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

Authors: Humaira Kalsoom; Muhammad Aamir Ali; Muhammad Idrees; Praveen Agarwal; Muhammad Arif
Journal: Mathematical Problems in Engineering
 (2021)

---

## Abstract

The main objective of this paper is to introduceIp,qϱ-derivative and Ip,qϱ-integral for interval-valued functions and discuss their key properties. Also, we prove the Ip,qϱ-Hermite–Hadamard inequalities for interval-valued functions is the development of p,qϱ-Hermite–Hadamard inequalities by using new defined Ip,qϱ-integral. Moreover, we prove some results for midpoint- and trapezoidal-type inequalities by using the concept of Pompeiu–Hausdorff distance between the intervals. It is also shown that the results presented in this paper are extensions of some of the results already shown in earlier works. The proposed studies produce variants that would be useful for performing in-depth investigations on fractal theory, optimization, and research problems in different applied fields, such as computer science, quantum mechanics, and quantum physics.


---

## Body

## 1. 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. This 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. This 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 interval-valued functions. For more details, see [9–13].Postquantum calculus theory, prefixed by thep,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 [16–18]. 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. Therefore, some scientists have combined classical inequalities with interval values to achieve several extensive inequalities, see [21, 22].The paper is summarized as follows. We review some basic properties of interval analysis in Section2. In Section 3, we put forward the concepts of Ip,qϱ-derivative and give some properties. Similarly, the concepts of Ip,qϱ-integral and some properties are presented in Section 4. In Sections 5 and 6, we give some new Ip,qϱ-Hermite–Hadamard-type inequalities and some results related to upper and lower bounds of Ip,qϱ-Hermite–Hadamard. Briefly, conclusion has been discussed in Section 7.


## 2. Preliminaries

Throughout this paper, we suppose that closed intervalKc=U=σ¯,σ¯|,σ¯,σ¯∈ℝ,σ¯≤σ¯. You can describe the length of interval σ¯,σ¯∈Kc as LU^:=σ¯−σ¯. In addition, we conclude that U seems to be positive if σ¯>0, and we present that all positive intervals belong to Kc.For some kind ofU^=σ¯,σ¯,V^=α¯,α¯∈Kc and β∈ℝ; then, we have the following properties:(1)U^+V^=σ¯,σ¯+α¯,α¯=σ¯+α¯,σ¯+α¯,βU^=βσ¯,σ¯=βσ¯,βσ¯,ifβ>0,0,ifβ=0,βσ¯,βσ¯ifβ<0.Definition 1.
(see [23]). For some kind of U^,V^∈Kc, we denote the H^-difference of U^ and V^ as the set W^∈Kc, and we have(2)U^⊖GV^=W^⇔iU^=V^+W^,oriiV^=U^+−W^.
It seems beyond controversy that(3)U^⊖GV^=σ¯−α¯,σ¯−α¯,ifLU^≥ℒV^,σ¯−α¯,σ¯−α¯,ifLU^<ℒV^.
Suppose that if we take a consentV^=α∈ℝ, then(4)U^⊖gV^=σ¯−α,σ¯−α.
The relation betweenU^ and V^ can be described by the relation of “⊆”:(5)U^⊆V^,ifα¯≤σ¯andσ¯≤α¯.
The distance of Hausdorff–PompeiuH^:Kc×Kc⟶0,∞ between U^ and V^ is denoted as H^U^,V^=maxσ¯−α¯,σ¯−α¯. The later result is that Kc,H^ is a complete metric space, as proven in [24].Definition 2.
Suppose that a continuous functionF^:ϱ,τ⟶Kc at ϖ0∈ϱ,τ if(6)H^F^ϖ,F^ϖ0⟶0,asϱ⟶ϱ0.
We denoteCϱ,τ,Kc and Cϱ,τ,ℝ to show the collection of all continuous interval- and real-valued functions on ϱ,τ, respectively.
For much more simple notations with interval analysis, see [23, 25, 26].
In this paper, the symbolsF^ and G^ are used to refer to functions with interval values. If a function F^:ϱ,τ⟶Kc and F^=U¯,U¯, then F^ is L-increasing (or L-decreasing) on ϱ,τ if LF^:ϱ,τ⟶0,∞ is increasing (or decreasing) on ϱ,τ. If LF^ is monotone on ϱ,τ, then F^ is L-monotone on ϱ,τ.


