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HOLOMORPHIC FUNCTIONS WITH IMAGE OF GIVEN LOGARITHMIC OR ELLIPTIC CAPACITY

Part of: Two-dimensional theory Geometric function theory

Published online by Cambridge University Press:  08 March 2013

DIMITRIOS BETSAKOS
DIMITRIOS BETSAKOS*
Affiliation:
Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece email [email protected]
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Abstract

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For holomorphic functions $f$ in the unit disk $ \mathbb{D} $ with $f(0)= 0$, we prove a modulus growth bound involving the logarithmic capacity (transfinite diameter) of the image. We show that the pertinent extremal functions map the unit disk conformally onto the interior of an ellipse. We prove a modulus growth bound for elliptically schlicht functions in terms of the elliptic capacity ${\mathrm{d} }_{\mathrm{e} } f( \mathbb{D} )$ of the image. We also show that the function ${\mathrm{d} }_{\mathrm{e} } f(r \mathbb{D} )/ r$ is increasing for $0\lt r\lt 1$.

Keywords

holomorphic functionlogarithmic capacityelliptic capacitycondenser capacityelliptically schlicht functionSchwarz’s lemmasymmetrizationextremal length

MSC classification

Secondary: 30C80: Maximum principle; Schwarz's lemma, Lindelöf principle, analogues and generalizations; subordination 30C85: Capacity and harmonic measure in the complex plane 31A15: Potentials and capacity, harmonic measure, extremal length
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 94 , Issue 2 , April 2013 , pp. 145 - 157
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

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