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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References
G. Choquet, “Theory of capacities,”Ann. Inst. Fourier,5, 131–295 (1951/52).
H. Renggli, “An inequality for logarithmic capacities,”Pacif. J. Math.,11, 313–314 (1961).
M. Klein, “Estimates for the transfinite diameter with applications to conformal mapping,”Pacif. J. Math.,22, 267–279 (1967).
I. P. Mityuk, “An upper estimate for the product of inner radii of domains, and covering theorems,”Izv. Vuz., Mat.,8, 39–47 (1987).
I. P. Mityuk, “A lower estimate for the sum of capacities of condensers,”Ukr. Mat. Zh.,41, 1442–1444 (1989).
I. P. Mityuk and A. Yu. Solynin, “Ordering of systems of sets and condensers in space,”Izv. Vuz., Mat.,8, 54–59 (1991).
G. M. Goluzin,Geometric Theory of Functions of a Complex Variable, Am. Math. Soc., Providence, Rhode Island (1969).
J. Ferrand, “A characterization of quasiconformal mappings by the behaviour of a function of three points,”Lect. Notes Math.,1351, 110–123 (1988).
P. Seittenranta, “Möbius-invariant metrics” (to appear).
S. Segawa, “Martin boundaries of Denjoy domains,”Proc. Am. Math. Soc.,103, 177–183 (1988).
A. Yu. Solynin, “Ordering of sets and comparison theorems,” Preprint POMI 5/1996; POMI, St. Petersburg (1996).
A. W. Marshall and I. Olkin,Inequalities: Theory of Majorization and Its Applications, New York (1979).
A. Yu. Solynin, “Application of polarization in proving functional inequalities,” Preprint POMI 10/1995; POMI, St. Petersburg (1995).
A. Yu. Solynin, “Functional inequalities via polarization,”Algebra Analiz,8, No. 6, 148–185 (1996).
F. Brock and A. Yu. Solynin, “An approach to symmetrization via polarization” (to appear).
M. Protter and H. Weinberger,Maximum Principle in Differential Equations, Prentice-Hall (1967).
M. Heins, “On a class of conformal metrics,”Nagoya Math. J.,21, 1–60 (1962).
A. Weitsman, “A symmetry property of the Poincaré metric,”Bull. London Math. Soc.,11, 295–299 (1979).
A. Beardon and Ch. Pommerenke, “The Poincaré metric of plane domains,”J. London Math. Soc.,18, 475–483 (1978).
V. G. Maz'ya,S. L. Sobolev Spaces, Springer-Verlag, Berlin-New York (1985).
D. A. Brannan and W. K. Hayman, “Research problems in complex analysis,”Bull. London Math. Soc.,21, 1–35 (1989).
J. A. Jenkins, “A uniqueness result in conformal mapping,”Proc. Am. Math. Soc.,22, 324–325 (1969).
A. Pfluger, “On a uniqueness theorem in conformal mapping,”Michigan Math. J.,23, 363–365 (1977).
D. Betsacos, “On certain harmonic measures on the unit disk,”Colloq. Math.,73, 221–228 (1997).
M. Essén and K. Haliste, “On Beurling's theorem for harmonic measure and the rings of Saturn,”Complex Variables, Theory Appl.,12, 137–152 (1989).
A. Beurling,The Collected Works of Arne Beurling (Complex Analysis, 1), BirkhÄuser (1989).
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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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DOI: https://doi.org/10.1007/BF02172470