## 3.Ip,qϱ-Derivative for Interval-Valued Functions

In this portion, we introduce theIp,qϱ-derivative concepts and give some properties. Firstly, let us study the p,q-derivative concept. Let any constant be 0<q<p≤1.Definition 3.
(see [15]). Let F^:ϱ,τ⟶ℝ and F^∈Cϱ,τ,ℝ; the p,q-derivative of function F^ at ϖ∈ϱ,pτ+1−pϱ is defined by(7)Dϱp,qF^ϖ=F^pϖ+1−pϱ−F^qϖ+1−qϱp−qϖ−ϱ,ϖ≠ϱ,Dϱp,qF^ϱ=limϖ⟶ϱ+Dp,qF^ϖ.
If, for allϖ∈ϱ,pτ+1−pϱ and Dϱp,qF^ϖ exists, then we called F^ as a q-differentiable on ϱ,τ. If ϱ=0 in (9), then D0p,qF^=Dp,qF^; then,(8)Dp,qF^ϖ=F^pϖ−F^qϖp−qϖ.
For more details, see [15].
Now, we are adding theIp,qϱ-derivative for the interval-valued functions and some related properties.Definition 4.
Suppose thatF^:ϱ,τ⟶Kc and F^∈Cϱ,τ,Kc, and the Ip,qϱ-derivative of F^ at ϖ∈ϱ,pτ+1−pϱ is denoted as(9)Dϱp,qF^ϖ=F^pϖ+1−pϱ⊖GF^qϖ+1−qϱp−qϖ−ϱ,ϖ≠ϱ,Dϱp,qF^ϱ=limϖ⟶ϱ+Dp,qF^ϖ,where Dp,qF^ is said Ip,qϱ-derivative of F^ denoted as(10)Dp,qF^ϖ=F^pϖ⊖GF^qϖp−qϖ.Theorem 1.
Suppose thatF^:ϱ,τ⟶Kc. F^ is Ip,qϱ-differentiable at ϖ∈ϱ,pτ+1−pϱ if and only if U¯ and U¯ are p,q-differentiable at ϖ∈ϱ,pτ+1−pϱ, and(11)Dϱp,qF^ϖ=minDϱp,qU¯ϖ,Dϱp,qU¯ϖ,maxDϱp,qU¯ϖ,Dϱp,qU¯ϖ.Proof.
SupposeF^ is Ip,qϱ-differentiable at ϖ; then, there exist G¯ϖandG¯ϖ such that Dϱp,qF^ϖ=G¯ϖ,G¯ϖ. According to Definition 4,(12)G¯ϖ=minU¯pϖ+1−pϱ−U¯qϖ+1−qϱp−qϖ−ϱ,U¯pϖ+1−pϱ−U¯qϖ+1−qϱp−qϖ−ϱ,G¯ϖ=maxU¯pϖ+1−pϱ−U¯qϖ+1−qϱp−qϖ−ϱ,U¯pϖ+1−pϱ−U¯qϖ+1−qϱp−qϖ−ϱ,Exist; then, equation (11) is proved by using the above derivatives.
Conversely, supposeU¯ and U¯ are p,q-differentiable at ϖ.
IfDϱp,qU¯ϖ≤Dϱp,qU¯ϖ, then(13)Dϱp,qU¯ϖ,Dϱp,qU¯ϖ=U¯pϖ+1−pϱ−U¯qϖ+1−qϱp−qϖ−ϱ,U¯pϖ+1−pϱ−U¯qϖ+1−qϱp−qϖ−ϱ=F^pϖ+1−pϱ⊖GF^qϖ+1−qϱp−qϖ−ϱ=Dϱp,qF^ϖ.
So,F^ is Ip,qϱ-differentiable at ϖ. Similarly, if Dϱp,qU¯ϖ≥Dϱp,qU¯ϖ, then Dϱp,qF^ϖ=Dϱp,qU¯ϖ,Dϱp,qU¯ϖ.
Show the above result in the next example.Example 1.
LetF^:ϱ,τ⟶Kc, taking F^ϖ=−ϖ,ϖ. It shows that F^ϖ is Ip,qϱ-differentiable. By Definition 4, if ϱ<0, we have(14)Dϱp,qF^0=1−pϱ,−1−pϱ⊖G1−qϱ,−1−qϱ1−q−ϱ=min1−pϱ−1−qϱp−q−ϱ,−1−qϱ+1−qϱp−q−ϱ,max1−pϱ−1−qϱp−q−ϱ,−1−pϱ+1−qϱp−q−ϱ=−1,1,and taking ϱ=0, then(15)D0p,qF^0=limϖ⟶0+−pϖ,pϖ⊖G−qϖ,qϖp−qϖ=minlimϖ⟶0+−pϖ+qϖp−qϖ,limϖ⟶0+pϖ−qϖp−qϖ,maxlimϖ⟶0+−pϖ+qϖp−qϖ,limϖ⟶0+pϖ−qϖp−qϖ=−1,1.
In the meantime, we realize thatU¯ϖ=−ϖ and U¯ϖ=ϖ are p,q-differentiable at 0. In the same way, taking ϱ<0, we have(16)Dϱp,qU¯0=1−pϱ−1−qϱp−q−ϱ=1,Dϱp,qU¯0=−1−pϱ+1−qϱp−q−ϱ=−1,and taking ϱ=0, then(17)D0p,qU¯0=limϖ⟶0+−pϖ+qϖp−qϖ=−1,D0p,qU¯0=limϖ⟶0+pϖ−qϖp−qϖ=1.
It show that ifϱ<0, then Dϱp,qF^0=Dϱp,qU¯0,Dϱp,qU¯0. And, if ϱ=0, then D0p,qF^0=D0p,qU¯0,D0p,qU¯0.
We include the following findings to more clearly explain the existence of the derivatives.Theorem 2.
LetF^:ϱ,τ⟶Kc. If F^ is Ip,qϱ-differentiable on ϱ,τ, then we have that(i)
Dϱp,qF^ϖ=Dϱp,qU¯ϖ,Dϱp,qU¯ϖ, for all ϖ∈ϱ,pτ+1−pϱ, if F^ is L-increasing(ii)
Dϱp,qF^ϖ=Dϱp,qU¯ϖ,Dϱp,qU¯ϖ, for all ϖ∈ϱ,pτ+1−pϱ, if F^ is L-decreasingProof.
First, supposeF^ is L-increasing and Ip,qϱ-differentiable is on ϱ,τ. For any ϖ∈ϱ,pτ+1−pϱ, we have pϖ+1−pϱ>qϖ+1−qϱ. Since LF^=U¯−U¯ is increasing, then we have(18)U¯pϖ+1−pϱ−U¯pϖ+1−pϱ−U¯qϖ+1−qϱ−U¯qϖ+1−qϱ>0,U¯pϖ+1−pϱ−U¯qϖ+1−qϱ>U¯pϖ+1−pϱ−U¯qϖ+1−qϱ.
