# Hermite–Hadamard and Fractional Integral Inequalities for Interval-Valued Generalizedp-Convex Function

Authors: Zhengbo Li; Kamran; Muhammad Sajid Zahoor; Huma Akhtar
Journal: Journal of Mathematics
 (2020)

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## Abstract

In the present paper, the new interval-valued generalizedp convex functions are introduced. By using the notion of interval-valued generalized p convex functions and some auxiliary results of interval analysis, new Hermite–Hadamard and Fejér type inequalities are proved. The established results are more generalized than existing results in the literature. Moreover, fractional integral inequality for this generalization is also established.


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## Body

## 1. Introduction

The theory of interval analysis introduced in numerical analysis by Moore in [1] had rapid development in last few decades. In computational problems, one can program a computer to find interval that contains the exact answer to the problems. Also, interval analysis provides rigorous enclosure of solution to the model equation. Moreover, the interval analysis is widely used in chemical and structured engineering, economics, control circuitry design, robotics, beam physics, behavioral ecology, constraint satisfaction, computer graphics, signal processing, asteroid orbits and global optimization [2], neural network output optimization [3], and many others. For interesting fundamental results, we refer [2, 4–8] to the readers.Since the convexity play a vital role not only in convex analysis but also in almost all branches of mathematics. The famous inequalities in convex analysis are Jensen type, Hermite–Hadamard type, Fejér type, Ostrowski type, etc. For deeper insight about these inequalities, we refer [9–16] and references therein.Furthermore, the definition of classical convexity enables us to tackle modern applied problems, because most of the problems are nonconvex in nature. Famous generalization of convexity are logarithmic convexity [16], p-convexity [17], η convexity [18], h-convexity [19], modified h-convexity [15], etc. For example, in [20], Nchama et al. used the CaputoFabrizio fractional integral and gave some new inequalities. For detailed applications of fractional calculus, we refer [21–28] to the readers and references therein.In order to introduce the main definition of this paper, let us recall few generalizations of convexity present in the literature.Definition 1 (see [17]).
An intervalI1 is p-convex set, if for any x1,x2∈I1, α1∈0,1, we have(1)α1x1p+1−α1x2p1/p∈I1,where p=2k1+1 or p=n1/m1, n1=2r1+1, m1=2t1+1, and k1,r1,t1∈N.Definition 2 (see [17]).
A mappingf defined from a p-convex set I1 to ℝ is said to be p-convex function, if(2)fα1p+1−α1x2p1/p≤α1fx1+1−α1fx2,for each x1,x2∈I1 and α1∈0,1 hold.Definition 3 (see [29]).
The mappingf defined from I1 to ℝ is said to be η-convex if(3)fα1x1+1−α1x2≤fx2+α1ηfx1,fx2holds with respect to η:B1×B1⟶B2 for appropriate B1,B2⊆ℝ, and for each x1,x2∈I1, α1∈0,1.Definition 4 (see [29]).
A mapping is nonnegatively homogeneous ifηαx1,αx2=αηx1,x2 for each x1,x2∈ℝ and α≥0.Definition 5 (see [30]).
A mappingf defined from a p-convex set I1 to ℝ is said to be generalized p convex function, if(4)fα1x1p+1−α1x2p1/p≤fx2+α1ηfx1,fx2holds for η:B1×B1⟶B2 be a bifunction for appropriate B1,B2⊆ℝ and for each x1,x2∈I1 and α1∈0,1.
Now, we present the concept of interval-valued generalizedp convex function.Definition 6.
A mappingf defined from a p-convex set I1 to ℝ is said to be interval-valued generalized p-convex function, if(5)fα1x1p+1−α1x2p1/p⊇fx2+α1ηfx1,fx2holds for η:B1×B1⟶B2 be a bifunction for appropriate B1,B2⊆ℝ and for each x1,x2∈I1 and α1∈0,1.
Here, forf¯=f¯ and p = 1, (5) is an η-convexity, for f¯=f¯ and ηx1,x2=x1−x2 (5) is p-convexity, and for p = 1 and ηx1,x2=x1−x2, (5) is classical convexity.
This article is in the direction of the concepts and some results discussed in [30], but now we use interval-valued generalized p-convex function instead of generalized p convex function. After this introduction, in Section 2, we develop some basic properties of interval-valued generalized p convex functions. InSection 3, we make some new inequalities like Hermite–Hadamard’s and Fejér type for interval-valued generalized p convex functions.


