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

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 $n=3$).