On normal criterion of meromorphic functions.(English)Zbl 0706.30024

This paper is a continuation of work of the author in [Sci. China, Ser. A 32, No.7, 782-791 (1989; Zbl 0687.30023)] which makes use of L. Zalcman’s heuristic principle for normality: a family of holomorphic functions which have a common property P in a domain D is (apt to be) a normal family in D if P cannot be possessed by a non-constant entire function in the finite plane. The author uses a modified version of Zalcman’s principle to establish the normality of the family of meromorphic functions $$f(z)$$ satisfying $$f'-af^ n\neq b$$, $$n\geq 4$$, $$a\neq 0$$, $$\infty$$, $$b\neq \infty$$ in a domain D. This conjecture of W. K. Hayman [Research problems in function theory (1967; Zbl 0158.063)] was proved for $$n\geq 5$$ by L. Xianjin [Sci. Sin. Ser. A 28 (1985)].
Reviewer: St.Dragosh

MSC:

 30D45 Normal functions of one complex variable, normal families

Keywords:

Bloch’s principle; normal family

Citations:

Zbl 0687.30023; Zbl 0158.063