Abstract: | Let ck, k=1,...,4, be arbitrary distinct points of . LetD be the family of all systems of simply connected domains in. By R(Dk, ck) we denote the conformal radius of the domain Dk, relative to the point ck. We prove that in the familyD one has the sharp inequality, (1) where a=( +i)/( –1), being the cross-ratio of the points c1,c2, c3, c4: E(–1, 1,a) is the continuum of least capacity containing the points –1,1,a. An explicit expression for capE(–1,1,a.) in terms of elliptic Jacobi functions has been obtained earlier by the author Tr. Mat. Inst. Akad. Nauk SSSR,94, 47–65, 1968]. On the basis of the well-known properties of continua of least capacity, one shows that the largest value of the right-hand side of (1) is attained for a=± i 3 and it is equal to 4–8/3·32. One gives all the configurations for which equality prevails in the obtained estimates.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 100, pp. 131–145, 1980. |