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 $|H_3(1)|\le 1/9$ is sharp for the class ${\mathcal{S}}_{\wp }^{\ast }=\{z{f}^{\prime }(z)/f(z)\prec \phi (z):=1+z{e}^z\}$. In addition, we also establish bounds for sixth and seventh coefficient, and $|H_4(1)|$ for functions in ${\mathcal{S}}_{\wp }^{\ast }$. 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.