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Geometric versions of Schwarz's lemma for quasiregular mappings

Author: Dimitrios Betsakos
Journal: Proc. Amer. Math. Soc. 139 (2011), 1397-1407
MSC (2010): Primary 30C65, 30C80
Published electronically: September 15, 2010
MathSciNet review: 2748432
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Abstract: We prove monotonicity and distortion theorems for quasiregular mappings defined on the unit ball $ \mathbb{B}^n$ of $ \mathbb{R}^n$. Let $ K_I(f)$ be the inner dilatation of $ f$ and let $ \alpha=K_I(f)^{1/(1-n)}$. Let $ m_n$ denote $ n$-dimensional Lebesgue measure and $ c_n$ be the reduced conformal modulus in $ \mathbb{R}^n$. We prove that the functions $ r^{-n\alpha}m_n(f(r\mathbb{B}^n))$ and $ r^{-\alpha}c_n(f(r\mathbb{B}^n))$ are increasing for $ 0<r<1$. These results can be viewed as variants of the classical Schwarz lemma and as generalizations of recent results by Burckel et al. for holomorphic functions in the unit disk.

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Dimitrios Betsakos
Affiliation: Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece

Keywords: Quasiregular mapping, Schwarz’s lemma, capacity, conformal modulus, extremal length, symmetrization, diameter.
Received by editor(s): February 16, 2010
Received by editor(s) in revised form: April 26, 2010
Published electronically: September 15, 2010
Communicated by: Mario Bonk
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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