Problem on extremal decomposition of the complex plane

  • Iryna Denega Institute of Mathematics, National Academy of Sciences of Ukraine
Keywords: inner radius of domain, non-overlapping domains, radial system of points, separating transformation, quadratic differential, Green’s function


The paper is devoted to one extremal problem in geometric function theory of complex variables associated with estimates of functionals defined on the systems of non-overlapping domains. We consider Dubinin’s problem of the maximum of product of inner radii of n non-overlapping domains containing points of the unit circle and the power of the inner radius of a domain containing the origin. The problem was formulated in 1994 in the work of Dubinin and then repeated in his monograph in 2014. Currently it is not solved in general. In this paper we generalized it to the case of the more general system of points and obtained a solution of this problem for some concrete values of n and y.


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