Subclass of analytic functions related with Miller-Ross-type Poisson distribution series

The purpose of the present paper is to find a necessary and sufficient condition for the Miller-Ross-type Poisson distribution series to be in the class $\mathcal{P}^{\ast}(\alpha,\beta,\gamma)$ of analytic functions with negative coefficients. Also, we investigate several inclusion properties of the classes of Janowski type close-to-starlike functions, Janowski type close-to-convex functions and Janowski type quasi-convex functions associated with the operator $\mathbb{I}_{\theta,\epsilon}^s$ defined by this distribution. Further, we consider an integral operator related to the Miller-Ross-type Poisson distribution series. Several corollaries and consequences of the main results are also considered.