Post-quantum Hermite–Hadamard type inequalities for interval-valued convex functions

*Correspondence: chuyuming@zjhu.edu.cn 4Department of Mathematics, Huzhou University, 313000, Huzhou, China Full list of author information is available at the end of the article Abstract In this research, we introduce the notions of (p,q)-derivative and integral for interval-valued functions and discuss their fundamental properties. After that, we prove some new inequalities of Hermite–Hadamard type for interval-valued convex functions employing the newly defined integral and derivative. Moreover, we find the estimates for the newly proved inequalities of Hermite–Hadamard type. It is also shown that the results proved in this study are the generalization of some already proved research in the field of Hermite–Hadamard inequalities.


Introduction
Many studies have recently been carried out in the field of q-analysis, starting with Euler due to a high demand for mathematics that models quantum computing q-calculus appeared as a connection between mathematics and physics. It has a lot of applications in different mathematical areas such as number theory, combinatorics, orthogonal polynomials, basic hypergeometric functions, and other sciences, quantum theory, mechanics, and the theory of relativity [1][2][3][4][5]. Apparently, Euler was the founder of this branch of mathematics by using the parameter q in Newton's work of infinite series. Later, Jackson was the first to develop q-calculus known as without limits calculus in a systematic way [2]. In 1908-1909, Jackson defined the general q-integral and q-difference operator [4]. In 1969, Agarwal described the q-fractional derivative for the first time [6]. In -1967 Al-Salam introduced q-analogues of the Riemann-Liouville fractional integral operator and q-fractional integral operator [7]. In 2004, Rajkovic gave a definition of the Riemann-type q-integral which was a generalization of Jackson q-integral. In 2013, Tariboon introduced mathematicians have done studies in q-calculus analysis, the interested reader can check [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25].
Inspired by these ongoing studies, we give the idea about the post-quantum derivative and integral in the setting of interval-valued calculus. We also prove some new inequalities of Hermite-Hadamard type and find their estimates.

Interval calculus
We give notation and preliminary information about the interval analysis in this section. Let the space of all closed intervals of R denoted by I c and K be a bounded element of I c , we have the representation where k, k ∈ R and k ≤ k. The length of the interval K = [k, k] can be stated as L(K) = kk. The numbers k and k are called the left and the right endpoints of interval K , respectively. When k = k, the interval K is said to be degenerate, and we use the form K = k = [k, k]. Also, we can say that K is positive if k > 0, or we can say that K is negative if k < 0. The sets of all closed positive intervals of R and closed negative intervals of R are denoted by I + c and Ic , respectively. The Pompeiu-Hausdorff distance between the intervals K and M is defined by is known to be a complete metric space (see [26]). The absolute value of K , denoted by |K|, is the maximum of the absolute values of its endpoints: |K| = max |k|, |k| . Now, we mention the definitions of fundamental interval arithmetic operations for the intervals K and M as follows: Scalar multiplication of the interval K is defined by The opposite of the interval K is where μ = -1. The subtraction is given by In general, -K is not additive inverse for K , i.e., K -K = 0. The definitions of operations cause a great many algebraic features which allows I c to be a quasilinear space (see [27]). These properties can be listed as follows (see [26][27][28][29][30]): (1) (Associativity of addition) (K + M) (Associativity law) λ(μK) = (λμ)K for all K ∈ I c and all λ, μ ∈ R, (9) (First distributivity law) λ(K + M) = λK + λM for all K, M ∈ I c and all λ ∈ R, (10) (Second distributivity law) (λ + μ)K = λK + μK for all K ∈ I c and all λ, μ ∈ R.
Definition 1 ([31]) For the intervals K and M, we state that the gH-difference of K and M is the interval T such that It looks beyond dispute that Particularly, if M = m ∈ R is a constant, we have Moreover, another set feature is the inclusion ⊆ that is defined by Throughout this paper, 0 < q < 1 and a function F ]. For condensation, interval-valued quantum calculus and interval-valued post-quantum calculus are denoted by Iq-calculus and I(p, q)-calculus, respectively.
In [4], Jackson gave the q-Jackson integral from 0 to b for 0 < q < 1 as follows: provided the sum converges absolutely.
defined by the expression (see [5]) On the other hand, recently, Lou et al. introduced the notions of I(q)-calculus. They gave the following definitions of I(q)-derivative and integral, and proved some inequalities of I(q)-Hermite-Hadamard type for interval-valued convex functions.

Definition 7 ([31]) For an interval-valued function
Remark 1 If we set a = 0 in (3.4), then we have Iq-Jackson integral defined by the following equation: In [34], Alp et al. gave the definition of Iq b -integral and proved inequalities of Hermite-Hadamard type for interval-valued convex functions by using Iq b -integral.

