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 Dal'nevost. Mat. Zh., 2005, Volume 6, Number 1-2, Pages 39–56 (Mi dvmg197)

On the preservation of generalized reduced modulus under some geometric transformations of domains in the plane

V. N. Dubinin, E. G. Prilepkina

Institute of Applied Mathematics, Far-Eastern Branch of the Russian Academy of Sciences

Abstract: We study necessary and sufficient conditions for the preservation of the generalized reduced modulus under the extension of domain or a part of its boundary. Also, we give such conditions for polarization and dissymmetrization. As applications, the descriptions of extremal configurations in the known inequalities for the inner radii, the Green functions and the Robin functions are obtained. Some of our results supplement the known boundary distortion theorems for univalent regular functions.

Key words: capacity of condenser, reduced modulus, polarization, dissymmetrization, Green function, Robin function, regular functions.

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Document Type: Article
UDC: 517.54, 517.956
MSC: Primary 31A15; Secondary 30C85

Citation: V. N. Dubinin, E. G. Prilepkina, “On the preservation of generalized reduced modulus under some geometric transformations of domains in the plane”, Dal'nevost. Mat. Zh., 6:1-2 (2005), 39–56

Citation in format AMSBIB
\Bibitem{DubPri05} \by V.~N.~Dubinin, E.~G.~Prilepkina \paper On the preservation of generalized reduced modulus under some geometric transformations of domains in the plane \jour Dal'nevost. Mat. Zh. \yr 2005 \vol 6 \issue 1-2 \pages 39--56 \mathnet{http://mi.mathnet.ru/dvmg197} \elib{http://elibrary.ru/item.asp?id=15484721} 

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This publication is cited in the following articles:
1. V. N. Dubinin, E. G. Prilepkina, “On variational principles of conformal mappings”, St. Petersburg Math. J., 18:3 (2007), 373–389
2. V. N. Dubinin, D. B. Karp, V. A. Shlyk, “Izbrannye zadachi geometricheskoi teorii funktsii i teorii potentsiala”, Dalnevost. matem. zhurn., 8:1 (2008), 46–95
3. E. G. Prilepkina, “Teoremy iskazheniya dlya odnolistnykh funktsii v mnogosvyaznykh oblastyakh”, Dalnevost. matem. zhurn., 9:1-2 (2009), 140–149
4. V. N. Dubinin, E. G. Prilepkina, “Distortion theorems for univalent meromorphic functions on an annulus”, Siberian Math. J., 51:2 (2010), 229–243
5. E. G. Prilepkina, “On composition principles for reduced moduli”, Siberian Math. J., 52:6 (2011), 1079–1091
6. E. G. Prilepkina, “Transfinite diameter with respect to Neumann function”, J. Math. Sci. (N. Y.), 200:5 (2014), 605–613
7. V. N. Dubinin, “On the reduced modulus of the complex sphere”, Siberian Math. J., 55:5 (2014), 882–892
8. E. G. Prilepkina, “On quadratic forms generated by the Neumann functions”, J. Math. Sci. (N. Y.), 207:6 (2015), 909–922
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