# Some Inequalities Related to Interval-Valuedηh-Convex Functions

Authors: Lei Chen; Muhammad Shoaib Saleem; Muhammad Sajid Zahoor; Rahat Bano
Journal: Journal of Mathematics
 (2021)

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## Abstract

Convexity plays an important role in many areas of mathematics, especially in the study of optimization problems where they are distinguished by a number of convenient properties. Our aim is to introduce a more extended version of convexity. In this paper, we introduced interval-valued generalizedηh convex function and proved Hermite–Hadamard-, Jensen-, and Ostrowski-type inequalities in this generalization. The presented results are generalizations of many existing results of literature.


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## Body

## 1. Introduction

In nonlinear programming and optimization theory, convexity plays an important role. There were three important areas of nonlinear analysis: monotone operator theory, convex analysis, and theory of nonexpensive mapping, but in early 1960, these theories get emerged. These areas got the attention of many researchers, and many connections have been identified between them over the past few years.It is always interesting and appreciable to generalize the definition of convexity from different aspects because it helps to tackle modern world problem. The famous generalizations of convexity areE-convex functions [1], α convex functions [2], ϕ-convex function [3], and convex vector [4].Consider a convex functionf1:J1⊆ℝ⟶ℝ, then the following inequality holds:(1)f1x1+y12≤1y1−x1∫x1y1f1xdx≤f1x1+f1y12,where x1,y1∈J1 and x1<y1.In literature, we can find many generalization of inequality (1) for different classes of convex functions, for example, Delavar and Dragomir [5] generalized (1) for η-convex functions and Varosanec in [6] generalized (1) for h−convexity. For other generalizations, we refer the readers to [7–9].In 1996, Moore in [10] introduced the interval analysis. Later on, Costa et al. [11] introduced generalized interval vector spaces and interval optimization. The convex functions in interval-valued calculus got attention of researchers due to its interesting geometric features and infimum properties [12–15]. This theory is also appealing for the applied engineers and programmes due to its applications in convex optimization [16–18]. To increase the reader’s interest in convex analysis and interval analysis, we recommend [6, 7, 19–25].In this paper, we introducedηh generalized convex function over interval-valued setting and proved Hermite–Hadamard-, Jensen-, and Ostrowski-type inequalities. The paper is organized as follows: in Section 2, we present some basic notions and definitions. Section 3 contains some basic properties interval-valued generalized ηh convex functions, and Section 4 contains Hermite–Hadamard-, Jensen-, and Ostrowski-type inequalities for interval-valued generalized ηh convex functions. The paper is concluded in Section 5.


