Properties of Bazilevic Functions Involving $q$-analogue of the Generalized $M$-series (European Journal of Pure and Applied Mathematics, 2025)
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, Bazilevič and $α$-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.
Keywords: Generalized $M$-series; Mittag-Leffler function; Generalized hypergeometric functions.; quantum calculus; Analytic and univalent functions; starlike and convex functions; Bazilevič function; Differential subordination