On sharp third Hankel determinant for certain starlike functions

TL;DR

This paper establishes sharp bounds for the third Hankel determinant of specific starlike functions, extending previous results to new parameter ranges.

Contribution

It provides the first sharp bounds for the third Hankel determinant for starlike functions of order α in specified intervals, expanding existing theoretical understanding.

Findings

Sharp bounds for the third Hankel determinant are derived.

Results extend previous bounds to new ranges of α.

The bounds are proven to be optimal.

Abstract

In this paper, we provide an estimation for the sharp bound of the third Hankel determinant of starlike functions of order $\alpha$, where $\alpha$ ranges in the interval $[0, 1/6]\cup \{1/2\}$ and thereby extending the result of Rath et al. (Complex Anal Oper Theory: No. 65, 16(5), 8 pp 2022).