New Post Quantum Analogues of Hermite–Hadamard Type Inequalities for Interval-Valued Convex Functions

Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China Zhejiang Province Key Laboratory of Quantum Technology and Device, Department of Physic, Hangzhou 310027, China Department of Mathematics, Anand International College of Engineering, Jaipur, India Department of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, Pakistan


Introduction
In mathematics, the quantum calculus is equivalent to usual infinitesimal calculus without the concept of limits or the investigation of calculus without limits (quantum is from the Latin word "quantus" and literally it means how much and in Swedish it is "Kvant"). Euler and Jacobi can be credited with establishing the basis of the modern understanding of quantum calculus, but these developments were recently applied in the field, bringing about tremendous development. is could be due to the fact that it acts as a connection between mathematics and physics. In 2002, the book [1] by Kac and Cheung presented some in-depth details of q-calculus. Later on, a few scholars have continued to establish the idea of q-calculus in a different direction of mathematics and physics. Jackson [2] created the concept of quantum-definite integrals in quantum calculus in the twentieth century.
is inspired many quantum calculus analysts, and several papers have been published in this field as a consequence. Ernst [3] developed the history of q-calculus and a new method for finding quantum calculus. Gauchman [4] derived integral inequalities in q-Calculus, which is a generalization of classical integral inequalities. In 2013, Tariboon et al. presented q-calculus principles over finite intervals, explored their characteristics, and applied impulsive difference equations in [5]. In 2015, Sudsutad et al. [6] proved quantum integral inequalities for convex functions. Shortly afterward, certain q-Hermite-Hadamard form inequalities are acquired by Alp in [7]. Recently, Lou et al. [8] presented basic properties of Iq-calculus and derived Iq-Hermite-Hadamard inequalities for convex intervalvalued functions. For more details, see [9][10][11][12][13].
Postquantum calculus theory, prefixed by the (p, q)-calculus, is a native q-calculus generalization. We deal with q-number with one base q in a recent development in the study of quantum calculus, but postquantum calculus includes p and q numbers with two independent p and q variables. Chakarabarti and Jagannathan [14] was the first to consider this. Inspired by the current research on Tunc and Gov [15], the definitions of (p, q)-derivatives and (p, q)-integrals have been adopted on finite intervals; interested readers are referred to [16][17][18]. A good deal of the book by Moore [19] is a narrative of the methods used by Moore to find an unknown variable and substitute it with an interval of real numbers and an arithmetic interval used in error analysis, which has a significant effect on the outcome of the calculation and automatic error analysis. It has been used extensively in several countries in recent days to address a variety of uncertain topics. In particular, Costa et al. [20] developed convex function understandings in the field of inequality and provided Jensen inequality in 2017 for the interval-valued functions. erefore, some scientists have combined classical inequalities with interval values to achieve several extensive inequalities, see [21,22]. e paper is summarized as follows. We review some basic properties of interval analysis in Section 2. In Section 3, we put forward the concepts of I(p, q) ϱ -derivative and give some properties. Similarly, the concepts of I(p, q) ϱ -integral and some properties are presented in Section 4. In Sections 5 and 6, we give some new I(p, q) ϱ -Hermite-Hadamard-type inequalities and some results related to upper and lower bounds of I(p, q) ϱ -Hermite-Hadamard. Briefly, conclusion has been discussed in Section 7.

Preliminaries
roughout this paper, we suppose that closed interval K c � U � [σ, σ]|, σ, σ ∈ R, σ ≤ σ . You can describe the length of interval [σ, σ] ∈ K c as L(U): � σ − σ. In addition, we conclude that U seems to be positive if σ > 0, and we present that all positive intervals belong to K c .
For some kind of U � [σ, σ], V � [α, α] ∈ K c and β ∈ R; then, we have the following properties: Definition 1 (see [23]). For some kind of U, V ∈ K c , we denote the H-difference of U and V as the set W ∈ K c , and we have It seems beyond controversy that Suppose that if we take a consent V � α ∈ R, then e relation between U and V can be described by the relation of "⊆": e later result is that (K c , H) is a complete metric space, as proven in [24].
In this paper, the symbols F and G are used to refer to functions with interval values. If a function F: If, for all ϖ ∈ [ϱ, pτ + (1 − p)ϱ] and ϱ D p,q F(ϖ) exists, then we called F as a q-differentiable on [ϱ, τ]. If ϱ � 0 in (9), then 0 D p,q F � D p,q F; then, For more details, see [15]. Now, we are adding the I(p, q) ϱ -derivative for the interval-valued functions and some related properties.
where D p,q F is said I(p, q) ϱ -derivative of F denoted as Proof. Suppose F is I(p, q) ϱ -differentiable at ϖ; then, there exist G(ϖ) and G(ϖ) such that ϱ D p,q F(ϖ) � [G(ϖ), G(ϖ)]. According to Definition 4, Exist; then, equation (11) is proved by using the above derivatives.
Conversely, suppose U and U are (p, q)-differentiable at ϖ.
If ϱ D p,q U(ϖ) ≤ 9 D p,q U(ϖ), then Show the above result in the next example.
Mathematical Problems in Engineering 3 and taking ϱ � 0, then In the meantime, we realize that U(ϖ) � − |ϖ| and U(ϖ) � |ϖ| are (p, q)-differentiable at 0. In the same way, taking ϱ < 0, we have and taking ϱ � 0, then We include the following findings to more clearly explain the existence of the derivatives. erefore, e other condition can be proved, similarly.
(i) Suppose F and G are L-increasing on [ϱ, τ]. Since F and G are I(p, q) ϱ -differentiable, we have that U, U, G, and G are (p, q)-differentiable and ϱ D p,q U ≤ 9 D p,q U, 9 D p,q G ≤ 9 D p,q G. (36) (iii) e case of F and G are both L-decreasing can be proved, similarly.

I(p, q) 9 -Integral for Interval-Valued Functions
In this section, we present the concepts of I(p, q) ϱ -integral and give some properties. Firstly, let us review the definition of (p, q) ϱ -integral.
Next, we give the concept of the I(p, q) ϱ -integral and discuss some basic properties.
Proof. e proof can be obtained by combining Definitions 5 and 6 and, hence, is omitted. (50) Proof. From Definition 6, we have that Proof. First, we have It implies that erefore, (57) is not true for all ϖ ∈ [0, 2].

I(p, q) 9 -Hermite-Hadamard Inequalities for Interval-Valued Functions
Now, we review the content of the convex interval-valued functions.