Sufficient conditions in the two-functional conjecture for univalent functions

TL;DR

This paper investigates the two-functional conjecture in univalent function theory, using the Loewner differential equation to establish sufficient conditions and compare them with necessary ones.

Contribution

It introduces new sufficient conditions for the conjecture using Loewner theory and analyzes their relation to existing necessary conditions.

Findings

Sufficient conditions are derived for the two-functional conjecture.

Comparison shows the sufficiency conditions are close to necessary conditions.

The results advance understanding of extremal univalent functions.

Abstract

The two-functional conjecture says that if a function f analytic and univalent in the unit disk maximizes Re{L} and Re{M} for two continuous linear functionals L and M, L is not equal to cM for any c>0, then f is a rotation of the Koebe function. We use the Loewner differential equation to obtain sufficient conditions in the two-functional conjecture and compare the sufficient conditions with necessary conditions.