Estimates of products of inner radii of non-overlapping domains in the complex plane

Abstract

This paper is devoted to one open extremal problem of Dubinin in geometric function theory of complex variables associated with estimates of a functional 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 obtained a solution of this problem for some concrete values of n and .

Keywords:

• Inner radius of domain
• non-overlapping domains
• separating transformation
• quadratic differential
• Green’s function

Notes

No potential conflict of interest was reported by the authors.