Therefore,(19)Dϱp,qF^ϖ=U¯pϖ+1−pϱ,U¯pϖ+1−pϱ⊖GU¯qϖ+1−qϱ,U¯qϖ+1−qϱp−qϖ−ϱ=U¯pϖ+1−pϱ−U¯qϖ+1−qϱp−qϖ−ϱ,U¯pϖ+1−pϱ−U¯qϖ+1−qϱp−qϖ−ϱ=Dϱp,qU¯ϖ,Dϱp,qU¯ϖ.
The other condition can be proved, similarly.Remark 1.
Letγ∈ϱ,pτ+1−pϱ be a given point. If F^ is L-increasing on ϱ,γ and L-decreasing on γ,τ, then Dϱp,qF^=Dϱp,qU¯,Dϱp,qU¯ on ϱ,γ and Dϱp,qF^=Dϱp,qU¯,Dϱp,qU¯ on γ,pτ+1−pϱ.Example 2.
Suppose that a functionℱ:0,1⟶Kc; then, we take ℱϖ=−ϖ2−1,ϖ2−2ϖ. We know that LF^=2ϖ2−2ϖ+1, and it shows that F^ is L-decreasing on 0,1/2 and L-increasing on 1/2,1. We know that U¯ϖ=−ϖ2−1 and U¯ϖ=ϖ2−2ϖ are p,q-differentiable on 0,1; then, we have(20)D0p,qU¯ϖ=−pϖ2−1+qϖ2+1p−qϖ=−2p,qϖ,D0p,qU¯ϖ=pϖ2+2pϖ−qϖ2−2qϖp−qϖ=2p,qϖ−2,D1/2p,qU¯ϖ=−pϖ+1/21−p2−1−−qϖ+1/21−q2−1p−qϖ−1/2=−2p,qϖ−12p−q,D1/2p,qU¯ϖ=pϖ+1/21−p2+2pϖ+1/21−p−qϖ+1/21−q2+2qϖ+1/21−q1−qϖ−1/2=2p,qϖ−2+1/2p−q,limϖ⟶1/2Dp,qU¯ϖ=−1,limϖ⟶1/2Dp,qU¯ϖ=−1.
Therefore,(21)Dϱp,qF^ϖ=−2p,qϖ,2p,qϖ−2,ifϖ∈0,12,−1,ifϖ=12,2p,qϖ2+12p−q,−2p,qϖ−12p−q,ifϖ∈12,1.Theorem 3.
Suppose thatF^:ϱ,τ⟶Kc is a Ip,qϱ-differentiable on ϱ,τ with C^=c¯,c¯∈Kc and β∈ℝ; then, functions F^+C^ and βF^ are Ip,qϱ-differentiable on ϱ,τ, such that Dϱp,qF^+C^=Dϱp,qF^ and Dϱp,qβF^=βDϱp,qF^.Proof.
For anyϖ∈ϱ,pτ+1−pϱ,(22)iDϱp,qF^ϖ+C^=F^pϖ+1−pϱ+C^⊖GF^qϖ+1−qϱ+C^p−qϖ−ϱ=ℱpϖ+1−pϱ⊖GF^qϖ+1−qϱp−qϖ−ϱ=Dϱp,qF^ϖ,iiDϱp,qβF^ϖ=βF^pϖ+1−pϱ⊖GβF^qϖ+1−qϱp−qϖ−ϱ=βF^pϖ+1−pϱ⊖GF^qϖ+1−qϱp−qϖ−ϱ=βDϱp,qF^ϖ.Theorem 4.
Suppose thatF^:ϱ,τ⟶Kc is a Ip,qϱ-differentiable on ϱ,τ. Let C^=c¯,c¯∈Kc; if LF^−LC^ has a constant sign on ϱ,τ, then functions F^⊖GC^ is Ip,qϱ-differentiable on ϱ,τ and Dϱp,qF^⊖GC^=Dϱp,qF^.Proof.
For anyϖ∈ϱ,pτ+1−pϱ,(23)Dϱp,qF^ϖ⊖GC^=F^pϖ+1−pϱ⊖GC^⊖GF^qϖ+1−qϱ⊖GC^p−qϖ−ϱ=F^pϖ+1−pϱ⊖GF^qϖ+1−qϱp−qϖ−ϱ=Dϱp,qF^ϖ.Theorem 5.
Suppose thatF^,G^:ϱ,τ⟶Kc. Let F^andG^ be Ip,qϱ-differentiable on ϱ,τ; then, F^+G^:ϱ,τ⟶Kc is Ip,qϱ-differentiable on ϱ,τ; then, we have the following properties:(i)
IfF^andG^ are equally L-monotonic on ϱ,τ, for all ϖ∈ϱ,pτ+1−pϱ, then(24)Dϱp,qF^ϖ+G^ϖ=Dϱp,qF^ϖ+Dϱp,qG^ϖ.(ii)
IfF^ and G^ are differently L-monotonic on ϱ,τ, for all ϖ∈ϱ,pτ+1−pϱ, then(25)Dϱp,qF^+G^ϖ=Dϱp,qF^ϖ⊖G−1Dϱp,qG^ϖ.Therefore, we have(26)Dϱp,qF^ϖ+G^ϖ⊆Dϱp,qF^ϖ+Dϱp,qG^ϖ.Proof.
(i)
Suppose that two functionsF^andG^ are Ip,qϱ-differentiable and L-increasing on ϱ,τ; taking U¯, U¯, G¯, and G¯ are p,q-differentiable,(27)Dϱp,qU¯≤Dϱp,qU¯,Dϱp,qG¯≤Dϱp,qG¯.(ii)
It shows thatU¯+G¯ and U¯+G¯ are p,q-differentiable functions on ϱ,τ, and also, F^+G^ is Ip,qϱ-differentiable on ϱ,τ and(28)Dϱp,qF^+G^=minDϱp,qU¯+Dϱp,qG¯,Dϱp,qU¯+Dϱp,qG¯,maxDϱp,qU¯+Dϱp,qG¯,Dϱp,qU¯+Dϱp,qG¯=Dϱp,qU¯+Dϱp,qG¯,Dϱp,qU¯+Dϱp,qG¯=Dϱp,qF^+Dϱp,qG^.(iii)
Correspondingly, bothF^ and G^ can be shown to be L-decreasing.(iv)
Suppose thatF^ is L-increasing and G^ is L-decreasing. Then,(29)Dϱp,qU¯≤Dϱp,qU¯,Dϱp,qG¯≥Dϱp,qG¯.(v)
Moreover,(30)Dϱp,qF^+G^=minDϱp,qU¯+Dϱp,qG¯,Dϱp,qU¯+Dϱp,qG¯,maxDϱp,qU¯+Dϱp,qG¯,Dϱp,qU¯+Dϱp,qG¯,(31)Dϱp,qF^⊖G−1Dϱp,qG^=Dϱp,qU¯,Dϱp,qU¯⊖G−1Dϱp,qG¯,Dϱp,qG¯=Dϱp,qU¯,Dϱp,qU¯⊖G−Dϱp,qG¯,−Dϱp,qG¯=minDϱp,qU¯+Dϱp,qG¯,Dϱp,qU¯+Dϱp,qG¯,maxDϱp,qU¯+Dϱp,qG¯,Dϱp,qU¯+Dϱp,qG¯.(vi)
We get (25) by comparing (30) with (31). Additionally,(32)Dϱp,qF^+Dϱp,qG^=Dϱp,qU¯+Dϱp,qG¯,Dϱp,qU¯+Dϱp,qG¯.(v)
