# Inclusions Involving Interval-Valued Harmonically Co-Ordinated Convex Functions and Raina’s Fractional Double Integrals

Authors: Bandar Bin Mohsin; Muhammad Uzair Awan; Muhammad Zakria Javed; Hüseyin Budak; Awais Gul Khan; Muhammad Aslam Noor
Journal: Journal of Mathematics
 (2022)

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## Abstract

The aim of this article is to obtain some new integral inclusions essentially using the interval-valued harmonically co-ordinated convex functions andκ-Raina’s fractional double integrals. To show the validity of our theoretical results, we also give some numerical examples.


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## Body

## 1. Introduction

A convex analysis is the branch of mathematics in which we study the properties of convex sets and convex functions. These classical concepts have a wide range of applications in both pure and applied sciences. For instance, every one is familiar with the role of convexity in the theory of optimization, operations research, mathematical economics, theory of means etc. In recent years the classical concepts of the convexity have been extended and generalized in different directions using novel and innovative ideas. For example, Dragomir [1] extended the notion of classical convex functions on the coordinates and introduced the class of co-ordinated convex functions. Iscan [2] introduced the notion of harmonically convex functions and observed that this class enjoys some nice properties which the convex functions have. Nikodem [3] introduced the class of interval-valued convex functions and discussed its properties. Zhao et al. [4] introduced the notion of interval-valued harmonically convex functions. For more details, interested readers are referred to the book [5].Another charming aspect of the theory of convexity is its relation with the theory of inequalities. Many inequalities which are known to us are direct consequences of the applications of the convexity property of the functions. In this regard, one of the most studied results is Hermite-Hadamard’s inequality. This inequality provides us with a necessary and sufficient condition for a function to be convex. It reads as,letΘ:ℐ=℘1,℘2⊆ℝ⟶ℝ be a convex function, then(1)Θ℘1+℘22≤1℘2−℘1∫℘1℘2Θxdx≤Θ℘1+Θ℘22.This result is one of the most significant results pertaining to the convexity property of the functions which has been studied extensively as well as intensively. In recent years this result has been extended and generalized in different ways using novel and innovative ideas. For example, Dragomir [1] obtained a new version of Hermite-Hdamard’s inequality by using the co-ordinated convexity property of the functions. Iscan [2] obtained Hermite-Hadamard’s inequality using the class of harmonic convex functions. Zhao obtained a similar result by using the interval-valued harmonically convex functions. Sarikaya et al. [6] have utilized the concepts of fractional calculus and obtained fractional analogues of Hermite-Hadamard’s inequality. For some more recent studies regarding Hermite-Hadamard’s inequality and its applications, see [7].On the other hand, the interval analysis, which is used in mathematics and computer models as one of the ways for resolving interval uncertainty, is an important material in mathematics. Despite the fact that this theory has a lengthy history dating back to Archimedes’ estimate of the circumference of a circle, a substantial research in this topic was not published until the 1950s. In 1966, Moore, the pioneer of interval calculus, released the first book [8] on interval analysis. Following that, a slew of researchers delved into the theory and applications of interval analysis. Many authors have recently focused on integral inequalities derived from interval-valued functions. Sadowska [9] discovered the Hermite-Hadamard inequality for set-valued functions, which is a more general form of interval-valued mappings:Theorem 1 (see [9]).
Suppose thatΘ:℘1,℘2⟶ℝI+ is interval-valued convex function such that Ft=Θ¯t,F¯Θ. Then, we have the following inequalities:(2)Θ℘1+℘22⊇1℘2−℘1∫℘1℘2Θxdx⊇Θ℘1+Θ℘22.
Many papers have been devoted to generalizations of the inclusions (2). For example; Budak et al. [10] proved the Hermite-Hadamard inclusion by using Riemann-Liouville fractional integrals of interval-valued functions. Some papers focused on to generalization of (2) by utilizing some kinds of general convexities [4, 11–13]. Many authors also proved the corresponding Hermite-Hadamard inclusions for interval-valued functions with two variables [14–17]. For interested readers, we refer to the following articles [18–20]. The main objective of this paper is to derive new fractional refinements of Hermite–Hadamard’s inclusion using the class of interval-valued co-ordinated harmonically convex functions.


## 2. Preliminaries

In this section, we discuss some preliminary concepts and results.A setC⊆ℝ is said to be convex, if(3)1−tx+ty∈C,∀x,y∈C,t∈0,1.A functionΘ:C⟶ℝ is said to be convex, if(4)Θ1−tx+ty≤1−tΘx+tΘy,∀x,y∈C,t∈0,1.Dragomir extended the notion of classical convexity and introduced the class of co-ordinated convex functions as follows.Definition 1 (see [1]).
LetΔ=℘1,℘2×℘3,℘4⊆ℝ×ℝ be a rectangle. A function Θ:Δ⟶ℝ be a co-ordinated convex, if(5)Θ1−t+ty,1−ru+rw≤1−t1−rΘx,u+r1−tΘx,w+t1−rΘy,u+trΘy,w,holds for all x,y,u,w∈Δ, t,r∈0,1.
The following result is the Hermite–Hadamard’s inequality pertaining to the co-ordinated convex functions.Theorem 2 (see [1]).
Suppose thatΘ:Δ⟶ℝ is co-ordinated convex function, then(6)Θ℘1+℘22,℘3+℘42≤1℘2−℘1℘4−℘3∫℘1℘2∫℘3℘4Θx,ydxdy≤Θ℘1,℘3+Θ℘1,℘4+Θ℘2,℘3+Θ℘2,℘44.
Iscan [2] introduced the notion of harmonically convex functions as follows.Definition 2 (see [2]).
A functionΘ:ℋ⊆0,∞⟶ℝ is said to be harmonically convex, if(7)Θxytx+1−ty≤1−tΘx+tΘy,∀x,y∈ℋ,t∈0,1.Proposition 1 (see [2]).
(1)
IfΘx⊂0,∞ is a nondecreasing function, then Θx is a harmonically convex function(2)
IfΘx⊂0,∞ is a nonincreasing harmonic convex function then Θx is a convex functionTheorem 3 (see [2]).
LetΘ:ℐ=℘1,℘2⊆0,∞⟶ℝ be a harmonically convex function, then(8)Θ2℘1℘2℘1+℘2≤℘1℘2℘2−℘1∫℘1℘2Θxx2dx≤Θ℘1+Θ℘22.
Noor et al. [21] introduced the class of co-ordinated harmonically convex functions and derived associated Hermite-Hadamard’s inequality.Definition 3 (see [21]).
Consider a rectangleΔ=℘1,℘2×℘3,℘4⊆0,∞×0,∞. A function Θ:Δ⟶ℝ is said to be co-ordinated harmonically convex function on Δ, if(9)Θxytx+1−ty,uwru+1−rw≤1−t1−rΘx,u+t1−rΘy,u+1−trΘx,w+trΘy,w.
wheneverx,y∈℘1,℘2,u,w∈℘3,℘4 and t,r∈0,1.Theorem 4 (see [21]).
Suppose thatΘ:Δ⟶ℝ is co-ordinated harmonically convex function, then(10)Θ2℘1℘2℘1+℘2,2℘3℘4℘3+℘4≤abcd℘2−℘1℘4−℘3∫℘1℘2∫℘3℘4Θx,yx2y2dxdy≤Θ℘1,℘3+Θ℘1,℘4+Θ℘2,℘3+Θ℘2,℘44.
We now discuss some preliminaries from interval analysis. The idea behind the interval analysis is the observation that if you compute a numberμ and a bound M on the total error in μ, as an approximation to some unknown number X, such that X−μ≤M, then no matter how you computed μ and M, you will know that X lies in the interval μ−M,μ+M.
LetK℘3 be a collection of all closed and bounded intervals of the form w∗,w∗ for all w∗,w∗∈R. If w∗,w∗≥0, then w∗,w∗ is called a positive interval. Furthermore, K℘3+=w∗,w∗,∀w∗≥0 is known as the set of all positive intervals.
We now discuss about interval arithmetics and related properties Unlike the usual arithmetics the interval addition, subtraction product and division is defined as follows:(11)u∗,u∗+w∗,w∗=u∗+w∗,u∗+w∗,u∗,u∗−w∗,w∗=u∗−w∗,u∗−w∗,u∗,u∗⋅w∗,w∗=r∗,r∗,where(12)r∗=minu∗w∗,u∗w∗,u∗w∗,u∗w∗,r∗=maxu∗w∗,u∗w∗,u∗w∗,u∗w∗.