## 2. Preliminaries

Throughout this paper,ℝI and ℝI+ denote the family of all intervals and positive intervals equipped with the algebraic operations “+” and “.,” respectively. The algebraic operations “+” and “.” are defined by(2)x1,x2+y1,y2=x1+y1,x2+y2,λ⋅x1,x2=λx1,λx2,if0≤λ,λx2,λx1,ifλ<0.Letf1,f2:V⟶ℝ are real functions such that f1x≤f2x for all x∈V, then the function F:V⊆ℝ⟶ℝI defined by(3)Fx=f1x,f2x,is said to be an interval-valued function.For any two intervalsx¯,x¯ and y¯,y¯, the Hausdorff distance is defined by(4)dx¯,x¯,y¯,y¯=maxx¯−y¯,x¯−y¯,and ℝI,d is called Hausdorff space.A set of numbersxi−1,yi,xii=1m is tagged partition P of x,y if(5)x=x0<x1<⋯<xm=y,and if xi−1≤yi≤xi for all i=1,2,…,m. Moreover, if we let Δxi=xi−xi−1 and Δxi≤δ for each i, then the partition is δ-fine. The family of all δ-fine partitions of x,y is represented by Pδ,x,y.GivenX∈Pδ,x,y, we define an integral sum of f:x,y⟶ℝI as follows:(6)Sf,X,δ,x,y=∑i=1mfyixi−xi−1.The four arithmetic operators (+, −, ., /) forr=r¯,r¯ and s=s¯,s¯ are defined as follows:(1)
r+s=r¯+s¯,r¯+s¯.(2)
r−s=r¯−s¯,r¯−s¯.(3)
r⋅s=minr¯.s¯,r¯.s¯,r¯.s¯,r¯.s¯,maxr¯.s¯,r¯.s¯,r¯.s¯,r¯.s¯.(4)
r/s=minr¯/s¯,r¯/s¯,r¯/s¯,r¯/s¯,maxr¯/s¯,r¯/s¯,r¯/s¯,r¯/s¯.Here,0∉s¯,s¯Now, we present some definitions and introduce interval-valued generalizedηh convex functions. From now to onward in this section, we take J1=r1,s1.Definition 1 (convex function [26]).
A functionf1:J1⟶ℝ is said to be convex, if(7)f1βx1+1−βy1≤βfx1+1−βfy1for all x1,y1∈J1 and β∈0,1.Definition 2 (η-convex function [5]).
A functionf1:J1⟶ℝ is said to be convex with respect to η:A1×A1⟶A2 for appropriate A1,A2⊆ℝ, if(8)f1βx1+1−βy1≤f1y1+βηf1x1,f1y1for all x1,y1∈J1 and β∈0,1.Example 1.
Consider a functionf1:ℝ+⟶ℝ+ defined by(9)f1x1=x1,if0≤x1≤1;1,ifx1>1.
Define(10)ηx1=x1+y1,ifx1≤y1,2x1+y1,ifx1>y1,for all x1,y1∈ℝ+, then f is an η-convex function but not convex.Definition 3 (h-convex function [6]).
Leth:I1⟶ℝ is a nonnegative function, where 0,1⊆I1 and h≠0. A function f1:J1⟶ℝ is said to be h-convex if f1 is nonnegative and(11)f1βx1+1−βy1≤hβf1x1+h1−βf1y1,for all x1,y1∈J1 and β∈0,1.
The class ofh-convex functions is denoted by SXh;J1.Definition 4 (modifiedh-convex function [27]).
Leth:I1⊆ℝ⟶ℝ be a nonnegative function, where 0,1⊆I1 and h≠0. A function f1:J1⊆ℝ⟶ℝ is said to be modified h-convex if(12)f1βx1+1−βy1≤hβf1x1+1−hβf1y1,for all x1,y1∈J1 and β∈0,1.Definition 5 (supermultiplicative function [28]).
A functionh:J1⟶ℝ is said to be supermultiplicative if(13)f1x1y1≥f1x1f1y1,for all x1,y1∈J1.Definition 6 (ηh-convex function [28]).
Leth:I1⟶ℝ be a nonnegative function, where 0,1⊆I1 and h≠0. A function f1:J1⟶ℝ is said to be ηh-convex if(14)f1βx1+1−βy1≤f1y1+hβηf1x1,f1y1for all x1,y1∈J1 and β∈0,1.Example 2.
Consider a functionf1:ℝ⟶ℝ defined by(15)f1x1=−x1,ifx1≥0;x1,ifx1<0.
Define a bifunctionη as ηx1,y1=−x1−y1, for all x1,y1∈ℝ− and ht=tk, where k≤1. Then, f1 is an ηh-convex function but not convex.Definition 7 (interval-valuedh-convex function [29]).
Leth:I1⟶ℝ be a nonnegative function, where 0,1⊆I1 and h≠0. A function f1:J1⟶ℝ is said to be interval-valued h-convex, if f1 is nonnegative and(16)f1βx1+1−βy1⊇hβf1x1+h1−βf1y1,for all x1,y1∈J1 and β∈0,1.
The class of interval-valuedh-convex functions is denoted by SXh;J1.Definition 8 (P-function [30]).
A functionf1:a1,b1⟶ℝ is said to be P-function if f1 is nonnegative and(17)f1βx1+1−βy1≤fx1+fy1,for all x1,y1∈J1 and β∈0,1.
Now, the definition of interval valuedηh-convex function is as follows.Definition 9.
(interval-valued generalizedηh-convex function). Let h:J1⟶ℝ be a nonnegative function, where 0,1⊆J1 and h≠0. Also, let η:A1×A1⟶A2 be a bifunction for appropriate A1,A2⊆ℝ. A function f1:J1⟶ℝ is said to be interval-valued generalized ηh convex if(18)f1βx1+1−βy1⊇f1y1+hβηf1x1,f1y1for all x1,y1∈J1 and β∈0,1.Remark 1.
Of course, (18) is η-convexity for hβ=β, modified h-convexity for ηx1,y1=x1−y1, and classical convex function for hβ=β and ηx1,y1=x1−y1 simultaneously.Remark 2.
If we takex1=y1 in (18), we get(19)hβηf1x1,f1y1≥0,for any x1,y1∈J1 and β∈0,1.Remark 3.
If we takeβ=1 and h1=1 in (18), we get(20)f1x1−f1y1≤ηf1x1,f1y1for any x1,y1∈J1. The second condition obviously implies the first.


