Quantum Hermite–Hadamard-type inequalities for functions with convex absolute values of second qb$q^{b}$-derivatives

In this paper, we obtain Hermite–Hadamard-type inequalities of convex functions by applying the notion of qb-integral. We prove some new inequalities related with right-hand sides of qb-Hermite–Hadamard inequalities for differentiable functions with convex absolute values of second derivatives. The results presented in this paper are a unification and generalization of the comparable results in the literature on Hermite–Hadamard inequalities.


Introduction
The Hermite-Hadamard inequality introduced by Hermite and Hadamard (see also [1] and [2, p. 137]) is one of the most well-known inequalities in the theory of convex functional analysis. It has an interesting geometrical interpretation with several applications.
These inequalities state that if f : I → R is a convex function on an interval I of real numbers and a, b ∈ I with a < b, then (1.1) Both inequalities hold in the reversed manner if f is a concave function. Note that the Hermite-Hadamard inequalities may be viewed as a refinement of the concept of convexity and follows from Jensen's inequality. Hermite-Hadamard inequalities for convex functions have received much attention in the recent years, and, consequently, a remarkable variety of refinements and generalizations have been obtained. Many well-known integral inequalities such as the Hölder, Hermite-Hadamard, Ostrowski, Cauchy-Bunyakovsky-Schwarz, Gruss, Gruss-Chebyshev, and other integral inequalities have been studied in the setup of q-calculus using the concept of classical convexity. For more results in this direction, we refer to [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18].
The purpose of this paper is to study Hermite-Hadamard-like inequalities for convex functions by applying the new concept of 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, we give a brief description of the concepts of q-calculus and some related works in this direction. In Sect. 3, we present the Hermite-Hadamard-type inequalities for the q b -integrals. We also study the relation between the results presented herein and comparable results in the literature. Section 4 contains some conclusions and further directions for the future research. We believe that the study initiated in this paper may inspire new research in this area.
Jackson [20] defined the q-Jackson integral of a given function f from 0 to b as follows: provided that the sum converges absolutely. Jackson [20] defined the q-Jackson integral of a given function over the interval [a, b] as follows:  [19]) given by Alp et al. [3] proved the following q a -Hermite-Hadamard inequalities for convex functions in the setting of quantum calculus.
In [3] and [23] authors established some bounds for the left-and right-hand sides of inequality (2.3).
On the other hand, Bermudo et al. [22] gave the following definition and obtained the related Hermite-Hadamard-type inequalities.
and 0 < q < 1, then we have the following q-Hermite-Hadamard inequalities: From Theorems 1 and 2 we obtain the following inequalities.
In this paper, we will also find some bounds for right-hand side of inequality (2.4).

New Hermite-Hadamard-type inequalities for quantum integrals
We now give some new Hermite-Hadamard-type inequalities for functions whose second q b -derivatives in absolute value are convex. We start with the following useful lemma.
Remark 1 If we take the limit as q → 1in Lemma 1, then we have as given in [25].

f is continuous and integrable on [a, b], then we have the following inequality, provided that
where 0 < q < 1.
Proof Taking the modulus in Lemma 1 and applying the convexity of | b D 2 q f |, we obtain which completes the proof.
Remark 2 Under the assumptions of Theorem 4 with the limit as q → 1 -, we have the following trapezoidal inequality: 2 where 0 < q < 1.
Proof Taking the modulus in Lemma 1 and applying the well-known power mean inequality, we have By the convexity of | b D 2 q f | p 1 we have