)) into itself.

P1'oof. Let (h) be a covering of f(A) by arcs of1[' and define Ak = Anf- 1(h). It follows from Theorem 10.3 and Theorem 9.21 that

and thus cA~(A) :::; C L A~(Ak) :::; L 'P(diamh)

k k


Exercises 10.3

Let / rnap l!)) conforrnally into C and let A be a Borel set on 1['.

1. If Jf'(z)J ;:::: a(1 _JzJ)b with a > O,b < 1 show that dirn/CA) ;:::: (b + 1)-1 dirnA.

2. Show that dirn/CA) ;:::: dirnA/(2.601- dirnA).

3. Let 9 rnap l!)) conforrnally onto a Jordan dornain and let Jg'(z)J ;:::: a(.::)(1 - JzW (1'(.::) < JzJ < 1) for every .:: > O. If /(l!))) c gel!))) and /(A) c 8g(l!))) show that dirn/CA) ;:::: dirnA.

4. Use Theorem 9.23 and Proposition 10.10 to prove that Aa(A) > 0 implies Aaj2 (f(A» > O.

The following two exercises are more difficult.

5. If / is conforrnal for all ( E A show that dirn / (A) = dirn A.

6. Let 0 < a ::; 1. Then

J/«() - /(r()J (1-)- 2(1;"') (1 _1_) ~ (1 - r)Jf'(1'()J < l' og 1 - l'

for (E1['\E, ro«()<r<1

where the exceptional set satisfies Aa(E) = 0 (Porn8T).

10.4 Zygmund Measures and the Angular Derivative

In Section 7.2 we defined a Zygmund measure f.L as a non-negative finite measure on 1[' such that /f.L(I) - J.t(I')/ :::; M1A(I) for all adjacent arcs I and I' of equallength. Let MI, M 2 , • .• denote suitable positive constants.

Theorem 10.12. Let J.t be a Zygmund measu1'e and let

1 1 log - log log log -

t t fo1' 0 < t :::; e _ee . (1) 'IjJ(t) = t


1f E is a Borel set on 1[' then

(2) A",(E) = 0 =? JL(E) = o.

The function 'ljJ in this result (Mak89b) is related to that of Theorem 10.6 but is not the same. Makarov has given an example of a nonzero Zygmund measure JL such that A",(E) < 00 where E is the support of JL. Hence it follows from Proposition 10.2 that 'ljJ in (2) cannot be replaced by any function <p with <p(t)N(t) -t 0 as t -t O.

Proof. By Proposition 7.3, the function

(3) g(Z)=21 f;+zdJL(() (zEIIJ)) 7rIIf.,,-z has positive real part and belongs to ß. Since dJL( () ~ 0 we see that

1 f 1-lz12 Reg(z) ~ -2 I( 12 dJL(() ,

7r I(z) - Z

where l(re iiJ ) = {e it : It - '!91 ::; 7r(1 - rH. Since I( - zl ::; (1 + 7r)(1 -Izl) for ( E I we conclude that

(4) JL(I(z)) = f dJL(O::; M2Ig(z)IA(I(z)). JI(Z)

For n = 1,2, ... and v = 1, ... , 2n, let Znv be the dyadic points defined by (6.5.14) and 1nv the corresponding dyadic ares. We have IZnvl = rn = 1- 2-n. Since

Ig(rn() - g(znv)1 ::; M3 for (E 1nv

by (4.2.5), we obtain from (4) that, for k = 1,2, ... ,

2n

1Ig(rn()12k-l dJL(() ::; Mt L(1 + Ig(znv)1)2k-lJL(Inv) T v=l

v=l

Using again (4.2.5) and then Theorem 8.9 we conclude that

(5) llg(r()1 2k- 1 dJL(() ::; M; (k! (log 1 ~ r) k + 1)

for 0 < r < 1 and k = 1,2, .... As in the proof of Theorem 8.10 we deduce that

10.4 Zygmund Measures and the Angular Derivative 237

(6) lim sup Ig( r() I ::::: MB for Il-almost all ( E 11' . r->l Vlog _1_ log log log _1_

1-r 1-r

Now let E be a Borel set on 11' with Il(E) > O. By (6) there exists E' c E with Il(E' ) > 0 such that

(7) for (E E', ro < r < 1

with some ro < 1. Let (1j ) be a covering of E' by arcs with diam 1j ::::: 1 - ro each of which intersects E' at some point (j. If r j is defined by diam 1j = 1-r j then 1j C 1(rj(j) and thus, by (4) and (7),

ll(Ij) ::::: 1l(1(rj(j» ::::: M 2 Ig(rj(j)IA(1{rj(j» ::::: M107jJ(diam1j )

so that A",(E' ) 2: Il(E' )/MlO > O. D

The angular derivative was studied in Section 4.3 and 6.5.

Theorem 10.13. Let f map][)) conformally into C. Then f has a finite angular derivative on a set E C 11' with A",(E) > 0 where 7jJ is given by (1).

This result (Mak89b, compare AnPi89) solves a problem on which much work has been done starting with Ha168; see in particular AnPi88 and also Ber92 for a related result in a more general context. The answer is best possible (Mak89b): There is f such that {( E 11' : f' (() #- 00 exists} has a-finite A",measure. The Anderson conjecture that

fo 11!"(r()1 dr < 00 für some (E 11'

is however still open; see And71.

Proof. The case that f has a nonzero angular derivative on a set E with A(E) > 0 is trivial because then A",(E) = 00 by Proposition 10.2. Note that there are conformal maps without any finite nonzero angular derivative (Proposition 4.12).

The other case is a consequence of Proposition 4.7 and the next result applied to the Bloch function 9 = - log f'. D

Theorem 10.14. 1f gE ß has a finite angular limit almost nowhere, then

Reg(r() ----+ +00 as r ----+ 1-, (E E

for some Borel set E C 11' with A",(E) > 0 where 7jJ is given by (1).

Proof. Let f map ][)) conformally onto some component of {w E ][)) : Re g( w) > O}; this set is non-empty because otherwise 9 would have finite radial limits


almost everywhere by Fatou's theorem and the Riesz uniqueness theorem. Then h = gof has positive real part and is also aBloch function as we see from (4.2.1) and (1 -lzI 2 )1f'(z)1 :::; 1 -lf(z)l2. Hence, by Proposition 7.3,

(8) 1 s+z h(z) = ib + -- dJt(s)

']["s-z for z E lDl

where Jt is a Zygmund measure on 11'. Suppose now that Jt is absolutely continuous. Then we can find a set

A C 11' with A(A) > 0 such that the radial limit h(() exists for ( E A and Re h( () > 0 because Jt is a finite measurej see Dur70, p.4. Thus g( w) ----+ h( () for w = f(r(), r ----+ 1 so that the Bloch function 9 has the angular limit h(() at f(() by Theorem 4.3. Since f(lDl) is a component of {Reg(w) > O} we see that f(() E 11', and since A(A) > 0 we conclude from Löwner's lemma that A(f(A)) > o. This contradicts our assumption that 9 has a finite angular limit almost nowhere on 11'.

Hence Jt is not absolutely continuous. It follows (HeSt69, p.296) that Jt(B) > 0 where B is the set on 11' such that the increasing function corresponding to Jt has an infinite derivative. If ( E B then, by (8),

(9) Reh(r() = hp(r(,s)dJt(S) ----+ +00 as r ----+ 1

which is proved similarly as in Dur70, p. 4. Hence Re g( w) ----+ +00 for w = f(r(), r ----+ 1 so that, by (4.2.19),

(10) Reg(rw)----++oo as r----+l, WEE=f(B).

Finally it follows from Jt(B) > 0 by Theorem 10.12 that A..p(B) > O. Since f(B) C 11' by (10) and the definition of fand since 'l/J satisfies the assumptions of Theorem 10.11 with a = 1/2, we obtain that A..p(f(B)) > O. 0

We now show that there are always many points where a conformal map is not twistingj compare Section 6.4.

Theorem 10.15. If f maps lDl conformally into ethen there is a set E c 11' with dim E = 1 such that

(11) sup I arg!,(r()1 <00, suplarg[f(r()-f(()ll<oo for (EE. r r

This result (Mak89a) is best possible: There exists f such that the set where (11) holds has A<p-measure 0 for every <p satisfying <p(t) = O(t") (t ----+ 0) for each a < 1.

Proof. For k = 7,8, ... we define !k by fk(O) = 0 and log f~ = k-lJog f' so that, by Proposition 1.2,

(1-lzI2)lff(z)/f~(z)1 :::; 6/k < 1 for z E lDl.

10.4 Zygmund Measures and the Angular Derivative 239

Hence fk maps ]])l conformally onto the inner domain of a Jordan curve Jk by the Becker univalence criterion. It follows from (4.2.4) that I log f~(z)1 :::; (3/k) 10g[I/(I-lzl)] +const so that fk satisfies a Hölder condition of exponent 1 - 3/k (Dur70, p.74).

Let Bk be the set of all w E Jk such that {tw : t > 1 }nJk = 0. A projection argument shows that A1(Bk ) > o. It follows (Exercise 10.2.2) that

dimf;l(Bk) :::: (1 - 3/k) dimBk :::: 1 - 3/k

so that E = Uk f;l(Bk) satisfies dimE:::: 1. The dimension is not decreased ifwe delete from E those points where the angular limit f(() does not exist or is infinite because these points form a set of zero capacity by Theorem 9.19.

Now let ( E E and thus ( E f;l(Bk) for some k. Then I arg[fk(r() -fd()] < 271" for 0 < r < 1 by the definition of Bk. As in the proof (ii) ::::} (iii) of Theorem 11.1 below, it can be deduced that I arg fHr() I is bounded. Hence I arg p(r()1 is bounded and thus also I arg[f(r() - f(()]I; see Ost35. D

In fact more is true (Mak89d, Mak8ge): If gE B then

(12) dim{ ( E 1l' : sup Ig(r()1 < oo}= 1. r

For conformal maps this means that

(13) dim{ ( E 1l' : sup I log !'(r() I < oo}= 1. r

This can be further generalized (Roh89): Let 9 E B have finite angular limits almost nowhere and consider a square grid of mesh size a > O. We prescribe a sequence (Qn) of closed squares of this grid with the only stipulation that Qn and Qn+l are adjacent; see Fig. 10.3. Then there exists a set E c 1l' with

(14) dimE:::: 1 - a- 1 KllgllB

(where K is an absolute constant) such that, for each ( E E, we can find rn --+ 1 with

(15)

In other words practically all types of behaviour occur in sets of dimension almost one. .

We will consider the subspace Bo in Section 11.2.

Exercises 10.4

Let f map ]])l conformally into C-

l. Suppose that f is conformal almost nowhere on 1l'. Show that the set where argf'(r() --+ +00 and the set where f'(() = 0 have positive A",-measure where 1/1 is given by (1).

2. Use Rohde's theorem stated above to show that there is a set E C 1l' with dirn E :::: 1 such that f' (() #- 00 exists and sUPr I arg f' (r() I < 00 for ( E E.


Fig.1O.3

10.5 The Size of the Boundary

In the definition of Hausdorff measure we allowed coverings by sets the diameter of which may vary a great deal. For many purposes coverings by sets (e.g. disks) of the same size is more natural and simpler to handle.

