Ukrainian Mathematical Journal

, Volume 70, Issue 9, pp 1477–1483

# Inequalities for the Inner Radii of Symmetric Disjoint Domains

• A. K. Bakhtin
• I. V. Denega
• L. V. Vygovskaya
Article

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].

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## Authors and Affiliations

• A. K. Bakhtin
• 1
• I. V. Denega
• 1
• L. V. Vygovskaya
• 1
1. 1.Institute of MathematicsUkrainian National Academy of SciencesKyivUkraine