Sharp Coefficient Bounds for Starlike Functions Associated with Cosine Function

Abstract

Let Scos* denote the class of normalized analytic functions f in the open unit disk D satisfying the subordination zf′(z)f(z)≺cosz. In the first result of this article, we find the sharp upper bounds for the initial coefficients a3, a4 and a5 and the sharp upper bound for module of the Hankel determinant |H2,3(f)| for the functions from the class Scos*. The next section deals with the sharp upper bounds of the logarithmic coefficients γ3 and γ4. Then, in addition, we found the sharp upper bound for H2,2Ff/2. To obtain these results we utilized the very useful and appropriate Lemma 2.4 of N.E. Cho et al., 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™ 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 by Hacet.

Keywords

• starlike functions
• subordination
• cosine functions
• initial coefficient bounds
• Hankel determinants
• logarithmic coefficients