  Recently, the subclass of starlike functions associated with exponential
function $e^z$, given by ${S}^*_e = \{f(z)\in {S}:{zf'(z)}/{f(z)} \prec e^z,
(z\in \mathbb{D}) \}$ was introduced and studied by Mendiratta $et$ $al.$
~\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 Carath\'{e}odory functions.
