Certain differential subordination results for univalent functions associated with $q$-Salagean operators

In this paper, we employ the concept of the $q$-derivative to derive certain differential and integral operators, $D_{q,\lambda }^n$ and $I_{q,\lambda }^n$, resp., to generalize the class of Salagean operators over the set of univalent functions. By means of the new operators, we establish the subclasses $M_{q,\lambda }^n$ and $D_{q,\lambda }^n$ of analytic functions on an open unit disc. Further, we study coefficient inequalities for each function in the given classes. Over and above, we derive some properties and characteristics of the set of differential subordinations by following specific techniques. In addition, we study the general properties of $D_{q,\lambda }^n$ and $I_{q,\lambda }^n$ and obtain some interesting differential subordination results. Several results are also derived in some details.