On some new quantum midpoint-type inequalities for twice quantum differentiable convex functions

Quantum calculus, sometimes called calculus without limits, is equivalent to the traditional infinitesimal calculus without the notion of limits. In the field of q-analysis, many studies have recently been carried out. Euler started this field because of the very high demand of mathematics that models quantum computing q-calculus appeared as a connection between physics and mathematics. It has applications in numerous areas of mathematics, such as combinatorics, number theory, basic hypergeometric functions, and orthogonal polynomials, and in fields of other sciences, such as mechanics, theory of relativity, and quantum theory [1–7]. Apparently, Euler was the founder of this branch of mathematics, by using the parameter q in Newton’s work on infinite series. Later, the q-calculus was first given by Jackson [8]. In 1908–1909, Jackson defined the general q-integral and q-difference operator [3]. In 1969, Agarwal described the q-fractional derivative for the first time [9]. In 1966–1967, Al-Salam introduced a q-analog of the RiemannLiouville fractional integral operator and q-fractional integral operator [10]. In 2004, Rajkovic gave a definition of the Riemann-type q-integral which generalized to Jackson q-integral. In 2013, Tariboon introduced D a q-difference operator [11].


Introduction
Quantum calculus, sometimes called calculus without limits, is equivalent to the traditional infinitesimal calculus without the notion of limits. In the field of q-analysis, many studies have recently been carried out. Euler started this field because of the very high demand of mathematics that models quantum computing q-calculus appeared as a connection between physics and mathematics. It has applications in numerous areas of mathematics, such as combinatorics, number theory, basic hypergeometric functions, and orthogonal polynomials, and in fields of other sciences, such as mechanics, theory of relativity, and quantum theory [1][2][3][4][5][6][7]. Apparently, Euler was the founder of this branch of mathematics, by using the parameter q in Newton's work on infinite series. Later, the q-calculus was first given by Jackson [8]. In 1908-1909, Jackson defined the general q-integral and q-difference operator [3]. In 1969, Agarwal described the q-fractional derivative for the first time [9]. In 1966-1967, Al-Salam introduced a q-analog of the Riemann-Liouville fractional integral operator and q-fractional integral operator [10]. In 2004, Rajkovic gave a definition of the Riemann-type q-integral which generalized to Jackson q-integral. In 2013, Tariboon introduced D a q -difference operator [11].
In this paper, motivated through q-calculus, we found bounds for q-midpoint integral inequalities. We first obtain an identity for twice q-differentiable functions. After that, through the derived identity, we attain some new outcomes for q-midpoint inequalities. With all this, we revealed the results we found in the classical analysis by using → − q 1 .

Preliminaries and definitions of q-calculus
Throughout this paper, let < a b and let < < q 0 1 be a constant. The definitions and theorems for q-derivative and q-integral of a function f on [ ] a b , are as follows.
, is characterized by the expression if it exists and it is finite.
In 2004, Rajkovic et al. [24] gave the following definition of the Riemann-type q-integral which was generalized to Jackson q-integral on [ ] a b , : Alp et al. [11] proved the q-Hermite-Hadamard inequality; in [25], the authors proved the same inequality by removing the differentiability assumptions as follows: , and < < q 0 1. Then, we have On the other hand, in [26], Bermudo et al. gave the following new definitions of quantum integral and derivative. In the same paper, the authors also proved a new variant of the quantum Hermite-Hadamardtype inequality linked to their newly defined quantum integral: , is given by , is given by , and < < q 0 1, then, q-Hermite-Hadamard inequalities are given as follows: We frequently used the following notations:    In this section, we will prove an equality which will help to obtain our main results.
, and ∈ ( , , then we attain the identity Proof. From Definition 4, we have the following equality: From (9) and the fundamental properties of q-integrals, we have Now, let us calculate the values of integrals I 1 and I 2 as follows: q qq q f q a q b

New midpoint-type inequalities for quantum integrals
In this section, we obtain some new midpoint-type inequalities for newly defined quantum integrals utilizing twice q-differentiable convex functions.
Theorem 3. Suppose that the assumptions of Lemma 5 hold. If| , , then we have the inequality This completes the proof. □ Remark 2. If we take the limit → − q 1 in Theorem 3, then we obtain the inequality   Authors contribution: All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.