## 3. Basic Results

In this section, we proved some basic results.Remark 4.
(operation which preserves generalizedηh convexity). Let f1,g1:J1⟶ℝ are two generalized ηh convex functions. Then, the following statements hold:(1)
Ifη is additive, then f1+g1:J1⟶ℝ is generalized ηh convex.(2)
Ifη is nonnegatively homogeneous, then for any β≥0, βf1:J1⟶ℝ is generalized ηh convex.Theorem 1.
Letg and h are two nonnegative functions defined on I with the property hβ⊇gβ. If f1 is interval valued ηh-convex function, then f1 is interval valued ηg-convex function.Proof.
For anyx1,y1∈J1 and β∈0,1, we have(21)f1βx1+1−βy1⊇f1y1+hβηf1x1,f1y1⊇f1y1+gβηf1x1,f1y1,which implies that f1 is interval-valued ηg-convex function.Theorem 2.
Letf1:r1,s1⟶ℝJ be an interval-valued function such that f1β=f1¯,f1¯. Then, f1∈SVηh,J1, iff f1¯∈SXηh,J1 and f1¯∈SXηh,J1.Proof.
Suppose thatf1∈SXηh,r1,s1,ℝJ+, for any x1,y1∈r1,s1 and β∈0,1, then we have(22)f1y1+hβηf1x1,f1y1⊆f1βx1+1−βy1,that is,(23)f1¯y1+hβηf1¯x1,f1¯y1,f1¯y1+hβηf1¯x1,f1¯y1⊆f1¯βx1+1−βy1,f1¯βx+1−βy.
It follows that(24)f1¯y1+hβηf1¯x1,f1¯y1≥f1¯βx1+1−βy1,f1¯y1+htηf1¯x1,y1≤f1¯βx1+1−βy1.
This shows thatf1¯∈SXηh,r1,s1,ℝJ+ andf1¯∈SVηh,r1,s1,ℝJ+.
Conversely, suppose thatf1¯∈SXηh,r1,s1,ℝJ+ and f1¯∈SVηh,r1,s1,ℝJ+, then from Definition 8 and inclusion (23), it follows that f1∈SXηh,r1,s1,ℝJ+.Theorem 3.
Letf1:r1,s1⟶ℝJ+ be an interval valued function such that f1β=f1¯β,f1¯β, then f1∈SVηh,J1 iff f1¯∈SXηh,J1 and f1¯∈SXηh,J1.Proof.
The proof is similar to that of Theorem2, so we omit it.


