We generalize some classical results in the theory of extreme problems for nonoverlapping domains. Extreme Problems for n-Ray Systems of PointsThe present work is devoted to the solution of new extreme problems for nonoverlapping domains with free poles on the rays and their generalization to some classes of open sets. The formation of this field in the geometric theory of functions of complex variable can be traced back to the classical work by Lavrent'ev [1] devoted to the solution of the problem of product of conformal radii of two mutually nonoverlapping domains. This problem attracts serious attention of numerous mathematicians. At present, the results and methods connected with problems of this kind belong to the well-known field in the geometric theory of functions of complex variable (see, e.g., [2][3][4][5][6][7][8][9][10][11][12][13][14]).We now present the main notation used in what follows. Let N and R be the sets of natural and real numbers, respectively, R + = ( , ) 0 ∞ , let C be a complex plane, and let C = C ∪ {∞} be its one-point compactification. The inner radius of a domain B ⊂ C relative a point a B ∈ is denoted by r B a ( , ). The logarithmic capacity of a set E is denoted by cap E (see, e.g., [2-4]). Also let χ( ) t : = 1 2 1Further, let n, m ∈ N. A system of points A n m , : = a k p , ∈ { } C , k = 1, n, p = 1, m, is called an ( , ) n m -ray system if the following relations hold for all k = 1, n and p = 1, m: 0 < a k,1 < … < a k m , < ∞, arg , a k 1 = arg , a k 2 = … = arg , a k m = : θ k , 0 = θ 1 < θ 2 < … < θ n < θ n +1 : = 2π.In the set of ( , ) n m -ray systems of points, we consider the quantities α k : = 1 1 π θ θ k k + − [ ] , k = 1, n, α n +1 : = α 1 , α 0 : = α n , k n k = ∑ 1 α = 2. Denote P A n m ( ) , = P k k n { } =1 , where P k : = { w ∈C: θ k < arg w < θ k +1 }, k = 1, n. Let z w k ( ) be a singlevalued branch of a multivalued analytic function z = − ( ) − / i e w i k k θ α 1, which univalently maps the domain P k onto the right half plane.For m = 1, an ( , ) n 1 -ray system of points is called an n-ray system of points. In this case, we introduce the following simpler notation: a k,1 = : a k , k = 1, n, A n,1 = : A n .
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