Linearization Models for Complex Dynamical Systems: Topics in Univalent Functions, Functional Equations and Semigroup Theory

Operator Theory: Advances and Applications Vol. 208 Founded in 1979 by Israel Gohberg Editors: Joseph A. Ball (Blacksb...

Author: Mark Elin | David Shoikhet

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Operator Theory: Advances and Applications Vol. 208 Founded in 1979 by Israel Gohberg

Editors: Joseph A. Ball (Blacksburg, VA, USA) Harry Dym (Rehovot, Israel) Marinus A. Kaashoek (Amsterdam, The Netherlands) Heinz Langer (Vienna, Austria) Christiane Tretter (Bern, Switzerland)

Associate Editors: Vadim Adamyan (Odessa, Ukraine) Albrecht Böttcher (Chemnitz, Germany) B. Malcolm Brown (Cardiff, UK) Raul Curto (Iowa City, IA, USA) Fritz Gesztesy (Columbia, MO, USA) Pavel Kurasov (Lund, Sweden) Leonid E. Lerer (Haifa, Israel) Vern Paulsen (Houston, TX, USA) Mihai Putinar (Santa Barbara, CA, USA) Leiba Rodman (Williamsburg, VI, USA) Ilya M. Spitkovsky (Williamsburg, VI, USA)

Subseries Linear Operators and Linear Systems Subseries editors: Daniel Alpay (Beer Sheva, Israel) Birgit Jacob (Wuppertal, Germany) André C.M. Ran (Amsterdam, The Netherlands)

Subseries Advances in Partial Differential Equations Subseries editors: Bert-Wolfgang Schulze (Potsdam, Germany) Michael Demuth (Clausthal, Germany) Jerome A. Goldstein (Memphis, TN, USA) Nobuyuki Tose (Yokohama, Japan) Ingo Witt (Göttingen, Germany)

Honorary and Advisory Editorial Board: Lewis A. Coburn (Buffalo, NY, USA) Ciprian Foias (College Station, TX, USA) J. William Helton (San Diego, CA, USA) Thomas Kailath (Stanford, CA, USA) Peter Lancaster (Calgary, AB, Canada) Peter D. Lax (New York, NY, USA) Donald Sarason (Berkeley, CA, USA) Bernd Silbermann (Chemnitz, Germany) Harold Widom (Santa Cruz, CA, USA)

Linearization Models for Complex Dynamical Systems Topics in Univalent Functions, Functional Equations and Semigroup Theory

Mark Elin David Shoikhet

Birkhäuser

L O L S

Linear Operators & Linear Systems

Authors: Mark Elin David Shoikhet Department of Mathematics ORT Braude College Karmiel 21982 Israel e-mail: [email protected] [email protected]

2010 Mathematics Subject Classiﬁcation: 37Fxx, 30C45, 47H20, 47B33, 30D05 Library of Congress Control Number: 2010924097

Bibliographic information published by Die Deutsche Bibliothek. Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliograﬁe; detailed bibliographic data is available in the Internet at http://dnb.ddb.de

ISBN 978-3-0346-0508-3 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microﬁlms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained.

© 2010 Springer Basel AG P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced from chlorine-free pulp. TCF∞ Printed in Germany ISBN 978-3-0346-0508-3

e-ISBN 978-3-0346-0509-0

987654321

www.birkhauser.ch

Contents Preface

ix

1 Geometric Background 1.1 Some classes of univalent functions . . . . . . . . . . . . . . 1.1.1 Starlike functions . . . . . . . . . . . . . . . . . . . . 1.1.2 Class S ∗ [0]. Nevanlinna’s condition . . . . . . . . . . 1.1.3 Classes S ∗ [τ ], τ ∈ Δ. Hummel’s representation . . . ˇ cek’s condition . . . . . . . . 1.1.4 Spirallike functions. Spaˇ 1.1.5 Close-to-convex and ϕ-like functions . . . . . . . . . 1.2 Boundary behavior of holomorphic functions . . . . . . . . 1.3 The Julia–Wolﬀ–Carath´eodory and Denjoy–Wolﬀ Theorems 1.4 Functions of positive real part . . . . . . . . . . . . . . . . .

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1 1 1 2 3 4 6 7 10 13

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17 17

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2 Dynamic Approach 2.1 Semigroups and generators . . . . . . . . . . . . . . . . . . . 2.2 Flow invariance conditions and parametric representations of semigroup generators . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Denjoy–Wolﬀ and Julia–Wolﬀ–Carath´eodory Theorems for semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Generators with boundary null points . . . . . . . . . . . . . 2.5 Univalent functions and semi-complete vector ﬁelds . . . . . .

. . .

19

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23 25 34

3 Starlike Functions with Respect to a Boundary Point 3.1 Robertson’s classes. Robertson’s conjecture . . . 3.2 Auxiliary lemmas . . . . . . . . . . . . . . . . . . 3.3 A generalization of Robertson’s conjecture . . . . 3.4 Angle distortion theorems . . . . . . . . . . . . . 3.4.1 Smallest exterior wedge . . . . . . . . . . 3.4.2 Biggest interior wedge . . . . . . . . . . . 3.5 Functions convex in one direction . . . . . . . . .

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39 39 41 44 46 46 49 56

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vi

Contents

4 Spirallike Functions with Respect to a Boundary Point 4.1 Spirallike domains with respect to a boundary point . . . 4.2 A characterization of spirallike functions with respect to a boundary point . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Subordination criteria for the class Spiralμ [1] . . . . . . . 4.4 Distortion Theorems . . . . . . . . . . . . . . . . . . . . . 4.4.1 ‘Spiral angle’ distortion theorems . . . . . . . . . . 4.4.2 Growth estimates for semigroup generators . . . . 4.4.3 Growth estimates for spirallike functions . . . . . . 4.4.4 Classes G(μ, β) . . . . . . . . . . . . . . . . . . . . 4.5 Covering theorems for starlike and spirallike functions . .

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63 63

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69 73 75 75 79 81 84 90

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95 95 99

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5 Kœnigs Type Starlike and Spirallike Functions 5.1 Schr¨ oder’s and Abel’s equations . . . . . . . . . . . . . . . . . . 5.2 Remarks on stochastic branching processes . . . . . . . . . . . 5.3 Kœnigs’ linearization model for dilation type semigroups. Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Valiron’s type linearization models for hyperbolic type semigroups. Embeddings . . . . . . . . . . . . . . . . . . . . . . 5.5 Pommerenke’s and Baker–Pommerenke’s linearization models for semigroups with a boundary sink point . . . . . . . . . . . . 5.5.1 Pommerenke’s linearization model for automorphic type mappings . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Baker–Pommerenke’s model for non-automorphic type self-mappings . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 Higher order angular diﬀerentiability at boundary ﬁxed points. A uniﬁed model . . . . . . . . . . . . . . . . . . 5.6 Embedding property via Abel’s equation . . . . . . . . . . . . .

. . . . . . . .

. . 103 . . 105 . . 112 . . 112 . . 116 . . 117 . . 119

6 Rigidity of Holomorphic Mappings and Commuting Semigroups 6.1 The Burns–Krantz theorem . . . . . . . . . . . . . . . . . . 6.2 Rigidity of semigroup generators . . . . . . . . . . . . . . . 6.3 Commuting semigroups of holomorphic mappings . . . . . . 6.3.1 Identity principles for commuting semigroups . . . . 6.3.2 Dilation type . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Hyperbolic type . . . . . . . . . . . . . . . . . . . . 6.3.4 Parabolic type . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

121 122 128 133 133 140 144 146

7 Asymptotic Behavior of One-parameter Semigroups 7.1 Dilation case . . . . . . . . . . . . . . . . . . . . 7.1.1 General remarks and rates of convergence 7.1.2 Argument rigidity principle . . . . . . . . 7.2 Hyperbolic case . . . . . . . . . . . . . . . . . . . 7.2.1 Criteria for the exponential convergence .

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153 154 154 157 159 159

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Contents . . . . .

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168 173 173 176 184

8 Backward Flow Invariant Domains for Semigroups 8.1 Existence . . . . . . . . . . . . . . . . . . . . . . 8.2 Maximal FIDs. Flower structures . . . . . . . . . 8.3 Examples . . . . . . . . . . . . . . . . . . . . . . 8.4 Angular characteristics of ﬂow invariant domains 8.5 Additional remarks . . . . . . . . . . . . . . . . .

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195 195 205 208 211 216

9 Appendices 9.1 Controlled Approximation Problems . . . . . . 9.1.1 Setting of approximation problems . . . 9.1.2 Solutions of approximation problems . . 9.1.3 Perturbation formulas . . . . . . . . . . 9.2 Weighted semigroups of composition operators

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221 221 221 223 231 240

7.3

7.2.2 Angular similarity principle Parabolic case . . . . . . . . . . . . 7.3.1 Discrete case . . . . . . . . 7.3.2 Continuous case . . . . . . 7.3.3 Universal asymptotes . . .

vii . . . . .

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Bibliography

247

Subject Index

257

Author Index

261

Symbols

263

List of Figures

265

Preface Interactions between Complex Analysis, Iteration Theory and Dynamical Systems have fascinated mathematicians for more than a century. Richness and beauty of these topics originated from the transparency of geometric ideas and depth of analytic methods forming a peculiarity of this area. Iteration algorithms and continuous dynamical processes are at the heart of many related ﬁelds, e.g., branching stochastic processes, composition operators, control theory and theory of linear operators in indeﬁnite metric spaces. For recent books in these topics see, for example, [27, 104, 127, 130, 2, 40] and [120]. Each iteration procedure generates a discrete dynamical system, which can be considered a semigroup with respect to an integer parameter. The study of the behavior of such semigroups in a domain in the complex plane C goes back to the classical works of Julia, Fatou, Denjoy and Wolﬀ and has been developed in diﬀerent directions (including higher dimensional generalizations and dynamics in hyperbolic metric spaces and operator algebras, see [120] and references therein). In parallel, and even earlier, the study of the local behavior of iterations near a ﬁxed point was based on linearization models provided by the functional equations of Schr¨ oder, Abel and Bottcher. In 1884 Kœnigs showed how to solve Schr¨ oder’s functional equation h (F (z)) = λh(z) in a neighborhood of an attractive ﬁxed point of F by using a limit scheme bearing his name. Thus, the iteration properties of a given mapping F can be understood by means of the iteration properties of the linear mapping λw. This approach has been developed over a period of about a century by the combined eﬀorts of many mathematicians (Valiron, Hadamard, Pommerenke, Baker and Cowen, among others). The key tool in these investigations is the ability to construct an intertwining mapping h for diﬀerent types of holomorphic functions as well as the analysis of its uniqueness and geometric and analytic properties. Although the theory of discrete dynamical systems was developed extensively, little was known about semigroups with respect to a continuous parameter. At the same time, continuous semigroups have various applications in Markov Stochastic

x

Preface

Processes [79, 126], Control Theory and Optimization [81]. For example, one of the central problems in Markov branching processes is to describe the asymptotic behavior of the process in a neighborhood of the so-called extinction probability which can be understood as the common attractive ﬁxed point of a corresponding semigroup. In 1978, Berkson and Porta [17] showed that each continuous semigroup {Ft }t≥0 of holomorphic self-mappings on the open unit disk Δ is everywhere differentiable with respect to its parameter. Hence the limit f (z) := lim

t→0+

z − Ft (z) t

(0.0.1)

exists and deﬁnes a holomorphic function f in Δ. This function is called the inﬁnitesimal generator of the semigroup. Furthermore, it can be shown (see [17], [115] and [130]) that given f , the semigroup can be reproduced by the solution of the Cauchy problem: ⎧ ⎨ ∂Ft (z) + f (F (z)) = 0, t ≥ 0, t ∂t ⎩ z ∈ Δ. F0 (z) = z, Moreover, in the same work Berkson and Porta showed that a holomorphic function f is a semigroup generator if and only if it admits the representation f (z) = (z − τ ) (1 − zτ ) p (z) ,

z ∈ Δ,

(0.0.2)

for some point τ ∈ Δ and some function p ∈ Hol (Δ, C) with Re p(z) ≥ 0, z ∈ Δ; and this representation is unique. Based on this fact, Siskakis investigated in [136] continuous analogs of Schr¨ oder’s and Abel’s functional equations: h(Ft (z)) = λt h(z)

(0.0.3)

h(Ft (z)) = h(z) + t.

(0.0.4)

and

It seems he was the ﬁrst to focus on the connection between their solutions and the semigroup generators. In particular, he showed that if a one-parameter continuous semigroup has an interior ﬁxed point, then Schr¨ oder’s equation (0.0.3) (or Abel’s equation (0.0.4) in the opposite case) should have the unique normalized solution. The linearization models provided by these equations allowed Siskakis to study weighted semigroups of composition operators as well as the Ces`aro and Volterra averaging operators on classical Hilbert and Banach spaces of analytic functions (Hardy spaces, Bergman spaces etc.). Linearization models for continuous semigroups attracted special attention in the last decade. Recently, Contreras, D´ıaz-Madrigal and Pommerenke in a series

Preface

xi

of works [32, 33, 34] used these models to study dynamical properties of evolution equation and the boundary behavior of Kœnigs functions. Also, we would like to emphasize that linearization models for discrete and continuous time dynamical systems are driving forces for the Composition Operator Theory on function spaces in the unit disk of the complex plane. Naturally, the eigenvalue problems for composition operators on Hardy, Bergman and Dirichlet spaces of analytic functions involve Schr¨ oder’s functional equations. They can be considered linearization models for self-mappings of the underlying domain (see, for example, the books [127, 18] and [40]). In fact, as a part of operator theory, research into composition with a ﬁxed function acting on a space of analytic functions is of recent origin, dating back to the mid-1960s, while investigation of continuous semigroups of composition operators started intensively from the work of Berkson and Porta at the beginning of the 1980s (see for example, the survey [136]). Recent developments of the Denjoy–Wolﬀ Theory and Julia–Carath´eodory boundary versions of the Schwarz–Pick Lemma in the context of the generation theory for semigroups of holomorphic mappings lead to an independent study of linearization models for complex dynamical systems lying at the interface of analytic function theory and operator theory. In addition, solutions of Schr¨ oder’s and Abel’s equations were applied to solve the Kœnigs embedding problem in [49] and the rigidity problem in the spirit of the Burns–Krantz theorem in [51, 53]. Diﬀerent aspects of the boundary rigidity problem inspired by Abel’s functional equation and its linearization model for holomorphic semigroups and generators can be integrated into the boundary interpolation theory for Schur’s functions on the unit disk. It turns out, that intertwining mappings for continuous semigroups have transparent geometric properties: they should be either starlike, spirallike, or convex in one-direction mappings. Thus these linearization models link to new questions in the classical Geometric Function Theory. Such questions have been investigated in [56, 5], as well as new covering and distortion theorems for spirallike and starlike functions studied in [47, 65, 132]. Finally, the nice geometry of linearization models enable the study of backward ﬂow invariant domains for continuous semigroups [68, 69]. This book is devoted to a systematic and detailed survey and treatment of linearization models for one-parameter continuous semigroups, functional equations, diﬀerent classes of univalent functions which serve intertwining mappings for these semigroups, and their applications to various problems of complex dynamics. We believe that the book will be useful to a wide readership with an interest in these areas. Although many results have been previously published in journal papers, we endeavor to facilitate the understanding of their proofs to students and nonspecialists.

xii

Preface

Some topics of this book have been given in special courses for students of ORT Braude College, Karmiel. More advanced issues were presented at the Seminars on Nonlinear Analysis at the Technion, Haifa, on Complex Analysis at the Bar–Ilan University, Ramat-Gan, on Operator Theory in the Banach Center of Polish Academy of Sciences, on Complex Analysis at the University Rome II Tor Vergata. We would like to express our thanks to our colleagues who have read the manuscript, made useful remarks and suggested improvements. We are grateful to Professor Lawrence Zalcman who greatly inﬂuenced the conceptual approaches applied here. We also beneﬁted from discussions with Professors Simeon Reich and Dov Aharonov. Many results included in the book were obtained due to collaboration with them. We are very thankful to Professors Filippo Bracci, Manuel Contreras and Santiago D´ıaz-Madrigal for their many useful discussions related to these topics. Finally, special thanks to Marina Levenshtein who browsed through the entire manuscript and made helpful corrections.

Chapter 1

Geometric Background 1.1 Some classes of univalent functions 1.1.1 Starlike functions Let D be a domain (open connected set) in the complex plane C. The set of all holomorphic functions on D will be denoted by Hol(D, C). The subset of Hol(D, C), which consists of all univalent functions (with no normalization), we denote by Univ(D). The set of all holomorphic self-mappings of D we denote by Hol(D). This set is a semigroup with respect to composition operation. The concept of univalent starlike functions was introduced by Alexander [7] in 1915 (see also Goodman [75] and Golusin [74]). Deﬁnition 1.1. A domain Ω ⊂ C is said to be starlike (with respect to the origin) if for every point w ∈ Ω, the linear segment joining w to zero: (0, w] := {tw : t ∈ (0, 1]} lies entirely in Ω. It is clear that if a domain Ω is starlike, then 0 ∈ Ω. Let D be a simply connected domain in C. Deﬁnition 1.2. A univalent function h ∈ Hol(D, C) is called a starlike function on D if h(D) is a starlike domain. • If 0 ∈ h(D), then h is said to be starlike with respect to an interior point. • If 0 ∈ ∂h(D), then h is said to be starlike with respect to a boundary point. In Fig. 1.1 we see images of starlike functions: (i) — with respect to an interior point, and (ii) — with respect to a boundary point. We will mostly consider the case where D = Δ = {z ∈ C : |z| < 1} is the open unit disk in C.

2

Chapter 1. Geometric Background

(i)

(ii) 2 6 1 4

–2

–1

1

2

0

2

–1 –4

–2

0

2

4

6

8

10

12

–2

–2

–4

–3

–6 –4

Figure 1.1: Images of starlike functions. For any starlike function on Δ with respect to an interior point there exists a unique point τ ∈ Δ such that h(τ ) = 0. If h is a starlike function on Δ with respect to a boundary point, then there exists a unique point τ ∈ ∂Δ and a sequence {zn }∞ n=1 ⊂ Δ converging to τ , such that h(τ ) := lim h(zn ) = 0. n→∞

In both cases we use the notation S ∗ [τ ] for the set of starlike functions (with respect to an interior or a boundary point) satisfying the condition h(τ ) = 0, τ ∈ Δ. The class S ∗ ⊂ S ∗ [0] of starlike functions normalized by the conditions h(0) = 0 and h (0) = 1 has been considered one of the ﬁrst objects of interest in the classical geometric function theory.

1.1.2 Class S ∗ [0]. Nevanlinna’s condition The following well-known theorem is due to Nevanlinna [106] (see also [74, 75]). Theorem 1.1. Let h ∈ Univ(Δ) and h(0) = 0. Then h ∈ S ∗ [0] if and only if Re

zh (z) > 0, h(z)

z ∈ Δ.

(1.1.1)

Remark 1.1. Actually, one can show that if a holomorphic function h satisﬁes (1.1.1) and is locally univalent at zero, that is, h (0) = 0, then h(0) = 0 and h ∈ Univ(Δ). Hence, in this case h ∈ S ∗ [0]. Using Nevanlinna’s condition we can easily check that if h ∈ S ∗ [0], then the image h(Δr ) of the disk Δr := {z : |z| < r} is a starlike domain for each r ∈ (0, 1). Stronger inequalities than Nevanlinna’s condition (1.1.1) led to the study of some interesting subclasses of S ∗ [0]. In particular, in 1936 Robertson [121]

1.1. Some classes of univalent functions

3

introduced the class Sλ∗ of starlike functions of order λ: Sλ∗ =

zh (z) h ∈ S ∗ : Re > λ, h(z)

z∈Δ ,

(1.1.2)

where λ ≥ 0. It is clear that for λ > 0, Sλ∗ ⊂ S0∗ = S ∗ and Sλ∗ = ∅ when λ > 1 because zh (z) = 1 for any function h ∈ S ∗ . lim z→0 h(z) Remark 1.2. Since h ∈ S ∗ is normalized by the condition h (0) = 1, it is easy to 1−λ h(z) ∈ Sλ∗ . check (using Remark 1.1) that h1 (z) := z z Maximum values of many functionals on the class S ∗ as well as on the class Univ(Δ) including Taylor coeﬃcients (Bieberbach problem) are attained for the so-called Koebe function: k(z) =

∞ z 2 3 n = z + 2z + 3z + . . . + nz + . . . = nz n . (1 − z)2 n=1

(1.1.3)

This function is in S ∗ . It maps Δ in a one-to-one manner onto the domain that consists of the entire complex plane except for the slit along the negative real axis from w = −∞ to w = −1/4. In link with Remark 1.2, the function kλ (z) =

z (1 − z)2(1−λ)

will be referred to as the λ-starlike Koebe function.

1.1.3 Classes S ∗ [τ ], τ ∈ Δ. Hummel’s representation τ −z , τ ∈ Δ, one can easily 1 − τ¯z ∗ describe the classes S [τ ] of univalent starlike functions such that h(τ ) = 0, τ ∈ Δ.

By using the M¨ obius transformation Mτ (z) =

Theorem 1.2. Let h ∈ Hol(Δ, C) be a locally univalent function on Δ. Then h belongs to S ∗ [τ ], τ ∈ Δ, if and only if the function q(z) =

h (z) (z − τ )(1 − z τ¯) h(z)

is a well-deﬁned holomorphic function on Δ with Re q(z) > 0.

4

Chapter 1. Geometric Background

A diﬀerent formulation of this fact was given in 1978 by Wald in [144] (see also [75] and [130]). However, this result cannot be useful when τ approaches the boundary of Δ because of degeneration of the M¨ obius transformation Mτ . Note also that if h ∈ S ∗ [τ ] and Dr (τ ) := {z ∈ C : |Mτ (z)| < r} (the so-called pseudohyperbolic disk centered at τ , see [40, 130]), then the image h(Dr (τ )) is a starlike domain for each r ∈ (0, 1). Another deep connection between classes S ∗ [0] and S ∗ [τ ], τ ∈ Δ, was established by Hummel [84, 85]; see also [130]. Theorem 1.3. Let h1 ∈ Hol(Δ, C), h1 (0) = 0, and h2 ∈ Hol(Δ, C) be two functions related by the formula: h2 (z) = Ψτ (z)h1 (z), where Ψτ (z) =

z ∈ Δ,

(1.1.4)

1 (z − τ )(1 − z τ¯) . · 2 1 − |τ | z

Then h1 ∈ S ∗ [0] if and only h2 ∈ S ∗ [τ ]. Moreover, the inverse transform can be deﬁned by the formula h1 (z) = Ψτ (Mτ (z))h2 (z). Furthermore, Hummel proved that if h ∈ S ∗ [τ ], τ ∈ Δ, then for each ε > 0, there exists ρ ∈ (0, 1) such that the function h satisﬁes a ‘weak’ Nevanlinna’s condition: zh (z) Re > −ε h(z) in the annulus ρ < |z| < 1; see [130] for details.

ˇ cek’s condition 1.1.4 Spirallike functions. Spaˇ The class of spirallike functions (sometimes called also logarithmically spirallike) is an important generalization of starlike functions. Deﬁnition 1.3. A domain Ω ⊂ C is said to be spirallike (with respect to the origin) if there is a number μ ∈ C with Re μ > 0 such that for every point w ∈ Ω the spiral curve {e−tμ w : t ≥ 0}

(1.1.5)

lies entirely in Ω. It is clear that if the number μ is real, then Ω is actually starlike. Since Re μ > 0 we have lim e−tμ w = 0 for any point w ∈ C and then 0 ∈ Ω t→∞ for each spirallike domain Ω.

1.1. Some classes of univalent functions

5

Deﬁnition 1.4. If a univalent function h on a simply connected domain D maps D onto a spirallike domain, we say that h is a spirallike function. • If for each point w ∈ h(D) the image h(D) contains the spiral curve (1.1.5), then h is said to be a μ-spirallike function. • If 0 ∈ h(D), then the function h is said to be spirallike with respect to an interior point. • If 0 ∈ ∂h(D), then the function h is said to be spirallike with respect to a boundary point. In Fig. 1.2 we see images of spirallike functions ((i) — with respect to an interior point, and (ii) — with respect to a boundary point) and spiral lines of the form we−(1+i)t which are contained in the images.

(i)

(ii)

1.5

1

3

0.5 2 –2

–1

1 1 –0.5

–1

1

–1.5

2

3

4

–1

–2

Figure 1.2: Images of spirallike functions. If a spirallike function h on the open unit disk Δ has no interior null point (i.e., h is spirallike with respect to a boundary point), there exists a unique point τ ∈ ∂Δ and a sequence {zn }∞ n=1 ∈ Δ converging to τ , such that h(τ ) := lim h(zn ) = 0. For a point τ ∈ Δ, the set of all spirallike functions h that satisfy

n→∞

h(τ ) = 0 will be denoted by Spiral[τ ]. It seems that Montel [105] was the ﬁrst mathematician who suggested the study of spirallike functions. A generalization of Nevanlinna’s condition was given ˇ cek [137]. His result can be reformulated as follows. by Spaˇ Theorem 1.4. Let h ∈ Univ(Δ) have the form h(z) = z +

∞ k=2

ak z k ,

(1.1.6)

6

Chapter 1. Geometric Background

(i.e., h(0) = 0 and h (0) = 1). Then h ∈ Spiral[0] if and only if

zh (z) Re e−iθ > 0, h(z)

z ∈ Δ,

(1.1.7)

for some θ ∈ (− π2 , π2 ). Moreover, h is μ-spirallike, where Re μ > 0, if and only if condition (1.1.7) holds for θ = arg μ. Similarly as for starlike functions, one can consider classes of functions satisfying a stronger condition than (1.1.7). To do so, we use the following notion. Deﬁnition 1.5. A function h ∈ Spiral[0] is said to be spirallike of order λ if for some θ ∈ R, it satisﬁes the following inequality:

−iθ zh (z) > λ ≥ 0, z ∈ Δ, (1.1.8) Re e h(z) and h (0) = 1.

(z) Since lim Re e−iθ zh = cos θ for any function h ∈ Spiral[0], the class of h(z) z→0

spirallike functions of order λ is empty for λ > cos θ.

1.1.5 Close-to-convex and ϕ-like functions The following class of holomorphic functions was introduced by Kaplan in 1952 [89]. Deﬁnition 1.6. A function h ∈ Hol(Δ, C) is called close-to-convex if there is a function h1 ∈ S ∗ [0] such that Re

zh (z) > 0. h1 (z)

Obviously, any starlike function h ∈ S ∗ [0] is close-to-convex. At the same time, one can show (see, for example, [75]) that any close-to-convex function belongs to Univ(Δ), but not necessarily to S ∗ [0]. Brickman [22] introduced another intensely studied generalization of classes S ∗ [0] and Spiral[0]. The generalization, in fact, includes all univalent functions preserving the origin. Deﬁnition 1.7. Let h ∈ Hol(Δ, C) satisfy h(0) = 0 and h (0) = 0. Let ϕ ∈ Hol(h(Δ), C) satisfy ϕ(0) = 0 and Re ϕ (0) > 0. Then h is a ϕ-like function if zh (z) Re > 0. ϕ(h(z))

1.2. Boundary behavior of holomorphic functions

7

If we set in this deﬁnition ϕ(w) = aw with a real positive number a, then the class of ϕ-like functions with this choice of ϕ coincides with the class S ∗ [0]. Setting ϕ(w) = μw with Re μ > 0, we get the subclass of Spiral[0] which consists of μ-spirallike functions. Furthermore, any close-to-convex function h ∈ Hol(Δ, C) relative to a starlike function h1 ∈ S ∗ [0] is also ϕ-like with ϕ(w) = h1 h−1 (w) . As a matter of fact, each univalent function that preserves the origin can be represented as a ϕ-like function with a suitable choice of ϕ. Theorem 1.5 ([22]). Let h ∈ Hol(Δ, C) satisfy h(0) = 0. Then h ∈ Univ(Δ) if and only if h is ϕ-like for some ϕ. Below we discuss a dynamic approach to the study of starlike, spirallike and, more generally, ϕ-like functions on Δ. This approach will clarify Theorem 1.5.

1.2 Boundary behavior of holomorphic functions Let f be a continuous function on the open unit disk Δ, and ζ be a boundary point of Δ. The limit lim− f (rζ) r→1

(if it exists) is called the radial limit of f at ζ. We say that f has the unrestricted limit A at the point ζ ∈ ∂Δ if the limit lim

z→ζ, z∈Δ

f (z) = A

exists ﬁnitely. In this case one can deﬁne f (ζ) = A and then f becomes continuous in Δ ∪ {ζ}. A diﬀerent notion of the limit at a boundary point is based on the so-called non-tangential approach regions at this point. Deﬁnition 1.8. For a point ζ on the unit circle ∂Δ and κ > 1, a non-tangential approach region at ζ is the set Γ(ζ, κ) = {z ∈ Δ : |z − ζ| < κ(1 − |z|)} .

(1.2.1)

The term “non-tangential” refers to the fact that Γ(ζ, κ) lies in a sector W in Δ (with vertex at ζ), which is symmetric about the radius to ζ; i.e., the boundary curve of Γ(ζ, κ) has a corner at ζ of angle less than π. Deﬁnition 1.9. We will say that a complex-valued function f on Δ has a nontangential (or angular) limit L at a point ζ ∈ ∂Δ if f (z) → L as z → ζ, z ∈ Γ(ζ, κ) for each κ > 1. In this case we write L = ∠ lim f (z). z→ζ

8

Chapter 1. Geometric Background

1

0.5

–1

–0.5

0.5

1

–0.5

–1

Figure 1.3: A non-tangential approach region and a Stolz angle at the point ζ = eiπ/6 . and r < 2 cos a, a Stolz angle at ζ ∈ ∂Δ is the set ¯ < a, |z − ζ| < r Da,r (ζ) = z ∈ Δ : | arg(1 − z ζ)|

For 0 < a <

π 2

(see Fig 1.3). It is clear that f ∈ Hol(Δ, C) has a non-tangential limit L at a point ζ if and only if f (z) → L as z → ζ, for each Stolz angle Da,r (ζ) at ζ. Once again one can set f (ζ) := ∠ lim f (z). z→ζ

Theorem 1.6 (Lindel¨ of ’s Theorem, [101]). Let ζ ∈ ∂Δ and let f ∈ Hol(Δ, C) be bounded on each non-tangential approach region at ζ. If for some continuous curve γ ∈ Δ ending at ζ, there exists the limit L = lim f (z), z→ζ

z ∈ γ,

then the angular limit ∠ lim f (z) = L z→ζ

also exists. Remark 1.3. For univalent functions the boundedness condition in this theorem can be omitted (see Corollary 2.17 [113]). That is, a univalent function f has an angular limit at the point ζ if and only if it has a limit along some curve ending at ζ. Moreover, both limits coincide. Deﬁnition 1.10. A holomorphic function f is said to have an angular derivative at a boundary point ζ ∈ ∂Δ, if the angular limit f (ζ) of f at ζ exists ﬁnitely and

1.2. Boundary behavior of holomorphic functions

9

the angular limit f (z) − f (ζ) =: f (ζ) z−ζ

∠ lim

z→ζ

(1.2.2)

exists. Theorem 1.7 (see [113], p. 79). Let f ∈ Hol(Δ, C) and ζ ∈ ∂Δ. Then the function f has a ﬁnite angular derivative f (ζ) at the point ζ if and only if f has the ﬁnite angular limit f (ζ) at ζ. In this case f (ζ) = ∠ lim f (z). z→ζ

Theorem 1.8 (see [113], p. 80). Let f be a univalent function (f ∈ Univ(Δ)), and let f have a ﬁnite angular limit f (ζ) at ζ ∈ ∂Δ. If lim

z→ζ, z∈Γ

f (z) − f (ζ) =A z−ζ

(1.2.3)

for some curve Γ ⊂ Δ ending at ζ, or if lim

z→ζ, z∈Γ

f (z) = A,

then f (ζ) = A exists. Deﬁnition 1.11. Let f ∈ Hol(Δ, C) and ζ ∈ ∂Δ. • The function f is called conformal at the boundary point ζ if the angular derivative f (ζ) exists ﬁnitely and f (ζ) = 0. • The function f is called isogonal at the point ζ if there is a ﬁnite angular limit f (ζ) and if ∠ lim arg z→ζ

f (z) − f (ζ) z−ζ

(1.2.4)

exists ﬁnitely. It is obvious that if f is conformal at the point ζ ∈ ∂Δ, then it is isogonal at ζ. Theorem 1.9 (see [113], p. 81). Let f ∈ Univ(Δ) and ζ ∈ ∂Δ. The function f is isogonal at ζ if and only if f and arg f have ﬁnite angular limits at ζ. In this case ∠ lim

z→ζ

(z − ζ)f (z) = 1. f (z) − f (ζ)

(1.2.5)

10

Chapter 1. Geometric Background The quotient Qf (ζ, z) :=

(z − ζ)f (z) f (z) − f (ζ)

(1.2.6)

is usually called the Visser–Ostrowski quotient. Condition (1.2.5) is called the Visser–Ostrowski condition. Note, however, that f ∈ Hol(Δ, C) satisfying the Visser–Ostrowski condition is not necessarily isogonal (see [113]). Deﬁnition 1.12. We say that f satisﬁes a generalized Visser–Ostrowski condition if Qf (ζ) := ∠ lim Qf (ζ, z) exists ﬁnitely and is diﬀerent from zero. z→ζ

1.3 The Julia–Wolﬀ–Carath´eodory and Denjoy–Wolﬀ Theorems In this section we continue to consider the boundary behavior of bounded holomorphic functions that are not necessarily univalent. Without loss of generality one can assume that F ∈ Hol(Δ), i.e., the lower bound for |F (z)| is the unity. First let F be an arbitrary continuous self-mapping of the open unit disk Δ. Recall that a point ζ ∈ Δ is said to be a ﬁxed point of F if F (ζ) = ζ. Since we have not assumed that the function F is continuous on the boundary of Δ, we need to extend the notion of ﬁxed point to include the case of a ﬁxed point on the unit circle. Namely, a point ζ ∈ Δ will be called a ﬁxed point of F if lim F (rζ) = ζ.

r→1−

(1.3.1)

Obviously, an interior ﬁxed point of a self-mapping F of Δ is a common ﬁxed 1 n+1 point of all iterates {F n }∞ = F ◦ F n. n=1 , F = F, F In addition, if a holomorphic self-mapping F has an interior ﬁxed point ζ, then the classical Schwarz–Pick Lemma that asserts F leaves invariant each closed z−ζ ≤ r , r < 1, centered at this point pseudohyperbolic disk Dr (ζ) := z : 1 − z ζ¯ (see, for example, [40, 130]), i.e., F (z) − ζ z − ζ ≤ 1 − F (z)ζ 1 − zζ . Moreover, if F is not the identity, then equality holds if and only if F is an elliptic automorphism of Δ. Hence, if F is not the identity mapping of Δ, then it has at most one ﬁxed point inside the disk. At the same time, holomorphic functions may have many ﬁxed points on the circle. It seems that the ﬁrst descriptions of the boundary behavior of holomorphic self-mappings of the open unit disk were given by Julia (Julia’s Lemma, [88]), Wolﬀ (Boundary Schwarz–Wolﬀ’s Lemma, [145]–[147]) and Carath´eodory (see [26]).

1.3. The Julia–Wolﬀ–Carath´eodory and Denjoy–Wolﬀ Theorems

11

Combining their results one obtains the following characterization of a ﬁxedpoint-free holomorphic self-mapping of the unit disk Δ. Theorem 1.10 (Julia–Wolﬀ–Carath´eodory Theorem). Let F ∈ Hol(Δ). Then the following are equivalent. (i) F has no ﬁxed points in Δ; (ii) there is a unique unimodular point τ ∈ ∂Δ such that α := ∠ lim

z→τ

F (z) − τ z−τ

exists with 0 < α ≤ 1; (iii) there is a unique unimodular point τ ∈ ∂Δ such that lim inf z→τ

1 − |F (z)| = α ≤ 1; 1 − |z|

(iv) there is a unique unimodular point τ ∈ ∂Δ such that sup z∈Δ

ϕτ (F (z)) = α ≤ 1, ϕτ (z)

where ϕτ (z) :=

|1 − z τ¯|2 |z − τ |2 = , z ∈ Δ. 1 − |z|2 1 − |z|2

(1.3.2)

Moreover, (a) the boundary points τ and the numbers α in (i)–(iv) are the same; (b) the non-tangential limit in (ii) can be replaced by the radial limit. Remark 1.4. This theorem can be considered a boundary version of the Schwarz– Pick Lemma. Geometrically it means that each horocycle |z − τ |2 Δr (τ ) = z ∈ Δ : ≤r 1 − |z|2 internally tangent to ∂Δ at τ is F -invariant (see Fig 1.4). As we already mentioned, a holomorphic self-mapping of Δ may have many boundary ﬁxed points. However, for a ﬁxed-point-free mapping F , condition (iv) means that there is a unique ﬁxed point τ such that all the disks internally tangent to ∂Δ at τ are invariant.

12

Chapter 1. Geometric Background

1

0.5

–1

–0.5

0.5

1

–0.5

–1

Figure 1.4: A horocycle and its image Remark 1.5. The number α (= F (τ )) in this theorem is sometimes called the Julia number of F . Note that condition (iv) implies the inequality: |z − τ |2 |F n (z) − τ |2 ≤ αn . n 2 1 − |F (z)| 1 − |z|2

(1.3.3)

So, if α < 1, then τ ∈ ∂Δ is an attractive boundary ﬁxed point of F , i.e., ∞ the semigroup {F n }n=0 of iterates of F converges to τ uniformly on each compact subset of Δ with the power rate of convergence in the sense of the non-euclidean “distance” ϕτ (z) deﬁned by (1.3.2). The question is: ♦ Whether this point is also attractive when α = 1? The aﬃrmative answer to this question is given in the next assertion, following Wolﬀ [145, 146, 147] and Denjoy [41]; see also [40, 127, 117, 130]. Theorem 1.11 (Denjoy–Wolﬀ Theorem). If F ∈ Hol(Δ) is not the identity and not an elliptic automorphism of Δ, then there is a unique point τ ∈ Δ such that the one-parameter discrete semigroup S = {F n }∞ n=0 of iterates of F converges to τ uniformly on compact subsets of Δ. The point τ ∈ Δ mentioned in Theorems 1.10 and 1.11 is called the Denjoy– Wolﬀ point for a self-mapping F . If τ ∈ ∂Δ, it is also variously called Wolﬀ ’s point or the sink point of F . It is clear that if F has no ﬁxed point in Δ then the limit point τ ∈ ∂Δ of its iterates must be Wolﬀ’s point of F .

1.4. Functions of positive real part

13

1.4 Functions of positive real part We have already seen that the classes S ∗ [τ ] and Spiral[0], as well as the classes of close-to-convex and ϕ-like functions, are characterized by a lower bound of the real part of some analytic expressions. Now let us recall some known properties of the class P of holomorphic functions in the open unit disk Δ of positive real part. Namely, P := {p ∈ Hol(Δ, C) : Re p(z) ≥ 0 for all z ∈ Δ} and P0 := {p ∈ P : p(0) = 1} . The class P0 is usually called the class of Carath´eodory. It should be noted that p ∈ P is not required to be univalent. In the same way that the Koebe function is essential for the class S ∗ [0], the linear fractional function C(z) :=

∞ 1+z = 1 + 2z + 2z 2 + 2z 3 + . . . = 1 + 2 zn 1−z n=1

is crucial for the class P. Sometimes this function is called the Cayley transform. It is univalent in Δ and maps Δ onto the right half-plane Π+ := {w ∈ C : Re w > 0}. The following remarkable fact was established by Herglotz in 1911 [83]. Theorem 1.12. Let p ∈ Hol(Δ, C). Then p ∈ P0 if and only if ¯ dσp (ζ), z ∈ Δ, p(z) = C(z ζ)

(1.4.1)

∂Δ

for some probability measure σp on the unit circle ∂Δ. form

It follows by (1.4.1) that each element of class P can be represented in the 1 + z ζ¯ p(z) = dσp (ζ) + i Im p(0) (1.4.2) 1 − z ζ¯ ∂Δ

with a positive measure σp on the unit circle ∂Δ. Formulas (1.4.1) and (1.4.2) are variously called the Herglotz or Riesz– Herglotz integral representation for classes P0 and P, respectively. Theorem 1.13 (Harnack’s inequality). If p ∈ P, then it satisﬁes the estimates: Re p(0)

1 + |z| 1 − |z| ≤ Re p(z) ≤ Re p(0) , 1 + |z| 1 − |z|

z ∈ Δ.

Other more advanced properties of the class P are given below. It follows by (1.4.2) that for each τ ∈ ∂Δ, the angular limit δ = (δp (τ )) := ∠ lim (1 − z τ¯)p(z) z→τ

(1.4.3)

14

Chapter 1. Geometric Background

exists ﬁnitely and is a real nonnegative number. We call this number the charge of p at the boundary point τ ∈ ∂Δ. Consider the class P + [τ ] = {p ∈ P : δp (τ ) > 0} of functions with the strictly positive charge at a point τ ∈ ∂Δ. By using the Julia–Wolﬀ–Carath´eodory Theorem (see Theorem 1.10) one can establish a somewhat more precise lower estimate for functions of the classes P and P + [τ ] than the right-hand side of Harnack’s inequality (1.4.3). Without loss of generality we assume that τ = 1. Theorem 1.14 (see [132]). A function p belongs to the class P if and only if the following inequality holds: Re p(z) ≥ a

1 − |z|2 , |1 − z|2

z∈Δ

for some a ≥ 0. Moreover, in this case the number a can be set as a = Thus, p belongs to P +[1] if and only if a can be found greater than zero. In particular, for z = x ∈ (−1, 1) we have δp (1) 1 + x 1+x ≥ Re p(x) ≥ · , 1−x 2 1−x where σp is the Riesz–Herglotz measure for p and |σp | = ∂Δ dσp (ζ). |σp |

Now consider the class P + [1, −1] = {p ∈ P : δ+ > 0 and δ− > 0} , where δ+ = δp (1) = ∠ lim (1 − z)p(z), z→1

δ− = δ p1 (−1) = ∠ lim

z→−1

First we note that it follows again from (1.4.2) that δ+ = δp (1) ≤ 2 |σp | = 2 Re p(0) and δ− = δ p1 (−1) ≤ 2 Re

1 . p(0)

1 ≤ 1, we ﬁnd that δ+ · δ− ≤ 4. p(0) In addition, as in the above lemma, we have 1 1 δ− 1 − |z|2 p(z) ≥ Re p(z) ≥ 2 · |1 + z| .

Since Re p(0) · Re

z+1 . p(z)

1 2 δp (1).

1.4. Functions of positive real part

15

1 · Re p(z) ≤ 1 we obtain Again, since p(z) |p(z)| ≤

2 |1 + z|2 · . δ− 1 − |z|2

Thus we can make the following assertion. Theorem 1.15 (see [132]). A function p ∈ P belongs to the class P + [1, −1] if and only if 1 − |z|2 |1 + z|2 ≥ |p(z)| ≥ Re p(z) ≥ a b 1 − |z|2 |1 − z|2 for some positive a and b. In this case the numbers a and b can be chosen to be 2 δ+ and b = a= . Moreover, 2 δ− δ+ · δ− ≤ 4, 1+z for some α > 0. and equality holds if and only if p(z) = αC(z) = α 1−z In particular, if z = x ∈ (−1, 1) we have for p ∈ P + [1, −1]:

2 1+x δ+ 1 + x ≥ Re p(x) ≥ . δ− 1 − x 2 1−x

Chapter 2

Dynamic Approach Dynamic approach to the study of starlike and spirallike functions is based on the following observation. Let f ∈ Univ(Δ), then f is spirallike if and only if there is μ ∈ C with Re μ > 0 such that e−μt f (Δ) ⊂ Δ for all t ≥ 0. Then, for each t ≥ 0, the function Ft (z) = f −1 e−μt f (z)

(2.0.1)

is a well-deﬁned holomorphic self-mapping of the open unit disk Δ. It is easy to verify that the family of these functions {Ft }t≥0 satisﬁes the so-called semigroup property (see Deﬁnition 2.1 below) and if f (τ ) = 0, τ ∈ Δ, then lim Ft (z) = τ for all z ∈ Δ. Moreover, as we will see, lim Ft (z) = τ ∈ ∂Δ if t→∞

t→∞

and only if the angular limit ∠ lim f (z) exists and equals zero. In the latter case z→τ

f is a spirallike (or starlike) function with respect to a boundary point. Therefore, to describe various properties of starlike and spirallike functions (with respect to an interior or to a boundary point), it should be natural to study the asymptotic behavior of one-parameter semigroups of holomorphic selfmappings of Δ.

2.1 Semigroups and generators Let A be either N+ := {0, 1, 2, . . .} or R+ := [0, ∞) and let D be a domain in C. Deﬁnition 2.1. A family S = {Ft }t∈A ⊂ Hol(D) of holomorphic self-mappings of D is called a one-parameter semigroup if (i) Ft+s = Ft ◦ Fs for all s and t in A; (ii) F0 (z) = z for all z ∈ Δ; that is, F0 is the identity mapping on D.

18

Chapter 2. Dynamic Approach • In the case when A = N+ , the family S = {F0 , F1 , F2 , . . . , Fn , . . .}, Fn ∈ Hol(D), is said to be a one-parameter discrete semigroup. • In the case when A = R+ , the family S = {Ft }t≥0 , Ft ∈ Hol(D), is said to be a one-parameter continuous semigroup if lim Ft (z) = Fs (z)

t→s

for all s ≥ 0 and z ∈ D.

If A = N+ , then a one-parameter discrete semigroup S = {Fn }∞ n=0 actually consists of iterates of F = F1 , because of conditions (i) and (ii), i.e., F0 = I, Fn = F ◦ Fn−1 , n = 1, 2, . . .. If A = R+ , then the only right continuity at zero of a semigroup, in fact, implies its continuity (right and left) on all of R+ = [0, ∞). Moreover, in this case the semigroup is diﬀerentiable on R+ with respect to the parameter t ≥ 0. For the one-dimensional case this result is due to Berkson and Porta [17] (see also [130]); for the higher-dimensional case see Abate[3] (see also Reich and Shoikhet [115, 118] for Banach spaces). Theorem 2.1 (see [130], pp. 63, 66). Let D be a simply connected domain in C. Let S = {Ft }t≥0 be a one-parameter semigroup of holomorphic self-mappings of D, such that for each z ∈ D, lim Ft (z) = z.

t→0+

(2.1.1)

Then for each z ∈ D there exists the limit lim

t→0+

z − Ft (z) = f (z), t

(2.1.2)

which is a holomorphic function on D. The convergence in (2.1.2) is uniform on each subset strictly inside D. Moreover, the semigroup S can be deﬁned as a (unique) solution of the Cauchy problem: ⎧ ⎨ ∂Ft (z) + f (F (z)) = 0, t ≥ 0, t ∂t (2.1.3) ⎩ F0 (z) = z, z ∈ D. So, a semigroup satisfying (2.1.1) is, in fact, diﬀerentiable in parameter, hence, continuous. Deﬁnition 2.2. The function f ∈ Hol(D, C) deﬁned by the limit (2.1.2) is called the (inﬁnitesimal) generator of the one-parameter continuous semigroup S = {Ft }t≥0 . Deﬁnition 2.3. A function f ∈ Hol(Δ, C) is called a semi-complete (respectively, complete) vector ﬁeld if for any initial point z ∈ Δ, the Cauchy problem (2.1.3) has the unique solution {Ft (z)} ⊂ Δ deﬁned for all t ≥ 0 (respectively, for all −∞ < t < ∞).

2.1. Flow invariance conditions and parametric representations

19

Thus, Theorem 2.1 asserts that a holomorphic function f on Δ is a semicomplete vector ﬁeld if and only if it is a generator of a one-parameter continuous semigroup of holomorphic self-mappings of Δ. The family of all holomorphic generators on D will be denoted by G(D). If D is convex, this set is a real cone in Hol(D, C) [115, 130]. Diﬀerent descriptions of the class G = G(Δ) (or semi-complete vector ﬁelds on Δ) can be found in [4, 6, 17, 76]. We will formulate some of them in the next section. Meanwhile observe that since a discrete semigroup is completely deﬁned by an arbitrary self-mapping of Δ, its elements are not necessarily univalent functions. At the same time, it follows by the uniqueness of the solution of the Cauchy problem (2.1.3) that each element of a continuous semigroup is a univalent function on Δ. Furthermore, an element of a semigroup S maps Δ onto Δ if and only if the generator f of S (deﬁned by (2.1.2)) is a complete vector ﬁeld. In this case the semigroup S can be extended to a one-parameter group that automatically consists of automorphisms of Δ, so that (Ft )−1 = F−t ,

−∞ < t < ∞.

The set of complete vector ﬁelds on D (that is, the set of group generators) is usually denoted by aut(D) (see, for example, [87], [142], [130]).

2.2 Flow invariance conditions and parametric representations of semigroup generators In this section we make references to some criteria that guarantee that a holomorphic function f ∈ Hol(Δ, C) is a semi-complete vector ﬁeld, or, which is the same, a semigroup generator on Δ. First, we suppose that a function f ∈ Hol(Δ, C) is continuous on the closed disk Δ. The following fact was established in [115] (see also [130] for details). Theorem 2.2 (Boundary ﬂow invariance condition). Let f ∈ Hol(Δ, C) have a continuous extension to Δ. Then f ∈ G if and only if Re f (z)¯ z ≥ 0 for all z ∈ ∂Δ. This theorem gives a simple criterion that makes a holomorphic function, which is continuous on the closed disk, a semigroup generator. However, there are holomorphic functions on the open disk Δ that have no continuous extension to Δ. To describe such generators we need interior ﬂow invariance conditions. Such a criterion was given in Aharonov et al [6]. Theorem 2.3. Let f ∈ Hol(Δ, C). Then f ∈ G if and only if Re f (z)¯ z ≥ Re f (0)¯ z (1 − |z|2 ), z ∈ Δ.

(2.2.1)

20

Chapter 2. Dynamic Approach

Moreover, the equality in (2.2.1) holds if and only if f is a complete vector ﬁeld, i.e., f is the generator of a group of automorphisms. Using this theorem and Harnack’s inequality (1.4.3), one can establish a more qualiﬁed ﬂow invariance condition, which can be considered a distortion theorem for the class G. Theorem 2.4 (see [6] and Proposition 3.5.3 [130]). A function f ∈ Hol(Δ, C) is a semi-complete vector ﬁeld if and only if Re f (0) ≥ 0 and the following inequality holds: 1 + |z| 2 z Re f (0)¯ z (1 − |z|2 ) + Re f (0) |z| ≥ Re f (z)¯ 1 − |z| 1 − |z| 2 ≥ Re f (0)¯ z (1 − |z|2 ) + Re f (0) (2.2.2) |z| 1 + |z| for all z ∈ Δ. Moreover, the equality in (2.2.2) holds if and only if Re f (0) = 0. In this case f is complete. Theorem 2.5 below is an inﬁnitesimal version of the Schwarz–Pick Lemma. To formulate it recall that the (hyperbolic) Poincar´e metric on Δ is a function ρ : Δ × Δ → R+ deﬁned by ρ(z, w) =

1 + |Mz (w)| 1 log , 2 1 − |Mz (w)|

where Mz (w) =

z−w . 1 − z¯w

(2.2.3)

Deﬁnition 2.4. A complex-valued continuous function f on Δ is called ρ-monotone if for each pair z, w ∈ Δ and for r ≥ 0 the following condition holds: ρ(z + rf (z), w + rf (w)) ≥ ρ(z, w),

(2.2.4)

whenever z + rf (z) and w + rf (w) belong to Δ. Theorem 2.5 (see [116]). Let f ∈ Hol(Δ, C). The following assertions are equivalent: (i) f ∈ G; (ii) f is a ρ-monotone function; (iii) the function f satisﬁes the inequality

Re

f (w)w¯ z¯f (w) + wf (z) f (z)¯ z + ≥ Re 2 2 1 − |z| 1 − |w| 1 − z¯w

(2.2.5)

for all z, w ∈ Δ. We have already mentioned that the class G is a real cone. It turns out that this cone is isomorphic to the cone C × P.

2.2. Flow invariance conditions and parametric representations

21

Theorem 2.6 (see [4]). A function f ∈ Hol(Δ, C) belongs to G if and only if it admits the following parametric representation f (z) = a − a ¯z 2 + zq(z),

(2.2.6)

where a ∈ C and q ∈ P. Moreover, f is complete if and only if Re q(z) = 0, i.e., q(z) ≡ ib, b ∈ R. Thus all complete vector ﬁelds are, in fact, polynomials of at most order 2 of the form f (z) = a − a ¯z 2 + ibz, a ∈ C, b ∈ R. Another useful parametric representation of the class G is due to Berkson and Porta [17] (see, also [4] and [130]). Theorem 2.7. A function f ∈ Hol(Δ, C) is of the class G if and only if there are a point τ ∈ Δ and a function p ∈ P such that f (z) = (z − τ )(1 − z τ¯)p(z).

(2.2.7)

Moreover, this representation is unique. It is clear that if τ in (2.2.7) is of modulus less than one, then it is a null point of f ∈ G. Moreover, the Berkson–Porta formula (2.2.7) shows that any function f ∈ G must have at most one null point inside Δ. In this case formula (2.2.7) can be considered a characterization of a subcone of G of generators that vanish at the point τ ∈ Δ: {f ∈ G : f (τ ) = 0} = f ∈ G : f (z) = (z − τ )(1 − z τ¯)p(z), p ∈ P . (2.2.8) If τ ∈ ∂Δ, then f has no interior null point in Δ. However, it turns out that f (τ ) := ∠ lim f (z) = 0, z→τ

i.e., f has at least one boundary null point τ ∈ ∂Δ. It can be shown, by using the Riesz–Herglotz integral representation (1.4.1) and the Berkson–Porta formula (2.2.7), that the angular derivative f (z) f (τ ) = ∠ lim = ∠ lim f (z) z→τ z − τ z→τ exists ﬁnitely and is a real nonnegative number. (The precise formulation of this fact is given below in Theorem 2.10.) Note once again that a function f ∈ G may have more than one boundary null point. For our purposes, we are especially interested in the class G[ζ], ζ ∈ Δ, deﬁned by (2.2.9) G[ζ] := f ∈ G : f (ζ) = 0 and f (ζ) exists ﬁnitely .

22

Chapter 2. Dynamic Approach

Observe that if ζ ∈ Δ, then it must be the unique interior null point of f and f (ζ) exists. Therefore, in this case G[ζ] coincides with the cone deﬁned in (2.2.8) with τ = ζ. In the case when ζ ∈ ∂Δ, one can show that for each f ∈ G[ζ], the angular derivative f (ζ) is a real number. In general, a point ζ ∈ ∂Δ is called a boundary regular null point of f ∈ Hol(Δ, C) if ∠ lim f (z) = 0 and ∠ lim f (z) exists ﬁnitely. z→ζ

z→ζ

Example 2.1. Consider √ two holomorphic functions f1 and f2 deﬁned by f1 (z) = z(1 − z n ) and f2 (z) = z 1 − z n . By Theorem 2.7, f1 and f2 belongs to G.Obvi ously, they have the same null point set 0, zk = e points f2 .

n−1 {zk }k=0

2πik n

, k = 0, 1, . . . , n − 1 . The

are boundary regular null points for f1 while they are irregular for

The following assertion is a continuous analog of the classical Julia–Carath´eodory Theorem (see, for example, [40, 127]). Without loss of generality one can assume that ζ = 1. Theorem 2.8 (see [132], cf. also [62, 64, 34, 59]). Let f ∈ G. The following assertions are equivalent: (i) f ∈ G[1], i.e., η = 1 is a boundary regular null point of f ; (ii) lim sup Re z→1

f (z) =: β > −∞; z−1

(iii) the function f admits the following representation f (z) = −(1 − z)2 q(z) +

β 2 z −1 , 2

(2.2.10)

where β ∈ R, q ∈ P and ∠ lim (z − 1)q(z) = 0. z→1

Remark 2.1. The above theorem with a stronger condition (see Section 2.4 below): f (z) =: β > −∞ z−1 instead of (ii) was proved in [132]. It was shown in [59] that the lower limit can be replaced by the upper limit. Observe also that the equivalence of conditions (ii) and (iii) implies that f (1) must be a real number that equals β. (ii ) lim inf Re z→1

Now comparing formulas (2.2.7) and (2.2.10) we see that τ in (2.2.7) can be equal to 1 if and only if β in (2.2.10) is nonnegative. In this case formula (2.2.7) becomes f (z) = −(1 − z)2 p(z), (2.2.11) and, consequently, representations (2.2.11) with p ∈ P of the form p(z) = q(z) + β 1+z and (2.2.10) are equivalent. 2 1−z

2.3. The Denjoy–Wolﬀ and Julia–Wolﬀ–Carath´eodory Theorems for semigroups23 The subclass of G of the generators for which f (ζ) := ∠ lim f (z) = 0 and β = f (ζ) := ∠ lim f (z) z→ζ

z→ζ

(2.2.12)

with Re β ≥ 0 will be denoted by G + [ζ] ⊂ G[ζ] (obviously, if ζ ∈ Δ, then the cones G + [ζ] and G[ζ] coincide). Finally, following [76] and [132] we introduce the subcone G[1, −1] by the formula (2.2.13) G[1, −1] := G + [1] G[−1]. Corollary 2.1 (see [132]). A function f ∈ Hol(Δ, C) is of the class G[1, −1] if and only if it admits the representation (2.2.10) with β ≥ 0 and q ∈ P such that ∠ lim

z→−1

q(z) z+1

exists ﬁnitely and ∠ lim (z − 1)q(z) = 0. z→1

Another representation of the class G[1, −1] was given by Goryainov in [76] (see Section 2.4 below). Since the classes G[1] and G + [1] play a crucial role in our study of the asymptotic behavior of one-parameter continuous semigroups, as well as in the study of starlike and spirallike functions with respect to a boundary point, we prove Theorem 2.8 simultaneously in Section 2.4. In addition, the class G[1, −1] is useful to describe unbounded starlike and spirallike functions.

2.3 The Denjoy–Wolﬀ and Julia–Wolﬀ–Carath´eodory Theorems for semigroups Recall that for a continuous semigroup S = {Ft }t≥0 , Ft ∈ Hol(Δ), a common ﬁxed point ζ ∈ Δ of its elements: Ft (ζ) = ζ

for all t ≥ 0

is called a stationary point of the semigroup. Thus a continuous semigroup has at most one stationary point in Δ. Moreover, it follows by the semigroup property that ζ ∈ Δ is a stationary point of S = {Ft }t≥0 if and only if Ft0 (ζ) = ζ at least for one t0 > 0. In addition, the uniqueness of the solution of the Cauchy problem (2.1.3) implies that a point ζ ∈ Δ is a stationary point of a continuous semigroup S = {Ft }t≥0 if and only if it is a null point of the semigroup generator f ∈ G. In this case f ∈ G[ζ], i.e., it has the form (2.2.7) with τ = ζ.

24

Chapter 2. Dynamic Approach

Since our main object of interest is starlike and spirallike functions with respect to a boundary point, we turn now to the asymptotic behavior of semigroups with no stationary point in Δ. For continuous semigroups an analog of the Denjoy–Wolﬀ Theorem can be found in [17, 117, 119, 130]. Theorem 2.9 (Denjoy–Wolﬀ Theorem for semigroups). Let S = {Ft }t≥0 ⊂ Hol(Δ) be a continuous semigroup of holomorphic self-mappings on Δ. If for at least one t0 the function Ft0 is not the identity and is not an elliptic automorphism of Δ, then there is a unique point τ ∈ Δ such that the semigroup {Ft }t≥0 converges to τ as t → ∞ uniformly on compact subsets of Δ. The point τ in Theorem 2.9 is called the Denjoy–Wolﬀ point of a continuous one-parameter semigroup S = {Ft }t≥0 . If τ ∈ ∂Δ, it is also variously called Wolﬀ ’s point or the sink point of a semigroup. The asymptotic behavior of continuous semigroups can be also described in terms of their generators. A continuous version of the Julia–Wolﬀ–Carath´eodory Theorem was established in [62]. Theorem 2.10. Let S = {Ft }t≥0 be a one-parameter continuous semigroup generated by f ∈ G. The following are equivalent: (i) f has no null point in Δ; (ii) f admits the Berkson–Porta representation f (z) = (1 − z τ¯)(z − τ )p(z) for some τ ∈ ∂Δ and Re p(z) ≥ 0 everywhere; (iii) there is a point τ ∈ ∂Δ such that f ∈ G + [τ ]; (iv) there is a point τ ∈ ∂Δ such that ∠ lim

z→τ

f (z) =β z−τ

exists and Re β ≥ 0; (v) there are a point τ ∈ ∂Δ and a real positive number γ, such that 2 |Ft (z) − τ |2 −tγ |z − τ | ≤ e . 1 − |Ft (z)|2 1 − |z|2

(2.3.1)

Moreover, (a) the points τ ∈ ∂Δ in (ii)–(v) are the same; (b) the number β in (iv) is, in fact, a nonnegative real number, which is the maximum of all γ ≥ 0 that satisfy (2.3.1).

2.4. Generators with boundary null points

25

Remark 2.2. Note that if f ∈ G[ζ], ζ ∈ ∂Δ, with f (ζ) = β ∈ R (not necessarily nonnegative), then the point ζ is also a common ﬁxed point for the semigroup S = ∂Ft {Ft }t≥0 generated by f . In addition, (ζ) = e−tβ and inequality (2.3.1) holds ∂z even if β < 0 (see [132, 64]). In particular, for the class G[1, −1] = G + [1] ∩ G[−1] we have the following estimates |z − 1|2 |Ft (z) − 1|2 ≤ e−tβ+ 2 1 − |Ft (z)| 1 − |z|2 and

|z + 1|2 |Ft (z) + 1|2 ≤ e−tβ− , 2 1 − |Ft (z)| 1 − |z|2

where

β+ = f (1) ≥ 0

and β− = f (−1) < 0.

We prove these facts (including Theorem 2.10), which are key points in our discussion in the next section. Here we just observe that in contrast to the case of interior null points, a boundary null point ζ of a generator f may be not a ﬁxed point of the generated semigroup if f (ζ) does not exist ﬁnitely (see Example 1, p. 104 in [130]).

2.4 Generators with boundary null points Our purpose in this section is an advanced study of the classes G[ζ], G + [ζ] and G[1, −1] deﬁned in Section 2.2. • By F[ζ] we denote the class of holomorphic self-mappings of Δ with a boundary regular ﬁxed point ζ ∈ ∂Δ, i.e., for each F of this class F (ζ) := (z)−ζ is ﬁnite. lim− F (rζ) = ζ and the angular derivative F (ζ) := ∠ lim F z−ζ z→ζ

r→1

The class of holomorphic self-mappings of Δ with the boundary Denjoy–Wolﬀ point τ ∈ ∂Δ (see Section 2.3) will be denoted by F + [τ ]. So, by Theorem 2.10 f ∈ G + [τ ] if and only if the semigroup S = {Ft }t≥0 generated by f belongs to the class F + [τ ], i.e., ﬁxes τ ∈ ∂Δ and this point is the Denjoy–Wolﬀ point for {Ft }t≥0 . In this case f has no null point in Δ, but may have other null points on the boundary ∂Δ. Without loss of generality we can set τ = 1. Note that the class G + [1] can be also described using the Berkson–Porta representation (2.2.7). Namely, f ∈ G + [1] if and only if it has the form: f (z) = −(1 − z)2 p(z), with p ∈ P.

(2.4.1)

26

Chapter 2. Dynamic Approach f (z) z→1 z−1

If for f ∈ G + [1] the angular derivative f (1) = ∠ lim

= 0 (or what is the

same due to (2.4.1), ∠ lim (1 − z)p(z) = 0), then f is said to be a parabolic type z→1

generator. The subcone of G + [1] of parabolic type generators will be denoted by Gp [1], i.e., f (z) =0 . Gp [1] = f ∈ G : f (1) = ∠ lim z→1 z − 1 If for f ∈ G + [1] the angular derivative f (1) is positive, then f is said to be a hyperbolic type generator. The subcone of G + [1] of hyperbolic type generators will be denoted by Gh [1], i.e., f (z) >0 . Gh [1] = f ∈ G : f (1) = ∠ lim z→1 z − 1 So, G + [1] = Gp [1] ∪ Gh [1]. A detailed study of semigroups generated by elements of classes Gp [1] and Gh [1] can be found in Sections 5 and 7 below. Goryainov in [76] considered an important class F[1, −1] = F + [1] ∩ F[−1] of self-mappings of Δ, which ﬁxes two boundary points 1 and −1 on ∂Δ in the sense that (2.4.2) ∠ lim F (z) = ±1, z→±1

and such that τ = 1 is the Denjoy–Wolﬀ point for F . The corresponding class of all generators of semigroups in F [1, −1] is denoted by GF [1, −1]. A result in [76] asserts that f ∈ GF [1, −1] if and only if it admits the representation 1 + h(z) , (2.4.3) f (z) = α(1 + z)(1 − z)2 zh(z) − 1 in which α ≥ 0 and h ∈ Hol(Δ) or h is a constant of modulus less or equal to 1. However, it is not clear whether the classes GF [1, −1] and G[1, −1] = G + [1] ∩ G[−1] (see (2.2.13)) coincide. The answer should be aﬃrmative if we show that f ∈ G[ζ] if and only if the semigroup S = {Ft }t≥0 generated by f belongs to F [ζ] for a boundary point ζ ∈ ∂Δ. Obviously, it is possible to use conjugation of a semigroup S ⊂ F[1, −1] by a M¨ obius transformation of the unit disk Δ onto itself to get a characterization of the class GF [−1, 1] of all generators f on Δ, such that the semigroup S generated by f belongs to the class F[−1, 1]. In other words, it ﬁxes −1 and 1 and such that τ = −1 is the Denjoy–Wolﬀ point for S. It is clear that GF [1, −1] ∩ GF [−1, 1] = {0} and for each pair f+ ∈ GF [1, −1] and f− ∈ GF [−1, 1] the sum f = f+ + f− ∈ GF [1, −1] ∪ GF [−1, 1]. Then the question is how to recognize to which class GF [1, −1] or GF [−1, 1] this element f belongs. Also note that this union contains the subclass Gaut [±1] = f (z) = α z 2 − 1 , α ∈ R

2.4. Generators with boundary null points

27

of all (quadratic) generators that vanish at the points 1 and −1. In fact, an element f ∈ Gaut [±1] if and only if it generates a group of hyperbolic automorphisms which ﬁxes these points. It is clear that all mentioned above classes belong to G[1]. So, the question is: ♦ Whether each element f ∈ G[1] generates a semigroup of the class F [1]? Note that, in general, the mere fact that f ∈ G vanishes at a boundary point ζ ∈ ∂Δ does not imply that ζ is necessarily a ﬁxed point of the semigroup S = {Ft }t≥0 ⊂ Hol(Δ) generated √ by f . Indeed, Example 5.1 in [115] shows that the function f (z) = z − 1 + 1 − z is of the class G and clearly vanishes at the boundary point ζ = 1. However, all elements of the semigroup S = {Ft }t≥0 generated by f are strictly less than 1 at this point, i.e., Ft (1) < 1 for all t ≥ 0. The point is that the angular derivative of this generator at τ = 1 does not exist ﬁnitely, so f does not belong to G[1]. We ﬁrst present a theorem for decomposition of the class G[1] by its subclasses Gaut [±1] and Gp [1] (cf., Theorem 2.8). Namely, we show that G[1] = Gp [1] ⊕ Gaut [±1]. As a consequence we derive at the fact that f ∈ G[1] if and only if S = {Ft }t≥0 generated by f belongs to F[1]. Theorem 2.11. A function f ∈ G[1] if and only if it admits the representation: f (z) = −(1 − z)2 p(z) + where p ∈ P and

β 2 (z − 1), 2

∠ lim (1 − z)p(z) = 0, z→1

(2.4.4)

(2.4.5)

and β ∈ R = (−∞, ∞). Moreover, τ = 1 is the Denjoy–Wolﬀ point of the semigroup S = {Ft }t≥0 generated by f if and only if β ≥ 0. In this case f has no null point in Δ. Proof. Suﬃciency. Let f ∈ Hol(Δ, C) admit representation (2.4.4). Then, the ﬁrst term f1 (z) = −(1 − z)2 p(z) is an element of G[1] due to the Berkson–Porta representation (see formulas (2.2.7) and (2.4.1)). The second term f2 (z) =

β 2 (z − 1) 2

is the generator of a group of automorphisms on Δ (see Theorem 2.6). Since G, hence G[1], is a real cone in Hol(Δ, C), the function f = f1 + f2 belongs to G[1].

