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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
MathSciNet review: 2898706
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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 \,.$

References [Enhancements On Off] (What's this?)

  • [A] L.V. Ahlfors, Conformal invariants: topics in geometric function theory. McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0357743 (50:10211)
  • [BCS] R.W. Barnard, L. Cole, and A. Yu. Solynin, Minimal harmonic measure on complementary regions. Comput. Methods Funct. Theory 2 (2002), 229-247. MR 2000559 (2004g:30037)
  • [D1] V.N. Dubinin, The product of inner radii of ``partially nonoverlapping'' domains. Questions in the metric theory of mappings and its application (Proc. Fifth Colloq. Quasiconformal Mappings, Generalizations Appl., Donetsk, 1976) (Russian), 24-31, ``Naukova Dumka'', Kiev, 1978. MR 558368 (81e:30029)
  • [D2] V.N. Dubinin, Symmetrization in the geometric theory of functions of a complex variable. Uspekhi Mat. Nauk 49 (1994), no. 1(295), 3-76 (Russian); translation in Russian Math. Surveys 49 (1994), no. 1, 1-79. MR 1307130 (96b:30054)
  • [J1] J.A. Jenkins, On the existence of certain general extremal metrics. Ann. of Math. (2) 66 (1957), 440-453. MR 0090648 (19:845g)
  • [J2] J.A. Jenkins, The method of the extremal metric (English summary). Handbook of complex analysis: geometric function theory, Vol. 1, 393-456, ed. by R. Kühnau, North-Holland, Amsterdam, 2002. MR 1966200 (2004a:30024)
  • [K] J. Krzyż, Circular symmetrization and Green's function (Russian summary). Bull. Acad. Polon. Sci. Ser. Sci. Math. Astr. Phys. 7 (1959), 327-330. MR 0107701 (21:6424)
  • [Kuz] G. V. Kuz'mina, Methods of the geometric theory of functions. II. Algebra i Analiz 9:5 (1997), 1-50 (Russian); translation in St. Petersburg Mathematical Journal, 1998, 9:5, 889-930. MR 1604397 (99c:30047a)
  • [PBM] A.P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integraly i ryady [Integrals and series]. Elementarnye funktsii [Elementary functions] (Russian), ``Nauka'', Moscow, 1981, 799 pp. MR 635931 (83b:00009)
  • [S] A. Yu. Solynin, Moduli and extremal metric problems. Algebra i Analiz 11 (1999), no. 1, 3-86 (Russian); translation in St. Petersburg Math. J. 11 (2000), no. 1, 1-65. MR 1691080 (2001b:30058)

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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

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

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.

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