On the Problem of Maximizing the Product of Powers of Conformal Radii Nonoverlapping Domains

A sharp estimate of the product (as usual, R(D,b) denotes the conformal radius of a domain D with respect to a point b D) in the family of all quadruples of nonoverlapping simply connected domains {D_k},b_k D_k,k=1,...,4, is obtained. Here, {b₁,...,b₄} are four arbitrary distinct points on is an arbitrary positive number. The proof involves the solution of the problem on maximizing a certain conformal invariant, which is related to the problem under consideration. Bibliography: 5 titles.