## 2. Basic Results

Here, we derive some operations which preserves interval-valued generalizedp convex function.Proposition 1.
Letf1 and f2 be two interval-valued generalized p convex functions:(1)
Ifη is additive, then f1+f2 is interval-valued generalized p convex(2)
Ifη is nonnegatively homogeneous, then λf1 is interval-valued generalized p convex for any λ≥0.Proof.
The proof is straightforward.Theorem 1.
Letf:r,s⟶RI+ be an interval-valued function such that fλ=f¯λ,f¯λ, then f∈SXη,p,r,s,RI+ iff f¯∈SXη,p,r,s,RI+ and f¯∈SVη,p,r,s,RI+.Proof.
Letf∈SXη,p,r,s,RI+, then for any x,y∈r,sλ∈0,1, we have(6)fy+tηfx,fy⊆fλxp+1−λyp,that is,(7)f¯y+tηf¯x,f¯y,f¯y+tηf¯x,f¯y⊆f¯λxp+1−λyp1/p,f¯λxp+1−λyp1/p.
It follows that(8)f¯y+tηf¯x,f¯y≥f¯λxp+1−λyp1/p,f¯y+tηf¯x,f¯y≤f¯λxp+1−λyp1/p.
This shows that(9)f¯∈SXη,p,r,s,RI+,f¯∈SVη,p,r,s,RI+.
Conversely, suppose that(10)f¯∈SXη,p,r,s,RI+,f¯∈SVη,p,r,s,RI+.
Then, it follows thatf∈SXη,p,r,s,RI+. This completes the proof.Theorem 2.
Letf:r,s⟶RI+ be an interval-valued function such that fλ=f¯λ,f¯λ, then f∈SVη,p,r,s,RI+ if f¯∈SVη,p,r,s,RI+ and f¯∈SXη,p,r,s,RI+.Proof.
The proof is similar to that of Theorem1.


