On the problem of V. N. Dubinin for symmetric multiply connected domains

Аннотация

Abstract

The problem of maximum of the functional

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

is considered. Here, \( \upgamma \in \left(0,n\right],{a}_0=0,\kern0.5em \left|{a}_k\right|=1,k=\overline{1,n},{a}_k\in {B}_k\subset \overline{\mathrm{\mathbb{C}}},\kern0.5em k=\overline{0,n},\kern0.5em {\left\{{B}_k\right\}}_{k=1}^n \) are pairwise non-overlapping 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 \( B\subset \overline{\mathrm{\mathbb{C}}} \) with respect to the point a ∈ B. For γ = 1 and n ≥ 2, the problem was formulated as an open problem by V. N. Dubinin in 1994. L. V. Kovalev solved the Dubinin problem in 2000. The article deals with finding the maximum of the functional I_n(γ) for γ > 1.