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

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.

### 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