A Conjecture on $H_3(1)$ For Certain Starlike Functions

TL;DR

This paper proves a sharp bound for the third Hankel determinant for a specific class of starlike functions and establishes bounds for higher coefficients and determinants, advancing the understanding of geometric function theory.

Contribution

It confirms a conjecture on the third Hankel determinant for a class of starlike functions and derives bounds for related coefficients and determinants, extending existing results.

Findings

Proved that |H_3(1)| q 1/9 for the class _{\u00a8}^*.

Established bounds for sixth and seventh coefficients.

Derived bounds for |H_4(1)| and for symmetric functions in Ma-Minda classes.

Abstract

We prove a conjecture concerning the third Hankel determinant, proposed in ``Anal. Math. Phys., https://doi.org/10.1007/s13324-021-00483-7", which states that $|H_3(1)|\leq 1/9$ is sharp for the class $\mathcal{S}_{\wp}^{*}=\{zf'(z)/f(z) \prec \varphi(z):=1+ze^z\}$. In addition, we also establish bounds for sixth and seventh coefficient, and $|H_4(1)|$ for functions in $\mathcal{S}_{\wp}^{*}$. The general bounds for two and three-fold symmetric functions related to the Ma-Minda classes $\mathcal{S}^*(\varphi)$ of starlike functions are also obtained.