On a product of the inner radii of symmetric multiply connected domains

Abstract

The article is devoted to the study of a quite general problem of the geometric theory of functions on an extreme decomposition of the complex plane. The problem of maximum of the functional

$I_n\left(\text{upgamma}\right)=r^{\text{upgamma}}(B_0,0)\prod _{k=1}^{n}r(B_k,a_k),$

where γ ∈ (0, 1], n ≥ 2, a₀ = 0,$\left|a_k\right|=1,k=\overline{1,n},a_k\in B_k\subset \bar{\mathrm{C}},k=\overline{0,n},{\left\{B_k\right\}}_{k=0}^n$ are pairwise disjoint domains, ${\left\{B_k\right\}}_{k=0}^n$ are symmetric domains with respect to the unit circle, and r(B, a) is the inner radius of the domain B ⊂ $\bar{\mathrm{C}}$ relative to the point a ∈ B, is considered.