Normal families: New perspectives

Zalcman, Lawrence

doi:10.1090/S0273-0979-98-00755-1

Normal families: New perspectives
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Bull. Amer. Math. Soc. 35 (1998), 215-230 Request permission

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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