28

Chapter 2. Dynamic Approach Note that due to (2.4.4) and (2.4.5) we have f (z) = ∠ lim f (z) = β. ∠ lim z→1 z − 1 z→1

Hence, z = 1 is the Denjoy–Wolﬀ point of S = {Ft }t≥0 if and only if β ≥ 0 by Theorem 2.10. Necessity. Let f ∈ G[1]. Observe, that by Theorem 2.5 each element of the class G satisﬁes the inequality

f (w)w¯ f (z)¯ z z¯f (w) + wf (z) + Re ≥ Re (2.4.6) 1 − |z|2 1 − |w|2 1 − z¯w for all z, w ∈ Δ. Now rewriting (2.4.6) in the form z¯ w ¯ w ¯ z¯ ≥ Re f (w) − − Re f (z) 1 − zw ¯ 1 − z¯w 1 − |w|2 1 − |z|2 and setting w = r ∈ (0, 1) we have f (r) r − z¯ r z¯ ≥ Re · . − Re f (z) 1 − |z|2 1 − zr r − 1 (1 − r¯ z )(1 + r) Now letting r to 1− and taking into account that lim f (r) = 0 and r→1−

lim−

r→1

we get

Re f (z)

f (r) = f (1), r−1

z¯ 1 − 1 − |z|2 1−z

≥ Re

f (1) , 2

(2.4.7)

or (after some manipulations): − Re

f (z) 1 1 − |z|2 Re f ≥ (1) . (z − 1)2 2 |1 − z|2

(2.4.8)

Setting q(z) = − and p(z) = q(z) −

f (z) (1 − z)2

1+z Re f (1), 2(1 − z)

(2.4.9)

(2.4.10)

we obtain from (2.4.8)–(2.4.10)

11+z Re f (1) f (z) = −(1 − z)2 p(z) + 21−z

(2.4.11)

2.4. Generators with boundary null points

29

with 1+z 1 Re f (1) · Re ≥ 2 1−z

1 − |z|2 1+z 1 − Re = 0. ≥ Re f (1) 2 |1 − z|2 1−z

Re p(z) = Re q(z) −

It is clear that (2.4.11) is equivalent to (2.4.4) with β = Re f (1). Now we want to show that f (1) is, in fact, a real number, or what is the same, that f (1) = β. To this end, we observe that by the Riesz–Herglotz formula (1.4.2) for each p ∈ P the limit δp (1) = ∠ lim (1 − z)p(z) z→1

exists and is a nonnegative real number (see, for example, [130]). Therefore, f (1) = ∠ lim

z→1

f (z) = β (= Re f (1)) z−1

by (2.4.4). This immediately implies (2.4.5). The theorem is proved.

Remark 2.3. It can be seen easily from the proof of Theorem 2.11 that actually the f (z) can be replaced by a formally weaker the existence of the limit β = ∠ lim z→1 z − 1 condition: f (r) lim inf > −∞. (2.4.12) Re r−1 r→1− Indeed, repeating the necessary part of the proof of Theorem 2.11, we see that f (r) . In turn, this (2.4.12) implies representation (2.4.4) with β = lim inf Re r−1 r→1− representation and the Riesz–Herglotz formula (1.4.2) applied to p(z) show that, in fact,

f (r) β lim− (1 − r)p(r) = lim− − (r + 1) r−1 2 r→1 r→1 exists and equals zero (cf., [62]). Finally, using Harnack’s inequality (1.4.3) for the function p, we get that the function h(z) = (1 − z)p(z) is bounded on each non-tangential approach region at the point z = 1. Thus, condition (2.4.5) follows from Lindel¨ of’s Theorem 1.6, and we are done. This observation leads to the following inﬁnitesimal version of the Julia– Carath´eodory Theorem.

30

Chapter 2. Dynamic Approach

Theorem 2.12. Let f ∈ G and let S = {Ft }t≥0 be the semigroup generated by f . Then f ∈ G[1] if and only if S ∈ F[1]. Moreover, if β = f (1) := ∠ lim f (z),

(2.4.13)

(Ft ) (1) := lim (Ft ) (z) = e−tβ .

(2.4.14)

z→1

then for each t ≥ 0

z→1

Proof. Suﬃciency. Let S = {Ft }t≥0 be the semigroup generated by f ∈ G. Suppose that S ∈ F[1], that is, for each t ≥ 0 there exists the angular limit Ft (z) − 1 < ∞. (2.4.15) αt := (Ft ) (1) = ∠ lim t→1 z−1 Then it follows again by the classical Julia–Carath´eodory Theorem (see [40, 127]) that |1 − Ft (z)|2 |1 − z|2 ≤ α . t 1 − |Ft (z)|2 1 − |z|2 Diﬀerentiating this inequality at t = 0+ and using the semigroup property we obtain 1 − αt = β > −∞, lim inf + t t→0 and

Re f (z)

z¯ 1 − 2 1 − |z| 1−z

≥ β ≥ 2.

Letting z = r and r tend to 1− , we get the inequality lim inf − r→1

β f (r) ≥ , r−1 2

which is equivalent (2.4.12). Remark 2.3 now implies the required inclusion: f ∈ G[1]. Necessity. Suppose that f ∈ G[1]. First we show that z = 1 is a common ﬁxed point for the semigroup S = {Ft }t≥0 . Indeed, if β = f (1), then f = f1 + f2 , where f1 (z) = −(1 − z)2 p(z) and f2 (z) = β2 (z 2 − 1) are elements of G[1]. (1)

In addition, as we mentioned above, if S1 = {Ft }t≥0 is the semigroup generated by f1 , then z = 1 is the Denjoy–Wolﬀ point of this semigroup; hence, (1)

ϕ(Ft (z)) ≤ ϕ(z),

(2.4.16)

2.4. Generators with boundary null points where ϕ(z) =

|1 − z|

31

2

1 − |z|2

.

(2)

Also, if S2 = {Ft }t≥0 is the semigroup (actually, group) generated by f2 , it follows by direct calculations that (2) ϕ Ft (z) ≤ e−βt ϕ(z). (2.4.17)

that

Now it follows by the semigroup product formula (see, for example, [118]) n (2) (1) Ft = lim F t ◦ F t , (2.4.18) n→∞

n

n

where (F ) denotes the n-fold iterate of a mapping F : Δ → Δ. Then, applying (2.4.16), (2.4.17) and (2.4.18), by induction we get: n

ϕ(Ft (z)) ≤ e−tβ ϕ(z), or, explicitly,

|1 − z|2 |1 − Ft (z)|2 ≤ e−tβ . 2 1 − |Ft (z)| 1 − |z|2

The latter inequality immediately implies that for each t ≥ 0 lim Ft (r) = 1,

r→1−

or, what is the same, because of the boundedness of Ft , ∠ lim Ft (z) = 1.

(2.4.19)

z→1

It remains to show that Ft (z) − 1 = e−tβ . z→1 z−1

(Ft ) (1) = ∠ lim

(2.4.20)

To this end we note that it follows by representation (2.4.4) and the Cauchy problem (2.1.3) that β ∂Ft (z) = (1 − Ft (z))2 p(Ft (z)) − ((Ft (z))2 − 1). ∂t 2 This implies that Ft (z) − 1 log =− z−1

0

t

β (1 − Fs (z)p(Fs (z))ds − 2

0

t

(Fs (z) + 1)ds.

Taking into account (2.4.5) we obtain (2.4.20). The theorem is proved.

32

Chapter 2. Dynamic Approach

The classes G[1], Gh [1] and G[1, −1] play a crucial role in the study of spirallike and starlike functions with respect to a boundary point. Another useful characterization of these classes can be given as follows. Theorem 2.13 (cf. [48]). Let f ∈ Gh [1], i.e., f ∈ G[1] with f (1) = ∠ lim f (z) = z→1

β+ > 0. Then for each c ∈ 0, β2+ there is a unique F = Fc ∈ F + [1] with ∠ lim F (z) =: F (1) = z→1

2c ≤ 1, β+

and such that f (z) = −c(1 − z)2

1 + F (z) . 1 − F (z)

(2.4.21)

Moreover, if f ∈ G[1, −1] ∩ Gh [1] with β− = ∠ lim f (z), then F ∈ F[1, −1], with z→1

∠ lim F (z) = − z→−1

β− . 2c

Conversely, if F ∈ F + [1] (respectively, F ∈ F[1, −1]), then for each c > 0 the function f deﬁned by (2.4.21) belongs to Gh [1] (respectively, f belongs to G[1, −1]∩ Gh [1], i.e., ∠ lim f (z) = 0 with f (1) > 0 and f (−1) < 0). z→±1

Proof. Let f ∈ Gh [1] ⊂ G + [1]. Then one can present f in the form f (z) = −(1 − z)2 p(z) with Re p(z) ≥ 0. For each c > 0 there is a unique holomorphic function F = Fc on Δ such that |F (z)| < 1, z ∈ Δ, and p(z) = c

1 + F (z) . 1 − F (z)

(2.4.22)

Furthermore, since β+ = f (1) = ∠ lim

z→1

we get β+ = c lim

r→1−

f (z) > 0, z−1

1−r (1 + F (r)) . 1 − F (r)

This relation shows that lim F (r) = 1

r→1−

and lim

r→1−

1−r β+ = > 0. 1 − F (r) 2c

(2.4.23)

2.4. Generators with boundary null points

33

Now it follows by the Julia–Carath´eodory Theorem that ∠ lim F (z) = ∠ lim z→1

z→1

1 − F (z) 2c = =: α+ > 0. 1−z β+

If c ∈ 0, β2+ , then α+ ≤ 1 and it follows by Wolﬀ’s Lemma (see, for example, [40] and [127]) that ζ = 1 is the Denjoy–Wolﬀ point for F . Now let f ∈ G[1, −1] and set β− = f (−1) = ∠ lim f (z). z→−1

Then ∠ lim

z→−1

1 + F (z) 1 f (z) = β− = ∠ lim −(1 − z)2 c z→−1 z+1 1 − F (z) 1 + z 1 + F (z) 1 = −4c∠ lim · ∠ lim . z→−1 z→−1 1 − F (z) 1+z

(2.4.24)

Again it follows by the Julia–Carath´eodory Theorem and (2.4.24) that ∠ lim

z→−1

1 + F (z) =: α− 1+z

exists ﬁnitely and 1 β− . 2c Reverse considerations complete our proof. α− = −

Corollary 2.2. Let f ∈ G[1, −1] with f (1) = β+ ≥ 0 and f (−1) = β− . Then −β− ≥ β+ . Moreover, the equality −β− = β+ is possible if and only if f is a generator of a group of hyperbolic automorphisms of Δ, i.e., f (z) =

β+ 2 (z − 1). 2

Proof. The assertion is obvious if β+ = 0 since β− < 0. Let us assume that β+ > 0 and −β− ∈ (0, β+ ). Then the function f1 deﬁned by f1 (z) = f (z) +

β− 2 (z − 1) 2

belongs to the class G[1, −1], because this class is a real cone. In addition, we have f1 (1) = β+ + β− > 0 while

f1 (−1) = β− − β− = 0.

34

Chapter 2. Dynamic Approach

Then f1 = 0 and both points 1 and −1 are Denjoy–Wolﬀ points of the semigroup generated by f . This contradiction implies that β+ ≤ −β− . Moreover, the same considerations show that β+ = −β− if and only if f1 (z) ≡ 0. Hence, f (z) has the form β+ 2 β− (1 − z 2 ) = (z − 1). (2.4.25) f (z) = 2 2 Thus, f belongs to Gaut [±1]. This complete our proof. Corollary 2.3 (cf., [76]). If F ∈ F[1, −1] then F (1) · F (−1) ≥ 1.

(2.4.26)

Furthermore, the equality F (1) · F (−1) = 1 is possible if and only if F is a hyperbolic automorphism of Δ. Proof. Indeed, if F ∈ F[1, −1], then it follows by Theorem 2.13 that f deﬁned by (2.4.21) belongs to G[1, −1]. In addition, (2.4.26) and (2.4.21) imply that F (1) · F (−1) = α+ · α− = −

2c β− β− · ≥1 =− β+ 2c β+

(see the previous corollary). If α+ · α− = 1, then −β− = β+ and by the same corollary we have f (z) = −c(1 − z)2

1 + F (z) β+ 2 = (z − 1). 1 − F (z) 2

Hereby, F must be of the form F (z) =

(z + 1) − α+ (1 − z) , (z + 1) + α+ (1 − z)

where α+ =

2c . β+

The assertion is proved.

2.5 Univalent functions and semi-complete vector ﬁelds To complete the auxiliary part of the work we now state some relations between starlike (spirallike) functions and semigroup generators. First we observe that each μ-spirallike (starlike, if μ is real) function h on D determines a family {Ft }t≥0 of holomorphic self-mappings of D deﬁned by Ft (z) = h−1 e−μt h(z) , Re μ > 0. Obviously this family forms a one-parameter continuous semigroup (see Fig. 2.1). Diﬀerentiating this equality at t = 0+ we obtain that μh(z) = h (z)f (z),

(2.5.1)

where f is the generator of the semigroup S = {Ft }t≥0 . As a matter of fact, it turns out that equation (2.5.1) is a criterion for a univalent function f to be spirallike. More precisely:

2.5. Univalent functions and semi-complete vector ﬁelds

35

(i)

(ii) 2

1

1.5 y 1

0.5

0.5 –1 –3

–2

–1

0

1

–0.5

0

0.5

2

1 x

–0.5 –0.5 –1

–1.5

–1

Figure 2.1: A semigroup deﬁned by a starlike function Theorem 2.14 (see [56, 54, 55]). Let a holomorphic function h be univalent on a simply connected domain D. Then h is a spirallike function if and only if there exist a semigroup generator f ∈ G(D) and a complex number μ with Re μ > 0 such that μh(z) = h (z)f (z),

z ∈ Δ.

(2.5.2)

In the case where μ is a real positive number, h is, actually, starlike. Corollary 2.4. Let D = Δ = {z ∈ C : |z| < 1} and f ∈ G. Assume that for some μ, Re μ > 0, equation (2.5.2) is fulﬁlled with some h ∈ Univ(Δ). The following assertions hold. (a) If f has an interior null point τ , then h is a spirallike (starlike) function with respect to an interior point with h(τ ) = 0. Moreover, h is starlike (spirallike) on each pseudohyperbolic disk Dr (τ ) ⊂ Δ centered at τ . (b) If f has no interior null point, then there exists a boundary point τ such that ∠ lim h(z) = ∠ lim f (z) = 0. z→τ

z→τ

In this case h is a spirallike (starlike) function with respect to a boundary point. Moreover, h is starlike (spirallike) on each horodisk Δr (τ ) ⊂ Δ internally tangent to ∂Δ at τ . Using these facts and the Berkson–Porta representation (2.2.7) of generators, one can easily obtain the classical Nevanlinna condition (1.1.1), as well as Wald’s ˇ cek’s condition (1.1.7) (see details in [58] and [130]). Theorem 1.2 and Spaˇ In general, a connection between univalent functions and semigroup generators is based on the notion of ϕ-like domains introduced by Brickman in [22] (see Section 1.1.5).

36

Chapter 2. Dynamic Approach

Deﬁnition 2.5. Let Ω ⊂ C be a domain containing w = 0, and let ϕ ∈ Hol(Ω, C) with ϕ(0) = 0 and Re ϕ (0) > 0. Then Ω is a ϕ-like domain if for any w ∈ Ω the Cauchy problem ⎧ ⎨ ∂v(t, w) + ϕ(v(t, w)) = 0, ∂t (2.5.3) ⎩ v(0, w) = w has a solution v(t, w) deﬁned for all t ≥ 0 such that v(t, w) ∈ Ω for all t ≥ 0. Theorem 2.15 ([22]). Let h ∈ Univ(Δ) satisfy h(0) = 0. Then the image h(Δ) is zh (z) > 0 for all z ∈ Δ. a ϕ-like domain if and only if h is ϕ-like, i.e., Re ϕ(h(z)) Remark 2.4. Note that the solution v = v(t, ·) of the Cauchy problem (2.5.3) is a semigroup of holomorphic self-mappings of Ω. Condition Re ϕ (0) > 0, in fact, means that w = 0 is an attractive common ﬁxed point of this semigroup. Furthermore, if Ω = h(Δ) for some h ∈ Univ(Δ), then as in Lemma 3.7.1 in [58], it easy to verify that the function f (z) =

ϕ(h(z)) h (z)

is the generator of the semigroup {u(t, ·)}t≥0 ⊂ Hol(Δ), deﬁned by u(t, z) = h−1 (v(t, h(z))) . Therefore, Deﬁnition 2.5 with Ω = f (Δ) can service a generalized notion of a ϕ-like function h even if h(0) = 0. To generalize the notion of starlike (or spirallike) function with respect to a boundary point, it would be preferable to omit the restriction 0 ∈ Ω = h(Δ). Namely, • Let h ∈ Univ(Δ) and let ϕ ∈ Hol(h(Δ), C) satisfy either (i) ϕ(0) = 0 and Re ϕ (0) > 0 when τ ∈ Δ, or (ii) ∠ lim ϕ(h(z)) = 0 and ∠ lim Re ϕ (h(z)) > 0 when τ ∈ ∂Δ. z→τ

z→τ

A function h with h(τ ) = 0 for τ ∈ Δ, is called a generalized ϕ-like function if for any z ∈ Δ the Cauchy problem ⎧ ⎪ ⎨ ∂u(t, z) + ϕ (h(u(t, z))) = 0, ∂t h (u(t, z)) (2.5.4) ⎪ ⎩ u(0, z) = z has a solution u(t, z) deﬁned for all t ≥ 0 such that {u(t, z), t ≥ 0} ⊂ Δ for all initial values z ∈ Δ.

2.5. Univalent functions and semi-complete vector ﬁelds

37

Substituting in this deﬁnition ϕ(w) = w (or, more generally, ϕ(w) = μw with Re μ > 0), we get the class of starlike (or spirallike) functions on Δ. Note that for the case h(0) = 0 our deﬁnition of generalized ϕ-like functions is equivalent to those given in [22]. At the same time, if h has no null point in Δ, setting ϕ(w) = w we obtain another class of univalent functions that are starlike with respect to a boundary point. We study this class in the next chapter.

Chapter 3

Starlike Functions with Respect to a Boundary Point Although starlike functions normalized by the condition h(0) = 0 have been studied intensively during the last century, until 1981 a few works only dealt with starlike functions with respect to a boundary point. The ﬁrst observation related to such functions was most likely that of Egerv` ary [46] who studied the mapping properties of the Ces` aro means of the partial sums of the geometric series z/(1 − z) = z + z 2 + z 3 + . . . + z n + . . .. He showed that the Ces` aro means of the ﬁrst order are starlike with respect to a boundary point. An intensive study of the class of starlike functions with respect to a boundary point was initiated by Robertson in his paper [123].

3.1 Robertson’s classes. Robertson’s conjecture Robertson [123] introduced the following classes of functions: Deﬁnition 3.1. Let the class G∗ consist of the constant function h(z) ≡ 1 and all functions h ∈ Univ(Δ) starlike with respect to a boundary point and normalized by h(0) = 1 and h(1) = lim− h(r) = 0. In addition, assume that for any h ∈ G∗ r→1

there is a real α such that Re eiα h(z) > 0,

z ∈ Δ.

Thus, G∗ \ {h(z) ≡ 1} is the subclass of S ∗ [1] consisting of the functions h that take values h(z), z ∈ Δ, in a half-plane that contains h(0) = 1 (see Fig. 3.1). Deﬁnition 3.2. Let G denote the class of functions h(z) = 1 + d1 z + d2 z 2 + . . . + dn z n + . . . ,

40

Chapter 3. Starlike Functions with Respect to a Boundary Point

2

y 1

–1

0

1

2

3

4

x

–1

–2

Figure 3.1: A function of Robertson’s class. holomorphic and non-vanishing in Δ, which satisfy

2zh (z) 1 + z Re + >0 h(z) 1−z

(3.1.1)

for all z ∈ Δ. Robertson [123] conjectured that G = G∗ .

(3.1.2)

Actually, he partially proved this relation. Namely, Robertson showed that G∗ ⊂ G ⊂ G∗ , where G∗ is the subset of G∗ consisting of functions holomorphic on the closed disk Δ. In [102] Lyzzaik completed the proof of Robertson’s conjecture (see also Section 3.3) and showed that (3.1.2) holds without any additional restriction. Note also that Robertson–Lyzzaik’s result means that inequality (3.1.1) is a characterization of those functions from S ∗ [1] that are equal to 1 at zero and have images in a wedge of angle π with its vertex at the origin. Furthermore, we will see below that the smallest wedge containing h(Δ) is exactly of angle π if and only if h satisﬁes the Visser–Ostrowski condition (see Section 1.2). In the same paper Robertson proved that if h ∈ G, h ≡ 1, then log h is a closeto-convex function relative to the starlike function −k(z), where k is the Koebe z function k(z) = (1−z) 2 . In other words, h ∈ G, h ≡ 1, satisﬁes the inequality:

h (z) < 0, Re (1 − z)2 h(z)

z ∈ Δ.

(3.1.3)

As a matter of fact, we will show below that a function h ∈ Univ(Δ) with h(1) = 0 is starlike with respect to a boundary point if and only if condition

3.2. Auxiliary lemmas

41

(3.1.3) holds. Moreover, if, in addition, h satisﬁes a generalized Visser–Ostrowski condition (z − 1)h (z) ∠ lim = μ, 0 < μ ≤ 1, z→1 h(z) then inequality (3.1.1) is equivalent to (3.1.3). To prove these statements as well as Robertson’s conjecture (3.1.2), we need some auxiliary assertions.

3.2 Auxiliary lemmas Following Silverman and Silvia [135] we deﬁne the following classes of functions. Deﬁnition 3.3. Let λ ∈ (0, 2]. Denote by G(λ)1 the class of functions h(z) = 1 + d1 z + d2 z 2 + . . . + dn z n + . . . , holomorphic and non-vanishing in Δ, which satisfy

2zh (z) 1 + z + >0 Re λh(z) 1−z

(3.2.1)

for all z ∈ Δ. In particular, the class G(1) is Robertson’s class G (see Deﬁnition 3.2). The ﬁrst result gives an integral representation for the class G(λ), which itself is interesting. Lemma 3.1 (see [56], cf. [135]). Let h ∈ Hol(Δ, C). Then h ∈ G(λ) if and only if it admits the following representation: λ ¯ h(z) = (1 − z) exp −λ log(1 − z ζ)dσ(ζ) (3.2.2) |ζ|=1

with some probability measure σ on the unit circle |ζ| = 1. Proof. First, suppose that a function h ∈ Hol(Δ, C) is represented by (3.2.2). We see at once that h(0) = 1. By simple calculation we get 2zh (z) 1 + z + =1+ λh(z) 1−z Since Re

|ζ|=1

2z ζ¯ dσ(ζ). 1 − z ζ¯

z ζ¯ 1 > − , we obtain inequality (3.2.1). ¯ 2 1 − zζ

1 The classes G , α ∈ [0, 1), originally introduced in [135], coincide with the classes G(λ) for α λ = 2(1 − α).

42

Chapter 3. Starlike Functions with Respect to a Boundary Point

2zh (z) 1 + z + , λh(z) 1−z which evidently belongs to the Carath´eodory class P0 and thus can be represented by the Riesz–Herglotz formula (1.4.1): Suppose now that h ∈ G(λ). Consider the function p(z) =

p(z) = |ζ|=1

1 + z ζ¯ dσ(ζ) 1 − z ζ¯

with a probability measure σ. This implies that h (z) = λh(z)

|ζ|=1

1 ζ¯ dσ(ζ). − 1 − z ζ¯ 1 − z

Integrating both sides of this equality we obtain (3.2.2)

(3.2.3)

Corollary 3.1. Let h ∈ Hol(Δ, C) and λ1 , λ2 ∈ (0, 2]. Then h ∈ G(λ1 ) if and only λ2

if the function h λ1 belongs to G(λ2 ). Lemma 3.2. Let h ∈ Hol(Δ, C) with h(0) = 1. Then h ∈ G(λ) if and only if there is a function h1 ∈ Sκ∗ , h1 (0) = 0, starlike of order κ = 1 − λ2 , such that h(z) =

(1 − z)λ h1 (z) . z

(3.2.4)

In particular, h ∈ G if and only if the function h1 (z) :=

zh(z) 1−z

∗ belongs to S1/2 .

Proof. Let h1 ∈ Sκ∗ be starlike of order κ ∈ [0, 1). It follows by the deﬁnition of the zh1 (z) −λ h (z) ∗ is of Carath´eodory’s class Sκ (see formula (1.1.2)) that the function 1 1−λ class P0 . Using the Riesz–Herglotz formula (1.4.1) we get the following representation for h1 : ¯ log(1 − z ζ)dσ(ζ) . (3.2.5) h1 (z) = z exp −λ |ζ|=1

Let h and h1 be connected by (3.2.4) and set λ = 2 − 2κ. Then by (3.2.2) and (3.2.5) we obtain our assertion. Lemma 3.3. Let 0 < λ1 < λ2 ≤ 2. Then G(λ1 ) ⊂ G(λ2 ).

3.2. Auxiliary lemmas

43

Proof. Let h ∈ G(λ1 ). We have h(z) = (1 − z)

λ1

exp −λ1

|ζ|=1

¯ log(1 − z ζ)dσ(ζ)

= (1 − z)λ2 exp −λ2

|ζ|=1

¯ σ(ζ) , log(1 − z ζ)d

where the probability measure σ on the unit circle is deﬁned by σ=

λ1 λ2 − λ1 σ+ δ λ2 λ2

(δ is the Dirac measure at the point ζ = 1). Then the assertion follows by the integral representation (3.2.2). Lemma 3.4. The set

⎧ λj ∞ ⎨" n ! 1−z : ⎩ 1 − z ζ¯j n=1 j=1

n

λj = λ, |ζj | = 1

j=1

⎫ ⎬ ⎭

is dense in G(λ) in the topology of uniform convergence on compact subsets of Δ. Proof. Replacing the integral in (3.2.2) by integral sums we have for any h ∈ G(λ): ⎡ ⎤ n h(z) = lim (1 − z)λ exp ⎣−λ log(1 − z ζ¯j )σj ⎦ n→∞

j=1

= lim (1 − z)λ n→∞

with

n *

n "

(1 − z ζ¯j )−λσj ,

j=1

σj = 1. Writing λj = λσj we complete the proof.

j=1

Lemma 3.5. Let h ∈ G(λ), h ≡ 1. Then the function F (z) := − log h(z), F (0) = 0, z is close-to-convex relative to the Koebe function k(z) = (1−z) 2 , i.e.,

2h

(z) Re (1 − z) h(z)

z(− log h(z)) = − Re < 0, k(z)

z ∈ Δ.

Proof. Using Lemma 3.1 (or formula (3.2.3) from its proof) we see that 1 − ζ¯ h (z) =λ dσ(ζ). (1 − z) −(1 − z)2 h(z) 1 − z ζ¯ ∂Δ Since the integrand takes values in the right half-plane, we have the required assertion.

44

Chapter 3. Starlike Functions with Respect to a Boundary Point The following result was originally formulated in [135] (see also [56]).

Lemma 3.6. Let h ∈ G(λ) and h(z) ≡ 1. Then the function 1−h is close-to-convex, hence h is univalent in Δ. Proof. Let h ∈ G(λ) and h(z) ≡ 1. Then h ∈ G(2) by Lemma 3.3. Now Lemma 3.2 zh(z) ∗ ∗ implies that the function h1 (z) = (1−z) 2 belongs to the class S0 = S . In turn, by Lemma 3.5 we have Re

h (z) z(1 − h(z)) = − Re(1 − z)2 >0 h1 (z) h(z)

for all z ∈ Δ. This completes our proof.

3.3 A generalization of Robertson’s conjecture In terms of our notation, Robertson’s conjecture is that the class G(1) coincides with the class G∗ of starlike functions with respect to a boundary point h whose images lie in a half-plane with h(0) = 1 and h(1) = 0. Following Silverman and Silvia [135] we show now that h ∈ Univ(Δ) with h(0) = 1 and h(1) = 0 is a starlike function with respect to a boundary point if and only if h ∈ G(λ) for some λ ∈ (0, 2]. Moreover, in this case the image h(Δ) lies in a wedge of angle λπ. In particular, if λ ∈ (0, 2] this proves Robertson’s conjecture. Theorem 3.1. If h ∈ G(λ), h ≡ 1, then h ∈ S ∗ [1], i.e., h is a starlike function with respect to a boundary point with h(1) = 0, and its image h(Δ) is contained in a wedge of angle λπ. Proof. Since h ∈ G(λ) and h ≡ 1, it follows from Lemma 3.5 that there exists a function p ∈ P of positive real part such that (1 − z)2 or, equivalently,

1 h (z) =− , h(z) p(z)

h(z) = h (z) · −(1 − z)2 p(z) .

It is clear that the function f deﬁned by f (z) = −(1−z)2 p(z) satisﬁes the Berkson– Porta representation (2.2.7) with τ = 1 ∈ ∂Δ. Consequently, by Theorem 2.7, f is a semi-complete vector ﬁeld. Moreover, f ∈ G + [1]. Therefore, Theorem 2.14 implies that h is a starlike function with respect to a boundary point and h(1) = 0. In other words, h ∈ S ∗ [1]. Further, by Lemma 3.4 one can approximate h ∈ G(λ) by functions of the form hn (z) :=

n " j=1

λj

(ωj (z))

,

where ωj (z) =

n 1−z , λj = λ, |ζj | = 1. 1 − z ζ¯j j=1

3.3. A generalization of Robertson’s conjecture

45

Each function ωj (z)maps theopen unit disk Δ onto a half-plane. Thus, for some βj ∈ R we have arg eiβj ωj (z) < π2 , j = 1, . . . , n. n * λj βj . Then for each z ∈ Δ, Let β = j=1

⎞ ⎛ n " λj ⎠ arg eiβ hn (z) = arg ⎝eiβ (ωj (z)) j=1 n n π λπ = λj arg eiβj ωj (z) < λj = . 2 2 j=1 j=1

Hence h(Δ) is contained in a wedge of angle λπ.

Theorem 3.2. If h ∈ S ∗ [1] and the image h(Δ) lies in a wedge of angle λπ, then h ∈ G(λ). We will prove this theorem by using a geometrical idea of Lyzzaik [102]. This idea is based on the existence of an approximation sequence consisting of starlike functions with respect to interior points and the Carath´eodory Theorem on kernel convergence. Other analytic approaches, which have a more constructive character for approximation problems, are given in Section 9.1. Proof. Since the image h(Δ) lies in a wedge of angle λπ, the function h0 (z) = 2 h(z) λ , h0 (0) = 1, is a well-deﬁned univalent function on Δ. Furthermore, h0 (1) = 0 and h0 is starlike with respect to a boundary point, i.e., h0 ∈ S ∗ [1]. For each n = 1, 2, . . ., let Dn be a starlike domain deﬁned by 1 , Dn = h0 (Δ) ∪ w ∈ C : |w| < n and let hn : Δ → Dn be the univalent (starlike) mapping of Δ onto Dn such that hn (0) = 1 and arg hn (0) = arg h0 (0). It follows by Carath´eodory’s theorem on kernel convergence that hn converges to h0 uniformly on compact subsets of Δ as n tends to inﬁnity. Because each hn (Δ) = Dn is a starlike domain that contains the origin, there is a point τn ∈ Δ such that hn (τn ) = 0, i.e., hn ∈ S ∗ [τn ]. Then by Theorem 1.3 there are starlike functions ψn ∈ S ∗ [0] that satisfy hn (z) =

ψn (z) (z − τn )(1 − τ¯n z), z

z ∈ Δ.

(3.3.1)

Since 1 = hn (0) = −τn ψn (0), we get hn (0) =

ψn (0) + ψn (0) 1 + |τn |2 . 2ψn (0)

Thus the convergence hn (0) → h0 (0) and the estimate |ψn (0)| ≤ 4|ψn (0)| imply that the sequence {ψn (0)} is uniformly bounded. Therefore, there is a subsequence

46

Chapter 3. Starlike Functions with Respect to a Boundary Point

of {ψn } that converges to a starlike function −ψ, ψ ∈ S ∗ [0]. One can assume that the corresponding subsequence of {τn } converges to a point τ ∈ Δ. Now it follows by (3.3.1) that −ψ(z) (z − τ )(1 − τ¯z), z ∈ Δ. h0 (z) = z Letting z approach 1 we conclude that τ = 1. Hence h0 (z) =

ψ(z) · (1 − z)2 . z

This equality implies that ψ (0) = 1, and then ψ ∈ S ∗ . Raising to the power λ/2, we obtain λ/2 ψ(z) (1 − z)λ . h(z) = z λ/2 ψ(z) Note that the function z is starlike of order κ = 1− λ2 . Using Lemma 3.2 z we conclude that h ∈ G(λ). This completes the proof.

3.4 Angle distortion theorems 3.4.1 Smallest exterior wedge We have already shown that a function h ∈ S ∗ [1], h(0) = 1, belongs to G(λ) if and only if its image h(Δ) lies in a wedge of angle λπ. This fact is a geometric explanation of Lemma 3.3. Now the question is: ♦ Given h ∈ S ∗ [1], ﬁnd the minimal λ ∈ (0, 2] such that h ∈ G(λ). In other words, one should determine the angle of the minimal wedge W ∗ (h) that contains h(Δ). Obviously, W ∗ (h) contains the point w = 1 because of the normalization. An additional question that arises in this context is the precise location of W ∗ (h), say, with respect to the real axis. Theorem 3.3 (cf. [56, 63]). Let h ∈ S ∗ [1], h(0) = 1. Then (i) h satisﬁes a generalized Visser–Ostrowski condition (see Deﬁnition 1.12) (z − 1)h (z) = κ, z→1 h(z)

∠ lim with 0 < κ ≤ 2; (ii) the radial limit

lim arg h(r) = θ∗

r→1−

exists and |θ∗ | < π;

3.4. Angle distortion theorems

47

(iii) the minimal wedge W ∗ (h), which contains the image h(Δ), is κπ κπ + θ∗ < arg w < + θ∗ . W ∗ (h) = w ∈ C : − 2 2

(3.4.1)

Consequently, h ∈ G(κ), and for any λ < κ, h ∈ G(λ). Proof. Suppose that W ∗ (h) is of angle κπ for some κ, 0 < κ ≤ 2. Then h ∈ G(κ) by Theorem 3.2. So, by Lemma 3.1, it can be represented as follows: ¯ h(z) = (1 − z)κ exp −κ log(1 − z ζ)dσ(ζ) (3.4.2) |ζ|=1

for some probability measure σ on the unit circle |ζ| = 1. We claim that the measure σ in this formula is mutually singular with the Dirac measure δ at the point ζ = 1 ∈ ∂Δ. Indeed, decomposing σ relative to δ, one can write σ = tσ + (1 − t)δ, where 0 ≤ t ≤ 1, and σ and δ are mutually singular probability measures. Thus tκ ¯ log(1 − z ζ)dσ(ζ) ; h(z) = (1 − z) exp −tκ |ζ=1

that is, h ∈ G(tκ). By Theorem 3.1, h(Δ) is contained in a wedge of angle tκπ. If t < 1, our supposition is contradicted. Therefore, t = 1, and the measures σ = σ and δ are mutually singular. Further, using (3.4.2) we see that (z − 1)h (z) 1 − ζ¯ =κ (3.4.3) ¯ dσ(ζ), z ∈ Δ. h(z) |ζ|=1 1 − z ζ Let {zn = xn + iyn }∞ n=1 be any sequence that tends to 1 non-tangentially, i.e., {zn }∞ lies in some Stolz angle n=1 Da,r (1) = {z ∈ Δ : | arg(1 − z)| < a, |z − 1| < r} . Consider the functions φn : ∂Δ → C, n = 1, 2, . . ., deﬁned by φn (ζ) :=

1 − ζ¯ , 1 − zn ζ¯

ζ ∈ ∂Δ.

Each function φn maps the unit circle ∂Δ onto the circle |ξ − cn | = |cn |, where cn = (1 − z¯n )/(1 − |zn |2 ). Hence, there is a natural number N independent of ζ ∈ ∂Δ such that 2 yn 1 + 1−x n 2 2 |φn (ζ)| ≤ 4|cn |2 = 4

2 ≤ 2(1 + a ) 2 yn 1 + xn − 1−xn

48

Chapter 3. Starlike Functions with Respect to a Boundary Point

for all n ≥ N . Using (3.4.3) and applying Lebesgue’s Bounded Convergence Theorem we obtain (z − 1)h (z) = κ lim φn (ζ)dσ(ζ) = κ, ∠ lim n→∞ |ζ|=1 z→1 h(z) i.e., h satisﬁes a generalized Visser–Ostrowski condition. To prove (ii) we note that by Lemma 3.1 arg h(z) = κ arg(1 − z) − arg 1 − z ζ¯ dσ(ζ) ,

(3.4.4)

|ζ|=1

and, consequently, for any r ∈ (0, 1), arg h(r) = −κ

|ζ|=1

¯ < Since | arg(1 − rζ)| that the radial limit

π 2,

arg 1 − rζ¯ dσ(ζ).

Lebesgue’s Bounded Convergence Theorem implies θ∗ := lim arg h(r) r→1−

exists and |θ∗ | <

κπ . Moreover, 2 θ∗ = −κ

|ζ|=1

arg 1 − ζ¯ dσ(ζ).

Combining the last equality with (3.4.4) we see that |arg h(z) − θ∗ | = κ arg(1 − z) − arg 1 − z ζ¯ dσ(ζ) + arg 1 − ζ¯ dσ(ζ) |ζ|=1 |ζ|=1 ¯ arg (1 − z)(1 − ζ) dσ(ζ) . ≤κ ¯ 1 − zζ |ζ|=1 Since the function

¯ (1 − z)(1 − ζ) takes values in the right half-plane, we have 1 − z ζ¯ |arg h(z) − θ∗ | <

κπ . 2

Taking into account that the minimal wedge W ∗ (h) is of angle κπ, we complete the proof. A consequence of Theorems 2.13, 2.14 and 3.3 is the following assertion.

3.4. Angle distortion theorems

49

Corollary 3.2 (cf. [97, 99]). Let h ∈ Hol(Δ, C), h(0) = 1. Then h ∈ S ∗ [1] if and only if it satisﬁes the condition h(z) = −

1 + F (z) (1 − z)2 h (z) , 4 1 − F (z)

where F ∈ Hol(Δ) is such that F (1) = 1 and F (1) = α ∈ (0, 1]. Moreover, the smallest wedge which contains h(Δ) is exactly of angle 2απ. Now it is natural to distinguish the classes of starlike functions that satisfy a generalized Visser–Ostrowski condition. Namely, for each μ ∈ (0, 2] we say that • a univalent function h belongs to the class Starμ [1] if it is of S ∗ [1], is normalized by h(0) = 1, and satisﬁes (z − 1)h (z) = μ. z→1 h(z)

Qh (1) := ∠ lim

Corollary 3.3. The class Starμ [1] consists of those univalent functions in G(μ) that do not belong to G(λ) for λ < μ; and G(μ) \ {h ≡ 1} =

!

Starλ [1].

λ≤μ

3.4.2 Biggest interior wedge Given a starlike function h ∈ S ∗ [τ ] (with respect to an interior or boundary point τ ∈ Δ), the following additional questions are seen to be natural: ♦ ﬁnd conditions which ensure that the image h(Δ) contains a wedge; ♦ ﬁnd the size and the location of a maximal wedge W∗ (h), which is contained in h(Δ). To answer these questions we need the following deﬁnition. Deﬁnition 3.4. We say that a function h ∈ S ∗ [τ ] belongs to the subclass S ∗ [τ, η], where η ∈ ∂Δ, η = τ , if the angular limit Qh (η) := ∠ lim

z→η

(z − η)h (z) h(z)

(3.4.5)

exists ﬁnitely and is diﬀerent from zero. In the case where τ = 1, we say that h ∈ Starμ,ν [1, η] if h ∈ Starμ [1] and the angular limit (3.4.5) at the boundary point η = 1 equals ν. Thus each function h ∈ S ∗ [τ, η] satisﬁes a generalized Visser–Ostrowski condition at the boundary point η (see Deﬁnition 1.12).

50

Chapter 3. Starlike Functions with Respect to a Boundary Point To proceed, we note that the inequality η = τ implies that for each h ∈ S ∗ [τ, η], ∠ lim h(z) = ∞. z→η

We require representation formulas for the classes of starlike functions S ∗ [τ ] and S ∗ [τ, η], respectively Starμ [1] and Starμ,ν [1, η]. For a boundary point w, denote by δw the Dirac measure (δ-function) at this point. Lemma 3.7 (cf. [75] and [64]). Let τ ∈ Δ and η ∈ ∂Δ, η = τ . Let h ∈ Hol(Δ, C) satisfy h(τ ) = 0. Then (i) h ∈ S ∗ [τ ] if and only if it has the form ⎡ h(z) = C(z − τ )(1 − z τ¯) · exp ⎣−2

⎤ ¯ σ(ζ)⎦ , log(1 − z ζ)d

(3.4.6)

∂Δ

where σ is an arbitrary probability measure on the unit circle and C = 0. (ii) Moreover, h ∈ S ∗ [τ, η] if and only if it has the form h(z) =

C(z − τ )(1 − z τ¯)(1 − z η¯)−2a · ⎡ ⎤ ¯ ⎦, · exp ⎣−2(1 − a) log(1 − z ζ)dσ(ζ)

(3.4.7)

∂Δ

where σ is a probability measure on the unit circle singular relative to δη , C = 0 and a ∈ (0, 1]. In this case, Qh (η) = −2a. Remark 3.1. The constant C in (3.4.6) can be chosen starting from a normalization of the functions under consideration. Because functions of the classes Starμ [1] are normalized by h(0) = 1, one sets C = 1. On the other hand, since a starlike function h is a solution of a linear homogeneous equation (see Theorem 2.14)), C arises in the integration process of this equation. Remark 3.2. In the case where τ = 0, formula (3.4.6) is the well-known representation of functions of the class S ∗ [0]; see, for example, [8]. For instance, the Koebe z function k(z) = (1−z) 2 is obtained by choosing the measure σ to be the Dirac δ-function at the point 1. On the other hand, for τ = 1, formula (3.4.6) follows by Lemma 3.1. In turn, formula (3.4.6) is more general than formula (3.2.2). Proof. (i) First, suppose that τ = 0, and let h be normalized by h(0) = 0 and h (0) = 1. Recall that Nevanlinna’s criterion (Theorem 1.1) asserts that h ∈ S ∗ [0] if and only if zh (z) q(z) := h(z)

3.4. Angle distortion theorems

51

has a positive real part. (Note that the same fact follows by Theorem 2.14, because of the Berkson–Porta representation of generators vanishing at the origin: g(z) = zp(z) with Re p(z) ≥ 0). Representing q by the Riesz–Herglotz formula, we write zh (z) 1 + z ζ¯ dσ(ζ) = h(z) 1 − z ζ¯ ∂Δ

with some probability measure σ. Integrating this equality, we get

¯ σ(ζ) . h(z) = z exp −2 log(1 − z ζ)d

(3.4.8)

∂Δ

This proves (3.4.6) for the case τ = 0. Now let h(τ ) = 0 for τ ∈ Δ, τ = 0. It follows from Hummel’s Theorem 1.3 that h ∈ S ∗ [τ ] if and only if z h(z) ∈ S ∗ [0]. (z − τ )(1 − z τ¯) Thus, (3.4.8) implies (3.4.6) for the interior location of τ . The reverse consideration and Hummel’s criterion show that if h satisﬁes (3.4.6) with τ ∈ Δ, it must be starlike. Finally, let τ ∈ ∂Δ. Following Lyzzaik [102] (see also the proof of Theorem 3.2) one can approximate h ∈ S ∗ [τ ] by a sequence {hn } of functions starlike with respect to interior points τn that converge to τ . Also, one can assume that hn (0) = h(0). Representing each function hn by (3.4.6), ⎡ ⎤ ¯ σn (ζ)⎦ , hn (z) = Cn (z − τn )(1 − z τ¯n ) · exp ⎣−2 log(1 − z ζ)d/ ∂Δ

we see that h(0) = hn (0) = −Cn τn . h(0) − τ .

Thus Cn → Since the set of all probability measures is compact, {/ σn } has a subsequence converging to some probability measure σ. Therefore, any function h ∈ S ∗ [τ ] has the form (3.4.6). To prove the converse assertion, we suppose that h has the form (3.4.6) with τ ∈ ∂Δ. Note that h is starlike if and only if the function ah(cz), a = 0, |c| = 1, is also starlike. Therefore, without loss of generality, one can assume that τ = 1 and h is normalized by h(0) = 1, i.e., ⎡ ⎤ ¯ σ(ζ)⎦ . h(z) = (1 − z)2 · exp ⎣−2 log(1 − z ζ)d ∂Δ

52

Chapter 3. Starlike Functions with Respect to a Boundary Point

Then h ∈ G(2) by Lemma 3.1. Now Theorem 3.1 implies that h is a starlike function with respect to a boundary point with h(1) = 0, i.e., h ∈ S ∗ [1]. This proves the ﬁrst assertion. (ii) Let σ = aδη + (1 − a)σ, 0 ≤ a ≤ 1, be the Lebesgue decomposition of σ relative to the Dirac measure δη , where the probability measures σ and δη are mutually singular. Using this decomposition, we rewrite (3.4.6) in the form of (3.4.7). Now we calculate h (z)(z − η) z→η h(z)

((z − τ )(1 − z τ¯)) 2a¯ η = ∠ lim (z − η) + z→η (z − τ )(1 − z τ¯) 1 − z η¯ ζ¯ +2(1 − a) ¯ dσ(ζ) ∂Δ 1 − z ζ ¯ − η) ζ(z = −2a + 2(1 − a) ∠ lim ¯ dσ(ζ). z→η ∂Δ 1 − z ζ

Qh (η) = ∠ lim

Noting that

(3.4.9)

(3.4.10)

¯ − η) |z − η| ζ(z 1 − z ζ¯ ≤ 1 − |z| ,

we see that the integrand in the last expression of (3.4.9) is bounded on each non-tangential approach region Γ(η, κ) = {z : |z − η| < κ(1 − |z|)} , κ ≥ 1, at the point η. Since the measures σ and δη are mutually singular, we conclude by the Lebesgue convergence theorem that the last integral in (3.4.9) is equal to zero, so Qh (η) = −2a.

Our proof is complete. Remark 3.3. Note, in passing, that for starlike functions with respect to ary point of the class Starμ,ν [1, η], formula (3.4.7) can be reﬁned in the way: A function h belongs to Starμ,ν [1, η] if and only if it admits the representation:

μ ν ¯ log(1 − z ζ)dσ(ζ) h(z) = (1 − z) (1 − z η¯) exp −(μ + ν)

a boundfollowing following

(3.4.11)

∂Δ

with some probability measure σ on the unit circle, which is singular relative to both δ1 and δη . Furthermore, −μ ≤ ν ≤ 0.

3.4. Angle distortion theorems

53

Theorem 3.4 (cf. [128]). Let h ∈ S ∗ [τ, η] with Qh (η) = ν. Let θ∗ := lim− arg h(rη). r→1

Then the image h(Δ) contains the wedge |ν|π W∗ = w ∈ C : | arg w − θ∗ | < 2

(3.4.12)

and contains no larger wedge with the same bisector. Proof. By Lemma 3.7, the function h has the form (3.4.7) with a = − ν2 . First, we show that the image h(Δ) contains the wedge W∗ deﬁned by(3.4.12). Since (as mentioned above) ∠ lim h(z) = ∞, for each α ∈ 0, π2 and each z→η

R > 0, there exists r > 0 such that |h(z)| > R,

(3.4.13)

whenever z ∈ Dα,r (η) := {z ∈ Δ : |1 − z η¯| ≤ r, | arg(1 − z η¯)| ≤ α} . Now the Lebesgue Bounded Convergence Theorem implies the existence of h(z) ¯ = arg C(η − τ )(1 − η¯ τ ) − 2(1 − a) lim arg(1 − z ζ)dσ(ζ). lim arg z→η z→η (1 − z η¯)ν ∂Δ

Alternatively, by formula (3.4.7), we have ¯ τ ) − 2(1 − a) lim arg(1 − rη ζ)dσ(ζ). θ∗ = lim− arg h(rη) = arg C(η − τ )(1 − η¯ − r→1

r→1

∂Δ

Therefore, lim arg

z→η

h(z) = θ∗ . (1 − z η¯)ν

Fix ε > 0. Decreasing r (if necessary), we have θ∗ − ε < arg

h(z) < θ∗ + ε (1 − z η¯)ν

for all z ∈ Dδ,r (η). So, for each point z belonging to the arc Γ := {z ∈ Δ : |1 − z η¯| = r, | arg(1 − z η¯)| ≤ δ} ⊂ Dδ,r (η), i.e., z = η(1 − reit ), |t| ≤ δ, we get θ∗ − ε − t|ν| < arg h(z) < θ∗ + ε − t|ν|.

54

Chapter 3. Starlike Functions with Respect to a Boundary Point

In particular, arg h(η(1 − reiδ )) < θ∗ + ε − δ|ν| and arg h(η(1 − re−iδ )) > θ∗ − ε + δ|ν|. Thus, the curve h(Γ) lies outside the disk |z| ≤ R and joins two points having arguments less than θ∗ + ε − δ|ν| and greater than θ∗ − ε + δ|ν|, respectively. Since h is starlike, we see that h(Δ) contains the sector {w ∈ C : |w| < R, | arg w − θ∗ | < δ|ν| − ε} . Because R and ε are arbitrary, one concludes {w ∈ C : | arg w − θ∗ | < δ|ν|} ⊂ h(Δ). Letting δ tend to

π 2,

we obtain |ν|π ⊂ h(Δ). W∗ = w ∈ C : | arg w − θ∗ | < 2

Further, since h is a starlike function, arg h(eiϕ ) is an increasing function in ϕ ∈ (arg η − π, arg η + π). Hence the limits lim

ϕ→(arg η)±

arg h(eiϕ )

exist. Let ϕn,+ → (arg η)+ and ϕn,− → (arg η)− be two sequences such that the values h(eiϕn,± ) are ﬁnite. Then, once again by Lemma 3.7, lim arg h(eiϕn,+ ) − arg h(eiϕn,− ) n→∞ = lim (arg(1 − eiϕn,+ η¯))ν − (arg(1 − eiϕn,− η¯))ν = |ν|π. n→∞

Therefore, the image contains no wedge of angle larger than |ν|π. Thus, the wedge W∗ deﬁned by (3.4.12) is the largest one contained in h(Δ). The proof is complete. It is also of interest to establish a converse assertion. Theorem 3.5. Let h ∈ S ∗ [τ ]. Suppose that the image h(Δ) contains an open wedge κπ W = w ∈ C : | arg w − θ| < ⊂ h(Δ), 2 which is maximal in the sense that there is no wedge W1 = W such that W ⊂ W1 ⊂ h(Δ). Then for some boundary point η, η = τ , the function h belongs to S ∗ [τ, η] and satisﬁes the generalized Visser–Ostrowski condition ∠ lim

z→η

(z − η)h (z) = −κ. h(z)

3.4. Angle distortion theorems

55

Proof. By our assumption, the curve := h−1 {w : arg w = θ} lies in Δ and joins the point τ with a boundary point η. Since W cannot be extended to a larger wedge lying in h(Δ), for each ε > 0 there are boundary points of the image h(Δ) belonging to the sectors κπ κπ ≤ arg w ≤ θ + +ε w: θ+ 2 2 and

w: θ−

κπ κπ − ε < arg w ≤ θ − . 2 2

+ + iφn Let {φ+ n } be a decreasing sequence such that φn → arg η, the values h e + 1 iφn − < θ + κπ exist and θ + κπ 2 ≤ arg h e 2 + n . Similarly, let {φn } be an increasing − 1 iφn exist and θ − κπ sequence such that φ− n → arg η, the values h e 2 − n < − arg h eiφn ≤ θ − κπ 2 . Since η = τ , it is possible to deﬁne on a neighborhood of η a one-valued continuous branch of the arg(z −τ ). Without loss of generality we suppose function − iφ+ iφ that all of the points e n , e n lie in this neighborhood. By Lemma 3.7 we have arg h(z) = arg C + arg ((z − τ )(1 − z τ¯)) − 2a arg(1 − z η¯) ¯ − 2(1 − a) arg(1 − z ζ)dσ(ζ), ∂Δ

where the probability measure σ is singular relative to δη , C = 0 and a ∈ [0, 1]. Note that h ∈ S ∗ [τ, η] if and only ifa = 0. − + Consider the expression arg h eiφn − arg h eiφn , which tends to κπ: + − arg h eiφn − arg h eiφn + + − − = arg (eiφn − τ )(1 − eiφn τ¯) − arg (eiφn − τ )(1 − eiφn τ¯) + − − 2a arg 1 − eiφn η¯ − arg 1 − eiφn η¯ + 1 − eiφn ζ¯ − 2(1 − a) dσ(ζ). (3.4.14) arg − 1 − eiφn ζ¯ ∂Δ The ﬁrst summand tends to zero while the second one tends to 2aπ. The third summand also tends to zero because the integrand is a bounded function tending to zero for each ζ ∈ ∂Δ, ζ = η. Hence by Lebesgue’s bounded convergence theorem the integral in (3.4.14) goes to 0. Then we obtain κπ = 2aπ. Again by Lemma 3.7 we conclude that h ∈ S ∗ [τ, η] with Qh (η) = −κ. The proof is complete.

56

Chapter 3. Starlike Functions with Respect to a Boundary Point

Corollary 3.4. Let h ∈ Starμ [1]. Then there is a one-to-one correspondence between wedges −νπ ⊂ h(Δ), W = w ∈ C : | arg w − θ| < 2 that cannot be extended to a bigger one lying in h(Δ) and boundary points η such that lim− h(rη) = ∞, lim− arg h(rη) = θ and h belongs to Starμ,ν [1, η] (see r→1

r→1

Fig. 3.2).

PS vS

Figure 3.2: Angle distortion.

3.5 Functions convex in one direction The class of functions we consider here has been studied by several mathematicians (see, for example, Hengartner and Schober [82], Ciozda [30, 31], Lecko [98]) as a subclass of functions deﬁned by Robertson in [122]. Deﬁnition 3.5. We say that a univalent function g ∈ Hol(Δ, C) normalized by g(0) = 0

(3.5.1)

is convex in the positive direction of the real axis if, for each z ∈ Δ and t > 0, g(z) + t ∈ g(Δ)

and

lim g −1 (g(z) + t) = τ ∈ ∂Δ.

t→∞

(3.5.2)

The class of these functions is denoted by Σ[τ ]. We ﬁrst study the following question: ♦ Given a function of the class Σ[τ ], ﬁnd the minimal horizontal strip that contains its image. The next question is also natural but more complicated:

3.5. Functions convex in one direction

57

♦ Characterize those functions convex in the positive direction of the real axis whose images contain a whole (two-sided) strip and ﬁnd the size (width) of this strip. We solve the latter problem for functions having minimal horizontal strips of ﬁnite size. The problem is still open for the general case. To proceed we need the following lemmata. Lemma 3.8 (cf. [82, 30, 31]). Let g be a univalent function normalized by (3.5.1). Then g ∈ Σ[τ ] if and only if Re ((τ − z)(1 − z τ¯)g (z)) > 0

for all z ∈ Δ.

(3.5.3)

Proof. Let g ∈ Σ[τ ]. By Deﬁnition 3.5, for each t ≥ 0 the holomorphic function Ft deﬁned by Ft (z) = g −1 (g(z) + t) maps the unit disk into itself. It is easy to verify that the family S = {Ft }t≥0 forms a continuous semigroup of holomorphic self-mappings of the unit disk. Diﬀerentiating this semigroup at t = 0+ we get ∂Ft (z) 1 , (3.5.4) f (z) := − =− ∂t t=0+ g (z) where f is the inﬁnitesimal generator of S (see, for example, Theorem 2.1). By (3.5.2) the point τ is the Denjoy–Wolﬀ point of S. Therefore, its generator can be represented by the Berkson–Porta formula (see Theorem 2.7): f (z) = (z − τ )(1 − z τ¯)p(z),

where Re p(z) ≥ 0.

(3.5.5)

Comparing (3.5.4) and (3.5.5) proves inequality (3.5.3). Conversely, suppose that g satisﬁes (3.5.3). Then the function p(z) =

1 (τ − z)(1 − z τ¯)g (z)

has a non-negative real part. z 1 · belongs to the class Σ[τ ]. ib τ − z If Re p(z) > 0, z ∈ Δ, then by a result of Berkson and Porta (see Theorem 2.7), the function f deﬁned by (3.5.5) is the generator of a semigroup S = {Ft }t≥0 of holomorphic self-mappings of the unit disk. This semigroup can be deﬁned by the Cauchy problem: ⎧ ⎨ ∂Ft (z) + f (F (z)) = 0, t ∂t ⎩ F0 (z) = z, z ∈ D. If p(z) = ib, b ∈ R, then g(z) =

58

Chapter 3. Starlike Functions with Respect to a Boundary Point

1 we get Substituting here f (z) = (z − τ )(1 − z τ¯)p(z) = − g (z) ∂Ft (z) = 1. ∂t Integrating the latter expression on the interval [0, t] we get g (Ft (z))

g(Ft (z)) = g(z) + t,

that is, g(z) + t ∈ g(Δ).

Since S has a Denjoy–Wolﬀ point at τ , it follows that lim g −1 (g(z) + t) = lim Ft (z) = τ. t→∞

t→∞

This completes our proof. Later on we concentrate on the case τ = 1 and consider the class Σ[1]. Lemma 3.9. Let g ∈ Σ[1]. Then the limit ∠ lim (1 − z)g (z) = μ z→1

(3.5.6)

is either a positive real number or inﬁnity. Proof. By Lemma 3.8 the function p(z) =

1 (1 − z)2 g (z)

is either a non-constant holomorphic function of positive real part or an imaginary constant. In the latter case, p(z) = ib, b ∈ R, and the assertion is evident. Otherwise, one can write 1 = (1 − z)p(z). (1 − z)g (z) It was shown in [62] that, for any function p of the class P, the angular limit ∠ lim (1 − z)p(z) exists and is a non-negative real number. This proves our asserz→1 tion. Deﬁnition 3.6. We say that a univalent function g belongs to the class Σμ [1] with 0 < μ ≤ ∞, if it is of the class Σ[1] and the limit (3.5.6) is equal to μ. 0 Σμ [1]. Thus by Lemma 3.9 we have Σ[1] = 0<μ≤∞

Example 3.1. The function g1 (z) = log

1+z 1−z

belongs to Σ1 [1], while the function g2 (z) = belongs to Σ∞ [1].

z 1−z

3.5. Functions convex in one direction

59

Deﬁnition 3.7. Let η ∈ ∂Δ, η = 1. We say that a univalent function g belongs to the class Σμ,ν [1, η] if it is of the class Σμ [1] and satisﬁes the conditions ∠ lim Re g(z) = −∞ z→η

and ∠ lim (z − η)g (z) = ν. z→η

Theorem 3.6.

(i) If g ∈ Σμ [1], μ ∈ (0, ∞), then the horizontal strip

πμ T ∗ = z : | Im z − a∗ | < 2

with

a∗ = lim Im g(r) r→1−

(3.5.7)

is the smallest one that contains the image g(Δ). If g ∈ Σ∞ [1], then there is no horizontal strip of ﬁnite width containing g(Δ); (ii) If g ∈ Σμ,ν [1, η], μ ∈ (0, ∞), then the horizontal strip πν T∗ = z : | Im z − a∗ | < 2

with

a∗ = lim− Im g(rη)

(3.5.8)

r→1

is contained in g(Δ), and there is no strip T = T∗ with T∗ ⊂ T ⊂ g(Δ). Consequently, ν ∈ (0, μ]. Conversely, if g ∈ Σμ [1], μ ∈ (0, ∞), and the image g(Δ) contains some open horizontal strip, then there is ν (0 < ν ≤ μ) and a point η ∈ ∂Δ such that g belongs to Σμ,ν [1, η] (see Fig. 3.3).

PS

vS

Figure 3.3: Width distortion.

60

Chapter 3. Starlike Functions with Respect to a Boundary Point

Proof. (i) Suppose that g ∈ Σμ [1], μ ∈ (0, ∞), i.e., the limit (3.5.6) is ﬁnite. By Lemma 3.8, the function p(z) = (1 − z)2 g (z) is of nonnegative real part. Hence, there exists a self-mapping ω(z) such that (1 − z)2 g (z) = 4μ Thus

1 − ω(z) . 1 + ω(z)

1 − ω(z) 1 + ω(z) (1 − z)g (z) = . 1−z 4 μ

Since the right-hand side of this equality is bounded on each non-tangential approach region at the point z = 1, one concludes that ∠ lim ω(z) = 1. Now we z→1

calculate 1 − ω(z) 1 + ω(z) (1 − z)h (z) 1 = ∠ lim = ∈ (0, 1). z→1 1 − z z→1 4 μ 2

∠ lim

Consider the function

1 h(z) = exp − g(z) . μ

It satisﬁes −

(3.5.9)

h (z) 1 − ω(z) (1 − z)2 = 4 . h(z) 1 + ω(z)

So, by a result of Lecko and Lyzzaik ([99]), the function h is univalent and starlike with respect to a boundary point. Since (z − 1)h (z) = 1, z→1 h(z)

∠ lim

we see that h ∈ Starλ [1] with λ = 1. By Theorem 3.3 the smallest wedge that contains the image h(Δ) is deﬁned by (3.4.1): π π W ∗ (h) = w ∈ C : − + θ∗ < arg w < + θ∗ , 2 2 where θ∗ = lim− arg h(r) = lim− − r→1

r→1

Im g(r) . μ

Therefore, the smallest horizontal strip containing the image of function g(z) = −μ log h(z), is deﬁned by (3.5.7). Suppose now that g ∈ Σ[1] and its image is contained in a horizontal strip of ﬁnite width πμ (0 < μ < ∞). In this case the function h deﬁned by (3.5.9) is univalent and starlike with respect to a boundary point. Moreover, since the minimal strip, which contains the image g(Δ), is of width πμ, the minimal wedge

3.5. Functions convex in one direction

61

containing h(Δ) is of angle π. Once again by Theorem 3.3, h ∈ Starλ [1] with λ = 1; hence it satisﬁes the Visser–Ostrowski condition (z − 1)h (z) = 1. z→1 h(z)

∠ lim Since

(z − 1)h (z) (1 − z)g (z) = , h(z) μ

and 0 < μ < ∞, we have that g ∈ Σμ [1]. (ii) As above, deﬁne a starlike function h by (3.5.9). It is clear that lim− h(rη) r→1

= ∞. Since h is univalent, we conclude that ∠ lim h(z) = ∞. Furthermore, z→η

∠ lim

z→η

(z − η)h (z) −(z − η)g (z) ν = ∠ lim =− . z→η h(z) μ μ

So, h ∈ Star1,−ν/μ [1, η]. As mentioned above, in this case −1 ≤ − μν ≤ 0; that is, ν ∈ [0, μ]. By Theorem 3.4 the wedge νπ with θ∗ = lim arg h(rη) W∗ = w : |arg w − θ∗ | < 2μ r→1− is contained in h(Δ), and there is no wedge W = W∗ with W∗ ⊂ W ⊂ h(Δ). Since g(z) = −μ log h(z), we have that the limit a∗ =

lim Im g(x) = −μθ∗

x→−1+

exists. In addition, the strip S∗ deﬁned by (3.5.8) is contained in g(Δ), and there is no strip S = S∗ with S∗ ⊂ S ⊂ g(Δ). Conversely, suppose that g ∈ Σμ [1], and its image g(Δ) contains an open strip. Then the function h deﬁned by (3.5.9) belongs to Starμ [1] and its image h(Δ) contains an open wedge. By Corollary 3.4 we conclude that there is a boundary point η and a positive number ν such that h ∈ Starμ,−ν [1, η]. In this case g(z) = −μ log h(z) belongs to Σμ,ν [1, η]. This completes our proof.

Chapter 4

Spirallike Functions with Respect to a Boundary Point In this chapter, we introduce the class of spirallike functions with respect to a boundary point. It turns out that, in fact, each such function is a complex power of a starlike function with respect to a boundary point. So, what we studied in the previous chapter forms the base for our present considerations. We start by looking at the images of spirallike functions with respect to a boundary point, i.e., spirallike domains, the boundary of which contains the origin.