Also, ifw is an interval without zero element, then(13)1w=1w∗,1w∗,
so quotient ofu and w is defined as follows:(14)uw=u⋅1w.
Note that, addition, subtraction, product, and quotient of intervals is again an interval.
LetℝI, ℝI+, and ℝI− are set of closed intervals, set of positive closed intervals and set of negative closed intervals, respectively, then following are some algebraic properties of intervals:(i)
u+v+w=u+v+w,∀u,v,w∈ℝI (Associativity of addition)(ii)
u+v=v+u,∀u,v∈ℝI (Commutative property)(iii)
u+0=u,∀u∈ℝI (Additive element)(iv)
u+w=v+w⇒u=v,∀u,vw∈ℝI (Cancellation law)
All of these properties also hold for multiplication. But distributive law does not hold for intervals.  The following example illustrates this fact.Example 1.
(15)u=−3,−2,y=−2,0,w=0,1,
then(16)u⋅v+w=−3,−2−2,0+0,1=−3,−2−2,1=−3,6.
But(17)u⋅v+u⋅w=−3,−2−2,0+−3,−20,1=0.6+0.3=0.9.
ForA1=u∗,u∗, A2=w∗,w∗∈K℘3, the inclusion ⊆ is given by the following equation:(18)u∗,u∗⊆w∗,w∗,iffw∗≤u∗,u∗≤w∗.
Now, we rewrite the subsequent distance of intervalsA1 and A2 is regarded as Hausdorff–Pompeiu distance as follows:(19)℘4A1,A2=℘4u∗,u∗,w∗,w∗=maxu∗−w∗,u∗−w∗.
We now discuss the integration of interval-valued functions.
IfB℘1,℘2 is set of all partitions of ℘1,℘2 and Bρ,℘1,℘2 be the set of all points P such that meshP<ρ, then Θ:℘1,℘2⟶ℝI is called interval Riemann integrable on ℘1,℘2, if there exist K∈ℝI and for each ϵ>0 there exists ρ>0 such that(20)℘4SΘ,P,ρ,k<ε,where SΘ,P,ρ is the Riemann sum of Θ with respect to P∈Bρ,℘1,℘2. (20) shows that K is the IR-integral of Θ and given by the following equation:(21)K=IR∫℘1℘2Θtdt.Theorem 5.
LetΘ:℘1,℘2⟶ℝ be an interval-valued function such that Θt=Θ∗,Θ∗,Θ∈IR℘1,℘2 if and only if Θ∗,Θ∗∈R℘1,℘2 and(22)IR∫℘1℘2Θtdt=R∫℘1℘2Θ∗tdt,R∫℘1℘2Θ∗tdt.
Now, we discuss interval-valued integration of double integrals and generalized Raina’s interval-valued integrals.
Suppose that the rectangleΔ=℘1,℘2×℘3,℘4 where ℘1<℘2 and ℘3<℘4. A group of numbers xi−1,μi,xii=1m is termed as a tagged partition P1 with respect to interval ℘1,℘2 if: P1:℘1=x0<x1<x2<⋯<xm=℘2 and if xi−1≤μi≤xi,i=i,2,3,…,m. Also, we consider Λxi=xi−xi−1, and if Λxi<δ for every i, then partition P1 is known as δ-fine. If Λδ,℘1,℘2 is collection of all δ-partitions of interval ℘1,℘2. Suppose that xi−1,μi,xii=1m belongs to δ-fine P1 regarding ℘1,℘2 and yj−1,λj,yjj=1n is another δ-fine P2 regarding ℘3,℘4, then the series of rectangles Λ=xi−1,xi×yj,yj are the partitions of Δ, and the points μi,λj∈Λ°. Furthermore, letting Λδ,Λ be the set of all δ-fine partitions of rectangle Δ along with P1×P2, in which P1∈Λδ,℘1,℘2 along with P2∈Λδ,℘3,℘4. We consider using the notationΛAi,j denote the area of the rectangle δi,j. Within every rectangle Λi,j, for 1≤i≤m and 1≤j≤n, taking into account arbitrary μi,λj and we have SΘ,P,δ,Δ=∑i=1m∑j=1nΘμi,λjΛAi,j, which is named as SΘ,P,δ,Δ is the integral sum of Θ in connection Λδ,Δ.Theorem 6.
Assume that the interval-valued functionΘ:Δ⟶ℝI. then Θ is called double integrable defined on rectangle Δ with interval-valued double integral U=∫∫ΔΘx,ydA, if for every ϵ>0 there exist δ>0 satisfying that(23)℘4SΘ,P,δ,Δ<ε,for eachP∈Pδ,Δ.Theorem 7.
Assume thatΘ:Δ⟶ℝI is interval-valued double integrable defined on Δ, then(24)∬ΔΘx,ydA=∫℘1℘2∫℘3℘4Θx,ydydx.
For more details, see [8, 22].
The class of interval-valued convex functions is defined as follows.Definition 4 (see [3]).
A functionΘ:℘1,℘2⟶ℝI+ is said to be interval-valued convex function, if the following inclusion holds:(25)Θ1−tx+ty⊇1−tΘx+tΘy,x,y∈℘1,℘2,where 0≤℘1<℘2 and t∈0,1.
Now, we recall the definition of interval-valued harmonically convexity, which is defined as follows.Definition 5 (see [4]).
LetΘ:℘1,℘2⟶ℝI+ be interval-valued harmonically convex function if the following inclusion holds:(26)Θ℘1℘21−tx+ty⊇tΘx+1−tΘy,x,y∈℘1,℘2,where 0≤℘1<℘2 and t∈0,1.
The following is the definition of an interval-valued co-ordinated convex function.Definition 6 (see [17]).
Suppose thatΘ:Δ⟶ℝI+ belongs to an interval-valued function regarding two variable form, in which Δ=℘1,℘2×℘3,℘4 is a given rectangle. We say that Θ is an interval-valued co-ordinated convex function, if(27)Θ1−tx+ty,1−ru+rw⊇1−t1−rΘx,u+r1−tΘx,w+t1−rΘy,u+trΘy,w,holds for every x,y,u,w∈Δ together with t,r∈0,1.Proposition 2.
Let functionΘ:℘1,℘2⟶ℝI+ be a interval-valued function with Θx=Θ∗x,Θ∗x. Then, Θx will be interval-valued convex function if and if only Θ∗x is a convex function and Θ∗x is a concave function.
We now recall some concepts from fractional calculus. The classical Riemann-Liouville fractional integrals are defined as:Definition 7 (see [23]).
LetΘ∈L1℘1,℘2. The Riemann-Liouville integrals J℘1+αΘ and J℘2−αΘ of order α>0 are defined by the following equation:(28)J℘1+αΘx=1Γα∫℘1xx−tα−1Θtdt,x>℘1,J℘2−αΘx=1Γα∫x℘2t−xα−1Θtdt,x<℘2.
Sarikaya et al. [6] have obtained the following fractional analogue of Hermite–Hadamard’s inequality:Theorem 8.
LetΘ:℘1,℘2⟶ℝ be a positive function with 0≤℘1<℘2 and Θ:L1℘1,℘2. If Θ is a convex function on ℘1,℘2, then(29)Θ℘1+℘22≤Γα+12℘2−℘1αJ℘1+αΘ℘2+J℘2−αΘ℘1≤Θ℘1+Θ℘22.
This motivated Iscan and Wu [24] and as result, they have obtained a fractional analogue of Hermite–Hadamard’s inequality essentially using the class of harmonically convex functions.
Mubeen and Habibullah [25] introduced the notion of κ-Riemann-Liouville fractional integrals as follows.Definition 8.
LetΘ∈L1℘1,℘2. The κ-Riemann-Liouville integrals J℘1+α,κΘ and J℘2−α,κΘ of order α>0 are defined by the following equation:(30)J℘1+α,κΘx=1κΓκα∫℘1xx−τα/κ−1Θτdτ,x>℘1,J℘2−α,κΘx=1κΓκα∫x℘2τ−xα/κ−1Θτdτ,x<℘2,where Γκα=∫0∞τα−1eτκ/κdτ, Re α>0,κ∈ℝ+ is the κ–gamma function which was introduced and studied in [26].