We obtainF^+G^ if it is L-increasing or L-decreasing; we can obtain(33)Dϱp,qF^ϖ+G^ϖ⊆Dϱp,qF^ϖ+Dϱp,qG^ϖ.(vi)
The opposite case, similarly, can be proved.Theorem 6.
LetF^,G^:ϱ,τ⟶Kc. If F^andG^ are Ip,qϱ-differentiable and LF^−LG^ has a constant sign on ϱ,τ, then F^⊖GG^:ϱ,τ⟶Kc is Ip,qϱ-differentiable on ϱ,τ, and one of the following cases holds:(i)
IfF^andG^ are equally L-monotonic on ϱ,τ, for all ϖ∈ϱ,pτ+1−pϱ, then(34)Dϱp,qF^ϖ⊖GG^ϖ=Dϱp,qF^ϖ⊖GDϱp,qG^ϖ.(ii)
IfF^ and G^ are differently L-monotonic on ϱ,τ, for all ϖ∈ϱ,pτ+1−pϱ, then(35)Dϱp,qF^⊖GG^ϖ=Dϱp,qF^ϖ+−1Dϱp,qG^ϖ.Proof.
We now assume thatLF^≥LG^ on ϱ,τ, and F^⊖GG=U¯−G¯,U¯−G¯.(i)
SupposeF^andG^ are L-increasing on ϱ,τ. Since F^andG^ are Ip,qϱ-differentiable, we have that U¯, U¯, G¯, and G¯ are p,q-differentiable and(36)Dϱp,qU¯≤Dϱp,qU¯,Dϱp,qG¯≤Dϱp,qG¯.(ii)
Then,U¯−G¯ and U¯−G¯ are p,q-differentiable functions on ϱ,τ. So, F^⊖GG^ is Ip,qϱ-differentiable on ϱ,τ and(37)Dϱp,qF^⊖GG^=minDϱp,qU¯−Dϱp,qG¯,Dϱp,qU¯−Dϱp,qG¯,maxDϱp,qU¯−Dϱp,qG¯,Dϱp,qU¯−Dϱp,qG¯=Dϱp,qU¯,Dϱp,qU¯⊖GDϱp,qG¯,Dϱp,qG¯=Dϱp,qF^⊖GDϱp,qG^.(iii)
The case ofF^ and G^ are both L-decreasing can be proved, similarly.(iv)
SupposeF^ is L-increasing and G^ is L-decreasing. From (i), we have that(38)Dϱp,qU¯≤Dϱp,qU¯,Dϱp,qG¯≥Dϱp,qG¯.(v)
ForLF^≥LG^, on the one hand,(39)Dϱp,qF^⊖GG^=minDϱp,qU¯−Dϱp,qG¯,Dϱp,qU¯−Dϱp,qG¯,maxDϱp,qU¯−Dϱp,qG¯,Dϱp,qU¯−Dϱp,qG¯=Dϱp,qU¯−Dϱp,qG¯,Dϱp,qU¯−Dϱp,qG¯.(vi)
On the other hand,(40)Dϱp,qF^+−1Dϱp,qG^=Dϱp,qU¯,Dϱp,qU¯+−1Dϱp,qG¯Dϱp,qG¯=Dϱp,qU¯,Dϱp,qU¯+−Dϱp,qG¯,−Dϱp,qG¯=Dϱp,qU¯−Dϱp,qG¯,Dϱp,qU¯−Dϱp,qG¯.(vii)
Comparing (39) with (40), we get (35). The opposite case, similarly, can be proved.Example 3.
LetF^,G^:0,2⟶Kc, given by F^ϖ=0,−ϖ2+2ϖ and G^ϖ=0,2ϖ2−4ϖ+3. Since LF^ϖ=−ϖ2+2ϖ and LG^ϖ=2ϖ2−4ϖ+3, then LF^ϖ≤LG^ϖ, for all ϖ∈0,2. We have that F^ϖ is L-increasing on 0,1 and L-decreasing on 1,2. G^ϖ is L-decreasing on 0,1 and L-increasing on 1,2.
Furthermore, we have thatF^ϖ+G^ϖ=0,ϖ2−2ϖ+3 and F^ϖ⊖GG^ϖ=−3ϖ2+6ϖ−3,0. Since LF^ϖ+G^ϖ=ϖ2−2ϖ+3 and LF^ϖ⊖GG^ϖ=3ϖ2−6ϖ+3, then F^ϖ+G^ϖ, F^ϖ⊖GG^ϖ are L-decreasing on 0,1 and L-increasing on 1,2. For all ϖ∈0,1, we get that(41)D0p,qF^ϖ=D0p,qU¯ϖ,D0p,qU¯ϖ=0,−2p,qϖ+2,D0p,qG^ϖ=D0p,qG¯ϖ,D0p,qG¯ϖ=22p,qϖ−4,0,D0p,qF^ϖ+G^ϖ=D0p,qU¯ϖ+G¯ϖ,D0p,qU¯ϖ+G¯ϖ=2p,qϖ−2,0,D0p,qF^ϖ⊖GG^ϖ=D0p,qU¯ϖ−G¯ϖ,D0p,qU¯ϖ−G¯ϖ=0,−32p,qϖ+6.
Then, from (31) and (40),(42)D0p,qF^ϖ⊖G−1D0p,qG^ϖ=0,−2p,qϖ+2⊖G−122p,qϖ−4,0=0,−1+qϖ+2⊖G0,−22p,qϖ+4=min0,2p,qϖ−2,max0,2p,qϖ−2=2p,qϖ−2,0,D0p,qF^ϖ+−1D0p,qG^ϖ=0,−2p,qϖ+2+−122p,qϖ−4,0=0,−2p,qϖ+2+0,−22p,qϖ+4=0,−32p,qϖ+6.
Furthermore, for allϖ∈1,2, similarly, we obtain that(43)D1p,qF^ϖ=D1p,qU¯ϖ,D1p,qU¯ϖ=−2p,qϖ−p−q+2,0,D1p,qG^ϖ=D1p,qG¯ϖ,D1p,qG¯ϖ=0,22p,qϖ+2p−q−4,D1p,qℱϖ+Gϖ=D1p,qU¯ϖ+G¯ϖ,D1p,qU¯ϖ+G¯ϖ=0,2p,qϖ+p−q−2,D1p,qF^ϖ⊖GG^ϖ=D1p,qU¯ϖ−G¯ϖ,D1p,qU¯ϖ−G¯ϖ=−32p,qϖ−3p−q+6,0,D0p,qF^ϖ⊖G−1D0p,qG^ϖ=−2p,qϖ−p−q+2,0⊖G−10,22p,qϖ+2p−q−4=−2p,qϖ−p−q+2,0⊖G−22p,qϖ−2p−q+4,0=min2p,qϖ+p−q−2,0,max2p,qϖ+p−q−2,0=0,2p,qϖ+p−q−2,D0p,qF^ϖ+−1D0p,qG^ϖ=−2p,qϖ−p−q+2,0+−10,22p,qϖ+2p−q−4=−2p,qϖ−p−q+2,0+−22p,qϖ−2p−q+4,0=−32p,qϖ−3p−q+6,0.
Obviously, we can see thatDϱp,qF^+G^ϖ=Dϱp,qF^ϖ⊖G−1Dϱp,qG^ϖ and Dϱp,qF^⊖GG^ϖ=Dϱp,qF^ϖ+−1Dϱp,qG^ϖ.


