Generalizations of some integral inequalities related to Hardy type integral inequalities via (p,q)$(p,q)$-calculus

In this paper, we study generalizations of some integral inequalities related to Hardy type integral inequalities via $(p, q)$
 (
 p
 ,
 q
 )
 -calculus. Many results obtained in this paper provide extensions of existing results in the literature. Furthermore, some examples are given to illustrate the investigated results.

inequality. Let us just mention that in 1920, G. H. Hardy [32] presented the following famous inequality for f being a non-negative integrable function and s > 1: which is now known as Hardy inequality. This inequality plays an important role in analysis and applications, see [33,34] for more details. The Hardy inequality has been studied by a large number of authors during the twentieth century. Over the last 20 years, a large number of papers have appeared in the literature which deals with the simple proofs, various generalizations and discrete analogue of Hardy inequality, see [35][36][37][38][39] for more details.
In 2014, L. Maligranda et al. [40] studied a q-analogue of Hardy inequality (1.1) and some related inequalities. It seems to be a huge new research area to study of these so called q-Hardy type inequalities. They obtained more general results on q-Hardy type inequalities. By taking q → 1, we obtain classical results on Hardy inequality (1.1). Next, L.-E. Persson and S. Shaimardan [41] studied some q-analogue of Hardy type inequalities for the Riemann-Liouville fractional integral operator; see [42,43] for more details.
In 1964, N. Levinson [44] presented the inequality respecting integration from a to b for 0 < a < b < ∞, f is a non-negative integrable function and s > 1, then In 2012, W.T. Sulaiman [45] gave a generalization and improvement for inequalities similar to Hardy inequality in the sense when f > 0 on [a, b] ⊂ (0, ∞) and 0 < k < 1 ≤ h, as follows: In 2013, B. Sroysang [46] presented a generalization for inequalities (1.3) and (1.4) with additional parameter m in the sense when f > 0 on [a, b] ⊂ (0, ∞), 0 < k < 1 ≤ h and m > 0, as follows: In 2014, B. Sroysang [47] presented a new kind of Hardy inequality and obtained a direct generalization of the original Hardy inequality. Next, K. Mehrez [48] studied some generalizations and new refined Hardy type inequalities by using Jensen's inequality and Chebyshev integral inequality, see [49][50][51][52] for more details.
In 2016, S. Wu, B. Sroysang and S. Li [53] investigated certain integral inequalities similar to the Hardy inequality. They generalized versions of some known results related to the Hardy inequality and gave some new integral inequalities of Hardy type by introducing a monotonous function and established the inequality for β being a non-negative real number, as follows: is non-increasing, and Inspired by this ongoing study, we establish the generalization of some integral inequalities related to Hardy type integral inequalities via (p, q)-calculus. Many results obtained in this paper provide extensions of other results given in previous papers. Furthermore, we give some examples to illustrate the investigated results.
First, we give some (p, q)-notation, which would appear in this paper. For any real number n, the (p, q)-analogue of n is defined by [n] p,q = p nq n pq (2.1) and [-n] p,q = -1 (pq) n [n] p,q . (2.2) If p = 1, then (2.1) reduces to which is q-analogue of n. It should be noted that a D p,q f (a) = lim x→a a D p,q f (x).
In Definition 2.1, if a = 0, then 0 D p,q f = D p,q f is defined by And, if p = 1, then D p,q f (x) = D q f (x), which is the q-derivative of the function f , and also if q → 1 in (2.5), then it reduces to a classical derivative. .
The proof of this theorem is given by [18].
If a = 0 in (2.6), then one can get the classical (p, q)-integral defined by The proofs of the following theorems are given in [18].
and α is a constant, then the following formulas hold:

Main results
In this section, we are going to establish the generalization of some integral inequalities related to Hardy type integral inequalities via (p, q)-calculus. The first result is presented as follows.
where r = 1/k and 1/k The proof is thus accomplished.

t)(ta) 1-1/r (xa) 1/r-2 a d p,q x a d p,q t.
By the assumption that the function (xa + γ )/g(x) is non-increasing and Theorem 2.
This proof is completed. Also, if q → 1, then (3.4) reduces to an inequality, which appeared in [53].

Corollary 3.1 If f is a non-negative function, γ is a positive real number, and r > 1, then
(2) If 0 < γ < a, we obtain the following inequality: Also, if p = 1 and q → 1, then (3.5) is reduced to (1.2).

Theorem 3.3 If f is a non-negative function, g is a positive function on
By the assumption that the function (xa + γ )/g(x) is non-increasing and Theorem 2.2(v), Hence, the inequality (3.6) is established.  f is a positive function on [a, b] ⊆ (0, ∞), r ≥ 1, and m > 0, then t (xa) r-1 a d p,q x a d p,q t.
The inequality (3.8) is proved. Also, if q → 1, then (3.9) reduces to an inequality, which appeared in [46]. x m This completes the proof.
By the assumption of the function g and Theorem 2.
which finishes the proof.
Theorem 3.7 If f , g are positive functions on [a, b] ⊆ (0, ∞) such that g is non-decreasing, 0 < r < 1, and m > 0, then (3.14) By the assumption of the function g and Theorem 2.2(v), we obtain The proof is accomplished. Also, if q → 1, then (3.15) reduces to the generalization of (1.6), which appeared in [53].

Conclusion
In the present paper, we use (p, q)-calculus to establish new integral inequalities related to Hardy type integral inequalities. Many existing results in the literature are reduced to special cases of our results when p = 1 and q → 1. The results of this paper are new and significantly contribute to the existing literature on the topic. In addition, we shall study these results in fractional (p, q)-calculus and conformable fractional (p, q)-calculus in the future.