Fourth Hankel determinant and logarithmic coefficients for starlike functions associated with the cosine function

Abstract

UDC 517.53, 517.54

Let $\mathcal{\mathcal{S}}_{\cos}^{\ast}$ be a class of normalized analytic functions $f$ in an open unit disk $\mathbb{D}$ satisfying the subordination $\dfrac{zf^{\prime}(z)}{f(z)}\prec\cos z.$ The aim of the present article is to find upper bounds for the module of some coefficients, of the fourth Hankel determinant $H_{4,1}(f)$ for the function class $\mathcal{\mathcal{S}}_{\cos}^{\ast},$ and of some functionals defined by using the expansion coefficients in Taylor series. Moreover, we also find the upper bounds for the fifth and sixth logarithmic coefficients obtained for the functions from the same class. Note that some of our results are the best possible. The results of our paper improve numerous versions of the results recently presented in [K. Marimuthu, J. Uma, and T. Bulboacă, Hacet. J. Math. Stat., 52, No. 3, 596 (2023)]. The tools used in the proofs have been recently obtained in [N. E. Cho, B. Kowalczyk, A. Lecko, and B. ´Śmiarowska, Filomat, 34, No. 6, 2061 (2020)]. They are combined with the results from [F. Carlson, Ark. Mat. Astr. Fys., 27A, No. 1, 8 (1939)], and with the method aimed at finding the extrema of real functions of many variables.