## 4. Main Results

Theorem 4.
(Hermite–Hadamard-type inequality). Letf1:r1,s1⟶ℝ be an interval valued generalized ηh convex function such that η is bounded above on f1r1,s1×f1r1,s1, i.e., ηx1,y1≤M. Then, we have(25)f1x1+y12−h12M,⊇1y1−x1∫x1y1f1xdx⊇fx1+f1y12+M∫01hβdβ.Proof.
Takeu=βx1+1−βy1 and v=1−βx1+βy1, then we have(26)x1+y12=u+v2.
So,(27)f1x1+y12=f1u+v2.
By definition of interval-valued generalizedηh convex functions, we have(28)f1x1+y12=f112βx1+1−βy1+121−βx1+βy1⊇f11−βx1+y1β+h12ηf1βx1+1−βb,f11−βx1+βy1,f1¯x1+y12,f1¯x1+y12⊇f1¯1−βx1+βy1+h12M,f1¯1−βx1+βy1+h12M,which follows that(29)f1¯x1+y12≤f1¯1−βx1+βy1+h12M,(30)f1¯x1+y12≥f1¯1−βx1+βy1+h12M.
Integrating (29) with respect to β on 0,1, we get(31)f1¯x1+y12−h12M≤1y1−x1∫x1y1f1¯xdx.
Now,(32)∫x1y1f1¯xdx=y1−x1∫01f1¯βx1+1−βy1dβ≤y1−x1f1¯y1+∫01hβηf1x1,f1y1dβ,1y1−x1∫x1y1f1¯xdx≤f1¯y1+M∫01hβdβ.
Similarly,(33)1y1−x1∫x1y1f1¯xdx≤f1¯x1+M∫01hβdβ.
Integrating (30) with respect to β on 0,1, we have(34)f1¯x1+y12−h12M≥1y1−x1∫x1y1f1¯xdx.
Now,(35)∫x1y1f1¯xdx=y1−x1∫01f1¯βx1+1−βy1dβ≥y1−x1f1¯y1+∫01hβηf1x1,f1y1dβ,implies(36)1y1−x1∫x1y1f1¯xdx≥f1¯y1+M∫01hβdβ.
Similarly,(37)1y1−x1∫x1y1f1¯xdx≥f1¯x1+M∫01hβdβ.
Adding (32) and (33), we get(38)1y1−x1∫x1y1f1¯xdx≤f¯x1+f1¯y12+M∫01hβdβ.
Adding (36) and (37), we get(39)1y1−x1∫x1y1f1¯xdx≥f¯x1+f1¯y12+M∫01hβdβ.
Combining (31) and (38), we get(40)f¯x1+y12−h12M≤1y1−x1∫x1y1f1¯xdx≤f1¯x1+f1¯y12+M∫01hβdβ.
Combining (34) and (39), we get(41)f¯x1+y12−h12M≥1y1−x1∫x1y1f1¯xdx≥f1¯x1+f1¯y12+M∫01hβdβ.
Combining (40) and (41), we get(42)f¯x1+y12−h12M,f1¯x1+y12−h12M⊇1y1−x1∫x1y1f1¯xdx,1y1−x1∫x1y1f1¯xdx,⊇f¯x1+f1¯y12+M∫01hβdβ,f1¯x1+f1¯y12+M∫01hβdβ,which leads us to our desired result (25).Remark 5.
If we chooseηx1,y1=x1−y1, hβ=β, and f1¯=f1¯ in (25), we get the classical Hermite–Hadamard-type inequality for convex function.Example 3.
Supposer1,s1=0,2 and hβ=β for all β∈0,1/2. Furthermore, suppose f1:r1,s1⟶ℝI+ be defined by(43)f1x=x2,10−ex,for all x∈0,2. Then, we have(44)f1a+b2−h12M⊇1b−a∫abfxdx⊇fa+fb2+M∫01htdt,which implies(45)f1x1+y12−h12M=1,7.2817.
Furthermore,(46)1y1−x1∫x1y1fxdx=1y1−x1∫x1y1x2,10−exdx=1.3333,6.8055,(47)f1x1+f1y12+M∫01hβdβ=a2,10−ea+b2,10−eb2+M∫01βdβ=2,5.8055.
Combining (45), (46), and (47), we have(48)1,7.2817⊇1.3333,6.8055⊇2,5.8055.
Consequently, we get Theorem4.Example 4.
Supposehβ=β for β∈0,1,r1,s1=−1,1. Also, suppose f1:r1,s1⟶ℝI+ be defined by f1λ=λp,4−eλp and p be an odd number, then we have(49)f1x1+y12−h12M=0,3which implies(50)1y1−x1∫x1y1f1xdx=1b−a∫x1y1λp,4−eλpdx,=0,4−12e−e−1,=0,2.8248,(51)f1x1+f1y12+M∫01hβdβ,=ap,4−eap+bp,4−ebp2+M∫01βdβ,=0,2.457.
