Refinements of quantum Hermite-Hadamard- type inequalities

Both inequalities hold in the reversed direction if f is concave. We note that the Hermite-Hadamard inequality may be regarded as a refinement of the concept of convexity and it follows easily from Jensen’s inequality. Hermite-Hadamard inequality for convex functions has received renewed attention in recent years, and a remarkable variety of refinements and generalizations have been studied. On the other hand, quantum calculus, sometimes called calculus without limits, is equivalent to traditional infinitesimal calculus without the notion of limits. In the field of q-analysis, many studies have recently been carried out. It has applications in numerous areas of mathematics such as combinatorics, number theory, basic hypergeometric functions, orthogonal polynomials, and in fields of other sciences such as mechanics, theory of relativity, and quantum theory [3–7]. Apparently, Euler invented this important branch of mathematics when he used the q parameter in Newton’s work on infinite series. Later, the q-calculus was first given by Jackson [3]. In 1908–1909, the general form of the q-integral and q-difference operator was defined by Jackson [6]. In 1969, for the first time Agarwal [8] defined the q-fractional derivative. In 1966–1967, Al-Salam [9] introduced a q-analog of the q-fractional integral and


Introduction
The Hermite-Hadamard inequality discovered by C. Hermite and J. Hadamard (see, e.g., [1], [2, p. 137]) is one of the most well-established inequalities in the theory of convex functions with a geometrical interpretation and many applications. These inequalities state that if → f I : is a convex function on the interval I of real numbers and ∈ ω ω I , 1 2 with < ω ω Both inequalities hold in the reversed direction if f is concave. We note that the Hermite-Hadamard inequality may be regarded as a refinement of the concept of convexity and it follows easily from Jensen's inequality. Hermite-Hadamard inequality for convex functions has received renewed attention in recent years, and a remarkable variety of refinements and generalizations have been studied. On the other hand, quantum calculus, sometimes called calculus without limits, is equivalent to traditional infinitesimal calculus without the notion of limits. In the field of q-analysis, many studies have recently been carried out. It has applications in numerous areas of mathematics such as combinatorics, number theory, basic hypergeometric functions, orthogonal polynomials, and in fields of other sciences such as mechanics, theory of relativity, and quantum theory [3][4][5][6][7]. Apparently, Euler invented this important branch of mathematics when he used the q parameter in Newton's work on infinite series. Later, the q-calculus was first given by Jackson [3]. In 1908-1909, the general form of the q-integral and q-difference operator was defined by Jackson [6]. In 1969, for the first time Agarwal [8] defined the q-fractional derivative. In 1966-1967, Al-Salam [9] introduced a q-analog of the q-fractional integral and q-Riemann-Liouville fractional. In 2004, Rajkovic gave a definition of the Riemann-type q-integral, which was generalized to the Jackson q-integral. In 2013, Tariboon introduced the D q ω 1 -difference operator [10]. In recent years, because of the importance of convexity in numerous fields of applied and pure mathematics, it has been significantly investigated. The theory of convexity and inequalities are strongly connected to each other, therefore, various inequalities can be established in the literature which are proved for convex, generalized convex, and differentiable convex functions of single and double variables, see, for example, [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29].
The general structure of this paper consists of five main sections including Introduction. In Section 2, we give some necessary important notations for concept q-calculus and we also mention some related works in the literature. In Section 3 we present some new Hermite-Hadamard-type inequalities for q ω 2 integrals. Some refinements of quantum Hermite-Hadamard-type inequalities are proved in Section 4. We also examine the relation between our results and inequalities presented in the earlier works. Finally, in Section 5, some conclusions and further directions of research are discussed. We note that the opinion and technique of this work may inspire new research in this area.

Preliminaries of q-calculus and some inequalities
In this section, we present some required definitions and related inequalities about q-calculus.
We have to give the following notation which will be used many times in the following sections (see [7]): 1 2 is characterized by the expression: if it exists and it is finite.

New Hermite-Hadamard-type inequalities for q ω 2 -integrals
In this section, we prove two new quantum Hermite-Hadamard inequalities for q ω 2 -integrals.
Proof. We can write the equation of the tangent line for the function F at the point follows: By Definition 4, we get which gives the first inequality in (5). The second inequality is the same as in Theorem 3. □ Remark 1. If we take the limit → − q 1 in Theorem 4, then the inequalities (5) reduce to (1).
Proof. Similar way as in Theorem 4, we can write tangent line for the function F at the point  as follows: By Definition 4, we get This gives the first inequality in (6). The second inequality is the same as in Theorem 3. □

Main results
In this section, we present the refinements of quantum Hermite-Hadamard inequalities for q ω 2 -integrals.
ω ω ϰ, , By (9) and (10), we have the first part of (7). □ Remark 3. If we take the limit → − q 1 in Theorem 6, then the inequalities (7) reduce to the following inequalities: Remark 4. If we take the limit → − q 1 in Corollary 1, then (11) reduces to the inequalities     which shows that ϕ is convex on [ ] 0, 1 . By applying Theorem 3 for the convex function ϕ on [ ] 0, 1 , we have the inequalities That is,  which are given in [33].
Proof. The proof of this theorem follows a similar procedure to that in Theorem 9 by using Theorem 4. □ are valid for all ∈ [ ] t 0, 1 .
Proof. The proof of this theorem follows a similar procedure to that in Theorem 9 by using Theorem 5. □ Remark 8. If we take the limit → − q 1 , then the inequalities (17), (20) and (21)  which are given in [33].