Abstract. Sharp estimates of product of inner radii for pairwise disjoint domains are obtained. In particular, we solve an extremal problem in the case of arbitrary finite number of rays containing arbitrary even number of free poles.
The paper of Lavrent'ev [1] was the beginning of geometrical theory of functions of the complex variable. He solved a problem on the product of conformal radiuses of two non-overlapping domains. In many papers (see [2] -[13]) the Lavrent'ev's result are generalized. In this paper are obtained the new results of this direction.
A. TargonskiiLet N, R and C be the sets of natural, real and complex numbers respectively. We define C := C {∞} and R + := (0, ∞). Let n, m, d ∈ N such that m = nd. Consider the set of natural numbers(1)The following system of pointsare called the generalized (n, d)-equiangular system of points on the rays, if the condition (1) is fulfilled and if for all k = 1, n, p = 1, m k the following relations are true:
In this work derivation of accurate estimate the production inner radius non-overlapping domains and open set. The problems arise such type in the first time in work [1]. It is late result this work generalize and strengthen in works [2 - 13]. In works [7, 8, 10] introduce the general systems points, the name n-radial systems points. In this work a success the draw generalize some results the work [7].
We solve the extremal problem of finding the maximum of the functional n Y kD1 m k Y pD1 r.B k;p ; a k;p /; where m k 2 N; n X kD1 m k D m; n; m 2 N; 0 < ja k;1 j < : : : < ja k;m k j < 1; arg a k;1 D arg a k;2 D : : : D arg a k;m k DW Â k ; k D 1; n; 0 D Â 1 < Â 2 < : : : < Â n < Â nC1 WD 2 ; and r.B; a/ is the inner radius of a domain B with respect to a point a 2 B: The points a k;p ; k D 1; n; p D 1; m k ; are not fixed. Some generalizations of these results are also considered.The aim of the present paper is to obtain exact estimates for the products of inner radii of collections of mutually nonoverlapping domains. For the first time, problems of this type were considered in [1], where, in particular, the problem of the product of conformal radii of two mutually nonoverlapping simply connected domains was posed and solved. This result attracted attention of experts in the geometric theory of functions of complex variables and stimulated numerous investigations aimed at a generalization of this result (see, e.g., [2][3][4][5][6][7][8][9][10][11][12][13][14][15]).At present, much attention is given to the study of extremal problems of nonoverlapping domains with free poles of the corresponding quadratic differentials (see, e.g., [3][4][5][6][7]). The collection of free poles forms a system of points of the complex plane whose geometric properties affect the possibility of the complete solution of a specific extremal problem. In the case where free poles are located on a certain fixed circle, several extremal problems were completely solved for nonoverlapping domains and their generalizations. In [7,8,10], more general systems of points (called n-ray systems) were introduced. In the present paper, we generalize the notion of an n-ray system of points.Let N and R be the sets of natural and real numbers, respectively, let C be the complex plane, let C D C S f1g be its one-point compactification or the Riemann sphere, and let R C D .0; 1/: Assume that n; m; d 2 N and m D nd: Consider all possible collections of natural numbers fm k g n kD1 such that n X kD1 m k D m:(1)
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