Normal Families: a Geometric Perspective
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Normal Families: a Geometric Perspective A. F. Beardon · D. Minda
Received: 7 November 2013 / Revised: 12 December 2013 / Accepted: 25 December 2013 © Springer-Verlag Berlin Heidelberg 2014
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. Keywords Normal families · Schwarz–Pick lemma · Lipschitz functions · Conformal metrics Mathematics Subject Classification (2000)
30 · 30D45 · 30F45 · 30C80 · 30G30
To the memory of our friend, Fred W. Gehring, for his many important contributions to mathematics. Communicated by Matti Vuorinen. A. F. Beardon Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK e-mail: [email protected] D. Minda (B) Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221-0025, USA e-mail: [email protected]
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A. F. Beardon, D. Minda
1 Introduction We begin with an overview of the early history of normal families. One of the earliest results on the uniform convergence of a sequence of holomorphic functions is attributed to Weierstrass: if f 1 , f 2 , . . . are holomorphic in a plane region , and if f n → f uniformly on , then f is holomorphic in . We refer the reader to [30] for a comprehensive survey of, and references to, Weierstrass’ work. The next major advance towards a theory of normal families was perhaps that of Stieltjes who, in 1894 in a paper on continued fractions [36], proved that if a sequence of holomorphic functions is uniformly bounded in a plane region , and converges uniformly on some non-empty, open subset of , then it converges, uniformly on each compact subset of , to a function holomorphic in . This may have been the first result in which convergence is obtained on a larger region than that covered by the given hypotheses. Next, in 1901 Osgood showed that for uniformly bounded sequences it is sufficient to assume that convergence occurs on a dense subset of [26] (see also [4]). In 1904 Porter [28] showed that it is sufficient for convergence to occur on a curve in , and then later, Vitali [38,39] and, independently, Porter [29], proved, again for uniformly bounded functions, that it is sufficient for the functions to converge on a sequence of points that converges to a point in (see also [16,18]). Note that these results imply
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