Hyperbolic geometric versions of Schwarz’s lemma

Betsakos, Dimitrios

doi:10.1090/S1088-4173-2013-00260-9

Hyperbolic geometric versions of Schwarz’s lemma
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by Dimitrios Betsakos

Conform. Geom. Dyn. 17 (2013), 119-132

DOI: https://doi.org/10.1090/S1088-4173-2013-00260-9

Published electronically: November 1, 2013

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Abstract:

Let $f$ be a holomorphic self-map of the unit disk $\mathbb {D}$. We prove monotonicity theorems which involve the hyperbolic area, the hyperbolic capacity, and the hyperbolic diameter of the images under $f$ of hyperbolic disks in $\mathbb {D}$. These theorems lead to distortion and modulus growth theorems that generalize the classical lemma of Schwarz and to geometric estimates for the density of the hyperbolic metric.

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