Fractional ( p , q ) -Calculus on Finite Intervals and Some Integral Inequalities

: Fractional q -calculus has been investigated and applied in a variety of ﬁelds in mathematical areas including fractional q -integral inequalities. In this paper, we study fractional ( p , q ) -calculus on ﬁnite intervals and give some basic properties. In particular, some fractional ( p , q ) -integral inequalities on ﬁnite intervals are proven.


Introduction
In mathematics, quantum calculus or q-calculus is the study of calculus without limits. In the early Eighteenth Century, the well-known mathematician Leonhard Euler (1707-1783) established q-calculus in the way of Newton's work for infinite series. Yet, q-calculus was known to be initiated by F. H. Jackson in 1910, who introduced the q-derivative and q-integral in [1] (see also [2]).
As a connection between the fields of mathematics and physics, q-calculus has played a significant role in physics phenomena; for instance, Fock [3] studied the symmetry of hydrogen atoms using the q-difference equation. Furthermore, in modern mathematical analysis, q-calculus has many applications such as combinatorics, orthogonal polynomials, basic hypergeometric functions, number theory, quantum theory, mechanics, and the theory of relativity; see also  and the references cited therein. The book by V. Kac and P. Cheung [25] covers the basic theoretical concept of q-calculus.
In 2013, being one of the most attractive areas, some new researchers are interested in q-calculus; in particular, J. Tariboon and S.K. Ntouyas [26] defined the q k -calculus and proved some of its significant properties. Next, J. Tariboon and S.K. Ntouyas [27] extended some of the important integral inequalities to q-calculus. Moreover, in 2016, J. Necmettin, Z.S. Mehmet, andİ.İmdat [28] proved the correctness of the left part of the q-Hermite-Hadamard inequality and the generalized q-Hermite-Hadamard inequality. With these results, many researchers have extended some important topics of q-calculus together with applications in many fields, such as q-integral inequalities; see [29][30][31][32][33][34][35][36][37] for more details.
Fractional calculus is the field of mathematical analysis that deals with the investigation and applications of integrals and derivatives of arbitrary order. In 2015, J. Tariboon, S.K. Ntouyas, and P. Agarwal [38] proposed a new q-shifting operator a Φ q (m) = qm + (1 − q)a for studying new concepts of fractional q-calculus. Furthermore, in 2016, since inequalities play a vital role in modern analysis, as well as mathematical analysis depends on [k] p,q = p k −q k p−q , k ∈ N, (1) [k] p,q ! = A q-shifting operator is defined as: where m ∈ R. For any positive integer k, we have: By computing directly, we get the following results.

Property 1.
For any m, n ∈ R and for all positive integers j, k, we have: For m, n ∈ R and k ∈ N ∪ {0}, the q-Pochhammer symbol or the q-shifted factorial is a q-analogue of the Pochhammer symbol, which is defined by: and the new power of q-shifting operator is defined by: More generally, if γ ∈ R, then: and: The (p, q)-derivative of function f is defined on [a, b] at t ∈ [a, b] as follows.
Definition 1 ([40]). Let f : [a, b] → R be a continuous function. Then: a D p,q f (a) = lim t→a a D p,q f (t) (9) is called the (p, q)-derivative of a function f at a.
In Definition 1, if p = 1, then a D p,q f = a D q f , which is the q-derivative of function f on [a, b], and if a = 0, then (8) reduces to the q-derivative of the function f on [0, b]; see [25,41] for more details.
Furthermore, if c ∈ (a, t), then the (p, q)-integral is defined by: Note that if a = 0 and p = 1, then (10) reduces to the q-integral of the function, which can be found in [26]. Theorem 1 ([40]). The following formulas hold for t ∈ [a, b] : Theorem 2 ([40]). If f , g : [a, b] → R are continuous functions, t ∈ [a, b] and λ ∈ R, then the following formulas hold: Let us define the new (p, q)-analogue of the power function a (m − n) k p,q with k ∈ N ∪ {0} and m, n ∈ R as the following: It is easy to see that: More generally, if α ∈ R, then: Property 2. For α > 0, the following formulas hold: Proof. (i) For α > 0, we have: To prove (ii), we use (i) and let n = a Φ k q/p (m) in (14); we have:  By letting u = q n p n+1 s + 1 − q n p n+1 a and using Definition 2, we get: The proof is complete.
and an equivalent definition of (16) was given in [56] as: where: . For s, t > 0, the definition of the (p, q)-beta function is defined by: and (18) can also be written as: see [43,69] for more details.