Now, we give generalized fractional integrals involving Raina’s functions for interval-valued function, which is stated as follows.Definition 9.
LetΘ⊆ℝ⟶ℝI be an interval-valued function and F be a Reimann interval-valued integrable function satisfying the condition Θx=Θ∗x,Θ∗x:(31)ρ,α,wI℘1+σ,kΘ℘2=∫℘1℘2℘2−tα/k−1ℛρ,w,ασ,kw℘2−tρftdt,℘2>x,ρ,α,wI℘2−σ,kΘ℘1=∫℘1℘2t−℘1α/k−1ℛρ,w,ασ,kwt−℘1ρftdt,℘1<x.
Using the definitions ofκ-Riemann-Liouville fractional integrals Noor et al. [21] have obtained κ-fractional analogue of Hermite-Hadamard’s inequality.
We now introduce theκ-Raina’s fractional double integrals, which are defined as follows.Definition 10.
LetΘ⊆ℝ2⟶ℝI be a given interval valued-valued functions in two variables, satisfying that Θt,s=Θ∗t,s,Θ∗t,s and let Θ be interval Riemann integrable on rectangle Δ, then(32)ρ,α,wI℘1+,℘3+σ,kΘ℘2,℘4=∫℘1℘2∫℘3℘4℘2−tα1/k−1℘4−sα2/k−1ℛρ1,w1,α1σ1,kw1℘2−tρ1ℛρ2,w2,α2σ2,kw2℘4−sρ2Θt,sdsdt,ρ,α,wI℘1+,℘4−σ,kΘ℘2,℘3=∫℘1℘2∫℘3℘4℘2−tα1/k−1s−℘3α2/k−1ℛρ1,w1,α1σ1,kw1℘2−tρ1ℛρ2,w2,α2σ2,kw2s−℘3ρ2Θt,sdsdt,ρ,α,wI℘2−,℘3+σ,kΘ℘1,℘4=∫℘1℘2∫℘3℘4t−℘1α1/k−1℘4−sα2/k−1ℛρ1,w1,α1σ1,kw1t−℘1ρ1ℛρ2,w2,α2σ2,kw2℘4−sρ2Θt,sdsdt,ρ,α,wI℘2−,℘4−σ,kΘ℘1,℘3=∫℘1℘2∫℘3℘4t−℘1α1/k−1s−℘3α2/k−1ℛρ1,w1,α1σ1,kw1t−℘1ρ1ℛρ2,w2,α2σ2,kw2s−℘3ρ2Θt,sdsdt.
Here,ρ=ρ1,ρ2,α=α1,α2,w=w1,w2 and ℛρ,λσ,kz is the k-Raina’s function which is defined as follows:(33)ℛρ,λσ,kz=ℛρ,λσ0,σ1,…z≔=∑i=0∞σikΓkρki+λzi,z∈ℂ,where ρ,λ>0, with bounded modulus z<M, and σ=σ0,σ1,…,σi,… is a bounded sequence of positive real numbers. For details, see [27].
In the following example, we give the numerical verification of Definition(10).Example 2.
We choose℘1,℘2×℘3,℘4=0,1×1,2 and Θt,s=st,s+t, then we have the following equation:(34)ρ,α,wI0+,1+σ,kΘ1,2=∫01∫121−tα1/k−12−sα2/k−1ℛρ1,w1,α1σ1,kw11−tρ1ℛρ2,w2,α2σ2,kw22−sρ2st,s+tdsdt.
Then, (34) can be written as follows:(35)ρ,α,wI0+,1+σ,kΘ1,2=ρ,α,wI0+,1+σ,kΘ∗1,2,ρ,α,w,I0+,1+σ,kΘ∗1,2,where(36)ρ,α,wI0+,1+σ,kΘ∗1,2=∫01∫121−tα1k−12−sα2k−1ℛρ1,w1,α1σ1,kw11−tρ1ℛρ2,w2,α2σ2,kw22−sρ2stdsdt=k2ℛρ1,w1,α1+2kσ1,kw1kℛρ2,w2,α2+kσ2,kw2+k2ℛρ2,w2,α2+2kσ2,kw2,(37)ρ,α,wI0+,1+σ,kΘ∗1,2=∫01∫121−tα1/k−12−sα2/k−1ℛρ1,w1,α1σ1,kw11−tρ1ℛρ2,w2,α2σ2,kw22−sρ2s+tdsdt=k2ℛρ1,w1,α1+kσ1,kw1kℛρ2,w2,α2+kσ2,kw2+k2ℛρ2,w2,α2+2kσ2,kw2+k3ℛρ1,w1,α1+2kσ1,kw1ℛρ2,w2,α2+2kσ2,kw2.
A combination of (35), (36), and (37) yields the required result.
The main objective of this article is to obtain some new integral inclusions essentially using the interval-valued harmonically co-ordinated convex functions andk-Raina’s fractional double integrals. To show the validity of our theoretical results, we also give some numerical examples. We hope that the ideas and the techniques of this paper will inspire interested readers.


## 3. Results and Discussions

In this section, we discuss our main results. First of all, we introduce the class of interval-valued harmonically co-ordinated convex functions.Definition 11.
LetΘ:℘1,℘2×℘3,℘4⟶ℝI+ be interval-valued harmonically co-ordinated convex function, if the following inclusion holds:(38)Θxy1−t+ty,uw1−s+sw⊇tsΘx,u+t1−sΘx,w+1−tsΘy,u+1−t1−sΘy,w,where x,y,u,w∈Δ and t,s∈0,1 and 0≤℘1<℘2,0≤℘3<℘4. By reversing the the above-given inclusion, we obtain the notion of harmonically interval-valued co-ordinated concave function.Lemma 1.
LetΘ:℘1,℘2×℘3,℘4⟶ℝI+ be co-ordinated interval-valued function. Then, Θ is co-ordinated interval-valued harmonically convex function if and only if there exist two harmonically convex function Θyx:℘1,℘2⟶ℝI+,Θxy=Θx,y and Θxy:℘3,℘4⟶ℝI+,Θyx=Θx,y.Proof.
Using the definition of co-ordinated harmonic convex functions the proof is straight forward.Theorem 9.
Suppose thatΘ:℘1,℘2×℘3,℘4⟶ℝI+ is a given interval-valued harmonically convex function defined over a rectangle ℘1,℘2×℘3,℘4 with 0≤℘1<℘2,0≤℘3<℘4 m and Θx,y=Θ∗x,y,Θ∗x,y. Then, the following inclusion holds:(39)Θ2℘1℘2℘1+℘2⊇12kℛρ,w,α+kσ,kw℘2−℘1/℘1℘2ρ℘1℘2℘2−℘1α/kα,ρ,wI1/℘2+σ,kΘ°g1℘1α,ρ,wI1/℘1−σ,kΘ°g1℘2⊇Θ℘1+Θ℘22.
IfΘx,y is harmonically interval-valued concave function then we have the following equation:(40)Θ2℘1℘2℘1+℘2⊆12kℛρ,w,α+kσ,kw℘2−℘1/℘1℘2ρ℘1℘2℘2−℘1α/kα,ρ,wI1/℘2+σ,kΘ°g1℘1α,ρ,wI1/℘1−σ,kΘ°g1℘2⊆Θ℘1+Θ℘22.Proof.
The proof is left for interested readers.Example 3.
We setρ=0,α=1 and k=1 with σ0=1 in Theorem 9. If Θx=x2,−6x2+12x+24 and ℘1,℘2=1,2, then(41)Θ2℘1℘2℘1+℘2=1.78,29.33,(42)℘1℘2℘2−℘1∫℘1℘2Θxx2dx=2,28.63,(43)Θ℘1+Θ℘22=2.5,27.
(41)–(43) implies that(44)1.78,29.33⊇2,28.63⊇2.5,27,which gives the verification of Theorem 9.Theorem 10.
SupposeΘ:℘1,℘2⟶ℝI+ be an interval-valued harmonically convex functions with 0≤℘1<℘2, satisfying Θt=Θ∗t,Θ∗t, then(45)Θ2℘1℘2℘1+℘2⊇12kℛρ,w,α+kσ,kw℘2−℘1/2℘1℘2ρ2℘1℘2℘2−℘1α1/kα,ρ,wI℘1+℘2/2℘1℘2+σ,kΘ°g1℘1+α,ρ,wI℘1+℘2/2℘1℘2−σ,kΘ°g1℘2⊇Θ℘1+Θ℘22.
