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On the asymptotic behavior of condenser capacity under Blaschke products and universal covering maps


Authors: Dimitrios Betsakos, Georgios Kelgiannis, Maria Kourou and Stamatis Pouliasis
Journal: Proc. Amer. Math. Soc. 147 (2019), 2963-2973
MSC (2010): Primary 30C85, 30J10, 30C80, 31A15
DOI: https://doi.org/10.1090/proc/14585
Published electronically: April 3, 2019
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Abstract: We prove an estimate for the capacity of the condenser $ (\mathbb{D},K_{r})$, $ r\in (0,1)$, where $ \mathbb{D}$ is the open unit disc and $ \{K_{r}\}$ is a compact exhaustion of the inverse image of a compact set under a Blaschke product $ B$, involving weighted logarithmic integral means of the Frostman shifts of $ B$. Also, we describe the asymptotic behavior of the capacity of condensers $ (\mathbb{D},E_{r})$, where $ E_{r}$ is a connected component of the inverse image of a closed disc with radius $ r$ under universal covering maps as $ r\to 0$.


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Additional Information

Dimitrios Betsakos
Affiliation: Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
Email: betsakos@math.auth.gr

Georgios Kelgiannis
Affiliation: Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
Email: gkelgian@math.auth.gr

Maria Kourou
Affiliation: Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
Email: mkouroue@math.auth.gr

Stamatis Pouliasis
Affiliation: Department of Mathematics and Statistics, Texas Tech University, Lubbock, Texas 79409
Email: stamatis.pouliasis@ttu.edu

DOI: https://doi.org/10.1090/proc/14585
Keywords: Condenser capacity, Lindel\"of principle, Blaschke products, universal covering maps
Received by editor(s): October 7, 2018
Published electronically: April 3, 2019
Additional Notes: This research has been co-financed by the Operational Program “Human Resources Development, Education and Lifelong Learning” and is co-financed by the European Union (European Social Fund) and Greek national funds.
Communicated by: Stephan Ramon Garcia
Article copyright: © Copyright 2019 American Mathematical Society