Simpson- and Newton-Type Inequalities for Convex Functions via (p,q)-Calculus

In this paper, we establish several new (p, q)-integral identities involving (p, q)-integrals by using the definition of a (p, q)-derivative. These results are then used to derive (p, q)-integral Simpsonand Newton-type inequalities involving convex functions. Moreover, some examples are given to illustrate the investigated results.

A convex function is defined as follows: A function f : [a, b] → R is convex if the inequality holds for all x, y ∈ [a, b] and t ∈ [0, 1].
The error estimations of Simpson-and Newton-type inequalities are as follows: Theorem 1 (Ref. [19]). If f : [a, b] → R is a four times continuously differentiable function on (a, b) and f (4) Theorem 2 (Ref. [23]). If f : [a, b] → R is a four times continuously differentiable function on (a, b) and f (4) So far, Simpson-and Newton-type inequalities have been studied in the form of a convex function using quantum calculus by many researchers, and the results of quantum calculus can be found in [26][27][28][29][30][31][32], and the references cited therein.
Quantum calculus or q-calculus is the study of calculus with no limits, beginning by studying Newton's infinite series, and was revealed by Euler (1707-1783). Then, many researchers, such as Gudermann (1798-1897), Weierstrass (1815-1897), and Heaviside (1850-1925) studied the properties of q-series. In 1910, the q-derivative and q-integral of a continuous function on the interval (0, ∞), based on the q-calculus of an infinite series, were defined by Jackson [33]. 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 of its various applications in mathematics and physics (see [34][35][36][37][38][39][40] for more details and the references cited therein).
Another generalization of q-calculus on the interval (0, ∞) is well-known as postquantum calculus or (p, q)-calculus. The (p, q)-calculus includes two-parameter quantum calculus (p and q-numbers) which are independent. In (p, q)-calculus, we obtain the qcalculus formula for the case of p = 1, and then get the classical formula for the case of q → 1. This generalization was first introduced by Chakrabarti and Jagannathan [51] in 1991. Then, Tunç et al. [52,53] presented new (p, q)-derivatives and q-integrals of a continuous function on a finite interval in 2016. Based on the definitions of (p, q)-calculus, many literatures have been published by many researchers (see [54][55][56][57][58] and the references cited therein).
In 2020, Budak et al. [50] presented Simpson-and Newton-type inequalities for convex functions via q-calculus. In this paper, we establish some new integral inequalities of Simpson-and Newton-type inequalities for convex functions via (p, q)-calculus to generalize and extend the results given in the above-mentioned report. Furthermore, we give some examples to investigate the main results.
The rest of the paper is organized as follows: In Section 2, we recall some basic knowledge and notations used in the next part. In Section 3, we give Simpson-type inequalities for a convex function via the (p, q)-calculus. In Section 4, we give Newton-type inequalities for convex function via (p, q)-calculus. In Section 5, we show some examples to illustrate the investigated results. In the final part, we summarize the conclusions.

Preliminaries
In this section, we give basic knowledge used in our work. Throughout this paper, let [a, b] ⊆ R be an interval, and let a < b and 0 < q < p ≤ 1 be constants.
Definition 1 (Refs. [52,53]). Suppose that f : [a, b] → R is a continuous function. Then, the (p, q) a -derivative of function f at t ∈ [a, b] is defined by The function f is said to be a (p, q) a -differentiable function on In Definition 1, if p = 1, then a D 1,q f (t) = a D q f (t), and Equation (1) reduces to which is the q a -derivative of function f defined in [a, b] (see [59][60][61] for more details). In addition, if a = 0, then 0 D q f (t) = D q f (t), and Equation (2) reduces to which is the q a -derivative of function f defined in [0, b] (see [62] for more details).
Definition 2 (Ref. [63]). Suppose that f : [a, b] → R is a continuous function. Then, the The function f is said to be a (p, In Definition 2, if p = 1, then b D 1,q f (t) = b D q f (t), and Equation (4) reduces to which is the q b -derivative of function f defined in [a, b] (see [64,65] for more details).
Definition 3 (Refs. [52,53]). Suppose that f : [a, b] → R is a continuous function. Then, the (p, q) a -integral of function The function f is said to be a (p, q) a -integrable function on If a = 0, then Equation (6) is the (p, q)-integral in [0, b], which can be expressed as Definition 4 (Ref. [63]). ] Suppose that f : [a, b] → R is a continuous function. Then, the The function f is said to be a (p, Theorem 3 (Ref. [63]). If f : [a, b] → R is a convex differentiable function on [a, b], then the (p, q) b -Hermite -Hadamard inequalities are as follows: Theorem 4 (Ref. [53]). If f , g : [a, b] → R are continuous functions and r, s > 0 with

Simpson-Type inequalities for (p, q)-Calculus
where Proof. From basic properties of the (p, q)-integral and the definition of φ(t), it follows that By Definition 2, we have Then, we obtain Similarly, we obtain Finally, we observe that Substituting Equations (15)- (17) in Equation (14), we have Multiplying the above equality with b − a, we obtain the required result. The proof is completed. [50].
where Λ i (p, q), i = 1, 2, 3, 4 are defined by Proof. Taking the absolute value of both sides of Theorem 5, we observe that If the first (p, q)-integral on the right side of the inequality is used to consider the convexity of | b D p,q f |, from the case when a = 0 of Lemma 1, then we obtain Similarly, we have Substituting Equations (20) and (21) into Equation (19), we obtain the required result.
, and r, s > 1 with 1/r + 1/s = 1, then Proof. Taking the absolute value of both sides of Theorem 5, we have By Theorem 4, we have Using Equation (7), we obtain Calculating the (p, q)-integral in Equation (25) and substituting the inequality Equation (26) into Equation (25), we have Similarly, we have Substituting Equations (27) and (28) into Equation (24), we obtain the required result.

Newton-Type Inequalities for (p, q)-Calculus
where Proof. From basic properties of the (p, q)-integral and the definition of ψ(t), it follows that The rest of this proof is similar to that of Theorem 5.
Theorem 11. Let f : [a, b] → R be a (p, q) b -differentiable function on (a, b). If | b D p,q f | is a convex function and a (p, q) b -integrable function on [a, b], and r, s > 1 with 1/r + 1/s = 1, then Proof. The proof of Theorem 11 is similar to that of Theorem 7. Hence it is omitted.
Proof. The proof of Theorem 12 is similar to that of Theorem 8. It is clear that which demonstrates the result described in Theorem 6.

Conclusions
In this work, we employed (p, q)-calculus to establish new integral inequalities related to Simpson-and Newton-type inequalities for convex functions. The results in this study were the generalization and extension of some previously proved research in the literature of Simpson-and Newton-type inequalities. In addition, some examples were displayed to verify our main results.