Some subordination results for certain class with complex order defined by Salagean type $q$-difference operator

The theory of the basic quantum calculus (that is, the basic $q$-calculus) plays important roles in many diverse areas of the engineering, physical and mathematical science. Making use of the basic definitions and concept details of the $q$-calculus, Govindaraj and Sivasubramanian [10] defined the Salagean type $q$-difference ($q$-derivative) operator. In this paper, we introduce a certain subclass of analytic functions with complex order in the open unit disk by applying the Salagean type $q$-derivative operator in conjunction with the familiar principle of subordination between analytic functions. Also, we derive some geometric properties such as sufficient condition and several subordination results for functions belonging to this subclass. The results presented here would provide extensions of those given in earlier works.