Let E be a bounded set in C and let N(c, E) denote the minimal number of disks of diameter c that are needed to cover E. Up to bounded multiplies it is the same as the number of squares of a grid of mesh size c that intersect E. We define the (upper) Minkowski dimension of E by

. . logN(c,E) mdlmE = hmsup ( / )

0--->0 log 1 c (1)

whereas dim E was the Hausdorff dimension.

Proposition 10.16. 11 E is any bounded set in ethen

(2) dimE< liminflogN(c,E) S mdimE. - 0--->0 log(l/c)

Proof. Let ß be any number greater than the limes inferior in (2). Then there are Cn --+ 0 such that N n = N( €n, E) < €:;;ß. If E is covered by the disks D 1 , ... , D Nn of diameter €n then, for Cl: > ß,

Nn

~)diamDk)a = Nn€~ < €~-ß --+ 0 as n --+ 00

k=l

and thus dimE S Cl: which implies (2). 0

Example. The (locally connected) continuum

00 [n3 / 4 ]

E=[O,l]UU U [;,(1+i);] n=l v=l

10.5 The Size of the Boundary 241

has u-finite linear measure so that dim E = 1. On the other hand, given c > 0 choose n such that 1/(2n) :::; c < l/n. No disk of diameter c can intersect two different segments [v/n, (1 + i)v/n]. Hence we need at least cln3/ 2 such disks to cover E so that N(c,E) ~ C2c-3/ 2 . It follows that mdimE ~ 3/2 > dimE = 1.

There are however many cases for which the Hausdorff and Minkowski dimensions are identicalj see e.g. Hut81 and MaVu87.

The Minkowski dimension of the boundary is related (Pom89) to the growth of the integral means of the derivative.

Theorem 10.17. Let f map ][J) conformally onto a bounded domain C and let 1 :::; p :::; 2. 1/

(3) N(c,oC) = O(c-P) as c --+ 0

then

(4) i 1!,(r(Wld(1 = 0 ((1 _ :)P-l log 1 ~ r) as r --+ 1.

1/ (4) holds and i/ C is a J ohn domain then

(5) as c--+O.

Note that (5) is slightly weaker than (3). Furthermore every (bounded) quasidisk is a John domain. Let ßf(P) be defined by (8.3.1).

Corollary 10.18. 1/ f maps][J) con/ormally onto a John domain C then

mdimoC=p

where p is the unique solution 0/ ßf(P) = p - 1.

Proo/. Since C is a John domain it follows from (8.3.1) and Theorem 5.2 (iii) that ßf(P + 0) :::; ßf(P) + qo (0) 0) for some q < 1. Hence ßf(P) = P - 1 has a unique solution P and

[ 1!,(r()IP+6ld(1 = 0((1 - r)1-P-q6-T/) = 0 ((1- r)1-P-6log _1_) Je l-r

for suitable ." > 0 so that (4) holds with p replaced by p + o. Therefore mdimoC:::; p+o by (5) and thus mdimoC:::; p. The converse follows similarly from the fact that (3) implies (4). 0

The assumption that Cis a John domain cannot be deleted as the following example shows: Let C be a bounded starlike domain with area oC > o. Then


mdimE = 2 but ßf(P) :::; P - 1 for every P 2: 1; see the proof of Theorem 8.4 and use that f is bounded.

Corollary 10.19. Let f(z) = Eanzn map ][)) conformally onto a bounded domain G with mdim OG = p. 1f P = 1 then

(6) an = O(n,,-I) (n --t (0) for every c > 0,

and if P > 1 then there is TJ = TJ(p) > 0 such that

(7) - O( -'1- I/P) an - n as n --t 00.

This statement (Pom89) is more precise than (8.4.5) which corresponds to P = 2; see also CaJ091.

Proof. The first part of Theorem 10.17 shows that ß f (p) :::; P - 1 so that ßf(1) = 0 if P = 1. If P > 1 we see from Proposition 8.3 and Theorem 8.5 that, for 0 < 8 < 1,

ßf(1) :::; P - 1 ßf(8) + 1 - 8 ßf(P) :::; P - 1 (382 + 783 + 1 - 8) < 1 - ~ - TJ(p) p-8 p-8 p-8 P

for some TJ(p) > 0 if 8 is chosen small enough. Now (6) and (7) follow from (8.4.2). 0

Proof of Theorem 10.17. Let M, MI ... be suitable constants. For n = 1,2, ... let Znv (v = 1, ... , 2n) denote the dyadic points (6.5.14) and 1nv the dyadic arcs (6.5.15).

We first assume that (3) holds. By Corollary 1.6 it suffices to prove (4) for the case that r = r n = 1 - 2-n . For fixed n = 1,2, ... we write C-I = 0 and

(8)

Let mk denote the number of points Znv (v = 1, ... , 2n) such that

(9)

Since df(z) is bounded we have mk = 0 for k 2: Mn. It follows from Corollary 1.6 and (l-lzl)If'(z)1 :::; 4df(z) that

(10)

by (8). Now let k 2: 1. We see from (3) that OG can be covered by

(11)

10.5 The Size of the Boundary 243

disks of diameter ck/3; we may assume that each disk contains a point of 8G. Let Vk be the union of these disks; see Fig. 10.4.

Consider an index v such that (9) holds. Then j(znv) (j. V k. Hence we can find a connected set Akv C ]j]) connecting [0, e2i1r (v-l)/2n j with [0,e2i1W/2nj such that j(Akv ) C 8Vk. It follows from Corollary 4.19 that df(znv) < M 4A(f(Akv )) + ck/3. Summing over all v such that (9) holds we see that mkck/6 = mk(ck-l - ck/3) < M4 L A(f(Akv )) :::; M4A(8Vk) :::; 7rM4Nkck/3

v

because the sets Akv are essentially disjoint. Hence mk < M5ckP = M 52-kp+n by (11) and (8) so that, by (10),

1211" 1!,(rneit)IP dt < M 7 2(p-2)n (2n + L 2kP2-kP+n) < M sn2(p-l)n

o l~k<Mn

which proves (4) because 1- rn = 2-n .

Fig.l0.4

Conversely we assume that (4) holds and that Gis a John domain. Let 0< c < df(O) be given and let qn denote the number of v E {l, ... , 2n } such that

(12)

where c will be chosen below. Since G is a John domain we see from Theorem 5.2 (iii) with z = 0 that qn = 0 for n > Mglog(c/c). Furthermore

by Corollary 1.6 and by (4) so that qn < M 12n( c/ c)P. It follows that

00

(13) L qn < M13 (C/c)P[10g(c/cW . n=l


Let ( E 1l'. Since e < d,(O) there exists r = r(() such that d,(r() = e and thus n = n( () and 11 = 1I( () such that

( E Inv , M1le < d,(znv) < M14e,

where we have again used Corollary 1.6. Hence (12) holds ifwe choose c = M 14 •

Since G is a John domain it follows from Corollary 5.3 that diamf(Inv ) < M 15d,(znv) so that f(Inv) lies in at most M 16 disks of diameter e. If we consider all ( E 1l' we deduce that N(e, oG) ~ M 16 Ln qn and (5) follows from (13). 0

Exercises 10.5

1. Show that the set indicated in Fig. 10.1 has the Minkowski dimension log 3/ log 4.

2. Let the curve C be the limit of polygonal curves Cn where Cn +1 is obtained from Cn by replacing each side S by psides of length aA(S) where ap > 1. Show that mdimC::::: logp/log(l/a).

3. Let j(z) = Eanzn map][} conformally onto G and let mdim8G ::::: 1.5. Show that an = O(nO.331-1).

4. Let J be the image of 1l' under an o:-quasiconformal map of C onto C. Show that

mdim J < 1 + K 0: 2 , 0 < 0: < 1

where K is an absolute constant (BePo87j use Exercise 8.3.4 and the Lehto majorant principle).

Chapter 11. Local Boundary Behaviour

11.1 An Overview

We first consider quasieircles J for whieh

la - wl + Iw - bl = (1 + o(l))la - bl as a, bE J, la - bl ~ 0

where w lies on J between a and b. The conformal maps f of JI)) onto the inner domain of J can be characterized by the property that

(1) (l-lzl)!,,(z)/!,(z) ~ 0 as z ~ 1,

Le. log f' lies in the little Bloch space Bo. Then we study the behaviour of a general conformal map f of JI)) near a

given point ( E '[' where f has a finite angular limit w. We study the case that w is well-accessible, Le. accessible through a curvilinear sector in G. This is a local version of the concept of a John domain. In particular we consider a local version of (1), Le.

(2) (z-()!,,(z)/f'(z)~O as z~(ineveryStolzangle.

This leads to a geometrie characterization of isogonality. A main tool in the first two sections is the Caratheodory kernel theorem about convergence of domain sequences.

The main tool in the last. two sections is the module of curve families introduced in Section 9.2. For the moment we only consider domains of the form

(3) G = {w + pei~ : 0 < P < c, 'IL (p) < {) < {) + (p)}

and use the notation indieated in Fig. 11.1. A key fact (RoWa76, JenOi77) is that modr(PbP2) is approximately additive if and only if f-l(T(r)) is approximately a circular arc in JI)). This is used to establish conditions for f to be conformal at a point. The simplest is the Warschawski condition (Corollary 11.11): If {)±(p) = ±(7r/2 - 8(p)) where 8(p) is increasing then f is conformal at ( if and only if l e

8(p)p-1 dp < 00.

o

246 Chapter 11. Loeal Boundary Behaviour

If {)± satisfies a certain regularity condition then, with f(z) = w + pe iiJ ,

as z --+ , where b is a finite constantj see Theorem 11.16. The Ahlfors integral in (4) plays an important role in a wider context.

In the literat ure these results are often stated for conformal maps between strip domains and for the boundary point 00 because the statement becomes simpler in this setting. We however consider 11} and a finite boundary point in accordance with the other chapters.

We can present only a selection of the great number of results that are known. Some of the early workers were Ahlfors, Ostrowski, Wolff, Warschawski and Ferrand, see e.g. GaOs49a, GaOs49b and LeF55. There are many interesting later publications, in particular aseries of papers by Rodin and Warschawski.

Fig. 11.1. The family r (PI, P2) consists of all curves in the shaded domain that separate the cireular ares T(pI) and T(p2)

11.2 Asymptotically Conformal Curves and Bo

1. The Jordan curve Je <C is called asymptotically conformal if

(1) la - wl + Iw - bl 1 max --+ wEJ(a,b) la - bl

as a, b E J, la, bl --+ 0

where J(a, b) is the smaller arc of J between a and bj note that the quotient is constant On ellipses with fod a and b. It follows from (5.4.1) that every asymptotically conformal curve is a quasicircle. Every smooth curve is asymptotically conformal but corners are not allowed.

We nOW give (Pom78) analytic characterizationsj see PoWa82, RoWa84 and Bec87 for quantitative estimates.

Theorem 11.1. Let f map 11} conformally onto the inner domain of the Jordan curve J. Then the following conditions are equivalent:

(i) J is asymptotically conformal;

(ii) (1-lzI2)f"(z)/f'(z) --+ 0 as Izl--+ 1-;

11.2 Asymptotically Conformal Curves and Bo 247

... I(z) - I«() - Iz - (I (m) (z _ ()f'(z) ~ 1 as Izi ~ 1-, ( E][)), l-lzl ~ a (a > 0).