## 4.Ip,qϱ-Integral for Interval-Valued Functions

In this section, we present the concepts ofIp,qϱ-integral and give some properties. Firstly, let us review the definition of p,qϱ-integral.Definition 5.
(see [15]). Let F^:ϱ,τ⟶ℝ and F^∈Cϱ,τ,ℝ; then, the expression p,q-integral is defined by(44)∫ϱγF^ϖdϱp,qϖ=p−qγ−ϱ∑n=0∞qnpn+1F^qnpn+1γ+1−qnpn+1ϱ,for all γ∈ϱ,pτ+1−pϱ.
Additionally, ifc∈ϱ,γ, then the definite p,q-integral on ϱ,τ is defined by(45)∫cγfϖdϱp,qϖ=∫ϱγF^ϖdϱp,qϖ−∫ϱcfϖdϱp,qϖ=p−qγ−ϱ∑n=0∞qnpn+1F^qnpn+1γ+1−qnpn+1ϱ−p−qc−ϱ∑n=0∞qnpn+1F^qnpn+1c+1−qnpn+1ϱ.
Note that ifϱ=0, then (44) reduces to the classical p,q-Jackson integral of a function F^ϖ, defined by ∫0γF^ϖd0p,qϖ=p−qγ∑n=0∞qn/pn+1F^qn/pn+1γ for ϖ∈0,∞. For more details, see [15].
Next, we give the concept of theIp,qϱ-integral and discuss some basic properties.Definition 6.
LetF^:ϱ,τ⟶Kc and F^∈Cϱ,τ,Kc; then, the expression Ip,qϱ-integral is defined by(46)∫ϱγF^ϖdϱp,qIx=p−qγ−ϱ∑n=0∞qnpn+1F^qnpn+1γ+1−qnpn+1ϱ,for all γ∈ϱ,pτ+1−pϱ.Theorem 7.
LetF^:ϱ,τ⟶Kc and F^∈Cϱ,τ,Kc. If c∈ϱ,γ, then we have that(47)∫ϱcF^ϖdϱp,qIϖ+∫cγF^ϖdϱp,qIϖ=∫ϱγF^ϖdϱp,qIϖ.Proof.
(48)∫ϱcF^ϖdϱp,qIϖ+∫cγF^ϖdϱp,qIϖ=p−qc−ϱ∑n=0∞qnpn+1F^qnpn+1c+1−qnpn+1ϱ+p−qγ−c∑n=0∞qnpn+1F^qnpn+1γ+1−qnpn+1c=p−qc−ϱ∑n=0∞qnpn+1U¯qnpn+1c+1−qnpn+1ϱ,p−qc−ϱ∑n=0∞qnpn+1U¯qnpn+1c+1−qnpn+1ϱ+p−qγ−c∑n=0∞qnpn+1U¯qnpn+1γ+1−qnpn+1c,p−qγ−c∑n=0∞qnpn+1U¯qnpn+1γ+1−qnpn+1c=p−qγ−ϱ∑n=0∞qnpn+1U¯qnpn+1γ+1−qnpn+1ϱ,p−qγ−ϱ∑n=0∞qnpn+1U¯qnpn+1γ+1−qnpn+1ϱ=p−qγ−ϱ∑n=0∞qnpn+1F^qnpn+1γ+1−qnpn+1ϱ=∫ϱγF^ϖdϱp,qIϖ.Theorem 8.
LetF^:ϱ,τ⟶Kc. If F^∈Cϱ,τ,Kc, then F^ is Ip,qϱ-integral if and only if U¯ and U¯ are p,q-integral over ϱ,τ. Moreover,(49)∫ϱγF^ϖdϱp,qIϖ=∫ϱγU¯ϖdϱp,qIϖ,∫ϱγU¯ϖdϱp,qϖ.Proof.
The proof can be obtained by combining Definitions5 and 6 and, hence, is omitted.Example 4.
LetF^:0,1⟶Kc, given by F^ϖ=ϖ2,ϖ. For 0<q<p≤1, we have(50)∫01F^ϖd0p,qIϖ=∫01ϖ2d0p,qϖ,∫01ϖd0p,qϖ=p−q∑n=0∞q3np3n+1,p−q∑n=0∞q2np2n+1=13p,q,12p,q.Theorem 9.
LetF^,G^:ϱ,τ⟶Kc and let β∈ℝ. If F^,G^∈Cϱ,τ,Kc, for ϖ∈ϱ,pτ+1−pϱ, then we have that(i)
∫ϱγF^ϖ+G^ϖdϱp,qϖ=∫ϱγF^ϖdϱp,qϖ+∫ϱγG^ϖdϱp,qϖ(ii)
∫ϱγβF^ϖdϱp,qϖ=β∫ϱγF^ϖdϱp,qϖProof.
From Definition6, we have that(51)i∫ϱγF^ϖ+G^ϖdϱp,qIϖ=p−qγ−ϱ∑n=0∞qnpn+1F^qnpn+1γ+1−qnpn+1ϱ+G^qnpn+1γ+1−qnpn+1ϱ=p−qγ−ϱ∑n=0∞qnpn+1F^qnpn+1γ+1−qnpn+1ϱ+p−qγ−ϱ∑n=0∞qnpn+1G^qnpn+1γ+1−qnpn+1ϱ=∫ϱγF^ϖdϱp,qϖ+∫ϱγG^ϖdϱp,qϖ,ii∫ϱγβℱϖdϱp,qIϖ=p−qγ−ϱ∑n=0∞qnpn+1βℱqnpn+1γ+1−qnpn+1ϱ=β∫ϱγℱϖdϱp,qϖ.Theorem 10.
LetF^,G^:ϱ,τ⟶Kc. If F^,G^∈Cϱ,τ,Kc, then(52)∫ϱγF^ϖdϱp,qIϖ⊖G∫ϱγG^ϖdϱp,qIϖ⊆∫ϱγF^ϖ⊖GG^ϖdϱp,qIϖ.
Moreover, ifLF^−LG^ has a constant sign on ϱ,τ, then(53)∫ϱγF^ϖdϱp,qIϖ⊖G∫ϱγG^ϖdϱp,qIϖ=∫ϱγF^ϖ⊖GG^ϖdϱp,qIϖ.Proof.
First, we have(54)∫ϱγminU¯−G¯,U¯−G¯dϱp,qϖ≤min∫ϱγU¯−G¯dϱp,qϖ,∫ϱγU¯−G¯dϱp,qϖ≤max∫ϱγU¯−G¯dϱp,qϖ,∫ϱγU¯−G¯dϱp,qϖ≤∫ϱγmaxU¯−G¯,U¯−G¯dϱp,qϖ.
It implies that(55)∫ϱγF^dϱp,qIϖ⊖G∫ϱγG^dϱp,qIϖ=min∫ϱγU¯−G¯dϱp,qϖ,∫ϱγU¯−G¯dϱp,qϖ,max∫ϱγU¯−G¯dϱp,qϖ,∫ϱγU¯−G¯dϱp,qϖ⊆∫ϱγminU¯−G¯,U¯−G¯dϱp,qϖ,∫ϱγmaxU¯−G¯,U¯−G¯dϱp,qϖ=∫ϱγF^⊖GG^dϱp,qIϖ.
Moreover,F^⊖GG^=U¯−G¯,U¯−G¯ if LF^≥LG, or F^⊖GG^=U¯−G¯,U¯−G¯ if LF^≤LG^. We now assume that LF^≥LG^ on ϱ,τ, and F^⊖GG^=U¯−G¯,U¯−G¯. So, we have ∫ϱγU¯−G¯dϱp,qϖ≤∫ϱγU¯−G¯dϱp,qϖ. It implies that(56)∫ϱγF^⊖GG^dϱp,qIϖ=∫ϱγminU¯−G¯,U¯−G¯dϱp,qϖ,∫ϱγmaxU¯−G¯,U¯−G¯dϱp,qϖ=∫ϱγU¯dϱp,qϖ,∫ϱγU¯dϱp,qϖ⊖G∫ϱγG¯dϱp,qϖ,∫ϱγG¯dϱp,qϖ=∫ϱγF^dϱp,qIϖ⊖G∫ϱγG^dϱp,qIϖ.Theorem 11.