IfΘ is interval-valued harmonically co-ordinated concave then the following relation holds:(46)Θ2℘1℘2℘1+℘2⊆12kℛρ,w,α+kσ,kw℘2−℘1/2℘1℘2ρ2℘1℘2℘2−℘1α1/kα,ρ,wI℘1+℘2/2℘1℘2+σ,kΘ°g1℘1+α,ρ,wI℘1+℘2/2℘1℘2−σ,kΘ°g1℘2⊆Θ℘1+Θ℘22.Proof.
The proof is obvious by definition and some computation as well.Theorem 11.
Suppose thatΘ,G:℘1,℘2⟶ℝI+ are two interval-valued harmonically convex functions with 0≤℘1<℘2, satisfying Θt=Θ∗t,Θ∗t and Gt=G∗t,G∗t, then we have the following equation:(47)℘1℘2℘2−℘1αI1/℘2+,ρ,w,ασ,kG°g1℘1Θ°g1℘1+I1/℘1−,ρ,w,ασ,kΘ°g1℘2G°g1℘2⊇H1℘1,℘22∑i=0∞σiwi℘1℘2/℘2−℘1ρiρki+α+2kΓkρki+α−2∑i=0∞σiwi℘1℘2/℘2−℘1ρiρki+α+kΓkρki+α+Θρ,w,α+kσ,kw℘1℘2℘2−℘1ρ+H2℘1,℘22∑i=0∞σiwi℘1℘2/℘2−℘1ρiρki+α+kΓkρki+α−2∑i=0∞σiwi℘1℘2/℘2−℘1ρiρki+α+2kΓkρki+α,where(48)H1℘1,℘2=Θ℘1G℘1+Θ℘2G℘2,H2℘1,℘2=Θ℘1G℘2+Θ℘2G℘1.Proof 4.
From the given hypothesis and interval-valued harmonically convexity property, we have the following equation:(49)Θ∗℘1℘21−t℘1+tbG∗℘1℘21−t℘1+tb≤t2Θ∗℘1G∗℘1+t1−tΘ∗℘1G∗℘2Θ∗℘2G∗℘1+1−t2Θ∗℘2G∗℘2,(50)Θ∗℘1℘2t℘1+1−t℘2G∗℘1℘2t℘1+1−t℘2≤1−t2Θ∗℘1G∗℘1+t1−tΘ∗℘1G∗℘2+Θ∗℘2G∗℘1+t2Θ∗℘2G∗℘2.
Adding (49), (50) and multiplying inequality by tα/k−1ℛρ,w,ασ,kw℘1℘2/℘2−℘1ρtρp, we obtain the following equation:(51)℘1℘2℘2−℘1αI1/℘2+,ρ,w,ασ,kG∗°g1℘1Θ∗°g1℘1+I1/℘1−,ρ,w,ασ,kΘ∗°g1℘2G∗°g1℘2≤2∑i=0∞σiwi℘1℘2/℘2−℘1ρiρki+α+2kΓkρki+α−2∑i=0∞σiwi℘1℘2/℘2−℘1ρiρki+α+kΓkρki+α+Θρ,w,α+kσ,kw℘1℘2℘2−℘1ρ×Θ∗℘1G∗℘1+Θ∗℘2G∗℘2+Θ∗℘1G∗℘2+Θ∗℘2G∗℘12∑i=0∞σiwi℘1℘2/℘2−℘1ρiρki+α+kΓkρki+α−2∑i=0∞σiwi℘1℘2/℘2−℘1ρiρki+α+2kΓkρki+α.
Similarly, we can calculate, using interval-valued harmonically convexity,(52)℘1℘2℘2−℘1αI1/℘2+,ρ,w,ασ,kΘ∗°g1℘1G∗°g1℘1+I1/℘1−,ρ,w,ασ,kΘ∗°g1℘2G∗°g1℘2≥2∑i=0∞σiwi℘1℘2/℘2−℘1ρiρki+α+2kΓkρki+α−2∑i=0∞σiwi℘1℘2/℘2−℘1ρiρki+α+kΓkρki+α+Θρ,w,α+kσ,kw℘1℘2℘2−℘1ρ×Θ∗℘1G∗℘1+Θ∗℘2G∗℘2+Θ∗℘1G∗℘2+Θ∗℘2G∗℘12∑i=0∞σiwi℘1℘2/℘2−℘1ρiρki+α+kΓkρki+α−2∑i=0∞σiwi℘1℘2/℘2−℘1ρiρki+α+kΓkρki+α.
Combination of (51) and (52) refers to relation (47).
This completes the proof.Theorem 12.
Suppose thatΘ:℘1,℘2⟶ℝI+ is a given interval-valued harmonically co-ordinated convex function with 0≤℘1<℘2 and Θx=Θ∗x,Θ∗x. If ϕ:℘1,℘2⟶ℝ be a non-negative, integrable, and symmetric function: ϕx=ϕ1/1/℘1+1/℘2−1/x then the following inclusion holds:(53)⊇α,ρ,wI1/℘2+σ,kΘ°g1℘1ϕ°g1℘1+α,ρ,wI1/℘1−σ,kΘ°g1℘2ϕ°g1℘2Θ2℘1℘2℘1+℘2,2℘3℘4℘3+℘4α,ρ,wI1/℘2+σ,kϕ°g1℘1+α,ρ,wI1/℘1−σ,kϕ°g1℘2⊇Θ℘1+Θ℘22α,ρ,wI1/℘2+σ,kϕ°g1℘1+α,ρ,wI1/℘1−σ,kϕ°g1℘2.Proof.
The proof is left for interested readers.
Firstly, we establish fractional Hermite-Hadamard inequality.Theorem 13.
Suppose thatΘ:℘1,℘2×℘3,℘4⟶ℝI+ is a given interval-valued harmonically co-ordinated convex function defined over a rectangle ℘1,℘2×℘3,℘4 with 0≤℘1<℘2,0≤℘3<℘4, and Θx,y=Θ∗x,y,Θ∗x,y, then the following inclusion holds:(54)Θ2℘1℘2℘1+℘2,2℘3℘4℘3+℘4⊇14k2ℛρ1,w1,α1+kσ1,kw1℘2−℘1/℘1℘2ρ1ℛρ2,w2,α2+kσ2,kw2℘4−℘3/℘3℘4ρ2℘1℘2℘2−℘1α1/k℘3℘4℘4−℘3α2/k×α,ρ,wI1/℘2+,1/℘4+σ,kΘ°g1℘1,1℘3+α,ρ,wI1/℘2+,1/℘3−σ,kΘ°g1℘1,1℘4+α,ρ,wI1/℘1−,1/℘4+σ,kΘ°g1℘2,1℘3+α,ρ,wI1/℘1−,1/℘3−σ,kΘ°g1℘2,1℘4⊇Θ℘1,℘3+Θ℘1,℘4+Θ℘2,℘3+Θ℘2,℘44.
IfΘx,y is an interval-valued harmonically co-ordinated concave function then we have the following equation:(55)Θ2℘1℘2℘1+℘2,2℘3℘4℘3+℘4⊆14k2ℛρ1,w1,α1+kσ1,kw1℘2−℘1/℘1℘2ρ1ℛρ2,w2,α2+kσ2,kw2℘4−℘3/℘3℘4ρ2℘1℘2℘2−℘1α1/k℘3℘4℘4−℘3α2/k×α,ρ,wI1/℘2+,1/℘4+σ,kΘ°g1℘1,1℘3+α,ρ,wI1/℘2+,1/℘3−σ,kΘ°g1℘1,1℘4+α,ρ,wI1/℘1−,1/℘4+σ,kΘ°g1℘2,1℘3+α,ρ,wI1/℘1−,1/℘3−σ,kΘ°g1℘2,1℘4⊆Θ℘1,℘3+Θ℘1,℘4+Θ℘2,℘3+Θ℘2,℘44.Proof.
