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The hyperbolic metric and geometric function theory. (English) Zbl 1208.30001

Ponnusamy, S. (ed.) et al., Proceedings of the international workshop on quasiconformal mappings and their applications, December 27, 2005–January 1, 2006. New Delhi: Narosa Publishing House (ISBN 81-7319-807-1/hbk). 9-56 (2007).
From the author’s Introduction:
“The material presented here is a selection of topics from the book (Beardon, A.F.; Minda, D., The hyperbolic metric in complex analysis) that relate to the Schwarz-Pick lemma. Our goal is to develop the main parts of geometric function theory by using the hyperbolic metric and other conformal metrics. This paper is intended to be both an introduction to the hyperbolic metric and a concise treatment of a few recent applications of the hyperbolic metric to geometric function theory.”
In the Sections 2–5, holomorphic self-maps of the unit disk \(\mathbb D\) are studied, using the hyperbolic metric. The unit disk with hyperbolic metric and hyperbolic distance is presented as a model of the hyperbolic plane. Pick’s fundamental invariant formulation of the Schwarz lemma is followed by various extensions of the Schwarz-Pick lemma for holomorphic self-maps of \(\mathbb D\), including a Schwarz-Pick Lemma for hyperbolic derivatives.
In Sections 6–9, holomorphic maps between simply connected proper subregions of the complex plane \(\mathbb C\) are investgated, using the hyperbolic metric and negatively curved metrics on simply connected regions. The hyperbolic metric is explicitly determined for a number of special simply connected regions, and estimates are provided for general simply connected regions. The Ahlfors lemma is established, which states the maximality of the hyperbolic metric in the family of metrics with curvature at most \(-1\).
In Sections 10–13, holomorphic maps between hyperbolic regions are studied, that is, between regions whose complement in the extended complex plane \(\mathbb C_\infty\) contains at least three points; and negatively curved metrics on such regions are studied. The planar uniformization theorem is used to transfer the hyperbolic metric from the unit disk to hyperbolic regions. The Schwarz-Pick lemma and the Ahlfors lemma are extend to this context. The final section offers some suggestions for further reading on topics not included in the article.
For the entire collection see [Zbl 1152.30001].
Reviewer: A. Neagu (Iaşi)

MSC:

30-02 Research exposition (monographs, survey articles) pertaining to functions of a complex variable
30C99 Geometric function theory
30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
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