Normal Families (universitext)

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 This Is The First Book Devoted Solely To The Subject Of Normal Families Of Analytic And Meromorphic Functions Since The 1927 Treatise Of Paul Montel. A Considerable Body Of Research Has Evolved Since Then, And This Text Provides A Comprehensive Treatment Of The Entire Theory. 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, The Riemann Mapping Theorem, To Many Modern Results On Meromorphic Functions Via The Bloch Principle. It Is This Latter Which Has Provided Considerable Impetus Over The Years To The Study Of Normal Families, And Continues To Serve As A Guiding Hand To Future Work. Numerous Applications Of The Normal Family Theory Are Discussed, Particularly Those Found In The Study Of Extremal Problems, Normal Functions, Harmonic Functions, Discontinuous Groups, And Complex Dynamical Systems. Only A Basic Knowledge Of Complex Analysis And Topology Is Assumed. All Other Necessary Material For The Study Of The Subject Is Included In The First Chapter. The Scope Of The Book Ranges From Advanced Undergraduate To Research Level. 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. Joel L. Schiff. Includes Bibliographical References (p. [215]-231) And Index. Front Matter....Pages i-xii Preliminaries....Pages 1-31 Analytic Functions....Pages 33-70 Meromorphic Functions....Pages 71-99 Bloch Principle....Pages 101-159 General Applications....Pages 161-207 Back Matter....Pages 209-236