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

TL;DR

This paper establishes sharp bounds for the third Hankel determinant in classes of starlike and q-starlike functions, improving previous bounds and identifying extremal functions to confirm sharpness.

Contribution

It provides the first sharp bounds for |H_3(1)| in these classes and generalizes existing results for q-starlike functions.

Findings

Sharp bounds for |H_3(1)| in starlike functions.

Sharp bounds for |H_3(1)| in q-starlike functions.

Identification of extremal functions confirming bound sharpness.

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 $\mathcal{S}^*$, of starlike functions and $\mathcal{SL}_q^*$, 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.