## 1 Introduction  
Quantum calculus is the study of calculus without limits and is sometimes called \(q\)-calculus. In \(q\)-calculus, we obtain the original mathematical formulas when \(q\) tends to one. The beginning of the study of \(q\)-calculus can be dated back to the era of [PERSON] (1707-1783), who first launched the \(q\)-calculus in the tracks of [PERSON]'s work on infinite series. Then, in the early Twentieth Century, [PERSON] [1] defined an integral, which is known as the \(q\)-Jackson integral, and studied it in a systematic way. The subject of \(q\)-calculus has many applications in the field of mathematics and other areas such as number theory, special functions, combinatorics, basic hypergeometric functions, orthogonal polynomials, quantum theory, mechanics, and the theory of relativity and physics. In recent years, the topic of \(q\)-calculus has increasingly interested many researchers. For more details, see [2; 3; 4; 5; 6; 7; 8; 9] and the references therein. Recently, [PERSON] and [PERSON] [10; 11; 12] studied the concept of \((p,q)\)-calculus over the intervals of \([a,b]\subset\mathbb{R}\). The \((p,q)\)-derivative and \((p,q)\)-integral were defined and some basic properties are given. Furthermore, they obtained some new result for the \((p,q)\)-calculus of several important integral inequalities. Currently, the \((p,q)\)-calculus is being investigated extensively by many researchers, and a variety of new results can be found in the literature [13; 14; 15; 16; 17; 18] and the references cited therein.  
Mathematical inequalities are important to the study of mathematics, as well as in other area of mathematics such as analysis, differential equations, geometry, etcetera.  
In 1893, [PERSON] [19] investigated one of the fundamental inequalities in analysis as:  
\[f\left(\frac{a+b}{2}\right)\leq\frac{1}{b-a}\int_{a}^{b}f(x)dx\leq\frac{f(a)+f( b)}{2}, \tag{1}\]  
which is now known as the Hermite-Hadamard inequality.  
In 2014, [PERSON] and [PERSON] [20] studied the extension to \(q\)-calculus on the finite interval of (1), which is called the \(q\)-Hermite-Hadamard inequality, and some important inequalities. Next, [PERSON] et al. [21] approved the \(q\)-Hermite-Hadamard inequality and then obtained generalized \(q\)-Hermite-Hadamard inequalities.  
In 2018, [PERSON] et al. [22] proved the left-hand side of the \((p,q)\)-Hermite-Hadamard's inequality of (1) through \((p,q)\)-differentiable convex and quasi-convex functions, and then, they gave some new \((p,q)\)-Hermite-Hadamard's inequalities.  
In 2019, [PERSON] et al. [23] established the \(q\)-calculus of Hermite-Hadamard inequalities for the double integral as:  
\[f\bigg{(}\frac{a+b}{2}\bigg{)}\leq\frac{1}{(b-a)^{2}}\int_{a}^{b}\int_{a}^{b}f (tx+(1-t)y)dxdy\leq\frac{f(a)+f(b)}{2}, \tag{2}\]  
which was given by [PERSON] [24]. Moreover, they obtained refinements of the Hermite-Hadamard inequality for \(q\)-differentiable convex functions.  
The aim of this paper is to present the \((p,q)\)-calculus of Hermite-Hadamard inequalities for double integrals (2) and refinements of the Hermite-Hadamard inequality. These are obtained as special cases when \(p=1\) and \(q\to 1\).  
Before we proceed to our main theorem, the following definitions and some concepts require some clarifications.