Some trapezoid and midpoint type inequalities via fractional (p,q)$(p,q)$-calculus

*Correspondence: nkamsi@kku.ac.th 1Department of Mathematics, Khon Kaen University, 40002 Khon Kaen, Thailand Full list of author information is available at the end of the article Abstract Fractional calculus is the field of mathematical analysis that investigates and applies integrals and derivatives of arbitrary order. Fractional q-calculus has been investigated and applied in a variety of research subjects including the fractional q-trapezoid and q-midpoint type inequalities. Fractional (p,q)-calculus on finite intervals, particularly the fractional (p,q)-integral inequalities, has been studied. In this paper, we study two identities for continuous functions in the form of fractional (p,q)-integral on finite intervals. Then, the obtained results are used to derive some fractional (p,q)-trapezoid and (p,q)-midpoint type inequalities.


Introduction
The ordinary calculus of Newton and Leibniz is well known to be investigated extensively and intensively to produce a large number of related formulas and properties as well as applications in a variety of fields ranging from natural sciences to social sciences. In the early eighteenth century, the well-known mathematician Leonhard Euler (1707-1783) established quantum calculus or q-calculus, which is the study of calculus without limits, in the way of Newton's work for infinite series. Later, F. H. Jackson initiated a study of qcalculus in a symmetrical manner in 1910 and introduced q-derivative and q-integral in [1], see [2] for more details.
Many physical and mathematical problems have led to the necessity of studying qcalculus; for instance, Fock [3] studied the symmetry of hydrogen atoms using the qdifference equation. In addition, in modern mathematical analysis, q-calculus has lots of applications such as combinatorics, orthogonal polynomials, basic hypergeometric functions, number theory, quantum theory, mechanics, and theory of relativity, see also  and the references cited therein. The book by Kac and Cheung [25] covers the basic theoretical concepts of q-calculus.

Preliminaries
In this section, we recall some well-known facts on fractional (p, q)-calculus, which can be found in [10,11,38,53,55]. Throughout this paper, let [a, b] ⊂ R be an interval with a < b, and 0 < q < p ≤ 1 be constants,

Property 2.3 ([41])
For α > 0, the following formulas hold: Obviously, a function f is (p, q)-differentiable on [a, 1 Furthermore, if p = 1 in (2.7), then it reduces to D q f , which is q-derivative of the function f , see [25,73] for more details.
If a = 0 and p = 1 in (2.8), then we have the classical q-integral, see [25] for more details.
and an equivalent definition of (2.9) is given in [56] as Obviously, p,q (t + 1) = [t] p,q p,q (t). For s, t > 0, the definition of the (p, q)-beta function is defined by and (2.11) can also be written as , (2.12) see [74,75] for more details.

Main results
In this section, we give two identities for continuous functions in the form of fractional Riemann-Liouville (p, q)-integral type which will be used to prove the fractional Riemann-Liouville (p, q)-trapezoid and (p, q)-midpoint type inequalities.
, then the following equality holds: Proof By simple computation and using Definition 2.3, we have and Thus the proof is completed.
then the following Riemann-Liouville fractional (p, q)-trapezoid type inequality holds: Proof Using Lemma 3.1 and the convexity of | a D p,q f |, we have This completes the proof.

Theorem 3.2 Let f : [a, b] → R be a continuous function, α > 0, and a D p,q f be
for r ≥ 0, then the following Riemann-Liouville fractional (p, q)-trapezoid type inequality holds:

10)
where B 1 and B 2 are given in Theorem 3.1 and Proof Using Lemma 3.1, the convexity of | a D p,q f | r , and the power mean inequality, we have Therefore, the proof is completed.
Proof Using Lemma 3.1, the convexity of | a D p,q f | r , and Hölder's inequality, we have This completes the proof.
Moreover, if p = 1, then (3.13) reduces to which appeared in [40]. Now we will prove the following lemma to obtain the Riemann-Liouville fractional (p, q)-midpoint type inequalities.

Lemma 3.2 Let f : [a, b] → R be a continuous function and
, then the following equality holds: Proof By direct computation and using Definitions 2.1 and 2.2, we have On the other hand, in Lemma 3.1, the following integral was given: Consequently, from (3.17) and (3.18), we have Therefore, the proof is completed.