Applications of Livingston-type inequalities to the generalized Zalcman functional

TL;DR

This paper derives sharp bounds for a generalized Zalcman coefficient functional across various classes of univalent functions, extending previous inequalities and addressing open conjectures in the field.

Contribution

It generalizes Livingston-type inequalities to the Zalcman functional for multiple classes of univalent functions and proves an asymptotic version of the Zalcman conjecture.

Findings

Sharp estimates for the generalized Zalcman functional.

Generalization of Ma's inequality for starlike functions.

Asymptotic validation of the Zalcman conjecture.

Abstract

We obtain sharp estimates for a generalized Zalcman coefficient functional with a complex parameter for the Hurwitz class and the Noshiro-Warschawski class of univalent functions as well as for the closed convex hulls of the convex and starlike functions by using an inequality from [6]. In particular, we generalize an inequality proved by Ma for starlike functions and answer a question from his paper [16]. Finally, we prove an asymptotic version of the generalized Zalcman conjecture for univalent functions and discuss various related or equivalent statements which may shed further light on the problem.