Estimates of the products of inner radii for partially nonoverlapping domains of the complex plane

Abstract

Two problems of description of extremal configurations maximizing a product of the inner radii of mutually nonoverlapping domains are studied. One of the problems is analyzed in a more general situation: instead of nonoverlapping domains, the domains under the condition of partial disjointness are considered. The well-known problem posed in the work by V. N. Dubinin in 1988 is solved by Theorems 1 and 2. We study also the problem of maximum of a functional with the additional condition of symmetry defined by the domain G₀. Theorems 3 and 4 give its partial solution.