Extremal problems of nonoverlapping domains with free poles on a circle

Bakhtin, A. K.

doi:10.1007/s11253-006-0118-1

Cited by 4 publications

(6 citation statements)

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“…, n, the domains Γ ∞ are mutually disjoint for all k = 1, n, p = 1, m, and q = 1, 2, and the corresponding points γ (q) k,p , 1, and −1 form a system of 2m + 2 points on the unit circle. In view of Theorem 2 in [6], this yields…”

Section: C(ε D a Nm )| := (Cap C(ε D A Nm )) −1mentioning

confidence: 82%

“…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].…”

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confidence: 99%

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

Bakhtin

V’yun

2007

Nonlinear Oscill

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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

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Section: C(ε D a Nm )| := (Cap C(ε D A Nm )) −1mentioning

confidence: 82%

mentioning

confidence: 99%

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

Bakhtin

V’yun

2007

Nonlinear Oscill

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“…Substituting these inequalities into (4) and using the fact that σ α The statement concerning the equality sign in Theorem 2 is verified by analogy with [5] (the proof of Theorem 2).…”

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confidence: 92%

“…Let with respect to the point a (all definitions used in the present paper can be found, e.g., in [5]; note only that, in contrast to [6], we understand the interior radius with respect to an infinitely remote point in the sense of the definition proposed in [2]). For a Borel set E ⊂ C , we denote by cap E its logarithmic capacity.…”

Section: Introductionmentioning

confidence: 99%

Some extremal problems in the theory of nonoverlapping domains with free poles on rays

Bakhtin

Таргонський

2006

Ukr Math J

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“…The collection of numbers { } ρ k k n =1 is uniquely determined by the system of points A n,2 (see also [13]). Let ) are nonempty and pairwise disjoint for all k = 1, n, then we say that the set D satisfies the generalized condition of nonoverlapping with respect to the system of points A n,2 (respectively, with respect to the system of points A n = { } a k k n =1 )) (see, e.g., [14,15]).…”

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confidence: 99%

Application of a separating transformation to estimates of inner radii of open sets

Bakhtin

V’yun

2007

Ukr Math J

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We obtain solutions of new extremal problems of the geometric theory of functions of a complex variable related to estimates for the inner radii of nonoverlapping domains. Some known results are generalized to the case of open sets. Extremal problems of the geometric theory of functions of a complex variable related to estimates for the inner radii of nonoverlapping domains and open sets arose in connection with Lavrent'ev's work [1], where the problem of the product of conformal radii of two mutually nonoverlapping domains was solved. This work inspired a series of studies of many authors (see, e.g., [2 -16]). Let n, m ∈ N. A system of points A n m , : = { } , : , , , a k n p m k p ∈ = = C 1 1 is called an ( , ) n m -ray system if the following relations hold for all k = 1, n and p = 1, m :arg , a k 1 = arg , a k 2 = … = arg , a k m = : θ k = : θ k n m A ( ) , , 0 = θ 1 < θ 2 < … < θ n < θ n+1 : = 2 π , a n p +1, : = a p 1, .We define the operation of multiplication of an ( , ) n m -ray system A n m , = { } , a k p by a number t > 0 in the following way: t A n m , : = { } , ta k p , k = 1, n, p = 1, m. In the case m = 1, an ( , ) n 1 -ray system of points is called an n-ray system and we use the following simple notation: a k,1 = : a k

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Extremal problems of nonoverlapping domains with free poles on a circle

Cited by 4 publications

References 12 publications

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

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

Some extremal problems in the theory of nonoverlapping domains with free poles on rays

Application of a separating transformation to estimates of inner radii of open sets

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