4.1 Spirallike domains with respect to a boundary point Deﬁnition 4.1 (cf. Deﬁnition 1.3). A simply connected domain Ω ⊂ C, 0 ∈ ∂Ω, is called a spirallike domain with respect to a boundary point if there is a number μ ∈ C, Re μ > 0, such that for any point w ∈ Ω the spiral curve {e−tμ w, t ≥ 0} is contained in Ω. If, in particular, μ is real, then Ω is a starlike domain with respect to a boundary point. Since we intend to study functions that map the unit disk Δ onto spirallike domains, the requirement for Ω to be simply connected is natural in view of the Riemann Mapping Theorem. For a simply connected domain Ω with 0 ∈ ∂Ω it is possible to deﬁne on Ω a one-valued branch of the function arg w. If, in addition, w0 = 1 ∈ Ω then we can choose this branch in such a way that arg w0 = 0. In this manner, for any number λ ∈ C the function wλ = exp [λ(ln |w| + i arg w)] is well deﬁned on Ω and attains the value 1 at the point w0 = 1. We will denote the set of all spirallike (respectively, starlike) domains with respect to a boundary

64

Chapter 4. Spirallike Functions with Respect to a Boundary Point

point that contain the point w0 = 1 by SP (respectively, by ST ). It is clear that ST ⊂ SP. To continue our discussion, we ﬁnd a proper method to measure the “angular size” of spirallike domains. This is done as follows: Let a domain Ω be spirallike t ≥ 0. Denote the (Ω ∈ SP), w ∈ Ω and

connected component of the set ψ ∈ R : e−μ(t−iψ) w ∈ Ω which contains the point ψ = 0 by Φμ (w, t) = (aμ (w, t), bμ (w, t)). In other words, aμ (w, t) = inf φ ≤ 0 : e−μ(t−iψ) w ∈ Ω for all ψ ∈ (φ, 0) , (4.1.1) (4.1.2) bμ (w, t) = sup φ ≥ 0 : e−μ(t−iψ) w ∈ Ω for all ψ ∈ (0, φ) . Proposition 4.1. Let Ω ∈ SP and μ be a complex number with Re μ > 0 such that the curve {e−tμ w, t ≥ 0} ⊂ Ω for all w ∈ Ω. Then the limit α(w) := lim (bμ (w, t) − aμ (w, t))

(4.1.3)

t→+∞

exists ﬁnitely. Moreover, this limit does not depend on a point w ∈ Ω, i.e., α(w) ≡ α = constant. In the particular case where μ ∈ R and Ω ∈ ST this limit is equal to the size θ of the minimal angle in which Ω lies divided by μ, i.e., α = θ/μ. Proof. Deﬁnition 4.1 implies that if e−μt0 eiμφ w ∈ Ω then for all t ≥ t0 the point e−μt eiμφ w is contained in Ω. Consequently, aμ (w, t) is decreasing and bμ (w, t) is increasing (with respect to t). So, the limit in (4.1.3) exists. To prove that α is ﬁnite, it is enough to show that the functions aμ (w, t) and bμ (w, t) are bounded. Fix t ≥ 0. We show that 2π Re μ bμ (w, t) ≤ (4.1.4) |μ|2 and aμ (w, t) ≥ −

2π Re μ |μ|2

(4.1.5)

in Ω.

This is clear if Im μ = 0. If φ = − 2π ∈ Φμ (w, t), then Ω contains the circle μ −μ(t−iψ) e w, ψ ∈ [φ, 0] centered at the origin and Ω is not simply connected. Thus, without loss of generality, assume that Im μ > 0. The spirallikeness of Ω implies that the curve Γ1 , deﬁned by

2π Im μ −μ(t+t1 ) Γ1 (t1 ) = e , w, t1 ∈ 0, |μ|2

lies in Ω. If inequality (4.1.4) is not satisﬁed, then φ = therefore, the curve Γ2 , deﬁned by Γ2 (ψ) = e−μ(t−iψ) w,

2π Re μ ∈ Φμ (w, t) and, |μ|2

ψ ∈ [0, φ] ,

4.1. Spirallike domains with respect to a boundary point

65

also lies in Ω. If this were true, the curve Γ2 Γ−1 1 lying in Ω, would wind once about the origin. This contradicts the simply connectedness of Ω, and condition (4.1.4) is proved. 2π Im μ . Once As the function aμ (w, t) is decreasing we can suppose that t > |μ|2 again the spirallikeness of Ω implies that the curve Γ3 , deﬁned by

2π Im μ Γ3 (t1 ) = e−μ(t+t1 ) w, t1 ∈ − , 0 , |μ|2 lies in Ω. If inequality (4.1.5) is not satisﬁed, then φ = − therefore, the curve Γ4 , deﬁned by Γ4 (ψ) = e−μ(t−iψ) w,

2π Re μ ∈ Φμ (w, t) and, |μ|2

ψ ∈ [φ, 0] ,

also lies in Ω. If this were the case, the curve Γ4 Γ−1 3 lying in Ω, would wind once about the origin. As above, this contradicts the simply connectedness of Ω, and condition (4.1.5) is also proved. Now we show that α(w) does not depend on w ∈ Ω. Let K be any compact, connected subset of Ω. For each point w0 ∈ K, there exists > 0 such that the neighborhood U (w0 , ) = e−μ(t−iψ) w0 , − < t < , − < ψ < is contained in Ω. Let w1 ∈ U (w0 , ). Then

ˆ where w ˆ = eiμψ w0 w1 = e−μt w,

with |t | < , |ψ | < .

By formulae (4.1.1) and (4.1.2) we have bμ (w, ˆ t) − aμ (w, ˆ t) = bμ (w0 , t) − aμ (w0 , t), and thus α(w) ˆ = α(w0 ). Furthermore, it is clear that bμ (w1 , t) − aμ (w1 , t) = bμ (w, ˆ t + t ) − aμ (w, ˆ t + t ). Hence, the limits as t → ∞ in both sides of the two latter equations coincide, that is, α(w1 ) = α(w0 ), so it is a constant function on U (w0 , ). Finding a ﬁnite covering system of neighborhoods U1 , U2 , . . . Un of K, we can conclude that α(w) ≡ constant on U1 ∪ U2 ∪ . . . ∪ Un ⊃ K, so α does not depend on w ∈ Ω. In the case when the domain Ω is starlike (i.e., μ ∈ R), the quantity b1 (w, t)− a1 (w, t) equals exactly the size of the circle arch of the radius e−tμ (which lies in Ω) divided by μ. The proposition is proved.

66

Chapter 4. Spirallike Functions with Respect to a Boundary Point

Deﬁnition 4.2. Let μ be a complex number with Re μ > 0. Also let Ω be a simply connected domain such that 0 ∈ ∂Ω. Ω will be called μ-spirallike (with respect to a boundary point) if for any point w ∈ Ω the following two conditions hold: (a) {e−tμ w, t ≥ 0} ⊂ Ω; (b) the limit α in (4.1.3) exists and is equal to π: α = lim (bμ (w, t) − aμ (w, t)) = π. t→+∞

The set of all μ-spirallike domains Ω ∈ SP will be denoted by μ- SP. It is clear that ST =

!

μ- SP .

μ∈(0,2]

We investigate some properties of μ-spirallike domains. Lemma 4.1. only if

(i) If Re μ > 0 and Ω is of the class SP, then Ω ∈ μ- SP if and 1 = Ω μ1 := z μ1 : z ∈ Ω ∈ 1- SP . Ω

1 is starlike. Moreover, Ω (ii) If there exists Ω ∈ μ- SP then μ = ν.

2

ν- SP, where μ, ν ∈ C with Re μ > 0, Re ν > 0,

Proof. In addition to formulae (4.1.1) and (4.1.2) let us write 1 , ˆ t) = inf φ : e−(t−iφ) w ˆ∈Ω a1 (w, 1 , b1 (w, ˆ t) = sup φ : e−(t−iφ) w ˆ∈Ω 1 1 where w ˆ = w μ ∈ Ω. 1 are one and the same, Since the inclusions e−μ(t−iφ) w ∈ Ω and e−(t−iφ) w ˆ∈Ω we have ˆ t) − a1 (w, ˆ t). bμ (w, t) − aμ (w, t) = b1 (w,

Thus the limits as t goes to +∞ in both sides of this equality are either equal to π or diﬀer from π. Assertion (i) is proved. In turn, (i) implies that the domains 1 1 1 1 Ω1 = Ω μ := z μ , z ∈ Ω and Ω2 = Ω ν := z ν , z ∈ Ω belong to the class 1-SP. This means that for any point w ∈ Ω we have w1 = 1 1 w μ ∈ Ω1 and w2 = w ν ∈ Ω2 . So, we see: any point w2 ∈ Ω2 if and only if the ν ν point w1 = w2μ lies in Ω1 . In other words, Ω1 = Ω2μ .

4.1. Spirallike domains with respect to a boundary point

67

Suppose now that arg μ = arg ν. Assertion (i) proved above implies that the domain Ω1 is of the class μν -SP, i.e., by Deﬁnition 4.2, it contains the following spiral, which goes around the origin: μ e−t ν w, t ≥ 0 ⊂ Ω1 , when w ∈ Ω1 . This contradicts the inclusion Ω1 ∈ 1-SP (see Proposition 4.1). So, arg μ = arg ν. Suppose now that |μ| = |ν|; for example, μ = Rν, R > 1. Again we have ν

1

Ω1 = Ω2μ = Ω2R . Since the domain Ω2 is contained in some wedge of angle π, then π the domain Ω1 is contained in the wedge of angle R < π and this contradicts the inclusion Ω1 ∈ 1-SP. Thus we have μ = ν. The proved Lemma 4.1 states that each spirallike domain (with respect to a boundary point) is μ-spirallike with a unique number μ, Re μ > 0. Now we show that μ cannot be arbitrary in the right half-plane. Proposition 4.2.

(i) If Ω1 ∈ 1-SP and |μ − 1| ≤ 1, then Ω = Ωμ1 ∈ μ- SP.

(ii) In case for some μ ∈ C there exists a Ω that belongs to μ- SP, then |μ−1| ≤ 1. Proof. Without loss of generality we assume that a domain Ω1 ∈ 1-SP lies in Π+ := {z ∈ C : Re z > 0}. First, we will show that for any 1-spirallike domain Ω1 ⊂ Π+ , the domain Ω = Ωμ1 is simply connected if |μ − 1| ≤ 1, μ = 0; that is, 1 1 Re ≥ . μ 2 Since the domain Ω1 is simply connected and 0 ∈ ∂Ω1 , then Ωμ1 is simply connected if and only if the mapping z → z μ is one-to-one on Ω1 . This means that for any w ∈ Ω1 the equation (4.1.6) wμ = z μ has no solution z ∈ Ω1 \ {w}. Suppose that z = reiψ , |ψ| < π2 , is the solution of the above equation. Substituting w = ρeiφ , |φ| < π2 , we rewrite (4.1.6) in the following form: μ ln ρ + iφ = μ ln r + iψ + 2πki, k ∈ Z \ {0}, or,

1 ln ρ + iφ − ln r − iψ φ−ψ − ln ρ + ln r = = +i . μ 2πki 2πk 2πk

π 1 1 1 < = ≤ . The latter inequality μ 2π|k| 2|k| 2 1 1 contradicts our supposition that Re ≥ . Thus the domain Ω = Ωμ1 is simply μ 2 connected. It is easy to see by Deﬁnition 4.1 that Ω ∈ SP. By Lemma 4.1, Ω ∈ μ- SP. Assertion (i) is proved. This equality implies that Re

68

Chapter 4. Spirallike Functions with Respect to a Boundary Point

To prove assertion (ii), we suppose that Ω ∈ μ- SP, where the number μ 1 1 1 1− satisﬁes Re < , and so Re = for some ∈ (0, 1). μ 2 μ 2 Given a point w ∈ Ω and t large enough, it follows by Deﬁnition 4.2 that ≤ bμ (w, t) − aμ (w, t) ≤ π. π 1− 2 /1 and φ /2 such that In other words, there exist values φ /

e−μ(t−iφj ) w ∈ Ω,

j = 1, 2,

and /1 . /2 − φ π (1 − ) ≤ φ

/1 , φ /2 , Thus, for t big enough and for all φ ∈ φ e−μ(t−iφ) w ∈ Ω.

(4.1.7)

/1 and φ2 = In particular, the points e−μ(t−iφ1 ) w and e−μ(t−iφ2 ) w, where φ1 = φ / φ1 + π (1 − ), belong to Ω. It follows by Deﬁnition 4.2 that e−μ(tj −iφj ) w ∈ Ω,

j = 1, 2,

for all t1 , t2 ≥ t. Hence we can choose the numbers t1 and t2 such that 1 1− t2 − t1 = +i . μ 2 2 This implies that e−μ(t1 −iφ1 ) w = e−μ(t2 −iφ2 ) w.

(4.1.8) 1

It follows from Lemma 4.1 that the domain Ω1 = Ω μ ∈ 1-SP. Thus the 2 2 μ domain Ω2 = Ω1 = Ω ∈ 2-SP. Therefore, a one-valued branch of the function arg w is well-deﬁned in the domain Ω2 . Further, equation (4.1.8) implies that e−2(t1 −iφ1 ) w2 = e−2(t2 −iφ1 ) e2πi(1−) w2 ∈ Ω2 , 2

(4.1.9)

where w2 = w μ ∈ Ω2 . Equality (4.1.9) means that for the same point of the simply connected domain Ω1 , its argument has two diﬀerent values. This is a contradiction that proves assertion (ii).

4.2. A characterization of spirallike functions with respect to a boundary point69

4.2 A characterization of spirallike functions with respect to a boundary point Now we adapt Deﬁnition 1.4 to ﬁt our purposes. Deﬁnition 4.3. A univalent function h : Δ → C on the unit disk Δ is said to be of class Spiral[1] (respectively, Spiralμ [1]) if (a) h(0) = 1 and h(1) := lim− h(r) = 0; r→1

(b) h(Δ) ∈ SP (respectively, h(Δ) ∈ μ-SP). Remark 4.1. It is clear that each function h ∈ Spiralμ (Δ) is μ-spirallike in the sense of Deﬁnition 1.4. Moreover, Lemma 4.1 implies that if h ∈ Spiralμ [1] is νspirallike, then arg ν = arg μ. This fact is no longer true for spirallike functions with respect to an interior point. Observe also that if μ is real then Spiralμ [1] = Starμ [1], i.e., a function h normalized by h(0) = 1, h(1) = 0 belongs to Spiralμ [1] with a real positive number μ if and only if h is univalent, its image h(Δ) is starlike with respect to the origin, and the smallest wedge containing h(Δ) is of angle πμ (see Proposition 4.1). Now we formulate the main result of this section. Theorem 4.1. Let h : Δ → C be a holomorphic function and h(0) = 1. Let μ ∈ C, |μ − 1| ≤ 1, μ = 0. The following assertions are equivalent. (I) h ∈ Spiralμ [1]. (II) h1 (z) = h(z)1/μ ∈ Spiral1 [1] = Star1 [1], i.e., h1 is starlike with respect to the boundary point h(1) = 0 function, and the smallest wedge that contains its image is of angle π. (III) The function h satisﬁes the following condition 2 zh (z) 1 + z · + > 0, Re μ h(z) 1−z

z ∈ Δ,

(4.2.1)

and it is possible to replace the number μ in this inequality with a number ν only if ν = Rμ, R > 1. zh(z) r is φ-spirallike of order cos φ − , where μ = (1 − z)μ 2 r eiφ , i.e., g is a univalent function satisfying the condition r −iφ zg (z) Re e > cos φ − , z ∈ Δ, (4.2.2) g(z) 2

(IV) The function g(z) :=

and it is possible to replace the number μ in this inequality with a number ν only if ν = Rμ, R > 1.

70

Chapter 4. Spirallike Functions with Respect to a Boundary Point

(V) The function h satisﬁes the following three conditions: (a) h is univalent in Δ; h(0) h(z) (b) Re μ z¯ ≥ Re μ z¯ (1 − |z|2 ); h (z) h (0) (c) ∠ lim

z→1

h (z)(z − 1) = μ. h(z)

Moreover, if h is a univalent function on Δ, satisfying one of the assertions (II)– (V) with some complex number μ, Re μ > 0, then μ lies in the disk |μ − 1| ≤ 1 and h ∈ Spiralμ [1]. Remark 4.2. Recall that if f (1) := ∠ lim f (z) exists, then one can deﬁne Qf (1, z) := z→1

f (z)(z − 1) , which is called the Visser–Ostrowski quotient (see (1.2.6)). Thus, f (z) − f (1) it follows by the above assertion V(c) that f ∈ Spiral[1] is starlike whenever ∠ lim Qf (1, z) is a real number. z→1

Proof. The proof of the theorem is done in several steps. Step 1 (I)⇐⇒(II). By Lemma 4.1, it is immediate that if h ∈ Spiralμ [1], then h1 (z) = h(z)1/μ ∈ Spiral1 [1], and if h1 (z) ∈ Spiral1 [1], then h(z) = h1 (z)μ ∈ Spiralμ [1]. Step 2 (III)⇐⇒(IV). This equivalence is veriﬁed by substituting g(z) = g(z) zh(z) in (4.2.1). Indeed, it is easy to see in (4.2.2) and h(z) = (1 − z)μ μ (1 − z) z that zg (z) |μ| 2 zh (z) 1 + z 2 · + = e−iφ · + − e−iφ , μ h(z) 1−z |μ| g(z) 2 and this equality proves our claim. Step 3 (II)⇐⇒(III). Let us assume that (II) holds, i.e., h1 = h1/μ ∈ Spiral1 [1]. Then by Corollary 3.3, h1 belongs to Robertson’s class G = G(1), so h1 satisﬁes the inequality 2zh1 (z) 1 + z Re + > 0, z ∈ Δ, (4.2.3) h1 (z) 1−z which coincides with (4.2.1). If inequality (4.2.1) holds for some ν ∈ C, then the function h1 satisﬁes the inequality Re

2μ zh (z) 1 + z · 1 + > 0. ν h1 (z) 1−z

Therefore, the function h2 (z) = h1 (z)μ/ν belongs to the class G. Thus Corollary 3.3 implies that h2 ∈ Spirall [1] for some positive number l ≤ 1. Consequently, h1 =

4.2. A characterization of spirallike functions with respect to a boundary point71 μ h2 (z)ν/μ ∈ Spiral lν [1]. Hence, by Lemma 4.1, lν μ = 1 or ν = l . As l ≤ 1, assertion μ (III) holds. Assume now that assertion (III) holds. By substituting h(z) = h1 (z)μ we get h1 ∈ G. Using Corollary 3.3 we obtain: h1 ∈ Spirall [1] with some l ≤ 1. Suppose that l < 1. Again, by Corollary 3.3, h1 ∈ G(l); that is, 2zh1 (z) 1 + z > 0, z ∈ Δ. + Re lh1 (z) 1−z

Returning to the function h(z) = h1 (z)μ we see that 2zh (z) 1 + z Re + > 0, lμh(z) 1−z which contradicts our assumption. Thus l = 1, i.e., h1 ∈ Spiral1 [1], and we are done. To proceed we note that by Deﬁnitions 4.2 and 4.3 the inclusion h ∈ Spiralμ [1] implies that, for any z ∈ Δ and t ≥ 0. e−tμ h(z) ∈ h(Δ). This means that the family S = {Ft }t≥0 , deﬁned by Ft (z) := h−1 e−tμ h(z) , is a semigroup of holomorphic self-mappings of Δ. Diﬀerentiating S at t = 0+ we ﬁnd its generator: ∂Ft (z) μh(z) . (4.2.4) f (z) = − = ∂t t=0+ h (z) Step 4 (I)=⇒(V). Let h be a μ-spirallike function. Then condition (a) of assertion (V) follows at once. Hence, as mentioned above, the function f (z) = μ

h(z) h (z)

belongs to G, and by Theorem 2.3 3 4 3 4 Re f (z)¯ z ≥ Re f (0)¯ z (1 − |z|2 ),

z ∈ Δ.

Thus we get inequality V(b). It remains to check condition V(c). As shown above, (I) is equivalent to (III) (Steps 1 and 3). Then for any ν of the form ν = Rμ, R > 1, the following inequality holds: 2zμ 1+z Re + > 0, z ∈ Δ. νf (z) 1 − z

72

Chapter 4. Spirallike Functions with Respect to a Boundary Point

Note also that this inequality no longer holds for other values of ν. By the Riesz–Herglotz formula there exists a probability measure σ on the unit circle such that 2zμ 1 + z ζ¯ 1+z dσ(ζ), z ∈ Δ, + = νf (z) 1 − z 1 − z ζ¯ |ζ|=1

or, equivalently,

μ(z − 1) = νf (z)

|ζ|=1

1 − ζ¯ dσ(ζ), 1 − z ζ¯

z ∈ Δ.

(4.2.5)

Note that the integral representation (4.2.5) is not valid in cases when ν is diﬀerent from Rμ, R > 1. Now set ν = μ. Decomposing σ with respect to the Dirac measure δ at the point ζ = 1 ∈ ∂Δ, one can write σ = (1 − a)σ1 + aδ, where 0 ≤ a ≤ 1, and σ1 and δ are mutually singular probability measures. If a = 0, equation (4.2.5) implies that 1 − ζ¯ μ(z − 1) dσ1 (ζ), z ∈ Δ, λ = (1 − a)μ, = λf (z) 1 − z ζ¯ |ζ|=1

which is valid only if 1 − a ≥ 1. This shows that a = 0 and σ = σ1 is singular with respect to δ. Let {zn } be any sequence in Δ non-tangentially convergent to 1. This means that there is a positive number K such that for all n = 1, 2, . . ., |1 − zn | < K. 1 − Re zn We now consider the functions fn : ∂Δ → C deﬁned by fn (ζ) :=

1 − ζ¯ , 1 − zn ζ¯

ζ ∈ ∂Δ.

It is easy to see that each function fn maps the unit circle ∂Δ onto the circle |w − cn | = |cn |, where cn = cn (ζ) =

1 − z¯n , 1 − |zn |2

Hence,

n = 1, 2, . . . .

2|1 − zn | < 2K 1 − Re zn for all n = big enough. Setting ν = μ in (4.2.5) and applying Lebesgue’s Bounded Convergence Theorem we obtain zn − 1 1 − ζ¯ = lim lim ¯ dσ(ζ) = 1. n→∞ f (zn ) n→∞ 1 − zn ζ |fn (ζ)| ≤ 2|cn| ≤

|ζ|=1

4.3. Subordination criteria for the class Spiralμ [1] Therefore,

73

h (z)(z − 1) μ(z − 1) = ∠ lim = μ, z→1 z→1 h(z) f (z)

∠ lim

and condition V(c) follows. Step 5 (V)=⇒(I). Note that by condition V(a) the image h(Δ) is a simply connected domain. By Theorem 2.3, condition V(b) implies that the function μh(z) belongs to G. Solving the Cauchy problem f (z) = h (z) ⎧ ⎪ ⎨ ∂Ft (z) + μ h(Ft (z)) = 0, ∂t h (Ft (z)) (4.2.6) ⎪ ⎩ F0 (z) = z, z ∈ Δ, we ﬁnd the generated semigroup S = {Ft }t≥0 , namely, Ft (z) := h−1 e−tμ h(z) ∈ Hol(Δ). Thus for all z ∈ Δ the curve {e−tμ h(z), t ≥ 0} is contained in h(Δ), i.e., h ∈ Spiralμ [1]. Assume that for some ν ∈ C with Re ν > 0 the function h belongs to Spiralν [1]. We have already seen in Step 4 that in this case ∠ lim

z→1

h (z)(z − 1) = ν. h(z)

Comparing this equality with (V c) we get ν = μ. This completes the proof of the theorem.

4.3 Subordination criteria for the class Spiralμ [1] In this section we use the well-known notion of subordination. Deﬁnition 4.4. A function h1 ∈ Hol(Δ, C) is said to be subordinate to h2 ∈ Hol(Δ, C) (h1 ≺ h2 ) if there exists a holomorphic function ω with |ω(z)| ≤ |z|, z ∈ Δ, such that h1 = h2 ◦ ω. The following description of spirallike functions of the class Spiral[0] is due to Ruscheweyh (see [124, Corollaries 1 and 2]). Lemma 4.2. Let g ∈ Hol(Δ, C) with g(0) = g (0) − 1 = 0. Let α ∈ (−π/2, π/2) and 0 ≤ β < cos α. Then

z g (z) > β, z ∈ Δ, (4.3.1) Re exp(iα) g(z) if and only if one of the following two conditions holds:

74

Chapter 4. Spirallike Functions with Respect to a Boundary Point

(a) for all u, v ∈ Δ we have ug(vz) ≺ vg(uz)

1 − uz 1 − vz

2(cos α−β) exp(−iα) ;

(4.3.2)

or (b) for all t ∈ (0, 2 cos α) the function g satisﬁes the inequality g z(1 − exp(iα)t) ≤ F (t, α, β)|g(z)| for all z ∈ Δ,

(4.3.3)

where

t F (t, α, β) = |1 − exp(iα)t| 1 − 2 cos α

2 cos α(β−cos α) .

(4.3.4)

Moreover, this bound is sharp. By using this result and Theorem 4.1 one can characterize the class Spiralμ [1] in terms of subordination. Indeed, to do this we just have to substitute g(z) = zh(z) in (4.3.2) and (4.3.3), where h ∈ Spiralμ [1]. By Theorem 4.1 we already (1 − z)μ know that g satisﬁes the inequality zg (z) |μ| > cos φ − , z ∈ Δ, Re e−iφ g(z) 2 where φ = arg μ if and only if h ∈ Spiralμ [1]. Setting α := −φ = − arg μ and β := cos φ −

|μ| , 2

we get: 2(cos α−β) = |μ|. Thus one can rewrite conditions (a) and (b) of Lemma 4.2 in the form: vzh(vz) |μ| exp(i arg μ) u 1 − uz h(vz)(1 − uz)μ (1 − vz)μ = ≺ , (4.3.2 ) uzh(uz) h(uz)(1 − vz)μ 1 − vz v (1 − uz)μ and

z(1 − exp(−iφ)t)h(z(1 − exp(−iφ)t)) ≤ F (t, −φ, β) zh(z) μ μ (1 − z(1 − exp(−iφ)t)) (1 − z) −|μ| cos φ zh(z) t . = |1 − exp(−iφ)t| 1 − (4.3.3 ) · 2 cos φ (1 − z)μ So, we have proved the following characterization of the class Spiralμ [1].

4.4. Distortion Theorems

75

Theorem 4.2. Let h : Δ → C be a holomorphic function and h(0) = 1. Let μ ∈ C, |μ − 1| ≤ 1, μ = 0 and φ = arg μ ∈ (−π/2, π/2). Then h ∈ Spiralμ [1] if and only if one of the following conditions holds: (a) for all u, v ∈ Δ

1 − uz 1 − vz

μ

h(vz) ≺ h(uz)

1 − uz 1 − vz

μ ;

or (b) for all t ∈ (0, 2 cos φ) − Re μ h(z(1 − e−iφ t)) 1 − z(1 − e−iφ t) μ t ≤ · 1− . h(z) 1−z 2 cos φ Furthermore, setting u = 0, v = 1 in Theorem 4.2, we obtain the following corollary. Corollary 4.1. If h ∈ Spiralμ [1], then μ μ 1 1 h(z) ≺ . 1−z 1−z In particular, if h ∈ Starμ [1] with μ ≤ 1 (hence, h ∈ G), then Re

h(z) 1 > . (1 − z)μ 2

For the class of starlike functions, in particular, we also have the following consequence of Theorem 4.2. Corollary 4.2. Let h : Δ → C be a holomorphic function and h(0) = 1. Let μ ∈ (0, 2). Then h ∈ Starμ [1] if and only if, for all t ∈ (−1, 1), μ h(zt) 1 − zt 2 . h(z) ≤ 1 − z · 1 + t

4.4 Distortion Theorems 4.4.1 ‘Spiral angle’ distortion theorems First we observe that by Theorem 4.1 each spirallike function h ∈ Spiral[1] = 0 μ∈Ω Spiralμ [1], Ω = {λ : |λ − 1| ≤ 1, λ = 0}, is a complex power of a starlike function with respect to a boundary point. Therefore, one can apply Theorems 3.3, 3.4 and 3.5 (see also Corollary 3.4) to give geometric characteristics for images of functions of the class Spiralμ [1]. More precisely, let λ ∈ Ω = {μ ∈ C : |μ − 1| ≤ π π 1, μ = 0} and θ ∈ − , be given. Deﬁne the function hλ,θ ∈ Hol(Δ, C) by 2 2 λ 1−z hλ,θ (z) = . 1 + e−2iθ z

76

Chapter 4. Spirallike Functions with Respect to a Boundary Point

Here and later on, we choose a single-valued branch of the analytic function wλ such that 1λ = 1. We will see that hλ,θ ∈ Spiralλ [1]. Deﬁnition 4.5. The set Wλ,θ = hλ,θ (Δ) is called a canonical λ-spiral wedge with midline lθ,λ = {w ∈ C : w = e(iθ+t)λ , t ∈ R} and vertex at the origin. To explain this deﬁnition, let us observe that h = hλ,θ is a solution of the diﬀerential equation λh(z) = h (z)f (z), normalized by the conditions h(0) = 1, h(1) = 0, where f is given by (z − 1) 1 + e−2iθ z . f (z) = 1 + e−2iθ Since f ∈ G + [1] with f (1) = 1 and λ ∈ Ω, it follows by Theorem 4.1 that h is a λ-spirallike function with respect to the boundary point h(1) = 0. Moreover, f is a generator of a one-parameter group of hyperbolic automorphisms of Δ having two boundary ﬁxed points z = 1 and z = −e−2iθ . Hence, for each w ∈ Wλ,θ , the −tλ spiral curve e w, t ∈ R belongs to Wλ,θ . Using notions of Section 4.1, one can see that a canonical λ-spiral wedge belongs to λ-SP. Finally, we see that for real λ ∈ (0, 2], the set Wλ,θ is a straight wedge (sector) of angle πλ, whose bisector is lθ,λ = {w ∈ C : arg w = θλ}. An immediate consequence of Theorems 3.3 and 4.1 is the following assertion. Proposition 4.3. If h ∈ Spiralμ [1], then the canonical spiral wedge Wμ,θ∗ , where θ ∗ := lim− arg h1/μ (r), r→1

is the smallest one that contains the image h(Δ). In other words, this proposition says that each spirallike function of class Spiralμ [1] is subordinate to hμ,θ ∗ . Analogously to Deﬁnition 3.4, we introduce the following subclasses of Spiral[τ ], τ ∈ Δ. Deﬁnition 4.6. We say that a function h ∈ Spiral[τ ] belongs to the subclass Spiral[τ, η], where η ∈ ∂Δ, η = τ , if the angular limit Qh (η) := ∠ lim

z→η

(z − η)h (z) h(z)

(4.4.1)

exists ﬁnitely and is diﬀerent from zero. In the case where τ = 1, we say that h ∈ Spiralμ,ν [1, η] if h ∈ Spiralμ [1] and the angular limit (4.4.1) at the boundary point η equals ν.

4.4. Distortion Theorems

77

To proceed we recall that by Deﬁnition 1.4, a spirallike function h (with respect to an interior or a boundary point) is μ-spirallike if for each point w ∈ h(Δ) its image h(Δ) contains the spiral curve {e−tμ w : t ≥ 0}. Theorem 4.3. Let h ∈ Spiral[τ ], τ ∈ Δ, be a μ-spirallike function on Δ. Then the image h(Δ) contains a canonical λ-spiral wedge with arg λ = arg μ

(4.4.2)

if and only if h ∈ Spiral[τ, η] for some η ∈ ∂Δ. Moreover, if Qh (η) = ν, then for some θ ∈ [0, 2π) the canonical spiral wedge W−ν,θ ⊂ h(Δ); and it is maximal in the sense that there is no spiral wedge Wλ,θ ⊂ h(Δ) with λ satisfying (4.4.2) that contains W−ν,θ properly. Proof. First, given h ∈ Spiral[τ, η], we construct h1 ∈ Spiral[1, −1], which is spirallike with respect to a boundary point whose image eventually coincides at ∞ with h(Δ). If τ ∈ ∂Δ, we just set h1 = h ◦ Φ, where Φ ∈ Aut(Δ) is an automorphism of Δ such that Φ(1) = τ and Φ(−1) = η. If τ ∈ Δ, we take any two points z1 = eiθ1 and z2 = eiθ2 such that w1 = h(z1 ) and w2 = h(z2 ) exist ﬁnitely and θ1 ∈ (arg η − , arg η), θ2 ∈ (arg η, arg η + ), so the arc (θ1, θ2 ) on the unit circle contains the point η. Since h is spirallike with respect to an interior point, it satisﬁes the equation βh(z) = h (z)f (z),

z ∈ Δ,

where f ∈ G + [τ ] and β = f (τ ), so arg μ = arg β. This means that for each w ∈ h(Δ) the spiral curve {e−tβ w, t ≥ 0} belongs to h(Δ). In turn, the curves 1 = {z = h−1 (e−tβ w1 ), t ≥ 0} and 2 = {z = h−1 (e−tβ w2 ), t ≥ 0} lie in Δ with ends in z1 and τ , and z2 and τ , respectively. Since z1 = z2 and the interior points of 1 and 2 are semigroup trajectories in Δ, these curves do not intersect except at their common end point z = τ. Consequently, the domain D bounded by 1 , 2 and the arc (θ1, θ2 ) is simply connected, and there is a conformal mapping Φ of Δ such that Φ(Δ) = D and Φ(−1) = η, Φ(1) = τ . Now deﬁne h1 = h ◦ Φ. It follows by our construction that h1 (Δ) ⊂ h(Δ) and h1 is spirallike with respect to a boundary point h1 (1) = 0. In addition, since Φ is conformal at the point z = −1, it satisﬁes the Visser–Ostrowski condition, and we have (z + 1)h1 (z) (z + 1)h (Φ(z))Φ (z) = ∠ lim z→−1 z→−1 h1 (z) h(Φ(z)) (z + 1)Φ (z) (Φ(z) − η)h (Φ(z)) = ∠ lim · ∠ lim z→−1 Φ(z) − η z→−1 h(Φ(z)) (Φ(z) − η)h (Φ(z)) . = ∠ lim z→−1 h(Φ(z))

∠ lim

(4.4.3)

Note also that Φ is a self-mapping of Δ mapping the point z = −1 to η and having a ﬁnite derivative at this point.

78

Chapter 4. Spirallike Functions with Respect to a Boundary Point

It follows by the Julia–Carath´eodory Theorem, (see, for example, [40, 127] and [130]) that if z converges to −1 non-tangentially, then Φ(z) converges nontangentially to η = Φ(−1). Then (4.4.3) implies that (z + 1)h1 (z) =ν z→−1 h1 (z)

Qh1 (−1) = ∠ lim

(4.4.4)

exists ﬁnitely if and only if h ∈ Spiral[τ, η] and Qh1 (−1) = Qh (η).

(4.4.5)

We claim that this last relation that h1 (Δ) contains a canonical (−ν) implies spiral wedge W−ν,θ for some θ ∈ − π2 , π2 . To this end, observe that h1 satisﬁes the equation βh1 (z) = h1 (z) · f1 (z), f (Φ(z)) is a generator of a semigroup of Δ with f1 (1) = 0 and Φ (z) f1 (1) = β1 for some β1 > 0 such that

where f1 (z) =

|β − β1 | ≤ β. Therefore, h1 is a complex power of the function h2 ∈ Hol(Δ, C) deﬁned by the equation β1 h2 (z) = h2 (z)f1 (z), h2 (1) = 0, (4.4.6) i.e., h1 (z) = hμ2 (z), where μ =

β β1

(4.4.7)

= 0, |μ − 1| ≤ 1, hence arg μ = arg β. 1/μ

On the other hand, if we normalize h2 by h2 (0) = h1 (0), equation (4.4.7) has a unique solution h2 , which is a starlike function with respect to a boundary point (h2 (1) = 0). Obviously, 1 1 Qh2 (−1) = Qh1 (−1) = Qh (η) . (4.4.8) μ μ Note that ν1 := Qh2 (−1) is a negative real number, while ν := Qh1 (−1) = ν1 μ is complex. Now it follows by Theorem 3.4 that the starlike set h2 (Δ) contains a straight wedge (sector) of a nonzero angle π|ν1 |. Namely, the maximal (straight) wedge W∗ ⊂ h2 (Δ) is of the form 5 6 −ν1 1−z W∗ = W−ν1 ,θ1 = w ∈ C : w = , z∈Δ , 1 + e−2iθ1 z

4.4. Distortion Theorems

79

with ν /ν

−ν1 θ1 = lim− arg h2 (−r) = lim− arg h11 (−r) r→1

r→1

1/ν

= ν1 · lim− arg h1 (−r) = ν1 θ∗ , r→1

where 1/ν

θ∗ = lim arg h1 (−r). r→1−

Therefore, W∗μ = W−ν1 μ,θ1 = W−ν,−θ∗ is contained in h1 (Δ); hence in h(Δ). Finally, it follows by (4.4.8) that λ := −ν = |ν1 |μ. This implies (4.4.2). Conversely, let h be a μ-spirallike function on Δ such that h(Δ) contains a canonical λ-spiral wedge Wλ,θ for some λ satisfying (4.4.2) and θ ∈ − π2 , π2 . Then for each w0 ∈ Wλ,θ , the curve := w ∈ C : w = e−tλ w0 , t ∈ R belongs to h(Δ). Hence the curve h−1 (l) ⊂ Δ joins the point τ ∈ Δ to a point η ∈ ∂Δ. Again, as in the ﬁrst step of the proof, one can ﬁnd a conformal mapping Φ ∈ Hol(Δ) with Φ(1) = τ, Φ(−1) = η such that h1 = h ◦ Φ is a μ-spirallike function with respect to a boundary point, h1 (1) = 0, and Wλ,θ ⊂ h1 (Δ) ⊂ h(Δ).

(4.4.9)

1/μ

Again the function h2 = h1 is starlike with respect to a boundary point, and h2 (Δ) contains the (straight) wedge W λ ,θ because of (4.4.9). μ

λ Setting = κ, we see by (4.4.2) that κ is positive real. Hence, h2 (Δ) contains μ a straight canonical wedge Wκ,θ with 0 < κ ≤ |ν1 |, where ν1 = Qh2 (−1) exists ﬁnitely, and W|ν1 |,θ is the maximal wedge contained in h2 (Δ). But, as before, we have ν = Qh (η) = μQh2 (−1) = μν1 . The latter relations show that ν is ﬁnite and λ must satisfy the conditions arg λ = arg μ = arg(−ν) and 0 < |λ| ≤ |ν|. So the wedge W−ν,θ is a maximal canonical spiral wedge contained in h(Δ) satisfying condition (4.4.2). The theorem is proved.

4.4.2 Growth estimates for semigroup generators In this section we obtain growth estimates for semigroup generators from the classes considered in Section 2.4.

80

Chapter 4. Spirallike Functions with Respect to a Boundary Point

Let f (z) ∈ G + [1] with f (1) = β+ > 0. It follows by (2.4.21) and Julia’s Lemma that for some F ∈ Hol(Δ) 2 1 + F (z) (4.4.10) |f (z)| = c|1 − z| 1 − F (z) 2 |1 − F 2 (z)| 2 1 − |F (z)| = c|1 − z|2 ≥ c|1 − z| |1 − F (z)|2 |1 − F (z)|2 2 2 c|1 − z| 1 − |z| β+ (1 − |z|2 ), ≥ · = z ∈ Δ. α+ |1 − z|2 2 Similarly, for x ∈ (−1, 1) we obtain 1 + F (x) 1 − F (x) β+ 1 − |F (x)|2 ≤ − (1 − x2 ) . = −c(1 − x)2 |1 − F (x)|2 2

Re f (x) = −c(1 − x)2 Re

(4.4.11)

On the other hand, if f ∈ G[1, −1], then 1 1 − F (z) 1 = f (z) c|1 − z|2 1 + F (z) 1 2 1 − |z|2 1 1 − |z|2 ≥ = − , c|1 − z|2 α− |1 + z|2 β− |1 − z 2 |2 or

2 β− 1 − z 2 . |f (z)| ≤ − 2 1 − |z|2

(4.4.12)

Similarly, for x ∈ (−1, 1), Re

1 − F (x) 1 1 Re =− (4.4.13) 2 f (x) c(1 − x) 1 + F (x) 2 (1 − x2 ) 2 1 − |F (x)|2 1 1 ≤ = . =− 2 2 2 2 c(1 − x) |1 + F (x)| |β− | (1 − x ) |β− | 1 − x2

Combining (4.4.10) and (4.4.12) we get the following assertion. Theorem 4.4 (A distortion theorem). Let f ∈ G[1, −1] with β+ = f (1) ≥ 0 and β− = f (−1) < 0. Then for z ∈ Δ, |β− | |1 − z 2 |2 β+ 1 − |z|2 ≤ |f (z)| ≤ . 2 2 1 − |z|2 In particular, if z is in the circular lens |1 − z 2 | ≤ M , ΥM = z ∈ Δ : 1 − |z|2

M ≥ 1,

(4.4.14)

4.4. Distortion Theorems then

81

M |β− | β+ |1 − z 2 | ≤ |f (z)| ≤ 1 − z2 . 2M 2 Similarly, by using (4.4.11)–(4.4.13) we obtain the following.

(4.4.15)

Theorem 4.5. Let f ∈ G + [1, −1]∩Gh [1] with f (1) = β+ > 0 and f (−1) = β− < 0. Then for all x ∈ (−1, 1), β+ β− (1 − x2 ) ≤ Re f (x) ≤ − (1 − x2 ) < 0, 2 2 and −

1 1 1 2 2 ≤ Re < 0. ≤ β+ 1 − x2 f (x) β− 1 − x2

4.4.3 Growth estimates for spirallike functions In turn, the above assertions enable us to get distortion theorems for unbounded starlike and spirallike functions with respect to a boundary point of the classes Spiralμ,ν [1, η] deﬁned in Deﬁnition 4.6. Recall that each function h ∈ Spiralμ,ν [1, η] satisﬁes generalized Visser– Ostrowski conditions (z − 1)h (z) =μ z→1 h(z)

Qh (1) := ∠ lim and

Qh (η) := ∠ lim

z→η

(z − η)h (z) =ν h(z)

with μ ∈ Ω := {λ : |λ − 1| ≤ 1,

λ = 0}

and 0 ≤ − μν ≤ 1 (see the proof of Theorem 4.3). In addition, the function f ∈ Hol(Δ, C), deﬁned by f (z) =

μh(z) , h (z)

(4.4.16)

is of class G + [1] with f (1) = 1 (see also [130]) and h (Ft (z)) = e−μt h(z), where {Ft }t≥0 is the semigroup generated by f (see Section 2.5). Now for h ∈ Spiralμ [1] we obtain, from (4.4.10) and (4.4.16), h (z) 1 2|μ| h(z) = |μ| |f (z)| ≤ 1 − |z|2 .

(4.4.17)

82

Chapter 4. Spirallike Functions with Respect to a Boundary Point

Integrating (4.4.17) we get |log h(z)| ≤ log

1 + |z| 1 − |z|

2|μ| .

(4.4.18)

Assume now that h ∈ Spiralμ,ν [1, η], i.e., ∠ lim h(z) = ∞ and z→η

ν := ∠ lim

z→η

(z − η)h (z) h(z)

is ﬁnite and diﬀerent from zero. Without loss of generality we suppose that η = −1. Then again from (4.4.16) we have: ∠

f (z) h(z) μ = μ lim = . z→η=−1 z + 1 z→−1 (z + 1)h (z) ν lim

(4.4.19)

Hence, ∠ lim f (z) = 0 and z→−1

∠ lim f (z) =: β−

(4.4.20)

z→−1

is ﬁnite, i.e., f ∈ G[1, −1]. Moreover, it follows by (4.4.19) and (4.4.20) that νβ− = μ

(4.4.21)

Re ν < 0.

(4.4.22)

h (z) 1 − |z|2 h(z) ≥ 2|ν| |1 − z 2 |2

(4.4.23)

and Then we get, by (4.4.14),

for all z ∈ Δ. Thus we have obtained the following distortion theorem for a function of the class Spiralμ,ν [1, −1]. Theorem 4.6. Let h ∈ Spiralμ,ν [1, −1]. Then for all z ∈ Δ the following estimate holds: 2 (1 − |z| )2 |ν| h (z) 2 |μ| ≤ ≤ 2 2 . h(z) |1 − z 2 | 1 − |z| In particular, if z is in the circular lens |1 − z 2 | ≤M , ΥM = z ∈ Δ : 1 − |z|2 then

M ≥ 1,

h (z) 1 2|ν| ≤ 2|μ|M . ≤ 2 M |1 − z | h(z) |1 − z|2

4.4. Distortion Theorems

83

In fact, we have shown also that if h ∈ Spiralμ,ν [1, −1], then f deﬁned by (4.4.16) belongs to G[1, −1] with f (1) > 0. Conversely, let f ∈ G[1, −1] be given with f (1) = β+ = 1. As we already know for each μ ∈ Ω := {|λ − 1| ≤ 1, λ = 0}, equation (4.4.16) has a unique univalent solution h satisfying conditions h(0) = 1 and h(1) := lim h(r) = 0. We r→1−

want to show that actually h ∈ Spiralμ,ν [1, −1] for some ν ∈ C with Re ν < 0. Indeed, because of relations (4.4.19)–(4.4.21), such ν exists, i.e., h satisﬁes a generalized Visser–Ostrowsi condition. Thus it remains to show that ∠ lim h(z) = z→−1

∞, or, by Lindel¨ of’s Theorem 1.6 that lim h(x) = ∞.

x→−1

To this end we return to formula (2.4.21): f (z) = −c(1 − z)2 and consider equation

1 + F (z) , 1 − F (z)

h1 (z) = h1 (z)f (z).

(4.4.24)

This equation has a unique univalent solution h1 satisfying conditions h1 (0) = 1 and h1 (1) = 0. Also, it is easy to see (comparing (4.4.16) and (4.4.24) that, in fact, h(z) = h1 (z)μ . Therefore, it is suﬃcient to prove our assertion for function h1 . By (2.4.21) and (4.4.24) we get, for x ∈ (−1, 1), − Re

h1 (x) 1 1 − F (x) = Re 2 h1 (x) c(1 − x) 1 + F (x) 1 − |F (x)|2 1 1 2 = ≥ . 2 c(1 − x) |1 + F (x)|2 |β− | 1 − x2

(4.4.25)

Now if x ∈ (−1, 0) we obtain x h (x) dx Re 1 − log |h1 (x)| = − Re log h1 (x) = − h1 (x) 0 x 2 dx 1 1+x 2 ≤ · log , = |β− | 0 1 − x2 |β− | 2 1−x and we have log |h1 (x)| → ∞ as x → −1. Note that inequality (4.4.26) is equivalent to |h1 (x)| ≥

1−x 1+x

1

| β− |

.

(4.4.26)

84

Chapter 4. Spirallike Functions with Respect to a Boundary Point

If x ∈ (0, 1), then (4.4.25) implies − log |h1 (x)| ≥ or

|h1 (x)| ≤

1−x 1 log |β− | 1+x

1−x 1+x

1

| β− |

.

Finally, we observe that if ν = −μ, then β− = −1 = −β+ , hence, f (z) = must be the generator of a group of hyperbolic automorphisms with ﬁxed points at z = 1 and z = −1, such that z = 1 is the Denjoy–Wolﬀ point, i.e., f (z) = β2+ z 2 − 1 . Thus, we have proved the following assertion. μh(z) h (z)

Theorem 4.7. A univalent function h on Δ belongs to the class Spiralμ,ν [1, −1] if and only if the function μh(z) f (z) = h (z) belongs to the class G + [1, −1] with f (1) > 0. Moreover, if μ is real (μ ∈ (0, 2]), then so is ν (ν ≤ −μ < 0) and h is, in fact, a starlike function on Δ satisfying the estimates: |h(x)| ≤ and

|h(x)| ≥

1−x 1+x

1−x 1+x

−ν ,

x ∈ [0, 1),

−ν ,

x ∈ (−1, 0].

The equality ν = −μ is possible if and only if μ 1−z h(z) = . 1+z

4.4.4 Classes G(μ, β) Here we establish a number of distortion results, namely, estimates for h and h for holomorphic functions h belonging to some subclasses of the class Spiralμ [1]. More precisely, we consider the following classes of functions. Deﬁnition 4.7. We say that a univalent function h ∈ Hol(Δ, C), h(0) = 1, belongs to the class G(μ, β) (cf., Deﬁnition 3.3 and Theorem 4.1), where μ ∈ Ω := {λ ∈ C : |λ − 1| ≤ 0, λ = 0} and β ∈ [0, 1), if h satisﬁes the inequality 2 zh (z) 1 + z Re · + > β, z ∈ Δ. (4.4.27) μ h(z) 1−z

4.4. Distortion Theorems

85

To study properties of these classes we start with an integral representation that generalizes Lemma 3.2.2. Theorem 4.8. Let h be a univalent function. Then h ∈ G(μ, β) if and only if h admits the following representation: ¯ log(1 − z ζ)dσ(ζ) , (4.4.28) h(z) = (1 − z)μ exp −μ(1 − β) |ζ|=1

where σ is a probability measure on the unit circle |ζ| = 1. Proof. First, suppose that the function h is represented by (4.4.28). Diﬀerentiating (4.4.28) and making a simple calculation, we check that inequality (4.4.27) holds. Suppose now that h ∈ G(μ, β). Consider the function

1 2 zh (z) 1 + z p(z) = + −β , 1 − β μ h(z) 1−z which obviously belongs to the Carath´eodory class and thus can be represented by the Riesz–Herglotz formula: p(z) = |ζ|=1

1 + z ζ¯ dσ(ζ), 1 − z ζ¯

where σ is a probability measure. This implies that

1 h (z) (1 − β)ζ¯ − = dσ(ζ). μh(z) 1−z 1 − z ζ¯ |ζ|=1 Integrating both sides of this equation we obtain (4.4.28).

Remark 4.3. It follows by Theorem 4.1, Theorem 4.8 and Deﬁnition 4.7 that any function h, spirallike with respect to a boundary point, has the integral representation μ ¯ log(1 − z ζ) dσ(ζ) h(z) = (1 − z) exp −μ |ζ|=1

for some μ ∈ Ω and a probability measure σ. Deﬁne another measure σ such that σ = βλ + (1 − β)σ, where λ is the normalized Lebesgue measure on the unit circle ∂Δ. Obviously, dσ = 1. Since the integral of the antiholomorphic function ∂Δ

¯ with respect to the Lebesgue measure λ is zero, we have that h belongs log(1 − z ζ) 1 (σ − βλ) is positive. to the class G(μ, β) if and only if the measure σ = 1−β The following assertion is an immediate consequence of Theorem 4.8.

86

Chapter 4. Spirallike Functions with Respect to a Boundary Point

Corollary 4.3. Let μ1 , μ2 ∈ Ω and β1 , β2 ∈ [0, 1). Let h ∈ Hol(Δ, C) be univalent. Then h ∈ G(μ1 , β1 ) if and only if the function μ2 (1−β2 )

h(z) = (1 − z)μ2 (β2 −β1 ) (h(z)) μ1 (1−β1 ) 1

belongs to G(μ2 , β2 ). In particular, h ∈ G(μ, β) if and only if h(z) = (h(z)) μ belongs to G(1, β). Corollary 4.4. Let μ1 , μ2 ∈ Ω, μ1 = rμ2 with r ≤ 1, and let β2 ≤ rβ1 . Then G(μ1 , β1 ) ⊂ G(μ2 , β2 ). Proof. Let h ∈ G(μ1 , β1 ). We have h(z) = (1 − z)

μ1

exp −μ1 (1 − β1 )

|ζ|=1

¯ log(1 − z ζ)dσ(ζ)

= (1 − z)μ2 exp −μ2 (1 − rβ1 )

|ζ|=1

¯ σ(ζ) , log(1 − z ζ)d

where σ is a probability measure on the unit circle deﬁned by σ=

r(1 − β1 ) 1−r σ+ δ, 1 − rβ1 1 − rβ1

with δ the Dirac measure at the point ζ = 1. By Theorem 1.1, h satisﬁes the inequality 2 zh (z) 1 + z + > rβ1 ≥ β2 . · Re μ2 h(z) 1−z

Therefore, h ∈ G(μ2 , β2 ). Corollary 4.5. The set ⎫ ⎧ μ ∞ ⎨ n ⎬ ! 1−z 7n : λ ≥ 0, λ = 1 − β, |ζ | = 1 j j j ¯ λj ⎭ ⎩ j=1 (1 − z ζj ) n=1

j=1

is dense in G(μ, β) in the topology of uniform convergence on compact subsets of Δ. Proof. Replacing the integral in (4.4.28) by approximating sums, we have for any h ∈ G(μ, β): ⎡ ⎤ n h(z) = lim (1 − z)μ exp ⎣−μ(1 − β) log(1 − z ζ¯j )σj ⎦ n→∞

= lim (1 − z)μ n→∞

j=1 n " j=1

(1 − z ζ¯j )−μ(1−β)σj ,

4.4. Distortion Theorems with

n *

87

σj = 1. Putting λj = (1 − β)σj , we complete the proof.

j=1

Corollary 4.6. Let h ∈ G(μ, β). Then

1 h(z) ≺ . μ (1 − z) (1 − z)μ(1−β)

Proof. Since the function log(1 − z) is convex, for any probability measure σ there exists an analytic function ω : Δ → Δ with ω(0) = 0 such that ¯ log(1 − z ζ)dσ(ζ) = log(1 − ω(z)). |ζ|=1

Using Theorem 4.8 we obtain the equality h(z) 1 ¯ = exp −μ(1 − β) log(1 − z ζ)dσ(ζ) = , (1 − z)μ (1 − ω(z))μ(1−β) |ζ|=1 which proves our assertion.

It turns out that for classes G(μ, β) one can improve the result of Proposition 4.3. μ 1−z , where Corollary 4.7. Let f ∈ G(ν, β). Then h ≺ hμ,θ ∗ (z) := 1 + e−2iθ ∗ z μ πν μ ≤ 1 and |θ∗ | < (1 − β). Consequently, ν is a real number satisfying β ≤ ν 2μ 1−β π . |θ∗ | < · min 1, 2 β Proof. Using representation (4.4.28) one can estimate the numbers μ and θ ∗ from Proposition 4.3 (see also Theorem 4.1). Speciﬁcally, ζ¯ h (r)(r − 1) = 1 − (1 − β)(1 − r) ¯ dσ(ζ). νh(r) ∂Δ 1 − rζ Thus

h (r)(r − 1) μ = lim− ≥ β. ν νh(r) r→1 Further, ν ¯ . arg(1 − rζ)dσ(ζ) |θ∗ | = lim (1 − β) r→1− μ ∂Δ

Thus the assertion follows.

Remark 4.4. For the case when h ∈ G(ν, β) with real ν (consequently, h ∈ Spiralμ [1] with real μ), Proposition 4.3 (see also Theorem 3.3) asserts that the image of a function h is contained in the wedge of angle μπ with the midline arg w = μθ. The last corollary implies that if h ∈ G(ν, β), then the angle cannot be less than πνβ, and the argument of the midline cannot be greater than π 2 ν(1 − β).

88

Chapter 4. Spirallike Functions with Respect to a Boundary Point

Now we are ready to establish desired estimates for h and h when h ∈ G(μ, β). In particular, given a point z ∈ Δ, we ﬁnd the set of values for some functionals on these classes. For Robertson’s class G (the class of starlike functions with respect to a boundary point having image in a half-plane) similar results were established by Todorov [141]. Theorem 4.9 (A distortion theorem for the class G(μ, β)). For each ﬁxed z ∈ Δ we have 1−z : h ∈ G(μ, β) = (1 + λz)1−β , |λ| ≤ 1 (4.4.29) (i) h(z)1/μ and (ii)

h (z) 1 + : h ∈ G(μ, β) μh(z) 1 − z (1 − β)¯ z 1−β . = w : w − ≤ 1 − |z|2 1 − |z|2

(4.4.30)

Furthermore, if h ∈ G(μ, β) and z ∈ Δ, z = 0, one of the relations 1−z 1−β ∈ (1 + λz) , |λ| = 1 h(z)1/μ and

1 h (z) + ∈ μh(z) 1 − z

(1 − β)¯ z 1−β w : w − = 1 − |z|2 1 − |z|2

holds only if h(z) =

(1 − z)μ ¯ (1−β)μ , (1 − z ξ)

ξ ∈ ∂Δ.

Proof. (i) By Theorem 4.8,

1−z ¯ log(1 − z ζ)dσ(ζ). log = (1 − β) h(z)1/μ ∂Δ

(4.4.31)

(4.4.32)

By the Carath´eodory Principle, 1−z log : h ∈ G(μ, β) h(z)1/μ ¯ ζ ∈ ∂Δ , = Conv (1 − β) log(1 − z ζ), where Conv denotes the convex hull. Since for each z ∈ Δ the function g(w) := (1 − β) log(1 − zw) maps Δ onto a 1−z strictly convex domain, this formula and (4.4.32) imply that the value log h(z) 1/μ

4.4. Distortion Theorems

89

belongs to the image g(Δ) except for the case when the measure σ is the Dirac δ-function at some boundary point ξ. Hence the assertion follows. (ii) Once again using representation (4.4.28) we get h (z) ζ¯ 1 + = (1 − β) ¯ dσ(ζ). μh(z) 1 − z ∂Δ 1 − z ζ ¯

ζ maps the For each ﬁxed z ∈ Δ, the function g : ∂Δ → C deﬁned by g(ζ) := 1−z ζ¯ z¯ 1 unit circle onto the circle w : w − = . Therefore, (4.4.30) 1 − |z|2 1 − |z|2 follows from the Carath´eodory Principle.

For real μ similar facts can be found in [29, 135, 42]. The following two assertions are immediate consequences of Theorem 4.9 (i). 0 1−z : h ∈ G(μ, β) = (1 + λ)1−β , |λ| ≤ 1 . Corollary 4.8. 1/μ z∈Δ h(z) Corollary 4.9. For each z ∈ Δ and h ∈ G(μ, β), 1−z ≤ (1 + |z|)1−β , (1 − |z|)1−β ≤ h(z)1/μ 1 − z arg ≤ (1 − β) arcsin |z|, h(z)1/μ

and

where for z = 0, equality is achieved only for the functions (4.4.31) at the points z = ±|z|ξ and z = |z|ξe±i arccos |z| , respectively. In particular, if μ is real, then |1 − z|μ |1 − z|μ ≤ |h(z)| ≤ . (1 + |z|)μ(1−β) (1 − |z|)μ(1−β) In the case when μ ∈ (0, 2] (i.e., h ∈ G(μ, β) is, in fact, starlike), one obtains the following estimate for h . Corollary 4.10. For each z ∈ Δ and h ∈ G(μ, β), μ ∈ (0, 2], β ∈ [0, 1), μ|1 − z|μ (1 − |z|2 )(1 + |z|)μ(1−β)

1 − z¯ 1 − z + β z¯ − 1 + β ≤ |h (z)| 1 − z¯ μ|1 − z|μ ≤ + β z¯ + 1 − β . (1 − |z|2 )(1 − |z|)μ(1−β) 1 − z

In particular, |h (z)| ≤

2μ|1 − z|μ . (1 − |z|2 )(1 − |z|)μ(1−β)

The proof follows from Theorem 4.9 (ii) and Corollary 4.9.

90

Chapter 4. Spirallike Functions with Respect to a Boundary Point

4.5 Covering theorems for starlike and spirallike functions In this section we prove a covering theorem for classes G(μ, β) of spirallike functions with respect to a boundary point. Note that if for some class of functions there exists a domain D such that D ⊂ h(Δ) for each function h of the class, and a function h0 ∈ Hol(Δ, C) maps the open unit disk onto D conformally, then the corresponding covering result can be expressed as the subordination h0 ≺ h (see Deﬁnition 4.4). Theorem 4.10. Let h ∈ G(μ, β). Denote h0 (z) = (1 − z)μβ . Then h0 ≺ h. This subordination is sharp since h0 ∈ G(μ, β). Proof. By Corollary 4.3, it is suﬃcient to prove this assertion for the case μ = 1. First we suppose that h ∈ G(1, β) has the form n 1−z , λj = 1 − β, |ζj | = 1. λj j=1 (1 − zζj ) j=1

h(z) = 7n

Let z0 = eiφ ∈ ∂Δ, z0 = ζ1 , ζ2 , . . . , ζn . Consider the two univalent convex functions on Δ, g0 (z) = log h0 (z) = β log(1 − z) and g(z) = log(1 − z0 ) − (1 − β) log(1 − z0 z). It is easy to see that g0 (z0 ) = g(1), i.e., this is their common boundary point. Furthermore, zg0 (z)|z=z0 =

−βz0 1 − z0

and

zg (z)|z=1 =

(1 − β)z0 , 1 − z0

i.e., the exterior normal vectors to the images g0 (Δ) and g(Δ) at this common point 2 have reverse directions. Because these functions are convex, g0 (Δ) g(Δ) = ∅. Thus, log h(z0 ) = log(1 − z0 ) −

n

λj log(1 − z0 ζj )

j=1

=

n 4 λj 3 log(1 − z0 ) − (1 − β) log(1 − z0 ζj ) 1−β j=1

belongs to g(Δ) and does not belong to g0 (Δ) = log h0 (Δ). Since the point h(z0 ) is an arbitrary ﬁnite boundary point of the image h(Δ), we have proved that the boundary ∂h(Δ) does not intersect h0 (Δ). This fact implies that h0 is subordinate to any function h ∈ G(1, β) that has the form described above. In light of Corollary 4.5 and by the Carath´eodory Kernel Convergence Theorem we deduce the assertion of the theorem.

4.5. Covering theorems for starlike and spirallike functions

91

Example 4.1. Consider the function h(z) =

1−z . (1 − 0.9z − 0.4iz)0.2(1 − 0.9z + 0.4iz)0.2

A simple calculation shows that h ∈ G(μ, β) with μ = 1 and β = 0.6. Hence by Theorem 4.10, its image h(Δ) contains the image h0 (Δ), where h0 (z) = (1 − z)0.6 (see Fig. 4.1).

0.8 0.6 0.4 0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

–0.2 –0.4 –0.6 –0.8

Figure 4.1: The image and the covered domain. Corollary 4.11. If h ∈ G(μ, β) with μ ∈ (0, 2] and β ∈ (0, 1), then the image of h covers the disk {w : |w − 1| < r(μβ)}, where 5 √ 1 + 22s − 2s+1 , when s ∈ (0, 1] . r(s) = 1, when s ∈ (1, 2] Proof. It suﬃces to ﬁnd the distance between the point z = 1 and the curve {(1 − z)s , z ∈ ∂Δ}, where s = μβ. Indeed, setting z = −eit , t ∈ (−π, π), consider the function 2s s 2 t t st it s as (t) = (1 + e ) − 1 = 2 cos + 1 − 2 2 cos cos . 2 2 2 This function is even in t, as increases on the segment 0 ≤ t ≤ π when s < 1, and as decreases on the same segment when s > 1. So, 5 as (0), when s ∈ (0, 1] . min as (t) = −π

92

Chapter 4. Spirallike Functions with Respect to a Boundary Point

Remark 4.5. In [29] Chen and Owa proved a covering theorem for starlike functions with respect to a boundary point. Using our notation, their result can be quoted as follows: If h ∈G(μ, β) for some μ, β ∈ [0, 1] then its image covers the disk w : |w − 1| < μβ . The radius of the covered disk, which was found in the last 4 corollary, is considerably larger than the one due to Chen and Owa. In Fig. 4.2 we compare these two radii.

1

0.8

0.6

r 0.4

0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

s

Figure 4.2: Radii of covered disks. The following assertion connects the classes G(μ, β) with spirallike functions with respect to an interior point. Proposition 4.4. Let β ∈ [0, 1), and let a complex number μ = r eiφ belong to Ω = {λ : |λ − 1| ≤ 1, λ = 0}. Then a univalent function h, h(0) = 1, belongs to the class G(μ, β) if and only if the function g, deﬁned by g(z) :=

zh(z) , (1 − z)μ

r(1 − β) is φ-spirallike of order cos φ − , i.e., g satisﬁes the condition 2 r(1 − β) zg (z) > cos φ − , z ∈ Δ. Re e−iφ g(z) 2

(4.5.1)

(4.5.2)

Remark 4.6. Note that the relation μ = r eiφ ∈ Ω implies that cos φ − r(1−β) ≥ 0. 2 ˇ By a result of Spaˇcek (see [75]), the function g is univalent in Δ. Proof. Let a function g be deﬁned by (4.5.1). Then r zh (z) μz −iφ zg (z) = 1+ + e g(z) μ h(z) 1−z r(1 − β) r 2zh (z) 1 + z + − β + e−iφ − . = 2 μh(z) 1−z 2 This equality implies the assertion.

4.5. Covering theorems for starlike and spirallike functions

93

Repeating the proof of Theorem 4.2 and using Proposition 4.4 and Lemma 4.2, one can conclude the following. Corollary 4.12. Let h : Δ → C be a holomorphic function and h(0) = 1. Let μ = reiφ ∈ Ω and β ∈ [0, 1). Then h ∈ G(μ, β) if and only if one of the following conditions holds: (a) for all u, v ∈ Δ,

1 − uz 1 − vz

μ

h(vz) ≺ h(uz)

1 − uz 1 − vz

(1−β)μ ;

or (b) for all t ∈ (0, 2 cos φ), − Re μ(1−β) h(z(1 − e−iφ t)) 1 − z(1 − e−iφ t) μ t · 1− ≤ . h(z) 1−z 2 cos φ Setting u = 0 and v = 1 in Corollary 4.12, we obtain an alternative proof of Corollary 4.6 above. The next assertion follows from Theorem 4.10 and Proposition 4.4.

φ−α) iφ e . Corollary 4.13. Let φ ∈ − π2 , π2 , α < cos φ, β ∈ 0, cosα φ , and μ = 2(cos 1−β

Foreach univalent function g normalized by g(0) = 0, g (0) = 1 and satisfying zg (z) > α, the image of the function Re e−iφ g(z)

1 − (1 − z)

1 β

g(z) z

1 μβ

covers the open unit disk, and, consequently, the image of the function 1

(1 − z) β

1

2 − (1 − z) β

g(z) z

1 μβ

g(z) z

1 μβ

covers the right half-plane. In particular, setting φ = 0, α = β = starlike of order

1 2,

we get that for each function g 2 g(z) contains the the image of the function 1 − (1 − z) z

open unit disk, and the image of the function half-plane.

1 , 2

(1 − z)2 g(z) contains the right 2z − (1 − z)2 g(z)

Chapter 5

Kœnigs Type Starlike and Spirallike Functions In this chapter we study linear-fractional models for discrete and continuous semigroups of holomorphic functions on the open unit disk Δ. This study is based on solutions of classical Schr¨ oder’s and Abel’s functional equations. It turns out that diﬀerent problems of the semigroup theory (like the embedding problem, asymptotic behavior and rigidity) closely connected to geometrical properties of the solutions of Schr¨ oder’s and Abel’s equations. In order to emphasize more important conceptual issues, we refer the reader to related sources for the proofs of some technical results for discrete time semigroups.

