Generalized Zalcman conjecture for convex functions of order $\alpha$

Abstract

Let $\mathcal S$ denote the class of all functions of the form $f(z)=z+a_2z^2+a_3z^3+\cdots$ which are analytic and univalent in the open unit disk $\ID$ and, for $\lambda >0$, let $\Phi_\lambda (n,f)=\lambda a_n^2-a_{2n-1}$ denote the generalized Zalcman coefficient functional. Zalcman conjectured that if $f\in \mathcal S$, then $|\Phi_1 (n,f)|\leq (n-1)^2$ for $n\ge 3$. The functional of the form $\Phi_\lambda (n,f)$ is indeed related to Fekete-Szeg\H{o} functional of the $n$-th root transform of the corresponding function in $\mathcal S$. This conjecture has been verified for a certain special geometric subclasses of $\mathcal S$ but the conjecture remains open for $f\in {\mathcal S}$ and for $n > 6$. In the present paper, we prove sharp bounds on $|\Phi_\lambda (n,f)|$ for $f\in \mathcal{F}(\alpha )$ and for all $n\geq 3$, in the case that $\lambda$ is a positive real parameter, where $ \mathcal{F}(\alpha )$ denotes the family of all functions $f\in {\mathcal S}$ satisfying the condition $${\rm Re } \left( 1+\frac{zf''(z)}{f'(z)}\right) > \alpha ~\mbox{ for $z\in \ID$}, $$ where $-1/2\leq \alpha <1$. Thus, the present article proves the generalized Zalcman conjecture for convex functions of order $\alpha$, $\alpha \in [-1/2,1)$.