# Inequalities for the Inner Radii of Symmetric Disjoint Domains

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We study the following problem: Let *a*_{0} = 0, ∣*a*_{1} ∣ = … = ∣ *a*_{n} ∣ = 1, \( {a}_k\in {B}_k\subset \overline{\mathbb{C}} \), where *B*_{0}, … , *B*_{n} are mutually disjoint domains and *B*_{1}, … , *B*_{n} 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*(*B*_{k}, *a*_{k}) is the inner radius of *B*_{k} with respect to *a*_{k}. For *γ* = 1 and *n* ≥ 2, the problem was solved by Kovalev. We solve this problem for *γ* ∈ (0, *γ*_{n}], *γ*_{n} = 0.38*n*^{2}, and *n* ≥ 2 under the additional assumption imposed on the angles between the neighboring lines of the segments [0, *a*_{k}].

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