5.1 Schr¨oder’s and Abel’s equations The circle of ideas related to functional equations has a very long history. In 1871, Schr¨ oder [125] considered the problem of fractional iteration and related it to the solution of various functional equations including the functional equation h ◦ F = λh,

(5.1.1)

which bears his name. Here F ∈ Hol (Δ) and λ ∈ C are given. Note, in passing, that Schr¨ oder’s equation can be considered the eigenvalue problem for the composition operator CF deﬁned by CF (h) = h ◦ F. In 1884, Kœnigs [93] constructed a solution of Schr¨ oder’s equation near a ﬁxed point τ ∈ Δ of F when F (τ ) = λ, |λ| = 0, 1. In the years since these classical works, much has been written; the interested reader might consult the bibliographies in [38], [95], [96] and [92].

96

Chapter 5. Kœnigs Type Starlike and Spirallike Functions

The setting in that work was the analysis of a contractive map, analytic in a neighborhood of a ﬁxed point. The solution h ∈ Hol (Δ, C) of the basic equation was obtained as a concrete limit, now frequently called Kœnigs’ function in this context. A classical analysis problem is, given a function F (z), ﬁnd a function Ft (z) with F1 (z) = F (z), satisfying the semigroup property: 5 Ft+s (z) = Ft (Fs (z)) , t, s ≥ 0, (5.1.2) F0 (z) = z. This problem is called the (semigroup) embedding problem. Deﬁnition 5.1. A holomorphic self-mapping F ∈ Hol(Δ) of the open unit disk Δ is said to be embeddable on a domain Ω ⊂ Δ if there is a continuous one-parameter semigroup S = {Ft }t≥0 ⊂ Hol(Ω) such that F |Ω = F1 . In the case where Ω = Δ we say that F is embeddable globally1 . In this situation F n = Fn , where as above F n denotes the n-fold iterate of F. So, the element Ft , t > 0, can be considered the fractional iterate of F. It is easy to see that each automorphism of Δ is always embeddable globally in a one-parameter group (of automorphisms) of Δ. Although for a given F ∈ Hol(Δ) the discrete semigroup {F n }∞ n=1 in general may not be embeddable in a continuous semigroup, at each z ∈ Δ fractional iterates Ft of F can be deﬁned for all t(= t(z)) large enough (see, for example, [38, 40]). Note that a necessary condition for F ∈ Hol(Δ) to be embeddable on Ω is the univalence of F on Ω (see, for example, [17] or [130]). Therefore, if F (ξ) = 0 at some point ξ ∈ Ω, then F is not embeddable. If F is analytic in a neighborhood of a ﬁxed point τ, F (τ ) = τ, and 0 < |F (τ )| < 1, Kœnigs (1884) showed how the problem can be solved. The limit lim

n→∞

F n (z) − τ =: h (z) (F (τ ))n

(5.1.3)

can be shown to exist for z near τ . Then h satisﬁes Schr¨oder’s functional equation (5.1.1) with λ = F (τ ), whence F (z) = h−1 [λh(z)] and F n (z) = h−1 [λn h(z)]. Moreover, the latter expression serves a local deﬁnition of Ft (z) when t is not an integer. If, for example, λ = F (τ ) is real, λ = 0, 1, and h ∈ Hol(Δ, C) is a univalent starlike function on Δ, then the family {Ft (·)}t≥0 deﬁned by Ft (z) = h−1 [λt h(z)]

(5.1.4)

forms a one-parameter continuous semigroup of holomorphic mappings globally on Δ and F1 (z) = F (z). 1 Sometimes,

one also says that F satisﬁes the Kœnigs embedding property; see Section 5.3.

5.1. Schr¨ oder’s and Abel’s equations

97

Another approach when F has no interior ﬁxed point is via Abel’s functional equation (1881): g (F (z)) = g(z) + c,

c ∈ C.

(5.1.5)

If g can be found to satisfy (5.1.5) and is convex in the direction c (i.e., for each z ∈ Δ and t > 0 we have g(z) + tc ∈ g(Δ), cf. Section 3.5), then one may set Ft (z) = g −1 [tc + g(z)] to embed F into a continuous semigroup. A summary of some of the classical work on the problem may be found in Cowen [38], Hadamard [78], Harris [79] and Siskakis [136]. As it turns out, both Schr¨ oder’s and Abel’s equations can be considered particular cases of a uniﬁed construction. Under very general conditions, a holomorphic mapping F of the disk Δ into itself can intertwine with a linear-fractional transformation Φ in the following sense. Deﬁnition 5.2. Suppose F is a holomorphic self-mapping of the unit disk Δ. If there exists a univalent mapping σ : Δ → C, and a linear-fractional mapping Φ such that Φ(Δ) ⊂ Δ, Φ (σ(Δ)) ⊂ σ(Δ), and σ ◦ F = Φ ◦ σ,

(5.1.6)

we call the pair (Φ, σ) a linear-fractional (conjugation) model for F (shortly, LFModel). The mapping Φ related to F by (5.1.6) is said to be conjugate to F by σ. For Schr¨ oder’s equation (5.1.1), one sets σ = h and Φ(w) = λw, w ∈ C, whilst for Abel’s equation (5.1.5) σ = g and Φ(w) = w + c, w ∈ C. In turn, the induction method naturally involves the study of the whole ∞ discrete semigroup {F n }n=1 of iterates of F via the conjugation σ ◦ F n = Φn ◦ σ.

(5.1.7)

If, in addition, the image σ(Δ) lies in a half-plane (or a disk) that is Φinvariant, then F can be intertwining with a linear-fractional transformation Ψ of the unit disk, deﬁned by the conjugation: ◦ F = Ψ ◦ ,

(5.1.8)

where is a self-mapping of Δ (cf. [21]). Since many problems involving iteration or solution of functional equations can be explicitly solved for linear fractional transformations and the intertwining relates these answers to solutions of the problems for a given function F , we obtain the information we seek. Regarding continuous-time semigroups, the following general observation is an important point in our further considerations.

98

Chapter 5. Kœnigs Type Starlike and Spirallike Functions

Theorem 5.1. Let f ∈ G generate a semigroup {Ft }t≥0 ⊂ Hol(Δ). Let σ : Δ → C be any univalent function on Δ with Ω = σ(Δ), and let ϕ be a holomorphic function on Ω. Then σ satisﬁes the diﬀerential equation ϕ(σ(z)) = σ (z)f (z),

z ∈ Δ,

(5.1.9)

if and only if ϕ is the generator on Ω of the semigroup {Φt }t≥0 ⊂ Hol(Ω) deﬁned by σ(Ft (z)) = Φt (σ(z)), t ≥ 0. (5.1.10) In other words, functional equation (5.1.10) and diﬀerential equation (5.1.9) are equivalent. Proof. The suﬃciency follows immediately, if we diﬀerentiate equation (5.1.10) at the point t = 0+ . Conversely, let σ satisfy (5.1.9) for some Φ ∈ Hol(Ω, C). Deﬁne a oneparameter semigroup ϕt : Ω → Ω by the formula Φt (w) = σ(Ft (σ−1 (w)),

t ≥ 0.

(5.1.11)

Now we calculate

dFt σ −1 (w) −1 dΦt (w) = σ Ft σ (w) · dt dt = −σ Ft σ −1 (w) · f Ft σ−1 (w) ,

t ≥ 0.

In turn, formulas (5.1.11) and (5.1.10) imply dΦt (w) = −σ σ−1 (Φt (w)) · f σ−1 (Φt (w)) dt = −ϕ(σ(σ −1 (Φt (w)))) = −ϕ(Φt (w)). So, ϕ ∈ Hol(Ω, C) is the generator of the semigroup {Φt}t≥0 and we are done. Remark 5.1. Note also that solutions of equation (5.1.9) deﬁne a generalized class of the so-called ϕ-like functions in the sense of Brickman [22] (see Sections 1.1.5 and 2.5). If, in particular, ϕ(w) = μ(w − w0 ) for some w0 ∈ C and μ ∈ C with Re μ > 0, a solution σ of equation (5.1.9) (if it exists) deﬁnes a spirallike function with respect to w0 (in this case w0 must lie in σ(Δ)). In turn, (5.1.10) is an equation of Yaglom’s type: σ(Ft (z)) = e−tμ σ(z) + (1 − e−tμ )w0 (see, [79]), and it becomes Schr¨oder’s equation if w0 = 0. Similarly, if in (5.1.9) we set ϕ(z) = c = const, a solution of equation (5.1.10) (if it exists) deﬁnes a function convex in one direction (see Section 3.5), and solves Abel’s functional equation (5.1.5).

5.2. Remarks on stochastic branching processes

99

To proceed we need the following classiﬁcation. Deﬁnition 5.3. A holomorphic self-mapping F of Δ is said to be of • dilation type if it has a ﬁxed point in Δ; • hyperbolic type if it has no ﬁxed point in Δ and has the angular derivative F (τ ) at its Denjoy–Wolﬀ point τ ∈ ∂Δ strictly less than 1: 0 < F (τ ) < 1; • parabolic type if it has no ﬁxed point in Δ and has the angular derivative F (τ ) = 1 at its Denjoy–Wolﬀ point τ ∈ ∂Δ. All hyperbolic type mappings are automorphic in the sense that their orbits are separated in the hyperbolic metric of Δ (meaning that, for each z ∈ Δ, the hyperbolic distance between successive points of the orbit {F n (z)} stays bounded away from zero). Parabolic type mappings fall into two subclasses: • automorphic type: those having an orbit that is separated in the hyperbolic metric of Δ; • non-automorphic type: those for which no orbit is hyperbolically separated. Remark 5.2. It can be shown that either all orbits are separated or none are separated; see details in Section 5.5. Traditionally, the intertwining map σ in (5.1.6) is constructed by a limiting process from F , as illustrated by Kœnigs [93] in 1884 for the Denjoy–Wolﬀ point inside the disk. The hyperbolic case was developed by Valiron [143]. In general, the boundary location of the Denjoy–Wolﬀ point τ (in automorphic as well as in nonautomorphic cases) has been investigated by Pommerenke [110] and Baker and Pommerenke [12]. A uniﬁed construction for parabolic type mappings is presented by Bourdon and Shapiro in [18] (for details see the recent paper [36] by Contreras, D´ıaz-Madrigal and Pommerenke). Additional information for all types of mappings can be found in Shapiro’s book [127] and the survey of Bourdon and Shapiro [18]. Iteration of functions for which F (a) = 0 was studied in [38] and [71] via Bottcher’s functional equation f (F (z)) = (f (z))k .

5.2 Remarks on stochastic branching processes A model for iterations, which is related to the theory of branching processes in Probability Theory, goes back to the classical Galton–Watson processes through the probability generating functions. The probability generating function for a ∞ * pk z k , where pk ≥ 0 and Galton–Watson process is the function F (z) = ∞ * k=0

k=0

pk = 1. (see, for example, [79]). The coeﬃcient pk is interpreted as the prob-

ability that an individual will have k oﬀspring and the k th coeﬃcient of F n is interpreted as the probability that there will be k oﬀspring in the nth generation.

100

Chapter 5. Kœnigs Type Starlike and Spirallike Functions

In other words, if the number of particles in the nth generation is a random variable Z(n), then one can show (see [10]) that its generating function F n (z) =

∞

pk (n)z k ∈ Hol(Δ),

n ∈ N,

(5.2.1)

k=0

with pk (n) = P {Z(n) = k | Z(0) = 1}

(5.2.2)

is the n-fold iterate of F 1 (z) = F (z). So, {F n }∞ n=1 is a discrete-time semigroup of holomorphic self-mappings of Δ. Questions concerning eventual population size are related to the asymptotic behavior of the iterates of F . We note that every probability generating function for the Galton–Watson process is an analytic function mapping Δ into itself. If we replace the set of indexes N = {0, 1, 2, . . .} by R+ = {t ∈ R : t ≥ 0}, a Markov branching process {Z(t), t ∈ R+ } can be considered a continuous-time analog of the Galton–Watson process and interpreted as the number of oﬀspring produced at any (continuous) time t ∈ R+ . We concentrate now on generating functions Ft , t ≥ 0, of continuous branching processes deﬁned by the same formulas (5.2.1)–(5.2.2) for all nonnegative t: Ft (z) =

∞

pk (t)z k ∈ Hol(Δ),

t ≥ 0,

(5.2.3)

k=0

where pk (t) are deﬁned by pk (t) = P {Z(t) = k | Z(0) = 1} . By deﬁnition pk (t) ≥ 0 and

∞ *

(5.2.4)

pk (t) ≤ 1. A branching process Z(t) is called

k=0

regular if the latter sum is equal to 1, and then the point z = 1 is a ﬁxed point for all of functions Ft , t ≥ 0. In this case S := {Ft }t≥0 is a continuous-time semigroup of holomorphic self-mappings of Δ. The diﬀerential equation ⎧ ⎨ ∂Ft + f (Ft ) = 0, t ≥ 0, ∂t (5.2.5) ⎩ F0 = z, z ∈ Δ in this situation is called the Kolmogorov backward equation (see [10]). It is known that generators of such semigroups have the following form: ∞ f (z) = a z − p˜n z n , n=0

5.2. Remarks on stochastic branching processes

101

where the number a > 0 and the so-called inﬁnitesimal probabilities p˜n satisfying ∞ * p˜n = 1 are the total data of the process. Thus, 0 ≤ p˜n and n=0

f (z) =

∞

pn z n ,

(5.2.6)

n=0

where p1 ≥ 0, pn ≤ 0 for n = 1 and

∞

pn = 0.

n=0

Representation (5.2.6) implies immediately that f (1) = 0. The smallest root q ∈ [0, 1] of the equation f (x) = 0 has a clear probability meaning: q = lim P {Z(t) = 0} , t→∞

and is the probability that the given process becomes extinct. The number q is called the extinction probability of the branching process. Since Ft (0) = P {Z(t) = 0} we have that q (= lim Ft (0)) is the Denjoy–Wolﬀ t→∞

point of the semigroup, so each generator of type (5.2.6) can be also represented by the Berkson–Porta formula (2.2.7) with τ = q. From the point of view of the theory of branching processes it is natural to assume that the ﬁrst moment (the expectation) E [Z(t)] =

∞

n˜ pn

n=1

for a suitable Markov process exists, i.e., the series this means that the ﬁrst derivative f (1) =

∞ *

∞ *

n˜ pn converges. In our terms

n=1

npn for a generator f having the

n=1

form (5.2.6) exists. In this case f ∈ G[1]. We already know (see Theorem 2.11) that for f ∈ G[1] the number β = f (1) is real. It is now natural to diﬀerentiate among the following three cases: β > 0, β = 0 and β < 0. If β = f (1) ≥ 0, then by Theorem 2.10, the point τ = 1 is the Denjoy– Wolﬀ point for the semigroup generated by f , and f can be represented by the Berkson–Porta formula: f (z) = −(1 − z)2 p(z), where Re p(z) ≥ 0. If, in addition, f has the form of (5.2.6), then the continuoustime branching process related to the semigroup S := {Ft }t≥0 generated by f has the extinction probability q = lim P {Z(t) = 0} = 1. t→∞

102

Chapter 5. Kœnigs Type Starlike and Spirallike Functions

In the case when β > 0, the semigroup generated by f converges to τ = 1 with exponential rate of convergence, namely |Ft (z) − 1| ≤ C(z) exp(−tβ) (see Section 7.2.1). In this situation all elements of the semigroup are of hyperbolic type and the corresponding branching process is called subcritical. If β = 0, the rate of convergence is more slow than exponential. In this case the semigroup elements are of parabolic type; in the theory of branching processes this case is called critical. If β = f (1) < 0, then the point ζ = 1 is not an attractive point of the semigroup generated by f . Hence, it follows by the Denjoy–Wolﬀ Theorem and the Berkson–Porta representation that f must have a diﬀerent null point τ = 1, which is the Denjoy–Wolﬀ point for the semigroup S generated by f . If, in addition, f is of the form (5.2.6), this point (the extinction probability q) must lie on the interval [0, 1); this case is called supercritical. Note also that in the last situation the Berkson–Porta representation is not related to the boundary null point ζ = 1 of f . However, formula (2.4.4) proven above, can be used for all values of β. In particular, we have the following consequence of Theorem 2.11. Corollary 5.1. Let β 2 (z − 1) 2 be an inﬁnitesimal generating function of a supercritical branching process with the ﬁrst moment β < 0. Then the extinction probability q is the unique, on the interval [0, 1), root of the equation f (z) = −(1 − z)2 p(z) +

p(x) =

β x+1 . 2 x−1

For convenience we summarize the above facts in the following table. 1.

β = f (1) β>0

2.

β=0

3.

β<0

General case hyperbolic, ζ = 1 is the Denjoy–Wolﬀ point parabolic, ζ = 1 is the Denjoy–Wolﬀ point there is an additional root of f (z) = 0 in Δ, ζ = 1 is not the Denjoy–Wolﬀ point

Probabilistic case subcritical case critical case supercritical case, there is 0 ≤ q < 1 such that f (q) = 0

The bibliographies of Harris [79] and Athreya and Ney [10] are guides to classical work on branching processes. Solutions of Schr¨ oder’s and Abel’s equations (as general linearization models) play a crucial role in the study of branching processes and their asymptotics (see Section 7.2.1). Here we also mention Yaglom’s equation gˆ(F (z)) = mˆ g(z) + 1 − m,

5.3. Kœnigs’ linearization model for dilation type semigroups. Embeddings 103 which is a combination of Schr¨ oder’s and Abel’s equations.

5.3 Kœnigs’ linearization model for dilation type semigroups. Embeddings Kœnigs’ approach to solution of Schr¨ oder’s equation is an important complement of assertion (i) of the Denjoy–Wolﬀ Theorem mentioned above and plays a crucial role in the study of asymptotic behavior of discrete complex dynamics. It shows that any holomorphic self-mapping of Δ that has a non-zero derivative F (τ ) of module less than 1 at a ﬁxed point τ can be “linearized” in the sense that the original mapping is analytically conjugate in a neighborhood of τ to its derivative mapping w → λw, where (necessarily) λ = F (τ ). Here we give Kœnigs’ solution of Schr¨ oder’s equation for holomorphic mappings F ∈ Hol(Δ) with an interior ﬁxed point. Theorem 5.2 (see [127, 130]). Suppose F is a holomorphic self-mapping of Δ that ﬁxes a point τ ∈ Δ, is not an (elliptic) automorphism of Δ, and has non-vanishing derivative λ at τ . Then there exists the limit F n (z) − τ =: h(z), n→∞ λn lim

(5.3.1)

which satisﬁes Schr¨ oder’s equation (5.1.1). Moreover, if F is univalent, then so is h. Remark 5.3. Note that in this case the iterates F n have the following asymptotic behavior: F n (z) − τ ∼ λn . If F has an interior ﬁxed point τ ∈ Δ with λ = F (τ ) = 0, the global embedding property of F can be described in terms of its Kœnigs function. Theorem 5.3. Let F ∈ Hol(Δ) be a univalent function on the open unit disk Δ with an interior ﬁxed point τ ∈ Δ, and λ = F (τ ). Let h ∈ Hol(Δ, C) be the Kœnigs function for F deﬁned by (5.3.1). The following assertions hold: (i) If for some domain Ω ⊂ Δ, Ω τ, the restriction h|Ω is a (− log λ)-spirallike function, then F is embeddable on Ω. (ii) There exists a domain Ω ⊂ Δ, τ ∈ Ω, which satisﬁes (i), whence F (Ω) ⊂ Ω and F is embeddable on Ω. (iii) If λ = F (τ ) is a real number, then Ω can be chosen as the pseudohyperbolic disk of non-euclidean radius tanh π4 : π Ω = z ∈ Δ : |Mτ (z)| < tanh , 4 where Mτ (z) =

τ −z 1−z τ¯

is the M¨ obius transformation of the unit disk Δ.

104

Chapter 5. Kœnigs Type Starlike and Spirallike Functions

Proof. (i) Let F ∈ Hol(Δ) be a univalent function on the open unit disk Δ with an interior ﬁxed point τ ∈ Δ, λ = F (τ ). Then by Theorem 5.2, its Kœnigs function h is also univalent. Suppose that h is a (− log λ)-spirallike function on Ω. Then one can deﬁne a semigroup S = {Ft }t≥0 ⊂ Hol (Ω) by the formula Ft (z) = h−1 λt h(z) , z ∈ Ω, t ≥ 0. On the other hand, by Schr¨ oder’s equation (5.1.1) we have F1 (z) = h−1 (λh(z)) = F (z) for all z ∈ Ω ⊂ Δ, so F is embeddable on Ω. (ii) Since h(τ ) = 0 and h (τ ) = 1, we have by the Koebe One-Quarter Theorem (see, for example, [75]) that the image h(Δ) contains the disk Δ1 := w : |w| < 14 . By the Schwarz–Pick Lemma |λ| ≤ 1, and we also have that for each w ∈ Δ1 and for each t ≥ 0 the point λt w also belongs to Δ1 . Deﬁne Ω = h−1 (Δ1 ). Since Δ1 = h(Ω) is λ-spirallike we have that F is embeddable on Ω. (iii) By Theorem 5.2, h is a univalent function. Deﬁne a univalent function h1 by h1 (z) =h (Mτ (z)). Then h1 (0) = 0 and h1 (0) = h (τ ) = 1. Hence h1 maps the disk Δ2 = w : |w| < tanh π4 onto a starlike domain (see, for example [74]). Since h(Ω) = h1 (Δ1 ) and λ = F (τ ) is real, we conclude by (i) that F is embeddable on Ω. Theorem 5.4. Let F ∈ Hol(Δ) be a univalent function on the open unit disk Δ with an interior ﬁxed point τ ∈ Δ, and λ = F (τ ) with 0 < |λ| < 1. Let h ∈ Hol(Δ, C) be the Kœnigs function for F deﬁned by (5.3.1). Then F is embeddable globally if and only if h is a univalent spirallike function on Δ satisfying the condition Re

(z − τ ) (1 − zτ ) h (z) < 0, h (z) log λ

z ∈ Δ.

(5.3.2)

Proof. If h satisﬁes (5.3.2), then h is (− log λ)-spirallike. So, by Theorem 5.3, F can be globally embeddable. Suppose now that F is embeddable globally in a one-parameter semigroup S = {Ft }t≥0 . By the Berkson–Porta formula the inﬁnitesimal generator f of S admits the representation f (z) = (z − τ )(1 − z τ¯)p(z), where p is a holomorphic function in Δ with a positive real part. Hence, f (τ ) = (1 − |τ |2 )p(τ ). On the other hand, it follows from the Cauchy problem (2.1.3) that λ = F (τ ) (= F1 (τ )) = exp (−f (τ )). So, log λ = −(1 − |τ |2 )p(τ ). We now deﬁne the function ⎫ ⎧ z ⎬ ⎨ 1 (1 − |τ |2 )p(τ ) − 1 dζ . h1 (z) := (z − τ ) exp ⎭ ⎩ ζ −τ (1 − ζ τ¯)p(ζ) τ

This function is well-deﬁned, because p does not vanish in Δ.

5.3. Valiron’s type linearization models

105

For a ﬁxed z ∈ Δ, the function ξ(t) = h1 (Ft (z)) is deﬁned on [0, ∞). By the chain rule, dξ(t) = −h1 (Ft (z))f (Ft (z)) = h1 (Ft (z)) log λ = ξ(t) log λ. dt Thus ξ(t) is a solution of the Cauchy problem: dξ(t) = ξ(t) log λ, dt

ξ(0) = h1 (z).

Solving this problem, we obtain ξ(t) = h1 (Ft (z)) = et log λ h1 F (z). Setting t = 1 we ﬁnd that h1 is a solution of Schr¨ oder’s equation. By the uniqueness theorem for Schr¨ oder’s equation (see, for example [127]), h(z) = ch1 (z). Therefore, h (z) h (z) − log λ = 1 = , h(z) h1 (z) (z − τ )(1 − z τ¯)p(z)

where Re p(z) > 0; that is, h satisﬁes (5.3.2).

Remark 5.4. Thus if f is the generator of a one-parameter semigroup S = {Ft }t≥0 such that f (τ ) = 0 for some τ ∈ Δ, then the limit (5.3.1) with F = F1 , F n = Fn exists if and only if Re μ > 0, where μ = f (τ ) and λ = exp(−μ). Moreover, in this case the following asymptotic formula holds: h (z) = lim etμ (Ft (z) − τ ) t→∞

(see, for example, [58]). This formula is a continuous analog of (5.3.1).

5.4 Valiron’s type linearization models for hyperbolic type semigroups. Embeddings Even in the discrete case the situation becomes more complicated if F ∈ Hol(Δ) has no ﬁxed point in Δ. If this is the case, the Denjoy–Wolﬀ Theorem 1.11 asserts that there is a unique boundary point τ ∈ ∂Δ such that lim F n (z) = τ,

n→∞

z ∈ Δ.

Without loss of generality one can set τ = 1. We have F (1) := ∠ lim F (z) = 1 and 0 < α = F (1) := ∠ lim F (z) ≤ 1. z→1

z→1

Recall that F is classiﬁed to be of hyperbolic type if α = F (1) < 1. If F is a generating function of a Galton–Watson branching process, this case is also called subcritical.

106

Chapter 5. Kœnigs Type Starlike and Spirallike Functions In this situation one can consider the limit lim

n→∞

1 − F n (z) = ϕ (z) , αn

z ∈ Δ.

This function (if it exists) is an analog of the Kœnigs function (5.3.1) for the boundary case. The problem is that ϕ might be identically zero and then cannot serve as a characterization of the asymptotic behavior of F as well as its embedding criteria. Valiron [143] showed that ϕ(z) is a well-deﬁned non-constant holomorphic function on Δ if and only if ∞ 1 − F n+1 (0) (5.4.1) 1 − F n (0) < ∞. n=1 Another somewhat more explicit necessary and suﬃcient condition was established by Pommerenke [113]: 1 1 α − (5.4.2) 1 − F (x) 1 − x dx < ∞. 0 In addition, he showed that ϕ must be conformal at the boundary point τ = 1, i.e., the angular limit ∠ lim ϕ(z) z−1 exists ﬁnitely and is not zero. z→1

Earlier Wolﬀ and Valiron suggested another normalization and proved the following limit result. Theorem 5.5 (see, for example, [143]). Let F ∈ Hol(Δ) be such that F (1) = 1 and α = F (1) < 1. Then the limit 1 − F n (z) = ψ (z) , n→∞ |1 − F n (0)| lim

z ∈ Δ,

always exists and satisﬁes Schr¨ oder’s equation ψ ◦ F = αψ. This function ψ is also sometimes called the Kœnigs function for F (see [49, 32]). Remark 5.5. Once the existence of the above limit is proved, it is easy to see that the limit 1 − F n (z) = h(z), z ∈ Δ, lim n→∞ 1 − F n (0) also exists with h(z) = κψ(z) for some κ, |κ| = 1. For our purpose sometimes it is more convenient to consider this function because of the normalization h(0) = 1. We call it the Kœnigs–Valiron function for F (or simply the (K-V)–function). Moreover, each other solution to Schr¨oder’s equation of positive real part must be a positive multiple of ψ (see [20]).

5.4. Valiron’s type linearization models

107

We will see that in our settings a hyperbolic type mapping F has the (semigroup) embedding property if and only if its (K-V)–function is starlike with respect to a boundary point (see Section 3) and belongs to the class Star1 [1]. Theorem 5.6. Let F ∈ Hol(Δ) be embeddable. Suppose that F has its Denjoy–Wolﬀ point at τ = 1 with α = F (1) < 1. Then for each ξ ∈ Δ the limit hξ (z) = lim

n→∞

1 − F n (z) 1 − F n (ξ)

exists and the Kœnigs–Valiron function h0 (z) = Star1 [1].

hξ (z) hξ (0)

is a function of the class

Proof. First we note that for a given point ξ ∈ Δ the family of functions hξ,n (z) =

1 − F n (z) 1 − F n (ξ)

(5.4.3)

is a normal family. Indeed, since −π < arg(1 − F n (z)) − arg(1 − F n (ξ)) < π, we have that each hξ,n cannot obtain values in the left-hand side of the real axis. Therefore, by Montel’s Theorem, {hξ,n }∞ n=0 is normal, i.e., there is a subsequence {hξ,nk }∞ which converges locally uniformly on Δ either to a holomorphic k=1 function hξ or to inﬁnity. However, the last case is impossible because of the obvious equality hξ,n (ξ) = 1.

(5.4.4)

Setting, in particular, ξ = 0 we have h0 (0) = 1.

(5.4.5)

Now it follows by the Hurwitz Theorem that h0 is either a univalent function on Δ, or a constant. To see that the last case is again impossible, we diﬀerentiate hξ,n (z) to obtain (F nk ) (z) (5.4.6) h (z) = lim nk k→∞ F (ξ) − 1 and

h (z) (F nk ) (z) = lim n . k→∞ F k (z) − 1 h(z)

(5.4.7)

Further, since F is embeddable, we consider the net {ht }t≥0 deﬁned by ht (z) =

1 − Ft (z) . 1 − Ft (ξ)

(5.4.8)

108

Chapter 5. Kœnigs Type Starlike and Spirallike Functions

Since Fn = F n , we have by the previous considerations that there is a sequence {htk }∞ k=1 that converges to h. We will show that, in fact, the net {ht } itself converges to h, and h is of class Star1 [1]. To do this, consider now the expression qt (z) =

(Ft ) (z) . Ft (z) − 1

(5.4.9)

If f is the inﬁnitesimal generator of the semigroup {Ft }t≥0, then ∂Ft (z) = −f (Ft (z)). ∂t

(5.4.10)

On the other hand, it is known (see, for example, [115]) that {Ft }t≥0 also satisﬁes the diﬀerential equation ∂Ft (z) ∂Ft (z) =− f (z). ∂t ∂z

(5.4.11)

Comparing (5.4.10) and (5.4.11) we obtain the functional equation f (Ft (z)) =

∂Ft (z) · f (z). ∂z

(5.4.12)

By using (5.4.9) and (5.4.12) we get now qt (z) =

f (Ft (z)) 1 · . f (z) Ft (z) − 1

(5.4.13)

Since τ = 1 is the Denjoy–Wolﬀ point for the semigroup {Ft }t≥0 , it follows by a continuous version of the Julia–Wolﬀ–Carath´eodory Theorem (see Section 1.3) that the angular derivative β := f (1) = ∠ lim f (z) = ∠ lim z→1

z→1

f (z) z−1

(5.4.14)

exists and is a nonnegative real number. Moreover, ∂Ft (z) ∂Ft (z) = e−tβ . = ∠ lim z→1 ∂z z=1 ∂z

(5.4.15)

Since F1 = F and α = F (1) < 1, we obtain from (5.4.15) β = f (1) = − ln α > 0.

(5.4.16)

In addition, it follows by (5.4.7), (5.4.9) and (5.4.13) that f (Ftk (z)) 1 h (z) = lim qtk (z) = lim . k→∞ h(z) f (z) k→∞ Ftk (z) − 1

(5.4.17)

5.4. Valiron’s type linearization models

109

Since F1 ∈ Hol(Δ) is of hyperbolic type, it follows by Lemma 2.66 in [40] that for each compact K Δ there is a non-tangential approach region containing all the iterates Fn (K). By using compactness arguments we have that {Ft (z)}t≥0 converges to τ = 1 non-tangentially for each z ∈ Δ. Thus the limit f (Ftk (z)) = f (1) = β > 0. k→∞ Ftk (z) − 1 lim

So, we have by (5.4.17) that the function h(z) deﬁned by (5.4.6) satisﬁes the diﬀerential equation βh(z) = h (z)f (z). (5.4.18) Finally, it follows by Theorem 2.14 that h is a starlike function with respect to a boundary point h(1) = 0 ∈ ∂Δ. Moreover, (5.4.18) and (5.4.16) imply that the Visser–Ostrowski condition (z − 1)h (z) =1 z→1 h(z)

Qh (1) := ∠ lim

h ∈ Star1 [1]. h(0) In addition, it follows by the uniqueness of the solution of (5.4.18) that the limit in (5.4.6) does not depend on the choice of a convergent sequence {hnk }∞ k=1 . Thus we arrive at

holds. In other words, h0 =

h0 (z) = lim

n→∞

1 − F n (z) 1 − Ft (z) = lim , 1 − F n (0) t→∞ 1 − Ft (0)

(5.4.19)

and we are done. The following is a converse assertion.

Theorem 5.7. Let F ∈ Hol(Δ) have the Denjoy–Wolﬀ point at τ = 1, and let its (K-V)–function h (= h0 ) be not a constant on Δ. Then (i) h(F (0)) = α = F (1) = ∠ lim F (z) < 1; z→1

(ii) if h ∈ Star1 [1], then F is embeddable. Proof. Since h is not constant, F must be of hyperbolic type (see [110, p. 440]). Hence α = F (1) < 1. We have, by (5.4.19), 1 − F n+1 (z) n→∞ 1 − F n (0) 1 − F n+1 (z) 1 − F n (F (0)) = lim · = h(F (0)) · h(z). n→∞ 1 − F n+1 (0) 1 − F n (0)

h(F (z)) = lim

110

Chapter 5. Kœnigs Type Starlike and Spirallike Functions

On the other hand, since F n (0) = zn converges to τ = 1 non-tangentially, we have by the Julia–Carath´eodory Theorem that F n (F (0)) − 1 F (zn ) − 1 = lim = F (1) =: α < 1. n→∞ n→∞ F n (0) − 1 zn − 1

h(F (0)) = lim

Thus h satisﬁes Schr¨oder’s equation h(F (z)) = αh(z).

(5.4.20)

Now assume that h ∈ Star1 [1]. Since h(Δ) is a starlike domain, one can deﬁne the semigroup S = {Ft }t≥0 by Ft (z) = h−1 (e−βt h(z)),

(5.4.21)

where β = − ln α > 0. Then it follows by (5.4.20) and (5.4.21) that F1 (z) = h−1 (e−β h(z)) = h−1 (αh(z)) = F (z).

The theorem is proved.

Remark 5.6. Actually, in the proof of Theorem 2 (ii) we used only the fact that the (K-V)–function h of F is a starlike function with respect to a boundary point. Using, in particular, Lemma 3.2, one can formulate the following characterization of embeddable hyperbolic type self-mappings of the unit disk. Corollary 5.2. Let F ∈ Hol(Δ) have the Denjoy–Wolﬀ point at τ = 1. The following are equivalent: (i) F (1) = ∠ lim F (z) < 1 and F is embeddable; z→1 3 4 (ii) there is λ ∈ 0, 12 such that the function s ∈ Hol(Δ, C) deﬁned by s(z) =

z · h(z), (1 − z)2−2λ

where h is the (K-V)–function for F , is a normalized starlike function of order λ, i.e., s ∈ Sλ∗ . Remark 5.7. The extremal case λ = 0 described in Corollary 5.2 (ii) is of special interest because it is connected to the Koebe function, the classical extremal function for class S ∗ . Indeed, for λ = 0 we have that assertion (ii) is equivalent to the following one: (ii ) the function s deﬁned by s(z) = k(z)h(z), where k(z) = origin.

z (1−z)2

is the Koebe function, is starlike with respect to the

5.4. Valiron’s type linearization models

111

Remark 5.8. We have seen already that for F ∈ Hol(Δ) with F (1) = 1, α = F (1) < 1, the (K-V)–function h(z) is a solution of Schr¨ oder’s functional equation h(F (z)) = αh(z). If F is embeddable into a semigroup {Ft }t≥0 generated by f ∈ G + [1], then h is also a unique solution of the equation h(Ft (z)) = e−tβ h(z),

t ≥ 0,

where β = f (1) = − log α, normalized by the condition h(0) = 1. Also h(1) := ∠ lim h(z) = 0. z→1

The uniqueness can be seen by the uniqueness of the solution of (5.4.18) represented by ⎛ z ⎞ βdz ⎠. h(z) = exp ⎝ (5.4.22) f (z) 0

Since (5.4.22) is equivalent (5.4.18), this representation of the (K-V)–function with some f ∈ G[1] with β = f (1) (= − ln α) > 0 is also suﬃcient for F to be embeddable. By using the representations of class G[1] given in Theorems 2.7 and 2.13, one can formulate the following criteria for a self-mapping F of hyperbolic type to be embeddable. Corollary 5.3. Let F ∈ Hol(Δ) with F (1) = 1, α = F (1) < 1, and let h ∈ Hol(Δ, C) be its (K-V)–function deﬁned by 1 − F n (z) . n→∞ 1 − F n (0)

h(z) = lim The following assertions are equivalent: (i) F is embeddable;

p(z) z→1 z−1

(ii) there exists p ∈ Hol(Δ, C) with Re p ≥ 0 and ∠ lim such that h admits the representation ⎛ z ⎞ βp(z)dz ⎠, h(z) = exp ⎝− (1 − z)2

= p (1) = (ln α)−1

β = − log α;

0

(iii) for any c ∈ (0, − 21 ln α) there is G ∈ Hol(Δ) with G(1) = 1 and γ = G (1) = ∠ lim G(z) = 2c β , β = − ln α, such that the (K-V)–function h admits the z→1 representation ⎞ ⎛ z 1 − G(z) 2 dz ⎠ . h(z) = exp ⎝− γ (1 − z)2 (1 + G(z)) 0

112

Chapter 5. Kœnigs Type Starlike and Spirallike Functions

Proof. The equivalence of assertions (i) and (ii) is just an interpretation of formula (5.4.22) in the spirit of the Berkson–Porta representation of class G[1]: f (z) = −(1 − z)2 q(z)

(5.4.23)

with Re q ≥ 0 everywhere. 1 Setting p(z) = q(z) , we get by (5.4.23) that 0 < β := f (1) = lim

z→1

1 f (z) (z − 1) = − lim =− = − ln α. z→1 p(z) z−1 p (1)

In turn, (5.4.23) implies that (ii) is equivalent to (iii) when we present q(z) in the form 1 + G(z) , q(z) = c 1 − G(z) where G ∈ Hol(Δ) is a self-mapping of Δ. Since c ∈ (0, β2 ) we have 0 < β = ∠ lim

z→1

f (z) (1 − z) = c · lim · (1 + G(z)). z→1 1 − G(z) z−1

Since |G(z)| < 1, z ∈ Δ, we have that the limit γ = ∠ lim

z→1

1 − G(z) 1−z

(5.4.24)

exists and γ = 2c < 1. β The reverse considerations complete our proof.

5.5 Pommerenke’s and Baker–Pommerenke’s linearization models for semigroups with a boundary sink point 5.5.1 Pommerenke’s linearization model for automorphic type mappings For a pair z, w in Δ by ρ(z, w) we denote the hyperbolic distance between z and w, where ρ(·, ·) is deﬁned by (2.2.3). Assume that F ∈ Hol(Δ) has a boundary Denjoy–Wolﬀ point. It can be shown that if for some point z0 ∈ Δ the orbit ∞ {F n (z0 )}n=0 satisﬁes the condition lim ρ F n+1 (z0 ), F n (z0 ) = 0, then the same n→∞ condition holds for all z ∈ Δ, i.e., lim ρ F n+1 (z), F n (z) = 0. n→∞

5.5. Linearization models

113

Deﬁnition 5.4. Given a parabolic mapping F ∈ Hol(Δ), one says that F is of automorphic type (or of positive hyperbolic step) if for some z ∈ Δ (hence for all z ∈ Δ) lim ρ F n+1 (z), F n (z) > 0. n→∞

Otherwise, if for some z ∈ Δ (hence for all z ∈ Δ) lim ρ F n+1 (z), F n (z) = 0, n→∞

F is said to be of non-automorphic type (or of zero hyperbolic step). Let F ∈ Hol(Δ) have the boundary Denjoy–Wolﬀ point τ = 1. We study a linearization model h(F (z)) = ϕ(h(z)),

(5.5.1)

where ϕ is an aﬃne mapping in Π+ = {w ∈ C : Re w ≥ 0} of the form ϕ(w) = aw + b and h : Δ → Π+ is a conjugation for F and ϕ. Theorem 5.8 (Pommerenke). Let F be a holomorphic self-mapping of Δ of automorphic type with F (1) = 1, F (1) = α ≤ 1. Then (i) the limit h(z) = lim hn (z), z ∈ Δ exists, where n→∞

1 + F n (z) 2i Im F n (0) hn (z) = d(F (0)) − n 1 − F (z) |1 − F n (0)|2

n

with

d(z) =

|1 − z|2 , 1 − |z|2

and h : Δ → Π+ with h(0) = 1; (ii) the function h satisﬁes the following (Abel–Schr¨ oder) functional equation h(F (z)) = where b = lim 2d(F n (0)) n→∞

1

1 h(z) + ib, α

1−F n+1 (0)

−

1 1−F n (0)

b ∈ R,

= 0.

Remark 5.9. As a matter of fact, if F is of non-automorphic type, then the limit h(z) = lim hn (z) also exists, but h(z) ≡ 1 and b in (ii) is zero. So one concludes n→∞ that F is of automorphic type if and only if h is not constant. Remark 5.10. If F is of hyperbolic type, one can show that the function h deﬁned in Theorem 5.8 is connected to the function 1 − F n (0) ˜ h(z) = lim n→∞ 1 − F n (z)

114

Chapter 5. Kœnigs Type Starlike and Spirallike Functions

(compare with the (K-V)-function deﬁned by (5.4.19)) by the equality ˜ h(z) = 1 + A(h(z) − 1), where A = lim

n→∞

2 1 − F n (0) 1 − |F n (0)|

2

.

Hence, number b in Theorem 5.8 is deﬁned by 2 Im F n (0) 1 1 . b = Im − 1 (A − 1) = − 1 lim n→∞ 1 − |F n (0)|2 α α Remark 5.11. We observe that originally Pommerenke’s limit scheme as well as Valiron’s limit scheme were considered for automorphic type self-mappings of the right half-plane. From a certain point of view this makes the model more transpar1+z ent. Indeed, using the Cayley transform C(z) = 1−z , which maps Δ onto Π+ , one can translate the problem for a holomorphic self-mapping φ : Π+ → Π+ deﬁned by φ = C ◦ F ◦ C −1 .

(5.5.2)

For our aims it is also sometimes useful to pass to the associated model in Π+ . This mapping φ : Π+ → Π+ deﬁned by (5.5.2) is called the iteration mapping associated with F : Δ → Δ (see [36]). In this case equation (5.5.1) becomes g(φ(w)) = ϕ(g(w)),

(5.5.3)

g(w) = h(C −1 (w)),

(5.5.4)

where

g ∈ Hol(Π+ ) is a conjugation for φ and ϕ. It turns out that technically, problem (5.5.3) is simpler than (5.5.1). Note that in our settings the point w = ∞ is the Denjoy–Wolﬀ point for φ, that is, φn converges to ∞ locally uniformly on Π+ and φ (∞) =

1 ≥ 1, F (1)

(5.5.5)

where φ (∞) is understood as the angular derivative at ∞: φ (∞) = ∠ lim

ω→∞

φ(ω) . ω

(5.5.6)

5.5. Linearization models

115

Finally, if κ = φ (∞) = can be rewritten in the form

1 F (1)

, then the boundary Schwarz–Wolﬀ Lemma

Re φ(ω) ≥ κ Re ω,

ω ∈ Π+ .

(5.5.7)

We also have that κ = inf

u>0

Re φ(u + iv) , u

(5.5.8)

where u + iv = ω ∈ Π+ . Moreover, if for some ω0 ∈ Π+ equality (5.5.7) holds, then φ must be an automorphism of Π+ . =κ≥ Let φ : Π+ → Π+ be such that φ(∞) = ∞ and φ (∞) := ∠ lim φ(w) w w→∞

p(w) w→∞ w

1, i.e., φ(w) = κw + p(w) with Re p ≥ 0 and ∠ lim

= 0.

If κ > 1, then φ (respectively, F in (5.5.2)) is of hyperbolic type. If κ = 1, then φ (respectively, F in (5.5.2)) is of parabolic type. As in the disk, hyperbolic type mappings are of automorphic type. At the same time, the class of parabolic mappings falls into two subclasses of automorphic type and non-automorphic type, see[110]. Let us explain more precisely what this means. Denote wn = φn (1) = un +vn i wn+1 − wn wn+1 − wn and consider qn = = . wn+1 + wn (wn+1 − wn ) + 2un Since |qn | is the pseudohyperbolic distance in Π+ , it follows by the Schwarz– ∞ Pick Lemma that the sequence {|qn |}n=0 is non-increasing: |qn | ≥ |qn+1 |, hence lim |qn | exists. n→∞

In this case a mapping φ : Π+ → Π+ is of automorphic type (respectively, non-automorphic type) if lim qn = 0 (respectively, lim qn = 0). n→∞ n→∞ The property of φ : Π+ → Π+ to be of automorphic type means that the pseudohyperbolic distance of its orbit at w = 1 (hence, at any w ∈ Π+ ) is separated from zero. In this case |arg wn | must tend to π/2. However, it is not clear whether the sequence un = Re wn is bounded. Note that by the Julia–Wolﬀ–Carath´eodory Theorem the sequence {un } is increasing. Its boundedness means that wn converges to inﬁnity in a sense strongly tangentially. We will see in Section 7.3 that under some smoothness conditions, the strongly tangential convergence of a semigroup is equivalent to its property to be of automorphic type. Now we consider the expression L(w0 ) := lim Re φn (w0 ) ∈ (0, +∞]. n→∞

It is just a computation to show that L(w0 ) < ∞ if and only if inf n

|1 − F n (z0 )|2 > 0, 1 − |F n (z0 )|2

116

Chapter 5. Kœnigs Type Starlike and Spirallike Functions

where F = C −1 ◦ φ ◦ C and z0 = C −1 (w0 ). In other words, the forward orbit {F n (z0 )}∞ n=0 converges to τ = 1 strongly tangentially in the sense: it lies outside a circle internally tangent to ∂Δ at τ = 1. The following assertion shows that the property of L(w0 ) to be ﬁnite (or, equivalently, the property of strongly tangential convergence) does not depend on the initial point of the orbit. Theorem 5.9 ([36]). Let φ ∈ Hol(Π+ ) be of parabolic type with the Denjoy–Wolﬀ point at ∞. Then for each pair w1 , w2 ∈ Π+ the following limit always exists and is positive: Re φ(w1 ) L (= L(w1 , w2 ) := lim > 0. n→∞ Re φ(w2 ) Consequently, if L(w0 ) < ∞ for some w0 ∈ Π+ , then L(w) < ∞ for each w ∈ Π+ . In addition, if F is of non-automorphic type (of zero hyperbolic step), then L = 1.

5.5.2 Baker–Pommerenke’s model for non-automorphic type self-mappings Now let F : Δ → Δ be of non-automorphic type, that is, F (τ ) = τ for some τ ∈ ∂Δ and ρ(F n+1 (z), F n (z)) → 0 as n → ∞ for some z (hence, for all z) in Δ. In this situation F must be of parabolic type, i.e., F (τ ) = 1. Again for simplicity we set τ = 1. As we already know, the limit function deﬁned in Theorem 5.8 is just a constant; therefore, this model does not work. Moreover, sometimes there is no way to construct a linear-fractional transformation of the unit disk that leaves it invariant and is conjugate to F. Baker and Pommerenke [12] constructed a linear-fractional model by using another limit function that is not necessarily a self-mapping of a disk or a halfplane. Their original model was again given for Π+ . We, however, formulate it for Δ, according to our object of interest. Theorem 5.10 (Baker–Pommerenke). Let F : Δ → Δ be a parabolic mapping of non-automorphic type with F (1) = F (1) = 1. Then the limit 1 − F n+1 (0) F n (z) − F n (0) · n+1 n→∞ 1 − F n (0) F (0) − F n (0)

g(z) = lim

exists, and g is a solution of Abel’s equation g(F (z)) = g(z) + 1. Remark 5.12. In each one of the following two cases: (a) the unrestricted limit lim

z→1

F (z) − 1 exists (so, it is equal to 1); z−1

(5.5.9)

5.5. Linearization models

117

(b) the orbit {F n (0)}∞ n=0 converges to 1 non-tangentially, the limit in (5.5.9) coincides with F n (z) − F n (0) n→∞ F n+1 (0) − F n (0)

(5.5.10)

g(z) = lim

(see also Sections 5.5.3 and 7.3.1 below). In particular, this limit scheme is applicable to the study of generating functions of regular branching processes. In this study not only the ﬁrst derivative at 1, F (1) = 1, has a probabilistic interpretation (the expectation); the second derivative F (1) plays role of the variance of the branching process. The following result indicates the strong relationship between the variance and the Kœnigs type function g deﬁned by (5.5.9) or (5.5.10). Theorem 5.11 (see [111]). Let the generating function F of a branching process be not of the form F (z) = F˜ (z m ), m ≥ 2. Assume that F (1) = 1 and F (1) ∈ 2 g (k) (0) tends to as k → ∞. If F (1) is ﬁnite, then [0, +∞]. Then k! F (1) lim (1 − z)g(z) =

z→1

2 . F (1)

Explicit uniform estimates in terms of p0 = F (0) = P (Z(1) = 0), instead of the variance F (1), are given in [37]. Proposition 5.1. Let g be deﬁned by (5.5.10), where F is the generating function of a branching process. Then for all x ∈ [0, 1), 1 1 ≥ + np0 , n 1 − F (x) 1−x and for all z ∈ Δ, |g(z)| ≤

1 . p0 (1 − |z|)

5.5.3 Higher order angular diﬀerentiability at boundary ﬁxed points. A uniﬁed model Deﬁnition 5.5. Let f ∈ Hol(Δ, C) and let τ ∈ ∂Δ. We say that f is of class p+ε CA (τ ), where p ∈ N ∪ {0} and ε ∈ [0, 1), if it admits the expansion f (z) =

p

aj (z − τ )j + γ(z),

z ∈ Δ,

j=0

where a0 , . . . , ap ∈ C and γ ∈ Hol(Δ, C) with ∠ lim

z→τ

γ(z) = 0. (z − τ )p+ε

We say that f ∈ C p+ε (τ ) if this limit is taken in the whole disk.

(5.5.11)

118

Chapter 5. Kœnigs Type Starlike and Spirallike Functions

It is clear that the numbers a0 , . . . , ap appearing in the above expression p+ε are necessarily unique. If f ∈ CA (τ ), it is natural to deﬁne (angular) boundary (j) derivatives f (τ ) = j!aj , j = 0, . . . , p. Moreover, it can be shown by induction n that f ∈ CA (τ ) (n > 1) if and only if the limit ∠ lim f (n) (z) := f (n) (τ ) exists z→τ

ﬁnitely (see, for example, [18] and [51]). Consider a mapping F ∈ Hol(Δ) with a boundary ﬁxed point τ ∈ ∂Δ. According to the Julia–Wolﬀ–Carath´eodory Theorem (see Section 1.3), we easily 1 (τ ) if and only if F has a ﬁnite angular derivative at τ . Clearly, see that F ∈ CA when this happens, the corresponding number a1 is exactly F (τ ). In particular, 1 parabolic and hyperbolic members of Hol(Δ) are of class CA (τ ) at their Denjoy– Wolﬀ point τ (see Theorem 1.10). 2 1 It is easy to verify that F belongs to CA (τ ) if and only if F ∈ CA (τ ) and the following angular limit exists ﬁnitely: L := ∠ lim

z→τ

F (z) − τ − F (τ )(z − τ ) . (z − τ )2

Now the corresponding numbers a1 and a2 are just a1 = F (τ ) and a2 = L = Note that if F is of parabolic type with the Denjoy–Wolﬀ point τ = 1 ∈ ∂Δ, 2 then F ∈ CA (1) if and only if the following angular limit exists ﬁnitely: F (τ ) . 2

∠ lim

z→1

F (z) − z . (z − 1)2

For such mappings Bourdon and Shapiro have suggested in [18] a uniﬁed model for both classes of automorphic and non-automorphic parabolic mappings. This method was studied and developed in detail by Contreras, D´ıaz-Madrigal and Pommerenke. Namely, they proved the following results (see [36]). Theorem 5.12. Let F ∈ Hol(Δ) be a parabolic self-mapping of automorphic type (of positive hyperbolic step) with the Denjoy–Wolﬀ point τ = 1 ∈ ∂Δ. Likewise, let φ be the associated mapping with F in Π+ . Then the following are equivalent: 2 (1) F belongs to CA (1).

(2) For some (hence, any) w0 ∈ Π+ , the normalized sequence gn (w) := φn (w) − φn (w0 ),

w ∈ Π+ ,

converges in the compact open topology to some non-constant mapping g ∈ Hol(Π+ , C) (of course, g depends on w0 ). If any of the above statements hold, then there exists a real number ν = 0, such that g ◦ φ = g + iν. Theorem 5.13. Let F ∈ Hol(Δ) be of non-automorphic type (of zero hyperbolic step) with the Denjoy–Wolﬀ point τ = 1 ∈ ∂Δ. Likewise, let φ be the map associated with F in Π+ . Then the following are equivalent:

5.6. Embedding property via Abel’s equation

119

2 (1) F belongs to CA (1) and Re F (1) > 0.

(2) For some (hence, any) w0 ∈ Π+ , the normalized sequence gn (w) := φn (w) − φn (w0 ),

w ∈ Π+ ,

converges in the compact open topology to a non-constant function g ∈ Hol(Π+ , C) (of course, g depends on w0 ) such that g ◦φ = g +a with Re a > 0. Combining these two assertions with Theorem 4.4 (p. 52) in [18] one can conclude the following. Theorem 5.14. Let F ∈ Hol(Δ) be a self-mapping of Δ of parabolic type with the Denjoy–Wolﬀ point τ = 1 ∈ ∂Δ and let F belong to C 3 (1) with F (1) = 0. Then the sequence of functions hn (z) =

1 + F n (z) 1 + F n (z0 ) − 1 − F n (z) 1 − F n (z0 )

converges in the compact open topology to a non-constant function h ∈ Hol(Δ, C) (of course, h depends on z0 ), which satisﬁes Abel’s equation h◦F =h+a with a = F (1). Moreover, (a) if F is automorphic, then Re a = 0. (b) if F is non-automorphic, then Re a > 0.

5.6 Embedding property via Abel’s equation We know that if τ = 1 is the Denjoy–Wolﬀ point of F ∈ Hol(Δ) (so, F (1) ≤ 1), then in both hyperbolic (F (1) < 1) and parabolic (F (1) = 1) cases, Abel’s functional equation g(F (z)) = g(z) + 1 (5.6.1) can be solved. This fact may be used to study the semigroup embedding problem. Theorem 5.15. Let F ∈ Hol(Δ), F (1) = 1, be not an automorphism. Then F is embeddable if and only if there exists a solution g of Abel’s functional equation (5.6.1) g(F (z)) = g(z) + 1, which is holomorphic in Δ function satisfying the condition 3 4 Re (1 − z)2 g (z) > 0 for all z ∈ Δ.

120

Chapter 5. Kœnigs Type Starlike and Spirallike Functions

Proof. Let F ∈ Hol(Δ) be embedded into a one-parameter semigroup {Ft }t≥0 ⊂ Hol(Δ). Its inﬁnitesimal generator can be represented in the form f (z) = −(1 − z)2 p(z), where p is a holomorphic function on Δ with a positive real part. In this case we deﬁne z dζ . g(z) = (1 − ζ)2 p(ζ) 0

Note that g (z) = and

1 , (1 − z)2 p(z)

3 4 1 Re (1 − z)2 g (z) = Re >0 p(z)

for z ∈ Δ. For ﬁxed z ∈ Δ we consider ζ(t) = g(Ft (z)), t ≥ 0. Now we derive ∂ dζ = g (Ft (z)) Ft (z) = g (Ft (z))(1 − Ft (z))2 p(Ft (z)) = 1. dt ∂t From here and the initial condition ζ(0) = g(z), it follows that g(Ft (z)) = g(z) + t. This equality means that g is the required solution of Abel’s equation (5.6.1). Conversely, suppose that g is holomorphic in Δ function satisfying equation (5.6.1) and Re[(1 − z)2 g (z)] > 0 for z ∈ Δ. Noting that for each c the function g(z) + c satisﬁes the same conditions, we can assume that g(0) = 0. Therefore, g ∈ Σ[1] by Lemma 3.8. This means that together with each point w0 , the image domain g(Δ) contains the ray w(t) = w0 + t, t ≥ 0. Consequently, the family {Ft }t≥0 , where Ft (z) = g −1 (g(z) + t), is well deﬁned on Δ and is a one-parameter continuous semigroup on Δ. Since F1 (z) = g −1 (g(z) + 1) = F (z), we are done. A detailed survey on Abel’s equation for discrete time semigroups of parabolic type has recently presented in [37] (see also [108]). The existence, uniqueness and construction problems are discussed there.

Chapter 6

Rigidity of Holomorphic Mappings and Commuting Semigroups The problem of ﬁnding conditions for a holomorphic mapping F to coincide identically with a given holomorphic mapping G when they behave similarly on some subset of their common domain of deﬁnition, has been studied by many mathematicians. The following assertions are classical: If F and G are holomorphic in Δ and F = G on a subset of Δ that has a non-isolated point, then F ≡ G on Δ (Vitali’s uniqueness principle). If F and G are holomorphic in Δ and continuous on Δ, and F = G on some arc γ of the boundary ∂Δ, then F ≡ G on Δ. From the point of view of complex dynamics it is natural to look for conditions on derivatives of F and G at speciﬁc ﬁxed points that will yield the conclusion that F ≡ G. If G = I is the identity mapping on Δ, then the classical Schwarz–Pick lemma asserts: Let F ∈ Hol(Δ) and let τ ∈ Δ be a ﬁxed point of F . Then the equalities F (τ ) = G(τ ) (= τ ) and F (τ ) = G (τ ) (= 1) imply that F ≡ G on Δ. The same conclusion holds for an arbitrary holomorphic mapping G on Δ if F commutes with G and satisﬁes the conditions F (τ ) = G(τ ) = τ ∈ Δ and F (τ ) = G (τ ) = 0 (see, for instance, [21]). Diﬀerent “identity principles” have recently been studied by several mathematicians under suitable boundary conditions (see, for example, [25, 13, 21, 139, 140]). Generally speaking, the following cases have been considered:

122 Chapter 6. Rigidity of Holomorphic Mappings and Commuting Semigroups (A) G is a holomorphic self-mapping of Δ of a speciﬁc form, e.g., the identity, a constant mapping (say, zero), an aﬃne mapping, an automorphism of Δ or a linear fractional transformation. (B) G is an arbitrary self-mapping of Δ and F commutes with G, i.e., F ◦ G = G ◦ F. (C) In addition, one can consider certain families of holomorphic mappings and look for conditions which ensure that they either coincide or have similar structures.

6.1 The Burns–Krantz theorem In this section, we give a brief review on rigidity of holomorphic functions related to the remarkable theorem of Burns and Krantz (see [25]). In the next section, we will show how to prove these results by using the generation theory for one-parameter semigroups. Let F ∈ Hol(Δ) and let τ ∈ Δ be an interior ﬁxed point of F . In this case, if F coincides with the identity mapping I at τ up to the ﬁrst derivative, i.e., F (τ ) = τ and F (τ ) = 1, then F ≡ I on Δ by the classical Schwarz–Pick lemma. Moreover, if |F (τ )| = 1 and F (τ ) = 1, then F is an elliptic automorphism of Δ. In the boundary case (τ ∈ ∂Δ) a uniqueness theorem in the spirit of the uniqueness part of the classical Schwarz lemma was established by Burns and Krantz [25]. We formulate their result as follows: Theorem 6.1. Let F ∈ Hol(Δ). If the unrestricted limit lim

z→1

F (z) − z = 0, (z − 1)3

then F ≡ I on Δ. This result was improved by Tauraso and Vlacci [140] (see also Bracci, Tauraso and Vlacci [21]). They have shown that it is suﬃcient to require the vanishing of such a limit at a boundary point τ only in the radial sense, i.e., F (rτ ) − rτ = 0. Furthermore, it has been shown by Kriete and MacCluer [94] lim− (1 − r)3 r→1 that the function F in this limit can be replaced with its real part. Namely, for Re F (r) − r τ = 1, the equality lim inf = 0 implies that F ≡ I on Δ. More (1 − r)3 r→1− recently, Baracco, Zaitsev and Zampieri [13] have proved that it is suﬃcient to consider such a limit only on a sequence in Δ. We quote their result as follows: Theorem 6.2. Let F ∈ Hol(Δ). If for some sequence {zn }∞ n=1 ⊂ Δ, converging non-tangentially to τ = 1, lim

n→∞

F (zn ) − zn = 0, (1 − zn )3

6.1. The Burns–Krantz theorem

123

then F ≡ I on Δ. A continuous version of the Burns–Krantz theorem is presented in [134]: Let F ∈ Hol(Δ) and let lim inf r→1−

Then μ ≥ 0 and

Re F (r) − r = μ. (1 − r)3

|F (z) − z|2 ≤ μ · K(z),

z ∈ Δ,

for some continuous function K : Δ → R+ . In view of these results, the following problem arises: ♦ ﬁnd conditions on the ﬁrst derivatives of a holomorphic self-mapping F of the unit disk at its boundary Denjoy–Wolﬀ point τ (say, τ = 1) which ensure that F is an aﬃne self-mapping of Δ. From the point of view of the Burns–Krantz theorem, it seems plausible to conjecture that such conditions (if they do exist) 3 might be the following: F ∈ CA (1), F (1) = α, F (1) = F (1) = 0, and then F (z) = αz + (1 − α). However, the following counterexample shows that this conjecture is false. 1 3 Example 6.1. Let F (z) = 12 (z + 1) + 20 (z − 1)4 . Then F ∈ Hol(Δ) ∩ CA (1) and τ = 1 is its Denjoy–Wolﬀ point. Moreover, F (1) = F (1) = 0. However, F is not aﬃne.

At the same time, if a function F ∈ Hol(Δ) is such that |1 − F (z)|2 α|1 − z|2 ≤ , 2 2 1 − |F (z)| α(1 − |z| ) + (1 − α)|1 − z|2

z ∈ Δ,

(6.1.1)

where α = F (1), then these conditions do imply that F is aﬃne. Note that, for a parabolic self-mapping F (i.e., if α = 1), this inequality holds automatically. 3 (τ ) be a holomorphic self-mapping of Theorem 6.3 ([134]). Let F ∈ Hol(Δ) ∩ CA Δ with the Denjoy-Wolﬀ point τ = 1 and let F (1) = F (1) = 0. Then F is an aﬃne mapping of the form F (z) = αz + 1 − α (α = F (1)) if and only it satisﬁes condition (6.1.1), i.e., kα , k > 0. (6.1.2) F (D(1, k)) ⊆ D 1, 1 + k(1 − α)

It follows from the Julia–Wolﬀ–Carath´eodory theorem that the geometrical condition (6.1.2) is equivalent to the fact that F maps the whole disk Δ into the α horocycle D 1, 1−α . Suﬃcient conditions regarding the behavior of a holomorphic self-mapping of Δ in a neighborhood of its boundary Denjoy–Wolﬀ point which ensure that it

124 Chapter 6. Rigidity of Holomorphic Mappings and Commuting Semigroups is an automorphism of Δ can be found in [36] and [140]. To formulate them, we 2 (z) (z) denote by SF the Schwarzian derivative of F , i.e., SF (z) := FF (z) − 32 FF (z) . 3 (1), then the Tauraso and Vlacci mentioned in [140] that if F ∈ Hol(Δ) ∩ CA equalities F (1) = 1, Re F (1) = 0 and Re SF (1) = 0 imply that F is nothing but the parabolic automorphism given by

F (z) = λ

z+a , 1 + az

ib , b = Im F (1) and a = 2−ib . A more general assertion, valid for all where λ = 2−ib 2+ib types of automorphisms of Δ, is pointed out in [36] by Contreras, D´ıaz-Madrigal and Pommerenke. 3 Theorem 6.4. Let F ∈ Hol(Δ)∩CA (τ ) be a holomorphic self-mapping of Δ with the Denjoy–Wolﬀ point τ ∈ ∂Δ and let Re [τ F (τ )] = α(α − 1). Then the Schwarzian derivative SF (τ ) = 0 if and only if F = Φ is the automorphism

Φ(z) := λ

z+a , 1 + az

F (τ ) τ F (τ ) − 2α2 and λ = . τ F (τ ) − 2α2 τ F (τ ) − 2α2 In particular,

where a =

(1) a nontrivial (i.e., F = I) holomorphic map F ∈ Hol(Δ) is a parabolic auto3 (τ ) and morphism if and only if there exists τ ∈ ∂Δ such that F ∈ CA F (τ ) = τ,

F (τ ) = 1,

Re(τ F (τ )) = 0,

and

SF (τ ) = 0;

(2) a holomorphic map F ∈ Hol(Δ) is a hyperbolic automorphism if and only if 3 (τ ) and there exist τ ∈ ∂Δ and α ∈ (0, 1) such that F ∈ CA F (τ ) = τ,

F (τ ) = α,

Re(τ F (τ )) = α(α − 1),

and

SF (τ ) = 0.

Remark 6.1. As a matter of fact, since Φ is the general form of automorphisms with their Denjoy–Wolﬀ point at τ , Theorem 6.4 can be reduced to Theorem 6.2 when we apply to F the automorphism Φ−1 . Since each automorphism of Δ is a linear-fractional transformation (LFT), the following more general problem also arises: ♦ ﬁnd conditions concerning the behavior of F ∈ Hol(Δ) in a neighborhood of a boundary point τ ∈ ∂Δ which ensure that F is an LFT, i.e., that it is of the form F (z) = az+b , with complex coeﬃcients a, b, c, d. cz+d Linear-fractional transformations are also of intrinsic interest because of their deep connections with and applications to Markov branching processes, the theory of composition operators, and operator theory in Pontryagin and Krein spaces. An answer to the above problem was independently given in [140] and [134].

6.1. The Burns–Krantz theorem

125

3 Theorem 6.5. Let F ∈ Hol(Δ) ∩ CA (1) with F (1) = 1, and let a := α(1 − α)), where α = F (1). Then

1 α2 (F (1)

+

(i) Re a ≥ 0;

1 if and only if (ii) F (Δ) ⊂ D 1, Re a F (D(1, k)) ⊆ D 1,

αk 1 + αk Re a

for all

k > 0;

(6.1.3)

(iii) if the inclusions in (ii) hold, then Re SF (1) ≤ 0 and Im SF (1) = 0. Moreover, SF (1) = 0 if and only if F is an LFT. Thus F is an automorphism of Δ if and only if SF (1) = 0 and Re a = 0, i.e., Re F (1) = α(α − 1).

