Remote Access Conformal Geometry and Dynamics
Green Open Access

Conformal Geometry and Dynamics

ISSN 1088-4173



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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 $|w|\leq \prod _{j}|z_{j}|,$ 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.

References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Conformal Geometry and Dynamics of the American Mathematical Society with MSC (2010): 30C80, 30C85, 31A15.

Retrieve articles in all journals with MSC (2010): 30C80, 30C85, 31A15.

Additional Information

Galatia Cleanthous
Affiliation: Department of Mathematics, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece
MR Author ID: 1032084

Athanasios G. Georgiadis
Affiliation: Department of Mathematical Sciences, Aalborg University, Fredrik Bajersvej 7G, 9220 Aalborg East, Denmark
MR Author ID: 911967

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