Sharp bounds of third Hankel determinant for a class of starlike functions and a subclass of $q$-starlike functions

Abstract: Following the trend of coefficient bound problems in Geometric Function Theory, in the present paper, we obtain the sharp bound of $|H_3(1)|$ for the class $\mathcal{S}<sup>*$, of starlike functions and $\mathcal{SL}_q<sup>*$, of $q$- starlike functions related with lemniscate of Bernoulli. Bound on the initial class is also an improvement over the existing known bound and the bound on the latter class generalizes the prior known outcome. Further, we determine the extremal functions to prove the sharpness of our results.