Separating transformation and extremal problems on nonoverlapping simply connected domains.

We consider the well-known problem of maximum of the functionalInγ=rγB0.0∏k=1nrBkak,<graphic></graphic>where B0, …, Bn are pairwise disjoint domains in ℂ¯<inline-graphic></inline-graphic>, a0 = 0, |ak| = 1, k=1,n¯<inline-graphic></inline-graphic>, are different points of the circle, γ ∈ (0, n], and r(B, a) is the inner radius of the domain B⊂ℂ¯<inline-graphic></inline-graphic> 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. [ABSTRACT FROM AUTHOR]