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Normal families: New perspectives. (English) Zbl 1037.30021
Summary: This paper surveys some surprising applications of a lemma characterizing normal families of meromorphic functions on plane domains [see the author, Am. Math. Mon. 82, 813–817 (1975; Zbl 0315.30036)]. These include short and efficient proofs of generalizations of (i) the Picard Theorems, (ii) Gol’dberg’s Theorem (a meromorphic function on 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 d2 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.

MSC:
30D45Bloch functions, normal functions, normal families
30-02Research monographs (functions of one complex variable)