Estimation of the products of the inner radii of domains with an additional symmetry condition

Abstract

In geometric function theory of complex variable extremal problems on non-overlapping domains are well-known classic direction. A lot of such problems are reduced to determination of the maximum of product of inner radii on the system of non-overlapping domains satisfying a certain conditions. Based on these elementary estimates a number of new estimates for functions realizing a conformal mapping of a disc onto domains with certain special properties are obtained. Estimates of this type are fundamental to solving some metric problems arising when considering the cor\-res\-pon\-dence of boundaries under a conformal mapping. Also, on the basis of the results concerning various extremal properties of conformal mappings, the problem of the representability of functions of a complex variable by a uniformly convergent series of polynomials is solved. In this paper, we consider the problem on maximum the products of the inner radii of $n$ disjoint domains with an additional symmetry condition that contain points of extended complex plane and the degree $\gamma$ of the inner radius of the domain that contains the zero point. An upper estimate for the maximum of this product is found for all values of $\gamma\in(0,\,n]$. The main result of the paper generalizes and strengthens the results of the predecessors [1-4] for the case of an arbitrary arrangement of points systems on $\overline{\mathbb{C}}$. In proving the main theorem, the arguments of proving of Lemma 1 [5] and the ideas of proving Theorem 1 [3] played a key role. We also established the conditions under which the structure of points and domains is not important. The corresponding results are obtained for the case when the points are placed on the unit circle and in the case of any fixed $n$-radial system of points.