Generalized Zalcman conjecture for convex functions of order α

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 ${{\mathbb{D}} }$ 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–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 it 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

$${\rm{Re}} \Big( 1+\frac{zf''(z)}{f'(z)} \Big) > \alpha \quad \mbox{for } z\in {\mathbb{D}} ,$$

where ${-1/2\leq \alpha < 1}$. Thus, the present article proves the generalized Zalcman conjecture for convex functions of order ${\alpha}$, ${\alpha \in [-1/2,1)}$.