If z E ][)) and a > 1 then {( E iij : Iz - (I ~ a(1 - IzlH contains a proper arc of,][, . Further equivalent conditions (BeP078, compare AgGe65) are

(iv) (1 _lzI2)2 S,(z) ~ 0 as Izl ~ 1-;

(v) I has a quasiconformal extension to C such that

{)I/{)I ~ 0 as Izl ~ 1 + . Oz {)z

Prool (i) =} (ii). Let (i) hold and IZnl ~ 1-. We write Zn = (nrn and 8n = d,(zn) = dist(f(zn), J). Taking a subsequence we may assume that n8n ~ 0 as n ~ 00. Let an and bn be the first and last points where the (positively oriented) Jordan curve J intersects the circle {w : Iw - I ( zn) I = n8n} and let Jn be the arc of J between an and bn; see Fig. 11.2. Since lan - bnl ~ 2n8n we may assume, by (1), that

(2) (Ian - wl + Iw - bnl)/Ian - bnl < 1 + 1/(4n4 ) for w E Jn .

With an = arg(bn -an) it follows that I Im[e-ian(w-an)]1 < 8n/n for w E Jn. Since Iw - l(zn)1 2: n8n for w E J \ Jn and 8n = dist(f(zn), J) we conclude that I Im[e-ian(f(zn) - an)]- 8nl < 8n/n so that

(-1-~) 8n < Im [e-ian(w - I(zn))] < (-1 +~) 8n for w E Jn .

The univalent functions

(3) hn(z) = 6:;;Ieißn [! (n:: :nnz )- !(zn)] (z E j[J))

satisfy hn(O) = 0 and we can determine ßn such that h~(O) > O. Taking again a subsequence we deduce from the above results that hn (][))) converges to a halfplane {w : Im[ei-rw] > -I} (-y E IR) with respect to 0 in the sense of kernel convergence; see Section 1.4. Hence it follows from Theorem 1.8 that

(4) hn(z) ~ 2z/(1 - ei-y z) as n ~ 00 locally uniformly in ][)).

Now I(][))) is a quasidisk and thus a John domain. Applying Theorem 5.2 (iii) to I we obtain from (3) after a short computation that

(1 + rnx)a+llh~(x)1 2: clh~(O)1 for - rn < x < 0

where a and c are positive constants. Hence we conclude from (4) that (1 + x)a+111 - ei -Yxl-2 2: 2c for -1 < x < 0 so that eh = -1 and thus

r ( 2) !"(zn) _ h~(O) i-y _ ,>n 1 - rn I'(zn) - 2rn + h~(O) ~ 2 + 2e - 0

as n ~ 00 which implies (ii).

248 Chapter 11. Local Boundary Behaviour

Fig.ll.2

(ii) =} (iii). Let Izl = r < 1 and 1(1 < 1, Iz - (I ::; a(1 - r). By (ii) there exists ro < 1 such that

1 f'(() 1 1 r' f"(t) 1 1

log f'(z) = }z f'(t) dt ::; elog l-lsl for ro < r < 1

where we have integrated over the non-euclidean segment from z to ( and where s = (z - ()/(1 - z(). Hence If'(()/f'(z) - 11 ::; (1-lsl)-e: - 1. Since Id(/dsl = 11-z(12 /(1- r2) ::; b(l- r) with b = (a + 2)2, we obtain by another integration that, for ro < r < 1,

1 f(z) - f(() - 11 < b(1 - r) 1181 [(1 - a)-e: - 1] da (z - ()f'(z) - Iz - (I 0

=b(l-r) (1-(I- lsl)l-e: -1) <~ 11 - z(1 (1 - e)lsl - 1 - e

because the last bracket increases with IsI- By continuity this estimate also holds for 1(1 ::; 1. Hence (iii) is true.

(iii) =} (i). Let Wj = f((j) E J for j = 1,2. If IW1 - w21 is small then J(W1, W2) corresponds to the smaller arc ofT between (1 and (2. We determine z E ][)) such that I(j - zl = 2(1 - Izl). It follows from (iii) that

f(z) = f(()+(z-()!,(z)(I+o(I)) as Izl ~ 1 uniformly in I(-zl ::; 2(1-lzl)

and thus Iw - wjl = 1f'(z)I(I( - (jl + 0(1(1 - (21)) for W E J(WbW2) and ( = f-l(W). If IW1 - w21 ~ 0 then 1(1 - (21 ~ 0 hence

IW1 - wl + Iw - w21 = 1(1 - (I + I( - (21 + 0(1(1 - (21) ~ 1 IW1 - W21 1(1 - (21 + 0(1(1 - (21)

because ( lies between (1 and (2 on the circle T. Hence (i) holds. 0

2. Let again Bo be the space of all Bloch functions 9 with (1 - IzI 2)g'(z) ~ 0 as Izl ~ 1. Then

(5) J asymptotically conformal {::? log!, E Bo

by Theorem 11.1.

11.2 Asymptotically Conformal Curves and Bo 249

Example. Let bk E C with Ibkl < 1/3 and bk ~ 0 as k ~ 00. The function f defined by f(O) = 0 and

(6) 00

log!,(z) = I)kZ2k for z E [J)

k=O

maps [J) onto a quasidisk by Proposition 8.14 and log!, E Bo by Proposition 8.12. Hence f([J)) is bounded by an asymptotically conformal Jordan curve J. If L'lbk 12 = 00 then log!, has almost nowhere a finite angular limit; compare (8.6.12). Hence f is conformal almost nowhere on 11' so that J has a tangent only on a set of zero linear measure by Theorem 5.5, Proposition 6.22 and Theorem 6.24. Furthermore f is twisting almost everywhere on 11' by Theorem 6.18. This shows that asymptotically conformal curves can be rather pathological.

Theorem 11.2. Let f map [J) conformally onto the inner domain of an asymptotically conformal curve. Then f is conformal on a set of Hausdorff dimension one.

This result of Makarov is, by (5), equivalent to the assertion that

(7) dim{ ( E 11' : 9 has a finite angular limit at (} = 1

for 9 E Bo; see Mak89d for the proof. It is furthermore best possible. The assertion that (7) holds for 9 E Bo is related to (10.4.12) for 9 E B

and in fact follows from it by the following transference principle (CaCuP088): For every 9 E Bo, there exists g* E B such that

(E 1I',sup Ig*(r()1 < 00 T

=> lim g(r() =I- 00 exists. r __ l

Now let 9 E Bo be an inner function, that is g([J)) c [J) and Ig«()1 = 1 for almost all ( E 11'; see Exercise 4. Then

dim{( E 11': g«() = w} = 1 for each w E [J).

This surprising result (Roh89, compare Hun88) incidentally shows that there exist uncountably many disjoint sets of dimension one on 11'.

In Section 7.2 we considered

(8)

for 9 E B. As in Theorem 7.2 it can be shown that his Zygmund-smooth, i.e. that

max Ih(eitz) + h(e-itz) - 2h(z)1 = o(t) as t ~ 0, Izl=l

if and only if 9 E Bo; see Zyg45. The problem of differentiability has now been completely solved (Mak89d, Mak8ge):


Let h be a Zygmund-smooth function defined on 1I'. If either h can be continuously extended to an analytic function in lIJ) (Le. (8) holds with 9 E 130 ),

or if h is real-valued, then

dim{ t : h( eit ) has a finite derivative} = 1.

There is however a complex-valued Zygmund-smooth function that is nowhere differentiable.

3. Finally we mention a property that is the opposite to asymptotic conformality. Let J be a quasicircle such that

(9) max la - wl + Iw - bl > 1 + Cl (a, bE J) wEJ(a,b) la - bl

for some Cl > O. This holds e.g. for the snowflake curve and for many Julia sets (see e.g. Bla84 , Bea91).

If f maps lIJ) conformally onto the inner domain of J then (Jon89)

(10) (1-lz*1) I ';:((;:j I > C2 for all z E lIJ) and some z* with I t~ z:* I 0 and p < 1, furthermore

(11) . loglf'(r()1

hm sup > 0 for almost all ( E 1I' . r--->l Jlog -l-log log log _1_

1-r 1-r

This is a lower estimate in the law of the iterated logarithmj compare Corollary 8.11 and (8.6.22). See Roh91 for conformal welding.

Exercises 11.2

1. Let f map lIJ) conformally onto the inner domain of the asymptotically conformal curve J. Show that f is a-Hölder-continuous for each a < 1 and deduce that J has Hausdorff dimension 1.

2. Show that (1) is equivalent to the condition

max Im[(w - a)j(b - a)] ~ 0 as a, bE J, la - bl ~ O. wEJ(a,b)

3. Prove that (ii) implies (iv) for all analytic locally univalent functions f. 4. Let 9 be a function in 130 that has an angular limit almost nowhere and let r.p map

lIJ) conformally onto a component of {z E lIJ) : Ig(z)1 < 1}. Use Löwner's lemma to show that {( E 1I' : Ir.p(() I = 1} has zero measure and deduce that go r.p is an inner function in 130 •

11.3 The Visser-Ostrowski Quotient 251

11.3 The Visser-Ostrowski Quotient

We assurne throughout this section that f maps j[]) conformally onto G c C and that f has the finite angular limit w = f(() at a given point ( E 1l'. Let again df(z) = dist(J(z), aG) for z E j[]) and let M 1 , M 2 , •.. denote suitable constants.

The point f(() is called well-accessible (for ( E 'lr) if there is a Jordan arc C C j[]) ending at ( such that

(1) diamf(C(z)) :s: M 1df (z) for z E C,

where C(z) is the arc of C from z to (. Every boundary point of a John domain is well-accessible by Corollary 5.3, and the not ion of a well-accessible point can be considered as a local version of the not ion of a John domain. The following characterization (Pom64) is a local version of Theorem 5.2.

Theorem 11.3. The following three conditions are equivalent:

(i) f(() is well-accessible; (ii) there exists Q: > 0 with

1!,(p()1 :s: M 2 If'(r()1 (~ = ~) a-1 for O:S: r :s: p < 1;

(iii) there exists Q: > 0 with

(1- p)a

If(p() - f(()1 :s: M3df (r() -l-r

for O:S: r :s: p < 1 .

Proof. First let (i) hold. We may assurne that ( = 1. If 0 < r < 1 and r is dose to 1 then, by Corollary 4.18, we can find a curve 8 from r to 'IT' that satisfies diamf(8) :s: M 4 df (r) and intersects C, say at z. It follows from (1) that diamf(C(z)) :s: M 1df (z) :s: M 1 diamf(8) so that the part of 8 from r to z together with C(z) forms a curve r from z to 1 with diamf(r) :s: (1 + Md diamf(8). Hence we see from the Gehring-Hayman theorem (4.5.6) that

(2) diam f([r, 1]) :s: M5 diam f(r) :s: M6 diam f(8) :s: M4 M 6 df(r) .

Now we proceed as in the proof of (ii) ::::} (iii) in Theorem 5.2. If r :s: x < 1 and O:S: y:S: I-x then If(x+iy)-f(I)I:s: M7df(r) by (2) and Corollary 1.6. Hence (5.2.8) is again true if Q: > 0 is suitably chosen. This implies (5.2.9), i.e. our assertion (ii).

