Abstract
Exact relations are established for the reduced modules of the families of curves separating certain connected subsets in arbitrary connected domains. New explicit expressions for the harmonic measure of a connected boundary subset are obtained as corollaries.
References
V. K. Dzyadyk,Introduction to the Theory of Uniform Approximations of Functions by Polynomials [in Russian], Nauka, Moscow (1977).
V. I. Belyi, “Conformai mappings and approximation of analytic functions in the regions with quasiconformal boundary,”Mat. Sb.,102 (144), No. 3, 331–336 (1977).
V. I. Belyi, “Development of the method of conformai invariants and quasiconformal quasi-invariants from the viewpoint of application to problems of polynomial approximation,” in:Approximation and Function Spaces, North Holland, Amsterdam-Oxford-New York (1981), pp. 102–121.
V. I. Belyi, “On the constants in the estimates of distance distortion under conformal mappings,” in:Dokl. Rasshir. Sem. Im. Vekua,1, No. 1, 36–39 1985).
G. M. Goluzin,Geometrical Theory of Functions of a Complex Variable [in Russian], Nauka, Moscow (1966).
O. Teichmüller, “Untersuchungen über konforme und quasikonforme Ablildung,”Deutsch. Math.,3, 621–678 (1983).
L. V. Ahlfors andA. Beurling, “Conformal invariants and function theoretic null-sets,”Acta Math. 83, 101–129 (1950).
J. Hersch, “Longueurs extrémales et théorie des fonctions,”Com. Math. Helv.,29, 301–337 (1955).
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 4, pp. 576–578, April, 1993.
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Belyi, V.I., Kravchuk, E.V. Explicit relations for the reduced module and harmonic measure. Ukr Math J 45, 625–628 (1993). https://doi.org/10.1007/BF01062956
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DOI: https://doi.org/10.1007/BF01062956