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Normal families and quasiregular mappings

Part of: Entire and meromorphic functions, and related topics Geometric function theory

Published online by Cambridge University Press:  23 October 2023

Alastair N. Fletcher Open the ORCID record for Alastair N. Fletcher [Opens in a new window]  and
Daniel A. Nicks Open the ORCID record for Daniel A. Nicks [Opens in a new window]
Alastair N. Fletcher
Affiliation:
Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL, USA (afletcher@niu.edu)
Daniel A. Nicks
Affiliation:
School of Mathematical Sciences, The University of Nottingham, University Park, Nottingham, UK (dan.nicks@nottingham.ac.uk)
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Abstract

Beardon and Minda gave a characterization of normal families of holomorphic and meromorphic functions in terms of a locally uniform Lipschitz condition. Here, we generalize this viewpoint to families of mappings in higher dimensions that are locally uniformly continuous with respect to a given modulus of continuity. Our main application is to the normality of families of quasiregular mappings through a locally uniform Hölder condition. This provides a unified framework in which to consider families of quasiregular mappings, both recovering known results of Miniowitz, Vuorinen and others and yielding new results. In particular, normal quasimeromorphic mappings, Yosida quasiregular mappings and Bloch quasiregular mappings can be viewed as classes of quasiregular mappings which arise through consideration of various metric spaces for the domain and range. We give several characterizations of these classes and obtain upper bounds on the rate of growth in each class.

Keywords

normal familiesquasiregular mappingsBloch mappingsnormal mappingsYosida mappings

MSC classification

Primary: 30D45: Bloch functions, normal functions, normal families
Secondary: 30C65: Quasiconformal mappings in $^n$, other generalizations
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 67 , Issue 1 , February 2024 , pp. 79 - 112
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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