Next if (ii) holds then (iii) follows by an integration. Finally if (iii) holds then (1) is true with C = [0, (] and M 1 = M 3 . 0

We consider the Visser-Ostrowski quotient

(3) q(z) __ (z - ()!,(z) r j[]) lOr z E .

f(z) - w


If Re q( r() ~ c > 0 for ro ::; r < 1 it is easy to see that I f (rC) - f () I strietly decreases. Hence it follows from (3) that

diamf([r(,(]) ::; If(r() - f()1 ::; c-1(1- r)Ij'(r()1

so that f() is well-accessible by (1). We saw in Proposition 4.11 that f satisfies the Visser-Ostrowski condition

(4) q(z) = (z - ()f'(z) _ 1 as z _ ( in every Stolz angle f(z) - w

if f is isogonal at (. If G is bounded by an asymptotieally conformal curve then (4) holds uniformly for all ( E '][' by Theorem 11.1 (iii). The conformal map f defined by (11.2.6) is isogonal almost nowhere but 8f(lD) is asymptotieally conformal. Hence (4) is much weaker than isogonality.

The next result is a partiallocal analogue of Theorem 11.1.

Proposition 11.4. The Visser-Ostrowski condition at ( is equivalent to

(5) (z - ()!"(z)/ j'(z) - 0 as z - ( in every Stolz angle.

Proof. If (4) holds then it follows from Proposition 4.8 that (z - ()q'(z) - 0 in every Stolz angle and thus

J"(z) q'(z) (z - () f'(z) = q(z) - 1 + (z - () q(z) - 0 as z - (.

The converse is shown as in the proof of (ii) '* (iii) in Theorem 11.1. 0

Quantitative results on the Visser-Ostrowski condition can be found e.g. in RoWa84. We now give a geometrie characterization (RoWa80); see Fig. 11.3.

Theorem 11.5. The Visser-Ostrowski condition (4) holds at ( if and only if there is a curve C c lD ending at ( such that

(6) f(C) : w + teia(t), 0< t ::; tl

and the foUowing conditions are both satisfied:

(A)

(B)

{w : Re[e-ia(t)(w - w)] > cl, Iw - wl < t/c:} c G

for 0< c: < 1, 0< t < to(c:);

there are points w[ E 8G with e-ia(t)(W[ - w) rv ±it as t - O.

Proof. We may assume that ( = 1 and w = O. (a) First let (4) be satisfied. It follows that t = If(r)1 is strietly decreasing.

With C = [0,1) we can thus parametrize f(C) as in (6) with a(t) = argf(r).


G

Fig.1l.3

We consider the univalent functions

(7)

for 0 < t :::; t l and write Gt = It(lj))). With 8 = (z + r)/(1 + rz) we see from (4) that

(8) -log --It(z) =--+ (1-8)---+0 d [1+Z ] 2 l+r 1'(8) dz 1 - z 1 - z2 (1 + rz)(1 - z) 1(8)

as r -+ Ilocally uniformly in z E lDl and therefore that It(z) -+ (l-z)/(I+z). Hence it follows from Theorem 1.8 that Gt converges to {Rew > O} with

respect to 1 in the sense of Caratheodory kernel convergence as t -+ O. Thus

{w : Rew > c, Iwl < l/c} C Gt for 0< t < to(c) ,

and there exist b;- E 8Gt with b;- -+ ±i as t -+ o. Since I(r) = teior(t) and G = {f(r)w : w E Gt } we conclude that (A) and (B) hold where w;- = teior(t)b;"

(b) Conversely let there exist a curve C satisfying (A) and (B). For rn -+ 1 we consider the univalent functions

(9) gn(z) = t~le-ior(tn) 1 (: : :nnz ) , tn = I/(rn)l·

Since 1(1) = 0 is well-accessible by (A), we see from df(r) :::; I/(r)1 and from Theorem 11.3 (iii) with p = r that Cl < Ig~(O)1 < MI where Cl and MI are positive constantS. Hence we obtain from (ii) and (iii) by a short calculation that

(10) Ig~(x)1 > 1 C2 (-rn< x < 1), Ign(x)1 < M 2(1- xya (0< X < 1). +rnx

It follows from (A) and (B) first that

a(,xt) - a(t) -+ 0 (t -+ 0) for each ,x > 0

and therefore second that the domains gn (lDl) converge to H = {Re w > O} as n -+ 00. Hence, by the Caratheodory kernel theorem,


gn (z) --+ g( z) as n --+ 00 loeally uniformly in ]])),

where g(]]))) = H and g( -1) = 00, g(l) = 0 by (10), furthermore Ig(O)1 = 1 by (9), so that g(z) = (l-z)/(l+z). We eondude that g~(z)/ g~(z) --+ -2/(1+z). Henee we see from (9) that, with s = (z + rn)/(l + rnz),

1(1 - s/"(s) 1 = 11- zlll + rnz'l 2rn + g~(z) 1--+ 0 /,(s) l+rn l+rnz g~(z)

as n --+ 00 loeally uniformly in]])) which implies (5) and thus eondition (4) by Proposition 11.4. 0

Next we give a geometrie eharaeterization of isogonality (Ost36; see e.g. Fer42, War67, JenOi77).

Theorem 11.6. The function / is isogonal at ( i/ and only i/ there is a curve Ce]])) ending at ( such that

(11) /(C):w+tei<>,O<t::;tl

satisfies (A) and (B) with a(t) == a.

Proof. (a) First we assume that / is isogonal at ( so that (4) is satisfied. We proeeed as in the proof of Theorem 11.5. Sinee / is isogonal we see from Proposition 4.11 that a( r) = arg f' (r) --+ 0 as r --+ 1. We finally replaee [0, 1) by a eurve C C Jl}) ending at 1 sueh that (11) holds, and (A) and (B) eontinue to hold if we replaee a(t) by a.

(b) Conversely, let (A) and (B) be satisfied with a(t) == a. We define gn as in (9) with 0 instead of a(tn ) and deduee as in the proof of Theorem 11.5 that gn(z) --+ (1 - z)/(l + z). Henee

arg f' (: + rn ) = 0 + arg [(1 + rnZ)2g~(z)] --+ 0 + 7r +rnz

as n --+ 00 loeally uniformly in ]])). Henee arg f' has the angular limit 0 + 7r at 1 so that / is isogonal by Proposition 4.11. 0

Finally we prove a regularity theorem (Ost36; see e.g. LeF55, Chapt. IV and EkWa69) that we will use in the next seetion. If / satisfies the VisserOstrowski eondition (4) then I/(r() - /(()I is deereasing for r dose to 1. Henee we see from Proposition 2.14 that, for small p > 0, there is a unique erosseut Q(p) of]])) that interseets [0, () and is mapped onto an are of 8D(f((), p) where p = I/(r() - /(01; see Fig. 11.4.

Theorem 11.7. I/ / satisfies the Visser-Ostrowski condition at ( then

(12) I( - zl '" 1 - r as z E Q(p) , I/(r() - wl = p --+ O.


/ I

I I

I I I I \ ,

\ ,

Fig.l1.4

_--_ f(Q(l'lI ","" ........... 1_-/

" .........

G

Proof. We may assume that , = 1 and w = O. Let e > 0, 0 < 6 < 1 and

(13) A(r) = {z E 11} : 11 - zl = 1- r, 1 - Izl ~ 6(1- r)} for 0< r < 1.

It follows from (4) by integration over part of A(r) that

(14) I fez) 1 - z I e l Z Idsl log - -log -- < - -- < e f(r) 1- r - 7r r 11- si for z E A(r)

ifr > rl(e,6) and thus e-clf(r)1 < If(z)1 < eclf(r)l. If z± are the endpoints of A(r) then

df(z±) < 2(1-lz±1) I (1- z±)f'(z±) I < 36 If(z±)1 11- z±1 f(z±)

by (4) and (13) if r > r2(e,6) > rl' Hence, by (13) and Corollary 4.18, there are non-euclidean segments S±(r) from z± to ']['nD(z±, K I 6(1-r)) such that

IJ(z) - J(z±)1 < K 2df(z±) < 38K2 IJ(z±)1 for z E S±(r)

where K I and K 2 are absolute constants. If 6 = 6(e) > 0 is chosen sufficiently small we conclude that

(15) B(r) = S-(r) U A(r) U S+(r) C {z E 11}: e-c > 11- zl < ec } 1-r

for r > r3(e) and that

(16) e-2c lf(r)1 < If(z)1 < e2c lf(r)1 for z E B(r).

Let r' = 1- e4c(1- r) and r" = 1-e-4c(1- r). As in (14) we deduce that If(r')1 > e2c lf(r)1 and IJ(r")1 < e-2c lf(r)1 for r > r4(e) > r3' Hence it follows from (16) that Q(p) n B(r') = 0 and Q(p) n B(r") = 0. Since r' < r < r" we therefore see from (15) that

e-5c < 11- zl/(l- r) < e5c for z E Q(p), r > r4(e). D


Exercises 11.3

1. If w is well-accessible show that infr Iq(r()1 > O.

2. Let G have a generalized cusp at w in the sense that

dist(w,ßG) = o(lw - wl) as w -+ w, wEG.

Show that the Visser-Ostrowski quotient q has the angular limit 0 at ( and deduce that

log[f(() - f(r()] = o(log[l/(l- rm as r -+ 1.

3. Let G be a Jordan domain with a corner of opening 7rQ at f((); see Section 3.4. Show that q has the angular limit Q at (.

4. Show that f satisfies the Visser-Ostrowski condition at 1 if and only if

[f(:::J-f(r)]/[(1-r 2 )f'(r)]-+1:Z as r-+l.

5. Construct a domain G such that f both is twisting and satisfies the VisserOstrowski condition at some (.

6. Let G be a quasidisk. Use Theorem 11.6 to give a geometrical proof of the fact (Theorem 5.5 (ii)) that f is isogonal at ( if and only if ßG has a tangent at f(().

11.4 Module and Conformality

Throughout this section we assurne again that J maps ]]J) conformally onto Ge C and has the angular limit w = J(() -I- 00 at a given point ( E 1I'.

We consider a system of circular crosscuts T(p) of G with

(1) T(p) c ßD(w, p) (0< p ::::; Po)

that forms a null-chain defining the prime end corresponding to (i more precisely if Pn "" 0 then (T(Pn)) is a null-chain (see Section 2.4). Then, by Theorem 2.15,

(2) Q(p) = J-1(T(p)) (0< p::::; Po)

are crosscuts of ]]J) that separate 0 from ( if Po is small enough. For 0 < Pl < P2 ::::; Po, let r(Pl, P2) denote the family of all crosscuts C of G that separate T(Pl) and T(P2)i see Fig. 11.5. In particular we have T(p) E r(pl,p2) for Pl 0

holds iJ and only iJ, Jor every c > 0, there exists 8 > 0 such that

11.4 Module and Conformality 257

..... '"

Fig.ll.5

\ "1) \

I I

,-

I I

I

/Iw-f(pll = l'2

We saw in Theorem 11.7 that (3) is a consequence ofthe Visser-Ostrowski condition. The module of a curve family was introduced in Section 9.2. It follows from Proposition 9.3 that

(5) modr(Pl, P3) ~ modr(Pl,P2) + modr(P2, P3)

holds for all PI < P2 < P3 :$ Po. Hence (4) means that mod r(Pb P2) is "approximately additive".

Proof. (a) First let (3) hold. The function

(6) 1 (+ z i

s=-log--+-11" (-z 2

maps II:» conformallY onto the strip 8 = {O < Ims < I} and Q(p) onto a crosscut A(p) of 8. Now (3) is equivalent to

(7) sup Re s - inf Re s ---+ 0 as P ---+ 0 where s E A(p) .

