**3. Applications**

The *q*-calculus has played an important role in the study of almost every branch of mathematics and physics, for example, in the theory of special functions, differential equations, combinatorics, analytic number theory, quantum theory, quantum group, special polynomials, numerical analysis, operator theory and other related theories. Quantum calculus is considered as one of the most active research areas in mathematics and physics. For more details, please refer to [22,36–41].

#### **4. Concluding Remarks and Observations**

In this paper, we introduce a new integral operator J *λ*+*p*−1 *q* in *q*-analog and define the class <sup>H</sup>*λp*,*<sup>q</sup>* (*<sup>α</sup>*, *μ*, *β*) of multivalent functions by using this operator. Several useful properties such as sufficiency criteria, distortion bounds, radius of starlikness and radius of convexity, extreme points, weighted mean and partial sum for this newly defined subclass of multivalent functions are investigated. In addition, we observe that, if we take some suitable parameters *p*, *q*, *α*, *μ*, *β* in the results involved, we ge<sup>t</sup> the corresponding properties for *q*-analog of differential and integral operators mentioned in the Introduction.

**Author Contributions:** Conceptualization, M.A.; Formal analysis, M.R.; Funding acquisition, H.T.; Investigation, Q.K.; Methodology, M.R.; Software, Q.K.; Supervision, M.R., G.S. and H.T.; Visualization, G.S.; Writing—original draft, Q.K.; Writing—review & editing, M.A., S.u.R.

**Funding:** The research conducted in this article was partly supported by the Natural Science Foundation of the People's Republic of China under Grants 11561001 and 11271045, the Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region under Grant NJYT-18-A14, the Natural Science Foundation of Inner Mongolia of the People's Republic of China under Grant 2018MS01026 and the Natural Science Foundation of Chifeng of Inner Mongolia.

**Conflicts of Interest:** The authors declare no conflict of interest.
