Normal families: New perspectives. (English) Zbl 1037.30021

Zalcman, Lawrence

Bull. Am. Math. Soc., New Ser. 35, No. 3, 215-230 (1998).

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 \(\mathbb{C}\) 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 \(d\geq 2\) 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.

MSC:

30D45	Normal functions of one complex variable, normal families
30-02	Research exposition (monographs, survey articles) pertaining to functions of a complex variable

Keywords:

normal families; Picard’s theorem; algebraic differential equations; Julia set; Bloch’s principle

Citations:

Zbl 0315.30036

References:

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