The second and third Hankel determinants for starlike MA--Minda subclass associated to quadratic polynomials

Let $\mathcal{A}$ denote the class of analytic functions such that $f(0)=0$ and $f^{\mathrm{\prime }}(0)=1$ in the unit disk $\mathbb{D}:=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }<1\}$. In this paper, we discuss the properties of a starlike subclass and compute its second and third Hankel determinants; where the class is defined as ${\mathcal{S}}^{\ast }(\phi ):=\{f\in \mathcal{A}:z{f}^{\mathrm{\prime }}(z)\mathrm{/}f(z)\prec \phi (z):=1+z+m\mathrm{/}n{z}^2,\text{ such that }2m\le n,\text{ where }m,n\in \mathbb{N}\}\mathrm{.}$ Furthermore, we show that the bounds are sharp by determining the extremal functions for the Hankel determinants.