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

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 $\mathrm{\mid }{H}_3(1)\mathrm{\mid }\le 1\mathrm{/}9$ is sharp for the class ${\mathcal{S}}_{\mathrm{\wp }}^{\ast }=\{z{f}^{\mathrm{\prime }}(z)\mathrm{/}f(z)\prec \phi (z):=1+z{e}^z\}$. In addition, we also establish bounds for sixth and seventh coefficient, and $\mathrm{\mid }{H}_4(1)\mathrm{\mid }$ for functions in ${\mathcal{S}}_{\mathrm{\wp }}^{\ast }$. The general bounds for two and three-fold symmetric functions related to the Ma-Minda classes ${\mathcal{S}}^{\ast }(\phi )$ of starlike functions are also obtained.