Estimates of the inner radii of non-overlapping domains

The paper is devoted to extremal problems of the geometric function theory of complex variable related with estimates of functionals defined on systems of non-overlapping domains. Till now, many such problems have not been solved, though some partial solutions are available. In the paper improved method is proposed for solving problems on extremal decomposition of the complex plane. The main results of the paper generalize and strengthening some known results in the theory of non-overlapping domains with free poles to the case of an arbitrary arrangement of systems of points on the complex plane

Generalization of some extremal problems on non-overlapping domains with free poles

Some results related to extremal problems with free poles on radial systems are generalized. They are obtained by applying the known methods of geometric function theory of complex variable. Sufficiently good numerical results for γ are obtained

Extremal decomposition of multidimensional complex space for five domains

The paper is devoted to one open extremal problem in the geometric function theory of complex variables associated with estimates of a functional defined on the systems of non-overlapping domains. We consider the problem of the maximum of a product of inner radii of n non-overlapping domains containing points of a unit circle and the power γ of the inner radius of a domain containing the origin. The problem was formulated in 1994 in Dubinin’s paper in the journal “Russian Mathematical Surveys” in the list of unsolved problems and then repeated in his monograph in 2014. Currently, it is not solved in general. In this paper, we obtained a solution of the problem for five simply connected domains and power γ ∈ (1, 2.57] and generalized this result to the case of multidimensional complex space

ESTIMATE OF THE MAXIMUM PRODUCT OF INNER RADII OF NON-OVERLAPPING DOMAINS

In this paper, the upper estimate for the maximum of the products of inner radii of mutually non-overlapping domains is obtained for any -radial system of points on the complex plane at all possible values of some parameter . The conditions under which the structure of points is not important in the proved results are established