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arXiv · 2208.02975

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

Neha Verma·S. Sivaprasad Kumar
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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.

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Neha Verma, S. Sivaprasad Kumar. 2022-08-05. A Conjecture on $H_3(1)$ For Certain Starlike Functions. https://arxiv.org/abs/2208.02975

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