Sharp estimates of products of inner radii of non-overlapping domains in the complex plane

Abstract

In the paper we study a generalization of the extremal problem of geometric theory of functions of a complex variable on non-overlapping domains with free poles: Fix any γ ∈ R + and find the maximum (and describe all extremals) of the functional [r (B 0 , 0) r (B ∞ , ∞)] γ Π n k=1 r (B k , a k ) , where n ∈ N, n >= 2, a 0 = 0, |a k | = 1, B 0 , B ∞ , {B k } n k=1 is a system of mutually non-overlapping domains, a k ∈ B k ⊂ C, k = 0, n, ∞ ∈ B ∞ ⊂ C, (r(B, a) is an inner radius of the domain B ⊂ C at a ∈ B). Instead of the classical condition that the poles are on the unit circle, we require that the system of free poles is an n-radial system of points normalized by some "control" functional. A partial solution of this problem is obtained.

Keywords

• inner radius of a domain
• non-overlapping domains
• radial system of points
• separating transformation
• quadratic differential
• Green’s function