The family of curves in 8 separating {Res = Al} and {Res = A2} has module A2 - Al by (9.2.5). Since the module is conformally invariant and monotone, we deduce that, for 0 < PI < P2 :$ Po,

where SI E A(Pl) and s2 E A(p2). Hence (7) shows that

whieh implies (4). (b) Conversely let (4) hold; see Fig. 11.6. We extend 8 by reflecting upon

{lms = I} and then upon {Ims = O} to obtain 8' = {11m si < 2}. The crosscut A' (p) of S' obtained from A(p) is now symmetrie with respect to lR..


Applying Proposition 9.4 twice we see that the family r'(Pl, P2) of curves in S' separating A' (pt) and A' (P2) satisfies mod r (Pl, P2) = 4 mod r' (Pl, P2)'

Now let Pl < P2 < P3 < po· By (9.2.7) there is an (exponential) conformal map of the part of {I Im si::; 2} between A' (P3) and A' (Pl) ( with the upper and lower sides identified) onto the annulus {I < Iwl < R} where R = exp[(1T/2)modr(Pl,P3)]. Then A'(P2) is mapped onto a Jordan curve A"(P2) separating {Iwl = I} and {Iwl = R}. If r{' and r~' denote the families of Jordan curves separating A" (P2) from these two circles then, by (4),

(21T)-110gR = r 1 modr(pl,P3) < modr{' + modr~' + c/4

so that, by Proposition 9.5,

(9) suplwl/inflwl<1+Kc1/3 where wEA"(P2)

for small c > O. Consider sequences (Pi,n) for j = 1,2,3 with Pln < P2n < P3n < Po and

P3n ----+ 0, Pln/ P2n ----+ 0, P2n/ P3n ----+ 0 as n ----+ 00. Let hn map JI]) conformally onto the part of S' between A'(Pln) and A'(P3n) translated by -Sn where Sn E lR.nA'(P2n), such that hn(O) = 0 and h~(O) > O. It follows that hn(JI])) converges to S' in the sense of Caratheodory kernel convergence. Hence Theorem 1.8 shows that hn converges locally uniformly in JI]) to the conformal map of JI])

onto S'; see GaHa91 for good estimates. Since h;;-1(A(P2n)) lies in a fixed compact subset of JI]) by our construction of S' and by (9), we conclude from (9) that (7) and therefore (3) is satisfied. D

A'(P1)

Fig.ll.6

In Section 4.3 we defined f to be conformal at ( E '[' if f has a finite nonzero angular derivative f'((). No strictly geometrie characterization of conformality is known but there is a characterization in terms of modules (RoWa76, JenOi77):

Theorem 11.9. The function f is conformal at ( E '[' if and only if

1 P2 (10) modr(Pl,P2)--log-----+0 as 0<Pl<P2----+0

1T Pl

and if furthermore there is a such that


(11) {w: I arg[w - w]- al < rr/2 - c, Iw - wl < 8} c G

for every c > 0 and sorne 8 = 8(c) > o.

Proof. First let f be conformal at (. Then f is isogonal at ( so that (11) follows from Proposition 4.10. Since isogonality implies the Visser-Ostrowski condition by Proposition 4.11, it follows from Theorem 11.7 that, as P = If(r() - wl ~ 0,

sup I (+ Z I rv inf I (+ z I rv 21f'(()1 where Z E Q(p). (-z (-z P

Using the map (6) we therefore obtain (10) easily from (9.2.5), Proposition 9.1 and Proposition 9.2.

Conversely assume that the conditions of the theorem are satisfied. It follows from (10) that (4) holds and thus (3) by Proposition 11.8. As in part (a) of the proof of Proposition 11.8 we deduce that (8) holds which, by (6), can be rewritten

(12) as PI< P2 ~ 0,

where Zj E Q(Pj) and thus Pj = If(Zj) - wl for j = 1,2. Hence we obtain from (10) that

(13) lim log I f(z) - t() I f=. 00 exists where Z E U Q(p). z-+( Z -

Using (11) and (13) it can be shown (see RoWa76 or JenOi77) that f is isogonal at (. Together with (13) this implies that log[(f(r() - f((»/(r( - ()] has a finite limit as r ~ 1 so that f' (() f=. 0,00 exists. D

There are geometrie characterizations of conformality for several important caseSj see e.g. Fer44, LeF55, Hali65, Eke71 and RoWa77. We restriet ourselves to the case that f maps lDl conformally onto G c lDl and that the angular limit w at ( E 11' satisfies w E 11'.

We consider sequences (Pn) with Pn '\. 0 as n ~ 00 and define 8n > 0 as the smallest number such that (see Fig. 11.7)

(14) {w E lDl: Pn+1 < Iw - wl ::; Pn, I arg(l- ww)1 < rr/2 - 8n} c G = f(lDl).

Theorem 11.10. 1f G c lDl and w E 11' then the following three conditions are equivalent.

(i) f is conformal at (;

(ii) L:n On(Pn;;.Pn+d < 00 for all (Pn) with L:n (pn~n+l f < 00;


(iii) there exists a sequence (Pn) such that

n n

s

Fig.ll.7. The definition of Dn

Proof. We change the definition of 8n by replacing (14) by

(15) {w E TI) : PnH < I: ~: I ~ Pn, larg : ~: I < ~ - 8n } C G.

It is easy to see that conditions (ii) and (iii) remain unchanged. The function

s = log[(w + w)/(w - w)]

maps TI) conformally onto the strip S = {I Im si < 7r /2}. If H c S is the image of G then 8n is the smallest number such that

(16) {s ES: (J'n ~ Res< (J'nH, Ilmsi< 7r/2 - 8n} C H

where (J'n = 10g(I/Pn); see Fig. 11.7. (i) =} (ii). Let r* be the family of curves in H connecting {Re s = (J'n}

and {Re s = (J'nH}. It follows from Proposition 9.1 and from (9.2.5) that modr* = l/modr(Pn,PnH). There exists Sn E 8H with Im Sn = 7r/2 - 8n, say. Let a = (J'nH - (J'n and define p*(s) = l/a for those s E H with (J'n < Re s < (J'nH for which either Im s < 7r /2 - 8n or dist( Sn, s) < a. Otherwise we define p*(s) = O. Then fcp*(w)ldwl ~ 1 for C E r* so that p* is admissible and therefore

Since 0 ~ 8n < 7r /2 for large n we obtain that

and thus, for large j < k,


k-l 7rmodr(pj,Pk) 2: L7rmodr(Pn,Pn+d

n=j k-l 00

2: ak - aj + .!. L Dn(an+l - an) - L(an+l - an)2; 7r n=j n=l

the eonvergenee of the last series is equivalent to the assumption that E(Pn - Pn+d2 p;:;-2 < 00. If (i) holds then (10) is true by Theorem 11.9 and it follows that

(17)

whieh is equivalent to EDn(Pn - Pn+d/Pn < 00.

(ii) =} (iii). We refer to RoWa77, p. 9 for the purely geometrie construction of (Pn).

(iii) =} (i). Let r be the family of erosseuts of H separating {Re s = a'} and {Re s = a"} where a' < a". Let

R; = { an ::; Re s ::; a n+l, 7r /2 - Dn ::; ± Im s < 7r /2} ,

Vn± = {an - Dn < Res< an+l + Dn,7r/2 - 2Dn < ±Ims < 7r/2}.

We define p(s) = 1/7r for s E {s ES: a' < Res< a"} \ U(Vn+ U Vn-) and p( s) = 2/ 7r for s E U(Vn+ U Vn-). If CEr then C intersects 8 R;; and 8 R;" for suitable m, n > k where k ---. 00 as a' ---. 00. It follows that

so that P is admissible for r. Henee

jUli 111"/2 a" _ a' 4 00

modr::; p(a + iT)2 dadT ::; + 2" L area (vn+ U Vn-) 0" -11" /2 7r 7r n=k

a" - a' 16 00 a" - a' = + 2" " Dn(an+l - an + 2Dn) = + 0(1) 7r 7r ~ 7r

n=k

as a" > a' ---. 00 beeause of (iii), eompare (17). Furthermore modr 2: (a" -a')/7r beeause H c S. Sinee modr(p',p") =

modr if a' = 10g(1/ p') and a" = 10g(1/ p") we eonclude that eondition (10) of Theorem 11.9 holds, and (11) is true beeause Dn ---.0 as n ---. 00. It follows that f is eonformal at (. D

We deduee a simple sufficient eondition (War32; see e.g. War67 and RoWa77 for generalizations) whieh is a loeal version of Theorem 3.5; eompare also Theorem 4.14.

Corollary 11.11. Let f map][]J conformally into][]J and let ( E 'lr, w = f(() E

'lr. We define


(18) Go = {(1- peiiJ)w : 0 < P < Po, l-al < rr/2 - O(p)} ,

where O(p) (0 < p < Po) is a continuous increasing function. 1f f(ID» ::J Go and if

(19) !oPO O(p)p-l dp < 00

then f is conformal at (. Conversely, if f(ID» = Go and if f is conformal at ( then (19) holds.

Proof. First suppose that f(lDl) ::J Go and that (19) is satisfied. We choose Pn such that o(Pn) = I/n for n ~ m and suitable m. Then On :::; O(Pn) :::; 2o(Pn+d in (14) and thus, since 0 is increasing,

Hence condition (iii) of Theorem 11.10 holds so that f is conformal at (. Conversely assume that f is conformal at ( and that f(lDl) = Go. We

choose Pn = I/n for n ~ m. Since 0 increases it follows that On = O(Pn) and thus

and this series converges by Theorem 11.10 (ii).

Exercises 11.4

1. Let cp be continuous and increasing with cp(O) = 0 and let

f(lDl) = {u + iv : -00 cp(luIH.

If f(() = 0 show that !'(() exists and furthermore that

I!'(()I < 00 <=> 11 u- 2cp(u)du < 00.

2. Let f map lDl conformally onto G with f(l) = 00. Suppose that

C \ Ge {u + iv : u > 0, lvi< uß }

where 0 :S ß < 1. Show that (compare Hay59b, p.100)

lim (1 - X)2 f(x) =1= 0,00 exists. ",....,1

D

11.5 The Ahlfors Integral 263

11.5 The Ahlfors Integral

Let 1 map ]]J) conformally onto G c ce and let 1 have the finite angular limit w at ( E 1I'. As in the preceding section, we consider the circular cross cuts

T(p) c 8D(w,p) (0< P::::: Po)

of G and the crosscuts Q(p} = 1-1(T(p)) of]]J). In addition, we write

(1) A(p) = A(T(p)) (0< p ::::: Po)

so that 0 < A(p) ::::: 27fp, furthermore

(2) B= U T(p) , A= U Q(p)=r1(B). o<p<po o<p<po

Again let r(pl, P2) be the family of crosscuts of G that separate T(pd and T(P2)' See Fig. 11.8.

Fig. 11.8. The notation of this section

Proposition 11.12. 110< Pl < P2 ::::: Po then

(3) r 2 dp modr(pl,p2) - lpl A(P)::::: 0

and the difference is decreasing in Pl.

Proof. Let <p be admissible for r(pl,P2)' Since T(p) E r(pl,P2) for Pl < r < P2 we conclude from (1) that

1::::: (r <P(W)ldW 1)2::::: A(p) r <p(w)2Idwl lT(p) lT(p)

by the Schwarz inequality and thus, with (2),


so that(3) follows from the definition of the module. Furthermore, if 0 < pi < PI then, by Proposition 9.3,

(4)

and the expression on the last line is non-negative by (3). o

The basic result (AhI30, Tei38) is the Ahllors distortion theorem.

