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 \({\alpha}\), \({\alpha \in [-1/2,1)}\).