Main Results
From Lemma 1, we shall give that which leads to a definition of the fractional (p, q)integral of the Riemann-Liouville type with the consideration of the n-time as follows: r 2 a f (r 1 ) a d p,q r 1a d p,q r 2 . . . a d p,q r n−2a d p,q r n−1 .
The function f in (20) will be assumed to be continuous on [a, b]. From (15), we have: Integrating n-times reduces (20) to a single (p, q)-integral on [a, b] as follows.
Definition 3. Let f be defined on [a, b], and let α > 0. The Riemann-Liouville fractional (p, q)integral is defined by: where α > 0 and v is the smallest integer greater than or equal to α.
The basic q-hypergeometric function is defined in [70] as: and q-Vandermonde reversing the order of summation is defined as: Theorem 3. If f is a continuous function on [a, b] and α, β > 0, then the Riemann-Liouville fractional (p, q)-integral has the following semi-group property: Proof. For t ∈ [a, b], we have: Applying the (p, q)-gamma function in (16), we obtain: Taking m = i + j and interchanging the order of summation, we obtain: On the other hand, and: Substituting (28) into (27), we get: Therefore, we obtain: By (24), we have: From (25), we obtain: Substituting (32) into (31), we get: which is the series representation of a I α+β p,q f (t). Therefore, (26) holds.
Throughout this paper, the variable s is shown inside the fractional integral, which is denoted as: , then the following formula holds: Proof. Applying Definition 2, Property 2, and (18) and (19), we have: The proof of Lemma 2 is complete.

Remark 1.
Define functions f , g : [a, b] → R by f (t) = t and g(t) = t 2 . For α > 0, we get: Next, we study some fractional (p, q) integral inequalities on finite intervals.
• The fractional (p, q)-Hermite-Hadamard integral inequalities on [a, p α b + (1 − p α )a]: Theorem 5. If f : [a, b] → R is a convex differentiable function and α > 0, then we have: Proof. From the left-hand side of the proof in Theorem 3 and Theorem 5.1 in [15,42], respectively, there is a one line support: Γ p,q (α) on both sides of (38), we obtain: Taking the fractional (p, q)-integration of order α > 0 with respect to s on (39), where s ∈ (a, p α b + (1 − p α )a), we have: Moreover, from the left-hand side of (40), we have: and similar to the computation of (41) above, we also get: By substituting (42) and (41) into (40), we obtain the first part of (37). On the other hand, from the proof of the right-hand side of Theorem 3 in [42], we have: Multiplying by a Φ p α (b) − a Φ q (s) (α−1) p,q /p ( α 2 ) Γ p,q (α) on both sides of (43) and taking the (p, q)-integral of order α > 0 with respect to s ∈ [a, p α b + (1 − p α )a], we obtain: By using the same computation as in (41) and (42) for the left-hand side of (44), we obtain: Substituting (45) into (44), we derive the second part of (37). Therefore, the proof of Theorem 5 is complete.

Conclusions
In this work, we studied the fractional (p, q)-calculus on finite intervals. We also gave some of its significant properties. Furthermore, we proved some fractional (p, q)-integral inequalities on finite intervals. For the ideas, as well as the techniques of this paper, we hope that it will inspire interested readers working in this field.