From the definition of a harmonically co-ordinated convex function and according to a hypothesis, we have the following equation:(56)Θ∗2℘1℘2℘1+℘2,2℘3℘4℘3+℘4≤14Θ∗℘1℘2t℘1+1−t℘2,℘3℘4s℘3+1−s℘4+Θ∗℘1℘2t℘1+1−t℘2,℘3℘41−s℘3+sdΘ∗℘1℘21−t℘1+tb,℘3℘4s℘3+1−s℘4+Θ∗℘1℘21−t℘1+tb,℘3℘41−s℘3+sd,(57)Θ∗2℘1℘2℘1+℘2,2℘3℘4℘3+℘4≥14Θ∗℘1℘2t℘1+1−t℘2,℘3℘4s℘3+1−s℘4+Θ∗℘1℘2t℘1+1−t℘2,℘3℘41−s℘3+sdΘ∗℘1℘21−t℘1+tb,℘3℘4s℘3+1−s℘4+Θ∗℘1℘21−t℘1+tb,℘3℘41−s℘3+sd.
Multiplying both sides of (56) by tα1/k−1sα2/k−1ℛρ1,w1,α1σ1,kw1℘2−℘1/℘1℘2ρ1tρ1ℛρ2,w2,α2σ2,kw2℘4−℘3/℘3℘4ρ2sρ2 and integration with respect to t,s in 0,1×0,1, then we have the following equation:(58)Θ∗2℘1℘2℘1+℘2,2℘3℘4℘3+℘4∫01∫01tα1/k−1sα2/k−1ℛρ1,w1,α1σ1,kw1℘2−℘1℘1℘2ρ1tρ1ℛρ2,w2,α2σ2,kw2℘4−℘3℘3℘4ρ2sρ2dsdt≤14∫01∫01tα1/k−1sα2/k−1ℛρ1,w1,α1σ1,kw1℘2−℘1℘1℘2ρ1tρ1ℛρ2,w2,α2σ2,kw2℘4−℘3℘3℘4ρ2sρ2Θ∗℘1℘2t℘1+1−t℘2,℘3℘4s℘3+1−s℘4+Θ∗℘1℘2t℘1+1−t℘2,℘3℘41−s℘3+sdΘ∗℘1℘21−t℘1+tb,℘3℘4s℘3+1−s℘4+Θ∗℘1℘21−t℘1+tb,℘3℘41−s℘3+sddsdt.
After simple computation, we have the following equation:(59)Θ∗2℘1℘2℘1+℘2,2℘3℘4℘3+℘4≤14k2ℛρ1,w1,α1+kσ1,kw1℘2−℘1/℘1℘2ρ1ℛρ2,w2,α2+kσ2,kw2℘4−℘3/℘3℘4ρ2℘1℘2℘2−℘1α1/k℘3℘4℘4−℘3α2/k×α,ρ,wI1/℘2+,1/℘4+σ,kΘ∗°g1℘1,1℘3+α,ρ,wI1/℘2+,1/℘3−σ,kΘ∗°g1℘1,1℘4+α,ρ,wI1/℘1−,1/℘4+σ,kΘ∗°g1℘2,1℘3+α,ρ,wI1/℘1−,1/℘3−σ,kΘ∗°g1℘2,1℘4.
From (57), we have the following equation:(60)Θ∗2℘1℘2℘1+℘2,2℘3℘4℘3+℘4≥14k2ℛρ1,w1,α1+kσ1,kw1℘2−℘1/℘1℘2ρ1ℛρ2,w2,α2+kσ2,kw2℘4−℘3/℘3℘4ρ2℘1℘2℘2−℘1α1/k℘3℘4℘4−℘3α2/k×α,ρ,wI1/℘2+,1/℘4+σ,kΘ∗°g1℘1,1℘3+α,ρ,wI1/℘2+,1/℘3−σ,kΘ∗°g1℘1,1℘4+α,ρ,wI1/℘1−,1/℘4+σ,kΘ∗°g1℘2,1℘3+α,ρ,wI1/℘1−,1/℘3−σ,kΘ∗°g1℘2,1℘4.
Combination of (71) and (60), we obtain the first inclusion of (54).
To prove our second inclusion, we use a definition of harmonically co-ordinated convexity.(61)Θ∗℘1℘21−t℘1+tb,℘3℘41−s℘3+sd≤tsF∗℘1,℘3+t1−sΘ∗℘1,℘4+1−tsF∗℘2,℘3+1−t1−sΘ∗℘2,℘4,(62)Θ∗℘1℘21−t℘1+tb,℘3℘4s℘3+1−s℘4≤t1−sΘ∗℘1,℘3+tsF∗℘1,℘4+1−t1−sΘ∗℘2,℘3+1−tsF∗℘2,℘4,(63)Θ∗℘1℘2t℘1+1−t℘2,℘3℘41−s℘3+sd,Θ∗℘1℘2t℘1+1−t℘2,℘3℘41−s℘3+sd,(64)Θ∗℘1℘2t℘1+1−t℘2,℘3℘4s℘3+1−s℘4≤1−t1−sΘ∗℘1,℘3+1−tsF∗℘1,℘4+t1−sΘ∗℘2,℘3+tsF∗℘2,℘4.
Adding (61)–(63) and (64), we have the following equation:(65)Θ∗℘1℘21−t℘1+tb,℘3℘41−s℘3+sd+Θ∗℘1℘21−t℘1+tb,℘3℘4s℘3+1−s℘4+Θ∗℘1℘2t℘1+1−t℘2,℘3℘41−s℘3+sd+Θ∗℘1℘2t℘1+1−t℘2,℘3℘4s℘3+1−s℘4≤Θ∗℘1,℘3+Θ∗℘1,℘4+Θ∗℘2,℘3+∗℘2,℘4.
Multiplying both sides of (65) by tα1/k−1sα2/k−1ℛρ1,w1,α1σ1,kw1℘2−℘1/℘1℘2ρ1tρ1ℛρ2,w2,α2σ2,kw2℘4−℘3/℘3℘4ρ2sρ2 and integration with respect to t,s in 0,1×0,1, then we have the following equation:(66)14k2ℛρ1,w1,α1+kσ1,kw1℘2−℘1/℘1℘2ρ1ℛρ2,w2,α2+kσ2,kw2℘4−℘3/℘3℘4ρ2℘1℘2℘2−℘1α1/k℘3℘4℘4−℘3α2/k×α,ρ,wI1/℘2+,1/℘4+σ,kΘ∗°g1℘1,1℘3+α,ρ,wI1/℘2+,1/℘3−σ,kΘ∗°g1℘1,1℘4+α,ρ,wI1/℘1−,1/℘4+σ,kΘ∗°g1℘2,1℘3+α,ρ,wI1/℘1−,1/℘3−σ,kΘ∗°g1℘2,1℘4≤Θ∗℘1,℘3+Θ∗℘1,℘4+Θ∗℘2,℘3+Θ∗℘2,℘44.
Using the harmonically co-ordinated concavity and similar argument as (66), we obtain the following inequality.(67)14k2ℛρ1,w1,α1+kσ1,kw1℘2−℘1/℘1℘2ρ1ℛρ2,w2,α2+kσ2,kw2℘4−℘3/℘3℘4ρ2℘1℘2℘2−℘1α1/k℘3℘4℘4−℘3α2/k×α,ρ,wI1/℘2+,1/℘4+σ,kΘ∗°g1℘1,1℘3+α,ρ,wI1/℘2+,1/℘3−σ,kΘ∗°g1℘1,1℘4+α,ρ,wI1/℘1−,1/℘4+σ,kΘ∗°g1℘2,1℘3+α,ρ,wI1/℘1−,1/℘3−σ,kΘ∗°g1℘2,1℘4≥Θ∗℘1,℘3+Θ∗℘1,℘4+Θ∗℘2,℘3+Θ∗℘2,℘44.
Combining (66) and (67), we obtain the second part of our required result. This completes the proof.
Now, we develop the new midpoint Hermite-Hadamard’s inequality in the sense of interval-valued harmonically co-ordinated convexity.Theorem 14.
