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Extremal decomposition of a multidimensional complex space for five domains

  • Yaroslav ZabolotniiEmail author
  • Iryna Denega
Article

Abstract

The paper is devoted to one open extremal problem in the geometric function theory of complex variables associated with estimates of a functional defined on the systems of non-overlapping domains. We consider the problem of the maximum of a product of inner radii of n non-overlapping domains containing points of a unit circle and the power γ of the inner radius of a domain containing the origin. The problem was formulated in 1994 in Dubinin’s paper in the journal “Russian Mathematical Surveys” in the list of unsolved problems and then repeated in his monograph in 2014. Currently, it is not solved in general. In this paper, we obtained a solution of the problem for five simply connected domains and power γ (1; 2:57] and generalized this result to the case of multidimensional complex space.

Keywords

Inner radius of the domain non-overlapping domains polycylindrical domain radial system of points separating transformation quadratic differential Green’s function 

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of Mathematics of the National Academy of Sciences of UkraineKyivUkraine

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