Four counterexamples to Bloch’s principle. (English) Zbl 0602.30040

Rubel, Lee A.

Proc. Am. Math. Soc. 98, 257-260 (1986).

Bloch’s heuristic principle asserts that if P is a property of analytic (meromorphic) functions such that every entire function (meromorphic function on \({\mathbb{C}})\) with the property must be constant, then the class of all analytic (meromorphic) functions in a region \(\Omega\) with the property is apt to be a normal family. L. Zalcman [Am. Math. Monthly 82, 813-817 (1975; Zbl 0315.30036)] established a restricted version of Bloch’s principle. This version yields many classical results, but there are also instances in which Bloch’s principle is valid and Zalcman’s version does not apply. The reviewer [Proc. Am. Math. Soc. 93, 443-447 (1985; Zbl 0533.30023)] indicated one such situation involving Miranda’s theorem. Therefore, it is natural to inquire whether there is a rigorous version of Bloch’s principle that is more general than Zalcman’s version. The author presents four counterexamples to Bloch’s principle for analytic functions. These counterexamples will be useful to anyone seeking to establish a rigorous version of Bloch’s principle.

Reviewer: D.Minda

MSC:

30D45

Normal functions of one complex variable, normal families

Keywords:

552.30015; Bloch’s heuristic principle

Citations:

Zbl 0315.30036; Zbl 0533.30023