Suppose thatΘ:℘1,℘2×℘3,℘4⟶ℝI+ is a given interval-valued harmonically co-ordinated convex function defined over a rectangle ℘1,℘2×℘3,℘4 with 0≤℘1<℘2,0≤℘3<℘4 and Θx,y=Θ∗x,y,Θ∗x,y, then the following inclusion holds:(68)Θ2℘1℘2℘1+℘2,2℘3℘4℘3+℘4⊇14k2ℛρ1,w1,α1+kσ1,kw1℘2−℘1/2℘1℘2ρ1ℛρ2,w2,α2+kσ2,kw2℘4−℘3/2℘3℘4ρ22℘1℘2℘2−℘1α1/k2℘3℘4℘4−℘3α2/k×α,ρ,wI℘1+℘2/2℘1℘2+,℘3+℘4/2℘3℘4+σ,kΘ∘g1℘11℘3+α,ρ,wI℘1+℘2/2℘1℘2+,℘3+℘4/2℘3℘4−σ,kΘ∘g1℘11℘4+α,ρ,wI℘1+℘2/2℘1℘2−,℘3+℘4/2℘3℘4+σ,kΘ∘g1℘21℘3+α,ρ,wI℘1+℘2/2℘1℘2−,℘3+℘4/2℘3℘4−σ,kΘ∘g1℘21℘4⊇Θ℘1,℘3+Θ℘1,℘4+Θ℘2,℘3+Θ℘2,℘44.
IfΘx,y is interval-valued harmonically co-ordinated concave function, then we have the following equation:(69)Θ2℘1℘2℘1+℘2,2℘3℘4℘3+℘4⊆14k2ℛρ1,w1,α1+kσ1,kw1℘2−℘1/2℘1℘2ρ1ℛρ2,w2,α2+kσ2,kw2℘4−℘3/2℘3℘4ρ22℘1℘2℘2−℘1α1/k2℘3℘4℘4−℘3α2/k×α,ρ,wI℘1+℘2/2℘1℘2+,℘3+℘4/2℘3℘4+σ,kΘ∘g1℘11℘3+α,ρ,wI℘1+℘2/2℘1℘2+,℘3+℘4/2℘3℘4−σ,kΘ∘g1℘11℘4+α,ρ,wI℘1+℘2/2℘1℘2−,℘3+℘4/2℘3℘4+σ,kΘ∘g1℘21℘3+α,ρ,wI℘1+℘2/2℘1℘2−,℘3+℘4/2℘3℘4−σ,kΘ∘g1℘21℘4⊆Θ℘1,℘3+Θ℘1,℘4+Θ℘2,℘3+Θ℘2,℘44.Proof.
From the definition of a harmonically co-ordinated convex function, we have the following equation:(70)Θ∗2℘1℘2℘1+℘2,2℘3℘4℘3+℘4≤14Θ∗2℘1℘2t℘1+2−t℘2,2℘3℘4s℘3+2−s℘4+Θ∗2℘1℘2t℘1+2−t℘2,2℘3℘42−s℘3+sdΘ∗2℘1℘22−t℘1+tb,2℘3℘4s℘3+2−s℘4+Θ∗2℘1℘22−t℘1+tb,2℘3℘42−s℘3+sd.
Multiplying both sides of (70) by tα1/k−1sα2/k−1ℛρ1,w1,α1σ1,kw1℘2−℘1/2℘1℘2ρ1tρ1ℛρ2,w2,α2σ2,kw2℘4−℘3/2℘3℘4ρ2sρ2 and integration with respect to t,s in 0,1×0,1, then we have the following equation:(71)Θ∗2℘1℘2℘1+℘2,2℘3℘4℘3+℘4∫01∫01tα1/k−1sα2/k−1ℛρ1,w1,α1σ1,kw1℘2−℘12℘1℘2ρ1tρ1ℛρ2,w2,α2σ2,kw2℘4−℘32℘3℘4ρ2sρ2dsdt≤14∫01∫01tα1/k−1sα2/k−1Θρ1,w1,α1σ1,kw1℘2−℘12℘1℘2ρ1tρ1ℛρ2,w2,α2σ2,kw2℘4−℘32℘3℘4ρ2sρ2Θ∗2℘1℘2t℘1+2−t℘2,2℘3℘4s℘3+2−s℘4+Θ∗2℘1℘2t℘1+2−t℘2,2℘3℘42−s℘3+sdΘ∗2℘1℘22−t℘1+tb,2℘3℘4s℘3+2−s℘4+Θ∗2℘1℘22−t℘1+tb,2℘3℘42−s℘3+sddsdt.
After simple computation, we have the following equation:(72)Θ∗2℘1℘2℘1+℘2,2℘3℘4℘3+℘4≤14k2ℛρ1,w1,α1+kσ1,kw1℘2−℘1/2℘1℘2ρ1ℛρ2,w2,α2+kσ2,kw2℘4−℘3/2℘3℘4ρ22℘1℘2℘2−℘1α1/k2℘3℘4℘4−℘3α2/k×α,ρ,wI℘1+℘2/2℘1℘2+,℘3+℘4/2℘3℘4+σ,kΘ∗∘g℘1+℘22℘1℘2,℘3+℘42℘3℘4+α,ρ,wI℘1+℘2/2℘1℘2+,℘3+℘4/2℘3℘4−σ,kΘ∗∘g1℘1,1℘4+α,ρ,wI℘1+℘2/2℘1℘2−,℘3+℘4/2℘3℘4+σ,kΘ∗∘g1℘2,1℘3+α,ρ,wI℘1+℘2/2℘1℘2−,℘3+℘4/2℘3℘4−σ,kΘ∗∘g1℘2,1℘4.
Similarly, we have the following equation:(73)Θ∗2℘1℘2℘1+℘2,2℘3℘4℘3+℘4≥14k2ℛρ1,w1,α1+kσ1,kw1℘2−℘1/2℘1℘2ρ1ℛρ2,w2,α2+kσ2,kw2℘4−℘3/2℘3℘4ρ22℘1℘2℘2−℘1α1/k2℘3℘4℘4−℘3α2/k×α,ρ,wI℘1+℘2/2℘1℘2+,℘3+℘4/2℘3℘4+σ,kΘ∗∘g℘1+℘22℘1℘2,℘3+℘42℘3℘4+α,ρ,wI℘1+℘2/2℘1℘2+,℘3+℘4/2℘3℘4−σ,kΘ∗∘g1℘1,1℘4+α,ρ,wI℘1+℘2/2℘1℘2−,℘3+℘4/2℘3℘4+σ,kΘ∗∘g1℘2,1℘3+α,ρ,wI℘1+℘2/2℘1℘2−,℘3+℘4/2℘3℘4−σ,kΘ∗∘g1℘2,1℘4.
By the combination of (72) and (73), we obtain first inclusion of (68).
To prove our second inclusion, we use a definition of harmonically co-ordinated convexity.(74)Θ∗2℘1℘22−t℘1+tb,2℘3℘42−s℘3+sd≤14tsΘ∗℘1,℘3+t2−sΘ∗℘1,℘4+2−tsF∗℘2,℘3+2−t2−sΘ∗℘2,℘4,(75)Θ∗2℘1℘22−t℘1+tb,2℘3℘4s℘3+2−s℘4≤14t2−sΘ∗℘1,℘3+tsF∗℘1,℘4+2−t2−sΘ∗℘2,℘3+2−tsF∗℘2,℘4,(76)Θ∗2℘1℘2t℘1+2−t℘2,2℘3℘42−s℘3+sd≤142−tsF∗℘1,℘3+2−t2−sΘ∗℘1,℘4+tsF∗℘2,℘3+t2−sΘ∗℘2,℘4,(77)Θ∗2℘1℘2t℘1+2−t℘2,2℘3℘4s℘3+2−s℘4≤142−t2−sΘ∗℘1,℘3+2−tsF∗℘1,℘4+t2−sΘ∗℘2,℘3+tsF∗℘2,℘4.
By adding (74)–(76) and (77), we have the following equation:(78)Θ∗2℘1℘22−t℘1+tb,2℘3℘42−s℘3+sd+Θ∗2℘1℘22−t℘1+tb,℘3℘4s℘3+2−s℘4+Θ∗2℘1℘2t℘1+2−t℘2,2℘3℘42−s℘3+sd+Θ∗2℘1℘2t℘1+2−t℘2,2℘3℘4s℘3+2−s℘4≤Θ∗℘1,℘3+Θ∗℘1,℘4+Θ∗℘2,℘3+∗℘2,℘4.
