On the generalized Zalcman functional $\lambda a_n^2-a_{2n-1}$ in the close-to-convex family

TL;DR

This paper investigates bounds for a generalized Zalcman functional in the close-to-convex function class, confirming the conjecture for specific cases and extending understanding of coefficient inequalities in univalent function theory.

Contribution

It provides new bounds for the generalized Zalcman coefficient functional in the close-to-convex class, including the case that settles an open problem for n=3.

Findings

Established bounds for |_n^2 - a_{2n-1}| in close-to-convex functions.

Confirmed the Zalcman conjecture for n=3 in the close-to-convex class.

Extended the inequality to a generalized form with parameter mbda.

Abstract

Let ${\mathcal S}$ denote the class of all functions $f(z)=z+\sum_{n=2}^{\infty}a_{n}z^{n}$ analytic and univalent in the unit disk $\ID$. For $f\in {\mathcal S}$, Zalcman conjectured that $|a_n^2-a_{2n-1}|\leq (n-1)^2$ for $n\geq 3$. This conjecture has been verified only certain values of $n$ for $f\in {\mathcal S}$ and for all $n\ge 4$ for the class $\mathcal C$ of close-to-convex functions (and also for a couple of other classes). In this paper we provide bounds of the generalized Zalcman coefficient functional $|\lambda a_n^2-a_{2n-1}|$ for functions in $\mathcal C$ and for all $n\ge 3$, where $\lambda$ is a positive constant. In particular, our special case settles the open problem on the Zalcman inequality for $f\in \mathcal C$ (i.e. for the case $\lambda =1$ and $n=3$).