Semigroups in geometrical function theory

✍ Scribed by David Shoiykhet

• Publisher: Springer
• Year: 2001
• Tongue: English
• Leaves: 233
• Edition: 1
• Category: Library

✦ Synopsis

This manuscript provides an introduction to the generation theory of nonlinear one-parameter semigroups on a domain of the complex plane in the spirit of the Wolff-Denjoy and Hille-Yoshida theories. Special attention is given to evolution equations reproduced by holomorphic vector fields on the unit disk. A dynamic approach to the study of geometrical properties of univalent functions is emphasized. The book comprises six chapters. The preliminary chapter and chapter 1 give expositions to the theory of functions in the complex plane, and the iteration theory of holomorphic mappings according to Wolff and Denjoy, as well as to Julia and Caratheodory. Chapter 2 deals with elementary hyperbolic geometry on the unit disk, and fixed points of those mappings which are nonexpansive with respect to the Poincar? metric. Chapters 3 and 4 study local and global characteristics of holomorphic and hyperbolically monotone vector-fields, which yield a global description of asymptotic behavior of generated flows. Various boundary and interior flow invariance conditions for such vector-fields and their parametric representations are presented. Applications to univalent starlike and spirallike functions on the unit disk are given in Chapter 5. The approach described may also be useful for higher dimensions. Audience: The book will be of interest to graduate students and research specialists working in the fields of geometrical function theory, iteration theory, fixed point theory, semigroup theory, theory of composition operators and complex dynamical systems.

✦ Table of Contents

Cover......Page 1
Title Page......Page 2
Copyright Page......Page 3
Contents......Page 4
Preface......Page 6
0.1 Notations and notions......Page 12
0.2 Holomorphic functions of a complex variable......Page 15
0.3 Convergence of holomorphic functions......Page 17
0.4 Metric spaces and fixed point principles......Page 18
1.1 Schwarz-Pick Lemma and automorphisms......Page 20
1.2 Boundary behavior of holomorphic self-mappings......Page 28
1.3 Fixed points of holomorphic self-mappings......Page 36
1.4 Fixed point free holomorphic self-mappings of A. The Denjoy-Wolff Theorem......Page 43
1.5 Commuting family of holomorphic mappings of the unit disk.......Page 47
2.1 The Poincare metric on 0......Page 50
2.2 Infinitesimal Poincare metric and geodesics......Page 55
2.3 Compatibility of the Poincare metric with convexity......Page 57
2.4 Fixed points of p-nonexpansive mappings on the unit disk......Page 63
3.1 One-parameter continuous semigroup of holomorphic and p-nonexpansive self-mappings......Page 70
3.2 Infinitesimal generator of a one-parameter continuous semigroup......Page 73
3.3 Nonlinear resolvent and the exponential formula......Page 78
3.4 Monotonicity with respect to the hyperbolic metric......Page 90
3.5 Flow invariance conditions for holomorphic functions......Page 94
3.6 The Berkson-Porta parametric representation of semi-complete vector fields......Page 106
4.1 Stationary points of a flow on 0......Page 112
4.2 Null points of complete vector fields......Page 115
4.3 Embedding of discrete time group into a continuous flow......Page 120
4.4 Rates of convergence of a flow with an interior stationary point......Page 124
4.5 A rate of convergence in terms of the Poincare metric......Page 131
4.6 Continuous version of the Julia-Wolff-Caratheodory Theorem......Page 135
4.7 Lower bounds for p-monotone functions......Page 146
5 Dynamical approach to starlike and spirallike functions......Page 164
5.1 Generators on biholomorphically equivalent domains......Page 165
5.2 Starlike and spirallike functions......Page 168
5.3 A generalized Visser-Ostrowski condition and fanlike functions......Page 174
5.4 An invariance property and approximation problems......Page 177
5.5 Hummel's multiplier and parametric representations of starlike functions......Page 183
5.6 A conjecture of Robertson and geometrical characteristics of fanlike functions......Page 187
5.7 Converse theorems on starlike, spirallike and fanlike functions......Page 197
5.8 Growth estimates for spirallike, starlike and fanlike functions......Page 205
5.9 Remarks on Schroeder's equation and the Koenigs embedding property......Page 209
Bibliography......Page 216
Author and Subject Index......Page 227
List of figures......Page 232