Separating transformation and extremal problems on nonoverlapping simply connected domains

Abstract

We consider the well-known problem of maximum of the functional

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

where B₀, …, B_n are pairwise disjoint domains in $\bar{\mathbb{C}}$, 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 $B\subset \bar{\mathbb{C}}$ 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.