On theorems of Hayman and Clunie. (English) Zbl 0974.30025

Pang, Xuecheng; Zalcman, Lawrence

N. Z. J. Math. 28, No. 1, 71-75 (1999).

The authors give a simple proof of the fact that if \(f\) is a transcendental entire function, all of whose zeros have multiplicity at least \(k\), then for each natural number \(n\), \(f^{(k)}f^n\) takes on every nonzero value infinitely often. This is a generalization of results by W. K. Hayman [Ann. Math., II Ser. 70, 9-42 (1959; Zbl 0088.28505) and J. Clunie [J. Lond. Math. Soc. 42, 389-392 (1967; Zbl 0169.40801)]. The authors also obtain corresponding results on normal families.

Reviewer: Martin Chuaqui (Santiago de Chile)

MSC:

30D35	Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
30D45	Normal functions of one complex variable, normal families

Keywords:

transcendental; entire function; zeros; multiplicity; normal families

Citations:

Zbl 0169.40801; Zbl 0088.28505