A new generalization of some quantum integral inequalities for quantum differentiable convex functions

In this paper, we offer a new quantum integral identity, the result is then used to obtain some new estimates of Hermite–Hadamard inequalities for quantum integrals. The results presented in this paper are generalizations of the comparable results in the literature on Hermite–Hadamard inequalities. Several inequalities, such as the midpoint-like integral inequality, the Simpson-like integral inequality, the averaged midpoint–trapezoid-like integral inequality, and the trapezoid-like integral inequality, are obtained as special cases of our main results.


Introduction
The study of q-calculus was initiated in the early 20th century after the work of Jackson (1910) who defined an integral later known as the q-Jackson integral (see [16,22,23,27,28]). In q-calculus, the classical derivative is replaced by the q-difference operator to deal with nondifferentiable functions. For more discussion on this subject, we refer to [8,21]. Applications of q-calculus can be found in various disciplines of mathematics and physics (see [13,26,36,45]).
The purpose of this paper is to prove several new quantum integral inequalities by applying the newly defined concept of a q b -integral. We also discuss the relation of our results with comparable results existing in the literature.
The organization of this paper is as follows: In Sect. 2, a brief description of the concepts of q-calculus and some related works in this direction are given. In Sect. 3, the bounds of Hermite-Hadamard-type inequalities for the q b -integrals are presented. In Sect. 4, several special cases of our main results are discussed. The relationship between the results presented herein and comparable results in the literature is also studied. Section 5 contains some conclusions and further directions for future research. We believe that the study initiated in this paper may inspire new research in this area.
Jackson [27] defined the q-Jackson integral of a given function f from 0 to b as follows provided that the sum converges absolutely. Jackson [27] defined the q-Jackson integral of a given function over the interval [a, b] as follows: is identified by the the following expression: If x = a, we define a D q f (a) = lim x→a a D q f (x) if it exists and is finite.
if it exists and is finite.
Alp et al. [11] proved the following q a -Hermite-Hadamard inequalities for convex functions in the setting of quantum calculus: In [11] and [33], the authors established some bounds for the left-and right-hand sides of the inequality (2.3).
On the other hand, Bermudo et al. [15] gave the following definition and obtained the related Hermite-Hadamard-type inequalities: and 0 < q < 1, then the q-Hermite-Hadamard inequalities are given as follows: From Theorems 1 and 2, one can obtain the following inequalities: For any convex function f : [a, b] → R and 0 < q < 1, we have We recall the following well-known inequality: Theorem 3 (Hölder's inequality, [12, p. 604]) Let x > 0, 0 < q < 1, p 1 > 1 be such that 1

Main results
In this section, we give some new estimates of Hermite-Hadamard-type inequalities for functions whose first q b -derivatives in absolute value are convex.
Let's start with the following useful lemma.
By applying identical transformation, we obtain By the equality (2.1), we obtain It follows from (2.1) and Definition 4 that Using Before we present our main inequalities, we first give some calculated quantum integrals:

) such that b D q f is continuous and integrable on [a, b], then we have the following inequality provided that
where 0 < q < 1.
Proof On taking the modulus in Lemma 1 and applying the convexity of | b D q f |, we obtain which completes the proof.

Theorem 5 Suppose that f : [a, b] ⊂ R → R is a q b -differentiable function on (a, b) and b D q f is continuous and integrable on
], then we have following inequality: where 0 < q < 1.
Proof Taking the modulus in Lemma 1 and applying the well-known power-mean inequality for quantum integrals, we have which completes the proof.
Remark 3 If we take q → 1in Theorem 5, then we have [25,Theorem 2.2]. on (a, b) where 0 < q < 1 and Proof Taking the modulus in the Lemma 1 and applying well-known Hölder's inequality for quantum integrals, we obtain Using the fact that | b D q f | p 1 is convex, we have which completes the proof.

Special cases
In this section, we discus special cases of our main results and the relationship between the results presented herein and comparable results in the literature.
Remark 5 From Lemma 1, we have following observations: (i) If we take ν = 0, then we obtain following new identity: (4.1) (ii) By setting ν = 1 in Lemma 1, we obtain following new identity: (4.2) (iii) Set ν = 1 [2] q to obtain the following new identity: Remark 6 In the following, we give the different variants of Lemma 1.