Problem on extremal decomposition of the complex plane

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

rγ(B0,0)∏k=1nr(Bk,ak),$${r^\gamma }({B_0},0)\prod\limits_{k = 1}^n r ({B_k},{a_k}),$$

where B₀, B₁, B₂,..., Bn, n ≥ 2, are pairwise disjoint domains in ¯𝔺, a₀ = 0, |ak| = 1, k=1,n¯$k = \overline {1,n}$and γ ∈ 2 (0; n], r(B, a) is the inner radius of the domain, B ⊂ ¯𝔺, with respect to a point a ∈ B. In the paper we consider a more general problem in which restrictions on the geometry of the location of points ak, k=1,n¯$k = \overline {1,n}$are weakened.