Normal Families

Abstract

Author is senior lecturer in mathematics and statistics at Auckland University. Ch. 1. Preliminaries -- 1.1. Basic Notation -- 1.2. Spherical and Hyperbolic Metrics -- 1.3. Normal Convergence -- 1.4. Some Classical Theorems -- 1.5. Local Boundedness -- 1.6. Equicontinuity -- 1.7. Elliptic Functions -- 1.8. Nevanlinna Theory -- 1.9. Ahlfors Theory of Covering Surfaces -- Ch. 2. Analytic Functions -- 2.1. Normality -- 2.2. Montel's Theorem -- 2.3. Examples -- 2.4. Vitali-Porter Theorem -- 2.5. Zeros of Normal Families -- 2.6. Riemann Mapping Theorem -- 2.7. Fundamental Normality Test -- 2.8. Picard, Schottky, and Julia Theorems -- 2.9. Sectorial Theorems -- 2.10. Covering Theorems -- 2.11. Normal Convergence of Univalent Functions -- Ch. 3. Meromorphic Functions -- 3.1. Normality -- 3.2. Montel's Theorem -- 3.3. Marty's Theorem -- 3.4. Compactness -- 3.5. Poles of Normal Families -- 3.6. Invariant Normal Families -- 3.7. Asymptotic Values -- 3.8. Linear Fractional Transformations -- 3.9. Univalent Functions -- Ch. 4. Bloch Principle -- 4.1. Robinson-Zalcman Heuristic Principle -- 4.2. Counterexamples -- 4.3. Minda's Formalization -- 4.4. The Drasin Theory -- 4.5. Further Results -- Ch. 5. General Applications -- 5.1. Extremal Problems -- 5.2. Dynamical Systems -- 5.3. Normal Functions -- 5.4. Harmonic Functions -- 5.5. Discontinuous Groups -- Appendix: Quasi-Normal Families.