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Conformal Geometry and Dynamics

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Growth and monotonicity properties for elliptically schlicht functions

Authors: Galatia Cleanthous and Athanasios G. Georgiadis
Journal: Conform. Geom. Dyn. 20 (2016), 116-127
MSC (2010): Primary 30C80, 30C85, 31A15
Published electronically: May 4, 2016
MathSciNet review: 3493301
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Abstract: Let $ f$ be a holomorphic function of the unit disc $ \mathbb{D},$ with $ f(\mathbb{D})\subset \mathbb{D}$ and $ f(0)=0$. Littlewood's generalization of Schwarz's lemma asserts that for every $ w\in f(\mathbb{D}),$ we have $ \vert w\vert\leq \prod _{j}\vert z_{j}\vert,$ where $ \{z_{j}\}_j$ are the pre-images of $ w.$ We consider elliptically schlicht functions and we prove an analogous bound involving the elliptic capacity of the image. For these functions, we also study monotonicity theorems involving the elliptic radius and elliptic diameter.

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Additional Information

Galatia Cleanthous
Affiliation: Department of Mathematics, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece

Athanasios G. Georgiadis
Affiliation: Department of Mathematical Sciences, Aalborg University, Fredrik Bajersvej 7G, 9220 Aalborg East, Denmark

Keywords: Circular symmetrization, condenser capacity, elliptic capacity, elliptically schlicht functions, Green function, modulus metric, monotonicity theorems, Schwarz lemma, upper bounds.
Received by editor(s): September 30, 2014
Received by editor(s) in revised form: February 15, 2016
Published electronically: May 4, 2016
Article copyright: © Copyright 2016 American Mathematical Society