Extremal decomposition of the complex plane with restrictions for free poles

The problems of extremal decomposition with free poles on a circle are well known in the geometric theory of functions of a complex variable. One of such problems is the problem of maximum of the functional [mathematical expression not reproducible], where [gamma] [member of] (0, n], [B.sub.0], [B.sub.1], [B.sub.2], ..., [B.sub.n], n [greater than or equal to] 2, are pairwise disjoint domains in [bar.C], [a.sub.0] = 0, [absolute value of [a.sub.k]] = 1, k = [bar.1, n] are different points of the circle, r(B, a) is the inner radius of the domain B [subset] [bar.C] relative to the point a [member of] B. We consider a more general problem, in which the restriction [absolute value of [a.sub.k]] = 1, k = [bar.1, n], is replaced by a more general condition. Keywords. Inner radius of a domain, disjoint domains, radial system of points, control functional, separating transformation, quadratic differential, Green's function.
The extremal problems of disjoint domains compose the known classical direction of the geometric theory of functions of a complex variable [1-26]. Many such problems are reduced to the determination [...]