# Ordering of sets, hyperbolic metrics, and harmonic measures

Article

- Received:

DOI: 10.1007/BF02172470

- Cite this article as:
- Solynin, A.Y. J Math Sci (1999) 95: 2256. doi:10.1007/BF02172470

- 5 Citations
- 34 Downloads

## 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.

## Copyright information

© Kluwer Academic/Plenum Publishers 1999