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In this paper we introduced the class SG* of analytic functions which is related with starlike functions and generating functionof Gregory coefficients. By using bounds on some coefficient functionals for the family of functions with positive real part, we obtain several sharp coefficient bounds on the first six coeffcients and also further sharp bounds on the corresponding Hankel determinants for functions in the class SG*. Additionally, the sharp bounds for logarithmic and inverse coefficients of functions belonging to the considered class SG* were estimated. Mathematics Subject Classification (2010). Priminary 30C45; Secondary 30C50, 30C80.

Let $\mathcal{A}$ be the class of functions $f(z)$ which are analytic in the open unit disk $\mathbb{U}$ with $f(0)=0$ and $f'(0)=1$. For the class $\mathcal{A}$, a new general class $\mathcal{A}_{k}$ is defined. With this general class $\mathcal{A}_{k}$, two interesting classes $\mathcal{S}_{k}^{\ast}(\alpha)$ and $\mathcal{K}_{k}(\alpha)$ concerning classes of starlike of order $\alpha$ in $\mathbb{U}$ and convex of order $\alpha$ in $\mathbb{U}$ are considered.

In the present paper, we obtain the upper bounds for the second Hankel determinant for certain subclasses of analytic and bi-univalent functions. Moreover, several interesting applications of the results presented here are also discussed.