Bulletin of the American Mathematical Society

Author(s): Lawrence Zalcman
Journal: Bull. Amer. Math. Soc. 35 (1998), 215-230.
MSC (1991): Primary 30D45; Secondary 30D35, 34A20, 58F23
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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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Retrieve articles in Bulletin of the American Mathematical Society with MSC (1991): 30D45, 30D35, 34A20, 58F23

Lawrence Zalcman
Affiliation: Department of Mathematics and Computer Science, Bar-Ilan University, Ramat-Gan 52900, Israel
Email: zalcman@macs.biu.ac.il

DOI: 10.1090/S0273-0979-98-00755-1
PII: S 0273-0979(98)00755-1
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
Copyright of article: Copyright 1998, American Mathematical Society

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