Geodesic
Ordering of sets, hyperbolic metric, and harmonic measure
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 14, Tome 237 (1997), pp. 129-147
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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. 
@article{ZNSL_1997_237_a10,
     author = {A. Yu. Solynin},
     title = {Ordering of sets, hyperbolic metric, and harmonic measure},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {129--147},
     year = {1997},
     volume = {237},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1997_237_a10/}
}
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A. Yu. Solynin. Ordering of sets, hyperbolic metric, and harmonic measure. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 14, Tome 237 (1997), pp. 129-147. http://geodesic.mathdoc.fr/item/ZNSL_1997_237_a10/