LetF^:ϱ,τ⟶Kc. If F^ is Ip,qϱ-differentiable on ϱ,τ, then Dϱp,qF^ is Ip,qϱ-integral. Moreover, if F^ is L-monotone on ϱ,τ, then(57)F^ϖ⊖GF^c=∫cϖDϱp,qF^γdϱp,qIγ,for allc∈ϱ,pϖ+1−pϱ.Proof.
IfF^ is Ip,qϱ-differentiable on ϱ,τ, then, from Theorem 1, it follows that U¯ and U¯ are p,qϱ-differentiable. Hence, Dϱp,qU¯ and Dϱp,qU¯ exist on ϱ,τ. Meanwhile, Dϱp,qU¯ and Dϱp,qU¯ are p,qϱ-integral. Therefore, Theorem 8 implies that Dϱp,qF^ is Ip,qϱ-integral. If ℱ is L-increasing on ϱ,τ, then Dϱp,qF^ϖ=Dϱp,qU¯ϖ,Dϱp,qU¯ϖ, for all ϖ∈ϱ,pτ+1−pϱ. Then, we have that(58)U¯ϖ−U¯c=∫cϖDϱp,qU¯γdϱp,qγ,U¯ϖ−U¯c=∫cϖDϱp,qU¯γdϱp,qγ.
It follows that(59)F^ϖ=F^c+∫cϖDϱp,qF^γdϱp,qIγ.
Sinceℱ is ℒ-increasing on ϱ,τ, by (2), then(60)F^ϖ⊖GF^c=∫cϖDϱp,qF^γdϱp,qIγ.
IfF^ is L-decreasing on ϱ,τ, then Dϱp,qF^ϖ=Dϱp,qU¯ϖ,Dϱp,qU¯ϖ, for all c∈ϱ,pϖ+1−pϱ. Then, we get that(61)∫cϖDϱp,qF^γdϱp,qIγ=∫cϖDϱp,qU¯γdϱp,qγ,∫cϖDϱp,qU¯γdϱp,qγ=U¯ϖ−U¯c,U¯ϖ−U¯c=U¯ϖ,U¯ϖ⊖GU¯c,U¯c=F^ϖ⊖GF^c.Remark 2.
We remark that ifF^ is L-increasing on ϱ,τ, then (57) is equivalent with(62)F^ϖ=F^c+∫cϖDϱp,qF^γdϱp,qIγ,and if F^ is L-decreasing on ϱ,τ, then (57) is equivalent with(63)F^ϖ=F^c⊖G−1∫cϖDϱp,qF^γdϱp,qIγ,for all ϖ∈ϱ,pτ+1−pϱ. Also, we remark that relation (57) can be false if F^ is not L-monotone on ϱ,τ. Indeed, let F^:0,2⟶Kc, given by. For c∈0,1 and ϖ∈1,2, we have that (see Example 3)(64)∫cϖD0p,qF^γdϱp,qIγ=∫c1D0p,qF^γdϱp,qIγ+∫1ϖD1p,qF^γdϱp,qIγ=0,c2−2c+1+−ϖ2+2ϖ−1,0=−ϖ2+2ϖ−1,c2−2c+1.
Then, we get that(65)F^ϖ⊖GF^c=min0,c−ϖc+ϖ−2,max0,c−ϖc+ϖ−2≠∫cϖD0p,qF^γdϱp,qIγ.
Therefore, (57) is not true for all ϖ∈0,2.Example 5.
LetF^:0,2⟶Kc, given by F^ϖ=0,ϖ2. Since F^ϖ is Ip,qϱ-differentiable and L-increasing on 0,2, then Dϱp,qF^ϖ is Ip,qϱ-integral and Dϱp,qF^ϖ=0,2p,qϖ. Let c=1∈0,ϖ; then,(66)F^ϖ⊖GF^1=0,ϖ2−1,∫1ϖD0p,qℱγd0p,qIγ=0,∫1ϖ2p,qγd0p,qγ=0,∫0ϖ2p,qγd0p,qγ−∫012p,qγd0p,qγ=0,ϖ2−1.Theorem 12.
(see [27]). Let F^:ϱ,τ⟶ℝ be a convex differentiable function on ϱ,τ. Then, the following inequalities holds for p,q-integrals:(67)F^qϱ+pτ2p,q≤1pτ−ϱ∫ϱpτ+1−pϱF^ϖdϱp,qϖ≤qF^ϱ+pF^τ2p,q.Theorem 13.
(see [27]). Let F^:ϱ,τ⟶ℝ be a convex differentiable function on ϱ,τ. Then, the following inequalities holds for p,q-integrals:(68)F^qτ+pϱ2p,q+p−qτ−ϱ2p,qF^′qτ+pϱ2p,q≤1pτ−ϱ∫ϱpτ+1−pϱF^ϖdϱp,qϖ≤pF^ϱ+qF^τ2p,q.Theorem 14.
(see [27]). Let F^:ϱ,τ⟶ℝ be a differentiable function on ϱ,τ and Dϱp,qF^ is continuous and integrable on ϱ,τ. If Dϱp,qF^ is convex function over ϱ,τ, then we have the following p,q-midpoint inequality:(69)∫ϱpτ+1−pϱF^ϖdϱp,qϖ−F^qϱ+pτ2p,q≤qτ−ϱDϱp,qF^τW1p,q+Dϱp,qF^ϱW2p,q+Dϱp,qF^τW3p,q+Dϱp,qF^ϱW4p,q,where(70)W1p,q=p32p,q33p,q,W2p,q=p23p,q−p32p,q33p,q,W3p,q=2p32p,q33p,q,W4p,q=p4+p3q+p2q2−2p32p,q33p,q,and 0<q<p≤1.Theorem 15.
(see [28]). Let F^:ϱ,τ⟶ℝ be a differentiable function on ϱ,τ and Dϱp,qF^ be continuous and integrable on ϱ,τ. If Dϱp,qF^ is convex function over ϱ,τ, then we have the following new p,q-trapezoidal inequality:(71)qℱϱ+pF^τ2p,q−1pτ−ϱ∫ϱpτ+1−pρF^ϖdϱp,qϖ≤qτ−ϱ2p,qDϱp,qF^τW5p,q+Dϱp,qF^ϱW6p,q,where(72)W5p,q=qp3−2+2p+2p2+2q+pq2+2p2−2p2p,q33p,q,W6p,q=22p,q−12p,q2−W5p,q.


