Problem of the maximum of the product of the conformal radii of nonoverlapping domains in a circle

Abstract

Let a_k, k=1,2,3, be distinct points of the circle U={z:¦z¦<1}, a_3+k=1/¯a_k, k=1,2,3. Let D₁,...,D₆ be a system of nonoverlapping simply connected domains D₁,...,D₆ on

,a_k ɛD_k, k=1,...,6. Let R(D_k,a_k) be the conformal radius of the domain D_k with respect to the point a_k. One formulates the following theorem. For any points a_k ɛU, k=1,2,3, and any system of the indicated domains one has the sharp inequality

One points out all the cases when equality prevails in (1). One indicates the main steps of the proof of this theorem.