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

Neha Verma; S. Sivaprasad Kumar

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

Complex Variables 2022-08-08 v1

Authors: Neha Verma , S. Sivaprasad Kumar

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.

Keywords

starlike functions univalent functions function spaces and inequalities

Cite

@article{arxiv.2208.02975,
  title  = {A Conjecture on $H_3(1)$ For Certain Starlike Functions},
  author = {Neha Verma and S. Sivaprasad Kumar},
  journal= {arXiv preprint arXiv:2208.02975},
  year   = {2022}
}

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