On Simpson type inequalities for generalized strongly preinvex functions via $(p,q)$-calculus and applications

1 Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand 2 Department of Mathematics, Faculty of Applied Science, King Mongkuts University of Technology North Bangkok, Bangkok 10800, Thailand 3 Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece 4 Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia


Introduction
Quantum calculus, also called q-calculus, is the study of calculus without limits. In the beginning study of the q-calculus, Newton's infinite series was established by Euler (17071783). Then, Jackson [1] relied on the knowledge of Euler to define q-derivative and q-integral of a continuous function on the interval (0, ∞), based on q-calculus of infinite series, in 1910. In q-calculus, the main objective is to obtain the q-analoques of mathematical objects recaptured by taking q → 1 − . In recent years, the q-calculus has attracted interest because it can be applied in various fields such as mathematics and physics, see [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] for more details and the references cited therein. In 2002, Kac and Cheung [17] summarized the basic theoretical concept of the q-calculus in their book.
In 2013, Tariboon and Ntouyas [18] defined the new q-derivative and q-integral of a continuous function on finite interval and proved their basic properties. Moreover, they investigated the existence and uniqueness results of initial value problems for first and second order impulsive q-difference equations. Then, these definitions have been studied in various inequalities, for example, Simpson type inequalities, Newton type inequalities, Hermite-Hadamard inequalities, Ostrowski inequalities, and Fejr type inequalities, see [19][20][21][22][23][24][25] for more details and the references cited therein.
Several generalizations and extensions of convexity have been studied via various techniques in several directions. In 1981, Hanson [40] presented that a significant generalization of convex functions was that of invex functions. His initial result inspired many literatures and expanded the role and applications of invexity in both areas of pure and applied sciences [41][42][43][44]. A class of convex functions, called a preinvex function, was presented by Ben-Israel and Mond [45] in 1986. Moreover, the basic properties of the preinvex functions and their role in optimization were introduced by Weir and Mond [46] in 1988. In recent years, these concepts have been studied by many researchers in various fields, see [47][48][49][50][51] for more details and some researchers have studied the preinvex functions via q-calculus, see [52][53][54][55][56][57] for more details. After that, Noor et al. [58] studied a new class of generalized convex functions, called a strongly preinvex function, in 2006. Then, Deng et al. [59] studied strongly preinvex functions via q-calculus of Simpson type inequalities in 2019.
Mathematical inequalities play important roles in the study of pure and applied mathematics [60][61][62]. One of the most interesting inequalities is Simpson type inequalities. Simpson's rules, developed by Simpson (17101761), are techniques for the numerical integration and the numerical estimation of definite integrals. Then, there were a lot of results on Simpsons type inequalities studied by many researchers, see [63][64][65][66][67][68][69][70][71] for more details and the references cited therein. Simpsons quadrature (Simpsons 1/3 rule) is formulated as follows: see [72] for more details. The estimation of Simpson inequality is as follows: ] → R is a four times continuously differentiable function on (a, b) and Motivated by the above mentioned reports, we propose to study some new properties of Simpson type inequalities for the generalized strongly preinvex functions via (p, q)-calculus.
The rest of the paper is organized as follows. In Section 2 contains some basic knowledge and notation used in the next sections. In Section 3, we give some properties of Simpson type inequalities via (p, q)-calculus. In Section 4, we display some examples to illustrate the applications of the (p, q)calculus for Simpson type inequalities. In the final section, we summarize our results.

Preliminaries
In this section, we give basic knowledge used in our work. Throughout this paper, let [a, b] ⊆ R be an interval with a < b and 0 < q < p ≤ 1 be constants.

Definition 2.2. [48]
A function f on the invex set K ⊂ R is said to be preinvex with respect to η : holds for all x, y ∈ K and λ ∈ [0, 1].

Definition 2.3. [58]
A function f on the invex set K ⊂ R is said to be strongly preinvex with respect to η : R × R → R, and modulus µ > 0, if holds for all x, y ∈ K and λ ∈ [0, 1].

Definition 2.4. [59]
A function f on the invex set K ⊂ R is said to be generalized strongly preinvex with respect to η : R × R → R, and modulus µ ≥ 0, if holds for all x, y ∈ K and λ ∈ [0, 1].
In Definition 2.4, if µ = 0, then the generalized strongly preinvex functions reduce to the preinvex functions as defined in Definition 2.2.
Definition 2.5. [27,28] If f : [a, b] → R is a continuous function and 0 < q < p ≤ 1, then the (p, q)-derivative of function f at t ∈ [a, b] is defined by The function f is said to be (p, In Definition 2.5, if p = 1, then a D 1,q f (t) = a D q f (t), and (2.5) reduces to which is the q-derivative of function f defined on [a, b], see [77][78][79] for more details. In addition, if a = 0, then 0 D q f (t) = D q f (t), and (2.6) reduces to which is the q-derivative of function f defined on [0, b], see [17] for more details. (2.9) In addition, If p = 1, then (2.9) reduces to which is the q-Jackson integral, see [17] for more details.

Main results
In this section, we establish a new (p, q)-integral identity. The defined identity is then used to derive the (p, q)-integral inequalities of Simpson type for generalized strongly preinvex function.
Proof. It is not difficult to see that where and Using Definitions 2.5, 2.6, and Theorem 2.1, we have Then, we find that Similarly, we have

From (3.4), we obtain
Multiplying the above equality with η(b, a), we obtain the required result. Therefore, the proof is completed.
Proof. Using Lemma 3.1 and Definition 2.4, we have + tη(b, a)) d p,q t, Using Definition 2.6, Theorem 2.1 and Lemma 2.1, we obtain the required result. Therefore, the proof is completed.

Applications
It is worth noting that if µ = 0 in Definition 2.4, then the generalized strongly preinvex functions reduce to the preinvex functions. Moreover, if η(x, y) = x − y, then the preinvex functions reduce to the convex functions. Here, we give some examples to illustrate the applications of our main results.
If η(b, a) = b − a, then from (4.1) we have the following Corollary. a (p, q)-integrable function and a convex function, then where A 1 (p, q), A 2 (p, q), A 4 (p, q), and A 5 (p, q) are given in Theorem 3.1.

4)
where A 1 (q), A 2 (q), A 4 (q), and A 5 (q) are given in Remark 3.2, which appeared in [59]. In addition, if q → 1 − in (4.4) and we use the basic properties of q-derivative and q-integral (see [17,48]) then we obtain the inequality which appeared in [80]. If | a D q f | is a convex function and a q-integrable function with 0 < q < 1, then Now, we give the following example to assert that the left side of (4.5) is correct, but the right side of (4.5) is not correct. 2 , so the left side of (4.5) becomes and the right side of (4.5) becomes (b − a) 12 This implies that 1 6 7 54 .
In the following, we show a new result involving (p, q)-integrals of equalities (4.8) and (4.9) for 0 < q < 1 3 and 0 < q < 5 6 , respectively. If p = 1, then we give the correct results of quantum integral inequalities.
Proof. Using Definition 2.6, Theorem 2.1 and Lemma 2.1, we have Similarly, we obtain Therefore, the proof is completed.
In the following, we provide a modified version involving (p, q)-integral of inequality (4.5).