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

DOI:
https://doi.org/10.1090/S0002-9939-2011-11109-2

Published electronically:
November 17, 2011

MathSciNet review:
2898706

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a continuum in the closed unit disk of the complex -plane which divides the open disk into pairwise nonintersecting simply connected domains such that each of the domains contains some point on a prescribed circle , It is shown that for some increasing function independent of and the choice of the points the mean value of the harmonic measures

**[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

Email:
dubinin@iam.dvo.ru

**Matti Vuorinen**

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

Email:
vuorinen@utu.fi

DOI:
https://doi.org/10.1090/S0002-9939-2011-11109-2

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.