Theorem 11.13. 110< PI < P2 ::; Po and ZI E Q(pI}, Z2 E Q(P2) then

exp(i~2 >.~p) dP) ::; exp[7I"mOdr(pI,p2)]

I Z2 - ( I I ZI - ( I < 32 inf Z2 + ( / sup ZI + ( ; (5)

il the last quotient is < 1 it is replaced by 1.

Prool. The first inequality follows at onee from (3). Furthermore we map][)) by (11.4.6) onto {O < Ims < 1}, reßeet uponlR and thenmap {I Imsl < 1} bye7r8 •

The part of][)) between Q(pI} and Q(P2) is mapped onto a doubly eonneeted domain and the image of CE r(Pb P2) is a Jordan eurve separating the two boundary eomponents. The seeond inequality (5) therefore follows easily from Teichmüller's estimate (9.2.19). See Ah173, p.77 for details. 0

Now we prove an asymptotic version (Eke67).

Theorem 11.14. 11 A is given by (2) then

(6) lim Iz - (I exp ( [PO \ 71"( ) dP) =I- 00 exists. z-+(,zEA J1f(z)-wl A P

Prool (RoWa76). It follows from Proposition 11.12 that

(7) mod r(Pb P2) - [p2 >.(p)-I dp ~ a as PI ~ 0 Jp1

where 0 ::; a ::; +00. First we eonsider the ease that a < +00. We obtain from (3) and (4) (with PbP2,PO instead of pi,pI,p2) that

(8) modr(Pb P2) - [p2 >.(p)-I dp ~ 0 as PI < P2 ~ O. Jp1

Henee (11.4.4) holds and therefore also (11.4.3) by Proposition 11.8. As in part (a) ofits proof we deduee that


as Pl < P2 ~ 0,

Zl E Q(pd , Z2 E Q(P2) .

Together with (8) this shows that the limit (6) exists and is positive. Now we eonsider the ease that a = +00. Then, with Pl = If(z) - wl,

as z ~ (, z E A by (7) and the seeond inequality (5). This proves (6), and we have shown moreover that the limit is positive if and only if a is finite. D

We now turn to domains of a special type in order to eomplement (3) by an upper estimate (War42, RoWa76).

Proposition 11.15. Let (see Fig. 11.1)

(9) G = {w + pei ..? : 0 < P < c, '1L(p) < {) < {)+(p)}

where the junctions {) ± are loeally absolutely eontinuous and satisfy

(10)

Then, for Pl < P2 < Po,

(11) l p2 1 dp 1 l p2 {)~ + {)~ mod r (pl, P2) :<:; {) _ {) - + -2 {) _ {) P dp .

Pi + - P Pi + -

In particular it follows that the limit a in (7) is finite. See e.g. Ah130, LeF55, JenOi71, RoWa79a and RoWa79b for further suffieient geometrie eonditions and RoWa83 for a neeessary and sufficient eondition.

Proof. We eonsider the non-analytic homeomorphism

(12) s = (7 + iT 1-+ W = W + exp[-(7 + iT{)+(e-O") + i(1 - T){)_(e-O")]

of S = {-loge< (7 < +00, 0 < T < 1} onto G. Writing

we see that

Idwl2 = e-20" I - (1 + i"p) d(7 + iO dTI 2

(14)

With sand w related by (12) we define


(15) cp( w) = (1 + 'lj;2)1/2ea"()-1 ;

note that e-a = Iw - wl = p. If C E r(Pb P2) then the corresponding curve connects the horizontal sides of S. Hence it follows from (14) that cp is admissible so that, by (9.2.2), (13) and (15),

modr(PbP2)::; l p2 1{}+ cp(w+pei {})2 p d{}dp Pl {}-

lp211 1 + p2(r{}/ + (1 - r){}~_Y = + drdp

Pl 0 p( {} + - {} - )

::; [p2 1 + p2({}~ + {}~)/2 dp. 0 1 Pl p( {} + - {} - )

Finally we present an asymptotic formula (War42) for f(z) as z --+ ( or rat her for the inverse function f-l(W) as w --+ w.

Theorem 11.16. Let G be defined by (9) with locally absolutely continuous {}± and let (10) be satisfied. Then, with f(z) = w + pei {},

t 11"" dt. {} - {}-(p) (16) log(z - () = - lp {}+(t) _ {}_(t) t + Z1I"" {}+(p) _ {}_(p) + b + 0(1)

as z --+ (, z E JI)), where -00 < Reb < +00 and 1mb = arg ( + 11""/2.

Proof. The real part of (16) can be written

( (J c 11"" dt) Reb 17) Iz - (I exp {} () {} () - --+ e

If(z)-wl + t - - t t as Z--+(,zEJI)).

We know from Theorem 11.14 that this limit always exists; in the present situation we have ,x(p) = p({}+(p) - {}_(p)), and A is identical with JI)) near (. Since (10) is satisfied we see from Proposition 11.15 that the limit a in (7) is finite so that the limit in (6) is positive as we remarked in the proof of Proposition 11.14. Hence (17) holds for some b E C.

We refer to War42 or RoWa76 for the assertion about the imaginary part. In fact it holds under a condition weaker than (10). 0

It is clear that Theorem 11.16 also holds if G has the form (9) only in a neighbourhood of w.

We can use Theorem 11.16 to treat the case of an outward pointing cusp left out in Theorem 3.9. Let G be given by (9) where

(18) {}+(p) - {}_(p) rv ap (p --+ 0), 1c {}~(p)2 dp < 00

with a > O. Then (10) holds and we obtain from (16) that log Iz - (I -(1I""a-1 + O(l))p-l and thus

(19) If(Z)-f(()I=(~+O(l))/loglz~(1 as z--+(, zEJI)).


Exercises 11.5

1. Construct a function f that is conformal at ( but satisfies A(p) = 27rp for almost all p E (0,1) and deduce that the limit in (6) is zero.

2. Suppose that f(() = f(C) = w. If f'(() = 0 use Theorem 11.14 to show that 1'((*) = 00.

In the last two exercises we assume that G is given by (9).

3. Let 0 < a :S 2 and ß < 1. If {}± is absolutely continuous and

show that, for some a E <C with a =I- 0,

f(z) - f(() as z ~ 1", Z E 1nI (z _ ()", ~ a .. llJI

(corner or inward-pointing CUSpj compare Theorem 3.9).

4. If {}+(p) - {}_(p) ~ 0 as 3-n < p < 2· 3-n , n ~ 00, show that

loglf(z)-wl ~O as z~(, ZEJI))j log Iz - (I

compare Exercise 11.3.2.

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Author Index

Agard, S.B. 247, 269 Ahlfors, L.V. V, 4, 17,20,32,43, 78,

86, 90, 94, 95, 109, 112, 114, 122, 123, 136, 155, 159, 162, 195, 199, 207, 208, 246, 263-265, 269, 271

Alexander, H. 136, 269 Anderson, J.M. 72, 74, 77, 79, 223,

237, 269, 270 Arazy, J. 79, 270 Astala, K. 95, 172, 270

Baernstein 11., A. 163, 171, 175, 183, 270

Bagemihl, F. 37, 270 Baiiuelos, R. 187, 270 Beardon, A.F. 82,92, 250, 270 Beckenbach, E.F. 179, 270 Becker, J. VI, 16, 92, 103, 121, 241,

244, 246, 247, 270, 271 Bellman, R. 179, 270 Berman, R.D. 133, 237, 271 Bers, L. 95, 121, 269, 271 Besicovitch, A.S. 126, 130, 131, 271 Beurling, A. 23, 40, 95, 109, 112, 123,

195, 196, 20~ 208, 215, 229, 26~ 271

Bieberbach, L. 8, 185, 271 Bingham, N.H. 187, 271 Binmore, K.G. 192, 271 Bishop, C.J. 79, 127, 144, 148, 153,

163, 213, 271 Blanchard, P. 82, 250, 271 Bloch, A. 72, 78 Bojarski, B.V. 95 Bonk, M. 78, 272 Brelot, M. 210, 272 Brennan, J.E. 173,179,232,272 Browder, A. 153,272

Caratheodory, C. 13, 14, 18, 20, 24, 30, 32, 34, 35, 40, 83, 245, 272

Carleman, T. 86, 218 Carleson, L. 127, 153, 184, 195, 213,

223, 229, 232, 242, 271, 272 Carmona, J.J. 249, 272 Choquet, G. 118, 208, 210, 272 Clunie, J. 72, 74, 77, 79, 118, 173,

178, 183, 269, 272 Coifman, R.R. 155, 168, 170, 171, 272 Collingwood, E.F. 32, 33, 37-40, 139,

141, 272, 273 Cufi, J. 249, 272

David, G. 155, 162, 163, 273 deBranges, L. 9, 184, 185, 273 Dieudonne, J. 59, 273 Douady, A. 95, 112, 114, 118, 273 Dufresnoy, J. 219, 273 duPlessis, N. 210, 273 Duren, P.L. 8, 11, 16, 50, 63, 64, 84,

92, 124, 134, 150, 155, 158-160, 171, 175, 178, 184, 185, 187, 238, 273

Durrett, R. 156, 273

Earle, C.J. 95, 112, 114, 118, 120, 273 Eke, B.G. 254,259,264, 273, 274 Epstein, C.L. 17,274

Faber, G. 152,274 Falconer, K.J. 126, 128, 130, 131,

222, 223, 225, 228, 274 Fatou, P. 139, 141, 274 Fefferman, C. 155, 168, 170-172, 272,

274 Fejer, L. 23 Fekete, M. 185, 195, 203, 205, 274 Feng, J. 173,177,274

292 Author Index

Fermindez, J.L. 163, 188, 274 Ferrand, J. 80, 83, 246, 254, 259, 267,

274,280 Fisher, S.D. 79, 270 FitzGerald, C.H. 103, 169, 274 Frankl, E. 40 Frostman, O. 209, 226, 275

Gaier, D. V, 23, 35, 42, 81, 105, 220, 258, 275, 279, 288

Garabedian, P.R. 9, 185, 275 Garnett, J.B. 46, 127, 153, 156, 159,

161, 168, 170, 171, 213, 271, 275 Gattegno, C. 246, 275 Gehring, F.W. 5, 17,72,88,92,94,

95, 103, 123, 156, 170, 233, 247, 251, 270, 275, 276

Gnuschke-Hauschild, D. 79, 189, 192, 276

Golusin, G.M. 9, 14, 78, 276 Graeber, U. VI Grötzsch, H. 195, 202 Grunsky, H. 124, 276

Hadamard, J. 66, 188, 204 Hag, K. 92, 103, 275 Hahn, H. 20 Haliste, K. 259, 276 Hall, R.L. 237, 276 Hallenbeck, D.J. 192, 276 Hamilton, D.H. 40, 95, 163, 217, 223,

235, 274, 276 Hardy, G.H. 50, 137, 174, 187, 192,

194,276 HausdorfI, F. 222, 224 Hawkes, J. 223, 276 Hayman, W.K. 9, 11, 35, 72, 73, 88,

90, 163, 175, 183, 185, 195, 206, 208, 210, 220, 225, 226, 229, 233, 251, 258, 262, 275-277

