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.


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