## 3. Hermite–Hadamard-Type Inequality for Interval-Valued Generalizedp Convex Function

In the following theorem, we present the Hermite–Hadamard type inequality for interval-valued generalizedp convex function.Theorem 3.
Letf:I⟶ℝ be an interval-valued generalized p convex function for ξ1,ξ2∈I with condition ξ1<ξ2, then we obtain the following inequality:(11)fξ1p+ξ2p21/p−p2ξ2p−ξ1p∫ξ1ξ2xp−1ηfξ1p+ξ2p−xp1/p,fxdx⊇pξ2p−ξ1p∫ξ1ξ2xp−1fxdx⊇fξ1+fξ22+14ηfξ1,fξ2+ηfξ2,fξ1.Proof.
Takeup=tξ1p+1−tξ2p and vp=1−tξ1p+tξ2p, it implies(12)ξ1p+ξ2p2=up+vp2.
So,(13)fξ1p+ξ2p21/p=fup+vp21/p.
By definition of interval-valued generalizedp convex functions, we have(14)fξ1p+ξ2p21/p=f12tξ1p+1−tξ2p1/pp+121−tξ1p+tξ2p1/pp1/p⊇f1−tξ1p+tξ2p1/p+12ηftξ1p+1−tξ2p1/p,f1−tξ1p+tξ2p1/p.
Now, by the definition of interval(15)fξ1p+ξ2p21/p=f¯ξ1p+ξ2p21/p,f¯ξ1p+ξ2p21/p,
we have(16)f¯ξ1p+ξ2p21/p,f¯ξ1p+ξ2p21/p,⊇f¯1−tξ1p+ξ2p1/p+12ηf¯tξ1p+1−tξ2p1/p,f¯1−tξ1p+tξ2p1/p,f¯1−tξ1p+tξ2p1/p+12ηf¯tξ1p+1−tξ2p1/p,f¯1−tξ1p+tξ2p1/p.
It follows that(17)f¯ξ1p+ξ2p21/p≤f¯1−tξ1p+tξ2p1/p,+12ηf¯tξ1p+1−tξ2p1/p,f¯1−tξ1p+tξ2p1/p,(18)f¯ξ1p+ξ2p21/p≥f¯1−tξ1p+tξ2p1/p+12ηf¯tξ1p+1−tξ2p1/p,f¯1−tξ1p+tξ2p1/p.Integrating (17) with respect to “x” on [0, 1], we get(19)f¯ξ1p+ξ2p21/p≤∫01f¯1−tξ1p+tξ2p1/pdt+12∫01ηf¯tξ1p+1−tξ2p1/p,f¯1−tξ1p+tξ2p1/pdt,
which implies(20)f¯ξ1p+ξ2p21/p−p2ξ2p−ξ1p∫ξ1ξ2xp−1ηf¯ξ1p+ξ2p−xp1/p,f¯xdx,≤pξ2p−ξ1p∫ξ1ξ2xp−1f¯xdx.
Now,(21)∫ξ1ξ2xp−1f−¯xdx=ξ2p−ξ1pp∫01f¯tξ1p+1−tξ2p1/pdt,≤ξ2p−ξ1ppf¯ξ2+∫01tηf¯ξ1,f¯ξ2dt,pξ2p−ξ1p∫ξ1ξ2xp−1f¯xdx≤f¯ξ2+∫01tηf¯ξ1,f¯ξ2dt.
Similarly,(22)pξ2p−ξ1p∫ξ1ξ2xp−1f¯xdx≤f¯ξ1+∫01tηf¯ξ2,f¯ξ1dt.
Adding (21) and (22), we obtain(23)pξ2p−ξ1p∫ξ1ξ2xp−1f¯xdx≤f¯ξ1+f¯ξ22+14ηf¯ξ1,f¯ξ2+ηf¯ξ2,f¯ξ1.
Now, Integrating (18) with respect to “x” on [0, 1], we get(24)f¯ξ1p+ξ2p21/p≥∫01f¯1−tξ1p+tξ2p1pdt+12∫01ηf¯tξ1p+1−tξ2p1/p,f¯1−tξ1p+tξ2p1/pdt,
