1st Edition

# Normal Families and Normal Functions

By Peter V. Dovbush, Steven G. Krantz Copyright 2024
268 Pages
by Chapman & Hall

268 Pages
by Chapman & Hall

Also available as eBook on:

This book centers on normal families of holomorphic and meromorphic functions and also normal functions. The authors treat one complex variable, several complex variables, and infinitely many complex variables (i.e., Hilbert space).

The theory of normal families is more than 100 years old. It has played a seminal role in the function theory of complex variables. It was used in the first rigorous proof of the Riemann mapping theorem. It is used to study automorphism groups of domains, geometric analysis, and partial differential equations.

The theory of normal families led to the idea, in 1957, of normal functions as developed by Lehto and Virtanen. This is the natural class of functions for treating the Lindelof principle. The latter is a key idea in the boundary behavior of holomorphic functions.

This book treats normal families, normal functions, the Lindelof principle, and other related ideas. Both the analytic and the geometric approaches to the subject area are offered. The authors include many incisive examples.

The book could be used as the text for a graduate research seminar. It would also be useful reading for established researchers and for budding complex analysts.

Preface

1 Introduction

2 A Glimpse of Normal Families

3 Normal Families in Cn

3.1 Definitions and preliminaries

3.2 Marty’s Normality Criterion

3.3 Zalcman’s Rescaling Lemma

3.4 Pointwise Limits of Holomorphic Functions

3.5 Montel’s Normality Criteria

3.6 Application of Montel’s Theorem

3.7 Riemann’s theorem

3.8 Julia’s Theorem

3.9 Schwick’s Normality Criterion

3.10 Grahl and Nevo’s Normality Criterion

3.11 Lappan’s Normality Criterion

3.12 Mandelbrojt’s Normality Criterion

3.13 Zalcman-Pang’s Lemma

4 Normal Functions in Cn

4.1 Definitions and Preliminaries

4.1.1 Homogeneous domains

4.2 Normal function in

4.3 Algebraic Operation in Class of Normal Function

4.4 Extension for Bloch and Normal Functions

4.5 Schottky’s Theorem in

4.5.1 Picard’s little theorem

4.6 K-normal functions

4.7 P-point sequences

4.8 Lohwater-Pommerenke’s Theorem in

4.9 The Scaling Method

4.10 Asymptotic Values of Holomorphic Functions

4.11 Lindelöf theorem in

4.12 Lindelöf principle in

4.13 Admissible Limits of Normal Functions in

5 A Geometric Approach to the Theory of Normal Families

5.1 Introduction

5.2 History

5.3 The Kobayashi/Royden Pseudometric and Related Ideas

5.4 The Ascoli-Arzelà Theorem and Relative Compactness

5.5 Some More Sophisticated Normal Families Results

5.6 Some Examples

5.7 Taut Mappings

5.8 Classical Definition of Normal Holomorphic Mapping

5.9 Examples

5.10 The Estimate for Characteristic Functions

5.11 Normal Mappings

5.12 A Generalization of the Big Picard Theorem

6 Some Classical Theorems

6.1 Preliminaries

6.2 Uniformly Normal Families on Hyperbolic Manifolds

6.3 Uniformly Normal Families on Complex Spaces

6.4 Extension and Convergence Theorems

6.5 Separately Normal Maps

7 Normal Families of Holomorphic Functions

7.1 Introduction

7.2 Basic Definitions

7.3 Other Characterizations of Normality

7.4 A Budget of Counterexamples

7.5 Normal Functions

7.6 Different Topologies of Holomorphic Functions

7.7 A Functional Analysis Approach to Normal Families

7.8 Many Approaches to Normal Families

8 Spaces that Omit the Values 0 and 1

8.1 Schwarz-Pick systems

8.2 The Kobayashi Pseudometric

8.3 The Integrated Infinitesimal Kobayashi Pseudometric

8.4 A Montel Theorem

9 Concluding Remarks

Bibliography

Index

### Biography

Peter V. Dovbush Dr. habil., Associate Professor, in Moldova State University, Institute of Mathematics and Computer Science.  He received his Ph.D. in Lomonosov Moscow State University in 1983 and Doctor of Sciences in 2003.  He has published over over 50 scholarly articles.

Steven G. Krantz is a Professor of Mathematics at Washington University in St. Louis.  He has previously taught at UCLA, Princeton University, and Penn State University.  He received his Ph.D. from Princeton University in 1974.  Krantz has directed 20 Ph.D. students and 8 Masters students. He has published over 130 books and over 300 scholarly articles.  He is the holder of the Chauvenet Prize and the Beckenbach Book Award and the Kemper Prize.  He is a Fellow of the American Mathematical Society.