  In this paper we introduced the class $\mathcal{S}_{G}^{\ast }$ of analytic
functions which is related with starlike functions and generating function of
Gregory coefficients. By using bounds on some coefficient functionals for the
family of functions with positive real part, we obtain for functions in the
class $\mathcal{S}_{G}^{\ast }$ several sharp coefficient bounds on the first
six coeffcients and also further sharp bounds on the corresponding Hankel
determinants.
