Sharp Estimates of Hankel Determinants for certain classes of convex univalent functions

Abstract

Let $\mathcal{A}$ denote the class of analytic functions $f$ such that $f(0)=0$ and $f^{\prime }(0)=1$ in the unit disk $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}.$ We examine the properties of the class $\mathcal{C}(\phi )$ defined as $\mathcal{C}(\phi ):=\left\{f\in \mathcal{A}:1+z{f}^{\prime \prime }(z)/f^{\prime }(z)\prec \phi (z):=1+z+m/n{z}^2,\text{ with }2m\le n,\text{ for }m,n\in \mathbb{N}\right\},$ and compute the sharp second and third Hankel determinants for the functions in $\mathcal{C}(\phi )$. Furthermore, we determine the extremal functions for the sharp estimates of the Hankel determinants.