Multiplying both sides of (78) by tα1/k−1sα2/k−1ℛρ1,w1,α1σ1,kw1℘2−℘1/℘1℘2ρ1tρ1ℛρ2,w2,α2σ2,kw2℘4−℘3/℘3℘4ρ2sρ2 and integration with respect to t,s in 0,1×0,1, then we have the following equation:(79)14k2ℛρ1,w1,α1+kσ1,kw1℘2−℘1/2℘1℘2ρ1ℛρ2,w2,α2+kσ2,kw2℘4−℘3/2℘3℘4ρ22℘1℘2℘2−℘1α1/k2℘3℘4℘4−℘3α2/k×α,ρ,wI℘1+℘2/2℘1℘2+,℘3+℘4/2℘3℘4+σ,kΘ∗∘g℘1+℘22℘1℘2,℘3+℘42℘3℘4+α,ρ,wI℘1+℘2/2℘1℘2+,℘3+℘4/2℘3℘4−σ,kΘ∗∘g1℘1,1℘4+α,ρ,wI℘1+℘2/2℘1℘2−,℘3+℘4/2℘3℘4+σ,kΘ∗∘g1℘2,1℘3+α,ρ,wI℘1+℘2/2℘1℘2−,℘3+℘4/2℘3℘4−σ,kΘ∗∘g1℘2,1℘4≤Θ∗℘1,℘3+Θ∗℘1,℘4+Θ∗℘2,℘3+Θ∗℘2,℘44.
Using the harmonically co-ordinated concavity and similar argument as (79), we obtain the following inequality:(80)14k2ℛρ1,w1,α1+kσ1,kw1℘2−℘1/2℘1℘2ρ1ℛρ2,w2,α2+kσ2,kw2℘4−℘3/2℘3℘4ρ22℘1℘2℘2−℘1α1/k2℘3℘4℘4−℘3α2/k×α,ρ,wI℘1+℘2/2℘1℘2+,℘3+℘4/2℘3℘4+σ,kΘ∗∘g℘1+℘22℘1℘2,℘3+℘42℘3℘4+α,ρ,wI℘1+℘2/2℘1℘2+,℘3+℘4/2℘3℘4−σ,kΘ∗∘g1℘1,1℘4+α,ρ,wI℘1+℘2/2℘1℘2−,℘3+℘4/2℘3℘4+σ,kΘ∗∘g1℘2,1℘3+α,ρ,wI℘1+℘2/2℘1℘2−,℘3+℘4/2℘3℘4−σ,kΘ∗∘g1℘2,1℘4≥Θ∗℘1,℘3+Θ∗℘1,℘4+Θ∗℘2,℘3+Θ∗℘2,℘44.
Combining (79) and (80), we obtain the second part of our required result.Example 4.
We setρ1=0=ρ2,α1=1=α2 and k=1 with σ0=1 in Theorem 13. If Θx,y=2x2y2,−4x2+12x+6−2y2+4y+2 and ℘1,℘2=℘3,℘4=1,2, then(81)Θ2℘1℘2℘1+℘2,2℘3℘4℘3+℘4=5.71,56.77,(82)℘1℘2℘2−℘1℘3℘4℘4−℘3∫℘1℘2∫℘3℘4Θx,yx2y2dydx=8,51.92,(83)Θ℘1,℘3+Θ℘1,℘4+Θ℘2,℘3+Θ℘2,℘44=12.5,42.
(81), (82), and (83) imply that(84)5.71,56.77⊇8,51.92⊇12.5,42.
which provides the verification of Theorem13.
Next, we establish some Hermite-Hadamard-Fejer type inequality as well.Theorem 15.
Suppose thatΘ:℘1,℘2×℘3,℘4⟶ℝI+ is a given interval-valued harmonically co-ordinated convex function defined over a rectangle ℘1,℘2×℘3,℘4 with 0≤℘1<℘2,0≤℘3<℘4 and Θx,y=Θ∗x,y,Θ∗x,y. If ϕ:℘1,℘2×℘3,℘4⟶ℝ be a non-negative, integrable, and symmetric function of two variables:(85)ϕx,y=ϕ11/℘1+1/℘2−1/x,y,ϕx,11/℘1+℘3/℘4−1/y,ϕ11/℘1+1/℘2−1/x,11/℘1+℘3/℘4−1/y,
then the following inclusion holds:(86)Θ2℘1℘2℘1+℘2,2℘3℘4℘3+℘4α,ρ,wI1/℘2+,1/℘4+σ,kϕ∘g1℘1,1℘3+α,ρ,wI1/℘2+,1/℘3−σ,kϕ∘g1℘1,1℘4+α,ρ,wI1/℘1−,1/℘4+σ,kϕ∘g1℘2,1℘3+α,ρ,wI1/℘1−,1/℘3−σ,kϕ∘g1℘2,1℘4⊇α,ρ,wI1/℘2+,1/℘4+σ,kΘ∘g1℘1,1℘3ϕ∘g1℘1,1℘3+α,ρ,wI1/℘2+,1/℘3−σ,kΘ∘g1℘1,1℘4ϕ∘g1℘1,1℘4ϕ∘g1℘1,1℘4+α,ρ,wI1/℘1−,1/℘4+σ,kΘ∘g1℘2,1℘3ϕ∘g1℘2,1℘3+α,ρ,wI1/℘1−,1/℘3−σ,kΘ∘g1℘2,1℘4ϕ∘g1℘2,1℘4⊇Θ℘1,℘3+Θ℘1,℘4+Θ℘2,℘3+Θ℘2,℘44α,ρ,wI1/℘2+,1/℘4+σ,kϕ∘g1℘1,1℘3+α,ρ,wI1/℘2+,1/℘3−σ,kϕ∘g1℘1,1℘4+α,ρ,wI1/℘1−,1/℘4+σ,kϕ∘g1℘2,1℘3+α,ρ,wI1/℘1−,1/℘3−σ,kϕ∘g1℘2,1℘4.
IfΘx,y is a interval-valued harmonically co-ordinated concave, then the above-given relation holds in a reverse direction.Proof.
We start our proof from relation (56) and multiplying both sides of it by tα1/k−1sα2/k−1ℛρ1,w1,α1σ1,kw1℘2−℘1/℘1℘2ρ1tρ1ℛρ2,w2,α2σ2,kw2℘4−℘3/℘3℘4ρ2sρ2ϕ℘1℘2/1−t℘1+tb,℘3℘4/1−s℘3+sd and taking integration with respect to t,s on 0,1×0,1, we have the following equation:(87)4Θ∗2℘1℘2℘1+℘2,2℘3℘4℘3+℘4∫01∫01tα1/k−1sα2/k−1ℛρ1,w1,α1σ1,kw1℘2−℘1℘1℘2ρ1tρ1ℛρ2,w2,α2σ2,kw2℘4−℘3℘3℘4ρ2sρ2ϕ℘1℘21−t℘1+tb,℘3℘41−s℘3+sddsdt≤∫01∫01tα1/k−1sα2/k−1ℛρ1,w1,α1σ1,kw1℘2−℘1℘1℘2ρ1tρ1ℛρ2,w2,α2σ2,kw2℘4−℘3℘3℘4ρ2sρ2ϕ℘1℘21−t℘1+tb,℘3℘41−s℘3+sd×Θ∗℘1℘2t℘1+1−t℘2,℘3℘4s℘3+1−s℘4+Θ∗℘1℘2t℘1+1−t℘2,℘3℘41−s℘3+sdΘ∗℘1℘21−t℘1+tb,℘3℘4s℘3+1−s℘4+Θ∗℘1℘21−t℘1+tb,℘3℘41−s℘3+sddsdt.
After simplification and making use of symmetry ofϕx,y, we obtain the following equation:(88)Θ∗2℘1℘2℘1+℘2,2℘3℘4℘3+℘4α,ρ,wI1/℘2+,1/℘4+σ,kϕ∘g1℘1,1℘3+α,ρ,wI1/℘2+,1/℘3−σ,kϕ∘g1℘1,1℘4+α,ρ,wI1/℘1−,1/℘4+σ,kϕ∘g1℘2,1℘3+α,ρ,wI1/℘1−,1/℘3−σ,kϕ∘g1℘2,1℘4≤α,ρ,wI1/℘2+,1/℘4+σ,kΘ∗∘g1℘1,1℘3ϕ∘g1℘1,1℘3+α,ρ,wI1/℘2+,1/℘3−σ,kΘ∗∘g1℘1,1℘4ϕ∘g1℘1,1℘4ϕ∘g1℘1,1℘4+α,ρ,wI1/℘1−,1/℘4+σ,kΘ∗∘g1℘2,1℘3ϕ∘g1℘2,1℘3+α,ρ,wI1/℘1−,1/℘3−σ,kΘ∗∘g1℘2,1℘4ϕ∘g1℘2,1℘4.
