Weakened problem on extremal decomposition of the complex plane

Abstract

The paper deals with the problem of the maximum of the functional r γ (B 0 , 0) n ∏ k=1 r (B k , a k) , where B 0 ,..., B n , n 2, are pairwise disjoint domains in C, a 0 = 0, |a k | = 1, k ∈ {1,. .. , n} and γ ∈ (0, n] (r(B, a) is the inner radius of the domain B ⊂ C with respect to a). We show that the functional attains its maximum at a configuration of the domains B k and the points a k possessing rotational n-symmetry. The proof is due to Dubinin [1] for γ = 1 and to Kuz'mina [3] for 0 < γ < 1. Subsequently, Kovalev [4] solved this problem for n 5 under the additional assumption that the angles between neighbouring line segments [0, a k ] do not exceed 2π/ √ γ.

On the Koebe Quarter Theorem for Polynomials

“…An alternative approach to the problem of estimating the functionals in complex domains can be found in [2,3,12,14]…”