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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

An extremal decomposition problem for harmonic measure


Authors: Vladimir N. Dubinin and Matti Vuorinen
Journal: Proc. Amer. Math. Soc. 140 (2012), 2441-2446
MSC (2010): Primary 30C85
Published electronically: November 17, 2011
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Abstract: Let $ E$ be a continuum in the closed unit disk $ \vert z\vert\le 1$ of the complex $ z$-plane which divides the open disk $ \vert z\vert < 1$ into $ n\ge 2$ pairwise nonintersecting simply connected domains $ D_k$ such that each of the domains $ D_k$ contains some point $ a_k$ on a prescribed circle $ \vert z\vert = \rho $, $ 0 <\rho <1\, , \, k=1,\dots ,n\,. $ It is shown that for some increasing function $ \Psi \,,$ independent of $ E$ and the choice of the points $ a_k,$ the mean value of the harmonic measures

$\displaystyle \Psi ^{-1}\left [ \frac {1}{n} \sum _{k=1}^{k} \Psi ( \omega (a_k,E, D_k))\right ] $

is greater than or equal to the harmonic measure $ \omega (\rho , E^* , D^*)\,,$ where $ E^* = \{ z: z^n \in [-1,0] \}$ and $ D^* =\{ z: \vert z\vert<1, \vert{\rm arg}\, z\vert < \pi /n\} \,.$ This implies, for instance, a solution to a problem of R. W. Barnard, L. Cole, and A. Yu. Solynin concerning a lower estimate of the quantity $ \inf _{E} \max _{k=1,\dots ,n} \omega (a_k,E, D_k)\,$ for arbitrary points of the circle $ \vert z\vert = \rho \,.$ These authors stated this hypothesis in the particular case when the points are equally distributed on the circle $ \vert z\vert = \rho \,.$

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Additional Information

Vladimir N. Dubinin
Affiliation: Institute of Applied Mathematics, Far-Eastern Branch of the Russian Academy of Sciences, Vladivostok, Russia
Email: dubinin@iam.dvo.ru

Matti Vuorinen
Affiliation: Department of Mathematics, University of Turku, Turku 20014, Finland
Email: vuorinen@utu.fi

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-11109-2
PII: S 0002-9939(2011)11109-2
Keywords: Harmonic measure, inner radius, extremal decomposition.
Received by editor(s): December 4, 2010
Received by editor(s) in revised form: January 6, 2011, and February 24, 2011
Published electronically: November 17, 2011
Additional Notes: The research of the first author was supported by the Far-Eastern Branch of the Russian Academy of Sciences, project 09-III-A-01-007
The second author was supported by the Academy of Finland, project 2600066611
Communicated by: Mario Bonk
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.