On the refinements of some important inequalities via (p,q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(p,q)$\end{document}-calculus and their applications

We establish some interesting refinements of the (p,q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(p,q)$\end{document}-Hölder integral inequality and the (p,q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(p,q)$\end{document}-power-mean integral inequality. As applications, we show that some existing (p,q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(p,q)$\end{document}-integral inequalities can be improved by the results obtained in this paper.

In [15], a different version of Hölder-İşcan inequality was provided as follows.
In the rest of this section, we review some preliminaries of (p, q)-calculus. Throughout this paper, let [a, b] ⊂ R, and let p, q be two constants such that 0 < q < p ≤ 1. The existence of (p, q)-derivatives and (p, q)-integrals is required, and the convergence of the corresponding series mentioned later in the proofs are assumed. Now we recall the theory of (p, q)-calculus. These concepts and results related to the (p, q)-derivative and (p, q)integral are mainly due to Tunç and Göv [30].
For x = a, we have a D p,q a 2 = lim x→a ( a D p,q x 2 ) = 2a.
Again, for x = a, we have a D 2 p,q a 2 = lim x→a ( a D 2 p,q x 2 ) = p + q.
This paper is mainly devoted to investigating (p, q)-integral inequalities via (p, q)calculus. For this purpose, we extend some of important integral inequalities of analysis to (p, q)-calculus such as Hermite-Hadamard, Hölder, and the power-mean integral inequalities. To present some applications of our main results, we establish an identity to express the difference between the middle part and the right-hand side of the analogue of (p, q)-Hermite-Hadamard inequality (1.2). Based on this identity, we give several estimates for (p, q)-integral inequalities via convexity. Meanwhile, we compare some of the derived results in this paper, in an interesting way, with the known works.

Main results
In this section, we establish several integral inequalities concerning the quantum version of (p, q)-Hermite-Hadamard inequality, the improved (p, q)-Hölder-İşcan integral inequality, and the refinement of (p, q)-power-mean integral inequality. The first result is as follows.
, and let m be an integer. Then we have Proof Using the identity ∞ n=0 (1 -q p ) q n p n = 1, 0 < q < p ≤ 1, Jensen's inequality for infinite sums, and Definition 1.3, we have that Using Definition 1.3 and the convexity of f , we get The proof is completed.
Proof First, by Definition 1.3 and the discrete Hölder inequality we obtain This completes the proof for the first part of inequality (2.2). For the second part, using Definition 1.3 again yields that Thus the proof of Theorem 2.2 is completed.
As a generalization of the (p, q)-Hölder inequality, we give the following improvement of (p, q)-power-mean integral inequality.

Using Definition 1.3 and the discrete Hölder inequality, we have that
× f q n p n+1 x + 1 -q n p n+1 a w q n p n+1 x + 1 - × f q n p n+1 x + 1 -q n p n+1 a w q n p n+1 x + 1 - This completes the proof for the first part of inequality (2.3). Now let us invoke Definition 1.3 again to prove the second part. Specifically, we have The proof of Theorem 2.3 is completed.

Applications
In this section, we present some interesting applications of the results developed in Sect. 2.
To this end, we consider the difference between the middle part and the right-hand side of the analogue of (p, q)-Hermite-Hadamard inequality (1.2) and propose the following lemma.  (1p) Proof From Definitions 1.1 and 1.2 we have Utilizing this calculation and Definition 1.3, we have After suitable arrangement, we get the desired result. Thus the proof is completed.
Proof Using Lemma 3.1, the (p, q)-Hölder-İşcan integral inequality, and the convexity of We obtain the desired inequality by noting that This ends the proof.
In Theorem 5.2 of [20], as q → 1 -, the authors obtained the following result.
The next result deals with the other case where | a D 2 p,q f | γ 2 is convex for γ 2 > 1.

Figure 1
Error curves of E 1 and E 2 on the variable γ 2

Conclusion
We extend some important integral inequalities of analysis to (p, q)-calculus, which include the Hermite-Hadamard, Hölder, and power-mean integral inequalities. As applications, for mappings with convex absolute values of the second derivatives, we derive certain analogue of (p, q)-Hermite-Hadamard inequalities based on the established (p, q)integral identity. By an interesting comparison it turns out that the results obtained in this paper are shaper than the existing results. With these ideas and techniques developed in this work, the interested readers can be inspired to explore this fascinating field of (p, q)integral inequalities.