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

Abstract

Let $\mathcal{S}$ denote the class of all functions of the form $f(z)=z+a_2{z}^2+a_3{z}^3+\cdots$ 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 $|{\Phi }_1(n,f)|\le (n-1{)}^2$ for $n\ge 3$. The functional of the form ${\Phi }_{\lambda }(n,f)$ is indeed related to Fekete-Szegő 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 $n\ge 3$, 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 $${\rm Re } \left( 1+\frac{zf''(z)}{f'(z)}\right) > α~\mbox{ for $z\in \ID$}, $$ where $-1/2\le \alpha <1$. Thus, the present article proves the generalized Zalcman conjecture for convex functions of order $\alpha$, $\alpha \in [-1/2,1)$.

12 pages; The article has been with a journal for publication