**1. Introduction**

The study of *q*-extension of calculus and *q*-analysis has attracted and motivated many researchers because of its applications in different parts of mathematical sciences. Jackson was one of the main contributors among all mathematicians who initiated and established the theory of *q*-calculus [1,2]. As an interesting sequel to [3], in which the *q*-derivative operator was used for the first time for studying the geometry of *q*-starlike functions, a firm footing of the usage of the *q*-calculus in the context of Geometric Function Theory was provided and the basic (or *q*-) hypergeometric functions were first used in Geometric Function Theory in a book chapter by Srivastava (see, for details, [4] (pp. 347 et seq.)). The theory of *q*-starlike functions was later extended to various families of *q*-starlike functions by Agrawal and Sahoo in [5] (see also the recent investigations on this subject by Srivastava et al. [6–11]). Motivated by these *q*-developments in Geometric Function Theory, many authors added their contributions in this direction which has made this research area much more attractive in works like [4,12].

In 2014, Kanas and R ˘aducanu [13] used the familiar Hadamad products to define a *q*-extension of the Ruscheweyh operator and discussed important applications of this operator in detail. Moreover, the extensive study of this *q*-Ruscheweyh operator was further made by Mohammad and Darus [14] and Mahmood and Sokół in [15]. Recently, a new idea was presented by Darus [16] that introduced a new differential operator called a generalized *q*-differential operator, with the help of *q*-hypergeometric functions where they studied some useful applications of this operator. For the recent extensions of different operators in *q*-analogue, see the work in [17–19]. The operator defined in [13] was extended further for multivalent functions by Arif et al. in [20] where they investigated its important applications. The aim of this paper is to define a family of multivalent *q*-starlike functions associated with circular domains and to study some of its useful properties.
