**1. Introduction**

Quantum calculus or *q*-calculus is often known as "calculus without limits" and was first developed by Jackson in the early twentieth century, but the history of quantum calculus can be traced back to some much earlier work done by Euler and Jacobi et al. (see [1]). Over the recent decade, the investigation of *q*-calculus has attracted the interest of many researchers, because it has been found to have a lot of applications in mathematics and physics. As is known to us, *q*-calculus can be treated as a bridge between mathematics and physics, it is a significant tool for researchers working in analytic number theory, noncommutative geometry, or theoretical physics. In quantum calculus, we obtain the *q*-analogues of mathematical objects which can be recaptured as *q* → 1<sup>−</sup>. It has been noticed that quantum calculus is a subfield of timescale calculus. Timescale calculus provides a unified framework for studying dynamic equations on both the discrete and continuous domains. In quantum calculus, we are concerned with a specific timescale, called the *q*-timescale (see [1–4]).

The concept of convexity has been extended in several directions, since these generalized versions have significant applications in different fields of pure and applied sciences. We only point out that convexity was recently used in differential geometry to completely classify ideal Casorati submanifolds in complex space forms (see [5–8]). One of the convincing examples on extensions of convexity is the introduction of invex function, which was introduced by Hanson [9]. This concept is particularly interesting from an optimization viewpoint, since it provides a broader setting to study the optimization and mathematical programming problems. Such optimization problems have recently been considered in Riemannian geometry by an original choice of a set of quadratic programming problems. Since then, some classes of generalized convex functions, such as the preinvex function, strongly *α*-invex function, and strongly *α*-preinvex function, were put forward successively, see [10–16].

In this paper, the quantum calculus and the strongly preinvex function are subtly linked together via integral inequalities. It is well known that the theory of inequality plays a fundamental role in pure and applied mathematics and has extensive applications. Apart from the larger number of research results of inequalities in classical analysis, there are considerable works on the study of inequalities for *q*-calculus, particularly the study of inequalities related to quantum integral (*q*-integral), for example, *q*-Hermite–Hadamard integral inequality, *q*-Cauchy–Schwarz integral inequality, *q*-Hölder integral inequality, *q*-Ostrowski integral inequality, etc. For more details, we refer the interested reader to [17–23] and the references cited therein.

The purpose of this paper is to establish several *q*-integral inequalities of Simpson-type via strongly preinvex functions. The classical Simpson inequality is described as follows:

$$\left| \frac{1}{6} \left[ \phi(a) + 4\phi\left(\frac{a+\beta}{2}\right) + \phi(\beta) \right] - \frac{1}{\beta-a} \int\_a^\beta \phi(\nu) d\nu \right| \le \frac{1}{1280} \left\| \phi^{(4)} \right\|\_{\infty} (\beta - a)^4,\tag{1}$$

where the mapping *φ* : [*α*, *β*] → *R* is four times continuously differentiable, and *φ*(4)-∞ = sup*ν*<sup>∈</sup>(*<sup>α</sup>*,*β*) |*φ*(4)(*ν*)| < ∞ (see [24]).

The paper is organized as follows: In Sections 2 and 3, we shall introduce some notions and properties on strongly preinvex functions and *q*-calculus. As an auxiliary result, we present an identity associated with *q*-integral. In Section 4, with the help of the auxiliary result, we will establish our main results. At the end of the paper, some examples are provided to illustrate the applications of our main results.
