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Hyperbolic metric and membership of conformal maps in the Bergman space

Published online by Cambridge University Press:  06 May 2020

Dimitrios Betsakos
Department of Mathematics, Aristotle University of Thessaloniki, 54124Thessaloniki, Greece e-mail:
Christina Karafyllia
Institute for Mathematical Sciences, Stony Brook University, Stony Brook, New York 11794-3651 e-mail:
Nikolaos Karamanlis*
Department of Mathematics, Aristotle University of Thessaloniki, 54124Thessaloniki, Greece e-mail:


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.

© Canadian Mathematical Society 2020

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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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