THE PROBLEM OF EXTREMAL DECOMPOSITION OF A COMPLEX PLANE WITH FREE POLES.

Although much research (f. e. [1], [3], [5], [7-15]) has been devoted to the extremal problems of a geometric function theory associated with estimates of functionals defined on systems of non-overlapping domains, however, in the general case the problems remain unsolved. The paper describes the problem of finding the maximum of a functional. This problem is to find a maximum of the product of inner radii of mutually non-overlapping symmetric domains with respect to a unit circle and the inner radius in some positive certain degree of the domain with respect to zero and description of extreme configurations. The topic of the paper is devoted to the study of the problem of the classical direction of the geometric theory of complex variable functions, namely, the extremal problems for non-overlapping domains. [ABSTRACT FROM AUTHOR]