Third Hankel Determinant For A Class Of Starlike Functions Associated With Exponential Function

Recently, the subclass of starlike functions associated with exponential function $e^z$, given by $S_e^{\ast }=\{f(z)\in S:z{f}^{\mathrm{\prime }}(z)\mathrm{/}f(z)\prec e^z,(z\in \mathbb{D})\}$ was introduced and studied by Mendiratta $et$ $al\mathrm{.}$ ~\cite{mendiratta2015subclass}. In this article, we obtain sharp bound for third Hankel determinant for the class $S_e^{\ast }$, by improving the already known results, in this direction. In our methodology, we use the newly obtained expression for the fourth coefficient of Carathéodory functions.