Normal Families

By: Joel L. Schiff

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Paperback | 15 April 1993

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A book on the subject of normal families more than sixty years after the publication of Montel's treatise Ler;ons sur les familles normales de fonc tions analytiques et leurs applications is certainly long overdue. But, in a sense, it is almost premature, as so much contemporary work is still being produced. To misquote Dickens, this is the best of times, this is the worst of times. The intervening years have seen developments on a broad front, many of which are taken up in this volume. A unified treatment of the classical theory is also presented, with some attempt made to preserve its classical flavour. Since its inception early this century the notion of a normal family has played a central role in the development of complex function theory. In fact, it is a concept lying at the very heart of the subject, weaving a line of thought through Picard's theorems, Schottky's theorem, and the Riemann mapping theorem, to many modern results on meromorphic functions via the Bloch principle. It is this latter that has provided considerable impetus over the years to the study of normal families, and continues to serve as a guiding hand to future work. Basically, it asserts that a family of analytic (meromorphic) functions defined by a particular property, P, is likely to be a normal family if an entire (meromorphic in

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Preface
Preliminaries	p. 1
Basic Notation	p. 1
Spherical and Hyperbolic Metrics	p. 2
Normal Convergence	p. 7
Some Classical Theorems	p. 9
Local Boundedness	p. 12
Equicontinuity	p. 13
Elliptic Functions	p. 16
Nevanlinna Theory	p. 21
Ahlfors Theory of Covering Surfaces	p. 29
Analytic Functions	p. 33
Normality	p. 33
Montel's Theorem	p. 35
Examples	p. 40
Vitali-Porter Theorem	p. 44
Zeros of Normal Families	p. 47
Riemann Mapping Theorem	p. 48
Fundamental Normality Test	p. 54
Picard, Schottky, and Julia Theorems	p. 56
Sectorial Theorems	p. 62
Covering Theorems	p. 64
Normal Convergence of Univalent Functions	p. 67
Meromorphic Functions	p. 71
Normality	p. 71
Montel's Theorem	p. 74
Marty's Theorem	p. 75
Compactness	p. 79
Poles of Normal Families	p. 79
Invariant Normal Families	p. 82
Asymptotic Values	p. 94
Linear Fractional Transformations	p. 96
Univalent Functions	p. 98
Bloch Principle	p. 101
Robinson-Zalcman Heuristic Principle	p. 101
Counterexamples	p. 107
Minda's Formalization	p. 108
The Drasin Theory	p. 113
Further Results	p. 142
General Applications	p. 161
Extremal Problems	p. 162
Dynamical Systems	p. 167
Normal Functions	p. 178
Harmonic Functions	p. 182
Discontinuous Groups	p. 194
Appendix: Quasi-Normal Families	p. 209
References	p. 215
Index	p. 233
Table of Contents provided by Blackwell. All Rights Reserved.

Normal Families

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