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**Normal families.**
*(English)*
Zbl 0770.30002

Universitext. New York: Springer-Verlag. xii, 236 p. (1993).

Normal families were introduced by Montel early this century and have since proved to be an important concept in complex analysis. This book is devoted to the study of various aspects of normal families. After some preliminaries in Chapter 1, normal families of analytic functions are discussed in Chapter 2. The chapter starts with the theorems of ArzelĂ - Ascoli, Montel’s theorem that a locally bounded family of analytic functions is normal, and the Vitali-Porter theorem. Then, as an application of normal families, the Riemann mapping theorem is proved. Next the fundamental theorem that the family of all analytic functions that omit two fixed values is normal is proved. As consequences the theorems of Picard, Schottky, and Julia are obtained.

Chapter 3 discusses normal families of meromorphic functions and includes for example Marty’s theorem and invariant normal families of meromorphic (and analytic) functions. Chapter 4 centers around the Bloch principle which asserts that a family of functions having the property \(P\) is (likely to be) normal if a function in the plane with property \(P\) reduces to a constant. It includes the Robinson-Zalcman formalizations of this principle as well as another formalization due to Minda. Some counterexamples to this principle are also presented. Next Drasin’s theory of normal families and Nevanlinna theory is developed. The chapter ends with a section “Further results”, where various criteria for a family to be normal are discussed.

The final chapter of the book describes some applications of normal families, for example it gives the basics of the Fatou-Julia iteration theory and discusses normal functions.

The book may be used for function theorists interested in normal families and their applications. I do, however, not quite agree with the cover text which says “Only a basic knowledge of complex analysis and topology is assumed. All other necessary material... is included in the first chapter”. For example, the Drasin theory requires quite a bit of Nevanlinna theory, and those not familiar with it can probably not learn it from the short introduction into it given in Chapter 1. A similar remark applies to Ahlfors’s theory of covering surfaces.

Chapter 3 discusses normal families of meromorphic functions and includes for example Marty’s theorem and invariant normal families of meromorphic (and analytic) functions. Chapter 4 centers around the Bloch principle which asserts that a family of functions having the property \(P\) is (likely to be) normal if a function in the plane with property \(P\) reduces to a constant. It includes the Robinson-Zalcman formalizations of this principle as well as another formalization due to Minda. Some counterexamples to this principle are also presented. Next Drasin’s theory of normal families and Nevanlinna theory is developed. The chapter ends with a section “Further results”, where various criteria for a family to be normal are discussed.

The final chapter of the book describes some applications of normal families, for example it gives the basics of the Fatou-Julia iteration theory and discusses normal functions.

The book may be used for function theorists interested in normal families and their applications. I do, however, not quite agree with the cover text which says “Only a basic knowledge of complex analysis and topology is assumed. All other necessary material... is included in the first chapter”. For example, the Drasin theory requires quite a bit of Nevanlinna theory, and those not familiar with it can probably not learn it from the short introduction into it given in Chapter 1. A similar remark applies to Ahlfors’s theory of covering surfaces.

Reviewer: W.Bergweiler (Aachen)