Journal of Mathematical Sciences

, Volume 95, Issue 3, pp 2256–2266 | Cite as

Ordering of sets, hyperbolic metrics, and harmonic measures

  • A. Yu. Solynin
Article

Abstract

We establish a series of inequalities which relate solutions to certain partial differential equations defined on a given system of open sets with similar solutions defined on the ordered system of sets. As a corollary, we prove a comparison theorem for the hyperbolic metric that allows us to interpret this metric as a Choquet capacity. Using a similar comparison theorem for harmonic measures, we give a solution to S. Segawa's problem on the set having the minimal harmonic measure among all compact sets that lie on the diameter of the unit disk and have a given linear measure. Bibliography: 26 titles.

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© Kluwer Academic/Plenum Publishers 1999

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  • A. Yu. Solynin

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