He, Z.X. V,277 Heinonen, J. 90, 163, 274, 277 Helson, H. 171 Hempel, J.A. 90, 121,277 Henrici, P. V, 42, 277 Herold, H. 179, 277 Hewitt, E. 131, 238, 277 Hille, E. 17,277 Hinkkanen, A. 46, 277 Hopf, H. 117

Hough, D.M. 58, 220, 277, 283 Hummel, J.A. 185, 277 Hungerford, G.J. 222, 226, 249, 277 Hutehinson, J.E. 241, 277

Jaenisch, S. 88, 277 Janiszewski, Z. 2 Jenkins, J.A. 9, 185, 195, 197, 217,

245, 254, 256, 258, 259, 265, 278 Jensen, G. 36, 284 Jerison, D.S. 97, 155, 165, 169, 278 John, F. 94, 96 Jones, P.W. 92, 127, 144, 147, 148,

153, 163, 170, 184, 213, 242, 250, 271, 272, 278

J!1lrgensen, V. 91, 93, 278 Julia, G. 71, 82, 184

Kahane, J .P. 193, 278 Kaplan, W. 68, 70, 278 Keldys, M.V. 159, 278 Kellogg, O.D. 42, 48, 49, 278 Kenig, C.E. 97, 155, 165, 169, 278 Kennedy, P.B. 195, 206, 208, 210,

225, 226, 229, 277 Keogh, F .R. 70, 278 Koebe, P. 8, 9, 13 Kokkinos, C.A. 220, 283 Kolmogorov, A.N. 127, 134, 151, 223,

278 Komatu, Y. 217, 279 Koppenfels, W.V. 42,58, 279 Krushkal', S.L. 95, 279 Kühnau, R. 86, 91, 124, 219, 279 Kuratowski, C. 133, 279 Kuz'mina, G.V. 195, 279

Landau, E. 23, 279 Landkopf, N.S. 195, 210, 225, 279 Langley, J. VI Laura, P. V, 286 Lavrentiev, M. 42, 137, 155, 159, 163,

169, 278, 279 Lehman, R.S. 58, 279 Lehto, O. 17, 71, 74, 76, 95, 112, 123,

124, 195, 202, 244, 279 Leja, F. 203, 205, 280 Lelong-Ferrand, J. see Ferrand, J. Lesley, F.D. 103, 169, 274 Lewandowski, Z. 70, 280

Lewy, H. 58 Lindelöf, E. 34, 35, 41, 44, 51, 280 Liouville, J. 5 Littlewood, J.E. 50, 137, 178, 183,

187, 192, 194, 276, 280 Lohwater, A.J. 33, 39, 40, 139, 141,

273 Löwner, K. 9, 72, 85, 280 Lusin, N.N. 139, 145 Lyons, T. 187, 280 Lyubich, M.Yu. 82, 280

MacGregor, T.H. 173,177,274 MacLane, G.R. 73, 280 Magnus, W. 175, 191, 280 Makarov, N.G. 70, 78, 92, 127, 147,

151, 152, 173, 178, 184-186, 188, 212, 219, 222, 223, 229, 231, 232, 234, 236-239, 241, 249, 278, 280, 281

Maiie, R. 95, 120, 281 Martio, O. 92, 96, 103, 123, 163, 241,

274, 275, 281 Matsumotu, K. 229, 232, 281 Mazurkiewicz, S. 20, 32, 281 McMillan, J.E. 40, 77, 127, 133, 142,

143, 145-147, 151, 202, 223, 281 Meyer, Y. 162 Milloux, H. 218 Milnor, J.W. 15, 117,281 Minda, D. 90, 91, 282 Minkowski, H. 223, 240 Moore, R.L. 36, 282 Muckenhoupt, B. 155, 170, 282 Murai, T. 192-194, 282 Myrberg, P.J. 205, 282

Nagel, A. 35, 282 Näkki, R. 40,90,92,103-105,277,

282 Nehari, Z. 17,42, 122, 282 Nevanlinna, R. 5, 86, 87, 196, 218,

225,282 Newman, M.H.A. 2, 16, 19, 20, 27,

282 Nishiura, T. 133, 271 Noshiro, K. 33, 282

Oberhettinger, F. 175, 191, 280 Ohtsuka, M. 32, 40, 282

Author Index 293

Oikawa, K. 245, 254, 256, 258, 259, 265,278

Osgood, B.G. 92, 276 Ostrowski, A. 59, 80, 81, 90, 239,

246, 251, 252, 254, 275, 282, 283 Oxtoby, J.C. 39, 133, 283 0xendal, B. 127, 283

Paatero, V. 63,283 Paley, R.E.A.C. 183, 280 Palka, B. 92, 104, 105, 282 Papamichael, N. 58, 220, 277, 283 Pederson, R.N. 9,283 Peetre, J. 79, 270 Pfluger, A. 196, 212, 283 Philipp, W. 193, 283 Pick, G. 185, 283 Piranian, G. 23, 32, 40, 93, 147, 148,

159, 273, 283 Pitt, L.D. 223, 237, 269, 270 Plessner, A.I. 126, 139, 284 Poincare, H. 117 P6lya, G. 137, 207, 276, 284 Prawitz, H. 11, 175, 284 Privalov, 1.1. 126, 134, 139, 140, 145,

215, 284 Przytycki, F. 229, 285

Rad6, T. 26,285 Reade, M.O. 107, 285 Reimann, H.M. 168, 170, 285 Riesz, F. 126, 134, 285 Riesz, M. 126, 134, 139, 285 Robertson, M.S. 70, 285 Rodin, B. V, 245, 246, 252, 256, 258,

259, 261, 264-266, 285 Rogers, C.A. 126, 133, 222, 285 Rohde, S. VI, 178, 191, 213, 214, 227,

229, 239, 249, 250, 285, 286 Royden, H.L. 121, 171 Rubel, L.A. 46, 286 Rudin, W. 35, 282 Ruscheweyh, S. 66, 286 Rychener, T. 168, 170, 285

Sad, P. 95, 120, 281 Saks, S. 127, 145, 150, 286 Samotij, K. 192, 276 Sarason, D. 172, 286 Sarvas, J. 96, 123, 281

294 Author Index

Schabat, B.W. 42, 279 Schiemanowski, H. VI Schiffer, M. 9, 185, 219, 275, 283, 286 Schinzinger, R. V, 286 Schober, G. 17, 70, 124, 286 Schoenflies, A.M. 25 Semmes, S.W. 172,286 Shapiro, J.H. 16, 35, 158, 159, 273,

282 Sheil-Small, T. 65-68, 118, 272, 286 Shields, A.L. 16, 46, 158, 159, 273,

286 Sidon, S. 192, 286 Smirnov, V. 155, 159 Smith, W. 92, 100, 286, 287 Soni, R.P. 175,191,280 Stallmann, F. 42, 58, 279 Staples, S.G. 92, 287 Stegenga, D.A. 78, 79, 92, 100, 286,

287 Stephenson, K. 78, 79, 287 Stout, W. 193, 283 Strebel, K. 195, 287 Stromberg, K. 131, 238, 277 Suffridge, T.J. 65, 287 Sullivan, D. V, 95, 120, 121, 281, 285,

287 Szegö, G. 171,185,207,274

Tammi, O. 185, 287 Tamrazov, P.M. 46, 287 Taylor, A.L. 46, 286 Teichmüller, O. 95, 201, 202, 264, 287 Thomas, D.K. 70, 287 Thurston, W.P. V, 95, 121, 287 Tihomirov, V.M. 223, 278

Trefethen, L.N. V, 42, 287 Tukia, P. 109, 112, 155, 165, 166, 287 Twomey, J.B. 35, 66, 287

Urbanski, M. 229, 285

Väisälä, J. 95, 103, 109, 110, 195, 282, 287, 288

Vercenko, J. 127, 134, 151 Virtanen, K.I. 71, 74, 76, 95, 112,

123, 195, 202, 279 Visser, C. 81, 251, 252, 288 Vuorinen, M. 195, 241, 281, 288

Walsh, J.L. 81, 288 Warby, M.K. 58, 283 Warschawski, S.E. 42, 45, 48, 49, 52,

101, 13~ 195, 245, 246, 252, 254, 256, 258, 259, 261, 264-266, 273, 284, 285, 288

WeiB, G. 122, 269 Weiss, M. 193, 288 Weitsman, A. 90, 288 Wermer, J. 153, 272 Whyburn, G.T. 23,288 Wigley, N.M. 58, 288 Wolff, J. 20, 71, 82, 246 Wolff, T.H. 147, 278 Wu, J.-M. 163, 277

Zdunik, A. 229, 285 Zinsmeister, M. 90, 104, 155, 162-

164, 170-172,270,274,278,288, 289

Zygmund, A. 45, 157-159, 192, 193, 235, 249, 289

Subject Index

~ 126,130

(Ap ) condition 170, 172 (Aoo ) condition 168, 169 a-dimensional measure 224

see A,,(E) absolute convergence 92, 100, 192 absolutely continuous 64, 127, 133,

134, 138, 146-148, 150, 151, 154 accessible 35 admissible 196 Ahlfors distortion theorem 264, 265 Ahlfors integral 246, 263-266 Ahlfors-regular 155, 162-164, 169,

171 ambiguous point 37, 40 analytic curve 41 analytic family 121 analytic functional 123 analytic set 132 see Souslin set Anderson conjecture 237 Anderson-Pitt theorem 223, 237 angle-preserving 4, 81, 84, 101

see also isogonal angular derivative 71, 79, 80, 82-84,

94, 101, 134, 141, 237, 239, 258 see also conformal at point

angular limit 6, 35, 76, 77, 79, 139, 144, 215, 237

approximatelyadditive 245, 257 area 4 area theorem 8 asymptotic expansion 55, 57, 58, 266 asymptotic value 35, 76, 77, 192, 193 asymptotically conformal 246,

248-250 asymptotically smooth 171

(Bq )-condition 156, 170

B 71-79,140,156-158,172,173,185, 186, 188, 190, 192, 237, 239

Ba 79, 158, 172, 188, 192, 194, 248-250

ßf(P) 173,176-179,182,191,231, 241

backward tangent 59, 65 Bagemihl ambiguous point theorem

37,59 barycentric extension 118

see Douady-Earle extension Becker univalence criterion 16, 121,

182, 190 Beltrami differential equation 95, 123,

247 Bessel function 190 Bieberbach conjecture 8, 185 bilipschitz map 165, 166, 168, 228 Bloch function 72, 78 see B BMOA 171, 172 BMO(1l') 171 Borel set 128, 131, 132, 208 boundary principle 16 bounded boundary rotation 63 bounded mean oscillation 171

see BMOA, BMO(1l') bounded on radius 77, 78, 239, 249 bounded univalent function 183-185,

194, 241, 242 box dimension 223 see Minkowski

dimension Brennan conjecture 173, 179, 182, 232

C 1 C',a 49,50 coo 42,50 C(f, (), CE(f, () 33 see cluster set Cantor ·set 228 capacitable 208

296 Subject Index

capacity 195, 196, 203-212, 214, 215, 217,219-222,225

Caratheodory kernel theorem 14, 22, 245, 247, 253

Caratheodory theorem 18, 24 Carleman-Milloux problem 218 central limit theorem 188, 193 chord-arc 163 see Lavrentiev circle in C 4 circle packings V close-to-convex 68-70, 106, 107, 172,

