On an extremal problem for nonoverlapping domains *

TL;DR

This paper investigates the extremal values of a functional defined on pairs of univalent functions in nonoverlapping domains, deriving bounds expressed through elliptic integrals using the method of internal variations.

Contribution

It introduces a new extremal problem for pairs of univalent functions in nonoverlapping domains and solves it using the method of internal variations, deriving bounds involving elliptic integrals.

Findings

The range of the functional is bounded by a curve defined via elliptic integrals.

The method of internal variations effectively solves the extremal problem.

Explicit bounds depend on parameters of the functional.

Abstract

The paper considers the problem of finding the range of functional I = J f (z 0), f (z 0), F ($\zeta$ 0), F ($\zeta$ 0) , defined on the class M of pairs functions (f (z), F ($\zeta$)) that are univalent in the system of the disk and the interior of the disk, using the method of internal variations. We establish that the range of this functional is bounded by the curve whose equation is written in terms of elliptic integrals, depending on the parameters of the functional I.