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

\( {I}_n\left(\upgamma \right)={r}^{\upgamma}\left({B}_0,0\right)\prod \limits_{k=1}^nr\left({B}_k,{a}_k\right), \)

where γ ∈ (0, 1], n ≥ 2, a₀ = 0,\( \left|{a}_k\right|=1,k=\overline{1,n},\kern0.5em {a}_k\in {B}_k\subset \overline{\mathrm{C}},k=\overline{0,n},{\left\{{B}_k\right\}}_{k=0}^n \) 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 ⊂ \( \overline{\mathrm{C}} \) relative to the point a ∈ B, is considered.