The second and third Hankel determinants for certain classes of functions

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