On post quantum estimates of upper bounds involving twice (p,q)$(p,q)$-differentiable preinvex function

*Correspondence: chuyuming2005@126.com 3Department of Mathematics, Huzhou University, Huzhou, China Full list of author information is available at the end of the article Abstract The main objective of this paper is to derive a new post quantum integral identity using twice (p,q)-differentiable functions. Using this identity as an auxiliary result, we obtain some new post quantum estimates of upper bounds involving twice (p,q)-differentiable preinvex functions.


Introduction and preliminaries
The quantum calculus is often regarded as calculus without limits, we obtain q-analogues of 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. This can be attributed to the fact that it serves as a bridge between mathematics and physics. It is also pertinent to mention here that quantum calculus is a subfield of time scale calculus. In quantum calculus, we are concerned with a specific time scale, called the q-time scale. In the twentieth century Jackson [8] introduced the notion of q-definite integrals in quantum calculus. This motivated many quantum calculus analysts, and consequently a number of articles have been written in this area. It is worth to mention here for interested readers that it is possible that sometimes more than one q-analogue exists. In [9] interested readers may find some basic and interesting details on some recent developments of basic theory of quantum calculus. While studying quantum calculus, Tariboon et al. [23] introduced the notions of q-derivatives and q-integrals on finite intervals and developed several new q-analogues of classical inequalities. This particular article inspired many researchers working in the field of inequalities, particulary inequalities involving convexity and its generalizations. Resultantly, several new quantum analogues of classical results have been obtained. For example, Noor et al. [21] obtained the quantum analogues of Hermite-Hadamard's inequality using the class of preinvex functions. Sudsutad et al. [22] and Noor et al. [20] obtained new quantum analogues of trapezium like inequalities involving q-differentiable convex functions. Noor et al. [19] obtained quantum analogues of Ostrowski's inequality. Zhang et al. [26] obtained a new generalized q-integral identity, and utilizing this as an auxiliary result, they have obtained several new q-analogues of classical inequalities. Liu and Zhuang [15] obtained certain new q-analogues of Hermite-Hadamard's inequality using two times q-differentiable convex functions. Alp et al. [3] obtained some new refined q-analogues of Hermite-Hadamard's inequality. For more details, see [4,10,11,13] A recent development in the study of quantum calculus is the introduction of post quantum calculus. In quantum calculus we deal with 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 [7]. For some interesting applications, see [1,2,6,12,17,18]. Motivated by the research work going on, Tunc and Gov [24] introduced the concepts of (p, q)-derivatives and (p, q)-integrals on finite intervals.
Since the appearance of this article, a number of new post quantum analogues of classical inequalities have been obtained. For example, Kunt et al. [14] obtained new post quantum analogues of Hermite-Hadamard's inequality. Luo et al. [16] obtained some new variants of parameterized (p, q)-integral inequalities using a generalized integral identity involving (p, q)-differentiable functions.
The main idea behind the study of this paper is to obtain a new general post quantum integral inequality using twice (p, q)-differentiable functions. We then establish some new estimates of post quantum bounds essentially using the class of preinvex functions. We hope that the ideas and techniques of this paper will inspire interested readers working in this field.
Before we move to our next section of the paper, let us recall the definitions of invex set and preinvex function. Definition 1.1 ([24]) Let K ⊆ R be a nonempty set such that a ∈ K, 0 < q < p ≤ 1, and let f : K → R be a continuous function. Then the (p, .

Definition 1.4 ([25]) Let K ⊆ R be an invex set with respect to the bivariate function
for all a, b ∈ K and t ∈ [0, 1].

Results and discussions
In this section, we derive our main results. First of all we derive our new post quantum integral identity involving twice (p, q)-differentiable function.
Proof It suffices to prove that Elaborated computation leads to Multiplying both sides of the above equality by pq 2 ζ 2 (b,a) p+q , we get the required result.  + pζ (b, a)) p,q f | is a preinvex function with respect to ζ .
Proof It follows from Lemma 2.1 and the property of the modulus together with the preinvexity of | a D 2 p,q f | that .

Theorem 2.3
Let 0 < q < p ≤ 1, r > 1, K ⊆ R be an invex set with respect to the bivariate function ζ : R × R → R, and f : K → R be a twice (p, q)-differentiable function on K • such that a D 2 p,q f is continuous and (p, q)-integrable on K. Then the inequality Proof From Lemma 2.1, Hölder's inequality, and the preinvexity of | a D 2 p,q f | r , we get Theorem 2.4 Let 0 < q < p ≤ 1, r, s > 1 with 1/r + 1/s = 1, K ⊆ R be an invex set with respect to the bivariate function ζ : R × R → R, and f : K → R be a twice (p, q)-differentiable function on K • such that a D 2 p,q f is continuous and (p, q)-integrable on K. Then the inequality qf (a) + pf (a + pζ (b, a)) Proof Using Lemma 2.1, Hölder's inequality, and the preinvexity of | a D 2 p,q f | r , we have Theorem 2.5 Let 0 < q < p ≤ 1, r > 1, K ⊆ R be an invex set with respect to the bivariate function ζ : R × R → R, and f : K → R be a twice (p, q)-differentiable function on K • such that a D 2 p,q f is continuous and (p, q)-integrable on K. Then one has qf (a) + pf (a + pζ (b, a)) p + q -1 p,q f | r is a preinvex function with respect to ζ , where Proof It follows from Lemma 2.1 and Hölder's inequality together with the preinvexity of Theorem 2.6 Let 0 < q < p ≤ 1, r, s > 1 with 1/r + 1/s = 1, K ⊆ R be an invex set with respect to the bivariate function ζ : R × R → R, and f : K → R be a twice (p, q)-differentiable function on K • such that a D 2 p,q f is continuous and (p, q)-integrable on K. Then the inequality qf (a) + pf ( a + pζ (b, a))  + pζ (b, a)) Theorem 2.7 Let 0 < q < p ≤ 1, r, s > 1 with 1/r +1/s = 1, K ⊆ R be an invex set with respect to the bivariate function ζ : R × R → R, and f : K → R be a twice (p, q)-differentiable function on K • such that a D 2 p,q f is continuous and (p, q)-integrable on K. Then qf (a) + pf (a + pζ (b, a)) p + q -