Multiplying both sides of (57) by. tα1/k−1sα2/k−1ℛρ1,w1,α1σ1,kw1℘2−℘1/℘1℘2ρ1tρ1ℛρ2,w2,α2σ2,kw2℘4−℘3/℘3℘4ρ2sρ2ϕ℘1℘2/1−t℘1+tb,℘3℘4/1−s℘3+sd and taking integration with respect to t,s on 0,1×0,1, we have the following equation:(89)4Θ∗2℘1℘2℘1+℘2,2℘3℘4℘3+℘4∫01∫01tα1/k−1sα2/k−1ℛρ1,w1,α1σ1,kw1℘2−℘1℘1℘2ρ1tρ1ℛρ2,w2,α2σ2,kw2℘4−℘3℘3℘4ρ2sρ2ϕ℘1℘21−t℘1+tb,℘3℘41−s℘3+sddsdt≥∫01∫01tα1/k−1sα2/k−1ℛρ1,w1,α1σ1,kw1℘2−℘1℘1℘2ρ1tρ1ℛρ2,w2,α2σ2,kw2℘4−℘3℘3℘4ρ2sρ2ϕ℘1℘21−t℘1+tb,℘3℘41−s℘3+sd×Θ∗℘1℘2t℘1+1−t℘2,℘3℘4s℘3+1−s℘4+Θ∗℘1℘2t℘1+1−t℘2,℘3℘41−s℘3+sdΘ∗℘1℘21−t℘1+tb,℘3℘4s℘3+1−s℘4+Θ∗℘1℘21−t℘1+tb,℘3℘41−s℘3+sddsdt.
After simplification, we obtain the following equation:(90)Θ∗2℘1℘2℘1+℘2,2℘3℘4℘3+℘4α,ρ,wI1/℘2+,1/℘4+σ,kϕ∘g1℘1,1℘3+α,ρ,wI1/℘2+,1/℘3−σ,kϕ∘g1℘1,1℘4+α,ρ,wI1/℘1−,1/℘4+σ,kϕ∘g1℘2,1℘3+α,ρ,wI1/℘1−,1/℘3−σ,kϕ∘g1℘2,1℘4≥1k2ℛρ1,w1,α1+kσ1,kw1℘2−℘1/℘1℘2ρ1ℛρ2,w2,α2+kσ2,kw2℘4−℘3/℘3℘4ρ2℘1℘2℘2−℘1α1/k℘3℘4℘4−℘3α2/k×α,ρ,wI1/℘2+,1/℘4+σ,kΘ∗∘g1℘1,1℘3ϕ∘g1℘1,1℘3+α,ρ,wI1/℘2+,1/℘3−σ,kΘ∗∘g1℘1,1℘4ϕ∘g1℘1,1℘4ϕ∘g1℘1,1℘4+α,ρ,wI1/℘1−,1/℘4+σ,kΘ∗∘g1℘2,1℘3ϕ∘g1℘2,1℘3+α,ρ,wI1/℘1−,1/℘3−σ,kΘ∗∘g1℘2,1℘4ϕ∘g1℘2,1℘4.
In the context of (88) and (90), the first inclusion preserve.
The proof of the second relation proceeded from (78) and multiplying both sides by tα1/k−1sα2/k−1ℛρ1,w1,α1σ1,kw1℘2−℘1/℘1℘2ρ1tρ1ℛρ2,w2,α2σ2,kw2℘4−℘3/℘3℘4ρ2sρ2ϕ℘1℘2/1−t℘1+tb,℘3℘4/1−s℘3+sd and taking integration with respect to t,s on 0,1×0,1, then we have the following equation:(91)∫01∫01tα1/k−1sα2/k−1ℛρ1,w1,α1σ1,kw1℘2−℘1℘1℘2ρ1tρ1ℛρ2,w2,α2σ2,kw2℘4−℘3℘3℘4ρ2sρ2ϕ℘1℘21−t℘1+tb,℘3℘41−s℘3+sdΘ∗℘1℘21−t℘1+tb,℘3℘41−s℘3+sd+Θ∗℘1℘21−t℘1+tb,℘3℘4s℘3+1−s℘4+Θ∗℘1℘2t℘1+1−t℘2,℘3℘41−s℘3+sd+Θ∗℘1℘2t℘1+1−t℘2,℘3℘4s℘3+1−s℘4dsdt.≤Θ∗℘1,℘3+Θ∗℘1,℘4+Θ∗℘2,℘3+∗℘2,℘4×∫01∫01tα1/k−1sα2/k−1ℛρ1,w1,α1σ1,kw1℘2−℘1℘1℘2ρ1tρ1ℛρ2,w2,α2σ2,kw2℘4−℘3℘3℘4ρ2sρ2×ϕ℘1℘21−t℘1+tb,℘3℘41−s℘3+sddsdt.
From the above-given inequality, we infer(92)α,ρ,wI1/℘2+,1/℘4+σ,kΘ∗∘g1℘1,1℘3ϕ∘g1℘1,1℘3+α,ρ,wI1/℘2+,1/℘3−σ,kΘ∗∘g1℘1,1℘4ϕ∘g1℘1,1℘4ϕ∘g1℘1,1℘4+α,ρ,wI1/℘1−,1/℘4+σ,kΘ∗∘g1℘2,1℘3ϕ∘g1℘2,1℘3+α,ρ,wI1/℘1−,1/℘3−σ,kΘ∗∘g1℘2,1℘4ϕ∘g1℘2,1℘4≤Θ∗℘1,℘3+Θ∗℘1,℘4+Θ∗℘2,℘3+Θ∗℘2,℘4α,ρ,wI1/℘2+,1/℘4+σ,kϕ∘g1℘1,1℘3+α,ρ,wI1/℘2+,1/℘3−σ,kϕ∘g1℘1,1℘4+α,ρ,wI1/℘1−,1/℘4+σ,kϕ∘g1℘2,1℘3+α,ρ,wI1/℘1−,1/℘3−σ,kϕ∘g1℘2,1℘4.
Now, we calculate the following inequality essentially using harmonically co-ordinated concavity ofΘ∗.(93)α,ρ,wI1/℘2+,1/℘4+σ,kΘ∗∘g1℘1,1℘3ϕ∘g1℘1,1℘3+α,ρ,wI1/℘2+,1/℘3−σ,kΘ∗∘g1℘1,1℘4ϕ∘g1℘1,1℘4ϕ∘g1℘1,1℘4+α,ρ,wI1/℘1−,1/℘4+σ,kΘ∗∘g1℘2,1℘3ϕ∘g1℘2,1℘3+α,ρ,wI1/℘1−,1/℘3−σ,kΘ∗∘g1℘2,1℘4ϕ∘g1℘2,1℘4≤Θ∗℘1,℘3+Θ∗℘1,℘4+Θ∗℘2,℘3+Θ∗℘2,℘4α,ρ,wI1/℘2+,1/℘4+σ,kϕ∘g1℘1,1℘3+α,ρ,wI1/℘2+,1/℘3−σ,kϕ∘g1℘1,1℘4+α,ρ,wI1/℘1−,1/℘4+σ,kϕ∘g1℘2,1℘3+α,ρ,wI1/℘1−,1/℘3−σ,kϕ∘g1℘2,1℘4.
(92) and (93) yield the required relation.


## 4. Conclusion

In the recent study, we have interrogated the Hermite-Hadamard’s and its Fejer type containments in connection with interval analysis. In order to investigate, we define the double integral Raina’s fractional integrals ink form, which are demonstrated by an example. Furthermore, we introduced the notion of interval-valued harmonically co-ordinated convex functions. Initially, we obtained generalized fractional inclusions in association with harmonically interval-valued convex functions and their numeric verifications, then we established some two dimensional Hermite-Hadamard’s and Hermite-Hadamard-Fejer like inclusions involving co-ordinated harmonically interval-valued convexity. As the theory of convexity and interval analysis has great applications in optimization and error analysis. I hope the technique of the paper will inspire the interested readers and will stimulate the further research in the following direction.



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