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A Schwarz lemma for meromorphic functions and estimates for the hyperbolic metric

Author: Alexander Yu. Solynin
Journal: Proc. Amer. Math. Soc. 136 (2008), 3133-3143
MSC (2000): Primary 30C80
Published electronically: May 5, 2008
MathSciNet review: 2407076
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Abstract: We prove a generalization of the Schwarz lemma for meromorphic functions $ f$ mapping the unit disk $ \mathbb{D}$ onto Riemann surfaces $ {\mathcal{R}}$ with bounded in mean radial distances from $ f(0)$ to the boundary of $ {\mathcal{R}}$. A new variant of the Schwarz lemma is also proved for the Carathèodory class of analytic functions having positive real part in $ \mathbb{D}$. Our results lead to several improved estimates for the hyperbolic metric.

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Additional Information

Alexander Yu. Solynin
Affiliation: Department of Mathematics and Statistics, Texas Tech University, Box 41042, Lubbock, Texas 79409

Keywords: Schwarz lemma, meromorphic function, hyperbolic metric, reduced module, polarization
Received by editor(s): April 30, 2007
Published electronically: May 5, 2008
Additional Notes: This research was supported in part by NSF grant DMS-0525339
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.