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.
Original language | English |
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Pages (from-to) | 2256-2266 |
Number of pages | 11 |
Journal | Journal of Mathematical Sciences |
Volume | 95 |
Issue number | 3 |
DOIs | |
State | Published - 1999 |