Separating transformation and extremal problems on nonoverlapping simply connected domains

Abstract

We consider the well-known 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 B₀, …, B_n are pairwise disjoint domains in \( \overline{\mathrm{\mathbb{C}}} \), a₀ = 0, |a_k| = 1, \( k=\overline{1,n} \), are different points of the circle, γ ∈ (0, n], and r(B, a) is the inner radius of the domain \( B\subset \overline{\mathrm{\mathbb{C}}} \) relative to the point a. In the case of simply connected domains for n=2, 3, and 4, we have obtained the solution of this problem for the maximum interval of values of the parameter γ.