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.
Author(s): Peter V. Dovbush, Steven G. Krantz
Edition: 1
Publisher: CRC Press
Year: 2024
Language: English
Pages: 268
Tags: holomorphic functions, meromorphic functions, normal functions, normal families
Cover
Half Title
Title Page
Copyright Page
Contents
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 Functions in Cn
4.3. Algebraic Operation in Class of Normal Function
4.4. Extension for Bloch and Normal Functions
4.5. Schottky’s Theorem in Cn
4.5.1. Picard’s little theorem
4.6. K-normal Functions
4.7. P-point Sequences
4.8. Lohwater-Pommerenke’s Theorem in Cn
4.9. The Scaling Method
4.10. Asymptotic Values of Holomorphic Functions
4.11. Lindelöf Theorem in Cn
4.12. Lindelöf Principle in Cn
4.13. Admissible Limits of Normal Functions in Cn
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
Alphabetical Index
