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Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)



Normal families: New perspectives

Author: Lawrence Zalcman
Journal: Bull. Amer. Math. Soc. 35 (1998), 215-230
MSC (1991): Primary 30D45; Secondary 30D35, 34A20, 58F23
MathSciNet review: 1624862
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Abstract: This paper surveys some surprising applications of a lemma characterizing normal families of meromorphic functions on plane domains. These include short and efficient proofs of generalizations of (i) the Picard Theorems, (ii) Gol’dberg’s Theorem (a meromorphic function on $\mathbb {C}$ which is the solution of a first-order algebraic differential equation has finite order), and (iii) the Fatou-Julia Theorem (the Julia set of a rational function of degree $d\ge 2$ is the closure of the repelling periodic points). We also discuss Bloch’s Principle and provide simple solutions to some problems of Hayman connected with this principle.

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Lawrence Zalcman
Affiliation: Department of Mathematics and Computer Science, Bar-Ilan University, Ramat-Gan 52900, Israel

Keywords: Normal families, Picard’s Theorem, algebraic differential equations, Julia set, Bloch’s Principle
Received by editor(s): October 15, 1997
Received by editor(s) in revised form: May 26, 1998
Article copyright: © Copyright 1998 American Mathematical Society