## 5.Ip,qϱ-Hermite–Hadamard Inequalities for Interval-Valued Functions

Now, we review the content of the convex interval-valued functions.Definition 7.
(see [21]). Suppose that F^:ϱ,τ⟶Kc. Take ℱ is convex if, for all ϖ1,ϖ2∈ϱ,τ and γ∈0,1, we have(73)F^γϖ1+1−γϖ2⊇γF^ϖ1+1−γF^ϖ2.
We useSXϱ,τ,Kc to represent the set of all convex interval-valued functions.Theorem 16.
(see [21]). Let F^:ϱ,τ⟶Kc+. Then, F^ is said to be convex if and only if U¯ is convex and U¯ is concave on ϱ,τ.Theorem 17.
LetF^=U¯,U¯:ϱ,τ⟶Kc+ be a differentiable interval-valued convex function; then, the following inequalities hold for the Ip,qϱ-integral:(74)F^qϱ+pτ2p,q⊇1pτ−ϱ∫ϱpτ+1−pϱF^ϖdϱp,qIϖ⊇qF^ϱ+pF^τ2p,q.Proof.
SinceF^=U¯,U¯:ϱ,τ⟶Kc+ is an interval-valued convex function, therefore U¯ is a convex function and U¯ is a concave function. So, from U¯ and inequality (67), we have(75)U¯qϱ+pτ2p,q≤1pτ−ϱ∫ϱpτ+1−pϱU¯ϖdϱp,qϖ≤qU¯ϱ+pU¯τ2p,q,and from concavity of U¯ and (67), we have(76)qU¯ϱ+pU¯τ2p,q≤1pτ−ϱ∫ϱpτ+1−pϱU¯ϖdϱp,qϖ≤U¯qϱ+pτ2p,q.
From (75) and (76), we obtain(77)U¯qϱ+pτ2p,q≤1pτ−ϱ∫ϱpτ+1−pϱU¯ϖdϱp,qϖ≤1pτ−ϱ∫ϱpτ+1−pϱU¯ϖdϱp,qϖ≤U¯qϱ+pτ2p,q,and hence, we have(78)F^qϱ+pτ2p,q⊇1pτ−ϱ∫ϱpτ+1−pϱF^ϖdϱp,qIϖ.
Also, from (75) and (76), we obtain(79)1pτ−ϱ∫ϱpτ+1−pϱU¯ϖdϱp,qϖ≤qU¯ϱ+pU¯τ2p,q≤qU¯ϱ+pU¯τ2p,q≤1pτ−ϱ∫ϱpτ+1−pϱU¯ϖdϱp,qϖ,and hence, we have(80)1pτ−ϱ∫ϱpτ+1−pϱF^ϖdϱp,qIϖ⊇qU¯ϱ+pU¯τ2p,q.
By combining (78) and (80), we obtain the required inequality which accomplishes the proof.Theorem 18.
LetF^=U¯,U¯:ϱ,τ⟶Kc+ be a differentiable interval-valued convex function on ϱ,τ; then, the following inequalities hold for the Ip,q-integral:(81)F^qτ+pϱ2p,q+p−qτ−ϱ2p,qF^′qτ+pϱ2p,q⊇1pτ−ϱ∫ϱpτ+1−pϱF^ϖdϱp,qIϖ⊇pF^ϱ+qF^τ2p,q.Proof.
SinceF^=U¯,U¯:ϱ,τ⟶Kc+ is an interval-valued convex function, therefore, U¯ is a convex function and U¯ is a concave function. Because of convexity of U¯ and from inequalities (68), we obtain that(82)U¯qτ+pϱ2p,q+p−qτ−ϱ2p,qU¯′qτ+pϱ2p,q≤1pτ−ϱ∫ϱpτ+1−pϱU¯ϖdϱp,qϖ≤pU¯ϱ+qU¯τ2p,q.
Now, using the fact thatU¯ is concave function and from inequality (68), we obtain that(83)U¯qτ+pϱ2p,q+p−qτ−ϱ2p,qU¯′qτ+pϱ2p,q≥1pτ−ϱ∫ϱpτ+1−pϱU¯ϖdϱp,qϖ≥pU¯ϱ+qU¯τ2p,q.
The rest part of the proof can be done by applying the same lines of previous theorem and considering inequalities (82) and (83). Thus, the proof is completed.Theorem 19.
LetF^=U¯,U¯:ϱ,τ⟶Kc+ be a differentiable interval-valued convex function on ϱ,τ; then, the following inequalities hold for the Ip,q-integral:(84)maxℳ1,ℳ2⊇1pτ−ϱ∫ϱpτ+1−pϱℱϖdϱp,qIϖ⊇pF^τ+qF^ϱ2p,q,where(85)ℳ1=F^qϱ+pτ2p,q,ℳ2=F^qτ+pϱ2p,q+p−qτ−ϱ2p,qF^′qτ+pϱ2p,q.Proof.
From inequalities (74) and (75), we have the required inequalities (84). Thus, the proof is finished.


