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

TL;DR

This paper derives sharp bounds for the third Hankel determinant of a specific class of starlike functions associated with the exponential function, improving previous results by analyzing the fourth coefficient of Carathéodory functions.

Contribution

It provides the first sharp bounds for the third Hankel determinant for the class of starlike functions related to e^z, using a new coefficient expression.

Findings

Sharp bounds for third Hankel determinant obtained

Improved upon previous results in the literature

New expression for the fourth coefficient of Carathéodory functions

Abstract

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.