Combining (49), (50), and (51), we have(52)0,3⊇0,2.8248⊇0,2.457.
Consequently, we get Theorem4.Theorem 5.
(Jensen-type inequality). Letf1:J1⟶ℝ be an ηh-convex function. Then, for x1,x2∈J1 and α1+α2=1, we have(53)f1α1x1+α2x2≤f1x2+hα1ηf1x1,f1x2.
Also, forx1,x2,…,xn∈I,∑i=1nαi=1, and Ti=∑j=1nαj, when n>2, we have(54)f1∑i=1nαixi=f1Tn−1∑i=1n−1αiTn−1xi+αnxn≤f1xn+hTn−1ηf1∑i=1n−1αiTn−1xi,f1xn.Theorem 6.
Letf1:J1⟶ℝ be an ηh-convex and η be nondecreasing, nonnegatively, and sublinear in first variable. If Ti=∑j=1nαj for j=1,…,n such that Tn=1, then(55)f1∑i=1nαixi⊇f1xn+∑i=1n−1hTiηf1xi,xi+1,…,xnwhere ηfxi,xi+1,…,xn=ηηfxi,xi+1,…,xn,fxn and ηf1x=f1x for all x∈J1.Proof.
Sinceη is nondecreasing, nonnegatively, and sublinear in the first variable, from (55), it follows that(56)f1∑i=1nαixi⊇f1xn+hTn−1ηf∑i=1n−2αiTn−1xi,f1f1xnf1¯∑i=1nαixi,f1¯∑i=1nαixi,⊇f1¯xn+hTn−1ηf1∑i=1n−2αiTn−1xi,f1¯f1xn,f1¯xn+hTn−1ηf1¯∑i=1n−2αiTn−1xi,f1¯f1xn,=f1¯xn+hTn−1ηf1¯Tn−2Tn−1∑i=1n−2αiTn−2xi+αn−1Tn−1xn−1,f1¯xn,⊇f1¯xn+hTn−1ηf1¯xn−1+hTn−2Tn−1ηf1¯∑i=1n−2αiTn−2xi,f1¯xn−1,f1¯xn⊇f1¯xn+hTn−1ηf1¯xn−1,f1¯xn+hTn−2ηηf1¯∑i=1n−2αiTn−2xi,f1¯xn−1,f1¯xn,⊇⋯⊇f1¯xn+hTn−1ηf1¯xn−1,f1¯xn+hTn−2ηηf1¯xn−2,f1¯xn−1,f1¯xn+⋯+hT1ηη…f1¯x1f1¯x2,f1¯x3…,f1¯xn−1,f1¯xn,=f1¯xn+hTn−1ηf1¯xn−1,xn+hTn−2ηf1¯xn−2,xn−1,xn+⋯+hT1ηf1¯x1,x2,…,xn−1,xn,=f1¯xn+∑i=1n−1hTiηf1¯xi,xi+1,…,xn.
Now,(57)f1¯xn=f1¯xn+hTn−1ηf¯Tn−2Tn−1∑i=1n−2αiTn−2xi+αn−1Tn−1xn−1,f1¯xn⊇f1¯xn+hTn−1ηf1¯xn−1+hTn−2Tn−1ηf1¯∑i=1n−2αiTn−2xi,f1¯xn−1,f1¯xn⊇f1¯xn+hTn−1ηf1¯xn−1,f1¯xn+hTn−2ηηf1¯∑i=1n−2αiTn−2xi,f1¯xn−1,f1¯xn⊇⋯⊇f1¯xn+hTn−1ηf1¯xn−1,f1¯xn+hTn−2ηηf1¯xn−2,f1¯xn−1,f1¯xn+⋯+hT1ηη…f1¯x1f1¯x2,f1¯x3…,f1¯xn−1,f1¯xn,=f1¯xn+hTn−1ηf1¯xn−1,xn+hTn−2ηf1¯xn−2,xn−1,xn+⋯+hT1ηf1¯x1,x2,…,xn−1,xn,=f1¯xn+∑i=1n−1hTiηf1¯xi,xi+1,…,xn.
From (56) and (57), we have (55).Remark 6.
If we choosehβ=β, ηx1,y1=x1−y1, and f1¯=f1¯ in (55), then we have the Jensen inequality derived in [28].Lemma 1.
Letf1:J1⊆ℝ⟶ℝ be a differentiable mapping on J∘, where x1,y1∈J1 and r1<s1. If f1′∈Lr1,s1, then the following equality holds:(58)f1x−1y1−x1∫x1y1f1udu=x−x12y1−x1∫01βf1′βx+1−βx1dβ−y1−x2y1−x1∫01βf1′βx+1−βy1dβ,for each x∈r1,s1.Theorem 7.
(Ostrowski-type inequality). Leth:I1⊆ℝ⟶ℝ be a nonnegative function. Also, let f1:J1⊆ℝ⟶ℝ be a differentiable mapping on J∘1 such that f1′∈Lr1,s1 with r1<s1 and hβ⩾β with ∫01h2βdβ<∞. If f1′ is an ηh-interval-valued convex function on I1 and f1′x≤M for all x∈r1,s1, then we have(59)f1x−1y1−x1∫x1y1fudu⊇Mx−x12+y1−x2y1−x1∫01hβdβ+η∗f1′x,f1′c∫01h2βdβ,where(60)η∗f1′x,f1′c=x−x12+y1−x2y1−x1ηf1′x,f1′x1+y1−x2+y1−x2y1−x1ηf1′x,f1′y1,and x1,y1∈r1,s1.Proof.