(6.1.4)

In this case, condition (ii) just means that F (Δ) ⊂ Δ, i.e., it holds automatically. Remark 6.2. Note also that the latter assertion gives conditions on a holomorphic self-mapping F of Δ at a boundary regular ﬁxed point τ ∈ ∂Δ, which is not necessarily the Denjoy–Wolﬀ point of F , to be a linear-fractional transformation. At the same time, if τ = 1 is the Denjoy–Wolﬀ point of F , then this theorem can be essentially improved. Namely, one can show (see [134]) that it is enough to require that condition (6.1.3) holds for at least one k > 0 to ensure that equality SF (1) = 0 implies that F must be an LFT. Moreover, if F is a hyperbolic type mapping (α < 1), then one can even require a weaker inclusion than (6.1.3) to obtain the same conclusion. Remark 6.3. Observe that if equality (6.1.4) does not hold and α > 1, then in general F can be either a hyperbolic automorphism of Δ or a dilation type mapping. Indeed, if τ = 1 is a repelling ﬁxed point (α > 1), then there is a point ζ ∈ Δ which is the Denjoy–Wolﬀ point of F . If ζ ∈ ∂Δ, then F (ζ) = α1 < 1 and F is an LFT having two boundary ﬁxed points. Hence, F must be a hyperbolic automorphism of Δ. In this case, Re a = 0. If ζ ∈ Δ, then Re a = 0. It can be shown that, in fact, Re a > 0 and each horocycle D(1, k) with k ≥ αα−1 is F Re a 2

α−1 invariant. Moreover, |1−ζ| 1−|ζ|2 = α Re a =: k0 , that is, the Denjoy–Wolﬀ point ζ lies on the boundary of the F -invariant horocycle D(1, k0 ). Hence the mapping F is a hyperbolic automorphism of the disk D(1, k0 ).

3 Remark 6.4. Actually, one can show that the condition F ∈ CA (1) in Theorems 6.3, 6.4 and 6.5 can be replaced by the existence of the ﬁnite radial derivatives at τ up to the third order.

126 Chapter 6. Rigidity of Holomorphic Mappings and Commuting Semigroups More general rigidity results in the spirit of Theorem 6.5 which ensure that F ∈ Hol(Δ) is a rational self-mapping of Δ are presented in [140]. Other generalizations of the Burns–Krantz theorem can be found in Arslan [9] and Chelst [28]. To proceed, we note that many rigidity principles for holomorphic functions (even not necessarily self-mappings of Δ) can be obtained by manipulating the Julia–Wolﬀ–Carath´eodory theorem. For example, the following fact is a simple consequence of this theorem (see [103]): If F ∈ Hol(Δ, Δ), then the conditions lim F (rτ ) = τ and lim F (rτ ) = 0 r→1−

at some τ ∈ ∂Δ imply that F ≡ τ .

r→1−

1 − F (z) , 1 + F (z) where Π+ := {z : Re z > 0} is the right half-plane and F ∈ Hol(Δ, Δ), we get the following assertion (cf. [103]): Applying this fact to the function p ∈ Hol(Δ, Π+ ) deﬁned by p(z) :=

Let f ∈ Hol(Δ, Π+ ). Then the equality ∠ lim

z→τ

f (z) =0 z−τ

for some τ ∈ ∂Δ implies that f ≡ 0. As a matter of fact, these two assertions follow immediately from Hopf’s lemma (see [44, p. 116]). The second assertion can also be stated in the following form: Let f, g ∈ Hol(Δ, C) be such that Re f (z) ≥ Re g(z), and ∠ lim

z→τ

z ∈ Δ,

f (z) − g(z) =0 z−τ

for some τ ∈ ∂Δ; then f ≡ g in Δ. A generalization of these facts can be given as follows. Denote by απ Aα := | arg w| < , α ∈ (0, 2] 2 the sector of opening απ with vertex at z = 0. Theorem 6.6 (see [103]). If f ∈ Hol(Δ, Aα ), then the equality ∠ lim

z→τ

for some τ ∈ ∂Δ implies that f ≡ 0.

f (z) =0 (z − τ )α

6.1. The Burns–Krantz theorem

127

Now we consider the class of functions F ∈ Hol(Δ, C) that satisfy the socalled boundary ﬂow-invariance condition lim sup Re(F (z)z) ≤ 1.

(6.1.5)

z→∂Δ

In particular, each holomorphic self-mapping F of Δ obviously satisﬁes condition (6.1.5). Theorem 6.7 (cf. [53]). Let F ∈ Hol(Δ, C) satisfy condition (6.1.5). If for some τ ∈ ∂Δ, F (z) − z = 0, ∠ lim z→τ (z − τ )3 then F ≡ I on Δ. This assertion is in a sense a generalization of the Burns–Krantz theorem. If, in addition, F can be continuously extended to the boundary ∂Δ, then by setting f = I − F , one can reformulate it as follows: Theorem 6.8. Let f ∈ Hol(Δ, C) ∩ C(Δ) be such that Re(f (z)z) ≥ 0,

(6.1.6)

and assume that for some τ ∈ ∂Δ, ∠ lim

z→τ

f (z) = 0. (z − τ )3

Then f ≡ 0. Note that inequality (6.1.6) is a criterion for a function f ∈ Hol(Δ, C)∩C(Δ) to be a semigroup generator. Therefore, it is natural to look for rigidity properties for generators which are not necessarily continuous on ∂Δ. We will tackle this question in the next section. An extension of Theorem 6.8 to a wider class of functions f ∈ Hol(Δ, C) which are continuous on Δ and satisfy for some α ∈ (0, 2] the boundary condition Re(f (z)z) ≥ |f (z)| cos

απ , 2

z ∈ ∂Δ,

(6.1.7)

was given in [53]. This inequality coincides with (6.1.6) when α = 1. Theorem 6.9. Let f ∈ Hol(Δ, C) be continuous on Δ and satisfy condition (6.1.7). Then the equality lim

z→τ z∈Δ

implies that f ≡ 0.

f (z) =0 (z − τ )2+α

for some

τ ∈ ∂Δ

(6.1.8)

128 Chapter 6. Rigidity of Holomorphic Mappings and Commuting Semigroups Observe that if F ∈ Hol(Δ), then the mapping f deﬁned by f (z) = z − F (z) belongs to the class G. Therefore any rigidity result for semigroup generators provides immediately a rigidity theorem for holomorphic self-mappings. Another useful relation between classes Hol(Δ) and G is the following consequence of the 1 + F (z) . In the next section, we Berkson–Porta formula: f (z) = (z − τ )(1 − z τ¯) 1 − F (z) prove some results for generators which imply, in turn, the Burns–Krantz Theorem 6.1 as well as its extensions given in Theorems 6.3–6.8.

6.2 Rigidity of semigroup generators The following consequence of the Berkson–Porta formula establishes a one-to-one correspondence between the classes G[0] and G + [1]. Lemma 6.1. Let f and g be two holomorphic functions on Δ connected by the formula k(z) · f (z) + g(z) = 0, (6.2.1) z is the Koebe function. Then f ∈ G + [1] if and only if where k(z) = (1 − z)2 g ∈ G[0]. This formula implies, in its turn, inﬁnitesimal versions of the classical Schwarz lemma and the Burns–Krantz rigidity theorem. Proposition 6.1. Let g ∈ G[0]. Then g(z) ≡ 0 if and only if g (0) = 0. Proof. If g (0) = 0, then we get by (6.2.1) that f (z) g(z) = 0, = − lim 2 z→0 (1 − z) z→0 z lim

i.e., f (0) = 0, which is impossible, when g = 0, since f has then no null point in Δ. Corollary 6.1. Let f ∈ G be such that f (0) = 0. Then f generates a group S = {Ft }t∈R of automorphisms of Δ. Moreover, if f (0) = 0, then S consists of hyperbolic automorphisms of Δ. Proof. Since f ∈ G, it can be represented as f (z) = f (0) − f (0)z 2 + zq(z), where Re q(z) ≥ 0 (see Theorem 2.6). At the same time, the function g ∈ Hol(Δ, C), deﬁned by (6.2.1), also belongs to the class G and satisﬁes the conditions g(0) = 0 and g (0) = 0. Hence g(z) = 0 identically, and we are done. 3 (1), then f (1) Proposition 6.2. Let f ∈ Gp [1] be such that f (1) = 0. If f ∈ CA is a nonnegative real number. Moreover, f (1) = 0 if and only if f (z) ≡ 0.

6.2. Rigidity of semigroup generators

129

f (z) = 0. Then again by z→1 (z − 1)2 (6.2.1), we obtain that ∠ lim g(z) = 0. Hence z = 1 is a boundary null point of Proof. Let f ∈ Gp [1] be such that f (1) = ∠ lim z→1

the generator g. Since g has an interior null point, it cannot belong to G + [1]. In other words, the limit f (1) g(z) zf (z) =− = −∠ lim 3 z→1 z − 1 z→1 (z − 1) 3

∠ lim

must be a negative number whenever g is not identically zero.

The last part of this section follows the results obtained in [134]. Theorem 6.10. Let f be the generator of a semigroup S = {Ft }t≥0 having the 3 Denjoy–Wolﬀ point τ = 1. Let f ∈ CA (1) and let g(z) = f (1)(z − 1) + 12 f (1)(z − 2 1) be its quadratic part. Then (i) g is the generator of a semigroup of linear-fractional transformations on Δ; (ii) if Re f (1) = 0, then f = g if and only if h := f − g belongs to the class G. Proof. Since f ∈ G + [1], it follows from the Berkson–Porta representation that f (z) = −(1 − z)2 p(z), where Re p(z) ≥ 0. 2 Since f ∈ CA (1), we can also write that p(z) =

f (1)(z − 1) + 12 f (1)(z − 1)2 f (z) h(z) = − , 2 −(1 − z) −(1 − z)2 (z − 1)2

h(z) = 0, or z→1 (z − 1)2

where ∠ lim

1 h(z) 1 = − f (1) − 1−z 2 (z − 1)2 1 f (1) 1 + z h(z) + [f (1) − f (1)] − = . 2 1−z 2 (z − 1)2

p(z) = f (1)

Now it follows from Theorem 1.14 that the function 1 1+z p1 (z) := p(z) − f (1) 2 1−z is of nonnegative real part. Therefore, 1 [f (1) − Re f (1)] = ∠ lim Re p1 (z) ≥ 0. z→1 2

130 Chapter 6. Rigidity of Holomorphic Mappings and Commuting Semigroups This implies that the function q(z) :=

1 1+z 1 f (z) + [f (1) − f (1)] 2 1−z 2

is also of nonnegative real part. On the other hand, 1 1 f (1)(z 2 − 1) − [f (1) − f (1)] (z − 1)2 = 2 2 1 = f (1)(z − 1) + f (1)(z − 1)2 = g(z). 2

−(1 − z)2 q(z) =

Now assertion (i) follows again from the Berkson–Porta representation and assertion (ii) is seen to be a direct consequence of Proposition 6.2. Now we formulate our main rigidity result. 3 (1) with β := f (1) satisfy representation Theorem 6.11. Let f ∈ Hol(Δ, C) ∩ CA 2 f (z) = −(1 − z) p(z) with 2 β 1 − |z| =: m ≥ 0. (6.2.2) inf Re p(z) − z∈Δ 2 |1 − z|2

Then f generates a semigroup S = {Ft }t≥0 of linear-fractional transformations if and only if the following two conditions hold: (i) f (1) − Re f (1) ≤ 2m; (ii) f (1) = 0. Moreover, in this case, m = 0 if and only if f is a generator of a group of automorphisms of Δ. Remark 6.5. We will see below that, in fact, under our setting, f (1) − Re f (1) = 2m ≥ 0. So, we have the following consequences of Theorem 6.11: 3 Corollary 6.2. Let f ∈ G(Δ) ∩ CA (1) with f (1) = 0. Then f is a generator of a group S of automorphisms of Δ if and only if

f (1) − Re f (1) = f (1) = 0.

(6.2.3)

In addition, if (6.2.3) holds, then the group S consists of hyperbolic automorphisms if and only if f (1) = 0. Otherwise (f (1) = 0), S consists either of parabolic automorphisms (Im f (1) = 0) or identity mappings (Im f (1) = 0, hence, f (1) = 0).

6.2. Rigidity of semigroup generators

131

Corollary 6.3. Let F be a self-mapping of Δ with the Denjoy–Wolﬀ point τ = 1, and let f ∈ G have the form f (z) = −(1 − z)2

1 + F (z) . 1 − F (z)

(6.2.4)

Then F is an automorphism of Δ if and only if f generates a group of hyperbolic automorphisms of Δ. Moreover, F is parabolic if and only if f (1) = 2. 3 (1) be of the form f (z) = −(1 − z)2 p(z). Then f Corollary 6.4. Let f ∈ G ∩ CA is a generator of a semigroup S of aﬃne self-mappings of Δ if and only if the following two conditions hold:

(i) Re p(z) ≥ 12 f (1); (ii) f (1) = f (1) = 0. In this case β := f (1) ≥ 0 and f (z) = β(z − 1). The suﬃcient part of Theorem 6.11 can be improved as follows: 3 Theorem 6.12. Let f ∈ Hol(Δ, C) ∩ CA (1) with β := f (1) have the form f (z) = 2 −(1 − z) p(z), where function p ∈ P satisﬁes (6.2.2): 2 β 1 − |z| inf Re p(z) − =: m ≥ 0. z∈Δ 2 |1 − z|2

Let g(z) = f (1)(z − 1) + 12 f (1)(z − 1)2 be its Taylor’s polynomial of degree 2 at z = 1. The following assertions hold: (i) f (1) − Re f (1) ≥ 0; (ii) if f (1) − Re f (1) ≤ 2m, then f (1) is a nonnegative real number and 4 3 Re (f (z) − g(z))(1 − z¯)2 1 |f (z) − g(z)|2 ≤ f (1) , z ∈ Δ. (6.2.5) 2 6 1 − |z| In particular, f (z) = g(z), i.e., g generates a semigroup of LFT’s which are selfmappings of Δ if and only if f (1) = 0. Let F ∈ Hol(Δ) be a holomorphic self-mapping of Δ with the ﬁxed point τ = 1, i.e., F (1) = 1 and 0 < α = F (1) < ∞. Then the function f deﬁned by (6.2.4) belongs to G. By using direct calculations, one proves the following assertion. 3 Lemma 6.2. The function f deﬁned by (6.2.4) belongs to the class CA (1) if and only if F belongs to this class. Moreover, f generates a hyperbolic type semigroup

132 Chapter 6. Rigidity of Holomorphic Mappings and Commuting Semigroups with 2 > 0, α = F (1), α (F (1))]2 − F (1) f (1) = 2 (F (1))2

1 1 [F (1) − α(α − 1)] , =2 − α α2 2 f (1) = − SF (1). α f (1) =

(6.2.6) (6.2.7)

(6.2.8)

We are now at the point to prove our main results. 3 (1) i.e., f admits the representaProof of Theorem 6.12. Let f ∈ Hol(Δ, C) ∩ CA tion

1 1 f (z) = f (1)(z − 1) + f (1)(z − 1)2 + + f (1)(z − 1)3 + γf (z), 2 3! where ∠ lim

z→1

γf (z) = 0. (z − 1)3

(6.2.9)

Then p(z) = −f (z)(1 − z)−2 is of the form p(z) = f (1)

1 1 1 − f (1) − f (1)(z − 1) + γp (z) 1−z 2 3!

where, by (6.2.9), γp (z) γf (z) = 0. = −∠ lim z→1 (z − 1) z→1 (z − 1)3

∠ lim We can also write p(z) =

1 1+z 1 f (1) + (f (1) − f (1)) + q(z), 2 1−z 2

where q(z) = −

1 f (1)(z − 1) + γp (z). 3!

1+z Noting that the function p1 (z) := p(z) − β2 1−z is of positive real part and setting 1 (6.2.10) b = (f (1) − f (1)) , 2 we get that

Re b = ∠ lim Re [p1 (z) − q(z)] = ∠ lim Re p1 (z) ≥ 0, z→1

z→1

(6.2.11)

6.3. Commuting semigroups of holomorphic mappings

133

which proves assertion (i) of the theorem. Assume now that Re b ≤ m. Then Re q(z) ≥ 0 and q (1) = −

1 f (1). 3!

1 and going back to the diﬀerence Applying now Theorem 1.14 to the function q(z) 2 f (z) − g(z) = −(1 − z) q(z), we obtain inequality (6.2.5). To complete the proof, we note that g(z) = f (1)(z − 1) + 12 f (1)(z − 1)2 can be written in the form

1 g(z) = −b(1 − z)2 + β(z 2 − 1). 2 It follows from the Berkson–Porta formula that the ﬁrst term of this sum is a generator of a parabolic type semigroup of Δ, while the second term is a generator of a group of hyperbolic automorphisms. Since G is a real cone, we get that g must belong to G. Finally, we have by (6.2.5) that f ≡ g if and only if f (1) = 0. Remark 6.6. Note that, actually, we have from (6.2.11) that Re b = inf p1 (z) = m ≥ 0. This proves the assertion in Remark 6.5.

z∈Δ

Since the necessary part of Theorem 6.11 is obvious, Theorem 6.11 is now a direct consequence of Theorem 6.12. By using Lemma 6.2 and Theorems 6.11 and 6.12, one obtains Theorems 6.3– 6.5.

6.3 Commuting semigroups of holomorphic mappings 6.3.1 Identity principles for commuting semigroups In this section, we apply the above results to establish conditions on semigroup generators that provide that the generated semigroups commute. Deﬁnition 6.1. We say that two semigroups S1 = {Ft }t≥0 and S2 = {Gt }t≥0 of holomorphic self-mappings on Δ are commuting if for each pair s, t ≥ 0, the elements Ft ∈ S1 and Gs ∈ S2 commute: Ft ◦ Gs = Gs ◦ Ft . Theorem 6.13. Let f and g be generators of one-parameter commuting semigroups Sf = {Ft }t≥0 and Sg = {Gt }t≥0 , respectively, and f (τ ) = 0 at some point τ ∈ Δ. (i) Let τ ∈ Δ. If f (τ ) = g (τ ), then f ≡ g. (ii) Let τ ∈ ∂Δ. Suppose f and g admit the following representations: f (z) = f (τ )(z − τ ) + . . . +

f (m) (τ ) (z − τ )m + γ1 (z) m!

(6.3.1)

134 Chapter 6. Rigidity of Holomorphic Mappings and Commuting Semigroups and g(z) = g(τ ) + g (τ )(z − τ ) + . . . +

g (m) (τ ) (z − τ )m + γ2 (z), m!

(6.3.2)

γ1 (z) γ2 (z) and tend to 0 as z → τ along some curve lying in (z − τ )m (z − τ )m Δ and ending at τ . If f (m) (τ ) = g (m) (τ ) = 0, then f ≡ g.

where

Remark 6.7. If τ ∈ ∂Δ is the Denjoy–Wolﬀ point of a semigroup generated by a mapping h ∈ G(Δ), then h admits the expansion h(z) = h (τ )(z − τ ) + o(z − τ ) when z → τ in each non-tangential approach region at τ and h (τ ) = ∠ lim h (z). z→τ

Moreover, in this case, h (τ ) is a nonnegative real number which is zero if and only if h generates a semigroup of parabolic type (see Theorem 2.10). Therefore, if f (or g) in Theorem 6.13 generates a semigroup of hyperbolic type with the Denjoy–Wolﬀ point τ ∈ ∂Δ, then the condition f (τ ) = g (τ ) is enough to provide that f ≡ g. Remark 6.8. As a matter of fact, if f and g have expansion (6.3.1) and (6.3.2) when z → τ in each non-tangential approach region at τ ∈ ∂Δ up to the third order m = 3, such that f (τ ) = g (τ ), f (τ ) = g (τ ) and f (τ ) = g (τ ), then f ≡ g. If, in particular, f (i) (τ ) = g (i) (τ ) = 0, i = 1, 2, 3, then both f and g equal zero identically by Proposition 6.2. Theorem 6.13 is a consequence of the following more general assertion. Deﬁne two linear semigroups {At }t≥0 and {Bt }t≥0 of composition operators on Hol(Δ, C) by At (h) = h ◦ Ft

and Bt (h) = h ◦ Gt ,

t ≥ 0.

(6.3.3)

The operators Γf and Γg deﬁned by Γf (h) = h f

and Γg (h) = h g

(6.3.4)

are their generators, respectively. Theorem 6.14 ([53]). Let f and g ∈ Hol(Δ, C) be generators of one-parameter semigroups Sf = {Ft }t≥0 and Sg = {Gt }t≥0 , respectively. Let At and Bt be deﬁned by (6.3.3). Then the following are equivalent: (i) Ft ◦ Gs = Gs ◦ Ft , s, t ≥ 0, i.e., the semigroups Sf and Sg are commuting; (ii) At ◦ Bs = Bs ◦ At , s, t ≥ 0, i.e., the linear semigroups {At }t≥0 and {Bt }t≥0 are commuting;

6.3. Commuting semigroups of holomorphic mappings

135

(iii) Γf ◦ Γg = Γg ◦ Γf , i.e., the linear semigroup generators Γf and Γg are commuting; (iv) the Lie commutator

[f, g] = f g − g f = 0;

(v) f = αg for some α ∈ C. Proof. Suppose that f ≡ 0. First we prove the equivalence of assertions (i) and (v). Let (i) hold. If f (τ ) = 0, τ ∈ Δ, then τ is the unique common ﬁxed point for the semigroup Sf generated by f , i.e., Ft (τ ) = τ for all t ≥ 0. If Ft and Gs are commuting for all s, t ≥ 0, then we have Gs (τ ) = Gs (Ft (τ )) = Ft (Gs (τ )) . Hence, it follows by the uniqueness of the ﬁxed point τ that Gs (τ ) = τ for all s ≥ 0, and so g(τ ) = 0. Consider the function h ∈ Hol(Δ, C) deﬁned by the diﬀerential equation μh(z) = h (z)f (z).

(6.3.5)

It is known that if μ = f (τ ), then equation (6.3.5) has a unique solution h ∈ Hol(Δ, C) normalized by the condition h (τ ) = 1. In addition, this function h solves Schroeder’s functional equation h (Ft (z)) = e−μt h(z)

(6.3.6)

(see Theorem 5.1). Now, for any s, t ≥ 0, we get, from (6.3.6), h (Gs (Ft (z))) = h (Ft (Gs (z))) = e−μt h (Gs (z)) . Let hs = h ◦ Gs . Then we have hs (Ft (z)) = e−μt hs (z).

(6.3.7)

Diﬀerentiating (6.3.7) at t = 0+ , we get μhs (z) = hs (z)f (z).

(6.3.8)

Comparing (6.3.5) and (6.3.8) implies hs (z) = λ(s)h(z) for some λ(s) ∈ C, or h (Gs (z)) = λ(s)h(z).

(6.3.9)

Since the left-hand side of the latter equality is diﬀerentiable in s ≥ 0, the scalar function λ(s) is diﬀerentiable too. Diﬀerentiating (6.3.9) at s = 0+ , we get λ (0)h(z) = −h (z)g(z).

(6.3.10)

136 Chapter 6. Rigidity of Holomorphic Mappings and Commuting Semigroups Note that h(τ ) = 0 while h(z) = 0 for all z ∈ Δ, z = τ . In addition, by Theorem 5.2, h is univalent. Hence, h (z) = 0 for all z ∈ Δ. Finally, we obtain from (6.3.5) and (6.3.10) that f (z) = αg(z),

where α = −

μ . λ (0)

Now, let us suppose that f has no null point in Δ. Then the function p : Δ → C given by z dζ p(z) = − (6.3.11) f (ζ) 0

is a well-deﬁned holomorphic function on Δ with p(0) = 0. Recall that the semigroup {Ft }t≥0 generated by f can be deﬁned by the Cauchy problem ⎧ ⎨ dFt (z) + f (F (z)) = 0, t ≥ 0, t dt (6.3.12) ⎩ F0 (z) = z, z ∈ Δ. 1 , we obtain Substituting here f (z) = − p (z) p (Ft (z)) dFt (z) = dt. Integrating the latter equality on the interval [0, t], we get that p is a solution of Abel’s functional equation p (Ft (z)) = p(z) + t.

(6.3.13)

Now, for any ﬁxed s ≥ 0, we have p (Gs (Ft (z))) = p (Ft (Gs (z))) = p (Gs (z)) + t. Once again, setting ps = p ◦ Gs , we have ps (Ft (z)) = ps (z) + t.

(6.3.14)

Diﬀerentiating (6.3.14) at t = 0+, we get ps (z) = −

1 , f (z)

(6.3.15)

and by (6.3.11), ps (z) = p(z) + κ(s), κ(s) ∈ C, or p (Gs (z)) = p(z) + κ(s).

(6.3.16)

Diﬀerentiating (6.3.16) at s = 0+ , we obtain the equality p (z) = −

κ (0) . g(z)

(6.3.17)

6.3. Commuting semigroups of holomorphic mappings

137

Comparing (6.3.15) and (6.3.17) gives f = αg

with

α=

1 . κ (0)

(6.3.18)

Now we prove that (v)⇒(i). Let f = αg for some α ∈ C. First we assume that g has an interior null-point τ ∈ Δ. In this case, there is a univalent solution of the diﬀerential equation μh(z) = h (z)g(z)

(6.3.19)

with some μ ∈ C, Re μ ≥ 0. Since f = αg, we have that h is also a solution of the equation νh(z) = h (z)f (z),

ν = αμ.

(6.3.20)

In turn, equations (6.3.19) and (6.3.20) are equivalent to Schroeder’s functional equations h (Gs (z)) = e−μs h(z), s ≥ 0 (6.3.21) and

h (Ft (z)) = e−νt h(z),

t ≥ 0,

ν = αμ,

(6.3.22)

respectively, where {Ft }t≥0 is the semigroup generated by f . Consequently, Ft (Gs (z)) = h−1 e−νt h (Gs (z)) = h−1 e−νt · e−μs h(z) = h−1 e−μs h (Ft (z)) = Gs (Ft (z)) for all s, t ≥ 0 and we are done. Now let us assume that g has a boundary null-point τ ∈ ∂Δ with g (τ ) ≥ 0. In this case, for each c ∈ C, c = 0, Abel’s equations p (Gs (z)) = p(z) + cs and p (Ft (z)) = p(z) + cαt have the same solution z p(z) = −c 0

dζ = −cα g(ζ)

z 0

dζ , f (ζ)

which is univalent on Δ. Once again we calculate Ft (Gs (z)) = p−1 (p (Gs (z)) + cαt) = p−1 (p(z) + cαt + cs) = p−1 (p (Ft (z)) + cs) = Gs (Ft (z)) .

138 Chapter 6. Rigidity of Holomorphic Mappings and Commuting Semigroups The implication (v)⇒(i) is proved. The equivalence of (i) and (ii) is obvious. To verify the equivalence of (iii) and (iv), we just calculate: Γf (Γg (h)) = h gf + h g f, Γg (Γf (h)) = h f g + h f g. Hence, Γf ◦ Γg = Γg ◦ Γf if and only if f g − g f = 0. Now, it is clear, that (v) implies (iv). Finally we prove the implication (iv) ⇒ (v). Obviously, (iv) implies that if f has no null points in Δ, then g also has no null points in Δ and, hence, (v) follows. If f (τ ) = 0 for some τ ∈ Δ, then also g(τ ) = 0. Using the Berkson–Porta formula, one can write f (z) = (z − τ )p(z) and g(z) = (z − τ )q(z), where p and q do not vanish in Δ. Now it follows that [f, g] = (z − τ )2 [p, q]. Hence, again we have p = aq, and hence f = ag for some a ∈ C, a = 0.

Proof of Theorem 6.13. First we note that, by Theorem 6.14, f = αg,

α ∈ C.

(6.3.23)

(i) Let f (τ ) = g (τ ) = 0. By Theorem 2.7, f admits the representation f (z) = (z − τ )(1 − τ¯z)p(z),

z ∈ Δ,

where p ∈ Hol(Δ, C), Re p(z) ≥ 0. Since f (τ ) = (1 − |τ |2 )p(τ ) = 0, we have p(τ ) = 0. It follows from the Maximum Principle that p ≡ 0. Hence, f ≡ 0 and by (6.3.23) also g ≡ 0. Assume now f (τ ) = g (τ ) = 0. Then it follows from (6.3.23) that α = 1 and so f ≡ g. (ii) In general, by (6.3.23) we have f (k) (τ ) = αg (k) (τ ), 0 < k ≤ m. Hence, the condition f (k) (τ ) = g (k) (τ ) = 0 for some 0 < k ≤ m implies that α = 1 and, consequently, f ≡ g. Let Sf = {Ft }t≥0 be the semigroup generated by f ∈ G(Δ). The set Z(Sf ) of all semigroups S = {Gt }t≥0 such that Ft ◦ Gs = Gs ◦ Ft ,

t, s ≥ 0,

is called the centralizer of Sf . It is clear that for each f ∈ G(Δ) the centralizer Z(Sf ) contains Sαf for all α ≥ 0. Therefore we will say that the centralizer of Sf is trivial when the inclusion S ∈ Z(Sf ) implies that S = Sαf for some α ≥ 0.

6.3. Commuting semigroups of holomorphic mappings

139

Proposition 6.3. Let f be the generator of a semigroup Sf = {Ft }t≥0 , and let τ = 1 be the Denjoy–Wolﬀ point of Sf . Then if one of the following conditions holds, the centralizer Z(Sf ) is trivial: (i) Sf is a hyperbolic type semigroup (f (1) > 0) which is not a group; (ii) f admits the expansion f (z) = a(z − 1)3 + o (z − 1)3

with

a = 0

when z → 1 in each non-tangential approach region at τ . The ﬁrst statement is based on the following simple lemma: Lemma 6.3. Let f and g be generators of two nontrivial (neither f nor g are identically zero) commuting semigroups Sf = {Ft }t≥0 and Sg = {Gt }t≥0 , respectively. Then Sf is of hyperbolic type if and only if Sg is. In this case f = αg with real α. Moreover, α < 0 implies that Sf and Sg are both groups of hyperbolic automorphisms having ‘opposite’ ﬁxed points, i.e., the attractive point for Sf is the repelling point for Sg and conversely. Proof. Since Sf and Sg are commuting, by Theorem 6.14, there exists α ∈ C such that f = αg. In our settings, α is not zero. Suppose that τ = 1 is the Denjoy–Wolﬀ point of Sf . Then f (1) = 0 and therefore also g(1) = 0. Now, since f (1) > 0 then g (1) = α1 f (1) exists ﬁnitely, and it must be a real number by Theorem 2.11. So must be α. Now let us assume that α is negative. Then g (1) = α1 f (1) < 0. Hence the semigroup Sg generated by g must have the Denjoy–Wolﬀ point σ ∈ Δ diﬀerent from τ = 1. It is clear that σ cannot be inside Δ since otherwise it must be a common ﬁxed point of both semigroups Sf and Sg because of the commuting property. So, σ ∈ ∂Δ and g (σ) ≥ 0 (see Theorem 2.10), then f (σ) = 0 and f (σ) ≤ 0. It follows by Corollary 2.2 that 0 < f (τ ) ≤ −f (σ),

(6.3.24)

and the equality is possible if and only if f is the generator of a group of hyperbolic automorphisms. From the same Corollary 2.2, we have the reversed inequality for g 0 ≤ g (σ) ≤ −g (τ ) that means

1 1 f (σ) ≤ − f (τ ). α α Comparing this inequality with (6.3.24) gives us that f (τ ) = −f (σ) > 0 and g (τ ) = −g (σ) < 0, which means that both f and g generate groups of hyperbolic automorphisms with opposite ﬁxed points. 0≤

140 Chapter 6. Rigidity of Holomorphic Mappings and Commuting Semigroups Remark 6.9. The last assertion of this lemma follows also by a result of Behan (see [14]). Indeed, let α < 0. Then the equality f (z) = αg(z) implies that g (τ ) exists and is a real negative number. So, the Denjoy–Wolﬀ point τ of the semigroup Sf cannot be the Denjoy–Wollf point of the semigroup Sg . Hence, by [14], we conclude that Sf and Sg are groups of hyperbolic automorphisms. Proof of Proposition 6.3. The statement (i) is a direct consequence of the previous lemma. To prove the second statement, we note that the number a is a nonnegative real number (see Theorem 7.18 below). On the other hand, since Sf and Sg commute, by Theorem 6.14 there is a number α ∈ C such that f = αg. Therefore, since α = 0 also g admits the expansion g(z) = and we have that also

a α

a (z − 1)3 + o (z − 1)3 , α

≥ 0. This implies that α is a nonnegative real number.

A natural question which arises in the context of the above results is: ♦ If two elements Fp and Gq of semigroups Sf = {Ft }t≥0 and Sg = {Gt }t≥0 commute for some positive p and q, do these semigroup Sf and Sg commute? We study this problem for dilation, hyperbolic, and parabolic cases separately. Our exposition is based on the paper [51].

6.3.2 Dilation type In the following theorems, the condition F1 ◦ G1 = G1 ◦ F1 can be replaced by the condition Fp ◦ Gq = Gq ◦ Fp for some p, q > 0. Theorem 6.15. Let S1 = {Ft }t≥0 and S2 = {Gt }t≥0 be two continuous semigroups on Δ generated by f and g, respectively, and assume that F1 ◦ G1 = G1 ◦ F1 . Suppose that f has an interior null point τ ∈ Δ. If S1 and S2 are not groups of automorphisms of Δ, then they commute. Proof. Since τ is an interior null point of the generator f , it is the unique interior ﬁxed point of the semigroup S1 (see [3]). The commutativity of F1 and G1 implies that τ is a ﬁxed point of G1 and, consequently, τ is a ﬁxed point of Gt for each t > 0. By our assumption, S1 and S2 are not groups of automorphisms of Δ. By the Schwarz–Pick lemma and the univalence of Ft and Gt on Δ, we have 0 < |Ft (τ )| < 1 and 0 < |Gt (τ )| < 1 for all t > 0. Therefore a result in [39] and the semigroup property implies that G1 ◦ Ft = Ft ◦ G1 for all t ≥ 0. Similarly, Gs ◦ Ft = Ft ◦ Gs for all s, t ≥ 0. Surprisingly, the case where S1 contains elliptic automorphisms is more complicated. First we prove that a semigroup commuting with a group of elliptic automorphisms has a speciﬁc form.

6.3. Commuting semigroups of holomorphic mappings

141

Proposition 6.4. Let S1 = {Ft }t≥0 be a nontrivial group of elliptic automorphisms of Δ with a common ﬁxed point at τ ∈ Δ, and let S2 = {Gt }t≥0 be a semigroup of self-mappings of Δ. Then S1 and S2 commute if and only if S2 is a semigroup of linear fractional transformations of the form Gt (z) = mτ (e−at · mτ (z)) for some a ∈ C, where mτ (z) =

(6.3.25)

τ −z 1−τ z .

Note that the function Gt deﬁned by equality (6.3.25) is a self-mapping of Δ if and only if Re a ≥ 0. Proof. Let S2 be of the form (6.3.25). Since both S1 and S2 are actually linear semigroups up to conjugation with mτ , they must commute. Conversely, suppose that Ft ◦ Gs = Gs ◦ Ft for all s, t ≥ 0. Let Ft (z) = eiϕt z, Gt = mτ ◦ Gt ◦ mτ . Then {Ft }t≥0 is a group of automorphisms of Δ with a ﬁxed point at zero, and {Gt }t≥0 is a semigroup of self-mappings of Δ with a ﬁxed point at zero. It is obvious that the semigroups {Ft }t≥0 and {Gt }t≥0 commute. Consequently, their generators g(z) and f (z) = −iϕz are proportional (see [53]). So g(z) = az for some a ∈ C. Therefore G(z) = e−at z and Gt (z) = mτ (e−at mτ (z)). We will see below that if S1 is a group of elliptic automorphisms the commutativity of F1 and G1 does not imply that the semigroups S1 and S2 commute. Nevertheless, in this case, one can still obtain some additional information about the semigroup S2 . The following assertions explain our claim. Theorem 6.16. Let F be an elliptic automorphism of Δ and let S2 = {Gt }t≥0 be a semigroup of self-mappings of Δ which are not automorphisms. Then the commutativity of F and G1 implies that F ◦ Gt = Gt ◦ F for all t ≥ 0. Proof. Let τ ∈ Δ be the common ﬁxed point of F and Gt , t ≥ 0. Then the function τ −z F is of the form F (z) = mτ (eiϕ mτ (z)), ϕ ∈ R, z ∈ Δ, where mτ (z) = 1−τ z. iϕ Let F (z) := e z and Gt (z) = mτ (Gt (mτ (z))). Then {Gt }t≥0 is a semigroup of self-mappings of Δ which are not automorphisms with its common ﬁxed point at zero. It is obvious that for each t > 0, F and Gt commute if and only if F and Gt commute. Hence, by our assumption, F ◦ G1 = G1 ◦ F or, which is one and the same, eiϕ G1 (z) = G1 (eiϕ z). It follows that for all n ∈ N, F ◦ Gn = Gn ◦ F , where Gn are the iterates of G1 , i.e., Gn = G1 ◦ Gn−1 . Since G1 is a self-mapping of Δ (which is not an automorphism) with a ﬁxed point at the origin, there exists a unique univalent solution h of Schr¨ oder’s functional equation h(G1 (z)) = αh(z),

with

α = G1 (0),

142 Chapter 6. Rigidity of Holomorphic Mappings and Commuting Semigroups normalized by h(0) = 0, h (0) = 1 (see Theorem 5.2). This solution is given by Gn (z) . n→∞ αn

h(z) = lim

Moreover, for all real positive t (see, for instance, [49]), h(Gt (z)) = αt h(z) . Therefore, h(F (Gt (z))) = h(eiϕ Gt (z)) = lim

n→∞

Gn (eiϕ Gt (z)) αn

iϕ

e Gn (Gt (z)) eiϕ Gn (z) iϕ iϕ t t = e h( G (z)) = e α h(z) = α lim t n→∞ n→∞ αn αn iϕ Gn (e z) = αt lim = αt h(eiϕ z) = h(Gt (eiϕ z)) = h(Gt (F (z))) n→∞ αn = lim

and, by the univalence of h, we get F ◦ Gt = Gt ◦ F for all t ≥ 0. Consequently, F and Gt commute for all t ≥ 0 as asserted. Corollary 6.5. Let S1 = {Ft }t≥0 be a group of elliptic automorphisms of Δ, i.e., Ft (z) = mτ (eiϕt mτ (z)), ϕ ∈ R, τ ∈ Δ, and let S2 = {Gt }t≥0 be a semigroup of self-mappings of Δ. Suppose that ϕ π is an irrational number and F1 and G1 = I commute. Then Gt (z) = mτ (e−at mτ (z)), a ∈ C, and, consequently, the semigroups S1 and S2 commute. Proof. Once again, we deﬁne the functions Ft = eiϕt z and Gt = mτ ◦ Gt ◦ mτ . The commutativity of F1 and G1 implies that F1 ◦ G1 = G1 ◦ F1 and, by Theorem 6.16, F1 ◦ Gt = Gt ◦ F1 for all t ≥ 0. Therefore Gt (einϕ z) = einϕ Gt (z) for all n ∈ N. Since the set {einϕ }n∈N is dense in the unit circle, Gt (λz) = λGt (z) for all λ with |λ| = 1 and z ∈ Δ, by the continuity of Gt on Δ. Fix 0 = z ∈ Δ and t > 0, and consider the analytic function q(λ) on the closed unit disk deﬁned by ⎧ Gt (λz) ⎪ ⎪ ⎪ , λ = 0, ⎨ λ q(λ) = (6.3.26) ⎪ Gt (λz) ∂ ⎪ ⎪ ⎩ lim = z Gt (w) , λ = 0. λ→0 λ ∂w w=0 This function is constant on the unit circle: q(λ) = Gt (z). Moreover, q(λ) = 0 for all λ ∈ Δ. Therefore, q(λ) = Gt (z) for all λ ∈ Δ. So for each z = 0 and t > 0, Gt (λz) = λGt (z). Consequently, this equality holds for all z ∈ Δ. Hence Gt is a linear function for each t > 0, i.e., Gt (z) = e−at z for some a ∈ C, Re a ≥ 0, and the assertion follows.

6.3. Commuting semigroups of holomorphic mappings

143

In contrast with this corollary, if ϕ π is a rational number, the semigroups S1 and S2 do not necessarily commute. The following example gives a large class of semigroups S2 = {Gt }t≥0 such that F1 ◦ Gt = Gt ◦ F1 for all t ≥ 0, but the semigroups S1 and S2 do not commute. 2π

Example 6.2. Let S1 = {Ft }t≥0 , where Ft (z) = ei n t z, n ∈ N, and let S2 = {Gt }t≥0 be the semigroup generated by g(z) = zp(z n ), where Re p(z) ≥ 0 for all z ∈ Δ. Then F1 ◦ Gt = Gt ◦ F1 for all t ≥ 0. Indeed, let u = u(t, z) := Gt (z). Then u is the unique solution of the Cauchy problem ⎧ ⎨ ∂u + up(un ) = 0, ∂t (6.3.27) ⎩ u(0, z) = z, z ∈ Δ, and, consequently,

Gt (z)

z

dς = −t ςp(ς n )

for all z ∈ Δ.

(6.3.28)

2π

Substituting ei n z instead of z, we get

Gt (ei 2π ei n

2π n

z)

z

dς = −t. ςp(ς n )

2π

Now substitute ς = ei n w:

Gt (ei

2π n

z)e−i

2π n

dw = wp(wn ei2π )

z

Gt (ei

2π n

z)e−i

z

2π n

dw = −t wp(wn )

(6.3.29)

for all z ∈ Δ. Equalities (6.3.28) and (6.3.29) imply that

Gt (z)

tion

i 2π n

Gt (e

−i 2π n

z)e

dw = 0, wp(wn )

z ∈ Δ.

(6.3.30)

By the uniqueness of the solution to the Cauchy problem (6.3.27), the equa u dw = −s, s ≥ 0, z ∈ Δ n) wp(w z

has the unique solution u = Gs (z) for each s ≥ 0. Thus, it follows from (6.3.30) that 2π 2π 2π 2π Gt (ei n z)e−i n = G0 (Gt (z)) = Gt (z). Hence, Gt (ei n z) = ei n Gt (z). Therefore, F1 commutes with Gt for all t ≥ 0. At the same time, if p is not a constant function, the semigroups do not commute because their generators are not proportional (see Theorem 6.14).

144 Chapter 6. Rigidity of Holomorphic Mappings and Commuting Semigroups Remark 6.10. If at least one element of a semigroup S2 = {Gt }t≥0 , say G1 , has the form G1 (z) = zφ1 (z n ), φ1 ∈ Hol(Δ), then all the elements have the same form: Gt (z) = zφt (z n ) for some functions φt ∈ Hol(Δ); so the semigroup generator can be represented as g(z) = zp(z n ) with Re p(z) ≥ 0. Indeed, the representation G1 (z) = zφ1 (z n ) is equivalent to the commuta2π tivity of G1 with F1 (z) = ei n z. By Theorem 6.16, each mapping Gt , t ≥ 0, must commute with F1 . Hence Gt (z) = zφt (z n ), φt ∈ Hol(Δ). Diﬀerentiating Gt at t = 0+ , we arrive at our claim.

6.3.3 Hyperbolic type We start this section with a result of Heins [80]. Lemma 6.4. Let F be a hyperbolic automorphism of Δ, and let G ∈ Hol(Δ), G = I, commute with F . Then G is also a hyperbolic automorphism of Δ. This result can be complemented by the following assertion which is of independent interest. Proposition 6.5. Let F and G be two commuting holomorphic self-mappings of Δ and assume that G is not the identity. If F is of hyperbolic type, then G is of hyperbolic type too. Proof. If F is a hyperbolic automorphism of Δ, then by Lemma 6.4, G is a hyperbolic automorphism of Δ. Let F be a holomorphic self-mapping of Δ which is not an automorphism of Δ. In this case, the mappings F and G have a common Denjoy–Wolﬀ point τ ∈ ∂Δ (see [14]). We have to show that G is of hyperbolic type, i.e., 0 < G (τ ) < 1. Suppose, to the contrary, that G is of parabolic type, i.e., G (τ ) = 1. Then, by a result in [39], G must be a parabolic automorphism. τ +z . Then Let ϕ := C ◦ F ◦ C −1 and ψ := C ◦ G ◦ C −1 , where C(z) = τ −z f and g are two commuting holomorphic self-mappings of the right half-plane Π+ = {z ∈ C : Re z > 0} with their common Denjoy–Wolﬀ point at inﬁnity. Moreover, ψ is a parabolic automorphism of Π+ while ϕ is a hyperbolic selfmapping of Π+ . Consequently, ϕ and ψ are of the forms (see [127]): ϕ(w) = cw + ΓF (w)

with c =

1 ΓF (w) > 1 and ∠ lim = 0, w→∞ F (τ ) w

and ψ(w) = w + ib with

b ∈ R \ {0} and w ∈ Π+ .

By a simple calculation and the commutativity of ϕ and ψn , we infer from the above representations that ϕ(w + nib) = ϕ(w) + nib,

w ∈ Π+ .

(6.3.31)

6.3. Commuting semigroups of holomorphic mappings

145

Hence, ϕ(w + nib) ϕ(w) nib = + , w + nib w + nib w + nib

w ∈ Π+ .

exists Letting n → ∞, we obtain that for each w ∈ Π+ , the limit lim ϕ(w+nib) n→∞ w+nib and equals 1. Fix w0 ∈ Π+ . Consider the curve := {w0 + it : t ∈ R, sgn t = sgn b}. We intend to show that the limit lim ϕ(z) exists and equals 1. z z→∞

To this end, ﬁx an arbitrary ε > 0 and take N ∈ N such that |w| 1 |ϕ(w) − w| + |w| and N > N> |b| ε |b| for all w ∈ [w0 , w0 + ib]. Then ϕ(z) z − 1 < ε for all z ∈ with sgn b · Im z > sgn b(Im w0 + N b). Indeed, if sgn b · Im z > sgn b(Im w0 + N b), then z = α + ikb for some α ∈ [w0 , w0 + ib] and k ≥ N . Hence, k|b| ≥ |α| and k >

1 |b|

|ϕ(α)−α| ε

+ |α| . Consequently, |α + ikb| >

|ϕ(α)−α| ε

. k|b| − |α| > Now using (6.3.31), we obtain that ϕ(α + kib) ϕ(α) − α ϕ(z) z − 1 = α + kib − 1 = α + ikb < ε. Thus,

lim

z→∞

ϕ(z) z

= 1. It now follows from Lindel¨ of’s theorem 1.6 that

∠ lim ϕ(z) z→∞ z

= 1, which contradicts our assumption. Therefore the mapping G is indeed of hyperbolic type.

Theorem 6.17. Let S1 = {Ft }t≥0 and S2 = {Gt }t≥0 be continuous semigroups on Δ generated by f and g, respectively, and assume that F1 ◦ G1 = G1 ◦ F1 and G1 = I. Suppose that f has a boundary null point τ ∈ ∂Δ, such that f (τ ) := ∠ lim f (z) > 0, i.e., the semigroup S1 is of hyperbolic type. Then the semigroups z→τ S1 and S2 commute. Proof. By our assumption, τ is the Denjoy–Wolﬀ point of the semigroup S1 . First we suppose that S1 consists of automorphisms of Δ. Since f (τ ) > 0, S1 consists of hyperbolic automorphisms of Δ and its generator f is of the form f (z) =

a1 (z − τ )(z − ς), τ −ς

where a1 is a positive real number and ς is the second common ﬁxed point of the semigroup S1 (see [16]).

146 Chapter 6. Rigidity of Holomorphic Mappings and Commuting Semigroups Again Lemma 6.4 and the commutativity of F1 and G1 imply that G1 (hence, Gt , t ≥ 0) are hyperbolic automorphisms of Δ. Moreover, S2 has the same ﬁxed points τ and ς; consequently, its generator g is of the form g(z) =

a2 (z − τ )(z − ς), ς −τ

where a2 is a non-zero real number. Hence, g(z) = − aa12 f (z), and by Theorem 6.14, the semigroups commute. Suppose now that the semigroup S1 consists of self-mappings of Δ that are not automorphisms. By a result in [14], τ is the common Denjoy–Wolﬀ point of S1 and S2 . Then by Lemma 6.4 and Proposition 6.5, S2 consists also of hyperbolic mappings which are not automorphisms. Now our theorem is seen to be a consequence of a result of Cowen (see [39]). Remark 6.11. Note that if S1 and S2 are commuting semigroups of hyperbolic (τ ) is a type generated by f and g, respectively, then f (z) = kg(z), where k = fg (τ ) real constant. (This constant is positive whenever the semigroups are not groups.) Therefore S1 and S2 coincide up to rescaling. In particular, if in Theorem 6.17 the derivatives F1 (τ ) = e−f (τ ) and G1 (τ ) = e−g (τ ) are equal, then Ft (z) = Gt (z) for all t ≥ 0 and z ∈ Δ.

6.3.4 Parabolic type We start with the following auxiliary result: Lemma 6.5. Let F, G ∈ Hol(Δ) be two commuting univalent parabolic mappings and let τ = 1 be the Denjoy–Wolﬀ point of F . If one of the following conditions (i) F, G ∈ C 2 (1), F (1) = 0, G (1) = 0; 2 (ii) F, G ∈ CA (1), G (1) = 0, Re F (1) > 0;

(iii) F, G ∈ C 3 (1), F (1) = G (1) = 0, F (1) = 0, G (1) = 0 holds, then there exists a univalent function σ ∈ Hol(Δ, C) such that σ◦F =σ+1

(6.3.32)

and σ◦G=σ+λ −1

with

(6.3.33)

, ϕ, ψ ∈ Hol(Π+ ), where Π+ = {z ∈ 1+z C : Re z > 0} and C is the Cayley transformation given by C(z) = . Then 1−z ϕ and ψ are commuting parabolic maps in Hol(Π+ ) having ∞ as their common Denjoy–Wolﬀ point.

Proof. Let ϕ = C ◦ F ◦ C

, ψ = C ◦G◦C

λ ∈ C, λ = 0. −1

6.3. Commuting semigroups of holomorphic mappings

147

Denote w0 := 1, wn0 := ϕn (1), n = 1, 2, . . ., wn := ϕn (w), and hn (w) :=

wn ∈ Π+ ,

wn − wn0 , 0 wn+1 − wn0

w ∈ Π+ .

Then hn ∈ Hol(Π+ , C), and the sequence {hn }∞ n=1 converges in the compact open topology to a holomorphic function h ∈ Hol(Π+ , C) such that h ◦ ϕ = h + 1 and the function σ := h ◦ C solves equation (6.3.32) (see [35]). Since F is univalent in Δ, the solution σ of Abel’s equation (6.3.32) is also univalent in Δ. Suppose that (i) holds. Then the following expansions of ϕ and ψ at ∞ are satisﬁed (see [18]): ϕ(w) = w + F (1) + γϕ (w), and

ψ(w) = w + G (1) + γψ (w),

lim γϕ (w) = 0

(6.3.34)

lim γψ (w) = 0.

(6.3.35)

w→∞

w→∞

Hence, hn (ψ(w)) = =

ϕn (ψ(w)) − wn0 ψ(ϕn (w)) − wn0 = 0 0 wn+1 − wn0 wn+1 − wn0 wn + G (1) + γψ (wn ) − wn0 wn − wn0 G (1) + γψ (wn ) = 0 + 0 0 0 0 wn+1 − wn wn+1 − wn wn+1 − wn0

= hn (w) +

G (1) + γψ (wn ) F (1) + γϕ (wn ) . · 0 F (1) + γϕ (wn ) wn+1 − wn0

Letting n → ∞, we obtain h(ψ(w)) − h(w) =

G (1) F (1) + γϕ (wn ) · lim . 0 F (1) n→∞ wn+1 − wn0

(6.3.36)

Repeating this calculation with ϕ instead of ψ, we ﬁnd that F (1) + γϕ (wn ) . 0 n→∞ wn+1 − wn0

h(ϕ(w)) = h(w) + lim

F (1) + γϕ (wn ) = 1. 0 n→∞ wn+1 − wn0

At the same time, h ◦ ϕ = h + 1. Hence lim Rewrite (6.3.36) as follows: h(ψ(w)) − h(w) = λ,

where

λ=

G (1) = 0 F (1)

and w ∈ Π+ .

Substituting h = σ ◦ C −1 and ψ = C ◦ G ◦ C −1 in the last equality, we get (6.3.33).

148 Chapter 6. Rigidity of Holomorphic Mappings and Commuting Semigroups If (ii) holds, then Theorem 14 in [36] implies that for each z ∈ Δ, the sequence {Fn (z)}∞ n=1 converges to 1 (and, consequently, {wn } converges to ∞) nontangentially. So, in this case, one can repeat the proof of item (i), replacing the unrestricted limits in (6.3.34) and (6.3.35) by the angular limits. Suppose now that (iii) holds. Then the following expansions of ϕ and ψ at ∞ hold (see [18]): ϕ(w) = w −

2 F (1) + Γϕ (w), 3 w+1

w→∞

ψ(w) = w −

2 G (1) + Γψ (w), 3 w+1

w→∞

and

lim Γϕ (w)w = 0

(6.3.37)

lim Γψ (w)w = 0.

(6.3.38)

Therefore

(1) + Γϕ (wn ) − 23 Fwn +1 ϕ(wn ) − wn0 wn − wn0 hn (ϕ(w)) = 0 = + 0 0 0 0 wn+1 − wn wn+1 − wn wn+1 − wn0

= hn (w) +

(1) + Γϕ (wn ) − 32 Fwn +1 0 wn+1 − wn0

.

Letting n → ∞, we obtain

h(ϕ(w)) = h(w) + lim

(1) − 23 Fwn +1 + Γϕ (wn ) 0 wn+1 − wn0

n→∞

.

On the other hand, h(ϕ(w)) = h(w) + 1. Hence,

lim

(1) − 32 Fwn +1 + Γϕ (wn )

n→∞

0 wn+1 − wn0

= 1.

(6.3.39)

Now using (6.3.38), we ﬁnd wn − ψ(wn ) − wn0 = hn (ψ(w)) = 0 0 wn+1 − wn

2 G (1) 3 wn +1 + Γψ (wn ) 0 wn+1 − wn0

− wn0

F (1) − 2 G (1) + Γψ (wn )(wn + 1) − 23 wn +1 + Γϕ (wn ) · = hn (w) + 32 . 0 wn+1 − wn0 − 3 F (1) + Γϕ (wn )(wn + 1)

Letting n → ∞ and using (6.3.39), we get h(ψ(w)) − h(w) = λ, Consequently, σ ◦ G − σ = λ.

w ∈ Π+ ,

where

λ=

G (1) = 0. F (1)

6.3. Commuting semigroups of holomorphic mappings

149

Following [35], we say that the function σ mentioned in the lemma is the Kœnigs intertwining function associated with F . Remark 6.12. The function σ in Lemma 6.5 is completely determined by the function F . It does not depend on G. So if the conditions of the lemma hold for the same function F and another function G1 ∈ Hol(Δ), then we have the equality σ ◦ G1 = σ + λ1 with the same function σ and a constant λ1 = 0. Theorem 6.18. Let S1 = {Ft }t≥0 and S2 = {Gt }t≥0 be two nontrivial continuous semigroups on Δ generated by f and g, respectively, and let F1 ◦ G1 = G1 ◦ F1 . Suppose that τ = 1 is the boundary null point of f such that f (1) = 0. If S1 , S2 ⊂ C 0 (1) and one of the following conditions (i) the semigroups S1 and S2 are of non-automorphic type; (ii) the unrestricted limits α := lim f (z) and α := lim g (z) exist and are z→1

z→1

diﬀerent from zero; holds, then the semigroups commute. Proof. Since τ = 1 is a boundary null point of f and f (1) = 0, it is the common Denjoy–Wolﬀ point of the semigroup S1 . The commutativity of F1 and G1 implies that τ = 1 is the Denjoy–Wolﬀ point of G1 (see [14]) and, consequently, it is also the common Denjoy–Wolﬀ point of the semigroup S2 . Suppose that condition (i) holds. Consider the holomorphic function σ deﬁned by −1 σ (z) = , σ(0) = 0. (6.3.40) f (z) It follows by the Berkson–Porta formula (see Theorem 2.7) that f can be presented in the form f (z) = −(1 − z)2 p(z), where Re p(z) ≥ 0. Thus Re

σ (z) ≥ 0, q (z)

z is a univalent convex function. Then the function σ is close-to1−z convex, hence univalent in the open unit disk Δ (see, for example, Theorem 2.17 in [43]). In addition, it can be shown (see, for example, [136] and [49]) by using (6.3.40) that the function σ satisﬁes the following functional equation:

where q(z) =

σ (Ft (z)) = σ(z) + t.

(6.3.41)

Deﬁne now a univalent function σ in the unit disk by σ = σ ◦ G1 . It follows from (6.3.41) with t = 1 and the commutativity of F1 and G1 that σ (F1 (z)) = σ(z) + 1.

150 Chapter 6. Rigidity of Holomorphic Mappings and Commuting Semigroups Then by Theorem 3.1 in [35], there exists a constant λ ∈ C such that σ = σ + λ, i.e., σ (G1 (z)) = σ(z) + λ. (6.3.42) Note that if λ = 0, the univalence of σ implies that G1 (z) ≡ z. In this case all the functions Gt , t ≥ 0, coincide with the identity mapping. Hence the semigroups S1 and S2 commute. Therefore we can suppose that λ = 0. Consider now the holomorphic function σ1 deﬁned by σ1 (z) =

−1 , g(z)

σ1 (0) = 0.

(6.3.43)

As above, σ1 is univalent and satisﬁes the functional equation σ1 (Gt (z)) = σ1 (z) + t. In particular, σ1 is the Kœnigs intertwining function associated with G1 , σ1 (G1 (z)) = σ1 (z) + 1.

(6.3.44)

Comparing equations (6.3.42) and (6.3.44) and using again Theorem 3.1 in [35], we obtain that σ = λσ1 +μ for some complex number μ. Diﬀerentiating the last equality, we obtain by (6.3.40) and (6.3.43), that g(z) = λf (z). So, by Theorem 6.14, the semigroups S1 and S2 commute. Suppose now that condition (ii) holds. It can be shown that for each t ≥ 0, lim Ft (z) = −αt

z→1

and

lim Gt (z) = −αt.

z→1

We have already seen in the proof of Lemma 6.5 that σ(G1 (z)) − σ(z) =

G1 (1) . F1 (1)

(6.3.45)

Since Re F1 (1) = 0 and Re G1 (1) = 0 (see Theorem 4.4 in [18]), it follows G (1) that F 1 (1) ∈ R \ {0}. Moreover, 1

Ft (1) = −αt

and Gt (1) = −αt,

t > 0,

where α = f (1) = 0 and α = g (1) = 0. So equality (6.3.45) has the form σ(G1 (z)) − σ(z) = p,

where

p :=

α . α

(6.3.46)

On the other hand, σ(Ft (z)) − σ(z) =

αt Ft (1) = = t for all t ≥ 0. F1 (1) α

(6.3.47)

6.3. Commuting semigroups of holomorphic mappings

151

First we suppose that p > 0. From (6.3.46) and (6.3.47) we have σ(G1 (z)) = σ(Fp (z)), z ∈ Δ, and by the univalence of σ on Δ, G1 (z) = Fp (z) for all z ∈ Δ. Hence, G1 ◦ Ft = Ft ◦ G1 for all t ≥ 0. Fix t > 0 and repeat these considerations with G1 , Ft , Gs and σ instead of F1 , G1 , Ft and σ, respectively. Namely, σ(Ft (z)) − σ(z) = and σ(Gs (z)) − σ(z) =

αt Ft (1) = >0 G1 (1) α

Gs (1) =s G1 (1)

for all s > 0.

Let s := αt α > 0. Then σ(Ft (z)) = σ(Gs (z)), z ∈ Δ. By the univalence of σ on Δ we have Ft (z) = Gs (z). Therefore Gs ◦ Ft = Ft ◦ Gs for all s > 0. Since t > 0 is arbitrary, it follows that the semigroups S1 = {Ft }t≥0 and S2 = {Gs }s≥0 commute. Let now p < 0. Then by (6.3.47), σ(F−p (z))−σ(z) = −p for all z ∈ Δ. Hence, by (6.3.46), σ(F−p (G1 (z))) − σ(G1 (z)) = σ(z) − σ(G1 (z)),

z ∈ Δ,

and, therefore, σ(F−p (G1 (z))) = σ(z),

z ∈ Δ.

By the univalence of σ on Δ, F−p (G1 (z)) = z. Consequently, F−p = G−1 on 1 G1 (Δ). Since F−p ∈ Hol(Δ), G−1 is well deﬁned on Δ and so G , as well as F , 1 −p 1 are automorphisms of Δ. Therefore, {Ft }t≥0 is a semigroup of automorphisms (see, for example, [3]). Consequently, it can be extended to a group SF = {Ft }t∈R and G1 = Fp−1 = F−p ∈ SF . In particular, G1 ◦ Ft = Ft ◦ G1 for all t ≥ 0. Fix t > 0. In a similar way, using the commutativity of Ft and G1 , one can show that the semigroup {Gs }s≥0 can be extended to a group SG = {Gs }s∈R and that Ft ◦ Gs = Gs ◦ Ft for all s, t ∈ R. Recently, Levenshtein and Reich [100] showed that condition (ii) in Theorem 6.18 may be weakened. For discrete time semigroups, diﬀerent interesting results on ﬁxed points of commuting mappings based on the construction of Poggi-Corradini [107] are presented in [19].

Chapter 7

Asymptotic Behavior of One-parameter Semigroups In this chapter, we continue an advanced study of the asymptotic behavior of discrete and continuous semigroups using some geometric properties of associated Kœnigs functions. Our main goal is to establish rates of convergence of continuous semigroups to their attractive ﬁxed point and to study angular characteristics of their trajectories at the Denjoy–Wolﬀ point and other boundary regular ﬁxed points (when they exist). It turns out that the three mutually exclusive classes of semigroups, although sharing some common features, also exhibit strikingly diﬀerent phenomena. For example, semigroups of parabolic type, which in general possess the most complicated behavior, are the simplest in the sense of their angular asymptotic characteristics. In particular, all such semigroups which have the same second derivative at their common Denjoy–Wolﬀ point share the same asymptote, irrespective of their initial points. In the hyperbolic case, the situation is completely diﬀerent. Although in this case each semigroup trajectory has an asymptote at its Denjoy–Wolﬀ point, this asymptote does depend on the initial point of the trajectory. Moreover, if two such semigroups behave similarly as t → ∞, they must coincide up to rescaling (see Section 7.2.2 below). In general, the trajectories of a semigroup of dilation type have no asymptotes (except in the case when the trajectories themselves are straight lines). Nevertheless, one can use the angular characteristics of such trajectories to establish an asymptotic rigidity result in this case too (see Section 7.1.2 below).

154

Chapter 7. Asymptotic Behavior of One-parameter Semigroups

7.1 Dilation case 7.1.1 General remarks and rates of convergence The classical Schwarz–Pick Lemma implies that if ζ ∈ Δ is an interior ﬁxed point of F ∈ Hol(Δ), i.e., F (ζ) = ζ, (7.1.1) then F leaves each pseudohyperbolic disk Dr (ζ) centered at ζ invariant. In other words, for each r ∈ (0, 1), F (Dr (ζ)) ⊆ Dr (ζ),

(7.1.2)

where ¯2 z−ζ |1 − z ζ| Dr (ζ) = z ∈ Δ : ¯ < r = z ∈ Δ : 1 − |z|2 < K , 1 − ζz

(7.1.3)

−1 K = 1 − |ζ|2 1 − r2 . In turn, this fact and Vitali’s Theorem show that a holomorphic self-mapping of Δ diﬀerent from the identity has at most one interior ﬁxed point in Δ (see, for example, Proposition 1.3.2 in [130]). However, simple examples show that holomorphic self-mappings of Δ may have many ﬁxed points on the circle ∂Δ. We begin with the dilation case when F ∈ Hol(Δ) has an interior ﬁxed point ζ ∈ Δ. An additional consequence of the Schwarz–Pick Lemma is that |F (ζ)| ≤ 1.

(7.1.4)

Moreover, equality in (7.1.2) or (7.1.4) holds if and only if F is an (elliptic) automorphism of Δ. On the other hand, if the strong inequality |F (ζ)| < 1 holds, then it follows by the Banach Contraction Principle (locally applying to Compact Con F ) and ∞ vergence Principle that for each z ∈ Δ, the sequence F (n) (z) n=0 (the orbit) converges to ζ as n goes to inﬁnity. This is a part of the Denjoy–Wolﬀ Theorem (see Section 1.3), which is helpful in the study of the asymptotic behavior of discrete-time semigroups deﬁned by iterates of a holomorphic self-mapping of Δ. To summarize we formulate the following assertion. Proposition 7.1. Let F ∈ Hol(Δ) have a ﬁxed point ζ ∈ Δ. Then (i) for each r ∈ (0, 1) and n = 0, 1, 2, . . . , the following invariance condition holds: F (n) (Dr (ζ)) ⊆ Dr (ζ); (ii) if F is not the identity, then the point ζ ∈ Δ is a unique ﬁxed point of F in Δ. Moreover, the following are equivalent:

7.1. Dilation case

155

∞ (a) For each z ∈ Δ, the sequence F (n) (z) n=0 of iterates converges to ζ as n goes to inﬁnity. (b) The mapping F is not an (elliptic) automorphism of Δ. (c) |F (ζ)| < 1. Combining this statement with Brouwer’s Fixed Point Principle , one gets the following suﬃcient condition of existence and uniqueness of an interior ﬁxed point for holomorphic self-mappings of the unit disk. Corollary 7.1. Suppose that F ∈ Hol(Δ) maps Δ strictly inside, i.e., for some r ∈ (0, 1), |F (z)| ≤ r

(7.1.5)

for all z ∈ Δ. Then F has a unique ﬁxed point ζ ∈ Δ, |ζ| ≤ r, and for each z ∈ Δ ∞ the orbit F (n) (z) n=0 converges to ζ as n goes to inﬁnity. In fact, (7.1.5) means that F ∈ Hol(Δ) is a strict contraction in the Poincar´e hyperbolic metric on Δ. So, Corollary 7.1 is also a consequence of the Banach Fixed Point Principle. At the same time, if condition (7.1.5) does not hold, the ∞ problem of ﬁnding a global rate of convergence of {Fn }n=1 to the interior ﬁxed point ζ ∈ Δ seems to be open. This problem, however, can be successfully solved for continuous semigroups (see Proposition 7.4 below). To proceed with continuous semigroups we want to trace a connection between the iteration theory of functions in one complex variable and the asymptotic behavior of solutions of ordinary diﬀerential equations governed by evolution problems. Therefore, our terminology is related to both these topics. Deﬁnition 7.1. A point ζ ∈ Δ is said to be a stationary point of a continuous semigroup S = {Ft }t>0 ⊂ Hol(Δ) if Ft (ζ) = ζ

(7.1.6)

for all t > 0. In other words, ζ ∈ Δ is a stationary point of S if it is a common ﬁxed point of all Ft ∈ S. Note that the family S = {Ft }t≥0 is commuting, that is Ft ◦ Fs = Fs ◦ Ft = Ft+s for all t, s ≥ 0. Hence, it follows by Shields’ theorem [129] that if each Ft had been continuously extended to ∂Δ, the boundary of Δ, then the stationary point set of S would not be empty. As a matter of fact, it is enough to require the existence of an interior ﬁxed point only for one t > 0 to ensure the existence of such a point for the whole semigroup. Indeed, if for at least one t > 0 the mapping Ft ∈ S has an interior

156

Chapter 7. Asymptotic Behavior of One-parameter Semigroups

ﬁxed point ζ ∈ Δ, then it is a unique ﬁxed point for Ft , and for each s ≥ 0 we have: Fs (ζ) = Fs (Ft (ζ)) = Ft (Fs (ζ)) = ζ, i.e., ζ is also a ﬁxed point of each element Fs ∈ S, s ≥ 0. Hence, this ﬁxed point is a unique stationary point of S. Using the chain rule and the Schwarz–Pick Lemma, it is easy to show that if ζ ∈ Δ is a stationary point for S = {Ft }t>0 , then (Ft ) (ζ) = e−kt with some complex k such that Re k ≥ 0. Naturally, the strategy now is to study the convergence of a semigroup to its stationary point. The foregoing fact is the ﬁrst step in the study of the asymptotic behavior of a continuous semigroup in Δ. Proposition 7.2 ([1, 91]). Let S = {Ft }t>0 ⊂ Hol(Δ) be a semigroup on Δ. Then this semigroup converges uniformly on compact subsets of Δ to a holomorphic mapping F ∈ Hol(Δ, C) if and only if, for at least one t0 , the sequence {Ft0 n }∞ n=0 converges uniformly on compact subsets of Δ. Moreover, if Ft0 is not the identity, then F is a constant with modulus less than or equal to 1. The fact that we have formulated above implies immediately a continuous analog of the Denjoy–Wolﬀ Theorem [17, 1, 91] (see Section 2.3). Let S = {Ft }t≥0 ⊂ Hol(Δ) be a ﬂow on Δ. If for at least one t0 the mapping Ft0 is not the identity and is not an elliptic automorphism of Δ, then the net {Ft }t≥0 converges to a constant ζ ∈ Δ as t → ∞ uniformly on each compact subset of Δ. Since every continuous semigroup S = {Ft }t≥0 of holomorphic self-mappings of Δ is diﬀerentiable in t ≥ 0, it is natural to describe its asymptotic behavior in terms of the generator f = lim 1t (I − Ft ). This becomes more desirable when such t→0

a semigroup is not given explicitly, but is deﬁned as the solution of the Cauchy problem: ⎧ ⎨ ∂u(t, z) + f (u(t, z)) = 0 ∂t (7.1.7) ⎩ u(0, z) = z ∈ Δ. Here we set Ft (z) = u(t, z). Note also that if f is holomorphic in a neighborhood of the point ζ ∈ Δ, then it follows by the uniqueness of the solution of the Cauch´ y problem that f (ζ) = 0 if and only if ζ is a stationary point of S = {Ft }t≥0 . In particular, an interior null point of a generator is a stationary point of the generated semigroup. However, this fact is no longer true for a boundary null point even when f is continuous (but not diﬀerentiable) on Δ. The following assertion is a nonlinear analog of Lyapunov’s Stability Theorem.

