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Fixed points of composite meromorphic functions and normal families

Published online by Cambridge University Press:  12 July 2007

Walter Bergweiler
Walter Bergweiler
Affiliation:
Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, D-24098 Kiel, Germany (bergweiler@math.uni-kiel.de)
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Abstract

We show that there exists a function f, meromorphic in the plane C, such that the family of all functions g holomorphic in the unit disc D for which f ∘ g has no fixed point in D is not normal. This answers a question of Hinchliffe, who had shown that this family is normal if Ĉ\f(C) does not consist of exactly one point in D. We also investigate the normality of the family of all holomorphic functions g such that f(g(z)) ≠ h(z) for some non-constant meromorphic function h.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 134 , Issue 4 , August 2004 , pp. 653 - 660
Copyright
Copyright © Royal Society of Edinburgh 2004

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