On Hermite-Hadamard Type Inequalities for Coordinated Convex Functions via (p,q)-Calculus

In this paper, we define (p,q)-integrals for continuous functions of two variables. Then, we prove the Hermite-Hadamard type inequalities for coordinated convex functions by using (p,q)-integrals. Many results obtained in this paper provide significant extensions of other related results given in the literature. Finally, we give some examples of our results.


Introduction
Quantum calculus or q-calculus is the modern name of the study of calculus without limits. It has been studied since the early eighteenth century. The famous mathematician, Euler, established q-calculus and, in 1910, F. H. Jackson [1] determined the definite qintegral known as the q-Jackson integral. Quantum calculus has many applications in mathematics and physics such as combinatorics, orthogonal polynomials, number theory, basic hypergeometric functions, quantum theory, mechanics, and theory of relativity, see for instance  and the references therein. The book by V. Kac and P. Cheung [24] covers the fundamental knowledge and also the basic theoretical concepts of quantum calculus.
The Hermite-Hadamard inequality is a classical inequality that has fascinated many researchers, stated as: If f : [a, b] → R is a convex function, then (1) Inequality (1) was introduced by C. Hermite [56] in 1883 and was investigated by J. Hadamard [57] in 1893. If f : [a, b] × [c, d] → R is a convex function for coordinates, then S. Dragomir [58] stated the Hermite-Hadamard type inequalities in 2001 as follows: In 2019, the Hermite-Hadamard type inequalities for coordinates via q-calculus was presented by M. Kunt et al. [27]: for all q 1 , q 2 ∈ (0, 1). Recently, S. Bermudo, P. Korus and J. E. N. Valdes [28] defined new b q-derivative, b q-integral and also gave the Hermite-Hadamard inequality via q-calculus by using such the definitions. Consequently, H. Budak, M. A. Ali and M. Tarhanachi [29] defined some new b q-integrals for coordinates and gave the following inequalities for all q 1 , q 2 ∈ (0, 1). Moreover, Yu-Ming Chu et al. [51] presented the definitions for new b (p, q)-derivatives, b (p, q)-integrals and gave the Hermite-Hadamard type inequality for convex functions by using (p, q)-calculus. Our present work was motivated by the above mentioned literatures, we propose to define new b (p, q)-integrals for coordinates and then extend the Hermite-Hadamard type inequality in q-calculus for coordinated convex functions to (p, q)-calculus for coordinated convex functions.
The a (p, q)-integral x a f (t) a d p,q t is defined by are convex for all x ∈ (a, b) and y ∈ (c, d).
A formal definition for coordinated convex functions may be stated as follows: Then the derivatives are given by Then the definite integral is given by Then the derivatives are given by Then the definite integral is given by For convenience, we call the integral defined in Definition 10 as L-L (Left-Left) integral. Next, we define another integrals for continuous functions of two variables.
which is a continuous function of two variables. Then by Definitions 9 and 10, for p 1 = p 2 = 3 4 and q 1 = q 2 = 2 4 , we obtain At the end of this section, we give some known theorems needed to prove our main results.

Main Results
In this section, we give new (p, q)-Hermite-Hadamard type inequalities for coordinated convex functions and verify them.
By (p 1 , q 1 ) a -integrating both sides of (10) on [a, p 1 a + (1 − p 1 )b], we have Similarly, let h y : [a, b] → R defined by h y (x) = f (x, y) be a convex function on [a, b]. Using the inequality (6) on [a, b], we have for all y ∈ [a, b].
Adding (11) and (13), we obtain which proves the second and the third inequalities of the theorem. Since , it follows from the first inequality of (10) that Since , it follows from the first inequality of (12) that Adding two inequalities above, we obtain the first inequality in the theorem. Finally, using the second inequality of (10) and (12), we have Combining the inequalities above, we get the last inequality in the theorem. This completes the proof. Remark 1. If p 1 = p 2 = 1, then (9) reduces to (3), which was appeared in [29].

Remark 2.
If p 1 = p 2 = 1, q 1 and q 2 tend to 1, then (9) reduces to (2), which was appeared in [58]. Proof. i.e., for all x ∈ [a, b]. By (p 1 , q 1 ) b -integrating both sides of (15) on [p 1 a + (1 − p 1 )b, b], we have for all y ∈ [a, b]. By (p 2 , q 2 ) c -integrating both sides of (17) on [c, Adding (16) and (18), we obtain which proves the second and the third inequalities of the theorem. Since , it follows from the first inequality of (15) that Since , it follows from the first inequality of (17) that Adding the two inequalities above, we obtain the first inequality in the theorem. Finally, using the second inequality of (15) and (17), we have Combining inequalities above, we get the last inequality in the theorem. This completes the proof.