which implies(25)f¯ξ1p+ξ2p21/p−p2ξ2p−ξ1p∫ξ1ξ2xp−1ηf¯ξ1p+ξ2p−xp1/p,f¯xdx,≥pξ2p−ξ1p∫ξ1ξ2xp−1f¯xdx.
Now,(26)∫ξ1ξ2xp−1f−¯xdx=ξ2p−ξ1pp∫01f¯tξ1p+1−tξ2p1/pdt,≥ξ2p−ξ1ppf¯ξ2+∫01tηf¯ξ1,f¯ξ2dt,pξ2p−ξ1p∫ξ1ξ2xp−1f¯xdx≤f¯ξ2+∫01tηf¯ξ1,f¯ξ2dt.
Similarly,(27)pξ2p−ξ1p∫ξ1ξ2xp−1f¯xdx≤f¯ξ1+∫01tηf¯ξ2,f¯ξ1dt.
Adding (26) and (27), we obtain(28)pξ2p−ξ1p∫ξ1ξ2xp−1f¯xdx≥f¯ξ1+f¯ξ22,+14ηf¯ξ1,f¯ξ2+ηf¯ξ2,f¯ξ1.
Combining (20) and (21), we obtain(29)f¯ξ1p+ξ2p21/p−p2ξ2p−ξ1p∫ξ1ξ2xp−1ηf¯ξ1p+ξ2p−xp1/p,f¯xdx,≤pξ2p−ξ1p∫ξ1ξ2xp−1f¯xdx,≤f¯ξ1+f¯ξ22+14ηf¯ξ1,f¯ξ2+ηf¯ξ2,f¯ξ1.
Combining (25) and (28), we obtain(30)f¯ξ1p+ξ2p21/p−p2ξ2p−ξ1p∫ξ1ξ2xp−1ηf¯ξ1p+ξ2p−xp1/p,f¯xdx,≥pξ2p−ξ1p∫ξ1ξ2xp−1f¯xdx,≥f¯ξ1+f¯ξ22+14ηf¯ξ1,f¯ξ2+ηf¯ξ2,f¯ξ1.
Equations (29) and (30) follows:(31)f¯ξ1p+ξ2p21/p−p2ξ2p−ξ1p∫ξ1ξ2xp−1ηf¯ξ1p+ξ2p−xp1/p,f¯xdx,f¯ξ1p+ξ2p21/p−p2ξ2p−ξ1p∫ξ1ξ2xp−1ηf¯ξ1p+ξ2p−xp1/p,f¯xdx,⊇pξ2p−ξ1p∫ξ1ξ2xp−1f¯xdx,pξ2p−ξ1p∫ξ1ξ2xp−1f¯xdx,⊇f¯ξ1+f¯ξ22+14ηf¯ξ1,f¯ξ2+ηf¯ξ2,f¯ξ1,f¯ξ1+f¯ξ22,+14ηf¯ξ1,f¯ξ2+ηf¯ξ2,f¯ξ1.which completely follows (11)Remark 1.
By puttingf¯=f¯ and p=1, (11) becomes Hermite–Hadamard type inequality for η-convexity [18].Remark 2.
By puttingf¯=f¯ and ηξ1,ξ2=ξ1−ξ2 in (11), we obtain Hermite–Hadamard type inequality for p-convexity [17].Remark 3.
By puttingf¯=f¯, p  = 1 and ηξ1,ξ2=ξ1−ξ2 in (11), we get classical Hermite–Hadamard type inequality for convex functions.Example 1.
Considerηx,y=x−y,ξ1,ξ2=−1,1 and f:r,s⟶R+ be defined by fλ=λp,4−eλp with p as an odd number, then we have(32)fξ1p+ξ2p21/p−p2ξ2p−ξ1p∫ξ1ξ2xp−1ηfξ1p+ξ2p−xp1/p,fxdx=0,3,(33)pξ2p−ξ1p∫ξ1ξ2xp−1fxdx=pξ2p−ξ1p∫ξ1ξ2xp−1xp,4−expdx,=p2∫−11xp−1xpdx,p2∫−11xp−14−expdx.
Putz=xp and simplify, we get(34)pξ2p−ξ1p∫ξ1ξ2xp−1fxdx=120,8−e1−e−1=0,4−e1−e−12,(35)fξ1+fξ22+14ηfξ1,fξ2+ηfξ2,fξ1,=ξ1p,4−eξ1p+ξ2p,4−eξ2p2+14fξ1−fξ2+fξ2−fa,=−1,4−e−1+1,4−e12=0,4−e1+e−12.
Combining (32), (34), and (35), we get(36)0,3⊇0,4−e1−e−12⊇0,4−e1+e−12.