Sincef1′ is interval-valued ηh convex function, so by using Lemma 1, we can write(61)f1x−1y1−x1∫x1y1f1udu=x−x12y1−x1∫01βf1βx+1−βx1dβ+y1−x2y1−x1∫01βf′xβ+1−βbdβ⊇x−x12y1−x1∫01βf′x1+hβηf1′x,f1′y1dβ+y1−x2y1−x1×∫01βf1′y1+hβηf1′x,f1′y1dβ,f1¯x−1y1−x1∫x1y1f1¯udu,f1¯x−1y1−x1∫x1y1f1¯udu⊇x−x12y1−x1∫01tf1′¯x1+hβηf1′¯x,f1′¯x1dβ+y1−x2y1−x1×∫01βf1′¯y1+hβηf1′¯x,f1′¯y1dβ,x−x12y1−x1∫01βf1′¯x1+hβηf′¯x,f′¯x1dβ+y1−x2y1−x1∫01βf1′¯y1+hβηf1′¯x,f1′¯y1dβ,.
Also,(62)f1¯x−1y1−x1∫x1y1f1¯udu≤x−x12y1−x1∫01βf1′¯x1+hβηf1′¯x,f1′¯x1dβ+y1−x2y1−x1×∫01βf1′¯y1+hβηf1′¯x,f1′¯y1dβ,f1¯x−1y1−x1∫x1y1f1¯udu≥x−x12y1−x1∫01βf1′¯x1+hβηf1′¯x,f1′¯x1dβ+y1−x2y1−x1×∫01βf1′¯y1+hβηf1′¯x,f1′¯y1dβ.
Moreover, we have(63)f1¯x−1y1−x1∫x1y1f1¯udu≤Mx−x12y1−x1∫01hβdβ+x−x12y1−x1∫01h2βηf1′¯x,f1′¯x1dβ+My1−x2y1−x1×∫01hβdβ+y1−x2y1−x1∫01h2βηf1′¯x,f1′¯y1dβ,(64)f1¯x−1y1−b1∫x1y1f1¯udu≤Mx−x12+y1−x2y1−x1∫01hβdβ+x−x12y1−x1ηf1′¯x,f1′¯x1∫01h2βdβ+y1−x2y1−x1ηf1′¯x,f1′¯x1×∫01h2βdβ.
Furthermore, we have(65)f1¯x−1y1−x1∫x1y1f1¯udu≥Mx−x12y1−x1∫01hβdβ+x−x12y1−x1∫01h2βηf1′¯x,f1′¯x1dβ+My1−x2y1−x1×∫01hβdβ+y1−x2y1−x1∫01h2βηf1′¯x,f1′¯y1dβ,(66)f1¯x−1y1−x1∫x1y1f1¯udu≥Mx−x12+y1−x2y1−x1∫01hβdβ+x−x12y1−x1ηf1′¯x,f1′¯x1∫01h2βdβ+y1−x2y1−x1ηf1′¯x,f1′¯x1×∫01h2βdβ.
By combining (64) and (66), we get(67)f1¯x−1y1−x1∫x1y1f1¯udu,f1¯x−1y1−x1∫x1y1f1¯udu⊇Mx−x12+y1−x2y1−x1∫01hβdβ+x−x12y1−x1ηf1′¯x,f1′¯x1∫01h2βdβ+y1−x2y1−x1×ηf1′¯x,f1′¯x1∫01h2βdβ,Mx−x12+y1−x2y1−x1∫01hβdβ+x−x12y1−x1ηf1′¯x,f1′¯x1∫01h2βdβ+y1−x2y1−x1ηf1′¯x,f1′¯x1∫01h2βdβ.
This gives our desired result (59).Remark 7.
If we choosehβ=β, ηx1,y1=x1−y1 and f1¯=f1¯ in (67), then we get the classical Ostrowski inequality for convex functions [31].


## 5. Conclusion

Convexity and interval analysis are mathematical tools that help to solve real-world problems more accurately; that is why, several generalizations of interval valued convex functions are the main objective of mathematicians. In this paper, we introduced generalized interval-valuedηh convex functions and presented different types of well-known inequalities for this proposed class of convex functions. It is worthy to mention here that our results are the extensions of many existing results (for example, [28, 31, 32]).



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*Source: 6617074-2021-04-01.xml*