Targonskii, A. L.

Zhytomyr State University Library

Targonskii A.L. (Zhitomir, Ukraine)Extremal problem on (2n, 2m− 1)-system points on the rays.For fix number n ∈ N system pointsA2n,2m−1 = {ak,p ∈ C : k = 1, 2n, p = 1, 2m− 1 },we will called on the (2n, 2m − 1)-system points on the rays, if at all k = 1, 2n,p = 1, 2m− 1 the relations are executed:(1)0 < |ak,1| < . . . < |ak,2m−1| <∞;arg ak,1 = arg ak,2 = . . . = arg ak,2m−1 =: θk;0 = θ1 < θ2 < . . . < θn < θn+1 := 2pi.Let’s consider system of angular domains:Pk = {w ∈ C : θk < argw < θk+1}, k = 1, 2n.Let D, D ⊂ C – arbitrary open set and w = a ∈ D, then D(a) the defineconnected component D, the contain point a. For arbitrary (2n, 2m − 1)-systempoints on the rays A2n,2m−1 = {ak,p ∈ C : k = 1, 2n, p = 1, 2m− 1 } and open setD, A2n,2m−1 ⊂ D the define Dk (as,p) connected component set D (as,p)⋂Pk, thecontain point as,p, k = 1, 2n, s = k, k + 1, p = 1, 2m− 1, an+1,p := a1,p.The open set D, A2n,2m−1 ⊂ D satisfied condition meets the condition of un-applied in relation to the system of points (2n, 2m − 1)-system points on the raysA2n,2m−1 if a condition is executed(2) Dk(ak,s)⋂Dk(ak+1,p) = ∅,k = 1, 2n, p, s = 1, 2m− 1 on all corners Pk.The define r(B; a) inner radius domain B ⊂ C with respect to a point a ∈ B.Subject of studying of our work are the following problem.Problem. Let n,m ∈ N, n ≥ 2, m ≥ 2, α ∈ R+. Maximum functional be foundI =n∏k=1m∏p=1rα (D, a2k−1,2p−1) ·n∏k=1m−1∏p=1r (D, a2k−1,2p)××n∏k=1m−1∏p=1rα (D, a2k,2p) ·n∏k=1m∏p=1r (D, a2k,2p−1) ,where A2n,2m−1 – arbitrary (2n, 2m − 1)-system points on the rays, satisfied con-dition (1), D – arbitrary open set, the satisfied condition (2), ak,p ∈ D ⊂ C, andall extremal the describe (k = 1, 2n, p = 1, 2m− 1).1

Extremal problem on (2n,2m−1)-system points on the rays

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