Sharp Coefficient Bounds for Starlike Functions Associated with Cosine Function ^†

Let ${\mathcal{S}}_{cos}^{*}$ denote the class of normalized analytic functions f in the open unit disk $\mathbb{D}$ satisfying the subordination ${\displaystyle \frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}}\prec cosz$. In the first result of this article, we find the sharp upper bounds for the initial coefficients $a_3$, $a_4$ and $a_5$ and the sharp upper bound for module of the Hankel determinant $|H_{2,3}\left(f\right)|$ for the functions from the class ${\mathcal{S}}_{cos}^{*}$. The next section deals with the sharp upper bounds of the logarithmic coefficients ${\gamma }_3$ and ${\gamma }_4$. Then, in addition, we found the sharp upper bound for $\left|H_{2,2}\left(F_f/2\right)\right|$. To obtain these results we utilized the very useful and appropriate Lemma 2.4 of N.E. Cho et al. [Filomat 34(6) (2020), 2061–2072], which gave a most accurate description for the first five coefficients of the functions from the Carathéodory’s functions class, and provided a technique for finding the maximum value of a three-variable function on a closed cuboid. All the maximum found values were checked by using MAPLE™ 2016 computer software, and we also found the extremal functions in each case. All of our most recent results are the best ones and give sharp versions of those recently published in [Hacet. J. Math. Stat. 52, 596–618, 2023].