## 6. Midpoint- and Trapezoidal-Type Inequalities forIp,qϱ-Integral

In this section, some new inequalities of midpoint and trapezoidal type for interval-valued functions are obtained.Theorem 20.
LetF^=U¯,U¯:ϱ,τ⟶Kc+ be a Ip,qϱ-differentiable function. If Dϱp,qU¯ and Dϱp,qU¯ are convex functions on ϱ,τ, then the following Ip,qϱ midpoint inequality holds for interval-valued functions:(86)dH1pτ−ϱ∫ϱqτ+1−ppF^ϖdϱp,qIϖ,F^qϱ+pτ2p,q,≤qτ−ϱDϱp,qF^τW1p,q+Dϱp,qF^ϱW2p,q,+Dϱp,qF^τW3p,q+Dϱp,qF^ϱW4p,q,where W1p,q-W4p,q are defined in Theorem 14 and dH is Pompeiu–Hausdorff distance between the intervals.Proof.
Using the definition ofdH distance between intervals, one can easily obtain that(87)dH1pτ−ϱ∫ϱpτ+1−pϱF^ϖdϱp,qIϖ,F^qϱ+pτ2p,q=dH1pτ−ϱ∫ϱpτ+1−pϱU¯ϖdϱp,qϖ,1pτ−ϱ∫ϱpτ+1−pϱU¯ϖdϱp,qϖ,U¯qϱ+pτ2p,q,U¯qϱ+pτ2p,q=max1pτ−ϱ∫ϱpτ+1−pϱU¯ϖdϱp,qϖ−U¯qϱ+pτ2p,q,1pτ−ϱ∫ϱpτ+1−pϱU¯ϖdϱp,qϖ−U¯qϱ+pτ2p,q.
Now, using the fact thatDϱp,qU¯ is a convex function and from inequality (69), we have(88)1pτ−ϱ∫ϱpτ+1−pϱU¯ϖdϱp,qϖ−U¯qϱ+pτ2p,q≤qτ−ϱDϱp,qU¯τW1p,q+Dϱp,qU¯ϱW2p,q+Dϱp,qU¯τW3p,q+Dϱp,qU¯ϱW4p,q.
Similarly, considering thatDϱp,qU¯ is convex on ϱ,τ and using inequality (69), we have(89)1pτ−ϱ∫ϱpτ+1−pϱU¯ϖdϱp,qϖ−U¯qϱ+pτ2p,q≤qτ−ϱDϱp,qU¯τW1p,q+Dϱp,qU¯ϱW2p,q+Dϱp,qU¯τW3p,q+Dϱp,qU¯ϱW4p,q.
So, from inequalities (88) and (89), we have(90)dH1pτ−ϱ∫ϱpτ+1−pϱF^ϖdϱp,qIϖ,F^qϱ+pτ2p,q=max1pτ−ϱ∫ϱpτ+1−pϱU¯ϖdϱp,qϖ−U¯%qϱ+pτ2p,q1pτ−ϱ∫ϱpτ+1−pϱU¯ϖdϱp,qϖ−U¯qϱ+pτ2p,q≤maxqτ−ϱDϱp,qU¯τW1p,q+Dϱp,qU¯ϱW2p,q+Dϱp,qU¯τW3p,q+Dϱp,qU¯ϱW4p,q,qτ−ϱDϱp,qU¯τW1p,q+Dϱp,qU¯ϱW2p,q+Dϱp,qU¯τW3p,q+Dϱp,qU¯ϱW4p,q=qτ−ϱDϱp,qF^τW1p,q+Dϱp,qF^ϱW2p,q+Dϱp,qF^τW3p,q+Dϱp,qF^ϱW4p,q,since(91)Dϱp,qF^ϱ=maxDϱp,qU¯ϱ,Dϱp,qU¯ϱ,Dϱp,qF^τ=maxDϱp,qU¯τ,Dϱp,qU¯τ.
Therefore, the proof is completed.Corollary 1.
If we setp=1 in Theorem 20, then we have the following new Iqϱ -midpoint inequality for interval-valued functions:(92)dH1τ−ϱ∫ϱτF^ϖϱdqIϖ,F^qϱ+τ2q≤qτ−ϱDϱqF^τW11,q+DϱqF^ϱW21,q+DϱqF^τW31,q+DϱqF^ϱW41,q,where DϱqU¯ and DϱqU¯ both are convex functions.Corollary 2.
If we setp=1 and q⟶1− in Theorem 20, then we have following midpoint inequality for interval-valued functions:(93)dH1τ−ϱ∫ϱτF^ϖdIϖ,F^ϱ+τ2≤τ−ϱF^′τW11,1+F^′ϱW21,1+F^′τW31,1+F^′ϱW41,1,where U¯′ϱ and U¯′ϱ both are convex functions.Theorem 21.
LetF^=U¯,U¯:ϱ,τ⟶Kc+ be a Ip,qϱ-differentiable function. If Dϱp,qU¯ and Dϱp,qU¯ are convex functions on ϱ,τ, then the following Ip,qϱ trapezoidal inequality holds for interval-valued functions:(94)dHqF^ϱ+pF^τ2p,q,1pτ−ϱ∫ϱpτ+1−pρF^ϖdϱp,qIϖ≤qτ−ϱ2p,qDϱp,qF^τW5p,q+Dϱp,qF^ϱW6p,q,where W5 and W6 are defined in Theorem 15 and dH is Pompeiu–Hausdorff distance between the intervals.Proof.
From definition ofdH distance between the intervals and inequality (71) and using the strategies that were followed in Theorem 21, one can easily obtain inequality (94).Corollary 3.
If we setp=1 in Theorem 21, then we have following new Iqϱ-trapezoidal inequality for interval-valued functions:(95)dHF^ϱ+pF^τ2q,1τ−ϱ∫ϱτF^ϖdϱqIϖ≤qτ−ϱ2qDϱqF^τW51,q+DϱqF^ϱW61,q,where DϱqU¯ and DϱqU¯ both are convex functions.Corollary 4.
If we setp=1 and q⟶1− in Theorem 21, then we have the following new trapezoidal inequality for interval-valued functions:(96)dHF^ϱ+F^τ2,1τ−ϱ∫ϱτF^ϖdIϖ≤τ−ϱ2F^′τW51,1+F^′ϱW61,1,where U¯′ϱ and U¯′ϱ both are convex functions.


## 7. Conclusions

In this work, the concept ofIp,qϱ-derivative and Ip,qϱ-integral are introduced and some fundamental properties are discussed. Furthermore, some new Ip,qϱ-Hermite–Hadamard type inequalities are established and we proved some results for midpoint- and trapezoidal-type inequalities by using the concept of Pompeiu–Hausdorff distance between the intervals. We intend to study the integral inequalities of fuzzy-interval-valued functions and some applications in interval optimizations by using Ip,qϱ-integral.



---

*Source: 5529650-2021-06-30.xml*