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Ordering of sets, hyperbolic metrics, and harmonic measures

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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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Dedicated to the 90th anniversary of G. M. Goluzin's birth

Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 237, 1997, pp. 129–147.

This research was partially supported by the Russian Foundation for Basic Research, grant 97-01-00259.

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Solynin, A.Y. Ordering of sets, hyperbolic metrics, and harmonic measures. J Math Sci 95, 2256–2266 (1999). https://doi.org/10.1007/BF02172470

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