To the problem of extremal partition of the complex plane

  • Iryna V. Denega
  • Bogdan A. Klishchuk


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 |ak| = 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.


Inner radius of domain nonoverlapping domains radial systems of points separating transformation quadratic differential Green function 


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Mathematics of the NAS of UkraineKievUkraine

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