Post-quantum trapezoid type inequalities

: In this study, the assumption of being di ﬀ erentiable for the convex function f in the ( p , q )-Hermite-Hadamard inequality is removed. A new identity for the right-hand part of ( p , q )-Hermite-Hadamard inequality is proved. By using established identity, some ( p , q )-trapezoid integral inequalities for convex and quasi-convex functions are obtained. The presented results in this work extend some results from the earlier research.


Introduction
Quantum calculus or briefly q-calculus is a study of calculus without limits. Post-quantum or (p, q)-calculus is a generalization of q-calculus and it is the next step ahead of the q-calculus. Quantum Calculus is considered an incorporative subject between mathematics and physics, and many researchers have a particular interest in this subject. Quantum calculus has many applications in various mathematical fields such as orthogonal polynomials, combinatorics, hypergeometric functions, number theory and theory of differential equations etc. Many scholars researching in the field of inequalities have started to take interest in quantum calculus during the recent years and the active readers are referred to the articles [2, 3, 7-9, 11, 12, 16, 17, 21, 24, 25, 27-29] and the references cited in them for more information on this topic. The authors explore various integral inequalities in all of the papers mentioned above by using q-calculus and (p, q)-calculus for certain classes of convex functions.
In this paper, the main motivation is to study trapezoid type (p, q)-integral inequalities for convex and quasi-convex functions. In fact, we prove that the assumption of the differentiability of the mapping in the (p, q)-Hermite-Hadamard type integral inequalities given in [12] can be eliminated. The relaxation of the differentiability of the mapping in the (p, q)-Hermite-Hadamard type integral inequalities proved in [12] also indicates the originality of results established in our research and these findings have some relationships with those results proved in earlier works.

Preliminaries
The basic concepts and findings which will be used in order to prove our results are addressed in this section.
Let I ⊂ R be an interval of the set of real numbers. A function f : I → R is called as a convex on I, if the inequality holds for every x, y ∈ I and t ∈ [0, 1]. A f : I → R known to be a quasi-convex function, if the inequality holds for every x, y ∈ I and t ∈ [0, 1].
The following properties of convex functions are very useful to obtain our results.
Definition 2.1. [19] A function f defined on I has a support at x 0 ∈ I if there exists an affine function for all x ∈ I. The graph of the support function A is called a line of support for f at x 0 . Perhaps the most famous integral inequalities for convex functions are known as Hermite-Hadamard inequalities and are expressed as follows: where the function f : I → R is convex and a, b ∈ I with a < b. By using the following identity, Pearce and Pečarić proved trapezoid type inequalities related to the convex functions in [18] and [6]. Some trapezoid type inequalities related to quasi-convex functions are proved in [1] and [9]. Lemma 2.3.
[6] Let f : I • ⊂ R → R be a differentiable mapping on I • (I • is the interior of I), a, b ∈ I • with a < b. If f ∈ L [a, b], then the following equality holds: Some definitions and results for (p, q)-differentiation and (p, q)-integration of the function f : [a, b] → R in the papers [12,22,23]. 3) the q-derivative of the function f defined on [a, b] (see [16,21,25,26]).
Remark 2.1. If one takes a = 0 in (2.3), then 0 D p,q f (t) = D p,q f (t) , where D p,q f (t) is the (p, q)derivative of f at t ∈ [0, b] (see [5,10,20]) defined by the expression  [15]) given by the expression the definite q-integral of the function f defined on [a, b] (see [16,21,25,26]). [20,22,23]). We notice that for a = 0 and p = 1 in (2.7) , is the definite q-integral of f over the interval [0, b] (see [15]). Quantum trapezoid type inequalities are obtained by Noor et al. [16] and Sudsutad [21] by applying the definition convex and quasi-convex functions on the absolute values of the q-derivative over the finite interval of the set of real numbers.
Lemma 2.4. Let f : [a, b] ⊂ R → R be a continuous function and 0 < q < 1. If a D q f is a q-integrable function on (a, b), then the equality holds: The (p, q)-Hermite-Hadamard type inequalities were proved in [12].
In this paper, we remove the (p, q)-differentiability assumption of the function f in Theorem 2.5 and establish (p, q)-analog of the Lemma 2.4 and Lemma 2.3. We obtain (p, q)-analog of the trapezoid type integral inequalities by applying the established identity, which generalize the inequalities given in [1,6,9,16,18,21].

Main results
Throughout this section let I ⊂ R be an interval, a, b ∈ I • (I • is the interior of I) with a < b (in other words [a, b] ⊂ I • ) and 0 < q < p ≤ 1 are constants. Let us start proving the inequalities (2.13), with the lighter conditions for the function f . Theorem 3.1. Let f : I → R be a convex function on I and a, b ∈ I • with a < b, then the following inequalities hold: Proof. Since f is convex function on the interval I, by Theorem 2.2 f is continuous on In the proof of the Theorem 2.5 the authors used the tangent line at the point of x 0 = qa+pb p+q . Similarly, using the inequality (3.2) and a similar method with the proof of the Theorem 2.5 we have (3.1) but we omit the details. Thus the proof is accomplished.
We will use the following identity to prove trapezoid type (p, q)-integral inequalities for convex and quasi-convex functions.
Proof. Since f is continuous on I • and a, b ∈ I • , the function f is continuous on [a, b]. Hence, clearly , t 0 is well defined and exists. Since is continuous on [0, 1] and from Definition 2.3.
is well defined and exists. By using (2.7) and (3.4), we get q n p n f q n p n b + 1 − q n p n a q n p n f q n p n b + 1 − q n p n a This completes the proof. We can now prove some quantum estimates of (p, q)-trapezoidal integral inequalities by using convexity and quasi-convexity of the absolute values of the (p, q)-derivatives.
Proof. Taking absolute value on both sides of (3.3), applying the power-mean inequality and by using the convexity of a D p,q f r for r ≥ 1, we obtain We evaluate the definite (p, q)-integrals as follows Making use of (3.7), (3.8) and (3.9) in (3.6), gives us the desired result (3.5). The proof is thus accomplished.
Theorem 3.6. Let f : I • ⊂ R → R be a continuous function on I • and a, b ∈ I • with a < b. If a D p,q f is continuous on [a, b], where 0 < q < p ≤ 1 and a D p,q f r is a quasi-convex function on [a, b] r ≥ 1, Proof. Taking absolute value on both sides of (3.3), applying the power mean inequality and using the quasi-convexity of a D p,q f r on [a, b] for r ≥ 1, we have that (1). If we let p = 1, then: (2). If we take p = 1 and letting q → 1 − , then: where γ 3 (p, q; s) is as defined in Theorem 3.4 and 1 r + 1 s = 1. Proof. Taking absolute value on both sides of (3.3), applying the Hölder inequality and using the quasi-convexity of a D p,q f r on [a, b] for r > 1, we have that   (1). If p = 1, then we obtain the inequality proved in [9, Theorem 2]: The authors would like to thank the referee for his/her careful reading of the manuscript and for making valuable suggestions.