Coefficient estimates for starlike and convex functions associated with cosine function

Abstract

This paper deals with the new classes $\mathcal{S}^{\ast}_{\cos}$ and $\mathcal{S}^{\,\mathbf{c}}_{\cos}$ of starlike and convex functions, respectively, associated with the cosine function. We give initial coefficient bounds for the first seven coefficients of the functions that belong to these classes, and we evaluate the upper bounds for the Hankel determinant of order three and four. We found the upper bound of Zalcman functional for the above mentioned classes for the cases $n=3$ and $n=4$, showing that the Zalcman conjecture holds for these values. Moreover, we determined lower and upper bounds for the difference $\vert a_{4}\vert-\vert a_{3}\vert$ of the coefficients for the functions that belong to these classes.