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

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 $\mathcal{S}<em>{\wp}^{{*}={zf&#39;(z)/f(z)} \prec \varphi(z):=1+ze<sup>z}$. In addition, we also establish bounds for sixth and seventh coefficient, and $|H_4(1)|$ for functions in $\mathcal{S}</em>{\wp}<sup>{*}$. The general bounds for two and three-fold symmetric functions related to the Ma-Minda classes $\mathcal{S}<sup>*(\varphi)$ of starlike functions are also obtained.