On Some New Simpson’s Formula Type Inequalities for Convex Functions in Post-Quantum Calculus

: In this work, we prove a new ( p , q ) -integral identity involving a ( p , q ) -derivative and ( p , q ) -integral. The newly established identity is then used to show some new Simpson’s formula type inequalities for ( p , q ) -differentiable convex functions. Finally, the newly discovered results are shown to be reﬁnements of comparable results in the literature. Analytic inequalities of this type, as well as the techniques used to solve them, have applications in a variety of ﬁelds where symmetry is important.


Introduction
During his lifetime, Thomas Simpson created important approaches for numerical integration and the estimation of definite integrals, which became known as Simpson's rule (1710-1761). J. Kepler, who made a comparable calculation roughly a century before Newton, is the inspiration for Kepler's rule. Estimations based exclusively on a three-step quadratic kernel are commonly referred to as Newton-type results because Simpson's technique incorporates the three-point Newton-Cotes quadrature rule.
(1) Simpson's quadrature formula (Simpson's 1/3 rule) (2) Simpson's second formula or Newton-Cotes quadrature formula (Simpson's 3/8 rule) The following estimation, known as Simpson's inequality, is one of many linked with these quadrature rules in the literature: Theorem 1. Suppose that F : [π 1 , π 2 ] → R is a four-times continuously differentiable mapping on (π 1 , π 2 ), and let F (4) ∞ = sup x∈(π 1 ,π 2 ) F (4) (x) < ∞. Then, one has the inequality 1 3 Many researchers have focused on Simpson-type inequality in various categories of mappings in recent years. Because convexity theory is an effective and powerful technique to solve a huge number of problems from various disciplines of pure and applied mathematics, some mathematicians have worked on the results of Simpson's and Newton's type in obtaining a convex map. The novel Simpson's inequalities and their applications in numerical integration quadrature formulations were presented by Dragomir et al. [1]. Furthermore, Alomari et al. [2] discovered a number of inequalities in Simpson's kind of s-convex functions. The variance of Simpson-type inequality as a function of convexity was then observed by Sarikaya et al. in [3]. Refs. [4][5][6] can be consulted for further research on this subject.
On the other hand, quantum and post-quantum integrals for many types of functions have been used to study many integral inequalities. The authors of [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] employed leftright q-derivatives and integrals to prove HH integral inequalities and associated left-right estimates for convex and coordinated convex functions. Noor et al. proposed a generalized version of quantum integral inequalities in their paper [22]. In [23], the authors demonstrated some parameterized quantum integral inequalities for generalized quasi-convex functions. In [24], Khan et al. used the green function to prove quantum HH inequality. For convex and coordinated convex functions, the authors of [25][26][27][28][29][30] constructed new quantum Simpson's and quantum Newton's type inequalities. Consult [31][32][33] for quantum Ostrowski's inequality for convex and co-ordinated convex functions. Using the left (p, q)-difference operator and integral, the authors of [34] expanded the results of [9] and demonstrated HH-type inequalities and associated left estimates. In [16], the authors discovered the right estimates of HH-type inequalities, as demonstrated in [34]. Vivas-Cortez et al. [35] recently generalized the results of [11] and used the right (p, q)-difference operator and integral to prove HH-type inequalities and associated left estimates.
We use the (p, q)-integral to establish some new post-quantum Simpson's type inequalities for (p, q)-differentiable convex functions, as inspired by recent research. The newly revealed inequalities are also shown to be extensions of previously discovered inequalities.
The structure of this article is as follows. The principles of q-calculus, as well as other relevant topics in this subject, are briefly discussed in Section 2. The basics of (p, q)-calculus, as well as some recent research in this topic, are covered in Section 3. In Section 4, we prove a new (p, q)-integral identity involving a (p, q)-derivative. Section 5 describes the Simpson's type inequalities for (p, q)-differentiable functions via (p, q)-integrals. It is also taken into account the relationship between the findings given here and similar findings in the literature. Section 6 finishes with some research suggestions for the future.
In [37], Jackson gave the q-Jackson integral from 0 to π 2 for 0 < q < 1 as follows: provided that the sum converges absolutely.
, then we have the following inequality for q π 2 -integrals: where 0 < q < 1 and

Post-Quantum Calculus and Some Inequalities
In this section, we review some fundamental notions and notations of (p, q)-calculus. The [n] p,q is said to be (p, q)-integers and expressed as: respectively, and expressed as: Definition 5 ([39]). The (p, q)-derivative of mapping F : [π 1 , π 2 ] → R is given as:

An Identity
In this section, we deal with an identity that is required to reach our major estimates. In the following lemma, we first build an identity based on a two-stage kernel. Lemma 2. Let F : [π 1 , π 2 ] → R be a differentiable function on (π 1 , π 2 ). If π 1 D p,q F is continuous and integrable on [π 1 , π 2 ], then one has the identity Proof. Using the fundamental properties of (p, q)-integrals and the definition of function Λ(s), we find that According to Definition 6, one must also have Now, if we substitute the above equation into (16), we obtain When the first integral on the right-hand side of (17) is calculated using Definition 8, it is discovered that If we look at the other integrals on the right-hand side of (17), we obtain Substituting the expressions (18)-(20) into (17), and later multiplying both sides of the resulting identity by pq(π 2 − π 1 ), the equality (15) can be captured.

Main Results
For (p, q)-differentiable convex functions, we prove some new Simpson's formula type inequalities in this section. For the sake of brevity, we start this section with certain notations that will be utilized in our new results.
Proof. We observe that when we take the modulus in Lemma 2, because of the modulus' characteristics, we have Using the convexity of π 1 D p,q F , we may calculate integrals on the right-hand side of (26) as follows: When we apply the equality (12) idea to the aforementioned post-quantum integrals, we obtain Thus, we obtain [6] p,q π 1 D p,q F (sπ 2 + (1 − s)π 1 ) d p,q s (28) We obtain the inequality (25) by placing (27) and (28) in (26). This completes the proof. Corollary 1. In Theorem 7, if we set p = 1, then we have the following new Simpson's type inequality for q-integrals: Remark 5. In Theorem 7, if we assume p = 1 and later take the limit as q → 1 − , then we obtain the following Simpson's type inequality: We obtain the desired inequality (29) by inserting (31)-(34) into (30), which completes the proof.
We obtain the needed inequality (35) by swapping (37) and (38) in (36). As a result, the proof is complete. Corollary 3. In Theorem 9, if we set p = 1, then we obtain the following new Simpson's type inequality for the q-integral:

Conclusions
In this investigation, we have proven different variants of Simpson's formula type inequalities for (p, q)-differentiable convex functions via post-quantum calculus. We conclude that the findings of this research are universal in nature and contribute to inequality theory, as well as applications in quantum boundary value problems, quantum mechanics, and special relativity theory for determining solution uniqueness. The findings of this study can be utilized in symmetry. Results for the case of symmetric functions can be obtained by applying the concept in Remark 4, which will be studied in future work. Future researchers will be able to obtain similar inequalities for different types of convexity and co-ordinated convexity in their future work, which is a new and important problem.