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Normal Families: a Geometric Perspective

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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.

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Acknowledgments

The authors would like to thank the African Institute for Mathematical Sciences (AIMS) for its hospitality during the period when this work was started. In addition, the second author is thankful for support through an AIMS Research Fellowship for Visiting Researchers that enabled him to visit AIMS during the period January 1–March 31, 2013. A Taft Faculty Release Fellowship awarded by the Taft Research Center of the University of Cincinnati provided the second author with leave January–May, 2013.

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Authors and Affiliations

  1. Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 0WB, UK

    A. F. Beardon

  2. Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH, 45221-0025, USA

    D. Minda

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  1. A. F. Beardon
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  2. D. Minda
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Correspondence to D. Minda.

Additional information

Communicated by Matti Vuorinen.

To the memory of our friend, Fred W. Gehring, for his many important contributions to mathematics.

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Beardon, A.F., Minda, D. Normal Families: a Geometric Perspective. Comput. Methods Funct. Theory 14, 331–355 (2014). https://doi.org/10.1007/s40315-014-0054-2

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