7.1. Dilation case

157

Proposition 7.3 (see [130]). Let f ∈ Hol(Δ, C) be the generator of a one-parameter semigroup S = {Ft }t≥0 with f (τ ) = 0 for some τ ∈ Δ. Then (i) Re f (τ ) ≥ 0; (ii) Re f (τ ) > 0 if and only if S = {Ft }t≥0 converges to τ uniformly on compact subsets of Δ. Note that if f ≡ 0 and Re f (τ ) = 0, then the semigroup S = {Ft }t≥0 is actually a group of elliptic automorphisms (see [130]). Combining this fact with the Berkson–Porta Theorem 2.7, we conclude that if f ∈ G(Δ) is not a generator of the form f (z) = (z − τ )(1 − τ¯z)a with τ ∈ Δ and Re a = 0, then the semigroup S = {Ft }t≥0 generated by f does not consist of elliptic automorphisms and converges uniformly on each compact subset of Δ to a point in Δ. If this point belongs to Δ, then it is a unique uniformly attractive stationary point of the semigroup, and the question of ﬁnding a rate of convergence arises naturally. We consider here the case when f (0) = 0, i.e., the origin is a stationary point of the semigroup S = {Ft }t≥0 . Proposition 7.4 (see [77], [114] and [130]). Let f ∈ G be a holomorphic generator with f (0) = 0 and k = Re f (0) > 0, and let S = {Ft }t≥0 be the semigroup generated by f. Then there exists c ∈ [0, 1] such that for all z ∈ Δ and t ≥ 0 the following estimates hold: 1 + c|z| 1 − c|z| (a) |z| exp(−k t) ≤ |Ft (z)| ≤ |z| · exp −k t ; 1 − c|z| 1 + c|z| (b) exp(−kt)

|z| |Ft (z)| |z| ≤ ≤ exp(−kt) . 2 2 (1 + c|z|) (1 − c|Ft (z)|) (1 − c|z|)2

Note that estimate (a) with c = 1 is due to Gurganus [77], while estimate (b) was established by Poreda [114]. A more general case, when f ∈ G with f (τ ) = 0 for some τ ∈ Δ which is not necessarily zero, can be obtained easily from the above theorem by using the M¨ obius transformation τ −z Mτ (z) = , z ∈ Δ. 1 − τz Some other rates of convergence in terms of the hyperbolic Poincar´e metric on Δ can be found in [57] and [58].

7.1.2 Argument rigidity principle Now we turn again to a continuous time semigroup of dilation type generated by f ∈ G[τ ] , τ ∈ Δ, f (τ ) = 0 and Re f (τ ) > 0. In this situation, the trajectories {Ft (z)}t≥0 are either straight lines (if Im f (τ ) = 0) or spirals around τ (if

158

Chapter 7. Asymptotic Behavior of One-parameter Semigroups

Im f (τ ) = 0). In the latter case one cannot consider the angular characteristic of the trajectories as in the hyperbolic or parabolic case (see Sections 7.2.2 and 7.3.3 below). Nevertheless, we are able to treat two (possibly diﬀerent) semigroups in the sense of their similar asymptotic behavior according to the following notation. Deﬁnition 7.2. Let Sf = {Ft }t≥0 and Sg = {Gt }t≥0 be two semigroups generated by f and g respectively, such that they have the same Denjoy–Wolﬀ point τ ∈ Δ. We say that these semigroups have similar asymptotic behavior if τ − Ft (z) lim arg = 0. (7.1.8) t→∞ τ − Gt (z) Theorem 7.1 ([60]). Let f and g be generators of semigroups Sf = {Ft }t≥0 and Sg = {Gt }t≥0 , respectively. Suppose that f (τ ) = g(τ ) = 0 for some τ ∈ Δ and Re f (τ )·Re g (τ ) > 0. In other words, Sf and Sg have the same (interior) Denjoy– Wolﬀ point τ ∈ Δ. Then they have similar asymptotic behavior, i.e., τ − Ft (z) lim arg =0 (7.1.9) τ →∞ τ − Gt (z) if and only if for some k ∈ C and

f (z) = kg(z)

(7.1.10)

Im f (τ ) = Im g (τ ).

(7.1.11)

Moreover, in this case, the semigroups Sf and Sg must commute. Proof. Set τ = 0. Again we consider Kœnigs type functions (see Remark 5.4) hf (z) = lim etβf Ft (z)

(7.1.12)

hg (z) = lim etβg Gt (z),

(7.1.13)

t→∞

and t→∞

where βf = f (0) and βg = g (0) (Re βf · Re βg > 0). Then, by (7.1.9), (7.1.12) and (7.1.13), we get that lim arg

t→∞

hf (z) Ft (z) = arg + lim arg e(βg −βf )t = 0. Gt (z) hg (z) t→∞

(7.1.14)

Setting μ := βg − βf , we have also that arg eμt = t Im μ.

(7.1.15)

Therefore, we obtain from (7.1.14) and (7.1.15) that (7.1.9) holds if and only if arg hf (z) = arg hg (z)

(7.1.16)

7.2. Hyperbolic case

159

and Im βf = Im βg . Since hf (0) = hg (0) = 0 and

hf (0)

=

hg (0)

(7.1.17)

= 0, we have from (7.1.16) that

hf (z) = hg (z) =: h(z),

z ∈ Δ,

(7.1.18)

and, consequently, f (z) = kg(z), βf βg

where k = . Now Theorem 6.14 implies that the semigroups Sf and Sg commute. The reverse consideration completes our proof.

Remark 7.1. Note that, in contrast to the hyperbolic case, the trajectories of a dilation type of semigroups that satisfy condition (7.1.9) are not necessarily the same. Moreover, they coincide if and only if Im βf = Im βg = 0

or βf = βg .

7.2 Hyperbolic case 7.2.1 Criteria for the exponential convergence Let f ∈ G[1], i.e., f is a semigroup generator with lim f (r) = 0 and β = f (1) := r→1−

f (r) exists (see (2.2.9)), and let S = {Ft }t≥0 be a semigroup generated by lim r→1− r − 1 f . It follows by Theorems 2.12 and 1.10 that |1 − z|2 |1 − Ft (z)|2 ≤ e−tβ . 2 1 − |Ft (z)| 1 − |z|2

(7.2.1)

In addition, lim Ft (z) = 1 if and only if β ≥ 0 (see Theorem 2.10). Moreover, if t→∞

β > 0, this estimate establishes the exponential rate of convergence of {Ft }t≥0 to the point τ = 1 in terms of a non-Euclidean distance deﬁned by d(z, τ ) =

|τ − z|2 , 1 − |z|2

where τ ∈ ∂Δ, z ∈ Δ (see details in Theorem 2.10). The natural problem that has arisen here (see, for example, [10, 79]) is ♦ Whether the norm convergence is exactly of exponential type, i.e., Ft (z) − 1 ∼ e−βt ? In other words, the question is

160

Chapter 7. Asymptotic Behavior of One-parameter Semigroups ♦ Whether the following limit K(z) := lim exp (tβ) (1 − Ft (z)) t→∞

exists and is not zero? The following well-known fact is a very important result in the study of the probability asymptotics of branching processes (see, for example, [126]): Let Ft , t ≥ 0, be the generating function of a subcritical continuous branching process. Then 1 − Ft (0) = Ke−βt (1 + o(1)) if and only if the (real) integral 1 0

βx + f (1 − x) dx xf (1 − x)

converges, where f is the inﬁnitesimal generator of the semigroup S = {Ft }t≥0 of the form (5.2.6). In this case, the integral is equal to − log K. As a matter of fact, we will see below that this result does not depend on the probability nature of the semigroup rather than on the boundary behavior of the Kœnigs–Valiron function h deﬁned by (5.4.19), which in the probability sense is closely related to the solution gˆ of the so-called Yaglom equation gˆ(Ft (z)) = mt gˆ(z) + 1 − mt .

(7.2.2)

Note that, for discrete-time semigroups this problem has been studied in [143] and [110]. It turns out that, for continuous-time semigroups the answer can be given in terms of the (K-V)–function h(z) = h0 (z) = lim

t→∞

1 − Ft (z) , 1 − Ft (0)

deﬁned in Chapter 5. First, we observe that, actually, in Theorem 5.6 we proved the following fact: Let f ∈ G[1] have a positive angular derivative β := f (1) > 0 and generate a semigroup of holomorphic self-mappings {Ft }t≥0 . Then the limit h(z) := lim

t→∞

1 − Ft (z) 1 − Ft (0)

(7.2.3)

exists and is a function of class Star1 [1]. Moreover, this function satisﬁes the diﬀerential equation βh(z) = h (z)f (z),

(7.2.4)

7.2. Hyperbolic case

161

which is equivalent to Schr¨oder’s functional equation h(Ft (z)) = λt h(z),

t ≥ 0,

(7.2.5)

with λt = e−tβ (see Theorem 5.1). We now need the following consequence of Proposition 4.13 in [113]. Lemma 7.1. Let φ be a univalent function on the unit disk Δ, and let g ∈ Hol(Δ, C) satisfy g(Δ) ⊂ φ(Δ). Given a boundary point ζ ∈ ∂Δ, suppose that the limits ∠ lim g(z) and lim φ(z) exist ﬁnitely and coincide. Suppose also that lim φ (z) = z→ζ

z→ζ

z→ζ

s = 0, ∞. If there is a sequence {zn } ⊂ Δ non-tangentially convergent to ζ such that g (zn ) → α ∈ C, then ∠ lim g (z) = α. z→ζ

Proof. First we deﬁne

g1 (z) := φ−1 ◦ g (z).

This function is a well-deﬁned self-mapping of Δ because of g(Δ) ⊂ φ(Δ). Moreover, by our supposition, the point ζ is a boundary ﬁxed point of g1 . Then, by Proposition 4.13 in [113], the limit ∠ lim g1 (z) = A ≤ ∞ z→ζ

exists. Now let γ : [0, ∞) → Δ be a continuous curve in Δ such that γ(t) → ζ non-tangentially as t → ∞ and γ(n) = zn . We already know that lim g (γ(t)) t→∞ 1

= A.

Hence, lim g1 (zn ) = lim g1 (γ(n)) = A.

n→∞

n→∞

By deﬁnition, g = φ ◦ g1 , and so g (z) = φ (g1 (z)) · g1 (z). Therefore, α = lim g (zn ) = lim (φ ◦ g1 ) (zn ) · g1 (zn ) = sA. n→∞

n→∞

Then A = ∞, and ∠ lim g (z) = ∠ lim (φ ◦ g1 ) (z) · g1 (z) = sA = α. z→ζ

The lemma is proved.

z→ζ

162

Chapter 7. Asymptotic Behavior of One-parameter Semigroups

Deﬁnition 7.3. Let f be a continuous function on the open unit disk Δ. We say 1 that the integral f (z)dz converges non-tangentially if the function 0

z ϕ(z) =

f (z)dz 0

has a non-tangential limit at the boundary point z = 1. Theorem 7.2 (cf., [66]). Let f ∈ Gh [1], i.e., f (z) = −(1 − z)2 p(z) + β/2(z 2 − 1),

(7.2.6) where β = f (1) > 0 and p ∈ Hol(Δ, C) with Re p > 0 and ∠ lim (1 − z)p(z) = 0. Let {Ft }t≥0 be the semigroup generated by f . Then

z→1

(I) the limit K(z) := lim exp (tβ) (1 − Ft (z)) t→∞

exists. (II) the limit function K is either identically zero or a univalent function with Re K > 0. (III) K(z) ≡ 0 if and only if one of the following assertions holds: (a) the integral 1 0

β(1 − z) + f (z) dz (1 − z)f (z)

converges non-tangentially; (b) the integral 1 p(z) dz 0

converges non-tangentially; (c) the (K-V)–function h deﬁned by (7.2.3) is conformal at the point z = 1. In this case K(z) = K(0)h(z). Proof. Any element Ft , t ≥ 0, of the semigroup is a univalent self-mapping of the unit disk Δ. Therefore, each function Kt (z) := exp (tβ) (1 − Ft (z))

7.2. Hyperbolic case

163

is univalent and satisﬁes Re Kt > 0. So, the family {Kt }t≥0 is normal. In addition, it follows by (7.2.1) that this family is compact: |1 − z|2 1 − |z|2 |1 − z|2 ≤ 1 − |z|2

|Kt (z)| ≤

1 − |Ft (z)|2 |1 − Ft (z)| |1 − z|2 1 − |Ft (z)| (1 + |Ft (z)|) ≤ 2 . |1 − Ft (z)| 1 − |z|2

Hence, there exists a convergent sequence {Ktn }∞ n=1 . Since {Ft }t≥0 converges to τ = 1 as t → ∞ non-tangentially, we see that −Ft (z) 1 f (Ft (z)) β Kt (z) = = → as t → ∞. Kt (z) 1 − Ft (z) f (z) Ft (z) − 1 f (z) Thus, each limit function K satisﬁes the following equation: βK(z) = K (z)f (z).

(7.2.7)

Since (7.2.7) is a homogeneous linear diﬀerential equation which coincides with (7.2.4), it follows that K is either equal to zero identically, or K(z) = αh(z) for some α = 0. Suppose that there exists a sequence {Ktn }∞ n=1 converging to a non-zero function K(z) = lim Ktn (z) = lim exp (tn β) (1 − Ftn (z)) = αh(z). n→∞

n→∞

In this case, K is a univalent function on the unit disk Δ. We now show that the angular derivative K (1) exists and equals −1. We already know that any limit function K satisﬁes the diﬀerential equation (7.2.7). Then, by Theorem 5.1, K is a solution of Schr¨ oder’s equation (7.2.5). Therefore, denoting wn = Ftn (z) for ﬁxed z ∈ Δ and using Schr¨oder’s equation (7.2.5), diﬀerential equation (7.2.7) and representation (7.2.6), one can calculate 0 = lim etn β (1 − wn ) − etn β K(wn ) n→∞

K (wn )f (wn ) tn β = lim e (1 − wn ) − n→∞ β 3 4 K (wn ) (1 − wn )2 p(wn ) − β/2(wn2 − 1) tn β (1 − wn ) + = lim e n→∞ β

K (wn )(1 − wn )p(wn ) K (wn )(wn + 1) = lim etn β (1 − wn ) 1 + + n→∞ β 2

)p(w ) (1 − w n n . = K(z) lim 1 + K (wn ) + K (wn ) n→∞ β

164

Chapter 7. Asymptotic Behavior of One-parameter Semigroups

So,

lim K (Ftn (z)) = −1.

n→∞

By Lemma 7.1, we conclude that the angular derivative K (1) exists and equals −1. This implies that if there exists a sequence {Ktn }∞ n=1 convergent to a nonzero function K, then the function h(z) = K(z)/α has the angular derivative h (1) = −1 α = 0, i.e., h is conformal at the point z = 1. Moreover, such a non-zero limit function is unique: K(z) = −h(z)/h (1) and h(z) =

1 1 lim Kt (z) = lim exp (tn β) (1 − Ftn (z)) . α n→∞ n α n→∞

Further, by (7.2.4) and (7.2.6), we get

(1 − z)p(z) 1+z h(z) = −h (z)(1 − z) + . β 2 Combining these equalities with (7.2.5), we simply calculate

1 1 Kt (z) − h(z) = lim Kt (z) − etβ h(Ft (z)) lim t→∞ α t→∞ α

1 + Ft (z) 1 (1 − Ft (z))p(Ft (z)) + h (Ft (z)) + = lim Kt (z) t→∞ α β 2 1 + h (Ft (z)) = 0, = lim Kt (z) t→∞ α because of h (1) = − α1 and {Kt }t≥0 is a compact family. So, we have proved (I), (II) and (III c). To proceed, we note that by (7.2.6), p(z) − β2 β(1 − z) + f (z) = . (1 − z)f (z) (1 − z)p(z) + β2 (1 + z) Since the denominator in the last fraction does not vanish in Δ and tends to β = 0 as z goes to 1 non-tangentially, we conclude that conditions (a) and (b) in assertion (III) of the theorem are equivalent. Since βh(z) = h (z)f (z), we have

β(1 − z) + f (z) h (z) 1 = + (1 − z)f (z) h(z) 1−z

for all z ∈ Δ,

and thus z 0

β(1 − u) + f (u) du = log (1 − u)f (u)

h(z) 1−z

.

(7.2.8)

7.2. Hyperbolic case

165

Therefore, condition (a) holds if and only if the function h is conformal at the boundary point z = 1, i.e., condition (c) holds. The proof of the theorem is complete. Another proof that K ≡ 0 if and only if h is conformal at the boundary point z = 1 was given in [131]. Based on this fact, one can prove the equivalence of assertions (a) and (c) of the theorem by the following simple argument. Using the identity 1 β(z − 1) − f (z) β = − , f (z) (z − 1)f (z) 1−z one can solve the Cauchy problem (2.1.3) as follows: βt = log(1 − z) − log(1 − Ft (z)) Ft (z)

− z

β(1 − u) + f (u) (1 − u)f (u)

du,

and, consequently, Ft (0) β(1 − u) + f (u) du. log exp(βt) (1 − Ft (0)) = − (1 − u)f (u)

(7.2.9)

0

It is known that for any point z ∈ Δ, the curve w = Ft (z), t ≥ t0 lies in a Stolz angle with the vertex at τ = 1. Therefore, if condition (a) holds, relation (7.2.9) implies that K(0) = 0. By the proof of the ﬁrst part of the theorem, K is a univalent function with Re K(z) > 0, z ∈ Δ. Conversely, suppose that K is a univalent function. It follows by (7.2.8) and (7.2.9) that 1 − Ft (0) . exp(βt) (1 − Ft (0)) = h(Ft (0)) Then K(0) = lim

t→∞

1 − Ft (0) . h(Ft (0))

By Proposition 4.9 in [113], the angular derivative h (1) = ∠ lim

z→1

h(z) z−1

exists and is diﬀerent from 0, i.e., h is conformal at the point z = 1.

166

Chapter 7. Asymptotic Behavior of One-parameter Semigroups

Remark 7.2. It is clear that the function gˆ(z) = 1−h(z) is a solution of the Yaglom ∞ * equation (7.2.2) with mt = e−βt . In addition, it is known that gˆ(z) = bk z k , k=0

where bk = lim P (Z(t) = k| Z(t) > 0). Therefore, our theorem can be easily t→∞ reformulated in terms of the generating function gˆ. Also inequality (7.2.1) evolves a similar problem on exponential convergence |τ − z|2 . Namely, the question in terms of the non-Euclidean distance d(z, τ ) = 1 − |z|2 is: ♦

Whether the limit lim etβ d(Ft (z), τ ) =: m(z)

t→∞

exists and is not zero? The following considerations show that, in fact, the non-tangential convergence of a hyperbolic type semigroup implies that d(Ft (z), τ ) ∼ |τ − Ft (z)|. Recall that for κ > 1 the set Γ(ζ, κ) = {z ∈ Δ : |z − ζ| < κ(1 − |z|)} is a non-tangential approach region at ζ ∈ ∂Δ. We already know that each hyperbolic type semigroup S = {Ft }t≥0 converges to its boundary Denjoy–Wolﬀ point non-tangentially (see the proof of Theorem 5.6). In other words, for each z ∈ Δ there is κ = κ(z) ∈ (1, ∞) such that |1 − Ft (z)| < κ(z) (1 − |Ft (z)|)

for all t ≥ 0,

(7.2.10)

or, which is same, for each z ∈ Δ the function t (z) =

1 − |Ft (z)| |1 − Ft (z)|

is bounded away from zero on [0, ∞), i.e., 1 ≥ t (z) >

1 > 0. κ(z)

At the same time, d(Ft (z), 1) = So, the question is:

1 |1 − Ft (z)| · . 1 + |Ft (z)| t (z)

(7.2.11)

7.2. Hyperbolic case

167

♦ Whether the limit (z) := lim t (z) exists? t→∞

We now show that the answer is aﬃrmative. To do this, we will use a modiﬁed Kœnigs–Valiron function deﬁned as follows. Recall that by Theorems 5.5 and 5.6, for ﬁxed w ∈ Δ the limit ψw (z) = lim

t→∞

1 − Ft (z) |1 − Ft (w)|

(7.2.12)

exists and is a univalent starlike function with respect to ψw (1) = 0 with Re ψw (z) > 0, z ∈ Δ. This function satisﬁes Schr¨ oder’s functional equation ψw (Ft (z) = e−tβ ψw (z). Lemma 7.2. Let S = {Ft }t≥0 ⊂ Hol(Δ) be a semigroup of hyperbolic type and let ψw be deﬁned by (7.2.12). Then (w) := lim t (w) = Re ψw (w). t→∞

(7.2.13)

Proof. Fix w ∈ Δ and consider the net of univalent functions 1 − Ft (z) 1 + Ft (w) · . gt (z) = 1 + Ft (z) 1 − Ft (w) It is clear that lim gt (z) = ψw (z).

t→∞

Hence Re gt (z) → Re ψw (z) as t → ∞. On the other hand, 2 1 − |Ft (w)| 1 + Ft (w) Re gt (w) = · 2 |1 + Ft (w)| 1 − Ft (w) =

1 − |Ft (w)| 1 + |Ft (w)| · → (w), |1 − Ft (w)| |1 + Ft (w)|

and we are done. This lemma and Theorem 7.2 imply the following assertion.

Corollary 7.2. Let f ∈ Gh [1], and let S = {Ft }t≥0 ⊂ Hol(Δ) be the semigroup generated by f . Then the limit m(z) := lim etβ d(Ft (z), 1) t→∞

exists. Moreover, m(z) = where K(z) is deﬁned in Theorem 7.2.

|K(z)| , 2(z)

168

Chapter 7. Asymptotic Behavior of One-parameter Semigroups

Finally, regarding the asymptotic behavior of a semigroup generated by g ∈ G[1, −1], we note that using Theorem 2.12, one can now derive the following rates of convergence. Corollary 7.3. Let g ∈ G[1, −1] with β+ = g (1) ≥ 0 and g (−1) = β− < 0. Then for z ∈ Δ etβ− [ϕ− (z)]−1 ≤ ϕ+ (Ft (z)) ≤ e−tβ+ ϕ+ (z), where ϕ+ (z) =

|1 − z|2 1 − |z|2

and

ϕ− (z) =

|1 + z|2 . 1 − |z|2

In particular, if z = x ∈ (−1, 1) then etβ−

1−x |1 − Ft (x)|2 −tβ+ 1 − x ≤ . 2 ≤ e 1+x 1+x 1 − |Ft (x)|

7.2.2 Angular similarity principle In contrast with the dilation case, for a semigroup with the boundary Denjoy–Wolﬀ point τ ∈ ∂Δ, a natural question is: ♦ Does the limit lim arg(τ − Ft (z)) exist? t→∞

For semigroups of hyperbolic type, the answer is aﬃrmative. Moreover, we will see that a kind of rigidity results for semigroups of holomorphic mappings for the hyperbolic case can be established also in terms of their asymptotic angular behavior. It turns out, that in contrast to the parabolic case, two semigroups of hyperbolic or dilation types having similar asymptotic behavior at their common Denjoy–Wolﬀ point coincide if and only if they have the same derivatives of the ﬁrst order at this point. In addition, we note that for hyperbolic type semigroups (in contrast to the parabolic case) the limiting tangent of their trajectories {Ft (z)}t≥0 does depend on the choice of the initial point z ∈ Δ. Moreover, each other semigroup having the same limiting tangents (depending on z ∈ Δ) must be the same semigroup up to a rescaling. We begin with the following angular asymptotic formula which is important by itself: Theorem 7.3 (see [60]). Let S = {Ft }t≥0 be a semigroup of hyperbolic type with the Denjoy–Wolﬀ point τ = 1 and let f be its generator. Then

z r dζ dζ − lim Im . (7.2.14) lim arg(1 − Ft (z)) = f (1) Im z→1 r→1 0 f (ζ) 0 f (ζ) Proof. Consider a net of holomorphic functions ht : Δ → C deﬁned as ht (z) = αt (1 − Ft (z)),

(7.2.15)

7.2. Hyperbolic case

169

where {αt }t≥0 is a net of complex numbers, such that the limit lim αt (1 − Ft (0)) = α

t→∞

(7.2.16)

exists and is diﬀerent from zero. Then ∂Ft (z) ∂ht (z) f (Ft (z)) = −αt = αt · . ∂z ∂z f (z)

(7.2.17)

Since each trajectory {Ft (z)}t≥0 , z ∈ Δ, converges to z = 1 non-tangentially, we get from (7.2.15) and (7.2.17) that 1 f (1) ∂ht (z) f (Ft (z)) : ht (z) = lim · = . t→∞ t→∞ 1 − Ft (z) f (z) ∂z f (z) lim

(7.2.18)

Furthermore, since Re(1 − Ft (z)) > 0, for all t ≥ 0 and z ∈ Δ, we have by (7.2.16) that the family {ht }t≥0 is normal. In addition, (7.2.18) implies that each of its limit functions satisﬁes the diﬀerential equation f (1) · h(z) = h (z) · f (z)

(7.2.19)

and the conditions h(1) = 0

and h(0) = α.

(7.2.20)

Therefore, the net {ht }t≥0 is convergent locally uniformly by the uniqueness of the solution of (7.2.19) with conditions (7.2.20). Further, we will prove our theorem by using a diﬀerent choice of the net {αt }t≥0 . First we set 1 αt = . (7.2.21) 1 − Ft (0) Then h deﬁned as h(z) = lim

t→∞

1 − Ft (z) 1 − Ft (0)

(7.2.22)

exists and satisﬁes equation (7.2.19) with h(0) = 1 − h(1) = 0. In addition, for all s ≥ 0, we have by (7.2.19) and the Cauchy problem that h(Fs (z)) = e−sf

(1)

· h(z).

(7.2.23)

Now let us set αt in (7.2.15) in place of αt deﬁned as αt =

1 . |1 − Ft (0)|

(7.2.24)

Show that lim αt (1 − Ft (0)) := α

t→∞

(7.2.25)

170

Chapter 7. Asymptotic Behavior of One-parameter Semigroups

exists and is diﬀerent from zero. Indeed, it follows by Valiron’s theorem [143] that there exists the limit 1 − Fn (z) = h(z) = 0. (7.2.26) lim n→∞ |1 − Fn (0)| Then we have h(z) = lim

n→∞

h(z) 1 − Fn (z) 1 − Fn (z) |1 − Fn (0)| = lim · = . n→∞ 1 − Fn (0) |1 − Fn (0)| 1 − Fn (0) h(0)

(7.2.27)

It follows now by (7.2.23) that, for all s ≥ 0, h(Fs (z)) = e−sf

(1)

h(z).

(7.2.28)

Fix any s ≥ 0 and consider the functions hn+s (z) =

1 − Fn+s (z) . |1 − Fn+s (0)|

We have 1 − Fn (Fs (z)) n→∞ |1 − Fn+s (0)| 1 − Fn (Fs (z)) |1 − Fn (0)| · = lim n→∞ |1 − Fn (0)| |1 − Fn (Fs (0))|

lim hn+s (z) = lim

n→∞

h(z) 1 = = h(z), = h(Fs (z)) · h(Fs (0)) h(0) because of (7.2.28) and h(0) = 1. Thus, this limit does not depend on s and is equal to h(z). By using the compactness argument, we have that lim αt (1 − Ft (0)) = h(0) = 0

t→∞

exists and is a constant of modulus 1. To end our proof, we note that since each nontrivial solution h ∈ Hol(Δ, C) of (7.2.19) with h(1) = 0 is of the form h(z) = ch(z), c = 0, we get by (7.2.24) and (7.2.25) that lim arg(1 − Ft (z)) = arg h(z) = t→∞

arg h(z) − arg c. Finally, observe that both h and h are starlike functions with respect to a boundary point z = 1. Moreover, since they satisfy the so-called Visser–Ostrovski condition (see [113]), (z − 1)h (z) (z − 1)h (z) z−1 = ∠ lim f (1) = 1, = ∠ lim z→1 z→1 z→1 h(z) f (z) h(z)

∠ lim

7.2. Hyperbolic case

171

/ , respectively, containing h(Δ) and we have that the smallest wedges W and W h(Δ) are half-planes passing through the origin. Since Re h(z) ≥ 0, z ∈ Δ, we / = Π+ = {z ∈ C : Re z > 0}. It follows now by a result in [63] that have that W iϕ / W = e W where ϕ = limr→1− arg h(r) = − arg c. The theorem is proved. Remark 7.3. In fact, we have shown in our theorem that lim arg(1 − Ft (z)) = arg h(z),

t→∞

where h is the solution of equation (7.2.19) such that h(Δ) lies in the right-half plane Π+ = {z ∈ C : Re z ≥ 0}. In addition, it was shown in [63] that h(Δ) is a starlike domain with respect to the boundary point h(1) =0, having a corner of opening at this point equal to π. Therefore, for each α ∈ − π2 , π2 , there is a point z ∈ Δ, such that limz→1 arg(1 − Ft (z)) = α. Thus, our Theorem 7.3 actually completes Theorem 8 in [32]. Theorem 7.4. Let f and g be generators of hyperbolic type semigroups Sf = {Ft }t≥0 and Sg = {Gt }t≥0 , respectively. Suppose that Sf and Sg have the same Denjoy–Wolﬀ point τ ∈ ∂Δ. Then τ − Ft (z) lim arg =0 t→∞ τ − Gt (z) if and only if f (z) = kg(z)

(7.2.29)

for some k > 0, i.e., the semigroups Sf and Sg coincide up to a rescaling. Proof. Set τ = 1. Let hf and hg be Kœnigs type functions for Sf and Sg deﬁned by 1 − Ft (z) hf (z) = lim (7.2.30) t→∞ |1 − Ft (0)| and hg (z) = lim

t→∞

1 − Gt (z) , |1 − Gt (0)|

(7.2.31)

respectively. Then we have by (7.1.8), (7.2.30) and (7.2.31) that 1 − Ft (z) lim arg t→∞ 1 − Gt (z) 1 − Ft (z) |1 − Gt (0)| 1 − Ft (0) · · = lim arg t→∞ |1 − Ft (0)| 1 − Gt (z) 1 − Gt (0) = arg hf (z) − arg hg (z) = 0. Since both hf (z) and hg (z) do not vanish in Δ, the latter equality means that hf (z) = α(z)hg (z),

172

Chapter 7. Asymptotic Behavior of One-parameter Semigroups

where α(z) ∈ R. However, α is holomorphic on Δ, and hence, is constant: α(z) ≡ α, z ∈ Δ. At the same time, we know from (7.2.30) and (7.2.31) that |hf (0)| = |hg (0)| = 1. Therefore, |α| = 1. In addition, (7.2.30) and (7.2.31) imply that both Re hf (z) and Re hg (z) are nonnegative. Consequently, α must be 1, so we have hf (z) = hg (z) =: h(z),

z ∈ Δ.

(7.2.32)

Denote βf = f (1) and βg = g (1). As we already mentioned, both βf and βg are positive real numbers. In this setting, we have that βf h(z) = h (z) · f (z) and βg h(z) = h (z) · g(z). These equalities imply (7.2.29) with k =

βf . βg

In other words,

Ft (z) = h−1 e−tβf h(z) = h−1 e−sβg h(z) = Gs (z), where s = t · k. This completes our proof.

As we have already mentioned, an important question related to rigidity is how to realize that a given semigroup consists of linear-fractional transformations of the unit disk. The following assertion answers this question in terms of the asymptotic behavior of semigroups. Corollary 7.4. Let Sf = {Ft }t≥0 be a semigroup of hyperbolic type generated by 2 f ∈ Hol(Δ, C). Assume that τ = 1 is the Denjoy–Wolﬀ point of Sf and f ∈ CA (1), and let g ∈ G be deﬁned by the quadratic part of f : 1 2 g (z) = f (1) (z − 1) + f (1) (z − 1) , 2 i.e., g is a generator of the semigroup (of hyperbolic type) Sg = {Gt }t≥0 of linearfractional transformations which map Δ into itself. Then f = g if and only if f (1) 1 − f (1) . lim arg (1 − Ft (z)) = − arg (7.2.33) t→∞ 1−z 2 Proof. Note that the semigroup Sg = {Gt }t≥0 of linear-fractional transformations generated by g, g(z) = f (1)(z − 1) + 12 f (1)(z − 1)2 , can be written as Gt (z) = C −1 etβ C(z) + A(etβ − 1) ,

7.3. Parabolic case where β = f (1), A = 1 − calculation that

173 1 β f (1)

and C(z) =

1+z 1−z .

Then we get by direct

1+z +A t→∞ 1−z 1 β 1−z − f = − arg (1) . = arg 1−z 2 β − 12 f (1)(1 − z) lim arg (1 − Gt (z)) = − arg

Applying now Theorem 7.4, we complete our proof.

Corollary 7.5. Let f and g be generators of semigroups Sf and Sg , respectively. Suppose that Sf and Sg have the same Denjoy–Wolﬀ point τ ∈ Δ and Re f (τ ) · Re g (τ ) > 0 (i.e., both semigroups are either of hyperbolic or dilation type). Then f = g if and only if Re f (τ ) = Re g (τ ) and τ − Ft (z) lim arg = 0. t→∞ τ − Gt (z)

7.3 Parabolic case Let now S = {Ft } be a discrete (t ∈ N) or continuous (t ∈ R+ ) semigroup of dFt0 (1) = 1. parabolic type. This means that for at least one t0 > 0 (hence for all t) dz In this situation, the study of the asymptotic behavior of the semigroup is much more delicate and complicated. As we have already mentioned, the class of parabolic self-mappings falls into two subclasses of automorphic (of positive hyperbolic step) and non-automorphic (of zero hyperbolic step) types. The problem is: ♦ Whether these classes have a certain speciﬁcation regarding the asymptotic behavior of discrete or continuous semigroups?

7.3.1 Discrete case This section is devoted to a brief discussion of some recent results for discrete-time semigroups deﬁned by iterations of a single self-mapping. The ﬁrst assertion we mention in this direction is given by Bourdon and Shapiro (see [18, p. 97]). Theorem 7.5. Suppose that F is of parabolic type, and that for some z ∈ Δ the discrete-time semigroup of iterates {F n (z)}∞ n=0 converges to τ = 1 non-tangentially. Then F is of non-automorphic type. In a sense, a converse assertion is due to Poggi-Corradini [109]. To formulate it, we again consider, in parallel with F ∈ Hol(Δ), its associated iteration function Φ ∈ Hol(Π+ ) given by Φ(z) = C(F (C −1 (z))). We use the following notion suggested by Contreras, D´ıaz-Madrigal and Pommerenke in [36].

174

Chapter 7. Asymptotic Behavior of One-parameter Semigroups

Deﬁnition 7.4. Given a mapping F ∈ Hol(Δ) (respectively, its associated iteration mapping Φ = C ◦ F ◦ C −1 ), we say that its orbit {F n }∞ n=1 (respectively, ∞ {Φn }n=1 ) is of ﬁnite shift if for some w ∈ Π+ (hence, for each w ∈ Π+ ), L(w) := lim Re Φn (w) < ∞. Otherwise, if L(w) = ∞, the orbit is said to be n→∞ of inﬁnite shift. Thus, the property of {Fn }∞ n=1 to be of ﬁnite shift is equivalent to the fact that it converges to the Denjoy–Wolﬀ point strongly tangentially. At the same time, there are examples of semigroups (even continuous time) which converge to their Denjoy–Wolﬀ points tangentially, but not strongly tangentially, i.e., their orbits are of inﬁnite shift. ∞ If the orbit {F n }n=1 is of ﬁnite shift, then for each pair z1 , z2 in Δ, d(F n (z1 )) < ∞, n→∞ d(F n (z2 )) lim

where d(z) =

|1 − z|2 . 1 − |z|2

Theorem 7.6 ([109] (see also, [36])). Let F ∈ Hol(Δ) be a parabolic self-mapping of Δ with the boundary Denjoy–Wolﬀ point τ = 1. If its orbits {F n (z)}∞ n=0 , z ∈ Δ, are of ﬁnite shift (strongly tangentially convergent), then F is of automorphic type (of positive hyperbolic step). Of course, the property of F to be of non-automorphic type in general does not imply the non-tangential convergence. So, the question now is: ♦ Whether the property of F to be of automorphic type is equivalent to tangential (strongly tangential) convergence? The answer is aﬃrmative under some smoothness conditions. To be more concrete, we recall that for τ ∈ ∂Δ a function f ∈ Hol(Δ, C) is p+ε of class CA (τ ), where p ∈ N ∪ {0} and ε ∈ [0, 1), if it admits the expansion f (z) =

p f j (τ )(z − τ )j j=0

γ(z) p+ε z→τ (z−τ )

where ∠ lim

j!

+ γ(z),

z ∈ Δ,

(7.3.1)

= 0; and we say that f ∈ C p+ε (τ ) when this expansion holds

p+ε (τ ). as z → τ unrestrictedly. Of course, if f ∈ C p+ε (τ ), we have that f ∈ CA

Theorem 7.7 ([36]). Let F ∈ Hol(Δ) be a parabolic self-mapping of Δ of automorphic type. The following are equivalent: 2 (1) The mapping F belongs to CA (1). ∞

(2) The mapping F is of ﬁnite shift, i.e., the sequence {F n }n=1 converges to τ = 1 strongly tangentially. Combining this result with Theorem 7.6 above, we get the following characteristic condition:

7.3. Parabolic case

175

Theorem 7.8. Let F ∈ Hol(Δ) be a parabolic-type self-mapping of Δ. The following are equivalent: ∞

(i) The mapping F is of ﬁnite shift, i.e., {F n }n=1 converges to τ = 1 strongly tangentially; 2 (1) and is of automorphic type (of positive (ii) The mapping F belongs to CA hyperbolic step).

Actually, for mappings smooth enough at the boundary Denjoy–Wolﬀ point τ , the property of the mapping F to be of automorphic (or non-automorphic) type can be recognized by using the value F (τ ). Theorem 7.9 ([18] p. 52). Let F ∈ Hol(Δ) have the Denjoy–Wolﬀ point τ = 1. Suppose that F ∈ C 2 (1) and F (1) = 1, i.e., F is of parabolic type. Then (a) Re F (1) ≥ 0; (b) if either F (1) = 0 or Re F (1) > 0, then F is of non-automorphic type; (c) conversely, if F ∈ C 3+ε (1), ε > 0, and F (1) is pure imaginary, i.e., F (1) = 0 and Re F (1) = 0, then F is of automorphic type. Corollary 7.6. If F ∈ C 3+ε (1), ε > 0, then F is of automorphic type if and only if F (1) is purely imaginary. In the case when F (1) = 0, Bourdon and Shapiro have presented the following criterion of non-tangential convergence for each orbit {F n (z)}∞ n=1 of a selfmapping F (see [18, p. 54]). Theorem 7.10. Let F ∈ Hol(Δ) have the Denjoy–Wolﬀ point τ = 1. Suppose that F ∈ C 2 (1) and F is of parabolic type. Assume that F (1) = 0. Then for each z ∈ Δ the orbit {F n (z)}∞ n=1 converges to τ = 1 non-tangentially if and only if Re F (1) > 0. This theorem may be partially completed by the following assertion, which is a direct consequence of a result in Tauraso [139, p. 950]. Theorem 7.11. Let F ∈ C 3 (1) be of parabolic type, and let F (1) = 0. Then ∞ {F n (0)}n=1 is of inﬁnite shift. Moreover, there is a subsequence nk ∈ N, such that {F nk (0)}∞ k=1 converges to τ = 1 non-tangentially. On the other hand, a suﬃcient condition for the orbit of F to be strongly tangentially convergent was given earlier in [18, p. 65]. Theorem 7.12. Let F ∈ C 3+ε (1) be a self-mapping of Δ of parabolic type, with Denjoy–Wolﬀ point τ = 1. Suppose that F (1) is pure imaginary. Then for each z the sequence {F n (z)}∞ n=1 converges to τ = 1 strongly tangentially, i.e., it lies outside some disk in Δ that is tangent to ∂Δ at 1.

176

Chapter 7. Asymptotic Behavior of One-parameter Semigroups

Note also that by Theorems 7.9 and 7.10, if F ∈ C 2 (1) and {F n }∞ n=1 converges non-tangentially to τ = 1, then F must be of non-automorphic type. As a matter of fact, Theorem 7.5 states that this assertion holds without any smoothness requirement. Table 7.1 presents a scheme that summarizes the relation between diﬀerent subclasses of parabolic type mappings and the asymptotic behavior of their orbits.

7.3.2 Continuous case Since each continuous (in t ≥ 0) semigroup S = {Ft }t≥0 is diﬀerentiable, its asymptotic behavior is much more “ordered” than that of discrete semigroups. It seems that a trajectory of the solution of the Cauchy problem may behave like a “snake”. In fact, however, we will see that some smoothness of a generator at the Denjoy–Wolﬀ point implies that the generated semigroup’s asymptotic behavior is rigid. Let S = {Ft }t≥0 be a continuous semigroup of holomorphic self-mappings of Δ of parabolic type having the Denjoy–Wolﬀ point at τ = 1. Thus, for its generator f , we have f (1) = 0 and f (1) = 0. So, f is of the form f (z) = −(1 − z)2 p(z) with Re p ≥ 0 and ∠ lim (1 − z)p(z) = 0. z→1

To study the asymptotic behavior of the semigroup S = {Ft }t≥0 , consider two positive functions εt (z) and δt (z) deﬁned as follows: εt (z) =

|1 − Ft (z)|2 , 1 − |Ft (z)|2

z ∈ Δ,

t ≥ 0,

δt (z) =

|1 − Ft (z)| , 1 − |Ft (z)|

z ∈ Δ,

t ≥ 0.

and

Deﬁnition 7.5. We say that a semigroup S = {Ft }t≥0 is strongly tangentially convergent if, for each z ∈ Δ, ε(z) := lim εt (z) > 0. t→∞

Note that by the Julia–Wolﬀ–Carath´eodory Theorem (see Theorem 2.10), the function εt (z) is nonincreasing in t for each z ∈ Δ; hence, the limit above exists. We recall also that a semigroup S = {Ft }t≥0 is said to be non-tangentially convergent if for each z ∈ Δ there exists 1 < δ < ∞ such that sup δt (z) ≤ δ.

7.3. Parabolic case

177

Table 7.1: Parabolic-type mappings

Automorphic type (positive h-step) lim ρ F n+1 (z), F n (z) > 0

n→∞

Non-automorphic type (zero h-step) lim ρ F n+1 (z), F n (z) = 0

n→∞

Non-tangential convergence Strongly tangential convergence Finite shift type lim d (F n (z)) > 0

Inﬁnite shift type lim d (F n (z)) = 0

n→∞

n→∞

2 (1) F ∈ CA

F (1) = 0

F ∈ C 3+ε (1) Re F (1) ≥ 0

Re F (1) > 0 Re F (1)

=0 and F (1) = 0

F (1) = 0

178

Chapter 7. Asymptotic Behavior of One-parameter Semigroups

In fact, this deﬁnition means that for each z ∈ Δ, there is a non-tangential approach region Γ(1, κ), κ = κ(z) > 1, such that the whole trajectory {Ft (z), t ≥ 0} lies in Γ(1, κ). Now, following a linearization model suggested by Pommerenke (see Section 5.5, Theorem 5.8), we introduce the function ht by the following formula:

2i Im Ft (0) 1 + Ft (z) ht (z) = εt (0) . − 1 − Ft (z) |1 − Ft (0)|2 Proposition 7.5. If S = {Ft }t≥0 is a semigroup of parabolic type, which converges to its Denjoy–Wolﬀ point non-tangentially, then lim ht (z) = 1. t→∞

Proof. Suppose that S = {Ft }t≥0 converges to τ = 1 non-tangentially. Then the expression |1 − Ft (0)| δt (0) := 1 − |Ft (0)| is bounded. In addition, f (1) = lim

t→∞

f (Ft (z)) = 0, 1 − Ft (z)

z ∈ Δ.

Consider the expression ωt (z) :=

|1 − Ft (0)| . 1 − Ft (z)

Since Re ωt (z) > 0 for all t ≥ 0 and z ∈ Δ, we see that {ωt }t≥0 is a normal family. In addition, the value ωt (0) is of modulus 1 for all t ≥ 0. Therefore, this family is not compactly divergent. ∞ Now, let {ωtk }k=1 be any sequence that converges to a function ω ∈ ∞ Hol(Δ, Π+ ). Consider the corresponding sequence {hk }k=1 deﬁned by

1 + Ftk (z) 2i Im Ftk (0) hk (z) = εtk (0) . − 1 − Ftk (z) |1 − Ftk (0)|2 Since ht (0) = 1 for all t ≥ 0 (hence for all tk ), it is enough to show that dhk (z) converges to zero as k → ∞. dz Indeed, by direct calculation we get |1 − Ftk (0)|2 f (Ftk (z)) 1 dhk (z) =2 · 2 2 dz 1 − |Ftk (0)| (1 − Ftk (z)) f (z) =

2 |1 − Ftk (0)| |1 − Ftk (0)| f (Ftk (z)) · , · f (z) 1 − |Ftk (0)|2 1 − Ftk (z) 1 − Ftk (z)

which converges to zero, and we are done.

7.3. Parabolic case

179

Corollary 7.7. Let S = {Ft }t≥0 be a one-parameter continuous semigroup with the boundary Denjoy–Wolﬀ point. If S = {Ft }t≥0 converges to this point nontangentially, then all its elements Ft , t ≥ 0, are either of hyperbolic or of nonautomorphic parabolic type. As we have mentioned, under some smoothness conditions on f at the Denjoy–Wolﬀ point, one can provide more complete information regarding the behavior of the semigroup at this point. Assume now that f (z) = −(1 − z)2 p(z) and a = − lim p(z)

(7.3.2)

z→1

exists ﬁnitely. This is equivalent to the fact that S is of parabolic type and f ∈ C 2 (1) with a = 12 f (1). The following assertion is needed for our characterization of the asymptotic behavior of semigroups of parabolic type: Lemma 7.3. Let S = {Ft }t≥0 ⊂ Hol(Δ) be a semigroup of parabolic type with a boundary Denjoy–Wolﬀ point τ ∈ ∂Δ. Suppose that its generator f ∈ G is of class C 2 (τ ). If f (τ ) := lim f (z) exists ﬁnitely, then for each t ≥ 0, Ft (τ ) := z→τ

∠ lim F (z) also exists and z→τ

Ft (τ ) = −2at,

(7.3.3)

1 f (τ ). 2 Proof. Recall that the semigroup elements solve the Cauchy problem (2.1.3). Differentiating the equality

where a =

∂Ft (z) + f (Ft (z)) = 0, z ∈ Δ, t ≥ 0, ∂t

(7.3.4)

two times with respect to z ∈ Δ, we get 2 ∂ ∂ 2 Ft (z) ∂Ft (z) ∂ 2 Ft (z) (F (z)) + f (F (z)) =0 + f t t ∂t ∂z 2 ∂z ∂z 2 for all z ∈ Δ and t ≥ 0. Deﬁne the functions p(z, t) := f (Ft (z)), q(z, t) := −f (Ft (z)) 2

u(z, t) := the form

(7.3.5)

∂Ft (z) ∂z

2 and

∂ Ft (z) , z ∈ Δ, t ≥ 0. It is clear that u(z, 0) = 0. Rewriting (7.3.5) in ∂z 2 ∂u(z, t) + p(z, t)u(z, t) = q(z, t), ∂t

we ﬁnd u(z, t) = e−

t 0

p(z,s)ds

·

z ∈ Δ,

t

q(z, s)e 0

s 0

t ≥ 0,

p(z,ς)dς

ds.

180

Chapter 7. Asymptotic Behavior of One-parameter Semigroups

Now we ﬁx t and let z tend to τ in the right-hand side of this equality. Since S consists of mappings of parabolic type, Ft (τ ) = τ and Ft (τ ) = 1 for all t ≥ 0. We conclude that lim p(z, t) = 0 and lim q(z, t) = −2a for each t > 0. Hence, z→τ

z→τ

t s − 0t p(z,t)ds p(z,ς)dς 0 · q(z, s)e ds lim e

z→τ

=e

−

t

0

lim p(z,s)ds

0 z→τ

·

t

lim q(z, s) · e

s lim 0 z→τ

p(z,ς)dς

0 z→τ

ds = −2at.

Therefore,

∂ 2 Ft (z) = −2at, 0 ≤ t < ∞. z→τ ∂z 2 The next theorem shows that if S = {Ft }t≥0 is strongly tangentially convergent, then the number a must be purely imaginary. Moreover, in the case when a = 0, each element of S is of non-automorphic type. lim

Theorem 7.13 ([133]). Let f ∈ G admit the representation f (z) = a(z − 1)2 + γ(z), γ(z) 2 z→1 (z−1)

where lim

= 0, and let S = {Ft }t≥0 be the semigroup (of parabolic type)

generated by f . Then: 1. The limit h(z) = lim ht (z), t→∞

where

2i Im Ft (0) 1 + Ft (z) − ht (z) = εt (0) 1 − Ft (z) |1 − Ft (0)|2

exists ﬁnitely, uniformly on each compact subset of Δ. 2. The following assertions are equivalent: (i) a = 0 and S is strongly tangential convergent to τ = 1, i.e., ε = lim εt (0) > 0; t→∞

(ii) the mapping F1 is of automorphic type; (iii) for each t > 0 the mapping Ft is of automorphic type; (iv) the limit function h(z) = lim ht (z) is not constant on Δ (i.e., h(z) = t→∞

1). Moreover, in this case Re a = 0, and h is a univalent function on Δ mapping Δ into Π+ which satisﬁes Abel’s equation h(Ft (z)) = h(z) + ibt with b = −ε Im f (1).

7.3. Parabolic case

181

Proof. Diﬀerentiating ht (z) at z ∈ Δ we have 2 f (Ft (z)) ∂ht (z) = εt (0) · . ∂z (1 − Ft (z))2 f (z) As we have already mentioned, it follows by the Julia–Wolﬀ–Carath´eodory Theorem (see, Section 2.3) that, for each z ∈ Δ the semigroup Ft (z) converges to τ = 1 as t → ∞, and the function 2

εt(z) =

|1 − Ft (z)|

1 − |Ft (z)|2

is nonincreasing. In particular, the limit ε = lim εt (z) exists and is a nonnegative t→∞ number. Also, by our assumptions, we have f (Ft (z)) = a. t→∞ (1 − Ft (z))2 lim

Hence, for each z ∈ Δ, we get lim

t→∞

1 ∂ht (z) = 2εa · . ∂z f (z)

Now observe that for each t ≥ 0 the image ht (Δ) lies in the right half-plane. Hence, the family {ht(z)} is a normal family of univalent functions on Δ. Yet for all t ≥ 0, we have also ht (0) = 1. Hence, by the Hurwitz Theorem each limit function h(z) of the net {ht(z)}t≥0 is either a univalent function on Δ or identically equal to 1. At the same time, the latter formula implies that h (z) =

2εa , f (z)

(7.3.6)

which means that all limit functions of this net must coincide. So, assertion 1 is proved. Now we prove assertion 2. First we note that the above considerations and formula (7.3.6) immediately imply that (iv) is equivalent to the fact that neither a nor ε are not zero. So, (iv) implies (i). In turn, (i) implies (ii) and (iii) by Theorem 7.8. Now the above Lemma 7.3 and Theorem 7.9 (b) imply that if (ii) (or (iii)) holds, then a = 0 and Re a = 0. Hence (i) and (ii) imply (iv). This proves the equivalence of (i)–(iv). To prove our last statement, we just integrate formula (7.3.6), and we get z h(z) = 2εa 0

dw +1. f (w)

182

Chapter 7. Asymptotic Behavior of One-parameter Semigroups On the other hand, we know that ⎧ ⎨ dFt (z) = −f (F (z)), t dt ⎩ F0 (z) = z;

hence, Ft (z)

dw = −t. f (w)

z

Now it follows that Ft (z)

h(Ft (z)) = 2εa 0

dw +1 f (w)

⎡ ⎤ Ft (z) z dw dw ⎥ ⎢ + = 2εa ⎣ ⎦ + 1 = h(z) − 2εat, f (w) f (w) 0

t ≥ 0.

z

Thus, for each t ≥ 0 the mapping Ft satisﬁes Abel’s functional equation h(Ft (z)) = h(z) − 2εat. On the other hand, by Pommerenke’s Theorem 5.8, we see that F = F1 satisﬁes Abel’s equation h(F (z)) = h(z) + ib for some b ∈ R, b = 0. Comparing these equations for t = 1 completes our proof. Remark 7.4. Note that once the existence of the limit lim ht (z) = h(z) has been t→∞

shown, it is enough to require that εt (0) in Deﬁnition 7.5 is bounded away from zero to provide the strong inequality ε(z) = lim εt (z) > 0 for all z ∈ Δ. Indeed, t→∞ in this case, we have εt (0) = εt (z) Re ht (z). Remark 7.5. If f ∈ G ∩ C 2 (1), then it follows by (7.3.2) that Re f (1) ≤ 0. In turn, the above theorem implies that if Re f (1) < 0, then all elements of S = {Ft }t≥0 are of non-automorphic type and ε(z) = lim εt (z) = 0 for all t→∞

z ∈ Δ,

i.e., the semigroup S does not converge to τ = 1 strongly tangentially.

7.3. Parabolic case

183

The question now is: ♦ Whether in this case there is at least one element t0 ∈ R+ , t0 = 0, such that the orbit {Ft0 n }∞ n=1 converges to τ = 1 non-tangentially? The answer is aﬃrmative. Indeed, applying Theorem 14 in [36] and Theorem 7.10 as in the above theorem and by using some simple compactness argument, one can easily obtain the following assertion: Theorem 7.14. Let f ∈ G(Δ) admit the representation f (z) = a(z − 1)2 + γ(z), γ(z) 2 z−1 (z−1)

where lim

z ∈ Δ,

(7.3.7)

= 0, and let S = {Ft }t≥0 be the semigroup generated by f . Then

the following assertions are equivalent: (i) Re a = 0; (ii) a = 0 and there is t0 ∈ R+ such that {Fnt0 }∞ n=0 converges to τ = 1 nontangentially; (iii) a = 0 and for each t ∈ R+ the orbit {Fnt }∞ n=0 converges to τ = 1 nontangentially; (iv) a = 0 and the semigroup S = {Ft }t≥0 converges to τ = 1 non-tangentially; (v) the limit function h(z) = lim

t→∞

1 + Ft (z) 1 + Ft (0) − 1 − Ft (z) 1 − Ft (0)

(7.3.8)

is univalent on Δ and satisﬁes Abel’s equation h(F (z)) = h(z) + b

(7.3.9)

with Re b > 0. Moreover, in this case, Re a < 0, b = −2a, and each element Ft , t ∈ R+ , of S is of non-automorphic type. Remark 7.6. Note that the condition a = 0 itself implies that the limit in (7.3.8) is a non-constant univalent function on Δ. However, if S consists of automorphic type mappings, this function satisﬁes Abel’s equation (7.3.9) with Re b = −2 Re a = 0. In contrast to the limit scheme in Theorem 7.13, the converse assertion does not hold in general. At the same time, if f ∈ C 3+ (1) and a is purely imaginary, then the semigroup S = {Ft }t≥0 consists of automorphic type mappings (see Theorem 7.9); hence, it converges to τ = 1 strongly tangentially.

184

Chapter 7. Asymptotic Behavior of One-parameter Semigroups

7.3.3 Universal asymptotes A detailed description of the asymptotic behavior of one-parameter semigroups of holomorphic self-mappings in the sense of angular characteristics of their trajectories was given in [32]. In particular, it was proven there (see Theorem 2.9) that if S = {Ft }t≥0 is of parabolic type, then angular characteristics of their trajectories do not depend on the initial point z ∈ Δ. Moreover, if for at least one z ∈ ∂Δ the trajectory {Ft }t≥0 converges to the Denjoy–Wolﬀ point τ ∈ ∂Δ tangentially, then all the trajectories converge tangentially to this point. An open question mentioned there is: ♦ What happens in the general case (when {Ft (z)}t≥0 converge to τ ∈ ∂Δ not necessarily tangentially)? In particular, does the limit lim arg(1 − τ Ft (z))

t→∞

exist? We will answer this question below for the case when the generator f of the semigroup is twice diﬀerentiable at the boundary Denjoy–Wolﬀ point τ ∈ ∂Δ. We just note that Theorem 2.9 in [32] implies that If for at least one point z ∈ Δ the limit lim arg (1 − τ Ft (z)) = θ exists, t→∞ then it exists for all z ∈ Δ and is equal to θ. We show inter alia that if f ∈ G(Δ) ∩ C 2 (τ ), then the angle θ is uniquely determined by the value f (τ ); so even diﬀerent semigroups of such type have the same asymptote for their trajectories whenever the second derivatives of their generators coincide. • We say that a function k ∈ Hol(Δ, C) is of class P (n) [τ ], n = 2, 3, . . . , τ ∈ h(z) for ∂Δ, if k has a pole of order (n − 1) at τ in the sense that k(z) = (z − τ )n−1 some h ∈ Hol (Δ ∪ {τ }, C) with h(τ ) = 0. Let S = {Ft }t≥0 be a semigroup of holomorphic self-mappings of Δ with the Denjoy–Wolﬀ point τ = 1, and let k ∈ P (n) [1]. For a pair (z, w) ∈ Δ × Δ = Δ2 , deﬁne a family σt : Δ2 × R+ → C of holomorphic functions by the formula σt (z, w) = k(Ft (z)) − k(Ft (w)),

(z, w) ∈ Δ2 , t ≥ 0.

(7.3.10)

It is clear that σt (z, w) = −σt (w, z) and σt (w, w) = 0,

t ≥ 0.

(7.3.11)

Theorem 7.15 (see [60]). Let S = {Ft }t≥0 be a semigroup of parabolic type generated by f ∈ Gp [1].

7.3. Parabolic case

185

(A) Fix w ∈ Δ and assume that for some n ≥ 2 and k ∈ P (n) [1] the limit lim σt (z, w) =: σ(z, w)

t→∞

(7.3.12)

exists locally uniformly on Δ. The following assertions hold: n (i) f ∈ CA (1) with f (m) (1) = 0, m = 1, 2, . . . , n − 1, and

(−1)n Re f (n) (1) ≤ 0.

(7.3.13)

(ii) f (n) (1) = 0 if and only if σ(·, w) is locally univalent on Δ. (iii) If f is not identically zero, then n < 4, i.e., n is equal to either 2 or 3. (iv) The limit σ in (7.3.12) exists for any choice of the function k ∈ P (n) [1] and satisﬁes Abel’s functional equation σ(Ft (z), w) = σ(z, w) −

1 (n) f (1)b · t n!

(7.3.14)

where b = lim (1 − n)(z − 1)n−1 k(z) = 0. z→1

(v) For all z ∈ Δ, we have f (n) (1) 1 , (7.3.15) = (−1)n−1 (n − 1) n−1 t→∞ t(1 − Ft (z)) n! 1 f (n) (1) n−1 1 n−1 lim = 2 Re (−1) (n − 1) , (7.3.16) 1 t→∞ t n−1 d(F (z), 1) n! t lim

where d(z, τ ), z ∈ Δ, τ ∈ Δ, is a non-Euclidean distance on Δ deﬁned 2 |z − τ | by d(z, τ ) = , and 1 − |z|2 ⎧ n = 2, ⎨ − arg(−f (1)) 0 n = 3, lim arg (1 − Ft (z)) = (7.3.17) t→∞ ⎩ arg(1 − z) n = 4, 5, . . . . n (1) and S (B) Conversely. If for some n = 2, 3, . . . , f ∈ C n (1) (or f ∈ CA converges to τ = 1 non-tangentially), then for some m ≤ n the limit in (7.3.12) exists for any choice of the function k ∈ P (n) [1].

Proof. Step 1. Suppose that for some w ∈ Δ and n = 2, 3, . . . , there is k ∈ P (n) [1], such that the net {σt (·, w)}t≥0 deﬁned by (7.3.10) converges to σ(·, w) as t goes to ∞. Then we have that ∂σ(z, w) ∂Ft (z) = lim k (Ft (z)) · . t→∞ ∂z ∂z

(7.3.18)

186

Chapter 7. Asymptotic Behavior of One-parameter Semigroups

Since k has a pole of order n − 1 at τ = 1, the function k (z) can be written in the form h(z) , (7.3.19) k (z) = (z − 1)n where h ∈ Hol(Δ ∪ {1}, C) and h(1) = 0. Also we know from the Cauchy problem that f (Ft (z)) ∂Ft (z) = , z ∈ Δ. (7.3.20) ∂z f (z) So, we have by (7.3.18)–(7.3.20) that ∂σ(z, w) f (Ft (z)) 1 = · lim h(Ft (z)) · ∂z f (z) t→∞ (Ft (z) − 1)n b f (Ft (z)) = , · lim f (z) t→∞ (Ft (z) − 1)n

(7.3.21)

where b = h(1) = lim (1 − n)(z − 1)n−1 k(z) = 0. z→1

(7.3.22)

Thus, the limit lim

t→∞

1 ∂σ(z, w) f (Ft (z)) · f (z) = (Ft (z) − 1)n b ∂z

(7.3.23)

exists ﬁnitely. Now we claim that this limit does not depend on z ∈ Δ and is equal to 1 (n) 1 f (1) := ∠ lim f (n) (z). n! n! z→1 Indeed, it follows by the Berkson–Porta formula that f (z) = −(z − 1)2 p(z), where Re p(z) ≥ 0, z ∈ Δ. Then we have that p(Ft (z)) f (Ft (z)) 1 ∂σ(z, w) = − lim lim =− · f (z) (7.3.24) t→∞ (Ft (z) − 1)n−2 t→∞ (Ft (z) − 1)n b ∂z exists ﬁnitely. Applying now the Lindel¨ of Theorem 1.6, one can show that f (z) z→1 (z − 1)m

f (m) (1) := m!∠ lim

for all m = 1, 2, . . . , n exists; and f (m) (1) = 0 for m = 1, 2, . . . , n − 1, while f (n) (1) = n!

1 ∂σ(z, w) · f (z). b ∂z

(7.3.25)

Moreover, Proposition 6.2 also implies that if f (hence p) is not identically zero, the number n must be less or equal to 3. For n = 2 we have f (1) = 2∠ lim

t→∞

f (z) = −∠ lim p(z). t→∞ (z − 1)2

7.3. Parabolic case

187

So, in this case Re f (1) ≤ 0. For n = 3 we get that f (z) p(z) . = −∠ lim 3 t→∞ (z − 1) t→∞ z − 1

f (1) = 3!∠ lim

So, f (1) is, in fact, a real positive number. (Actually, we will show below that this fact holds in a more general case even f (1) = 0). Finally, observe that the ∂σ(z, w) is not zero. This complete the limit (7.3.24) is not zero if and only if ∂z proof of assertions (i)–(iii). Step 2. As soon as we know that for some n = 1, 2, . . . , the limit lim

t→∞

1 f (Ft (z)) = f (n) (1) n (Ft (z) − 1) n!

(7.3.26)

exists, repeating Step 1 for any other choice of the function k ∈ P (n) [1] we see that each limit function of the net (7.3.10) must satisfy the diﬀerential equation (7.3.25) with the initial condition (7.3.11). Hence they must coincide. Noting that by the Cauchy problem Ft (z)

0

dξ = f (ξ)

z 0

dξ + f (ξ)

Ft (z)

z

dξ = f (ξ)

z 0

dξ − t, f (ξ)

we get the equality σ(Ft (z), w) − σ(0, w) = σ(z, w) − σ(0, w) − ct, where c =

1 (n) (1)b, n! f

which proves assertion (iv) as well as assertion (B).

Step 3. Now by using again (7.3.26) and the Cauchy problem we have that the average 1 t

t 0

1 dFs (z) =− n (Fs (z) − 1) t

t 0

1 f (Fs (z)) ds → − f (n) (1). n (Fs (z) − 1) n!

