Generalized Zalcman conjecture for convex functions of order $α$

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 $λ>0$, let $Φ_λ(n,f)=λa_n^2-a_{2n-1}$ denote the generalized Zalcman coefficient functional. Zalcman conjectured that if $f\in \mathcal S$, then $|Φ_1 (n,f)|\leq (n-1)^2$ for $n\ge 3$. The functional of the form $Φ_λ(n,f)$ 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 $|Φ_λ(n,f)|$ for $f\in \mathcal{F}(α)$ and for all $n\geq 3$, in the case that $λ$ is a positive real parameter, where $\mathcal{F}(α)$ 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 $-1/2\leq α<1$. Thus, the present article proves the generalized Zalcman conjecture for convex functions of order $α$, $α\in [-1/2,1)$.

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