## 4. Fejér-Type Inequality for Interval-Valued Generalizedp Convex Function

Now, we develop Fejér type inequality for interval-valued generalizedp convex functions.Theorem 4.
Letfandg be nonnegative interval-valued generalized p convex functions ξ1,ξ2∈Iξ1<ξ2 such that fg∈L1ξ1,ξ2, then(37)pξ2p−ξ1p∫ξ1ξ2xp−1fxgxdx⊇Mξ1,ξ2+12Nξ1,ξ2,where(38)Mξ1,ξ2=fξ2gξ2+13ηfξ1,fξ2ηgξ1,gξ2,Nξ1,ξ2=fξ2ηgξ1,gξ2+gξ2ηfξ1,fξ2.Proof.
Sincef and g are interval-valued generalized p convex functions, we have(39)ftξ1p+1−tξ2p1/p⊇fξ2+tηfξ1,fξ2,gtξ1p+1−tξ2p1/p⊇gξ2+tηgξ1,gξ2,for all t∈ξ1,ξ2. Since f and g are nonnegative,(40)ftξ1p+1−tξ2p1/pgtξ1p+1−tξ2p1/p,⊇fξ2gξ2+tfξ2ηgξ1,gξ2+tgξ2ηfξ1,fξ2,+t2ηfξ1,fξ2ηgξ1,gξ2.
By the definition of interval, we have(41)f¯tξ1p+1−tξ2p1/pg¯tξ1p+1−tξ2p1/p,f¯tξ1p+1−tξ2p1/pg¯tξ1p+1−tξ2p1/p,⊇f¯ξ2g¯ξ2+tf¯ξ2ηg¯ξ1,g¯ξ2+tg¯ξ2ηf¯ξ1,f¯ξ2+t2ηf¯ξ1,f¯ξ2ηg¯ξ1,g¯ξ2,f¯ξ2g¯ξ2+tf¯ξ2ηg¯ξ1,g¯ξ2+tg¯ξ2ηf¯ξ1,f¯ξ2+t2ηf¯ξ1,f¯ξ2ηg¯ξ1,g¯ξ2.
It follows(42)f¯tξ1p+1−tξ2p1/pg¯tξ1p+1−tξ2p1/p,≤f¯ξ2g¯ξ2+tf¯ξ2ηg¯ξ1,g¯ξ2+tg¯ξ2ηf¯ξ1,f¯ξ2,+ηf¯ξ1,f¯ξ2ηf¯ξ1,f¯ξ2dt.(43)f¯tξ1p+1−tξ2p1/pg¯tξ1p+1−tξ2p1/p,≤f¯ξ2g¯ξ2+tf¯ξ2ηg¯ξ1,g¯ξ2+tg¯ξ2ηf¯ξ1,f¯ξ2,+t2ηf¯ξ1,f¯ξ2ηf¯ξ1,f¯ξ2dt.
Integrating (42) over 0,1, we obtain the following inequality:(44)∫01f¯tξ1p+1−tξ2p1/pg¯tξ1p+1−tξ2p1/pdt,≤∫01f¯ξ2g¯ξ2dt+∫01tf¯ξ2ηg¯ξ1,g¯ξ2dt+∫01tg¯ξ2ηf¯ξ1,f¯ξ2dt,+∫01t2ηf¯ξ1,f¯ξ2ηf¯ξ1,f¯ξ2dt.
Settingx=tξ1p+1−tξ2p1/p, we get(45)pξ2p−ξ1p∫ξ1ξ2xp−1f¯xg¯xdx,≤f¯ξ2g¯ξ2+12f¯ξ2ηg¯ξ1,g¯ξ2+12g¯ξ2ηf¯ξ1,f¯ξ2,+13ηf¯ξ1,f¯ξ2ηg¯ξ1,g¯ξ2.
Integrating (43) over (0, 1), we get(46)∫01f¯tξ1p+1−tξ2p1/pg¯tξ1p+1−tξ2p1/pdt,≥∫01f¯ξ2g¯ξ2dt+∫01tf¯ξ2ηg¯ξ1,g¯ξ2dt+∫01tg¯ξ2ηf¯ξ1,f¯ξ2dt,+∫01t2ηf¯ξ1,f¯ξ2ηf¯ξ1,f¯ξ2dt.
Settingx=tξ1p+1−tξ2p1/p, we get(47)pξ2p−ξ1p∫ξ1ξ2xp−1f¯xg¯xdx,≤f¯ξ2g¯ξ2+12f¯ξ2ηg¯ξ1,g¯ξ2+12g¯ξ2ηf¯ξ1,f¯ξ2,+13ηf¯ξ1,f¯ξ2ηg¯ξ1,g¯ξ2.
Using inequality, (47) we get(48)pξ2p−ξ1p∫ξ1ξ2xp−1f¯xg¯xdx,pξ2p−ξ1p∫ξ1ξ2xp−1f¯xg¯xdx,⊇f¯ξ2g¯ξ2+12f¯ξ2ηg¯ξ1,g¯ξ2+12g¯ξ2ηf¯ξ1,f¯ξ2,+13ηf¯ξ1,f¯ξ2ηg¯ξ1,g¯ξ2,f¯ξ2g¯ξ2+12f¯ξ2ηg¯ξ1,g¯ξ2,+12g¯ξ2ηf¯ξ1,f¯ξ2+13ηf¯ξ1,f¯ξ2ηg¯ξ1,g¯ξ2,pξ2p−ξ1p∫ξ1ξ2xp−1fxgxdx,⊇fξ2gξ2+12fξ2ηgξ1,gξ2+12gξ2ηfξ1,fξ2,+13ηfξ1,fξ2ηgξ1,gξ2.
Then, we obtain the inequality (37).Remark 4.
If we putf¯=f¯, p=1 and ηx,y=x−y in (37), then it reduces to classical convex functions.


