On the one extremal problem on the Riemann sphere

. 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.


Introduction
This paper belongs to the theory of extremal problems on classes of non-overlapping domain, which is a separate direction in geometric theory of functions of a complex variable.The begin of these investigations associated with the paper of M. A. Lavrent'ev [1] in 1934.He found the maximum of some functional with respect to two simply connected domains with two fixed points.We note that this result was needed him for applying to some aerodynamics problems.In 1947, G. M. Goluzin solved a similar problem for three fixed points on the complex plane [2].Then the topic began to evolve rapidly.In this connection we may recall papers of many authors, including Y. E. Alenitsina, M. A. Lebedev, J. Jenkins, P. M. Tamrazov, P. P. Kufareva and others.Using the idea of P. M. Tamrazov, in 1975 G. P. Bakhtin solved first the problem with so-called "free poles" on the unit circle, see, e.g., [3].
An important step for the development of this topic was papers of V. N. Dubinin.He developed a new method of research that is method of piecewise-separating transformation.He also first solved numerous of extremal problems for an arbitrary but fixed multi connected non-overlapping domains (see, e.g., [4], [5], [6]).Now this type of extremal problems is used for investigations in holomorphic dynamics.
In the present paper we use the mentioned about Bakhtin's method.

Theory
Let N, R and C be the sets of natural, real and complex numbers respectively.Let C := C ∪ {∞} be the Riemannian sphere and R + := (0, ∞).
We fix a number n ∈ N. A system of points A n = {a k ∈ C : k = 1, n}, we will called the n-equiangular system of points on rays, if for all k = 1, n, the following relations are satisfied: We introduce the following "managing" functionals for an arbitrary n-equiangular system of points on rays where χ(t) = 1 2 (t + 1 t ), t ∈ R + .Let us consider a system of angular domains: Let {B k , B ∞ } be an arbitrary non-overlapping domains such that The value of r(B, a) := exp (h B,a (z)) the define of inner radius domain B ⊂ C with respect to a point a ∈ B (see [4], [5], [6], [11], [12], [13]).We use the concept of a quadratic differential.Recall that a quadratic differential on a Riemann surface S is a map for some function φ U defined on z(U ).
Suppose that two charts z : U → C and w : V → C on S overlap, and let be the transition function.If φ is represented both as φ U (z)dz 2 and φ V (w)dw 2 on U ∩ V , then we have One way to say this is that quadratic differentials transform under pull-backs by the square of the derivative.As the main results associated with it can be found in [14].

Results
In this paper we investigate the following problems.
We intend to find a maximum of the functional and to describe all its extremals, if A n be an arbitrary n-equiangular system of points on rays satisfying the condition (1), and {B k , B ∞ } be an arbitrary set of non-overlapping domains, k=1 be an n-equiangular system of points on rays, and be an arbitrary set of non-overlapping domains.Then we have the inequality ) .
The equality sign holds, if points {a k } and domains B k , B ∞ are the poles and the circular domains of the quadratic differential where When α = n 2 we obtain the result follows.
Let also A n = {a k } n k=1 be an n-equiangular system of points on rays, and be an arbitrary set of non-overlapping domains.Then we have the inequality ) .
The equality sign holds, if points {a k } and domains B k , B ∞ are the poles and the circular domains of the quadratic differential Proof of Theorem.The proof leans on a method of the piece-dividing transformation developed by V. Dubinin (see [4], [5], [6]).
The function realizes univalent and conformal transformations of domain P k on the right half-plane Rez > 0 for all k = 1, n.
Taking into account the ratio of the last assertion of from ( 6), we have: where D ∞ , D 1 , D 2 -the circular domains of the quadratic differential International Journal of Advanced Research in Mathematics Vol. 4 In the quadratic differentials (8) make the change from the formula (3).
We obtain the differential (2).Using the relation (7) we finally obtain where points {a k } and domains B k , B ∞ are the poles and the circular domains of the quadratic differential (2).The theorem is proved.