Normality, Quasinormality and Periodic Points

Abstract

AbstractLet M > 1 be a positive number. Let be a family of holomorphic functions f in some domain D ⊂ ℂ for which there exists an integer k = k(f) > 2 such that |(fk)′(ζ)| ≤ Mk for every periodic point ζ of period k of f in D. We show first that is quasinormal of order at most one in D. This strengthens a result of W. Bergweiler. Secondly, for the case M = 1, we prove that is normal in D if there exists a positive number K < 3 such that | f(η)| ≤ K for each f ∈ and every fixed point η of f in D. This improves a result of M. Esséen and S. J. Wu. We also construct an example which shows that the condition |f’(η)| ≤ K < 3 can not be replaced by | f′(η) | < 3.