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Conformal Geometry and Dynamics
Conformal Geometry and Dynamics
ISSN 1088-4173

 

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
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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
Email: betsakos@math.auth.gr

DOI: http://dx.doi.org/10.1090/S1088-4173-2013-00260-9
PII: S 1088-4173(2013)00260-9
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