Targonskii, A. L.

Zhytomyr State University Library

Targonskii A.L. (Zhitomir, Ukraine)Extremal problem with variable quantity of points on rays.Let n,m, d ∈ N, m = nd. Let’s consider a set of natural numbers {mk}nk=1 suchthat(1)n∑k=1mk = m.System pointsAn,d :={ak,p ∈ C : k = 1, n, p = 1,mk},the satisfied condition (1), the define generalized (n, d)-equiangular with variablequantity of points on rays, if at all k = 1, n and p = 1,mk realize relation:(2)0 < |ak,1| < . . . < |ak,mk | <∞;arg ak,1 = arg ak,2 = . . . = arg ak,mk =2pin (k − 1).Let r(B, a) – inner radius domain B ⊂ C with respect to a point a ∈ B.Subject of studying of our work are the following problems.Problem 1. Let n,m, d ∈ N, m = nd, n ≥ 2. To find a maximum(r (B0, 0) · r (B∞,∞))n24 ·n∏k=1mk∏p=1r(Bk,p, ak,p),rn24 (B0, 0) ·n∏k=1mk∏p=1r(Bk,p, ak,p),rn24 (B∞,∞) ·n∏k=1mk∏p=1r(Bk,p, ak,p),n∏k=1mk∏p=1r(Bk,p, ak,p),where An,d = {ak,p} – arbitrary generalized (n, d)-equiangular with variable quan-tity of points (2), and {Bk,p} – arbitrary set partially non-overlapping domains, or,somewhat, partially non-overlapping domains, ak,p ∈ Bk,p ⊂ C, and to describe allextremals (k = 1, n, p = 1,mk).Problem 2. Let n,m, d ∈ N, m = nd, n ≥ 2. To find a maximum(r (D, 0) · r (D,∞))n24 ·n∏k=1mk∏p=1r(D, ak,p), rn24 (D, 0) ·n∏k=1mk∏p=1r(D, ak,p),rn24 (D,∞) ·n∏k=1mk∏p=1r(D, ak,p),n∏k=1mk∏p=1r(Bk,p, ak,p),where An,d = {ak,p} – arbitrary generalized (n, d)-equiangular with variable quan-tity of points (2), and D – arbitrary open set, ak,p ∈ D ⊂ C, and to describe allextremals (k = 1, n, p = 1,mk).1

Extremal problem with variable quantity of points on rays

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