Sufficient and Necessary Conditions for Generalized Distribution Series on Comprehensive Subclass of Analytic Functions

In this paper, we demonstrate a relationship between a generalized distribution series and a comprehensive subclass of analytic functions. The primary aim of this study is to determine a necessary and sufficient condition for the generalized distribution series E?? (?, z) to belong to the inclusive subclass ??(Q3,Q2,Q1,Q0). Necessary and sufficient conditions are also given for the generalized distribution series E?? (?, z)? and the integral operator J? ? (z) to be in the inclusive subclass ??(Q3,Q2,Q1, 0). Further, we provide a number of corollaries, which improve the existing ones that are available in some recent studies. The results presented here not only improve the earlier studies, but also give rise to a number of new results for particular choices of Q3,Q2,Q1 and Q0.