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Ahlfors-Beurling conformal invariant and relative capacity of compact sets

Authors: Vladimir N. Dubinin and Matti Vuorinen
Journal: Proc. Amer. Math. Soc. 142 (2014), 3865-3879
MSC (2010): Primary 30C85
Published electronically: July 10, 2014
MathSciNet review: 3251726
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Abstract: For a given domain $ D$ in the extended complex plane $ \overline {\mathbb{C}}$ with an accessible boundary point $ z_0 \in \partial D$ and for a subset $ E \subset {D},$ relatively closed w.r.t. $ D,$ we define the relative capacity $ {\rm rel cap}{\,}E$ as a coefficient in the asymptotic expansion of the Ahlfors-Beurling conformal invariant $ r(D\setminus E,z)/r(D, z)$ when $ z$ approaches the point $ z_0\,.$ Here $ r(G,z)$ denotes the inner radius at $ z$ of the connected component of the set $ G$ containing the point $ z\,.$ The asymptotic behavior of this quotient is established. Further, it is shown that in the case when the domain $ D$ is the upper half plane and $ z_0=\infty $, the capacity $ {\rm rel cap}{\,}E$ coincides with the well-known half-plane capacity $ {\rm hcap}{\,}E\,.$ Some properties of the relative capacity are proven, including the behavior of this capacity under various forms of symmetrization and under some other geometric transformations. Some applications to bounded holomorphic functions of the unit disk are given.

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Additional Information

Vladimir N. Dubinin
Affiliation: Far Eastern Federal University, Vladivostok, Russia
Address at time of publication: Institute of Applied Mathematics, Vladivostok, Russia

Matti Vuorinen
Affiliation: Department of Mathematics and Statistics, University of Turku, Turku 20014, Finland

Keywords: Conformal invariant, inner radius, holomorphic function, Schwarzian derivative.
Received by editor(s): March 10, 2012
Received by editor(s) in revised form: September 10, 2012, September 27, 2012, and November 30, 2012
Published electronically: July 10, 2014
Additional Notes: The research of the first author was supported by the Russian Foundation for Basic Research, project 11-01-00038
The research of the second author was supported by the Academy of Finland, project 2600066611
Communicated by: Jeremy Tyson
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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