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Conformal Geometry and Dynamics

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Hyperbolic geometric versions of Schwarz’s lemma

Author: Dimitrios Betsakos
Journal: Conform. Geom. Dyn. 17 (2013), 119-132
MSC (2010): Primary 30C80, 30C85, 30F45, 30H05
Published electronically: November 1, 2013
MathSciNet review: 3126908
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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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Additional Information

Dimitrios Betsakos
Affiliation: Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
MR Author ID: 618946

Keywords: Holomorphic function, Schwarz lemma, hyperbolic metric, hyperbolic area, hyperbolic capacity, hyperbolic diameter, condenser, symmetrization.
Received by editor(s): June 20, 2013
Received by editor(s) in revised form: September 14, 2013
Published electronically: November 1, 2013
Article copyright: © Copyright 2013 American Mathematical Society