Quadratic differentials in problems of the product of inner radii of nonoverlapping domains

Abstract

We study applications of the theory of quadratic differentials to the solution of new extremal problems of nonoverlapping domains with free poles on rays and generalize known results to a certain class of open sets. This paper is devoted to the solution of new extremal problems of nonoverlapping domains with free poles on rays and their generalization to certain classes of open sets. This branch of the geometric theory of functions of a complex variable arose in connection with classical Lavrent'ev's work [1], where the problem of the product of conformal radii of two mutually nonintersecting domains was solved. This problem aroused interest of many mathematicians. At present, the results and methods related to problems of this type form a well-known branch of the geometric theory of functions of a complex variable (see, e.g., [2][3][4][5][6][7]). For the first time, the fundamental role of quadratic differentials as a universal tool for the solution of extremal problems of the geometric theory of functions was noted by Teichmuller [8]. He formulated a principle according to which a solution of every extremal problem of this type is associated with a certain quadratic differential. This principle has later been expressed in the form of a so-called "general coefficient theorem," which was formulated and proved by Jenkins [9]. The method of quadratic differentials and its applications were further developed by Tamrazov [10], who have also substantially complemented the Jenkins theorem indicated above.Let us formulate the main results of the present paper. Assume that N is the set of natural numbers, R is the set of real numbers, R + = (0, ∞), C is the complex plane, and C = C ∪ {∞} is its one-point compactification. Let r(B, a) denote the inner radius of the domain B ⊂ C with respect to a point a ∈ B (see, e.g., [2,9,11]), let cap E denote the logarithmic capacity of the set E (see, e.g., [2,11]), and letAssume that n, m ∈ N. In the plane C, we consider an (n, m)-ray system of points A n,m = {a k,p }, k = 1, n, p = 1, m, such thatarg a k,1 = arg a k,2 = . . . = arg a k,m := θ k , 0 = θ 1 < θ 2 < . . . < θ n < θ n+1 = 2π.For every (n, m)-ray system, we define a collection of values

A.K. Bakhtin. Scientific legacy

“…Under the O. K. Bakhtin supervision A. L. Targonskyi (2006) [12][13][14][41][42][43][44][45], V. E. Vyun (2008) [17][18][19], R. V. Podvisotskii [5,11], I. Y. Vygovska (2012) [46][47][48], I. V. Denega (2013) [7][8][9], Y. V. Zabolotnyi (2014) [20,22,26,27], L. V. Vyhivska (2019) [10,15,16], I. Y. Dvorak (2019) [10,21] defended their candidate theses.…”