On certain subclasses of analytic functions associated with Poisson distribution series

Abstract

In this paper, we find the necessary and sufficient conditions, inclusion relations for Poisson distribution series $\mathcal{K}(m,z)=z+{\sum }_{n=2}^{\infty }\frac{m^{n-1}}{(n-1)!}{e}^{-m}{z}^n$ belonging to the subclasses $\mathcal{S}(k,\lambda )$ and $\mathcal{C}(k,\lambda )$ of analytic functions with negative coefficients. Further, we consider the integral operator $\mathcal{G}(m,z)={\int }_0^z\frac{\mathcal{F}(m,\zeta )}{\zeta }d\zeta$ belonging to the above classes.