## 5. Fractional Hermite–Hadamard-Type Inequalities for Interval-Valued Generalizedp Convex Functions

The fractional inequalities has applications in every field of science and engineering. The new fractional integral inequalities in analysis are always appreciable. Because of the wide applications of Hermite–Hadamard-type inequalities and fractional integrals, many researchers extended their studies to Hermite–Hadamard-type inequality involving fractional integral inequalities. For fractional integral inequalities for interval-valued function, we suggest the reader to refer [31, 32].Definition 7 (see [33–35]).
Letϕ∈La,b. The right-hand side and left-hand side RiemannLiouville fractional integral of order α>0 with b>a>0 are defined by(49)Ja+αϕx=1Γα∫axx−kα−1ϕkdk,x>a,Jb−αϕx=1Γα∫xbk−xα−1ϕkdk,x<b,respectively, where Γα is the Gamma function defined as ΓΓα=∫0∞e−kkα−1dk.
It is to be noted thatJα+0ϕx=Jb−0ϕx=ϕx
Reimann integral is reduced as classical integral forα=1.Definition 8.
letp∈ℝ/0. A function w:a,b⊂0,∞⟶ℝ is said to be p-symmetric with respect to ap+bp/21/p if wx=wap+bp−xp1/p and holds for all x,y∈I=a,b.
Following lemma will help us in obtaining our fractional integrals inequalities which can be found in [36].Lemma 1.
Letp∈ℝ/0,α>0 and w:a,b⊂0,∞⟶ℝ is integrable, p-symmetric with respect to ap+bp/21/p:(1)
Ifp>0,(50)Jap+αwogbp=Jbp−αwogap=12Jap+αwogbp+Jbp−αwogap.withgx=x1/p,x∈ap,bp.(2)
Ifp<0,(51)Jbp+αwogap=Jap−αwogap=12Jbp+αwogbp+Jap−αwogbp,with gx=x1/p,x∈bp,ap.Now, we are ready to develop the Fractional Hermite–Hadamard-type inequalities for interval-valued generalizedp convex functions.Theorem 5.
Letϕ:I⟶ℝ be generalized p convex function and provided η is bounded above on ϕI×ϕI and ϕ∈La,b. Then, following fractional integral inequality holds, if p∈ℝ/0 and p>0:(52)ϕap+bp21/p−2Mη,⊇Γα+1bp−apα21−αJap+bp/2+αϕogbp+Jap+bp/2−αϕogap,⊇ϕa+ϕb2+2Nη.Proof.
Letϕ be a generalized p convex function with p≥0 and η is bounded above by Mη.
Takex=kap+1−kbp1/p and y=kbp+1−kap1/p.
Since(53)ϕxp+yp21/p−Mη⊇ϕx+ϕy2+Mη,(53) becomes(54)ϕap+bp21/p−Mη⊇ϕkap+1−kbp1/p2+ϕkbp+1−kap1/p2+Mη.
Multiplying both sides of (54) by kα−1 and then integrating the resulting inequality with respect to k over 0,1/2, we obtain(55)∫01/2ϕap+bp21/pkα−1dk−∫01/2Mηkα−1dk,⊇12∫01/2ϕkap+1−kbp1/pkα−1dk+12∫01/2ϕkbp+1−kap1/pkα−1dk,+∫01/2Mηkα−1dk.
By definition of RiemannLiouville integrable function withgx=x1/p, we obtain(56)ϕap+bp21/p−2Mη⊇Γα+1bp−apα21−αJap+bp/2+αϕogbp+Jap+bp/2−αϕogap.which is the left-hand side of theorem (56).
To prove the right-hand side, we takex=kap+1−kbp1/p and y=kbp+1−kap1/p:(57)ϕkap+1−kbp1/p⊇ϕb+kηϕa,ϕb,(58)ϕkbp+1−kap1/p⊇ϕa+kηϕb,ϕa.
Adding the (57) and (58) and multiplying the resulting inequality with 2kα−1 and integrating with respect to k over 0,1/2 we obtain(59)2∫01/2ϕkbp+1−kap1/pkα−1dk+2∫01/2ϕkbp+1−kap1/pkα−1dk,⊇2∫01/2ϕakα−1dk+2∫01/2ϕbkα−1dk,+ηϕb,ϕa+ηϕa,ϕa2αα+1.
By definition of RiemannLiouville integrable function, we get(60)2Γαbp−apαJap+bp/2+αϕogbp+Jap+bp/2−αϕogap⊇ϕaα2α−1+ϕbα2α−1+αNηα+12α−2.
Rearranging the above inequality, we get the right-hand side:(61)Γα+1bp−apα21−αJap+bp/2+αϕogbp+Jap+bp/2−αϕogap⊇ϕa+ϕb2+2Nη.
This completes the proof.Remark 5.
If we putp=1,ηx,y=x−y, and ϕ¯=ϕ¯, then we will get Hermite–Hadamard-type inequality for fractional function for classical convex function [37].


## 6. Conclusions

The convex functions and fractional calculus play an important role in applied sciences [38–43]. Here, the new interval-valued generalized convex functions are introduced. By using the notion of interval-valued generalized p convex functions and some auxiliary results of interval analysis, some new Hermite–Hadamard- and Fejér-type inequalities are presented. Our results can be considered as generalization of many existing results. Moreover, fractional integral inequality for this generalization is also established.



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