Integrating the left-hand side of this formula we get (7.3.15). Rewriting this formula in the form 1 f (n) (1) n−1 n−1 lim = 2 (−1) (n − 1) 1 t→∞ t n−1 (1 − F (z)) n! t 1 + Ft (z)

and taking the real part in both sides, we get (7.3.16). Formula (7.3.17) is also a consequence of (7.3.15). The theorem is proved.

188

Chapter 7. Asymptotic Behavior of One-parameter Semigroups

Remark 7.7. Observe that even for a diﬀerent choice of functions k ∈ P (n) [1] the functions σ obtained by (7.3.12) coincide up to a complex factor. Namely, if k1 and k2 belong to P (n) [1], then the corresponding functions σ1 and σ2 are connected by the formula σ2 = σ1 lim kk21 (z) (z) . If n = 2 then an appropriate choice of k may be z→1

1+z made by the Cayley transform k(z) = C(z) = 1−z . In this case the limit scheme (7.3.10) is a continuous analog of the discrete scheme used in [18]. If n = 3, then an 2 appropriate choice of k may be made, for example, by the function k(z) = (C(z)) z or by the Koebe function k(z) = (z−1) 2.

Since our theorem is too long because of its generality and actually is relevant only for the cases n = 2 or n = 3 we formulate the corresponding consequence separately. Theorem 7.16. Let S = {Ft }t≥0 be a parabolic type semigroup generated by f ∈ 2 G ∩ CA (1) with the Denjoy–Wolﬀ point τ = 1, i.e., f (1) = f (1) = 0. Assume that f (1) a := 2 = 0. Then: (i) Re a ≤ 0. (ii) If for at least one z ∈ Δ the trajectory {Ft (z)}t≥0 converges to the point τ = 1 non-tangentially, then Re a < 0. Moreover, in this case for all z ∈ Δ we have lim arg(1 − Ft (z)) = − arg(−a), (7.3.27) t→∞

i.e., all trajectories {Ft (z)}t≥0 converge to τ = 1 non-tangentially. In other words, there is one and the same straight line passing through τ = 1, which does not depend on the initial point z ∈ Δ, and such that these trajectories are tangential to at the point τ = 1. Conversely. If Re a < 0 and f ∈ C 2 (1), then all the trajectories {Ft (z)}t≥0 , z ∈ Δ, converge to τ = 1 non-tangentially. Corollary 7.8. Suppose that conditions of Theorem 7.16 hold and f ∈ C 2 (1). Then Re a = 0 if and only if each trajectory {Ft (z)}t≥0 , z ∈ Δ, converges to τ = 1 tangentially. Moreover, in this case for all z ∈ Δ, lim arg(1 − Ft (z)) =

t→∞

π (sgn Im f (1)). 2

Thus, if f ∈ Hol(Δ, C) is a generator of a parabolic type semigroup with the Denjoy–Wolﬀ point τ ∈ ∂Δ and f (τ ) = 0, then either all trajectories converge to τ tangentially, or all trajectories converge to τ non-tangentially in such a way that each domain bounded by two trajectories has a cusp at the point τ. In fact, these assertions are a consequence of the following comparative result on the asymptotic behavior of two related semigroups.

7.3. Parabolic case

189

2 Theorem 7.17. Let f ∈ G ∩ CA (1) be of the form

f (z) = a(z − 1)2 + γ(z), where ∠ lim

z→1

(7.3.28)

γ(z) =0 (z − 1)2

and a = 0. The following assertions hold: (i) The function g ∈ Hol(Δ, C) deﬁned by the ﬁrst term of (7.3.28), g(z) = a(z − 1)2 ,

(7.3.29)

is a generator of a parabolic type semigroup Sg = {Gt }t≥0 of linear-fractional transformations, hence Re a ≤ 0; (ii) If the semigroup Sf = {Ft } generated by f converges to τ = 1 non-tangentially, or f ∈ C 2 (1) then 1 − Ft (z) = 1. t→∞ 1 − Gt (z) lim

(7.3.30)

Proof. Assertion (i) of Theorem 7.17 as well as assertion (i) of Theorem 7.16 are direct consequences of Theorem 6.10 and the Berkson–Porta formula (2.1.2). Further, it follows by the Cauchy problem, the Berkson–Porta representation and (7.3.29) that semigroups Sf = {Ft }t≥0 and Sg = {Gt }t≥0 satisfy the following equations: dFt (z) − p(Ft (z))dt = 0 (1 − Ft (z))2 and

dGt (z) + adt = 0, (1 − Gt (z))2

respectively. Integrating these equations we get 1 = 1 − Ft (z) and

Then

t p(Fs (z))ds + 0

1 1−z

1 1 = −at + . 1 − Gt (z) 1−z 1 − Ft (z) = 1 − Gt (z)

−a + 1 t

t 0

1 1 t 1−z

p(Fs (z))ds +

(7.3.31)

. 1 1 t 1−z

190

Chapter 7. Asymptotic Behavior of One-parameter Semigroups

It is clear that if for at least one z ∈ Δ the trajectory {Ft (z)}t≥0 converges to the point τ = 1 non-tangentially, or f ∈ C 2 (1), then p(Fs (z)) converges to −a, as s → ∞. Therefore, for such z ∈ Δ we have that 1 lim t→∞ t and

t p(Fs (z))ds = −a

(7.3.32)

0

1 − Ft (z) = 1. t→∞ 1 − Gt (z)

(7.3.33)

lim

This proves assertion (ii) of Theorem 7.17. In turn, (7.3.33) implies that in both cases {Ft (z)}t≥0 converges to τ = 1 non-tangentially, or for f ∈ C 2 (1)), lim arg(1 − Ft (z)) = lim arg(1 − Gt (z)) = − arg(−a).

t→∞

t→∞

This completes the proof of assertion (ii) of Theorem 7.16, as well as Corollary 7.8 of this theorem. The following assertion establishes asymptotic behavior of parabolic type |2 . semigroups in terms of a non-Euclidean distance on Δ deﬁned by d(z, τ ) = |z−τ 1−|z|2 We show that if S = {Ft }t≥0 converges to τ = 1 non-tangentially, then d(Ft (z), 1) ∼ |Ft (z) − 1| ∼

1 . t

Corollary 7.9. Let Sf = {Ft }t≥0 be a parabolic type semigroup generated by f ∈ 2 (1). Assume that {Ft }t≥0 converges to τ = 1 non-tangentially. Then G ∩ CA 2 f (1)

(7.3.34)

1 > 0. Re f (1)

(7.3.35)

lim t (1 − Ft (z)) = −

t→∞

and

2

lim t

t→∞

|1 − Ft (z)|

2

1 − |Ft (z)|

=−

Proof. As above denote by Sg = {Gt }t≥0 the semigroup of linear-fractional transformations generated by g ∈ G, where g(z) = 12 f (1) · (z − 1)2 . Then (7.3.34) follows by (7.3.31) and (7.3.30). Consequently, we have that lim

t→∞

1 1 + Ft (z) = −f (1). t 1 − Ft (z)

Taking here the real part in both sides we get the equality 2

lim

t→∞

which proves our assertion.

1 1 − |Ft (z)| = − Re f (1) t |1 − Ft (z)|2

7.3. Parabolic case

191

To study a critical case where f (1) = 0, we will assume that f is three times diﬀerentiable at the boundary Denjoy–Wolﬀ point τ = 1. We will see below that in this case all trajectories are either tangential to the real axis or do not converge (i.e., Fτ (z) ≡ z, t ≥ 0). Moreover, in this situation (when f is three times diﬀerentiable) we can obtain additional information, even f (1) = 0, but Re f (1) = 0. 3 Theorem 7.18. Let f ∈ G ∩ CA (1) and let f (1) = f (1) = Re f (1) = 0. The following assertions hold:

(i) Im f (1) = 0 and Re f (1) ≥ 0. (ii) Re f (1) = 0 if and only if f is a generator of automorphisms of Δ: 1 f (z) = i Im f (1)(z − 1)2 . 2 In particular, f (z) ≡ 0 (or equivalently, Ft (z) ≡ z, t ≥ 0) if and only if Im f (1) = 0. (iii) If f ∈ C 3 (1) and f (1) = 0, then Re f (1) = 0 (i.e., Re f (1) > 0) if and only if (7.3.36) lim arg(1 − Ft (z)) = 0, t→∞

that is, all trajectories {Ft (z)}t≥0 , z ∈ Δ, are tangential to the real axis. Proof. Under our setting we can write f (z) =

1 1 i Im f (1)(z − 1)2 + f (1)(z − 1)3 + γf (z), 2 3!

where ∠ lim

z→1

γf (z) = 0. (z − 1)3

(7.3.37)

(7.3.38)

On the other hand, f admits the Berkson–Porta representation (2.2.7). Then we have from (7.3.37) and (7.3.38) that, 1 1 p(z) = − i Im f (1) − f (1)(z − 1) + γp (z), 2 3!

(7.3.39)

where

γp (z) = 0. z−1 Deﬁne a positive real part function p1 ∈ Hol(Δ, C) by ∠ lim

z→1

1 1 p1 (z) := p(z) + i Im f (1) = − f (1)(z − 1) + γp (z). 2 3! Since p1 (1) = 0 we have from (7.3.39) and (7.3.40) that p1 (1) := ∠ lim

z→1

p1 (z) 1 = − f (1). z−1 3!

(7.3.40)

192

Chapter 7. Asymptotic Behavior of One-parameter Semigroups

At the same time we know that p1 (1) must be a nonpositive real number. Therefore, Im f (1) = 0, and 1 1 p (1) = − f (1) = − Re f (1) ≤ 0. 3! 3! Assertion (i) is proved. If now Re f (1) = 0, then p1 must be identically zero, hence f must be of the form f (z) = 12 i Im f (1)(z − 1)2 . It follows directly by solving the Cauchy problem that in this case f generates a group of parabolic automorphisms of Δ. It is clear now that, in particular, f is identically zero if and only if Im f (1) = 0. This proves assertion (ii). If f (1) = 0, but Re f (1) is possibly diﬀerent from zero, then we can calculate (by using again the Cauchy problem) that

1 γf (Ft (z)) dFt (z) dt f (1) + = (1 − Ft (z))3 3! (Ft (z) − 1)3 or 1 1 2 = f (1) + 2 t(1 − Ft (z)) 3 t

t 0

1 γf (Fs (z))ds + . 3 (Fs (z) − 1) t(1 − z)2

The latter equality implies (7.3.36) if and only if Re f (1) = 0, i.e., Re f (1) > 0, and we are done. Exactly as in Corollary 7.9 one can obtain the following additional information on the asymptotic behavior of semigroups. Corollary 7.10. Let Sf = {Ft }t≥0 be a parabolic type semigroup generated by f ∈ 3 G ∩ CA (1) with f (1) = 0. Then : 1 f (1) (7.3.41) = lim √ t→∞ 3 t(1 − Ft (z)) and 1 =2 lim √ t→∞ t d(1, Ft (z))

:

f (1) . 3

(7.3.42)

Remark 7.8. By Corollary 7.8, if the semigroup generator f belongs to the class C 2 (1), then all of the trajectories {Ft (z)}t≥0 converge to the Denjoy–Wolﬀ point τ = 1 tangentially if and only if Re f (1) = 0. At the same time, there is a more precise characteristic of the asymptotic behavior of parabolic type semigroups. Namely, it may happen that for each z ∈ Δ there is a horodisk D(τ, k) := {ζ ∈ Δ : d(τ, ζ) < k} , k = k(z), internally tangent to the unit circle at the point τ , such that the trajectory {Ft (z)}t≥0 lies outside D(τ, k). In this case we say that the semigroup S converges to τ strongly tangentially. Otherwise, if d (τ, Ft (z)) tends to zero, we say that S converges to τ weakly tangentially.

7.3. Parabolic case

193

Transfer a given semigroup in the right half-plane by the formulas Ft (w) = C ◦ Ft ◦ C −1 (w), where C(z) =

1+z 1−z

Re w > 0,

(7.3.43)

is the Cayley transform. Then Julia’s Lemma implies that

Re Ft (w) is an increasing function in t. The tangential convergence of the semiIm Ft (w) is unbounded when t goes to inﬁnity. group means that the function Re Ft (w) Roughly speaking, the semigroup converges tangentially when | Im Ft (w)| grows faster than Re Ft (w). Moreover, the semigroup {Ft }t≥0 converges strongly tangentially if and only if the function Re Ft (w) is bounded for each w, Re w > 0. For this reason, strongly tangentially convergent semigroups were referred to in [36] as semigroups of ﬁnite shift; and weakly tangentially convergent semigroups — as semigroups of inﬁnite shift. An open problem is: ♦ Let a semigroup generator f be of the class C 2 (1) and satisfy f (1) = f (1) = Re f (1) = 0, Im f (1) = 0 (hence, the generated semigroup converges tangentially). How to recognize a type of the semigroup convergence? In particular, if f ∈ C m (1), m > 3, one can obtain using Theorem 4.15 in [18] that the tangential convergence must be strongly tangential. So, it is natural to ask: Does there exist a semigroup which converges tangentially but not strongly tangentially? In the following example we show that the answer is aﬃrmative. Example 7.1. Consider a semigroup generator f deﬁned by the Berkson–Porta formula f (z) = −(1 − z)2 p(z) with : 1+i 1+z . p(z) = 2 1−z Let S = {Ft }t≥0 be the semigroup generated by f . We claim that S converges not strongly tangentially. Proof. Let the semigroup S = {Ft }t≥0 be deﬁned by (7.3.43). Diﬀerentiating at √ t = 0+ we ﬁnd its generator f (w) = −(1+i) w. Now we solve the Cauchy problem ⎧ ⎨ ∂u(t, w) − (1 + i);u(t, w) = 0, ∂t ⎩ u(0, w) = w, Re w > 0, and obtain

Ft (w) =

√ 1+i t+ w 2

2 =

√ it2 + w + (1 + i)t w. 2

Separating the real and imaginary parts we see that √ Re Ft (w) = Re w + t Re (1 + i) w , √ t2 + Im w + t Im (1 + i) w . Im Ft (w) = 2

194

Chapter 7. Asymptotic Behavior of One-parameter Semigroups Im Ft (w)

tends to inﬁnity as t → ∞, i.e., S (hence, S) converges Re Ft (w) tangentially. On the other hand, Re Ft (w) tends to inﬁnity, so this tangential convergence is not strongly. Obviously,

Concerning individual parabolic type self-mappings of the unit disk (not necessarily embeddable into a one-parameter semigroup), we know by Theorem 4.15 in [18] that if F ∈ C 3+ε with F (1) = 0, then the following assertions are equivalent: (1) each orbit {F n (z)}∞ n=0 converges tangentially; (2) each orbit {F n (z)}∞ n=0 converges strongly tangentially; (3) Re F (1) = 0. On the other hand, by Theorem 6.2 in [18], there are self-mappings not belonging to the class C 3 (1) such that their orbits converge non-tangentially. Comparing these facts with the example above, one can conclude that, to attain strongly tangential convergence, we need a certain smoothness of the semigroup’s elements (or its generator). In this context, the following statement seems to be plausible. Conjecture 7.1. Let a semigroup generator f be of the class C 3 (1) and satisfy f (1) = f (1) = Re f (1) = 0, Im f (1) = 0. Then the generated semigroup converges strongly tangentially.

Chapter 8

Backward Flow Invariant Domains for Semigroups 8.1 Existence In this section, we study conditions which ensure the existence of backward ﬂow invariant domains for semigroups of holomorphic self-mappings of a simply connected domain D ⊂ C. More precisely, the problem is the following. ♦ Given a one-parameter continuous semigroup S ⊂ Hol(D), ﬁnd a simply connected subset Ω ⊂ D (if it exists) such that S ⊂ Aut(Ω). Deﬁnition 8.1. Let S = {Ft }t≥0 be a semiﬂow on Δ. A domain Ω ⊂ Δ is called a (backward) ﬂow-invariant domain (shortly, FID) for S if S ⊂ Aut(Ω). Recall that the set of all semigroup generators (i.e., semi-complete vector ﬁelds) on D is denoted by G(D), and the set of complete vector ﬁelds on D is denoted by aut(D). In these terms, our problem can be rephrased as follows. ♦ Given f ∈ G(D), ﬁnd a domain Ω (if it exists) such that f ∈ aut(Ω). Let now D = Δ be the open unit disk in C. In this case, G = G(Δ) is a real cone in Hol(Δ, C), while aut(Δ) ⊂ G(Δ) is a real Banach space (see, for example, [118]). Moreover, by the Berkson–Porta representation formula (2.2.7), a function f belongs to G if and only if there is a point τ ∈ Δ and a function p ∈ Hol(Δ, C) with positive real part (Re p(z) ≥ 0 everywhere) such that f (z) = (z − τ )(1 − zτ )p(z).

(8.1.1)

For τ ∈ Δ, we use the notation G ∗ [τ ] for a subcone of G[τ ] of generators f deﬁned by (8.1.1) for which Re f (τ ) > 0. (8.1.2)

196

Chapter 8. Backward Flow Invariant Domains for Semigroups

In the case τ ∈ Δ, the inclusion f ∈ G ∗ [τ ] means that the semigroup of holomorphic self-mappings of Δ generated by f has the Denjoy–Wolﬀ point at τ and does not consist of elliptic automorphisms. In the case τ ∈ ∂Δ, G ∗ [τ ] consists of hyperbolic type generators, i.e., coincides with Gh [τ ]. We solve the problem mentioned above for the class G ∗ [τ ] of generators. Following an analogy with (2.2.13), we also introduce the subcone G ∗ [τ, η], where τ ∈ Δ and η ∈ ∂Δ, η = τ , by the formula G ∗ [τ, η] := G ∗ [τ ] G[η]. (8.1.3) So, f ∈ G ∗ [τ ] belongs to G ∗ [τ, η] if f (η) = ∠ lim f (z) = 0 and γ = ∠ lim f (z) z→η

z→η

exists ﬁnitely. It follows by Theorem 2.8 that in this case γ must be a real negative number (below, in Lemma 8.2, we specify a more precise result). Theorem 8.1 (see [68], cf., [69]). Let S = {Ft }t≥0 be a semiﬂow on Δ generated by f ∈ G ∗ [τ ], for some τ ∈ Δ with f (τ ) = β, Re β > 0. The following assertions are equivalent. (i) f ∈ G ∗ [τ, η] for some η ∈ ∂Δ. (ii) There is a nonempty (backward) ﬂow invariant domain Ω ⊂ Δ, so S ⊂ Aut(Ω). (iii) For some α > 0, the diﬀerential equation αϕ (z)(z 2 − 1) = 2f (ϕ(z))

(8.1.4)

has a locally univalent solution ϕ with |ϕ(z)| < 1 when z ∈ Δ. Moreover, in this case ϕ is univalent and is a Riemann mapping of Δ onto a ﬂow invariant domain Ω. This theorem can be completed by the following result. Theorem 8.2. Let S = {Ft }t≥0 be a semiﬂow on Δ generated by f ∈ G ∗ [τ ], for some τ ∈ Δ with f (τ ) = 0 and f (τ ) = β, Re β > 0. The following assertions hold. (a) If f ∈ G ∗ [τ, η] for some η ∈ ∂Δ with γ = ∠ lim f (z), then for each α ≥ −γ, z→η

equation (8.1.4) has a univalent solution ϕ such that ϕ(1) = τ, ϕ(−1) = η and Ω = ϕ(Δ) is a (backward) ﬂow invariant domain for S. In addition, τ = lim Ft (z) ∈ ∂Ω, z ∈ Ω, and lim Ft (z) = η ∈ ∂Δ ∩ ∂Ω for each z ∈ Ω. t→∞

t→−∞

(b) If Ω ⊂ Δ is a nonempty (backward) ﬂow invariant domain, then it is a Jordan domain such that τ ∈ ∂Ω, and there is a point η ∈ ∂Ω ∩ ∂Δ such that lim Ft (z) = η whenever z ∈ Ω, ∠ lim f (z) = 0 and ∠ lim f (z) =: γ exists t→−∞

z→η

z→η

with γ < 0. In addition, there is a conformal mapping ϕ of Δ onto Ω which satisﬁes equation (8.1.4) with some α ≥ −γ.

8.1. Existence

197

(c) Conversely, if for some α > 0, the diﬀerential equation (8.1.4) has a locally univalent solution ϕ ∈ Hol(Δ), then it is, in fact, a conformal mapping of Δ onto the ﬂow invariant domain Ω = ϕ(Δ) such that ϕ(1) = τ ∈ ∂Ω and ϕ(−1) = η for some η ∈ ∂Δ ∩ ∂Ω. In addition, f (η) = 0 and f (η) = γ with 0 > γ ≥ −α. The main tool of the proof of our theorems is a linearization method for semigroups which uses Kœnigs type starlike and spirallike functions on Δ (see Section 5). Recall that by Theorem 2.14, a univalent function h is spirallike (starlike) of the class Spiral[τ ] (of the class Star[τ ]) if and only if there is a semigroup generator f ∈ G ∗ [τ ] such that μh(z) = h (z)f (z)

(8.1.5)

for some μ with Re μ > 0. Obviously, if τ ∈ Δ, the number μ must be equal to f (τ ). If τ ∈ ∂Δ, then by Theorem 9.4, |μ − β| ≤ β. Moreover, by Theorem 5.1, this function h satisﬁes Schr¨oder’s equation h(Ft (z)) = e−μt h(z).

(8.1.6)

We call this function h the spirallike (starlike) function associated with f . Since we deal with generators having additional null points on the boundary, we need to recall Deﬁnition 4.6 (see also Deﬁnition 3.4): • For τ ∈ Δ and η ∈ ∂Δ, η = τ , we say that a function h ∈ Spiral[τ ] (h ∈ Star[τ ]) belongs to the subclass Spiral[τ, η] (S ∗ [τ, η]) if the angular limit Qh (η) := ∠ lim

z→η

(z − η)h (z) h(z)

exists ﬁnitely and is diﬀerent from zero, i.e., h satisﬁes a generalized Visser– Ostrowski condition at the boundary point η. To proceed, we note that the inequality η = τ implies that, for each h ∈ Spiral[τ, η], ∠ lim h(z) = ∞. z→η

The following fact is an immediate consequence of 8.1.5. Lemma 8.1. Let h ∈ Spiral[τ ] and f ∈ G ∗ [τ ] be connected by (8.1.5). Then h belongs to Spiral[τ, η] if and only if f ∈ G[τ, η]. In this case, Qh (η) =

μ f (η)

.

198

Chapter 8. Backward Flow Invariant Domains for Semigroups

Lemma 8.2. Let f ∈ G ∗ [τ, η] for some τ ∈ Δ (which is the Denjoy–Wolﬀ point for the semiﬂow S generated by f ) with β = f (τ ) > 0 and some η ∈ ∂Δ, such that f (η) := ∠ lim f (z) = 0 and γ = f (η) = ∠ lim f (z) exists ﬁnitely. z→η

z→η

The following assertions hold. (i) If τ ∈ Δ, then γ < − 12 Re β. (ii) If τ ∈ ∂Δ, then γ ≤ −β < 0 and the equality γ = −β holds if and only if f ⊂ aut(Δ) or, which is the same, S ⊂ Aut(Δ) consists of hyperbolic automorphisms of Δ. Proof. (i) Let τ ∈ Δ. Then f ∈ G ∗ [τ ] admits the representation f (z) = (z − τ )(1 − z τ¯)p(z) with Re p(z) > 0, z ∈ Δ and β (= f (τ )) = (1 − |τ |2 )p(τ ). Assume that, for some η ∈ ∂Δ, f (η) := ∠ lim f (z) = 0 z→η

and γ = ∠ lim

z→η

f (z) z−η

exists ﬁnitely. Then ∠ lim p(z) = 0, and z→η

γ = η|η − τ |2 · p (η), where

p(z) . z→η z − η To ﬁnd an estimate for p (η), we introduce a function p1 of positive real part by the formula p1 (z) = (1 − |τ |2 )p(m(z)), p (η) = ∠ lim

where

τ −z 1 − z τ¯ is the M¨obius transformation (involution) taking τ to 0 and 0 to τ . Thus m(z) =

2

p1 (0) = (1 − |τ | )p(τ ) = β; and, setting η1 = m(η), we have p1 (η1 ) = (1 − |τ |2 )p (η) · m (η1 ) =

1 − |τ |2 · p (η) = −(1 − η¯ τ )2 p (η). m (η)

8.1. Existence

199

On the other hand, introducing the function q = 1/p1 we have ∠ lim (z − η1 )q(z) = −η1 δq (η1 ), z→η1

where δq (η1 ) = ∠ lim (1−zη1 )q(z) is the charge of q at the point η1 . Consequently, z→η1

p1 (η1 ) = ∠ lim

z→η1

=

p1 (z) 1 = ∠ lim z→η1 (z − η1 )q(z) z − η1

−η1 = −(1 − η¯ τ )2 p (η). δq (η1 )

Hence p (η) = and γ=

η1 (1 − η¯ τ )2 δq (η1 )

η|η − τ |2 η1 1 . · 2 (1 − η¯ τ) δq (η1 )

Since by Theorem 1.14 δq (η1 ) ≤ 2 Re q(0) = |γ| ≥

2 2 = , we have Re p1 (0) Re β

1 Re β. 2

Note that equality is impossible since otherwise q (and hence p1 and p) are constant. But ∠ lim p(z) = 0, which means that p(z) ≡ 0. z→η1

This proves assertion (i). (ii) Let now τ ∈ ∂Δ. In this case, we know already that β = f (τ ) = ∠ lim f (z) > 0. z→τ

Without loss of generality, let us assume that τ = 1 and η = −1. In other words, we assume that f ∈ G ∗ [1, −1]. We have to show that in that case γ = ∠ lim f (z) ≤ −β, and equality holds if and only if f is a complete vector ﬁeld. z→−1

Indeed, suppose to the contrary that γ > −β. Then the function g ∈ Hol(Δ, C) deﬁned by γ g(z) = f (z) + (z 2 − 1) 2 belongs to the class G ∗ [1, −1], because this class is a real cone. In addition, g (1) = β + γ > 0, while

g (−1) = γ − γ = 0.

200

Chapter 8. Backward Flow Invariant Domains for Semigroups

Then both points 1 and −1 are sink points of the semigroup generated by g, which is impossible. This contradiction shows that γ ≤ −β. If γ = −β, then g (1) = g (−1) = 0, and consequently g vanishes identically; hence γ f (z) = − (z 2 − 1). 2 Thus f belongs to aut(Δ), and the ﬂow S = {Ft }t∈R consists of hyperbolic automorphisms of Δ. The lemma is proved. Now we are ready to prove our theorems. Since Theorem 8.2 is a compliment of Theorem 8.1, we give their proofs simultaneously. Proofs of Theorems 8.1 and 8.2. We prove implications (i)=⇒(ii)=⇒(iii)=⇒(i) of Theorem 8.1 successively, while assertions (a), (b) and (c) of Theorem 8.2 will be obtained in the process. Let S = {Ft }t≥0 be a semiﬂow on Δ generated by f ∈ G ∗ [τ ] with β = f (τ ), Re β > 0. Let h ∈ Hol(Δ, C) be the associated spirallike (starlike) function on Δ deﬁned by equation (8.1.5) with μ = β. Then h satisﬁes Schr¨ oder’s equation (8.1.6) h(Ft (z)) = e−tβ h(z)

(8.1.7)

for all t ≥ 0 and z ∈ Δ. Step 1 ((i)=⇒(ii)). If now f ∈ G ∗ [τ, η] for some η ∈ ∂Δ, that is f (η) = ∠ lim f (z) = 0 and γ = f (η) = ∠ lim f (z) exists ﬁnitely, then by Lemma 8.1 z→η

z→η

the function h belongs to the class Spiral[τ, η] with Qh (η) = ∠ lim

z→η

(z − η)h (z) β = . h(z) γ

Since γ = 0 (actually, γ < 0), Qh (η) is ﬁnite.

−β -spiral γ wedgeW ⊂ h(Δ) with vertex at the origin such that for each w ∈ W the spiral curve e−tβ w, t ∈ R belongs to W . Deﬁne the simply connected domain Ω ⊂ Δ by In turn, Theorem 4.3 implies that there is a nonempty canonical

Ω = h−1 (W ). Then the family Ft : Ω → Ω, Ft (z) = h−1 e−tβ h(z) ,

z ∈ Ω, t ∈ R,

forms a ﬂow (one-parameter group) of holomorphic self-mappings of Ω. Comparing the latter formula with (8.1.7), we see that for t ≥ 0, Ft (z) = Ft (z) whenever z ∈ Ω −1 < and ( Ft |Ω ) = F −t . Thus S ⊂ Aut(Ω).

8.1. Existence

201

Step 2 ((ii)=⇒(iii)). Let again S = {Ft }t≥0 be a semiﬂow generated by f ∈ G ∗ [τ ] so that lim Ft (z) = τ ∈ ∂Δ and

t→∞

Re β > 0,

where β = f (τ ),

(8.1.8)

and let Ω ⊂ Δ be a simply connected domain such that S ⊂ Aut(Ω). Let ψ : Δ → Ω be any Riemann conformal mapping of Δ onto Ω. Consider the ﬂow {Gt }t∈R ⊂ Aut(Δ) deﬁned by Gt (z) = ψ −1 (Ft (ψ(z))), t ∈ R.

(8.1.9)

In this case, ψ is a conjugation for Gt and Ft for each t ∈ R, i.e., ψ(Gt (z)) = Ft (ψ(z)),

z ∈ Δ, t ∈ R.

(8.1.10)

Denote by g ∈ aut(Δ) the generator of {Gt }t∈R : g(z) = lim

t→0

z − Gt (z) . t

Then by (8.1.10), ψ satisﬁes the diﬀerential equation ψ (z) · g(z) = f (ψ(z)).

(8.1.11)

First we show that the family {Gt }t∈R ⊂ Aut(Δ) consists of hyperbolic automorphisms or, which is the same, that it contains neither elliptic nor parabolic automorphisms. Indeed, suppose {Gt }t∈R contains an elliptic automorphism. Then there is a point a ∈ Δ such that Gt (a) = a for all t ∈ R; hence g(a) = 0 and Re g (a) = 0. By (8.1.11), f (ψ(a)) = 0; and thus ψ(a) = τ . On the other hand, diﬀerentiating (8.1.11) with respect to z and setting z = a, we get g (a) = f (τ ). Hence Re f (τ ) = 0, which contradicts (8.1.8). Thus {Gt } has no interior ﬁxed point in Δ; hence there are boundary points ζ1 and ζ2 such that lim Gt (z) = ζ1 ∈ ∂Δ, z ∈ Δ, (8.1.12) t→∞

and lim Gt (z) = ζ2 ∈ ∂Δ,

t→−∞

z ∈ Δ.

(8.1.13)

To show that the family {Gt }t∈R does not contain a parabolic automorphism it is suﬃcient to prove that ζ1 = ζ2 . To this end, we again consider the associated spirallike (starlike) function h deﬁned by equation (8.1.5) with μ = β and normalized by the conditions h(τ ) = 0, h (τ ) = 1 if τ ∈ Δ or by h(τ ) = 0 and h(0) = 1 if τ ∈ ∂Δ. Deﬁne h0 ∈ Hol(Δ, C) by h0 (z) = h(ψ(z)). (8.1.14)

202

Chapter 8. Backward Flow Invariant Domains for Semigroups

Since h satisﬁes Schr¨oder’s equation (8.1.7), it follows from (8.1.10) that for all t ≥ 0, h0 (Gt (z)) = h(ψ(Gt (z)) = h(Ft (ψ(z)) = e−tβ h(ψ(z)) = e−tβ h0 (z). Since the mapping Gt ∈ Hol(Δ) is an automorphism of Δ for each t ∈ R+ , we have, in fact, h0 (Gt (z)) = e−tβ h0 (z) (8.1.15) for all t ∈ R. From (8.1.15) we conclude that h0 (hence, h) is a univalent spirallike (starlike) function on Δ. Moreover, (8.1.15) and Corollary 2.17 in [113] imply that ∠ lim h0 (z) = 0, z→ζ1

while ∠ lim h0 (z) = ∞. z→ζ2

Thus ζ1 = ζ2 , and it follows that {Gt }t∈R consists of hyperbolic automorphisms. Now observe that W = h0 (Δ) is a spirallike (starlike) wedge with vertex at the origin belonging to h(Δ). Since all the points of ∂h(Δ) are admissible, ψ = h−1 ◦ h0 is a homeomorphism of Δ onto Ω; hence ∂Ω is a Jordan curve. Now (8.1.10) implies that lim ψ(Gt (z)) = lim Ft (ψ(z)) = τ

(8.1.16)

lim ψ(Gt (z)) = lim Ft (ψ(z)) = η

(8.1.17)

t→∞

t→∞

and t→−∞

t→−∞

for some η ∈ Δ. Applying again Corollary 2.17 in [113], we obtain ψ(ζ1 ) := lim ψ(z) = τ

(8.1.18)

ψ(ζ2 ) := ∠ lim ψ(z) = η.

(8.1.19)

z→ζ1

and z→ζ2

Thus η = τ and, moreover, η ∈ ∂Δ. Indeed, if η is an interior point of Δ, then η = ψ(ζ2 ) = ψ(Gt (ζ2 )) = Ft (ψ(ζ2 )) = Ft (η),

t ≥ 0,

i.e., it must be an interior ﬁxed point for all Ft ∈ S, t ≥ 0, which is impossible. So τ ∈ ∂Ω by (8.1.18), and η ∈ ∂Δ ∩ ∂Ω by (8.1.19). To show that equation (8.1.4) has a locally univalent (even univalent) solution ϕ ∈ Hol(Δ) for some α > 0, we use a M¨obius transformation m ∈ Aut(Δ) such

8.1. Existence

203

that m(1) = ζ1 and m(−1) = ζ2 . Then ψ1 = ψ ◦ m is a conformal mapping of Δ onto Ω with normalization ψ1 (1) = τ,

ψ1 (−1) = η.

For s ∈ (−1, 1), deﬁne another conformal mapping ϕs of Δ onto Ω by z−s , −1 < s < 1. ϕs (z) := ψ1 1 − zs Clearly ϕs (1) = ψ1 (1) = τ and ϕs (−1) = ψ1 (−1) = η. Note also that = {z = ϕs (0) (= ψ1 (−s)), s ∈ [−1, 1])} is a continuous curve joining the points z = 1 and z = −1, and so 1 = {z = h(ϕs (0))(= h(ψ1 (−s))), s ∈ [−1, 1]} is a continuous curve joining h(τ ) = 0 and h(η) = ∞. Hence, there exists s ∈ (−1, 1) such that |h(ϕs (0))| = 1. Thus there exists a homeomorphism ϕ(= ϕs ) of Δ onto Ω holomorphic in Δ such that ϕ(1) = τ, ϕ(−1) = η and h(ϕ(0)) = eiθ for some θ ∈ R. Since the mapping ψ in our previous consideration was arbitrary, we can replace it by ϕ. In this case, the “new” ﬂow {Gt }t∈0 deﬁned by Gt (z) = ϕ−1 (Ft (ϕ(z))) is a one-parameter group of hyperbolic automorphisms of Δ having the ﬁxed points z = 1 and z = −1 on ∂Δ. In turn, its generator g ∈ Hol(Δ, C) must have the form α 2 (z − 1) , 2

g(z) =

(8.1.20)

where α = g (1) > 0. Hence, equation (8.1.11) (with ϕ in place of ψ) becomes (8.1.4) αϕ (z)(z 2 − 1) = 2f (ϕ(z)).

(8.1.21)

Combining this with (8.1.5), we show that α must be greater than or equal to −γ > 0. Namely, deﬁning h0 ∈ Spiral[1] as in (8.1.14) by h0 (z) = h(ϕ(z)),

(8.1.22)

we have from (8.1.21) and (8.1.5) that βh0 (z) =

α 2 (z − 1)h0 (z) 2

(8.1.23)

with h0 (0) = h(ϕ(0)) = eiθ for some θ ∈ [0, 2π). Solving this equation, we obtain h0 (z) = eiθ

1−z 1+z

β/α (8.1.24)

204

Chapter 8. Backward Flow Invariant Domains for Semigroups

β with α = g (1). It is clear that the image h0 (Δ) is a canonical α -spiral wedge. On the other hand, by Theorem 4.3, the maximal spiral wedge contained in h(Δ) is W−ν,θ , where

ν = ∠ lim

z→η

β(z − η)h (z) z−η β = ∠ lim = . z→η h(z) f (z) γ

(8.1.25)

Thus γ is ﬁnite and ϕ = h−1 ◦ h0 is a well-deﬁned self-mapping of Δ if and only if α ≥ −γ. This completes the proof of the implication ((ii)=⇒(iii)) of Theorem 8.1, as well as assertions (a) and (b) of Theorem 8.2. Note in passing that we have also proved the implication (ii)=⇒(i) of Theorem 8.1. Step 3 ((iii)=⇒(i)). Suppose now that ϕ ∈ Hol(Δ) is locally univalent and satisﬁes (8.1.4) for some α ∈ R+ . Solving this diﬀerential equation explicitly, we get ϕ(z) z dw 2dz 1−z α = = log . (8.1.26) 2 1+z ϕ(0) f (w) 0 z −1 Since ϕ (z) = 0, z ∈ Δ, we have by (8.1.4) that there is no z ∈ Δ such that ϕ(z) = τ . So if is a curve joining 0 and z, the curve ϕ() joining ϕ(0) and ϕ(z) does not contain τ . Consider now the diﬀerential equation (8.1.5) with initial data h(ϕ(0)) = 1. Separating variables in this equation, we see that

ϕ(z)

β ϕ(0)

dw = f (w)

h(ϕ(z))

1

dh = log(h(ϕ(z)). h

(8.1.27)

Comparing (8.1.26) with (8.1.27), we have β log (h(ϕ(z)) = log α or

1−z 1+z

,

β 1−z α h(ϕ(z)) = . 1+z 6 5 β 1−z α : z ∈ Δ is a subset of h(Δ), so This equality implies that the set 1+z this set is diﬀerent from C \ {0}. It follows by Lemma 4.1 and Proposition 4.2 that β/α 1−z β is univalent on Δ. Its in this case α − 1 ≤ 1, the function h0 := 1+z image W = h0 (Δ) is a spiral wedge with vertex at the origin. So, by Theorem 4.3, there is a point η ∈ ∂Δ such that h(η) = ∞ and Qh (η) exists ﬁnitely with β arg Qh (η) = arg β and α ≤ |Qh (η)|.

8.2. Maximal FIDs. Flower structures

205

Finally, we note that ϕ(z) = h−1 (h0 (z)) is, in fact, a univalent function on Δ. Now, applying Lemma 8.1 with μ = β, we complete the proof of the implication (iii)=⇒(i) of Theorem 8.1, as well as assertion (c) of Theorem 8.2. Theorems 8.1 and 8.2 are proved.

8.2 Maximal FIDs. Flower structures Deﬁnition 8.2. A (backward) ﬂow invariant domain (FID) Ω ⊂ Δ for S is said to be maximal if there is no Ω1 ⊃ Ω, Ω1 = Ω, such that S ⊂ Aut(Ω1 ). Theorem 8.3. Let f ∈ G ∗ [τ, η] for some τ ∈ Δ, η ∈ ∂Δ with γ = f (η) < 0 , and let ϕ be a (univalent) solution of (8.1.4) with some α ≥ −γ normalized by ϕ(1) = τ and ϕ(−1) = η. The following assertions are equivalent: (i) Ω = ϕ(Δ) is a maximal FID; (ii) α = −γ; (iii) ϕ is isogonal at the boundary point z = −1. Proof. We already know by (8.1.22) and (8.1.24) that ϕ = h−1 ◦ h0 , where h is β/α 1−z iθ with the spirallike (starlike) function associated to f and h0 (z) = e 1+z β = f (τ ), Re β > 0 and α ≥ −γ. We also know that h0 (Δ) is a canonical spiral wedge W β ,θ∗ with some θ∗ . So, α

h0 (Δ) = W β ,θ∗ ⊂ h(Δ). α Thus Ω = ϕ(Δ) = h−1 W β ,θ∗ is maximal FID if and only if the canonical spiral α wedge W β ,θ∗ is maximal. In turn, by Theorem 4.3, this wedge W β ,θ∗ is maximal if α

α

β = −ν. Comparing this fact with (8.1.25), we obtain the equivalence and only if α of assertions (i) and (ii) of the theorem. We prove the equivalence of assertions (ii) and (iii) for the case where τ = 1. Namely, let f1 ∈ G ∗ [1, η] with f1 (1) = β1 > 0 and f1 (η) = γ1 < 0. Let ψ be a univalent solution of equation (8.1.4), i.e.,

αψ (z)(z 2 − 1) = 2f1 (ψ(z))

(8.2.1)

for some α ≥ −γ1 , normalized by ψ(1) = 1, ψ(−1) = η. Substituting in formula (8.1.22) the explicit form of h0 (see (8.1.24)) and the integral representation (3.4.7) with τ = 1 for the spirallike function h and taking into account that Qh (η) = ν = βγ11 (cf. (8.1.25)), we get (ψ(z) − 1)(1 − ψ(z)¯ η )β1 /γ1 β /α

1−z 1 ¯ · exp − (2 + β1 /γ1 ) log(1 − ψ(z)ζ)dσ(ζ) = C1 1+z ∂Δ

206

Chapter 8. Backward Flow Invariant Domains for Semigroups

or ψ(z) − η C2 (1 − z)γ1 /α = (z + 1)−1−γ1 /α · z+1 (1 − ψ(z))γ1 /β1

2γ1 + β1 ¯ · exp log(1 − ψ(z)ζ)dσ(ζ) . β1 ∂Δ Note that one can choose an analytic branch of the multivalued function (1 − z)γ1 /α C2 . We denote this branch by χ(z). It is a continuous function (1 − ψ(z))γ1 /β1 which does not vanish at the point z = −1. Hence its argument is a well-deﬁned continuous function at this point. Thus ψ(z) − η z+1 2γ1 + β1 −1−γ1 /α ¯ + arg χ(z) + = arg (z + 1) arg(1 − ψ(z)ζ)dσ(ζ). β1 ∂Δ

arg

Applying the Lebesgue Bounded Convergence Theorem, we conclude that the limit of the last summand as z → −1 exists ﬁnitely. Therefore, the function ψ is isogonal at the point z = −1 if and only if the limit lim arg (z + 1)−1−γ1 /α z→−1

exists. Obviously, this happens if and only if the exponent vanishes, i.e., α = −γ1 . Now let τ ∈ Δ be arbitrary, and let f ∈ G ∗ [τ, η] with f (τ ) = β, Re β > 0, and f (η) = γ < 0. Let ϕ be a univalent solution of equation (8.1.4) for some α ≥ −γ, normalized by ϕ(1) = τ, ϕ(−1) = η. Denote by h the spirallike function associated to f , that is, h satisﬁes equation (8.1.5) with μ = β. As above, let h0 be the function β/α which maps the disk Δ onto a spiral wedge, namely, h0 (z) = eiθ 1−z , such 1+z that ϕ = h−1 ◦ h0 . Find a conformal mapping Φ of Δ such that Φ(1) = τ, Φ(−1) = η, and h1 = h ◦ Φ is a spirallike function with respect to a boundary point. Note here that the domain D = Φ(Δ) has a corner of opening π at the point η because Φ maps a circular arc containing z = −1 onto a circular arc which contains z = η. By exists. Hence Φ is isogonal at the Theorem 3.7 of [113], the limit lim arg Φ(z)−η z+1 z→−1

point −1. Moreover, by Theorem 1.9, the function Φ satisﬁes the Visser–Ostrowski condition ∠ lim

z→−1

Φ(z) − η = 1. z+1

(8.2.2)

Now write ϕ = h−1 ◦ h0 = Φ ◦ h−1 1 ◦ h0 = Φ ◦ ψ,

(8.2.3)

8.2. Maximal FIDs. Flower structures

207

where ψ = h−1 1 ◦ h0 . One sees that ψ(−1) := lim + ψ(s) = lim + h−1 1 (h0 (s)) = −1 s→−1

and

s→−1

ψ(1) := lim− ψ(s) = lim− h−1 1 (h0 (s)) = 1. s→1

s→1

Using this notation, we have arg

Φ(z) − η Φ(ψ(z)) − η ψ(z) + 1 = arg + arg . z+1 ψ(z) + 1 z+1

(8.2.4)

Thus (8.2.2) and (8.2.4) imply that ϕ is isogonal at the point η if and only if ψ is isogonal at the point z = −1. Now we check that function ψ satisﬁes equation (8.2.1). We have seen already (Φ(z)) that βh1 (z) = h1 (z)f1 (z), where f1 ∈ G ∗ [1, −1] is deﬁned by f1 (z) = fΦ (z) . Using (8.2.2), we get f1 (−1) = ∠ lim

z→−1

Φ(z) − η f1 (z) f (Φ(z)) = ∠ lim · = γ. z→−1 Φ(z) − η (z + 1)Φ (z) z+1

Furthermore, 1 h (ψ(z))f1 (ψ(z)) β 1 h (z) 1 (h1 (ψ(z))) f1 (ψ(z)) = 0 f1 (ψ(z)). = β ψ (z) βψ (z)

h0 (z) = h1 (ψ(z)) =

β/α in the last equality and diﬀerentiating, we see Substituting h0 (z) = eiθ 1−z 1+z that equation (8.2.1) holds. But we have already shown that in that case ψ (hence, ϕ) is isogonal if and only if α = −f1 (−1) = −γ. This completes the proof. Remark 8.1. In general, a maximal FID for S need not be unique. Theorem 8.3 states that if S = {Ft }t≥0 is generated by f ∈ G ∗ [τ ], then its FID is not empty if and only if there is a point η ∈ ∂Δ, such that f (η) = ∠ lim f (z) = 0 and z→η

f (η) = ∠ lim f (z) exists ﬁnitely with f (η) < 0. This point η is a repelling ﬁxed z→η t (z) point for S = {Ft }t≥0 , namely, Ft (η) = η and ∂F∂z = e−tf (η) > 1 (see z=η

Theorem 2.12). Moreover, there is a one-to-one correspondence between maximal ﬂow invariant domains for S and such repelling ﬁxed points. Theorem 8.4. Let f ∈ G ∗ [τ, ηk ] for some sequence {ηk } ∈ ∂Δ, i.e., f (ηk ) = 0 and γk = f (ηk ) > −∞. The following assertions hold.

208

Chapter 8. Backward Flow Invariant Domains for Semigroups

(i) There is δ > 0 such that γk < −δ < 0 for all k = 1, 2, . . .. (ii) For each a < −δ < 0 there is at most a ﬁnite number of the points ηk such that a ≤ γk < −δ. Consequently equation (8.1.4) has a (univalent) solution ϕ ∈ Hol(Δ) for each α ≥ − max{γk } > −δ. (iii) If ϕk is a solution of (8.1.4) normalized by ϕk (1) = τ, ϕk (−1) = ηk with α = γk and Ωk = ϕk (Δ) (i.e., Ωk are maximal), then for each pair Ωk1 and Ωk2 such that ηk1 = ηk2 either Ωk1 ∩ Ωk2 = {τ } or Ωk1 ∩ Ωk2 = l, where l is a continuous curve joining τ with a point on ∂Δ. Proof. Assertions (i) and (ii) of the theorem are direct consequences of Lemma 8.2. To prove assertion (iii), we ﬁrst note that the inclusion τ ∈ ∩k ∂Ωk follows by assertion (a) of Theorem 8.2. Also observe that for each pair k1 and k2 such that ηk1 = ηk2 , the set Ωk1 ,k2 = Ωk1 ∩ Ωk2 is empty. Indeed, otherwise Ωk1 ,k2 is a FID for S. Hence, it must contain a point η ∈ ∂Ωk1 ,k2 ∩ ∂Δ such that η = ∠ lim Ft (z) t→−∞

whenever z ∈ Ωk1 ,k2 . Hence we should have a contradiction η = ηk1 = ηk2 . Let us suppose now that for a pair k1 and k2 there is a point z0 = τ , z0 ∈ Δ, such that z0 ∈ ∂Ωk1 ∩ ∂Ωk2 . Then the whole curve = {z ∈ Δ : z = Ft (z0 ), t ≥ 0} ending at τ must belong to both Ωk1 and Ωk2 , hence to l ⊂ ∂Ωk1 ∩ ∂Ωk2 , since Ωk1 ∩ Ωk2 = ∅. Finally, we have that f ∈ Hol(Δ, C) is locally Lipschitzian. Therefore, if ζ ∈ Δ is an interior end point of , ζ = τ, then there is δ > 0 such that the Cauchy problem ⎧ ⎨ ∂u(t, ζ) + f (u(t, ζ)) = 0, t ≥ 0, ∂t ⎩ u(0, ζ) = ζ, has a solution u(t, ζ) (= Ft (ζ)) for all t ∈ [−δ, ∞); and the curve 1 = {z ∈ Δ : z = u(t, ζ), t ∈ [−δ, ∞)} also belongs to ∂Ωk1 ∩ ∂Ωk2 . But 1 properly contains , which is impossible. So ζ must belong to ∂Δ. The proof is complete.

8.3 Examples We illustrate the content of our theorems in the following examples.

8.3. Examples

209

Example 8.1. Consider a generator f ∈ G ∗ [0] deﬁned by f (z) = z(1 − z n ),

n ∈ N.

Solving the Cauchy problem, we ﬁnd Ft (z) = √ n

ze−t . 1 − z n + z ne−nt

1

1

y

0.5

–1

–0.5

0

0.5

1

–1

0.5

0

–0.5

0.5

1 x

–1

–0.5

–0.5

–0.5

–1

–1

1

1

0.5

0.5

0

0.5

1

–1

–0.5

0

–0.5

–0.5

–1

–1

0.5

1

Figure 8.1: Example 8.1, n = 1, 2, 3, 5. 2πik

In this case, f has n additional null points ηk = e n , k = 1, 2, . . . , n, on the unit circle with ﬁnite angular derivative γ = f (ηk ) = −n. So the generated semiﬂow has n repelling ﬁxed points, and there are n maximal ﬂow invariant domains. One

210

Chapter 8. Backward Flow Invariant Domains for Semigroups

can show that the functions

: ϕk (z) = e

2πik n

n

1−z 2

are the solutions of (8.1.4) with α = n satisfying ϕk (1) = 0 and ϕk (−1) = ηk which 2 map Δ onto n FID’s Ωk (for n = 2, these domains form a lemniscate) with Ωi Ωj = {0} when i = j. The family {Ft }t∈R forms a group of automorphisms of each one of these domains. See Figure 8.1 for n = 1, 2, 3 and 5. For n = 1, for instance, it can be seen explicitly that Ft (ϕ(z)) is well-deﬁned for all t ∈ R and tends to η = 1 when t → −∞. Example 8.2. Consider a generator f ∈ G ∗ [1] deﬁned by f (z) = −

(1 − z)(1 + z 2 ) . 1+z

Solving the Cauchy problem, we ﬁnd Ft (z) =

; (1 + z 2 )e2t − (1 − z) 2(1 + z 2 )e2t − (1 − z)2 . (1 + z 2 )e2t − (1 − z)2

1

0.5

–1

–0.5

0

0.5

1 x

–0.5

–1

Figure 8.2: Example 8.2. The ﬂow generated by f (z) = − (1−z)(1+z 1+z ﬂow-invariant domains.

2

)

and two

Since f has the two additional null points η1,2 = ±i ∈ ∂Δ with ﬁnite angular derivative γ = f (±i) = −2, the generated semiﬂow has two repelling ﬁxed points. Thus, there are two maximal ﬂow invariant domains Ω1 and Ω2 . One can show that these domains Ω2j coincide with the upper and the lower half-disks (see Figure 8.2). So we have Ω1 Ω2 = {−1 < x < 1}. In each of these two domains, the family {Ft }t∈R is well deﬁned and forms a group of automorphisms.

8.4. Angular characteristics of ﬂow invariant domains

211

The following example shows that a maximal ﬂow invariant domain may be even dense in the open unit disk. Example 8.3. Let f ∈ G ∗ [0] be given by f (z) = z

1−z . 1+z

In this case, τ = 0 and η = 1. Also, we have f (0) = 1 and f (1) = − 21 . Solving equation (8.1.4) with α = 12 , one can write its solution in the form ϕ(z) = 2 z 1−z −1 h (h0 (z)), where h is the Koebe function h(z) = and h0 (z) = . (1 − z)2 1+z We shall see below that each solution of (8.1.4) has a similar representation. Thus ϕ maps Δ onto the maximal ﬂow invariant domain Ω = ϕ(Δ) = Δ \ {−1 ≤ x ≤ 0}; see Figure 8.3. (All the pictures were obtained by using the vector ﬁeld drawing tool in Maple 9.)

1

y

–1

–0.5

0.5

0

0.5

1 x

–0.5

–1

Figure 8.3: Example 8.3. The ﬂow generated by f (z) = z 1−z and the dense ﬂow1+z invariant domain.

8.4 Angular characteristics of ﬂow invariant domains In Theorem 8.3 above we have proved that for a univalent solution ϕ of (8.1.4) with some α ≥ −γ, the image Ω = ϕ(Δ) is a maximal FID if and only if α = −γ and if and only if ϕ is isogonal at the boundary point z = −1. Repeating its proof for arbitrary α ≥ −γ, one can complete this result and obtain the following angular description of ﬂow invariant domains (not necessary maximal).

212

Chapter 8. Backward Flow Invariant Domains for Semigroups

Theorem 8.5 (see [60]). Let f ∈ G and let η ∈ ∂Δ be a boundary regular null point of f, diﬀerent from the Denjoy–Wolﬀ point τ ∈ Δ of the generated semigroup S = {Ft }t≥0 , i.e., f (z) < 0. γ = f (η) := ∠ lim z→η z − η Let ϕ ∈ Hol(Δ) satisfy the diﬀerential equation (8.1.4) with some α ≥ −γ such that ϕ(1) = τ and ϕ(−1) = η, and let Ω = ϕ(Δ) be a (backward) ﬂow-invariant domain for the semigroup S. Assume that β = Re f (τ ) > 0. Then π|γ| at the point η = ϕ(−1) ∈ ∂Δ ∩ ∂Ω; α (ii) If τ ∈ ∂Δ, then the opening of the corner of Ω at the point τ = ϕ(1) ∈ ∂Ω∩∂Δ πβ is . α Corollary 8.1. Let S = {Ft }t≥0 ⊂ Hol(Δ) be a semigroup of hyperbolic type. Suppose that S has a backward ﬂow-invariant domain Ω ⊂ Δ. If ∂Ω has a corner of opening π at the Denjoy–Wolﬀ point τ ∈ ∂Δ, then S consists of hyperbolic automorphisms of Δ. (i) ∂Ω has a corner of opening

Proof. Let S = {Ft }t≥0 ⊂ Hol(Δ) be a semigroup of hyperbolic type, and let f be its generator. Since S has a backward ﬂow invariant domain Ω ⊂ Δ, there is a f (z) boundary regular null point η ∈ ∂Ω∩∂Δ of f such that γ = f (η) := ∠ lim < z→η z − η 0. By Theorem 8.5, the opening of the corner of ∂Ω at the point τ ∈ ∂Ω ∩ ∂Δ is πβ πβ α , where β = f (1) and α ≥ −γ. At the same time, by our assumption, α = π. Hence, β = α ≥ −γ. We claim that the last inequality is possible if and only if β = −γ and the maximal backward ﬂow invariant domain is all of Δ. In other words, in this case f generates a group of hyperbolic automorphisms of Δ. Indeed, let us set τ = 1 and η = −1 and assume that −γ ∈ (0, β). Then the function g deﬁned by g(z) = f (z) +

γ 2 (z − 1) 2

belongs to the class G + [1], because this class is a real cone. In addition, we have g (1) = β + γ > 0 while

g (−1) = γ − γ = 0.

Then g = 0 and both the points 1 and −1 are the Denjoy–Wolﬀ points of the semigroup generated by g, which is impossible. This contradiction implies that β ≤ −γ. Moreover, the same considerations show that β = −γ if and only if g(z) = 0. Hence f (z) has the form f (z) =

β γ (1 − z 2 ) = (z 2 − 1). 2 2

8.4. Angular characteristics of ﬂow invariant domains

213

Thus f generates a group of hyperbolic automorphisms of Δ. This completes our proof. Theorem 8.6. Let f ∈ G(Δ) have a boundary null point η ∈ ∂Δ such that the angular limit f (z) γ := f (η) = ∠ lim z→η z − η exists and is a negative real number. Let S = {Ft }t≥0 be the semigroup generated by f . Suppose that Ω is the maximal ﬂow-invariant domain such that η ∈ ∂Ω, and ϕ is a Riemann mapping of Δ onto Ω which can be deﬁned by the diﬀerential equation (8.1.4). Then for each w ∈ Ω, w γdζ 1 + ψ (w) = Im , (8.4.5) lim arg (1 − η¯Ft (w)) = arg t→−∞ 1 − ψ (w) ϕ(0) f (ζ) where ψ = ϕ−1 : Ω → Δ and the integration path lies in Ω. The right-hand side of (8.4.5) does not depend on the choice of a solution ϕ of (8.1.4). Proof. Consider the family of functions {ht }t≥0 deﬁned by ht (z) =

1 − η¯Ft (ϕ(z)) , |1 − η¯Ft (ϕ(0))|

z ∈ Δ.

(8.4.6)

This is a normal family since Re ht (z) > 0. Therefore by Montel’s Theorem, each sequence in this family contains a subsequence {htk }∞ k=0 , tk → ∞, which converges locally uniformly on Δ either to a holomorphic function h or to inﬁnity. However, the last case is impossible because of the obvious equality |ht (0)| = 1. Now it follows by the Hurwitz Theorem that h is either a univalent function on Δ, or a constant. To see that the last case is again impossible, we diﬀerentiate ht (z) to obtain −¯ η Ftk (ϕ(z))ϕ (z) h (z) = lim k→∞ |1 − η ¯Ftk (ϕ(0))| and −¯ ηFtk (ϕ(z))ϕ (z) h (z) = lim . (8.4.7) k→∞ h(z) 1 − η¯Ft (ϕ(z)) Since

f (Ft (z)) = Ft (z) for all t > 0, we obtain by (8.1.4) and (8.4.7) that f (z) −¯ η f (Ftk (ϕ(z))) ϕ (z) h (z) = lim k→∞ f (ϕ(z)) (1 − η h(z) ¯Ftk (ϕ(z))) 2f (Ftk (ϕ(z)) = lim k→∞ γ(z 2 − 1) (η − Ftk (ϕ(z))) −2 f (Ftk (ϕ(z)) = · lim . 2 γ(z − 1) k→∞ Ftk (ϕ(z)) − η

214

Chapter 8. Backward Flow Invariant Domains for Semigroups

Now we claim that Ftk (ϕ(z)) converges non-tangentially. Indeed, for each point w ∈ Ω, consider a ﬂow-invariant domain Ω1 deﬁned in the following way. Let Ωmax be the maximal ﬂow-invariant domain which contains w. The trajectory {Ft (w) : t ∈ R} divides Ωmax into two subdomains. Let Ω1 be one of them. By Theorem 8.1, the Riemann mapping ϕ onto Ω1 is a solution of the diﬀerential equation (8.1.4) with some α ≥ −γ. Since this domain is not maximal, α > −γ. Now Proposition 8.5 implies that the opening of the corner of ∂Ω1 at the point η is less than π. Hence, {Ft (w) : t ∈ R} ⊃ Ftk (ϕ(z)) converges non-tangentially to η as t → −∞. f (z) Because of the equality ∠ lim = γ, we conclude that the limit function z→η z − η h must satisfy the diﬀerential equation 2 h (z) = . h(z) 1 − z2 1+z . Since Re ht (z) > 0 for all z ∈ Δ 1−z . Therefore, for each and |ht (0)| = 1, we necessarily have h(0) = 1, so h(z) = 1+z 1−z convergent sequence {htk }∞ , k=1 Solving this equation we get h(z) = h(0) ·

lim htk (z) = lim

k→∞

k→∞

1+z 1 − η¯Ftk (ϕ(z)) = . |1 − η¯Ftk (ϕ(0))| 1−z

Since the limit does not depend on the choice of {htk }∞ k=1 , we obtain that lim

t→−∞

1 − η¯Ft (ϕ(z)) 1+z = . |1 − η¯Ft (ϕ(0))| 1−z

Hence lim arg (1 − η¯Ft (ϕ(z))) = arg

t→−∞

1+z , 1−z

or, denoting by ψ : Ω → Δ the inverse mapping to ϕ, lim arg (1 − η¯Ft (w)) = arg

t→−∞

1 + ψ(w) , 1 − ψ(w)

w(= ϕ(z)) ∈ Ω.

Note that one can rewrite (8.1.4) in the form γ 2ψ (w) = . f (w) 1 − ψ 2 (w) Hence, integrating this equation, we see that w 1 + ψ(w γdζ log = , 1 − ψ(w) w0 f (ζ)

(8.4.8)

8.4. Angular characteristics of ﬂow invariant domains

215

where ψ(w0 ) = 0, or, which is the same, w0 = ϕ(0). Extracting the imaginary part, we prove (8.4.5). To complete the proof, we note that the mapping ϕ is well-deﬁned by (8.1.4) up to composition with an automorphism of the unit disk preserving the points z = 1 and z = −1. Denoting by ϕ1 another Riemann mapping of Δ onto Ω, we have z+s for some s ∈ (−1, 1). ϕ1 (z) = ϕ 1 + zs In this case the mapping ψ1 , inverse to ϕ1 , has the form ψ1 (w) = Consequently,

ψ(w) − s . 1 − sψ(w)

(1 − s)(1 + ψ(w)) 1 + ψ1 (w) = , 1 − ψ1 (w) (1 + s)(1 − ψ(w))

and the argument of this quotient does not depend on s.

Corollary 8.2. Let Sf = {Ft }t≥0 and Sg = {Gt }t≥0 be two semigroups of holomorphic self-mappings on Δ with the same Denjoy–Wolﬀ point τ ∈ Δ and a common boundary regular repelling ﬁxed point η ∈ ∂Δ. The following assertions are equivalent. (i) The semigroups coincide up to rescaling, i.e., there is a positive real number a such that Ft = Gat for all t ≥ 0. (ii) There is a nonempty open set D ⊂ Δ, where Sf and Sg have a similar angular asymptotic behavior, i.e., lim arg

t→−∞

1 − η¯Ft (w) =0 1 − η¯Gt (w)

for all w ∈ D.

(iii) There is a common backward ﬂow-invariant domain Ω for both Sf and Sg with η ∈ ∂Ω. Proof. As above, denote the semigroup generators by f and g, respectively. Step 1. If assertion (i) holds, then f = ag. If it is the case, then Sf and Sg have the same ﬂow-invariant domains; and Theorem 8.6 implies that lim arg (1 − η¯Ft (w)) = lim arg (1 − η¯Gt (w))

t→−∞

t→−∞

for all w in any of their common ﬂow-invariant domain which contains η. So, (i)⇒(ii). Step 2. Assume now that assertion (ii) holds. Denote by Ω1 and Ω2 the maximal ﬂow-invariant domains for Sf and Sg , respectively, the boundary of which

216

Chapter 8. Backward Flow Invariant Domains for Semigroups

contains η. Let ϕ1 and ϕ2 be univalent mappings of Δ onto Ω1 and Ω2 , respectively, normalized by ϕi (1) = τ and ϕi (−1) = η, i = 1, 2; and let ψ1 and ψ2 be their inverse mappings ϕ1 and ϕ2 , respectively. By formula (8.4.5), for all w ∈ D, arg

1 + ψ2 (w) 1 + ψ1 (w) = arg . 1 − ψ1 (w) 1 − ψ2 (w)

By the Maximum Principle for harmonic functions, we conclude that there exists a positive constant K such that 1 + ψ1 (w) 1 + ψ2 (w) =K 1 − ψ1 (w) 1 − ψ2 (w) on D. Therefore, ψ1 (w) =

1−K 1+K ψ2 (w) 1−K 1+K

ψ2 (w) − 1−

for all

w ∈ D.

is real and of modulus less than 1. Hence, Note here that the number s = 1−K 1+K using the inverse functions, we have for all z ∈ D1 := ψ2 (D), z+s . ϕ1 (z) = ϕ2 1 + zs z+s By uniqueness this equality holds for all z ∈ Δ. Noting that the mapping 1+zs is an automorphism of the unit disk, we obtain that Ω1 = ϕ1 (Δ) = ϕ2 (Δ) = Ω2 , that is (iii) holds.

Step 3. Let now Sf and Sg have a common backward ﬂow-invariant domain Ω. Denote by ϕ a univalent mapping of Δ onto Ω such that ϕ(1) = τ and ϕ(−1) = η. Then by equation (8.1.4), we have for all z ∈ Δ, ϕ (z)(z 2 − 1) =

2f (ϕ(z)) 2g(ϕ(z)) = , αf αg α

where αf ≥ −f (η) and αg ≥ −g (η). In other words, we have f (w) = αfg g(w) on Ω. Again by the uniqueness, this equality holds for all z ∈ Δ. Hence, the generated semigroups coincide up to rescaling. The proof is complete.

8.5 Additional remarks Let F ∈ Hol(Δ) be a single self-mapping of Δ which can be embedded into a continuous semigroup, i.e., there is a semiﬂow S = {Ft }t≥0 such that F = F1 . In this case, all the fractional iterations Ft of F have the same collection of boundary

8.5. Additional remarks

217

ﬁxed points for all t ≥ 0 (see [33]). In turn, Theorem 8.1 asserts the existence of backward fractional iterations of F deﬁned on a FID Ω whenever F has a repelling boundary ﬁxed point η, i.e., A = F (η) = lim F (z) > 1.

(8.5.9)

z→η

As a matter of fact, for a single mapping which is not necessarily embedded into a semiﬂow (not even necessarily univalent on Δ), the existence of backward integer iterations under condition (8.5.9) was proved in [107]. This fact has provided the existence of conjugations near repelling points. More precisely, the main result in [107] asserts that z−a A−1 and G(z) = , then there is ϕ ∈ Hol(Δ) with A+1 1 − az ϕ(1) = 1 which is a conjugation for F and G, i.e., • if η = 1, a =

ϕ(G(z)) = F (ϕ(z)).

However, for the case in which F can be embedded into a continuous semigroup S = {Ft }, it is not clear whether ϕ is a conjugation for the whole semiﬂow S and the ﬂow produced by G. It is natural to expect a more precise result under stronger requirements. A direct consequence of the proof of our Theorem 8.1 is the following assertion for conjugations. Proposition 8.1. Let F ⊂ Hol(Δ) be embedded into a semiﬂow S = {Ft }t≥0 of hyperbolic type and let η ∈ ∂Δ be a repelling ﬁxed point of F with A = F (η) > 1. Then for each B ≥ A and the automorphism G(= GB ) ∈ Aut(Δ) deﬁned by G(z) =

z+b , 1 + zb

where b = B−1 , there is a homeomorphism ϕ(= ϕB ) of Δ, ϕ ∈ Hol(Δ), such that B+1 ϕ(η) = −1 and ϕ(G(z)) = F (ϕ(z)), z ∈ Δ. Moreover, for all t ∈ R and w ∈ ϕ(Δ), the ﬂow {Ft (w)}t∈R is well-deﬁned with F1 = F and Ft (ϕ(z)) = ϕ(Gt (z)), for all t ∈ R, where Gt (z) =

z + 1 + e−αt (z − 1) , z + 1 − e−αt (z − 1)

t ∈ R,

with α = log B. In addition, ϕB (Δ) ⊆ ϕA (Δ), with ϕA (Δ) = ϕB (Δ) if and only if A = B.

