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Hyperbolic metric and membership of conformal maps in the Bergman space

Part of: Riemann surfaces Geometric function theory Spaces and algebras of analytic functions

Published online by Cambridge University Press:  06 May 2020

Dimitrios Betsakos ,
Christina Karafyllia  and
Nikolaos Karamanlis
Dimitrios Betsakos
Affiliation:
Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece e-mail: betsakos@math.auth.gr
Christina Karafyllia
Affiliation:
Institute for Mathematical Sciences, Stony Brook University, Stony Brook, New York 11794-3651 e-mail: christina.karafyllia@stonybrook.edu
Nikolaos Karamanlis*
Affiliation:
Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece e-mail: betsakos@math.auth.gr
*
e-mail: nikaraman@math.auth.gr
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Abstract

We prove that for $0<p<+\infty $ and $-1<\alpha <+\infty ,$ a conformal map defined on the unit disk belongs to the weighted Bergman space $A_{\alpha }^p$ if and only if a certain integral involving the hyperbolic distance converges.

Keywords

Bergman spacehyperbolic distanceGreen functionconformal mapping

MSC classification

Primary: 30H20: Bergman spaces, Fock spaces
Secondary: 30F45: Conformal metrics (hyperbolic, Poincaré, distance functions) 30C35: General theory of conformal mappings

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 1 , March 2021 , pp. 174 - 181
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Copyright
© Canadian Mathematical Society 2020

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Footnotes

This research was co-financed by Greece and the European Union (European Social Fund-ESF) through the Operational Programme “Human Resources Development, Education and Lifelong Learning 2014–2020” in the context of the project “Angular derivatives and the hyperbolic metric” (MIS 5047551).

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