Normal families and quasiregular mappings

Fletcher, Alastair N.; Nicks, Daniel A.

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.

• Journal Article Type: Article
• Acceptance Date: Sep 22, 2023
• Online Publication Date: Oct 23, 2023
• Publication Date: 2024-02
• Deposit Date: Dec 13, 2023
• Publicly Available Date: Apr 24, 2024
• Journal: Proceedings of the Edinburgh Mathematical Society
• Print ISSN: 0013-0915
• Electronic ISSN: 1464-3839
• Publisher: Cambridge University Press
• Peer Reviewed: Peer Reviewed
• Volume: 67
• Issue: 1
• Pages: 79-112
• DOI: https://doi.org/10.1017/s0013091523000640
• Keywords: Normal families; quasiregular mappings; Bloch mappings; normal mappings; Yosida mappings
• Public URL: https://nottingham-repository.worktribe.com/output/26539002
• Publisher URL: https://www.cambridge.org/core/journals/proceedings-of-the-edinburgh-mathematical-society/article/abs/normal-families-and-quasiregular-mappings/7FCE05870A4A1B5AA513991AB7B76AFA
• Additional Information: Copyright: © The Author(s), 2023. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.