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.

Beardon 和 Minda 根据局部均匀 Lipschitz 条件给出了全纯和亚纯函数正常族的表征。在这里，我们将这种观点推广到更高维度的映射族，这些映射族相对于给定的连续性模量是局部一致连续的。我们的主要应用是通过局部均匀 Hölder 条件来求拟正则映射族的正态性。这提供了一个统一的框架，在其中考虑拟正则映射族，既恢复了 Miniowitz、Vuorinen 等人的已知结果，又产生了新的结果。特别地，正则拟正则映射、Yosida拟正则映射和Bloch拟正则映射可以被视为拟正则映射的类别，它们是通过考虑域和范围的各种度量空间而产生的。我们给出了这些类别的几个特征，并获得了每个类别增长率的上限。