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Let ak, k=1,...,4, be given distinct points of C. Let Dk, k=1,...,4, be a system of simply connected domains in the closed plane such that ak?Dk, k=1,...,4, DknDl=for k,l=1, ...,4, k?l. Let R(Dk,ak) be the conformal radius of the domain Dk relative to the point ak. In this paper we obtain an explicit expression for the maximum of the product in the family of all indicated system of domains in terms of elliptic functions. In the proof one makes directly use of the property of the extremal family of domains of the problem under consideration in terms of the associated quadratic differential.

Maximum of the product of the conformal radii of four nonoverlapping domains