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Normality, Quasinormality and Periodic Points

Published online by Cambridge University Press:  11 January 2016

Jianming Chang
Jianming Chang*
Affiliation:
Department of Mathematics Changshu Institute of Technology, Changshu, Jiangsu 215500, P. R. China jmchang@cslg.edu.cn
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Abstract

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Let 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.

Keywords

30D4530D0537F1037C25

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 195 , 2009 , pp. 77 - 95
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2009

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