Definition 8 For an interval-valued function
.
Then the Iq b -Hermite-Hadamard inequality is expressed as follows:

I(p, q)-calculus
In this section, the notions and results about the (p, q)-calculus are reviewed, and we are interested in introducing the concepts of I(p, q)-calculus. The [n] p,q is said to be (p, q)-integers and expressed as [n] p,q = p nq n pq with 0 < q < p ≤ 1. The [n] p,q ! and n k ! are called (p, q)-factorial and (p, q)-binomial, respectively, and expressed as On the other hand, Tunç and Göv gave the following new definitions of (p, q)-derivative and integrals.
For x = a, we state a D p,q f (a) = lim x→a a D p,q f (x) if it exists and is finite.
On the other hand, Ali et al. gave the following new definition of (p, q)-derivative and integral, and proved some related inequalities.

Definition 13 ([37]) For a continuous function
if it exists and is finite.
Now, we are able to introduce the concepts of I(p, q) b -derivative and integrals.

I(p, q) b -derivative
if it exists and is finite.
Remark 2 If we choose p = 1 in (4.10), then we have Iq b -derivative defined as follows: . From Definition 14, we have that and To prove conversely, we suppose that F and F are (p, q) b -differentiable at x ∈ [a, b]. Then we have two possibilities If b D p,q F(x) ≤ b D p,q F(x), then we have following relation: and by applying the similar concepts, we can prove

Theorem 8 Let F = [F, F] → I c be an I(p, q) b -differentiable function on [a, b]. Then the following equalities hold for all x ∈ [a, b]:
Proof To prove the first equality, we suppose that F is I(p, q) b -differentiable and Ldecreasing on [a, b]. So, we have Since L(f ) is increasing, then we have With the similar steps, the second equality can be done. Proof The proof can be easily done using Definition 14, hence we leave the proof for the readers. Proof The proof can be easily done using Definition 14, hence we leave the proof for the readers.

Definition 15 For a continuous interval-valued function
with 0 < q < p ≤ 1.
Remark 3 If we set p = 1 in (4.12), then we have the definition of Iq b -integral that we reviewed in the last section.
The following theorem provides us a relation between I(p, q) b -integral and (p, q) bintegral.

Theorem 11 Let F = [F, F] : [a, b] → I c be a continuous function on [a, b], the function F is I(p, q) b -integrable on [a, b] if and only if F and F are
Proof The proof of Theorem 13 can be easily done by using Theorems 15 and 7.

Hermite-Hadamard inequalities for I(p, q) b -integral
In this section, we review the concept of interval-valued convex functions and prove inequalities of Hermite-Hadamard type for an interval-valued convex function by using the newly defined I(p, q)-integral. for all x, y ∈ [a, b] and t ∈ (0, 1), we have F is a convex function [a, b] and F is a concave function on [a, b].

Theorem 14 A function F = [F, F] : [a, b] → I + c is said to be interval-valued convex if and only if
c be a differentiable interval-valued convex function, then the following inequalities hold for the I(p, q) b -integral: is an interval-valued convex function, therefore F is a convex function and F is a concave function. So, from F and inequality (4.6), we have and from the concavity of F and (4.6), we have From (5.2) and (5.3), we obtain and hence, we have Also, from (5.2) and (5.3), we obtain and hence, we have By combining (5.4) and (5.5), we obtain the required inequality, which accomplishes the proof.
is an interval-valued convex function, therefore F is a convex function and F is a concave function. Because of the convexity of F, from inequalities (4.7), we obtain that Now, using the fact that F is a concave function, and from inequality (4.7), we obtain that The rest of the proof can be done by applying the same lines of the previous theorem and considering inequalities (5.7) and (5.8). Thus, the proof is completed.
, then the following inequalities hold for the I(p, q) b -integral:

Midpoint and trapezoidal type inequalities for I(p, q) b -integral
In this section, some new inequalities of midpoint and trapezoidal type for interval-valued functions are obtained.
where A 1 (p, q)-A 4 (p, q) are defined in Theorem 5 and d H is a Pompeiu-Hausdorff distance between the intervals.
Proof Using the definition of d H distance between intervals, one can easily obtain that Now, using the fact that | b D p,q F| is a convex function, and from inequality (4.8), we have Similarly, considering that | b D p,q F| is convex on [a, b] and using inequality (4.8), we have So, from inequalities (6.2) and (6.3), we have Therefore, the proof is completed.

Corollary 1
If we set p = 1 in Theorem 18, then we have the following new q-midpoint inequality for interval-valued functions: where | b D q F| and | b D q F| both are convex functions.  Proof From the definition of d H distance between the intervals and inequality (4.9), and using the strategies followed in the last theorem, one can easily obtain inequality (6.4).

Corollary 3
If we set p = 1 in Theorem 19, then we have the following new q-trapezoidal inequality for interval-valued functions: where | b D q F| and | b D q F| both are convex functions.

Conclusion
In this study, we have introduced the notions of (p, q)-derivative and integral for intervalvalued functions and discussed their basic properties. We have proved some new Hermite-Hadamard type inequalities for interval-valued convex functions by using newly given concepts of (p, q)-derivative and integral. Moreover, we have proved midpoint and trapezoidal estimates for newly established (p, q)-Hermite-Hadamard inequalities. It is an interesting and new problem that the upcoming researchers can establish Simpson type inequalities, Newton type inequalities, and Ostrowski type inequalities for interval-valued functions by employing the techniques of this research in their future work.