New post quantum analogues of Ostrowski-type inequalities using new definitions of left–right (p,q)$(p,q)$-derivatives and definite integrals

The main objective of this paper is to introduce a new more elegant notion of left–right (p,q)-derivative and definite integrals. To show the significance of these concepts, we discuss some of their basic properties. A new generalized (p,q)-integral identity is also obtained. Utilizing this identity as an auxiliary result we then obtain our main results using the concept of n-polynomial convex functions.


Introduction
Integral inequalities play a significant role in both pure and applied sciences because of their wide applications in mathematics and physics, as well as many other natural and human social sciences, while convexity theory has remained an important tool in the establishment of the theory of integral inequalities. The Hermite-Hadamard inequality [13], as a member of the family of integral inequalities, is a classical inequality that has long fascinated numerous mathematical researchers, which can be stated as follows: This inequality provides us a necessary and sufficient condition for a function to be convex, it also gives us an estimate of the integral average for a continuous convex function on an interval. Recently, the improvements, generalizations, and variants for the Hermite-Hadamard inequality have been the subject of much research. The left-hand side of the Hermite-Hadamard inequality can be estimated by the Ostrowski [29] inequality, which reads as with the best possible constant 1/4 if f : In recent years, several successful attempts have been made in obtaining the variants and applications of Ostrowski inequality. For example, Dragomir and Rassias [12] provided many interesting results on its applications in numerical integration.
In the past few years, a variety of novel approaches have been utilized by researchers in generalizing the classical inequalities. One of those approaches is utilizing the concepts of quantum calculus instead of ordinary calculus. It is very well known to everyone that quantum calculus is calculus without limits. In quantum calculus we often establish the q-analogues of classical mathematical objects which can be recaptured by taking q → 1 -. Historically the subject of quantum calculus can be traced back to Euler and Jacobi, but in recent decades it has experienced a rapid development [16]. Consequently, new generalizations of the classical concepts of quantum calculus have been introduced and investigated in the literature. Tariboon and Ntouyas [34] introduced the quantum calculus concepts on finite intervals, obtained several q-analogues of classical mathematical objects, and opened a new venue of research. For instance, they have obtained the q-analogue of Hölder's integral inequality, q-analogue of Hermite-Hadamard's inequality, q-trapezoid inequality, q-Ostrowski inequality, q-Cauchy-Bunyakovsky-Schwarz inequality, and qanalogue of Grüss-Cebysev inequality, etc. This motivated other researchers and, as a result, numerous novel results pertaining to quantum analogues of classical mathematical results have been introduced in the literature. For example Noor et al. [28] and Sudsutad et al. [33] obtained some more q-analogues of trapezoid-like inequalities for first order qdifferentiable convex functions, and Liu and Zhuang [25] derived some new q-analogues of trapezoid-like inequalities involving second order q-differentiable convex functions. Alp et al. [2] obtained a corrected q-analogue of Hermite-Hadamard's inequality. Zhang et al. [38] obtained a new generalized q-integral identity and obtained several new q-analogues for first order q-differentiable convex functions. Noor et al. [27] utilized the concepts of quantum calculus and obtained some new q-analogues of the Ostrowski-type inequality. These new analogues reduce to the original results if q → 1 -. For some details from the application point of view of quantum differential and integral operators, see [1, 3-10, 14, 15, 17-21, 24, 30-32, 37]. Recently Kunt and Baidar [22] introduced some new concepts of quantum calculus, namely left-right quantum derivatives and definite integrals. Using these new definitions, the authors have obtained some new q-analogues of classical integral inequalities. Another significant generalization of quantum calculus is the post-quantum calculus. In quantum calculus we deal with a q-number with one base q, however, post-quantum calculus includes p-and q-numbers with two independent variables p and q. This was first considered by Chakarabarti and Jagannathan [11]. Tunc and Gov [36] introduced the concepts of (p, q)-derivatives and (p, q)-integrals on finite intervals as follows.
The main purpose of the article is to establish some new post-quantum estimates of the Ostrowski inequality by the use of new modified definitions of left-right (p, q)-derivatives and definite integrals. But before we start to proceed towards the main results, we need to recall some basic concepts and previously known results from the quantum and postquantum calculus. .
Recently, several researchers have utilized these concepts in obtaining some new (p, q)analogues of classical inequalities. For example, Kunt et al. [23] and Luo et al. [26] obtained new refinements of the Hermite-Hadamard inequality using the concepts of postquantum calculus.

New definitions
We now define some new concepts, examples, and their basic properties. Definition 2.1 Let f : I → R be a continuous function and let t ∈ I and 0 < q < p ≤ 1. Then the right (p, q)-derivative on I of function f at t is defined as .
Proof We leave the details of the proof of parts I and II as these are obvious.
The second equation can be obtained in a similar way by interchanging the functions f and g.
This completes the proof.
We now define right-(p, q)-quantum integral as right-(p, q)-antiderivative of F(t) by using the following shifting operator: where F(t) is the (p, q)-antiderivative of f . Applying mathematical induction to (2.1), we have From Definition 2.1, we have Making the use of u = pt + (1p) Hence Applying the formula of expansion of geometric series to (2.1), we obtain Thus We now define right (p, q)-integral on a finite interval as: Definition 2.4 Let f : I → R be a continuous function. Then for 0 < q < p ≤ 1, the right-

Theorem 2.5 Let f , g : [a, b] → R be arbitrary functions and λ ∈ R, then we have
Proof The proofs of claims I and II are obvious. III. Using Definitions 2.1 and 2.4, we have q n p n f q n p n s + 1 -

IV. Using Definitions 2.1 and 2.4, we have
q n p n s + 1 -

V. From claim III of Theorem 2.3, we have
By integrating over [a, b] and using (2.4), we have b a f qt + (1q) This completes the proof.
Theorem 2.6 Let f : I → R be a convex and (p, q)-integrable function with 0 < q < p ≤ 1, then Proof It is obvious that (2.6) Similarly, q n p n+1 f q n p n+1 a + 1 - Since f is convex on I, we have for all t ∈ I.

A key lemma
The following auxiliary result will play a significant role in the development of our next results. f : [a, b] → R be a (p, q)-differentiable function on (a, b) with a < b. If   a D p,q f is integrable on [a, b] and 0 < q < p ≤ 1, then

Lemma 3.1 Let
A direct computation gives q n p n+1 f px q n p n+1 + 1p q n p n+1 a Similarly, Thus we have which leads to the desired identity (3.1).
In order to prove our next results, we need the definition of n-polynomial convex functions which was introduced and studied by Toplu et al. [35]. 1t s f (y).
Proof Using Lemma 3.1 and the fact that | a + D p,q f | and | b -D p,q f | are n-polynomial convex functions, we have 1t s 0 + d p,q t