To the problem of extremal partition of the complex plane

Abstract

We consider one of the classical problems of the geometric theory of functions of a complex variable on a maximum of the functional

\( {\left[r\left({B}_0.0\right)r\left({B}_{\infty },\infty \right)\right]}^{\upgamma}\prod \limits_{k=1}^nr\left({B}_k,{a}_k\right), \)

where n ∈ ℕ, n ≥ 2, γ ∈ ℝ⁺, \( {A}_n={\left\{{a}_k\right\}}_{k=1}^n \) is a system of points such that |a_k| = 1, a₀ = 0, B₀, B_∞, \( {\left\{{B}_k\right\}}_{k=1}^n \) is a system of pairwise nonoverlapping domains, \( {a}_k\in {B}_k\subset \overline{\mathrm{\mathbb{C}}} \), \( k=\overline{0,n} \), \( \infty \in {B}_{\infty}\subset \overline{\mathrm{\mathbb{C}}} \), r(B, a) is the inner radius of the domain \( B\subset \overline{\mathrm{\mathbb{C}}} \) with respect to the point a ∈ B. We have analyzed this problem under some weaker restrictions on pairwise nonoverlapping domains.