Properties of Bazilevic Functions Involving $q$-analogue of the Generalized $M$-series

Abstract

With primary motive to unify and extend the various well-known studies, we define a new family of differential operator using the $q$-analogue of the generalized $M$-series. The generalized $M$-series unifies two well-known and extensively used special functions namely {\em generalized hypergeometric function} and {\em Mittag-Leffler function}. Making use of the defined operator, we define a new family of analytic functions expressed as a combination two differential characterizations. The combination of differential characterizations involving the operator not only unifies studies of starlike, convex, Bazilevi\v{c} and $\alpha$-convex function classes, it extends to new classes.  Estimates involving the initial coefficients of the functions, which belong to the defined function class are our main results. Some examples along with graphs have been used to establish the inclusion and closure properties.