Some ( p , q ) -Integral Inequalities of Hermite–Hadamard Inequalities for ( p , q ) -Differentiable Convex Functions

: In this paper, we establish a new ( p , q ) b -integral identity involving the ﬁrst-order ( p , q ) b derivative. Then, we use this result to prove some new ( p , q ) b -integral inequalities related to Hermite– Hadamard inequalities for ( p , q ) b -differentiable convex functions. Furthermore, our main results are used to study some special cases of various integral inequalities. The newly presented results are proven to be generalizations of some integral inequalities of already published results. Finally, some examples are given to illustrate the investigated results.


Introduction
In mathematics, the study of calculus without limits is called quantum calculus (briefly called q-calculus), and was first studied by Euler (1707-1783), introducing the number in the q-infinite series defined by Newton (also called Newton's infinite series). In the early 20th century, Jackson [1] relied on the concept of Euler to define the q-integral and qderivative (well-known q-Jackson integral and q-Jackson derivative) over the interval (0, ∞). In q-calculus, we obtain the q-analoques of mathematical objects that can be recaptured by taking q → 1. In recent years, q-calculus has had numerous applications in various disciplines of physics and mathematics; see [2][3][4][5][6][7][8][9][10] and the references cited therein for more details.
In 2013, Tariboon and Ntouyas [11] presented the q a -integral and q a -derivative over finite intervals and also investigated the existence and uniqueness results of initial value problems for the first-and second-order impulsive q a -difference equations. In 2020, Bermudo et al. [12] introduced the q b -integral and q b -derivative over finite intervals and also proved some of their basic properties. Recently, the topic of q-calculus has been applied in various integral inequalities, for example, Simpson-and Newton-type inequalities [13], Hanh inequalities [14], Ostrowski inequalities [15], Fejér-type inequalities [16], Hermite-Hadamard-like inequalities [17], Hermite-Hadamard inequalities [18], and the references cited therein. In particular, Hermite-Hadamard inequalities have also been studied by using q-calculus for convex functions by many researchers; see [12,[19][20][21][22][23][24][25] and the references cited therein for more details.
Definition 1 ([27,28]). Let Ψ : [a, b] → R be a continuous function. Then, the (p, q) a -derivative of Ψ at x is given by lim x→a a D p,q Ψ(x), i f x = a. (2) The function Ψ is called (p, q) a -differentiable function on [a, b] if a D p,q Ψ(x) exists for all x ∈ [a, (b − a)/p + a].
Note that if p = 1 and a D 1,q Ψ(x) = a D q Ψ(x), then (2) is reduced as follows: which is the well-known q a -derivative of Ψ on [a, b]; see [48,49] for more details. Moreover, if a = 0 and 0 D q Ψ(x) = D q Ψ(x), then (3) is reduced as follows: which is the well-known q a -derivative of Ψ on [0, b]; see [47] for more details.

Definition 3 ([27]
). Let Ψ : [a, b] → R be a continuous function. Then, the (p, q) a -integral of Ψ at x is given by The function Ψ is called (p, q) a -integrable function on Note that if a = 0, then (5) is reduced as follows: which appears in [28]. Moreover, if p = 1, then (6) is reduced as follows which is the well-known q-Jackson integral; see [1] for more details.

Main Results
In this section, we prove (p, q) b -integral inequalities related to Hermite-Hadamard inequalities for which the first-order (p, q) b -derivatives in absolute value are convex functions. We define . The (p, q)-integral identity is as follows: Proof. Using Definition 2, we have Applying an identical transformation, we obtain Using (6) and (11), we obtain Similarly, we obtain and Substituting (13) to (15) in (12), we obtain the required (p, q) b -integral identity. Therefore, the proof is completed.

Remark 4.
From Corollary 2, we have the new (p, q) b -integral identities as follows: (i) If we take ν = p/ [2] p,q , then (19) leads to the midpoint-type identity as follows: which was proposed by Aamir Ali et al. in [29]. (ii) Taking ν = p/ [2] p,q , then (20) leads to the Simpson-like integral identity as follows: (iii) If we set ν = p/ [2] p,q , then (21) leads to the averaged midpoint-trapezoid-type integral identity as follows: (iv) By setting ν = p/ [2] p,q , then (22) leads to the trapezoid-type integral identity as follows: Theorem 3. Suppose that Ψ : [a, b] → R is a (p, q) b -differentiable function on I 1 such that b D p,q Ψ is continuous and integrable functions on where Λ i (p, q, γ, ν), i = 1, 2, 3, and Θ j (p, q, γ, ν), j = 1, 2, 3, are given by Proof. Taking the absolute value of both sides of (10), using Lemma 1 and applying the convexity of | b D p,q Ψ|, we obtain which completes the proof.

Remark 7.
From Corollary 3, we have the new (p, q) b -integral inequalities as follows: (i) If we take γ = 0, then (28) leads to the midpoint-type integral inequality as follows: (ii) Taking γ = 1/3, then (28) leads to the Simpson-like integral inequality as follows: which appears in [46]. (ii) We obtain the Simpson-like integral inequality as follows: which appears in [46]. Moreover, if q → 1, then (34) is reduced as follows: which appears in [51]. (iii) We obtain the averaged midpoint-trapezoid-like integral inequality as follows: which appears in [46]. Moreover, if q → 1, then (35) is reduced as follows: which appears in [52].
(iv) We obtain the trapezoid-type integral inequality as follows: which appears in [52].
where Θ 2 (p, q, γ, ν) is given in Theorem 3 and ∆ 1 (p, q, γ, ν) is defined by Proof. Taking the absolute value of both sides of (10) and using the power-mean inequality for (p, q)-integrals, we obtain Applying the convexity of | b D p,q Ψ| r , we have which completes the proof.
Theorem 5. Suppose that Ψ : [a, b] → R is a (p, q) b -differentiable function on I 1 such that b D p,q Ψ is a continuous and integrable function on I 2 with γ, ν ∈ [0, 1]. If | b D p,q Ψ| r for r > 1 with 1/s + 1/r = 1 is a convex function on [a, b], then where Proof. Taking the absolute value of both sides of (10) and using Theorem 1, we obtain Applying the convexity of | b D p,q Ψ| r , we have 1 and the right side of (27) becomes It is clear that 0.0337 ≤ 1.3868, which demonstrates the result described in Theorem 4.

Conclusions
In this work, we established some new estimates of (p, q) b -integral inequalities related to Hermite-Hadamard inequalities for which the first-order (p, q) b -derivatives in absolute value are convex functions. The main results in this study were proven to be generalizations of some previously proved results of q b -integral inequalities related to Hermite-Hadamard inequalities for q b -differentiable convex functions. Furthermore, the obtained results were used to study some special cases, namely the midpoint-type integral inequality, Simpson-like integral inequality, averaged midpoint-trapezoid-type integral inequality, and trapezoid-type integral inequality. Examples were given to illustrate the investigated results.