177,231 cluster set 33, 34, 38, 39, 139 coefficient estimates 8, 183-185, 188,

194, 242, 244 Coifman-Fefferman condition 168

see (Aoo )

Collingwood maximality theorem 39, 144

Collingwood symmetry theorem 37, 38

component 2, 27, 28, 136 conformal at point 80-82, 84, 101,

126, 142-145, 235, 245, 249, 258, 259, 262

conformal invariance 197 conformal map 4 conformal maps in higher

dimensions 5 conformal naturalness 112 conformal parametrization 43, 65 conformal welding 25, 110, 153, 154,

250 connected set 1 continuity of derivative 42, 44, 45, 48,

49 continuity of map 18, 20, 22, 25, 45,

104, 192 continuity theorem 18, 20 continuous at point 6, 35 continuum 1 contraction 130, 204 convergence see power series convex function 65, 66, 70, 84, 103 convex in a direction 70 corner 51, 52, 57-59, 65, 267 corner at 00 54, 55 crosscut 27, 28 crowding 127, 220 curvature 43, 57

curve 2, 20, 29 curve family 196 cusp 43, 51, 65, 93, 256, 266, 267 cut point 23, 24

]I} 1 ]I}* 1

D(a,r} 1 df(Z} 9, 87, 92, 97, 100, 108, 216, 251 d#(z,w} 1 degenerate prime end 32 derivative at point 48, 101

see also angular derivative diamE 1 diam#E 1 diameter distortion 88, 90, 97, 103,

104, 108, 235, 251 differential inequality 173, 179, 180 dim E 222 see Hausdorff dimension Dini-continuous 46, 47 Dini-smooth 42, 48, 50, 52, 54, 84,

154, 163 Dirichlet algebra 153 Dirichlet problem 23 distance to the boundary 9 see d f (z ) domain 1 domain sequences 13-15, 22, 26, 137,

138 Douady-Earle extension 112, 118-120,

166 dyadic points 148, 232, 236

embedding theorem 123 end of a curve 3 entropy dimension 223 see Minkowski

dimension Epstein univalence criterion 17 equicontinuous 22, 112 equilibrium measure 205, 208, 210,

215 equivalent null-chains 30 Evans function 211, 215 extremallength 199 see module g-cover 129, 223

J#(z} 7 FO'-set 131, 145, 209, 211, 215 Fatou's theorem 139 Fekete points 203, 205, 210, 211, 225 Fekete polynomial 203, 205

first category 39, 141, 144 fixed point 84 forward tangent 59, 65, 151 fractal 222

G6-set 131 rE(r) 212 r(Pl,P2) 256,263 Gehring-Hayman theorem 72,88 geometrie-arithmetie mean

inequality 137, 169 Green's function 205-207, 209, 210 Grötzsch module theorem 202

HP see Hardy space Hadamard convolution 66 Hadamard determinant inequality 204 Hadamard gaps 188 see lacunary

series Hahn-Mazurkiewicz theorem 20 halfopen curve 3, 76 Hardy's identity 174 Hardy space 11, 13, 134, 154, 159,

163, 171, 172, 175, 176, 192 harmonie measure 72,84-88, 127,

151-154, 218 Hausdorff dimension 153, 222, 224,

225, 228, 229, 231, 235, 238-240, 249

Hausdorff measure 224 see A"" A ... Helson-Szegö theorem 171 Hölder condition 10, 47, 49, 50, 58,

65, 100, 105, 112, 228, 250, 251 Hölder domain 92, 93, 103 Hölder inequality 170 homeomorphic extension 25, 112, 121,

154 hyperbolic geodesie 91, 93 hyperbolic metrie 6, 90

I(p) 32, 34, 35, 38-40, 68 impression of prime end 32 see I(p) injective 4, 24, 27, 38, 120 inner capacity 208 see also capacity inner domain 2 inner function 249, 250 integral means of derivative 156, 160,

164, 165, 170-172, 174, 176, 183, 191, 223 see also ß,(p)

Subject Index 297

integral means of function 11, 174-176

irregular set 130 isogonal 80-82, 84, 101, 143, 193,

252,254 iteration theory 141, 184, 229, 250

Janiszewski's theorem 2 John domain 94, 96-103, 107, 125,

164, 169, 241, 245, 251 Jordan curve 2, 18, 24 Jordan curve theorem 2 Jordan domain 2,24 Julia set see iteration theory Julia-Wolff lemma 71, 82 junction point 36

K.-quasiconformal 95, 121-123, 125, 202, 219, 244

Kellogg-Warschawski theorem 42, 49, 151

kernel convergence 13 kind (prime end) 39, 40 Koebe distortion theorem 9, 10, 13,

124 Koebe function 8 Koebe transform 9, 105, 106, 120, 256 Kolmogoroff-Vercenko theorem 127,

134

A. 157 see Zygmund class A",(E) 222, 224-229, 235 A~(E) 134, 222, 225, 232 A(E) 2, 129 see length and linear

measure A(Zl, Z2), AlIli(Zl, Z2) 6 A ... (E) 223-226, 229, 236, 237, 239 A~(E) 225, 234 AG (Wl, W2) 91 see hyperbolic metric AO (Wl, W2) 92 see quasihyperbolic

metrie lacunary series 174, 188-194,249 Lavrentiev domain 155, 156, 163-165,

168-170, 172 law of the iterated logarithm 173,

186-188, 193, 194, 229, 237, 250 left-hand cluster set 37 Legendre function 175 Lehto majorant principle 123, 124 Lehto-Virtanen theorem 71, 74

298 Subject Index

length 2, 65, 130, 133, 134, 136, 151, 154, 163, 196, 211

length distortion 20, 72, 87, 88, 90, 112, 137, 142, 147, 163, 164, 172, 215, 217, 220

linear measure 129-131, 195 see also length and a-finite measure

linearly connected 94, 103-107, 160, 164, 168, 169

linearly invariant linearly measurable

134, 150 Liouville theorem

106, 107, 182 129, 131, 133,

5 Lipschitz see bilipschitz little Bloch space 79 see Ba localization 53, 251 locally connected 18-20, 35, 51 locally univalent 4 logarithmie capacity 203 see capacity logarithmic measure 229 Löwner's lemma 72, 85, 196, 217,

223, 234 Lusin-Privalov construction 139, 145

MacLane class 73 Makarov compression theorem 127,

147, 152 Makarov dimension theorem 222, 229,

231, 235 maximum principle for B 74 McMillan sector theorem 127, 146,

151 McMillan twist theorem 127, 142 mdimE 240 see Minkowski

dimension measurable 128 see linearly

measurable measure 128 metrie outer measure 128, 224, 225 minimum principle for B 74 Minkowski dimension 223, 240-242,

244 Möb 5 Möb(!Dl) 5 mod r 196 see module module 195-202, 212, 245, 256, 258,

263 modulus of continuity 46, 48, 50 Moore triod theorem 36

Muckenhoupt condition 170 see (Ap ), (Bq)

multiply connected domain V, 86, 90, 92

N(c, E) 240 nearly everywhere 195, 196, 209, 210,

215 Nehari univalence criterion 17, 122 non-euclidean metrie 6, 90 non-euclidean segment 6, 91 normal function 71, 73, 76 null-chain 29 numerieal methods V, 42, 58

w(z, A) 84 see harmonie measure w(z,A, G) 86 w(8,ip,A), w*(8,ip,A) 46 see

modulus of continuity open Jordan arc 2 order of family 106 outer domain 203 out er linear measure 129 see A(E) out er measure 128, 224

II(p) 33-35, 39 Pfiuger's theorem 196, 212 Plessner point 139, 144 Plessner's theorem 126, 139 Poincare-Hopf theorem 117 Poincare metric 90 point of density 131, 142 Poisson integral formula 42 Poisson kernel 43 power series 23, 35, 92, 100, 124, 183,

189, 194 prime end 30, 32, 34, 38-40, 69, 221 prime end theorem 18, 30 principal point 33 principle of domain extension 86 Privalov uniqueness theorem 126,

140, 215 probability measure 205 projection 128, 130, 207

quasieircle 107, 109, 123, 152, 250 see also quasidisk

quasieircle theorem 94, 109, 110, 114, 123

quasiconformal curve 107 see quasicircle

quasiconformal extension 94, 112-114, 120, 166, 168, 21~ 247

quasiconformal map 95, 112, 123, 195 quasidisk 94, 107, 109, 110, 114,

121-124, 144, 152, 163, 190, 246, 250,256

quasihyperbolic metric 92 quasismooth 163 see Lavrentiev quasisymmetrie 95, 109-113, 119,

120, 166

radial limit 11, 35, 66, 68, 71, 76, 141, 194

rectifiable curve 2, 45, 65, 134, 136, 144, 145, 148, 155, 162, 163 see also length

rectifiably accessible 90, 192, 215, 220 reflection principle 4, 41 regular curve 162 see Ahlfors-regular regular outer measure 131 regulated domain 42, 59-63, 65, 163 regulated function 59 Riemann mapping theorem 4 Riesz-Privalov theorem 126, 134 Riesz theorem 134 see Riesz-Privalov

theorem right-hand cluster set 38 rotation of the sphere. 1

S 8, 185 SI(Z) 7 see Schwarzian derivative E 8, 184, 185, 219 u-finite measure 129, 145, 147, 151,

154, 224, 237 Schoenflies theorem 25 Schottky's theorem 121 Schwarz-Christoffel formula 42, 65 Schwarz integral formula 42, 52, 62 Schwarzian derivative 7, 13, 17,43,

107, 122, 124, 182, 247 Sect(f) 127, 144-147, 152 sectorially accessible 103, 127, 144,

151, 169 see also Sect(f) semicontinuous 205 separated 2 sequences see domain sequences simply connected 4

Subject Index 299

singular 64, 151, 153, 159 Smirnov condition 159 see Smirnov

domain Smirnov domain 155, 156, 159-163,

169,231 smooth curve 42-46 snowflake curve 111, 244, 250 Souslin set 132, 208 spherical derivative 7 spherical metric 1 starlike function 66-68, 70, 149, 151,

152, 163, 172, 177, 210, 211, 215, 232, 241

stochastic process 156, 186, 193, 223 Stolz angle 6 subadditivity 50, 128, 198, 208, 221 symmetrie prime end 38, 40

1l' 1 tangent 51, 59, 81, 84, 101, 127, 138,

151, 153, 256 tangent direction angle 43, 46, 71,

127 tangential approach 35 Teichmüller module theorem 201, 202 transfinite diameter 203 see capacity triod 36 twisting 127, 141, 142, 144, 151, 238,

249,256 two-constants theorem 87

uniformly locally connected 22, 23 univalence criteria 15-17, 25, 190 univalent function 4 see conformal

map universal Teichmüller space 95 unrestricted limit 6, 35, 101, 134, 136 upper semicontinuous 205, 229

vanishing mean oscillation 172 see VMOA

variable domains see domain sequences

Visser-Ostrowski condition 81, 252, 254, 255

Visser-Ostrowski quotient 247, 251, 254,256

VMOA 172, 192

300 Subject Index

Warschawski condition 245, 261 weight see (Ap ), (Bq) condition well-accessible 251, 256 Wolff's lemma 20

Zygmund dass 157, 249 Zygmund measure 157, 158, 235 Zygmund-smooth 158, 159, 249, 250

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