Extremal decomposition of the complex plane with restrictions for free poles

Abstract

The problems of extremal decomposition with free poles on a circle are well known in the geometric theory of functions of a complex variable. One of such problems is the problem of maximum of the functional

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

where γ ∈ (0, n], B₀, B₁, B₂,...,B_n, n ≥ 2, are pairwise disjoint domains in \( \overline{\mathrm{C}},{a}_0=0,\left|{a}_k\right|=1,k=\overline{1,n} \) are different points of the circle, r(B, a) is the inner radius of the domain B ⊂ \( \overline{\mathrm{C}} \) relative to the point a ∈ B. We consider a more general problem, in which the restriction \( \left|{a}_k\right|=1,k=\overline{1,n}, \) is replaced by a more general condition.