Inequalities for the Inner Radii of Symmetric Disjoint Domains

Abstract

We study the following problem: Let a0 = 0, ∣a1 ∣ = … = ∣ an ∣ = 1, $$ {a}_k\in {B}_k\subset \overline{\mathbb{C}} $$ , where B0, … , Bn are mutually disjoint domains and B1, … , Bn are symmetric about the unit circle. It is necessary to find the exact upper bound for the product $$ {r}^{\gamma}\left({B}_0,0\right){\prod}_{k=1}^nr\left({B}_k,{a}_k\right) $$ , where r(Bk, ak) is the inner radius of Bk with respect to ak. For γ = 1 and n ≥ 2, the problem was solved by Kovalev. We solve this problem for γ ∈ (0, γn], γn = 0.38n2, and n ≥ 2 under the additional assumption imposed on the angles between the neighboring lines of the segments [0, ak].