The second and third Hankel determinants for certain convex subclass 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 the present paper, we consider $\mathcal{C}(\ddot{I}†):=\left\{f\in \mathcal{A}:1+z{f}^{\prime \prime }(z)/f^{\prime }(z)\prec \ddot{I}†(z):=(1+z/2{)}^2\right\}$, as subclass of convex functions and compute the sharp second and third Hankel determinants for functions in $\mathcal{C}(\ddot{I}†)$.