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The result of M.A. Lavrentiev on the product of conformal radii of two non-overlapping simply connected domains has been generalized and strengthened. A method that allowed new estimates for the products of the inner radii of mutually non-overlapping domains to be obtained has been proposed. Keywords. Mutually non-overlapping domains, Green's function, inner radius of the domain, transfinite diameter., 1. Introduction An important place in the geometric theory of functions of complex variable is occupied by the direction associated with the study of extreme problems on various classes of [...]

We consider the well-known problem of the geometric theory of functions of a complex variable on non-overlapping domains with free poles on radial systems. The main results of the present work strengthen and generalize several known results for this problem. Keywords. Inner radius of a domain, non-overlapping domains, extremal problems on non-overlapping domains, radial systems of points, separating transformation, 'control' functional, problem of V. N. Dubinin., Introduction The extremal problems on non-overlapping domains with free poles compose the well-known classical trend of the geometric theory of functions of a complex variable [1-31]. Many of such problems [...]

Problems on extremal decomposition of the complex plane with free poles located on an (n, m)-ray system of points are studied. A method that allowed us to obtain new upper bounds for the maximum of the products of the inner radii of mutually non-overlapping domains is proposed. Keywords. Inner radius of a domain, mutually non-overlapping domains, Green's function, quadratic differential, transfinite diameter, theorem on minimization of an area, Cauchy inequality., (Presented by O. A. Dovgoshey) Dedicated to the century of the birthday of G. D. Suvorov 1. Notations and definitions Let N and R be the sets of natural and [...]

We consider the problem of estimation of the functional [mathematical expression not reproducible], where r([B.sub.k], [a.sub.k]) is the inner radius of a domain [B.sub.k] relative to the point [a.sub.k], under the condition [a.sub.0] = 0, [absolute value of [a.sub.k]] = 1, k = [bar.1,n], [a.sub.k] [member of] [B.sub.k] [subset] [bar.C], where the domains [B.sub.k] [intersection] [B.sub.p] = 0, k [not equal to] p, k, p = [bar.0,n], and the domains [B.sub.k], k = [bar.1,n], possess a symmetry relative to a unit circle. In some partial cases, this problem was solved in [2-5]. The present work is devoted to the study of the problem for [gamma] [member of] (1, [n.sup.1/3]] and n [greater than or equal to] 14. Keywords. Inner radius of a domain, non-overlapping domains, extremal problems, separating transformation, quadratic differential, Green's function., Let N and R be the sets of natural and real numbers, respectively, let C be the complex plane, and let [bar.C] = C [universal] {[infinity]} be the extended complex [...]

We consider the well-known problem of maximum of the functional [mathematical expression not reproducible], where [B.sub.0], ..., [B.sub.n] 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, [gamma] [member of] (0,n], and r(B, a) is the inner radius of the domain B [subset] [bar.C] relative to the point a. In the case of simply connected domains for n=2, 3, and 4, we have obtained the solution of this problem for the maximum interval of values of the parameter [gamma]. Keywords. Inner radius of a domain, nonoverlapping domains, radial systems of points, control functional, separating transformation, quadratic differential, Green function., Presented by V. Ya. Gutlyanskii Let C be the complex plane, let [bar.C] = C [union] {[infinity]} be a one-point compactification of the complex plane or the Riemann sphere, N, [...]

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 [...]

In the geometric theory of functions of complex variables, we consider V.N. Dubinin's extremal problem associated with estimates of a functional defined on a system of nonoverlapping domains and obtain a particular solution of this problem. Keywords. Nonoverlapping domains, inner radius, quadratic differential., 1. Notations and definitions Among the classical directions of the geometric theory of functions of complex variable, the noticeable place is occupied by the solution of extreme problems on the [...]

Within the geometric theory of functions, we study one of the classical problems of extreme decomposition of a complex plane. Keywords. Inner radius of a domain, non-overlapping domains, radial systems of points, separating transformation, quadratic differential, Green function., 1. Introduction The extreme problems of non-overlapping domains compose a well-known classical direction in the geometric theory of functions of a complex variable [1-9]. Many similar problems are reduced to [...]