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Normal Families: a Geometric Perspective

Normal Families: a Geometric Perspective In this largely expository paper we present an alternative to the common practice of discussing normal families of analytic maps in terms of the Euclidean metric and equicontinuity. Indeed, in most cases the hyperbolic metric and the Schwarz–Pick Lemma are available, and then equicontinuity is redundant and is replaced by a much stronger Lipschitz condition that is expressed in terms of conformally invariant metrics. Here, we discuss normal families in terms of (not necessarily analytic) maps that satisfy types of uniform Lipschitz conditions with respect to various conformal metrics, especially the hyperbolic and spherical metrics. A number of classical results for normal families of analytic maps extend to these broader classes of (not necessarily analytic) functions that satisfy types of uniform Lipschitz conditions. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

Normal Families: a Geometric Perspective

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Publisher
Springer Journals
Copyright
Copyright © 2014 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/s40315-014-0054-2
Publisher site
See Article on Publisher Site

Abstract

In this largely expository paper we present an alternative to the common practice of discussing normal families of analytic maps in terms of the Euclidean metric and equicontinuity. Indeed, in most cases the hyperbolic metric and the Schwarz–Pick Lemma are available, and then equicontinuity is redundant and is replaced by a much stronger Lipschitz condition that is expressed in terms of conformally invariant metrics. Here, we discuss normal families in terms of (not necessarily analytic) maps that satisfy types of uniform Lipschitz conditions with respect to various conformal metrics, especially the hyperbolic and spherical metrics. A number of classical results for normal families of analytic maps extend to these broader classes of (not necessarily analytic) functions that satisfy types of uniform Lipschitz conditions.

Journal

Computational Methods and Function TheorySpringer Journals

Published: Feb 19, 2014

References