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 ???(?,?) to belong to the inclusive subclass ??(?3,?2,?1,?0) . Necessary and sufficient conditions are also given for the generalized distribution series ???(?,?)? and the integral operator ???(?) to be in the inclusive subclass ??(?3,?2,?1,0) . 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 ?3,?2,?1 and ?0 .