218

Chapter 8. Backward Flow Invariant Domains for Semigroups Our approach to construct conjugations is diﬀerent from that used in [107].

Further, following the work of Baker [11], Karlin and McGregor [90] considered the local embedding problem of holomorphic functions with two ﬁxed points into a continuous group. In particular, they studied a class L of functions holomorphic in the extended complex plane C except for an at most countable closed set in C and proved the following result. • Let F be a function of class L with two ﬁxed points z0 and z1 , such that the segment [z0 , z1 ] is in the domain of regularity of F and is mapped onto itself. Assume that 0 < |F (z0 )| < 1 < |F (z1 )| and that for z in the open segment (z0 , z1 ), F (z) = z, F (z) = 0. Then there is a continuous one-parameter group {Ft }t∈R of functions with common ﬁxed points z0 and z1 and invariant segment [z0 , z1 ] such that F1 (z) = F (z) if and only if F (z) is a linear fractional transformation on C. First we note that the condition that F map [z0 , z1 ] into itself implies that F (z) is real on this segment. Suppose now that F is linear fractional, F (z) ≡ z, and let z0 and z1 be its ﬁnite ﬁxed points, z0 = z1 . The following simple assertion can be obtained by using the linear model of mappings having two ﬁxed points 0 and ∞ and applying the Julia–Carath´eodory theorem. Lemma 8.3. The following are equivalent. (i) There is an open disk D such that either z0 ∈ ∂D and z1 ∈ D, or z0 ∈ D and z1 ∈ ∂D, which is F -invariant. (ii) Each open disk D such that z0 ∈ D and z1 ∈ / D is F -invariant. (iii) The segment [z0 , z1 ] is F -invariant and |F (z0 )| ≤ 1. (iv) If a = F (z0 ) then 0 < a < 1. Since Schr¨oder’s equation h(F (z)) = F (z0 )h(z),

(8.5.10)

with linear-fractional F has a linear-fractional solution h, we have that h is starlike; hence F can be embedded into a one-parameter semigroup {Ft }t∈R on each disk D containing z0 and such that z1 ∈ / D. This disk is Ft -invariant for all t ≥ 0. In turn, for the embedding property into a continuous group, we obtain the following assertion by using our Theorems 8.1 and 8.2 and Theorem 1 in [90]. Proposition 8.2. Let F be a function of class L with two diﬀerent ﬁxed points z0 and z1 . Assume that 0 < |F (z0 )| < 1 < |F (z1 )|, and that for all z in the open segment (z0 , z1 ), F (z) = z, F (z) = 0. The following assertions are equivalent. (i) For each open disk D such that z0 ∈ D and z1 ∈ / D, there is a semiﬂow S = {Ft }t≥0 with F1 = F such that S ⊂ Hol(D).

8.5. Additional remarks

219

(ii) For each domain Ω bounded by two circles passing through z0 and z1 , there is a one-parameter ﬂow S = {Ft }t∈R such that S ⊂ Aut(Ω) and F = F1 . (iii) The function F is linear fractional with 0 < F (z0 ) < 1. Consequently, in this case, for any disk D such that z0 ∈ D and z1 ∈ ∂D, the maximal (backward) ﬂow-invariant domain is the disk Ω ⊂ D whose boundary passes through z0 and is internally tangent to ∂D at z1 .

Chapter 9

Appendices 9.1 Controlled Approximation Problems In this section, we use autonomic dynamical systems to study approximation problems for starlike and spirallike functions with respect to a boundary point and for some related classes of functions. Since such systems are time-independent, their solutions form one-parameter semigroups of holomorphic self-mappings of the open unit disk.

9.1.1 Setting of approximation problems 1. A perturbation problem for spirallike function. The ﬁrst approximation problem we discuss here can be described as follows. Let f ∈ G + [1], and let h ∈ Spiral[1] be a spirallike function which, by Theeorem 2.14, satisﬁes the equation μh(z) = h (z)f (z).

(9.1.1)

Consider the perturbed equation μτ hτ (z) = hτ (z)fτ (z), where fτ ∈ G[τ ], i.e., fτ ∈ G, fτ (τ ) = 0 for some τ ∈ Δ, and μτ = fτ (τ ), are such that fτ converges to f locally uniformly on Δ when τ goes to 1 unrestrictedly. (Note that such a perturbation is always possible, see, for example, the Berkson– Porta representation (2.2.7).) We ask: ♦ Does the net {hτ } converge to h as τ → 1? The following simple example shows that, in general, the answer is negative. Example 9.1. Let f (z) = (z − 1) ∈ G + [1]. Then the function h(z) = 1 − z satisﬁes equation (9.1.1) with μ = 1: h(z) = h (z)f (z).

222

Chapter 9. Appendices

(z − τ )(1 − zτ ) , τ ∈ (0, 1). Obviously, fτ ∈ G[τ ] (by Theorem 1−z 2.7), and fτ converges to f as τ → 1− . Consider the perturbed problem Deﬁne now fτ (z) =

μτ hτ (z) = hτ (z)fτ (z),

(9.1.2)

where μτ = fτ (τ ) = 1 + τ, fτ → f as τ → 1− . Then the function 1

hτ (z) =

(τ − z)(1 − zτ ) τ τ

is a solution of equation (9.1.2) satisfying hτ (0) = h(0) = 1. Letting τ tend to the boundary point 1, we obtain that the limit function lim hτ (z) = (1 − z)2 ,

τ →1−

which is diﬀerent from h(z). At the same time if we choose fτ ∈ G[τ ] in a diﬀerent way, say fτ (z) = z − τ , we see that hτ = 1 − τz deﬁned as a solution of equation (9.1.2) converges to h. Thus one can consider the following perturbation problem. ♦ For any τ ∈ Δ, ﬁnd a perturbed function fτ ∈ G[τ ] converging to f as τ goes to 1 unrestrictedly and such that the solution of (9.1.2) converges to the original solution of equation (9.1.1) uniformly on compact subsets of Δ. Geometrically, an aﬃrmative answer to this question would give us a constructive method for approximation of spirallike (starlike) functions with respect to a boundary point by spirallike (starlike) functions with respect to interior points. We solve this problem as follows. Given τ ∈ Δ, we ﬁnd a transformation Φτ : Hol(Δ, C) → Hol(Δ, C) which takes h ∈ Spiral[1] to hτ ∈ Spiral[τ ], τ ∈ Δ (i.e., hτ = Φτ (h)), and such that Φτ (h) tends to h when τ tends to 1 (see Theorem 9.4). To do this, we need ﬁrst to consider some approximation and interpolation problems for the classes G and P which are of independent interest. 2. An approximation problem for generators. As we already saw in Section 5.4 a solution of the diﬀerential equation βh(z) = h (z)f (z),

β = f (1) > 0,

belongs to the class Star1 [1]. Therefore, by Proposition 4.2, if a solution h of equation (9.1.1) isunivalent, then μ must lie in the region Ω = w = 0 : |w − β| ≤ β, or |w + β| ≤ β . Actually, the instability phenomenon we have seen in Example 9.1 above follows from the fact that μτ = fτ (τ ) does not necessarily converge to μ as τ → 1 even if μ = f (1). Therefore, one can pose the following problem on controlled approximation for functions in the class Gh [1] ⊂ G[1].

9.1. Controlled Approximation Problems

223

♦ Let f ∈ Gh [1] with f (1) = β > 0. For τ ∈ Δ and given μ ∈ Ω+ := {w = 0 : |w − β| ≤ β}, ﬁnd a net {fτ }, fτ ∈ G[τ ], converging locally uniformly to f as τ → 1 unrestrictedly and such that μτ = fτ (τ ) converges to μ. 3. An approximation problem for semigroups. For each f ∈ G ∗ [τ ] with Re f (τ ) > 0, τ ∈ Δ, the generated semigroup S = {Ft }t≥0 converges to the point τ as t → ∞. Moreover, the rate of convergence of the semigroup can be estimated by the derivative f (τ ) of the generator f at the point τ (for the boundary case τ ∈ ∂Δ, this fact follows by Theorem 2.10; for the general case τ ∈ Δ, we refer to [57], see also [130]). Note that the number μ in equation (9.1.1) is not necessarily real, while the angular derivative β = f (1) is a nonnegative real number. So, in light of the instability phenomenon mentioned above, the following question seems to be natural. ♦ Let f ∈ Gh [1] generate the semigroup S = {Ft }t≥0 . Suppose that a net {fτ }, fτ ∈ G[τ ], τ ∈ Δ, converges to f locally uniformly as τ → 1. Do the corresponding elements Ft,τ , t ≥ 0, of the semigroups Sτ generated by fτ converge to Ft for all t ≥ 0 as τ tends to 1? Here the diﬃculty is that the value Ft,τ (z) must lie in a neighborhood of τ ∈ Δ for any z ∈ Δ and for t suﬃciently large, while Ft (z) is close to 1. Nevertheless, we shall show that for each r ∈ (0, 1) and T > 0, the net {Ft,τ (z)} converges to Ft (z) uniformly on the set [0, T ] × (rΔ). 4. An approximation problem for functions with positive real part. By the Berkson–Porta representation (see Theorem 2.7), each cone G + [τ ], τ ∈ ∂Δ, or G[τ ], τ ∈ Δ, can be parameterized by elements of the cone P of functions with positive real part. It turns out that the approximation problems above lead us to the following interpolation question about the approximation of functions of class P. Recall that a holomorphic function f ∈ Hol(Δ, C) is said to be conformal at a boundary point τ ∈ ∂Δ if its angular derivative f (τ ) exists ﬁnitely and f (τ ) = 0. ♦ Let q ∈ P be conformal at the boundary point 1. For τ ∈ Δ and given ϕ ∈ − π2 , π2 , does there exist a net {qτ } ⊂ P converging locally uniformly to q as τ → 1 unrestrictedly and such that arg qτ (τ ) = ϕ for all τ ∈ Δ.

9.1.2 Solutions of approximation problems Recall that by the Riesz–Herglotz formula (1.4.2) p(z) = ∂Δ

1 + zζ dσp (ζ) + i Im p(0), 1 − zζ

224

Chapter 9. Appendices

the angular limit δp (τ ) = ∠ lim (1 − zτ )p(z) z→1

(9.1.3)

exists and is a nonnegative real number. We call this number the charge of the function p ∈ P at the boundary point τ ∈ ∂Δ (see Section 1.4 for details). It follows by Theorem 1.14 that 1 − |z|2 Re p(z). z∈Δ |1 − z|2

δp (τ ) = 2 inf

As above, we denote by P + [τ ] the subclass of P consisting of functions with strictly positive charges at τ ∈ ∂Δ, so, p ∈ P + [1] if and only if δp (1) > 0. Thus, p ∈ P + [1] if and only if the function q ∈ P deﬁned by q = p1 is conformal at τ = 1 with q(1) = 0. Moreover, in this case, q (1) is a negative real number. The following assertion, which gives a solution to the problem in Section 9.1.1 (4), is the key for our further considerations. Theorem 9.1. Let q ∈ P be conformal at τ = 1 with q(1) = 0. Then for all τ ∈ Δ and each γ ∈ C such that Re γ ≥

α , 2

where α = −q (1) (> 0) ,

(9.1.4)

there exist functions {qτ }τ ∈Δ ⊂ P converging to q uniformly on compact subsets of Δ when τ tends to 1 unrestrictedly and such that qτ (τ ) = γ(1 − |τ |2 ) → 0 as τ → 1. In particular, if γ is real and γ ≥

α , the values qτ (τ ) are real numbers. 2

Proof. For γ ∈ C, consider the function r(z) =

zq(z) + γz 2 − γ¯ − 2iz Im γ . −(1 − z)2

(9.1.5)

Clearly, r ∈ Hol(Δ, C) and r(0) = γ¯. We claim that r ∈ P if and only if γ satisﬁes inequality (9.1.4). Indeed, consider the function f (z) = zq(z) + γz 2 − γ¯ − 2iz Im γ, which is the numerator of (9.1.5). It follows by Theorem 2.6 that f ∈ G. In addition, f (1) = ∠ lim f (z) = γ − γ¯ − 2i Im γ = 0 z→1

(9.1.6)

9.1. Controlled Approximation Problems

225

and f (z) f (1) = ∠ lim z→1 z − 1 γz 2 − γ¯ − 2iz Im γ = q (1) + 2 Re γ. = q (1) + ∠ lim z→1 z−1

Thus, f (1) = 2 Re γ − α ≥ 0

(9.1.7)

if and only if condition (9.1.4) holds. If it is the case, the Julia–Wolﬀ–Carath´eodory Theorem 2.10 implies that the point τ = 1 is the Denjoy–Wolﬀ point of the semigroup generated by f . On the other hand, by the uniqueness of the Berkson– Porta representation (see Theorem 2.7) of the class G, the inequality in (9.1.7) is equivalent to f (z) Re ≥ 0. −(1 − z)2 Comparing (9.1.7) with (9.1.5) and (9.1.6) proves our claim. Now for τ ∈ Δ, deﬁne the function qτ on Δ by qτ (z) =

4 13 (z − τ ) (1 − z τ¯) r(z) + γ¯ τ − γ τ¯z 2 + 2iz Im γ z

(9.1.8)

or qτ (z) =

1 gτ (z), z

(9.1.9)

where gτ (z) = (z − τ ) (1 − z τ¯) r(z) + γ¯ τ − γ τ¯z 2 + 2iz Im γ.

(9.1.10)

Since r(0) = γ¯, we have gτ (0) = 0. Hence, qτ is holomorphic in Δ. Moreover, since G is a real cone, we have by Theorems 2.6 and 2.7 that gτ ∈ G for all τ ∈ Δ. Consequently, gτ ∈ G[0]. gτ (z) Again by Theorem 2.7, Re qτ (z) = Re ≥ 0 for all z ∈ Δ. z In addition, since 1 (9.1.11) q(z) = g(z), z where 2

g(z) = − (1 − z) r(z) − γz 2 + γ¯ + 2iz Im γ,

(9.1.12)

we have by (9.1.9) and (9.1.10) that qτ (z) converges to q(z) as τ tends to 1 unrestrictedly. Finally, direct calculation shows that qτ (τ ) = γ 1 − |τ |2 .

226

Chapter 9. Appendices

Example 9.2. Consider the function q(z) = 1 − z, which obviously has positive real part. Then the function r deﬁned by (9.1.5) with γ = 1 has the form r(z) =

1 . 1−z

Substituting this function into formula (9.1.8), we ﬁnd the approximating functions qτ (z) = τ¯(1 − z) +

|1 − τ |2 . 1−z

We see that qτ → q as τ tends to 1 unrestrictedly. (1) (i) Choosing the sequence of real numbers τn = 1 − n1 , we get the approxi(1) mating sequence qn of functions with positive real part. In Figure 9.1, we see the (1) (1) (1) images of the unit circle ∂Δ under the approximating functions q1 , q2 and q4 (1) as well as the image of q. It is worth noting that q1 (Δ) = {w : Re w > 12 } and (1) (1) (1) that the images qn (Δ) increase in the sense that qn (Δ) ⊂ qn+1 (Δ). As n tends to inﬁnity, these images tend to the right half-plane, while the original function q is bounded. (Note that the Carath´eodory Kernel Theorem is not applicable here since the functions qn are not univalent.) (2) (2) (ii) Choosing another sequence τn converging to 1, say, τn = 1 − 3(1−i) n , (2) we get a diﬀerent approximating sequence qn of positive real part functions. In (2) (2) Figure 9.2, we see the images of the unit circle ∂Δ under the functions q4 , q6 (2) and q12 . Once again, in spite of the boundedness of q, for n large enough the (2) image qn (Δ) almost covers the right half-plane. Corollary 9.1. Let p ∈ P + [1] with the charge δp (1) = β > 0. Then for all τ ∈ Δ and each μ ∈ C such that |μ − β| ≤ β, μ = 0, (9.1.13) there exist functions {pτ }τ ∈Δ ⊂ P which converge to p uniformly on compact subsets of Δ as τ tends to 1 and such that μ pτ (τ ) = → ∞ as τ → 1. (9.1.14) 1 − |τ |2 In particular, if μ is real and 0 < μ ≤ 2β, the values pτ (τ ) are real numbers. Proof. Setting q(z) =

1 , p(z)

we have q ∈ P with q(1) = 0 and q (1) = ∠ lim

z→1

q(z) 1 1 = ∠ lim = − (:= −α) . z→1 (z − 1)p(z) z−1 β

9.1. Controlled Approximation Problems

227

3

0.6

2

0.4

1

0.2

0

0.3

0.4

0.5

0.6

0

0.7

–1

–0.2

–2

–0.4

–3

–0.6

0.2

0.4

0.6

0.8

1

1

0.8

0.6 0.5

0.4

0.2

0

0.2

0.4

0.6

0.8

1

1.2

0

1.4

0.5

1

1.5

2

–0.2

–0.4

–0.5

–0.6

–0.8

–1

(1)

(1)

(1)

Figure 9.1: Example 9.2 (i), the images of q1 , q2 , q4

and of q.

Using Theorem 9.1 for each γ ∈ C such that 2 Re γ ≥

1 β

(9.1.15)

and all τ ∈ Δ, one can ﬁnd qτ ∈ P converging to q as τ → 1 and such that qτ (τ ) = γ 1 − |τ |2 . Setting now pτ (z) =

1 qτ (z)

and γ = μ1 , we obtain that (9.1.13) is equivalent to (9.1.15) and pτ converges locally uniformly to p.

228

Chapter 9. Appendices

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0.2

0 0.2

0.4

0.6

0.8

1

1.2

1.4

0 –0.2

–0.5

–0.4 –0.6

–1

–0.8 –1

–1.5

–1.2 –2

–1.4

1

0.5

0

0.5

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0

–0.5

–0.5

–1

–1

0.5

1

(2)

(2)

1.5

2

(2)

Figure 9.2: Example 9.2 (ii), the images of q4 , q6 , q12 . Finally, pτ (τ ) = and we are done.

1 μ 1 = = , 2 qτ (τ ) γ (1 − |τ | ) 1 − |τ |2

It is now easy to prove the following theorem, which solves the problem in Section 9.1.1 (2). Theorem 9.2. Let f ∈ G satisfy the conditions f (1) = 0 and f (1) = β > 0, i.e., f ∈ Gh [1]. Then for each μ ∈ Ω+ = {w = 0 : |w − β| ≤ β} and for all τ ∈ Δ, there exist generators {fτ } with fτ ∈ G[τ ] such that fτ (τ ) = μ and fτ converges to f uniformly on compact subsets of Δ as τ tends to 1 unrestrictedly. In particular, setting μ = β we have fτ (τ ) = β ∈ (0, ∞) for all τ ∈ Δ.

9.1. Controlled Approximation Problems

229

Proof. By the Berkson–Porta formula (2.2.7), f must be of the form f (z) = − (1 − z)2 p(z),

(9.1.16)

where p ∈ P + [1] with δp (1) = β > 0. By Corollary 9.1, for each μ satisfying (9.1.13) and τ ∈ Δ, one can ﬁnd μ and such that pτ converges to p as τ → 1. pτ ∈ P with pτ (τ ) = 1 − |τ |2 Now we construct fτ ∈ G by the Berkson–Porta representation (2.2.7) fτ (z) = (z − τ ) (1 − z τ¯) pτ (z).

Since fτ (τ ) = 1 − |τ |

2

pτ (τ ), we obtain our assertion.

In some sense a converse assertion is also true. Theorem 9.3. Let f ∈ Gh [1] with f (1) = 0 and f (1) = β > 0, and let fn ∈ G[τn ], τn ∈ Δ, n = 1, 2, . . .. Suppose that fn converges to f uniformly on compact subsets of Δ. Then (i) the sequence {τn } converges to 1; (ii) if the sequence μn := fn (τn ) converges to μ = 0, then μ must satisfy condition (9.1.13), i.e., |μ − β| ≤ β. Proof. (i) For each n = 1, 2, . . . , the function fn ∈ G[τn ] has the form fn (z) = (z − τn ) (1 − zτn ) pn (z) with Re pn (z) ≥ 0 everywhere. Since Δ is a compact subset of C and {pn } is a normal family on Δ, one can choose a subsequence {nk } ⊂ N such that τnk converges to τ ∈ Δ and pnk converges either to p ∈ P or p = ∞. In any case, the sequence {fnk } ⊂ G converges to either f (z) = (z − τ ) (1 − z τ¯) p(z) with Re p(z) ≥ 0 and τ ∈ Δ or to inﬁnity. Since the latter case is impossible, the uniqueness of the Berkson–Porta representation and (9.1.16) imply that τ = 1 and p = p. (ii) Assume now that μn := fn (τn ) converges to μ ∈ C, μ = 0, and consider the diﬀerential equations μn hn (z) = hn (z)fn (z), normalized by the conditions hn (0) = 1. This initial value problem has a unique solution which is a univalent function on Δ spirallike with respect to an interior point and hn (τn ) = 0 ∈ hn (Δ). Explicitly, hn can be written as

z dz hn (z) = exp μn . (9.1.17) 0 fn (z) Now for each r ∈ (0, 1), one can ﬁnd n0 ∈ N such that for all n > n0 , the points τn do not belong to the closed disk Δr = {|z| ≤ r < 1}. In other words, for all

230

Chapter 9. Appendices

n > n0 the functions fn do not vanish in this disk. Then the functions hn deﬁned by (9.1.17) converge to a function h ∈ Hol(Δr , C) uniformly on this disk. Since r is arbitrary, we have that actually h ∈ Hol(Δ, C) and has the form

z dz . (9.1.18) h(z) = exp μ 0 f (z) In addition, it follows by Hurwitz’s theorem that h is either a univalent function h (z) on Δ or a constant. The latter case is impossible because of the equality = h(z) μ , which follows by (9.1.18) and μ = 0. f (z) On the other hand, we already know that the initial value problem β h(z) = h (z), f (z),

h(0) = 1

has also the unique solution

h(z) = exp β 0

z

dz , f (z)

(9.1.19)

which is a starlike function with respect to the boundary point h(1) = 0 . Comparing (9.1.18) and (9.1.19), we obtain

μβ h(z) = h(z) . In addition, h satisﬁes the Visser–Ostrowski condition: ∠ lim

z→1

(z − 1)h (z) h(z)

= 1.

Now Theorem 3.3 implies that the smallest wedge which contains h(Δ) is exactly of β 1 angle π. Since h is univalent, we get that Re ≥ , which is equivalent to (9.1.13). μ 2 As a consequence of Theorems 3.2 and 3.3, we obtain the following result. Theorem 9.4. Let f ∈ Gh [1] with f (1) = β > 0. Then the initial value problem λh(z) = h (z)f (z),

h(0) = 1

(9.1.20)

has a univalent solution h ∈ Hol(Δ, C) if and only if the complex number λ belongs to the set Ω = Ω+ ∪ Ω− , where Ω± = {ω ∈ C : |ω ∓ β| ≤ β, ω = 0} . Moreover,

9.1. Controlled Approximation Problems

231

• for each μ ∈ Ω+ and each τ ∈ Δ, there are hτ ∈ Spiral[τ ] which converge to the function μ h1 (z) = h λ (z) as τ tends to 1 unrestrictedly. In particular, if λ ∈ Ω+ , then choosing μ = λ, we ﬁnd hτ ∈ Spiral[τ ] which converge to the original function h as τ → 1. • for each μ ∈ Ω− and for each τ ∈ Δ, there are meromorphic functions hτ with a unique simple pole at τ and such that hτ converges to the holomorphic function μ h(z) = h λ (z) (9.1.21) as τ tends to 1 unrestrictedly. Proof. Take any μ ∈ Ω+ and any τ ∈ Δ. By Theorem 9.2, one can choose generators fτ ∈ G[τ ] such that the net {fτ } converges uniformly on compact subsets to f and satisﬁes the conditions fτ (τ ) = 0 and fτ (τ ) = μ. Then, as in the proof of the second part of Theorem 9.3, one shows that functions hτ ∈ Spiral[τ ] deﬁned by

z dz hτ = exp μ 0 fτ (z) converge to a univalent function

z dz (9.1.22) h1 (z) = exp μ 0 f (z) which satisﬁes the equation μh1 (z) = h1 (z)f (z).

(9.1.23)

If now λ ∈ Ω+ , then setting μ = λ, we see that h = h1 must be univalent. If λ ∈ Ω− , then setting μ = −λ, and comparing diﬀerential equations (9.1.20) and (9.1.23), we see that h = h−1 1 . Since h1 (z) = 0, z ∈ Δ, h is a well-deﬁned univalent function on Δ. In addition, it is clear that h−1 1 is a locally uniform limit of meromorphic functions h−1 τ with poles at τ . Conversely, assume that for some λ ∈ C, λ = 0, equation (9.1.20) has a univalent solution in Δ. If Re λ > 0, then setting μ = β in (9.1.22), we see as in β the proof of Theorem 9.3 that the image of the function h1 (z) = h λ (z) must lie in the wedge of the angle π, which is the smallest one containing h1 (Δ). Then, by Proposition 4.2, we have λ ∈ Ω+ . If Re λ < 0, then the same considerations show that −λ ∈ Ω+ , and we are done.

9.1.3 Perturbation formulas

Theorem 9.5. Let f ∈ Gh [1] with f (1) = β > 0 and let λ ∈ Ω+ = w ∈ C : |w − β| ≤ β, w = 0 . Assume that h ∈ Hol(Δ, C) is the solution of equation (9.1.20) λh(z) = h (z)f (z)

232

Chapter 9. Appendices

normalized by the conditions h(0) = 1 and h(1) = 0. Then for each μ ∈ Ω+ and for each τ ∈ Δ, the function hτ ∈ Hol(Δ, C) deﬁned by μ

hτ (z) = [h(z)] λ

(z − τ )(1 − z τ¯)μ/¯μ −(1 − z)1+μ/¯μ

is univalent on Δ and belongs to the class Spiral[τ ] with hτ (τ ) = 0 and hτ (0) = τ. If, in particular, μ = λ, then hτ converges to h whenever τ tends to 1. Thus h is a univalent function on Δ spirallike (starlike) with respect to a boundary point with h(1) = 0. Proof. Let h ∈ Hol(Δ, C) be a solution of the diﬀerential equation (9.1.20) with −(1 − z)2 f ∈ Gh [1] deﬁned by f (z) = , where q(z) Re q(z) > 0 and ∠ lim

z→1

1−z = β. q(z)

Then h (z) q(z) . = λh(z) −(1 − z)2

(9.1.24)

As in the proof of Theorem 9.1, for γ complex with 2 Re γ ≥ r ∈ P + [1] by r(z) =

zq(z) + (γz + γ¯)(z − 1) . −(1 − z)2

1 β

we deﬁne

(9.1.25)

Comparing (9.1.24) and (9.1.25), we have r(z) =

zh (z) γ z + γ¯ + . λh(z) 1−z

(9.1.26)

Once again, for each point τ ∈ Δ, we deﬁne qτ (z) =

4 13 (z − τ )(1 − z τ¯)r(z) + γ¯ τ − γ τ¯z 2 + (γ − γ¯ )z . z

Then the diﬀerential equation μhτ (z) = hτ (z)(z − τ )(1 − z τ¯)

1 qτ (z)

(9.1.27)

1 ∈ Ω+ . Moreover, hτ ∈ Spiral[τ ] γ hτ (z) is wellby Theorems 2.7 and 2.14. Since hτ (τ ) = 0, the function gτ (z) := z−τ has a holomorphic solution hτ if and only if μ =

9.1. Controlled Approximation Problems

233

deﬁned. Now, using (9.1.26) and (9.1.27), we calculate

h (z) γz + γ¯ γ¯ gτ (z) =μ + − gτ (z) λh(z) z(1 − z) z(1 − z τ¯)

h (z) γ γ¯ γ¯ (−¯ τ) =μ + + + λh(z) 1 − z 1−z 1 − z τ¯ 1 μ¯ γ μ¯ γ (−¯ τ) μh (z) + + + . = λh(z) 1−z 1−z 1 − z τ¯ Then μ

μ λ

hτ (z) = (z − τ )gτ (z) = C (h(z)) ·

(z − τ )(1 − z τ¯) μ¯ μ

(1 − z)1+ μ¯

with some constant C. So, to satisfy the normalization hτ (0) = τ , we must set C = −1. The proof is complete. Example 9.3. Consider the starlike function with respect to a boundary point h(z) = (1 − z)0.8 . It satisﬁes equation (9.1.20) with f (z) = z − 1 and λ = 0.8: 0.8h(z) = h (z) · (z − 1). Setting μ = λ = 0.8, we see that by Theorem 9.5, h can be approximated by the functions (τ − z)(1 − z τ¯) . hτ (z) = (1 − z)1.2 (1)

(i) Choosing, in particular, the sequence of real numbers τn = 1 − n3 , we get (1) the approximating sequence hn of starlike functions with respect to (diﬀerent) interior points. In Figure 9.3, we see the images of h as well as the images of the (1) (1) (1) approximating functions h6 , h10 and h30 . Note that in this case, the intersection 2∞ (1) n=0 hn (Δ) contains the left half-plane, while the image h(Δ) lies in the right half-plane. (2) (2) (ii) Choosing another sequence τn converging to 1, say, τn = 1 − (1 − i) n3 , (2) we get a diﬀerent approximating sequence hn of starlike functions with respect to interior points. In Figure 9.4, we see the images of the approximating functions (2) (2) (2) (2) h6 , h10 and h30 . Once again, all of the images hn (Δ) contain the left half-plane. (3)

(iii) On the other hand, setting μ = 1 + i and choosing τn = 1 − n3 , we (3) ﬁnd the sequence hn of spirallike functions with respect to interior points, which converges to the function 1+i

h(z) = (h(z)) 0.8 = (1 − z)1+i ,

234

Chapter 9. Appendices

0.8 1 0.6

0.4 0.5 0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0

1.6

0.5

0.6

0.7

0.8

0.9

–0.2 –0.5 –0.4

–0.6 –1 –0.8

1

1

0.5

0.5

0

0.4

0.6

0.8

1

1.2

0

–0.5

–0.5

–1

–1

0.2

0.4

0.6

0.8

(1)

1

(1)

1.2

1.4

1.6

(1)

Figure 9.3: Example 9.3 (i), the images of h, h6 , h10 and h30 . which is spirallike with respect to a boundary point. In Figure 9.5, one can see images of a number of approximating functions and the image of h. Using Theorem 9.4 above, one can prove the uniform stability of a semigroup generated by f ∈ Gh [1] under any perturbation of its generator (see Problem in Section 9.1.1 (3)). Let {Ft }t≥0 be a semigroup generated by f ∈ Gh [1] and let fn ∈ G[τn ], τn ∈

9.1. Controlled Approximation Problems

2.5

235

2

2 1.5 1.5 1 1 0.5 0.5

0

0.2

0.4

0.6

0.8

1

0

1.2

0.2

0.4

0.6

0.8

1

1.2

–0.5 –0.5 –1

1 0.8 0.6 0.4 0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

(2)

(2)

–0.2 –0.4 –0.6

(2)

Figure 9.4: Example 9.3 (ii), the images of h6 , h10 and h30 .

236

Chapter 9. Appendices

1

0.5

0.5

–0.5

0.5

1

–0.5

1.5

0.5

1

1.5

2

0

0

–0.5 –0.5

–1 –1 –1.5 –1.5

1

1

0.5 –1

0.5

–0.5

0.5

1

1.5

2

2.5

–1

1

2

3

0 –0.5

–0.5

–1

–1

–1.5 –1.5 –2 –2 –2.5

(3)

(3)

(3)

Figure 9.5: Example 9.3 (iii), the images of h6 , h10 , h30 and h.

Δ, bea sequence of generators converging to f uniformly on compact subsets of (n) (n) is a semigroup generated by fn , then it follows that Ft (z) → τn Δ. If Ft t≥0

for each z as t tends to ∞. In addition, since fn and f are locally Lipschitzian on Δ, the uniqueness property of the Cauchy problem implies that for each a ∈ Δ, there exist r > 0 and T (= T (r)) such that for each z from the circle |z − a| < r (n) and each t ∈ [0, T (r)], the sequence of values Ft (z) converges to Ft (z) as n tends to inﬁnity. The question is: ♦ Whether this convergence is uniform on each compact subset of Δ × R+ , in other words, whether for each 0 < r < 1 and each 0 < T < ∞ which does not

9.1. Controlled Approximation Problems

237

(n)

depend on r, the sequence Ft converges to Ft uniformly on the set Δr × [0, T ], where Δr = {z ∈ C : |z| ≤ r}? We now show that the answer is aﬃrmative. Theorem 9.6. Let {Ft }t≥0 be a semigroup generated by f ∈ Gh [1], and let fn ⊂ G be a sequence of generators converging to f uniformly on compact subsets of Δ such that fn(τn ) = 0 with τn ∈ Δ. Assume also that the closure of the set

{μn = fn (τn )} does not contain the origin. Then the sequence

(n)

Ft

(z)

of t≥0

semigroups generated by fn converges to Ft (z) uniformly on compact subsets of Δ × R+ . Proof. By passing to a subsequence, if necessary, we may suppose that the sequence μn = fn (τn ) is convergent, say, to μ ∈ C. It follows from our assumption that μ = 0 and from Theorem 9.3 that μ must satisfy the condition |μ − β| ≤ β. Then the sequence {hn } deﬁned by

z dz hn (z) = exp μn 0 fτn (z) converges to the function h1 deﬁned by (9.1.22). In addition, we have (n)

Ft

−tμn e (z) = h−1 hn (z) . n

(n)

Thus Ft (z) converges uniformly on compact subsets of R+ × Δ to a function G : R+ × Δ → Δ deﬁned by −tμ e h1 (z) . G(t, z) = h−1 (9.1.28) 1 On the other hand, we already know that if h is a solution of the equation βh(z) = h (z)f (z), then Ft (z) = h−1 e−tβ h(z) (9.1.29) and

μ

h1 = h β .

(9.1.30)

Then we get from (9.1.28) and (9.1.30) that μ

μ

h β (G(t, z)) = e−tμ h β (z) or h (G(t, z)) = e−tβ h(z). The latter equation and (9.1.29) imply that G(t, z) = Ft (z) for all z ∈ Δ and t ∈ R+ .

238

Chapter 9. Appendices

Since Gh [1] is a real cone, we can always consider equation (9.1.20) with f normalized by f (1) = 1. Then Deﬁnition 4.3 can be reformulated as follows: A function h ∈ Hol(Δ, C) belongs to the class Spiralλ [1] if it satisﬁes the equation λh(z) = h (z)f (z),

(9.1.31)

where f ∈ Gh [1] with f (1) = 1 and λ ∈ Ω+ = {w : |w − 1| ≤ 1, w = 0}. Theorem 9.7. Let λ ∈ Ω+ = {w : |w − 1| ≤ 1} , λ = 0. Then a function h ∈ Hol(Δ, C) belongs to the class Spiralλ [1] if and only if it admits the representation

λ 2λ h∗ (z) , (9.1.32) h(z) = (1 − z) z where h∗ is a starlike function of class S ∗ which satisﬁes the condition zh∗ (z) 1 − |z|2 Re 2 inf = 1. |1 − z|2 h∗ (z) Proof. Let h ∈ Spiral[1] be λ-spirallike. Then it satisﬁes the equation 1 λh (z) = h(z)f (z) = −h(z)(1 − z)2 · , q(z) where f (z) ∈ Gh [1] with f (1) = 1 (or, which is the same, q (1) = −1 and q ∈ P). By Theorem 9.5, the functions hτ ∈ Hol(Δ, C) deﬁned by μ

μ λ

hτ (z) = h (z)

(z − τ )(1 − z τ¯) μ¯ μ

−(1 − z)1+ μ¯

are spirallike with respect to an interior point (h(τ ) = 0) and satisfy the equations μhτ (z) = hτ (z)(z − τ )(1 − z τ¯)

1 , qτ (z)

where qτ ∈ P is deﬁned by the formula qτ (z) =

4 1 3 (z − τ )(1 − z τ¯)r(z) + γ¯τ − γ τ¯z 2 + 2iz Im γ , z

where r(z) = 1

hλ ·

zq(z) + γz 2 − γ¯ − 2iz Im γ 1 , γ= . 2 −(1 − z) μ

If, in particular, we set μ = 1 and τ = 0, then we get γ = 1; and h0 (z) = z ∗ −(1−z)2 , hence h∗ = −h0 , is of the class S and satisﬁes the equation h∗ (z) = h∗ (z)f0 (z) = h∗ (z) ·

z , q0 (z)

9.1. Controlled Approximation Problems

239

where q0 (z) = r(z). Since zq(z) z2 − 1 + ∠ lim = 1, z→1 z − 1 z→1 z − 1

δr (1) = ∠ lim (1 − z)r(z) = ∠ lim z→1

we obtain Re q0 (z) = Re

1 1 − |z|2 zh∗ (z) ≥ . h∗ (z) 2 |1 − z|2

Conversely, let h ∈ Hol(Δ, C) admit the representation h(z) = (1 − z)2λ

h∗ (z) z

λ ,

where h∗ satisﬁes the required condition. Diﬀerentiating logarithmically, we obtain

1 h (z) 2 1 zh∗ (z) 2 1 =− + −1 =− + [q0 (z) − 1] . γ h(z) 1 − z z h∗ (z) 1−z z Consider the function q ∈ Hol(Δ, C) deﬁned by q(z) = −

(1 − z)2 h (z) . λ h(z)

We have to show that q ∈ P and q (1) = −1. This will imply that h satisﬁes 1 equation (9.1.31) with f (z) = −(1 − z)2 q(z) . Indeed, if g(z) = zq(z), we have g(z) = 1 − z 2 − (1 − z)2 q0 (z). Both terms in the right-hand side of this equality are in G. Hence g also belongs to G, since G is a real cone. Thus q must belong to P by the Berkson–Porta formula (2.2.7) with τ = 0. Finally, a direct calculation shows that q(z) 1 − z2 = ∠ lim + ∠ lim (1 − z)q0 (z) = −1, z→1 z − 1 z→1 z − 1 z→1

q (1) = ∠ lim

and we are done.

Deﬁne S ∗ as the subclass of S ∗ of all starlike functions h ∈ Hol(Δ, C), h(0) = 0, which satisfy the condition 2 inf

zh (z) 1 − |z|2 Re = 1. 2 |1 − z| h(z)

Corollary 9.2. For each λ ∈ Ω+ = {w : |w − 1| ≤ 1, w = 0}, the class S ∗ is homeomorphic to the class Spiralλ [1].

240

Chapter 9. Appendices

Using representation (9.1.32), one checks easily the following characterization of spirallike functions with respect to a boundary point (cf., Theorem 4.1). Corollary 9.3 (Generalized Robertson condition). A function h ∈ Hol(Δ, C) normalized by the condition h(0) = 1 is of class Spiral[1] if and only if, for some λ ∈ Ω+ = {w : |w − 1| ≤ 1, w = 0},

2z h (z) 1 + z + ≥ 0. Re λ h(z) 1−z

9.2 Weighted semigroups of composition operators Another useful application of the results given above is the following description of the spectrum of the inﬁnitesimal generator of a one-parameter semigroup of composition operators in the inﬁnite dimensional Frech´et space E = Hol(Δ, C). For f ∈ G, deﬁne a linear operator Γf on E by the formula (Γf h) (z) := h (z)f (z).

(9.2.33)

Recall that the spectrum σ (Γf ) of the operator Γf is the set of all complex numbers λ ∈ C for which λI − Γf is not continuously invertible, where I is the identity operator on Hol(Δ, C). The point spectrum σp (Γf ) of Γf is the subset of σ (Γf ) which consists of its eigenvalues, i.e., σp (Γf ) = {λ ∈ C : (λI − Γf ) h = 0 for some h = 0} . In this case, a nontrivial solution of the equation (λI − Γf ) h = 0 is called an eigenvector of Γf corresponding to eigenvalue λ. Generally speaking, in inﬁnite dimensional spaces, the spectrum of a linear operator does not coincide with its point spectrum. However, since f ∈ G + [1] does not vanish in Δ, we have σp (Γf ) = σ(Γf ) = C. This, for example, implies immediately that the linear semigroup generated by Γf (and hence, the nonlinear semigroup S = {Ft }t≥0 generated by f ) cannot be holomorphically extended into a sector containing the positive real axis (see, for example, [148, Chapter IX.10]). Over the past few decades a lot of works address a wide range of topics concerning properties of composition operators on classical Banach spaces of analytic functions. In particular, composition operators have been studied extensively in the setting of Hardy or Bergman spaces on Δ (see [40] and [127]). In the case of an interior null point of a generator, the eigenvalues of Γf form a discrete set, while in case of the boundary Denjoy–Wolﬀ point, the spectrum of Γf may be essentially larger. Indeed, as we have already seen in Theorem 9.4, the eigenfunction h corresponding to the eigenvalue 2β, where β = f (1), is univalent and starlike with respect to a boundary point. Then h ∈ H p for each p < 12 (see, for example, [72]).

9.2. Weighted semigroups of composition operators

241 2β

Hence, for each positive λ the function h1 (z) := (h(z)) λ ∈ H q when qλ < β. Obviously, h1 is an eigenfunction corresponding to λ. Therefore, for any Hardy β q space H we have σ(Γf ) ⊃ 0, q . This, inter alia, motivates consideration of the univalence of eigenfunctions of composition operators on the (locally convex) Fr´echet space Hol(Δ, C). In a slightly more general setting, we describe the structure of the spectrum of weighted composition operators in the context of the k-valency of the corresponding eigenfunctions. Namely, we use the following deﬁnition. Deﬁnition 9.1 (see [75], p. 89). A function f meromorphic in a domain D is said to be k-valent in D if for each w0 (inﬁnity included) the equation f (z) = w0 has at most k roots in D (where the roots are counted in accordance with their multiplicity) and if there is some w1 such that the equation f (z) = w1 has exactly k roots in D. Let E be a space of meromorphic functions in Δ, and let S = {Ft }t≥0 be a semigroup of holomorphic self-mappings of Δ generated by f ∈ Gh [1], f (1) = β. For a suitable w ∈ E, one can deﬁne a (weighted) composition semigroup of linear operators Tt : E → E, t ≥ 0, by the formula Tt (h)(z) =

w(Ft (z)) h(Ft (z)). w(z)

This semigroup is generated by the operator Γf deﬁned by Γf h = h f + hf

w , w

h ∈ Hol(Δ, C).

(9.2.34)

When w ≡ 1, this reduces to the unweighted semigroup of composition operators {Ct }t≥0 generated by Γf , where Ct (h)(z) := h(Ft (z)). It is clear that for each λ in the point spectrum σp (Γf ), the eigenspace Eλ 0 corresponding to λ is one-dimensional. For k ∈ N {∞}, denote by σ (k) the subset of σp (Γf ) such that for each λ ∈ σ(k) the function wh is k-valent whenever h ∈ Eλ . Theorem 9.8. Let f ∈ Gh [1], f (1) = β, and let the operator Γf : E → E be deﬁned by (9.2.34). Then the spectrum σp Γf is the whole complex plane C. Moreover, ! σ (k) , σp Γf = k∈N∪{∞}

where for k ∈ N, σ(k) = (kΩ+ ∪ kΩ− ) \ ((k − 1)Ω+ ∪ (k − 1)Ω− )

242

Chapter 9. Appendices

with Ω± = {ω = 0 : |ω ∓ β| ≤ β}, and σ (∞) = {λ ∈ C : Re λ = 0} . In addition, for each λ = 0 and for each element h of the eigenspace corresponding to λ, the following modiﬁed Robertson inequality holds:

2β zw (z) 2β zh (z) 1 + z + + > 0. (9.2.35) Re λ h(z) 1−z λ w(z) Figure 9.6 illustrates the sets σ (k) described in Theorem 9.8.

Figure 9.6: The sets σ(k) of k-valence of eigenfunctions. Proof. The eigenvalue problem for the operator Γf is the diﬀerential equation h f + hf

w = λh. w

(9.2.36)

To solve it, we denote by h∗ the starlike function with respect to a boundary point which satisﬁes βh∗ = h∗ f

(9.2.37)

and is normalized by h∗ (0) = 1, h∗ (1) = 0 (h∗ is the eigenfunction of Γf corresponding to the eigenvalue β). Then (9.2.36) and (9.2.37) imply that λ h∗ (wh) = , wh β h∗ and hence w(z)h(z) = a (h∗ (z))

λ/β

,

a ∈ C.

(9.2.38)

9.2. Weighted semigroups of composition operators

243

For a given eigenvalue λ, (9.2.38) describes the eigenspace corresponding to λ. So it is enough to prove our assertion for the case w ≡ 1 or, which is the same, Γ f = Γf . To this end, let λ ∈ C. First suppose that Re λ > 0 and λ ∈ kΩ+ \ (k − 1)Ω+ . Then λ1 = λk belongs to Ω+ . Thus, by Theorem 9.4, the solution h1 of the initial value problem h1 (z)f (z) = λ1 h1 (z),

h1 (0) = 1,

is univalent. Set h(z) = h1 (z)k . Evidently, this function satisﬁes the equation Γf h(z) (= h (z)f (z)) = λh(z), and is at most k-valent. To show that h is k-valent, consider the univalent solution h2 of the initial value problem h2 (z)f (z) = 2βh2 (z),

h2 (0) = 1.

This function is starlike with respect to a boundary point, and the smallest wedge which contains the image h2 (Δ) is exactly of angle 2π. Therefore, for each > 0, there is real ψ and r > 0 small enough such that the image h2 (Δ) contains the curve Im λ Im λ r exp φ exp iφ − i log r : ψ ≤ φ ≤ ψ + 2π(1 − ) . Re λ Re λ Re λ Since λ ∈ (k −1)Ω+ , one can choose < 1− 2β(k−1) . Then there are k diﬀerent |λ|2 points z0 , z1 , . . . , zk−1 in Δ for which Im λ Im λ h2 (zm ) = r exp φm exp iφm − i log r , Re λ Re λ 2β Re λ , m = 0, 1, . . . , k − 1. φm = ψ + 2πm |λ|2

Note that h(z) = h2 (z)λ/2β . Now a simple calculation shows that h(zm ) does not depend on m = 0, 1, . . . , k − 1, i.e., h is k-valent and λ ∈ σ (k) . The case Re λ < 0 is reduced to the previous one by replacing h by 1/h. Finally, suppose Re λ = 0. As usual, denote by {Ft }t≥0 the semigroup generated by f . Thus, the function u(t, z) := Ft (z) satisﬁes the Cauchy problem ⎧ ⎨ ∂u(t, z) + f (u(t, z)) = 0, ∂t ⎩ u(0, z) = z ∈ Δ. Solving this problem for z = 0, we have u(t,0) t du =− dt = −t. f (u) 0 0

244

Chapter 9. Appendices

Since the function 1/f is holomorphic on the open unit disk Δ, we conclude that, for each point z which belongs to the curve Λ := {u(t, 0) : t ≥ 0} (joining the origin with the boundary point z = 1), the integral z du f (u) 0 takes real values which tend to −∞ as z ∈ Λ tends to 1. Returning to the eigenvalue problem Γf (h)(z) (:= h (z)f (z)) = λh(z) and separating variables in this diﬀerential equation, we have z z dh(z) λdz = , h(z) 0 0 f (z)

so that log

h(z) h(0)

z

=λ 0

dz . f (z)

Thus, for each point z ∈ Λ, h(z) = h(0) exp (−λt) , where z = u(t, 0). We claim that this function is inﬁnite-valent. When λ = 0, this is evident. Suppose λ = 0; then λ = ia, a = 0. Choosing the sequence 2πn , 0 ∈ Λ ⊂ Δ for any ﬁxed t0 ≥ 0, we see immediately that the zn = u t0 + a value h(zn ) does not depend on n. Now, let h∗ be the solution of the diﬀerential equation (9.2.37) normalized ∗ by h (0) = 1, h∗ (1) = 0. As in the proof of Theorem 9.3, we conclude that the image of the function h∗ (Δ) must lie in a wedge of angle π. Since h∗ is starlike with respect to a boundary point, it follows from a result of Lyzzaik [102] (see also Theorem 3.2 above) that

2zh∗ (z) 1 + z Re > 0. + h∗ 1−z Substituting (9.2.38) into the latter inequality, we get (9.2.35). The proof is complete. Combining this assertion with previous results, one can obtain additional geometrical information on (characteristics of) eigenfunctions. Example 9.4. For each a ∈ (0, 1), we rewrite (9.2.35) in the form

2aβ zh (z) 1 + z 1+z 2aβ zw (z) Re + > Re (1 − a) − . λ h(z) 1−z 1−z λ w(z)

(9.2.39)

9.2. Weighted semigroups of composition operators

245

Suppose that w(z) = (1 − z)δ for some δ ∈ C. Let c be a positive number less than 1. In this case, for each λ such that λδ is c , a simple calculation shows that the right-hand side in (9.2.39) real with λδ < − aβ is greater than c. Indeed, for each z ∈ ∂Δ,

2aβ zw (z) 1 2aβδ 2aβδz 1+z − = + Re − Re (1 − a) 1−z λ w(z) |1 − z|2 λ λ 2aβδ 1−z aβδ =− Re > c. ≥− λ |1 − z|2 λ Letting a → 1− , we conclude that

2β zh (z) 1 + z Re + > c. λ h(z) 1−z Therefore, by Theorem 4.10, the image h(Δ) of the eigenfunction h covers cλ the image of the function (1 − z) β . Example 9.5. Consider now the (nonanalytic) weight w(z) = mula (9.2.38), each eigenfunction has the form h(z) = a

(1−z)2 z

. By for-

λ z (h∗ (z)) β . (1 − z)2

In this situation, for all positive λ less than 2β, inequality (9.2.35) becomes λ 1 − |z|2 zh (z) > 1− > 0, Re h(z) 2β |1 − z|2 i.e., all eigenfunctions are starlike with respect to an interior point (h(0) = 0). In particular, the function h0 (z) =

z h∗ (z) (1 − z)2

is the eigenfunction corresponding to the eigenvalue λ = β. Since h∗ ∈ Spiral1 [1], Theorem 9.7 implies that h0 is a starlike function of class S ∗ [0] which satisﬁes the condition zh0 (z) 1 − |z|2 = 1. Re 2 inf |1 − z|2 h0 (z) Finally, we observe that the above theorem implies the following nice characterization of multivalent starlike functions with respect to a boundary point suggested by Bshouty and Lyzzaik (see [23] and [24]; cf. also [97] and [99]). Theorem 9.9. Let h ∈ Hol(Δ, C) satisfy the equation −(1 − z)2

h (z) 1 − ω(z) =4 , h(z) 1 + ω(z)

(9.2.40)

246

Chapter 9. Appendices

where ω ∈ Hol(Δ) is a holomorphic self-mapping of Δ with the boundary ﬁxed point τ = 1 and positive multiplier α = ∠ lim

z→1

1 − ω(z) . 1−z

Then (i) h(Δ) is a starlike domain; (ii) h is a k-valent function if and only if k − 1 < α ≤ k. In particular, if τ = 1 is the Denjoy–Wolﬀ point for ω (whence 0 < α ≤ 1), then h is univalent and the smallest wedge which contains h(Δ) is of angle 2απ, i.e., h ∈ Star2α [1]. Proof. Let p(z) =

α 1 + ω(z) . 2 1 − ω(z)

This function has positive real part. Moreover, ∠ lim (1 − z)p(z) = ∠ lim z→1

z→1

α 1−z (1 + ω(z)) = 1. 2 1 − ω(z)

Deﬁne a generator f ∈ G(Δ) by the Berkson–Porta formula f (z) = −(1 − z)2 p(z). It is clear that f ∈ Gh [1] with β = f (1) = 1. Since h satisﬁes the equation 2αh(z) = h (z)f (z), and α is positive, applying Theorem 9.8 with λ = 2α and β = 1, we obtain the result. If τ = 1 is the Denjoy–Wolﬀ point for ω, i.e., 0 < α ≤ 1, then (z − 1)h (z) = 2α ≤ 2; z→1 h(z)

∠ lim

and the assertion follows by Theorem 3.3. The proof is complete.

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Subject Index Abel’s equation, 95, 97, 102, 116, 119, 180, 185 Abel–Schr¨oder equation, 113 angular derivative, 8, 21, 25, 114 angular limit, 7, 8 angular similarity principle, 168 asymptotic behavior, 156, 179 similar, 158, 215 automorphism, 124, 125, 155 elliptic, 12, 24, 154, 156 hyperbolic, 27, 76, 124, 125, 144 parabolic, 124 Banach Fixed Point Principle, 155 Berkson–Porta representation, 21, 24, 25, 44, 112 Bottcher’s equation, 99 Boundary Schwarz–Wolﬀ’s Lemma, 10, 115 branching process, 99 continuous, 100 Galton–Watson, 99, 100, 105 regular, 100 Brouwer Fixed Point Principle, 155 Burns–Krantz Theorem, 122 continuous version, 123 Carath´eodory Kernel Convergence Theorem, 45 Carath´eodory Principle, 89 Carath´eodory’s class, 13 Cauchy problem, 18, 23, 36, 73 Cayley transform, 13, 114 centralizer, 138 trivial, 138

Ces`aro means, 39 charge, 14 circular lens, 82 commuting semigroups, 133 composition operator, 95, 240 weighted, 241 convergence exponential, 102, 159 non-tangential, 109, 166, 173, 175, 176, 183 strongly tangential, 116, 174, 176, 192 weakly tangential, 192 covering theorem, 90 Denjoy–Wolﬀ point, 12 boundary, 25 of a semigroup, 24, 27 Denjoy–Wolﬀ Theorem, 12, 154 for semigroups, 24, 156 Dirac measure, 47, 50, 72 distortion theorem ‘spiral angle’ distortion, 75 angle distortion, 46 for generators, 20, 80 for the class G(μ, β), 88 eigenvalue problem, 95 embedding problem, 96, 119 extinction probability, 101 ϕ-like domain, 36 ϕ-like function, 6 generalized, 36

258 ﬁxed point, 10, 155 attractive, 12 boundary, 11, 12 boundary regular, 25, 125 common, 10, 25 free mapping, 11 interior, 10, 155 repelling, 207, 215 ﬂow invariance condition, 19 boundary, 19, 127 interior, 19 ﬂow invariant domain, 195, 196 angular characteristics, 211 maximal, 205 function k-valent, 241 close-to-convex, 6 conformal at point, 9 convex in one direction, 56, 97 ϕ-like, 6, 98 isogonal at point, 9 ρ-monotone, 20 spirallike, 5 starlike, 1 univalent, 1

Subject Index horocycle, 11, 35 Hummel’s representation, 3, 4 Hurwitz Theorem, 107, 181 instability phenomenon, 223 Julia number, 12 Julia–Carath´eodory Theorem, 30, 33, 78, 110 continuous analog, 22 inﬁnitesimal version, 29 Julia–Wolﬀ–Carath´eodory Theorem, 11, 14, 115, 118 continuous version, 24, 108 Kœnigs embedding property, 96 Kœnigs function, 96, 103, 104, 106, 149, 197 Kœnigs–Valiron function, 106, 107, 160, 162, 167 Koebe function, 3 λ-starlike, 3 Koebe One-Quarter Theorem, 104 Kolmogorov backward equation, 100

generating function, 99, 117 generator, 18, 19, 24 of a group of automorphisms, 76 of hyperbolic type, 26 of parabolic type, 26 group generator, 20 of automorphisms, 19, 20, 130 of elliptic automorphisms, 141 of hyperbolic automorphisms, 33, 130, 198 of parabolic automorphisms, 130 growth estimates for semigroup generators, 79 for spirallike functions, 81

Lebesgue decomposition, 52 Lebesgue measure, 85 Lebesgue’s Bounded Convergence Theorem, 48, 53, 72 Lie commutator, 135 Lindel¨ of’s Theorem, 8, 29 linear-fractional model, 97 linear-fractional transformation, 124, 125, 129 linearization model, 102 of Baker and Pommerenke, 116 of Kœnigs, 103 of Pommerenke, 112 of Valiron, 105 uniﬁed for parabolic mappings, 118

Harnack’s inequality, 13, 20 Hopf’s lemma, 126

M¨ obius transformation, 3, 103 Montel Theorem, 107

Subject Index Nevanlinna’s condition, 2, 35 non-tangential approach region, 7, 8 non-tangential limit, 7, 8, 11 null point, 21 boundary regular, 22, 212 perturbation formula, 231 Poincar´e metric, 20 pole at a boundary point, 184 power convergence, 12 pseudohyperbolic disk, 4, 10, 154 radial limit, 7, 11 rate of convergence, 12 Riemann Mapping Theorem, 63 Riesz–Herglotz measure, 14 Riesz–Herglotz representation, 13, 21, 29, 51 rigidity argument principle, 157 of holomorphic functions, 122 principles, 127 Robertson’s class, 39, 70 Robertson’s conjecture, 39, 40 generalized, 44 Schr¨ oder’s equation, 95, 102, 106, 110, 111, 161, 167, 197 Schwarz–Pick Lemma, 10, 104, 115, 121, 122, 154 boundary version, 11 inﬁnitesimal version, 20 Schwarzian derivative, 124 self-mapping aﬃne, 123 embeddable globally, 96 embeddable on a domain, 96 of automorphic type, 99, 113, 115, 174 of dilation type, 99 of hyperbolic type, 99, 115, 144 of non-automorphic type, 99, 113, 115, 116, 173 of parabolic type, 99, 115, 146

259 of positive hyperbolic step, 113 of zero hyperbolic step, 113 semigroup, 17 asymptotic behavior, 23, 153 continuous, 18, 19, 24, 34 discrete, 12, 18, 19, 100, 154 generator, 18, 19, 35 of composition operators, 240 of dilation type, 103, 140, 154 of hyperbolic type, 102, 105, 145, 179 of non-automorphic type, 149, 179 of parabolic type, 102, 134, 153, 173, 176, 178, 179, 184, 188 sink point, 12 of a semigroup, 24 ˇ cek’s condition, 4, 35 Spaˇ spirallike domain, 4 μ-spirallike, 66 with respect to a boundary point, 63 spirallike function, 4, 5, 35 μ-spirallike, 5, 7, 34 associated, 197 of order λ, 6 with respect to a boundary point, 5, 35, 63 with respect to an interior point, 5, 35 starlike domain, 1, 4 with respect to a boundary point, 63 starlike function, 1, 35 of order λ, 3 with respect to a boundary point, 1, 35, 39, 44 with respect to an interior point, 1, 35 stationary point, 23, 155 Stolz angle, 8 subordination, 73 subordination criteria, 73

260 unrestricted limit, 7 vector ﬁeld complete, 18–21, 195 semi-complete, 18–20, 34, 44, 195 Visser–Ostrowski condition, 10, 40, 206 generalized, 10, 41, 46, 48, 49, 54, 81, 197 Visser–Ostrowski quotient, 10, 70 Vitali’s uniqueness principle, 121 Wald’s Theorem, 35 wedge biggest interior, 49 canonical λ-spiral, 76 smallest exterior, 46 Wolﬀ’s Lemma, 33 Wolﬀ’s point, 12 of a semigroup, 24 Yaglom’s equation, 98, 102, 160

Subject Index

Author Index Abate, M., 18, 247 Aharonov, D., xii, 19, 247 Alexander, J. W., 1, 247 Arslan, M., 126, 247 Athreya, K. B., 102, 247 Baker, I. N., ix, 99, 116, 218, 247 Baracco, L., 122, 248 Behan, D., 248 Berkson, E., x, 18, 21, 248 Bourdon, P. S., 99, 118, 173, 175, 248 Bracci, F., xii, 122, 248 Brickman, L., 6, 35, 98, 248 Bshouty, D., 245, 248 Burns, D. M., 122, 248 Carath´eodory, C., 10, 248 Carleson, L., 248 Chelst, D., 126, 248 Chen, M., 92, 248 Choczewski, B., 253 Ciozda, K., 56, 248 Contreras, M. D., x, xii, 99, 118, 124, 173, 249 Cowen, C. C., ix, 97, 249 D´ıaz-Madrigal, S., x, xii, 99, 118, 124, 173, 249 Denjoy, A., ix, 12, 249 Duren, P., 249 Egerv`ary, E., 39, 249 Gamelin, T. W., 248, 251 Garnett, J. B., 251 Ger, R., 253

Golusin, G. M., 1, 252 Goodman, A. W., 1, 252 Goryainov, V. V., 23, 26, 250, 252 Gurganus, K. R., 157, 252 Hadamard, J., ix, 97, 252 Harris, T. E., 97, 102, 252 Heins, M. H., 144, 252 Hengartner, W., 56, 252 Herglotz, G., 13, 252 Hummel, J. A., 4, 252 Julia, G., ix, 10, 252 Kaplan, W., 6, 252 Karlin, S., 218, 252 Khatskevich, V., 253 Kindermann, L., 253 Kœnigs, G., ix, 95, 96, 99, 253 Krantz, S. G., 122, 248 Kriete, T. L., 122, 253 Kuczma, M., 253 Lecko, A., 56, 60, 253 Levenshtein, M., xii, 151, 250, 253 Lindel¨ of, E., 8, 29, 253 Lyzzaik, A., 40, 45, 51, 60, 244, 245, 248, 253 MacCluer, B. D., 122, 249, 253 McGregor, J., 218, 252 Migliorini, S., 253 Milnor, J., 253 Montel, P., 5, 253 Nevanlinna, R., 2, 253

262 Ney, P. E., 102, 247 Owa, S., 92, 248 Poggi-Corradini, P., 173, 253 Pommerenke, Ch., ix, x, 99, 106, 113, 116, 118, 124, 173, 178, 247, 249, 254 Poreda, T., 157, 254 Porta, H., x, 18, 21, 248 Reich, S., xii, 18, 151, 247, 250, 251, 253, 254 Robertson, M. S., 2, 39, 40, 56, 254 Ruscheweyh, S., 73, 254 Schober, G., 56, 252 Schr¨ oder, E., ix, 95, 254 Sevastyanov, B. A., 254 Shapiro, J. H., 99, 118, 173, 175, 248, 255 Sheil-Small, T., 255 Shields, A. L., 255 Silverman, H., 41, 44, 255 Silvia, E. M., 41, 44, 255 Siskakis, A., x, 97, 255 ˇ cek, L., 5, 92, 255 Spaˇ Tauraso, R., 122, 124, 175, 248, 250, 255 Todorov, P., 88, 255 Valiron, G., ix, 99, 106, 256 Vlacci, F., 122, 124, 248, 253, 255 Wald, J. K., 4, 256 Wolﬀ, J., ix, 10, 12, 106, 256 Yacobzon, F., 251 Zaitsev, D., 122, 248 Zalcman, L., xii, 251 Zampieri, G., 122, 248

Author Index

Symbols aut(D), 19

Γ(ζ, κ), 7

CF , 95 C p+ε (τ ), 117 p+ε CA (τ ), 117

hλ,θ , 75 Hol(D), 1 Hol(D, C), 1

Da,r (ζ), 8 Dr (τ ), 4 δt , 176 δp (τ ), 14 Δr (τ ), 11

k, 3 kλ , 3

εt , 176 F + [ζ], 25 F [1, −1], 26 F[ζ], 25 ϕτ , 11 G, 39 G, 19 G∗ , 39 G∗ , 40 G ∗ [τ ], 195 G ∗ [τ, η], 196 G[1, −1], 23 Gaut [±1], 27 G(D), 19 GF [1, −1], 26 Gh [1], 26 G(λ), 41 G(μ, β), 84 Gp [1], 26 G[ζ], 21 G + [ζ], 23

Mτ (z), 3 μ- SP, 66 Ω+ , 223 P, 13 P + [1, −1], 14 P + [τ ], 14 P0 , 13 P (n) [τ ], 184 Π+ , 13 Qf (ζ), 10 Qf (ζ, z), 10 ρ(z, w), 20 S∗, 2 S ∗ , 239 S ∗ [τ ], 2 S ∗ [τ, η], 49 Sλ∗ , 3 SF , 124 S ∗ [τ ], 3 Σμ [1], 58 Σμ,ν [1, η], 59 Σ[τ ], 56

264 SP, 64 Spiral[τ ], 5 Spiral[1], 69 Spiralμ [1], 69 Spiralμ,ν [1, η], 76 Spiral[τ, η], 76 ST , 64 Starμ [1], 49 Starμ,ν [1, η], 49 Univ(D), 1 ΥM , 82 W∗ (h), 49 Wλ,θ , 76 W ∗ (h), 47 Z(S), 138

Symbols

List of Figures 1.1 1.2 1.3 1.4

Images of starlike functions. . . . . . . Images of spirallike functions. . . . . . A non-tangential approach region and ζ = eiπ/6 . . . . . . . . . . . . . . . . . A horocycle and its image . . . . . . .

. . . . . . . . point . . . . . . . .

8 12

2.1

A semigroup deﬁned by a starlike function . . . . . . . . . . . . . .

35

3.1 3.2 3.3

A function of Robertson’s class. . . . . . . . . . . . . . . . . . . . . Angle distortion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Width distortion. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40 56 59

4.1 4.2

The image and the covered domain. . . . . . . . . . . . . . . . . . Radii of covered disks. . . . . . . . . . . . . . . . . . . . . . . . . .

91 92

8.1 8.2

Example 8.1, n = 1, 2, 3, 5. . . . . . . . . . . . . . . . . . . . . . . . 209 2 ) Example 8.2. The ﬂow generated by f (z) = − (1−z)(1+z and two 1+z ﬂow-invariant domains. . . . . . . . . . . . . . . . . . . . . . . . . . 210 1−z Example 8.3. The ﬂow generated by f (z) = z 1+z and the dense ﬂow-invariant domain. . . . . . . . . . . . . . . . . . . . . . . . . . 211

8.3

(1)

(1)

. . . . . . . . . . . . . . . . a Stolz angle . . . . . . . . . . . . . . . .

(1)

. . . . at . . . .

9.1 Example 9.2 (i), the images of q1 , q2 , q4 and of q. (2) (2) (2) 9.2 Example 9.2 (ii), the images of q4 , q6 , q12 . . . . . (1) (1) (1) 9.3 Example 9.3 (i), the images of h, h6 , h10 and h30 . . . (2) (2) (2) 9.4 Example 9.3 (ii), the images of h6 , h10 and h30 . . . . (3) (3) (3) 9.5 Example 9.3 (iii), the images of h6 , h10 , h30 and h. . 9.6 The sets σ (k) of k-valence of eigenfunctions. . . . . . .

. . . . . .

. . . . the . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

2 5

227 228 234 235 236 242