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Semigroups in Geometrical Function Theory

By: D. Shoikhet

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Hardcover | 31 July 2001

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264 Pages

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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 PoincarA(c) 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.

Industry Reviews

`The book is well written and elegantly structured. Also it subject is interesting and the figures included are suggestive. It is useful to student and postgraduate students who are interested in complex dynamic systems, as well as to all researchers specialized in complex analysis and differential equations.'
Zbl. MATH, 98:30001

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You Can Find This Book In

Non-Fiction Mathematics Calculus & Mathematical Analysis Differential Calculus & Equations Complex Analysis Functional Analysis & Transforms Groups & Group Theory Geometry Analytic Geometry

Preface	p. vii
Preliminaries	p. 1
Notations and notions	p. 1
Holomorphic functions of a complex variable	p. 4
Convergence of holomorphic functions	p. 6
Metric spaces and fixed point principles	p. 7
The Wolff-Denjoy theory on the unit disk	p. 9
Schwarz-Pick Lemma and automorphisms	p. 9
Boundary behavior of holomorphic self-mappings	p. 17
Fixed points of holomorphic self-mappings	p. 25
Fixed point free holomorphic self-mappings of [Delta]. The Denjoy-Wolff Theorem	p. 32
Commuting family of holomorphic mappings of the unit disk	p. 36
Hyperbolic geometry on the unit disk and fixed points	p. 39
The Poincare metric on [Delta]	p. 39
Infinitesimal Poincare metric and geodesics	p. 44
Compatibility of the Poincare metric with convexity	p. 46
Fixed points of [rho]-nonexpansive mappings on the unit disk	p. 52
Generation theory on the unit disk	p. 59
One-parameter continuous semigroup of holomorphic and [rho]-nonexpansive self-mappings	p. 59
Infinitesimal generator of a one-parameter continuous semigroup	p. 62
Nonlinear resolvent and the exponential formula	p. 67
Monotonicity with respect to the hyperbolic metric	p. 79
Flow invariance conditions for holomorphic functions	p. 83
The Berkson-Porta parametric representation of semi-complete vector fields	p. 95
Asymptotic behavior of continuous flows	p. 101
Stationary points of a flow on [Delta]	p. 101
Null points of complete vector fields	p. 104
Embedding of discrete time group into a continuous flow	p. 109
Rates of convergence of a flow with an interior stationary point	p. 113
A rate of convergence in terms of the Poincare metric	p. 120
Continuous version of the Julia-Wolff-Caratheodory Theorem	p. 124
Lower bounds for [rho]-monotone functions	p. 135
Dynamical approach to starlike and spirallike functions	p. 153
Generators on biholomorphically equivalent domains	p. 154
Starlike and spirallike functions	p. 157
A generalized Visser-Ostrowski condition and fanlike functions	p. 163
An invariance property and approximation problems	p. 166
Hummel's multiplier and parametric representations of starlike functions	p. 172
A conjecture of Robertson and geometrical characteristics of fanlike functions	p. 176
Converse theorems on starlike, spirallike and fanlike functions	p. 186
Growth estimates for spirallike, starlike and fanlike functions	p. 194
Remarks on Schroeder's equation and the Koenigs embedding property	p. 198
Bibliography	p. 205
Author and Subject Index	p. 216
List of figures	p. 221
Table of Contents provided by Syndetics. All Rights Reserved.

Semigroups in Geometrical Function Theory

At a Glance

Hardcover

Industry Reviews

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How to return your order

Defective items

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