The second and third Hankel determinants for certain classes of functions

Abstract

Let $\mathcal{A}$ denote the class of analytic functions such that $f(0)=0$ and $f^{\prime }(0)=1$ in the unit disk $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}.$ In this paper, we consider ${\mathcal{S}}^{\ast }(\ddot{I}†):=\left\{f\in \mathcal{A}:z{f}^{\prime }(z)/f(z)\prec \ddot{I}†(z):=(1+z/2{)}^2\right\}$, a subclass of starlike functions and we compute the sharp second and third Hankel determinants for the functions in ${\mathcal{S}}^{\ast }(\ddot{I}†)$. Furthermore, we determine the extremal functions for the coefficient bounds of the functions belonging to ${\mathcal{S}}^{\ast }(\ddot{I}†)$.