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 Dal'nevost. Mat. Zh., 2002, Volume 3, Number 2, Pages 150–164 (Mi dvmg125)

The generalized reduced modulus

V. N. Dubinin, N. V. Eyrikh

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

Abstract: The boundary reduced moduli of the digons and triangles are the essential part of the extremal metric method. They have many applications in the geometric theory of functions of a complex variable. In the present paper, we use the capacity approach to extend these concepts to the concepts of the boundary reduced modulus of the polygons, with any number of vertexs. Moreover, we connect the concept of the boundary reduced modulus with the inner reduced modulus. The correctness of the definition of the generalized reduced modulus is proved. We consider the special cases of reduced modulus, the behavior of the reduced modulus under the extension of the sets and under the conformal mappings of the sets. The principle of the symmetry and the formulae for some reduced moduli are obtained. We prove the new composition principles for the generalized reduced moduli. These principles generalize some theorems about the separating transformation and about the extremal partitioning of the domains.

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

Citation: V. N. Dubinin, N. V. Eyrikh, “The generalized reduced modulus”, Dal'nevost. Mat. Zh., 3:2 (2002), 150–164

Citation in format AMSBIB
\Bibitem{DubEyr02} \by V.~N.~Dubinin, N.~V.~Eyrikh \paper The generalized reduced modulus \jour Dal'nevost. Mat. Zh. \yr 2002 \vol 3 \issue 2 \pages 150--164 \mathnet{http://mi.mathnet.ru/dvmg125} \elib{http://elibrary.ru/item.asp?id=1402939} 

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This publication is cited in the following articles:
1. V. N. Dubinin, “Generalized condensers and the asymptotics of their capacities under degeneration of some plates”, J. Math. Sci. (N. Y.), 129:3 (2005), 3835–3842
2. N. V. Eirikh, “O privedennom module I. P. Mityuka”, Dalnevost. matem. zhurn., 4:2 (2003), 167–181
3. V. N. Dubinin, N. V. Eyrikh, “Some applications of generalized condensers to analytic function theory”, J. Math. Sci. (N. Y.), 133:6 (2006), 1634–1647
4. V. V. Aseev, O. A. Lazareva, “On the continuity of the reduced modulus and the transfinite diameter”, Russian Math. (Iz. VUZ), 50:10 (2006), 8–16
5. V. V. Aseev, “Obobschennyi privedennyi modul v prostranstvennykh zadachakh emkostnoi tomografii”, Dalnevost. matem. zhurn., 7:1-2 (2007), 17–29
6. V. N. Dubinin, D. A. Kirillova, “On extremal decomposition problems”, J. Math. Sci. (N. Y.), 157:4 (2009), 573–583
7. V. V. Aseev, “Ned sets on a hyperplane”, Siberian Math. J., 50:5 (2009), 760–775
8. E. G. Prilepkina, “On composition principles for reduced moduli”, Siberian Math. J., 52:6 (2011), 1079–1091
9. V. N. Dubinin, “On the reduced modulus of the complex sphere”, Siberian Math. J., 55:5 (2014), 882–892
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