# Hermite-Hadamard and Schur-Type Inequalities for Stronglyh-Convex Fuzzy Interval Valued Functions

Authors: Putian Yang; Shiqing Zhang
Journal: Journal of Function Spaces
 (2022)

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## Abstract

The concept of fuzzy theory was developed in 1965 and becomes an acknowledged research subject in both pure and applied mathematics and statistics, showing how this theory is highly applicable and productive in many applications. In the present study, we introduced the definition of fuzzy interval valued stronglyh-convex function and investigated some of its properties. We established Hermite-Hadamard and Schur-type inequalities for the class of fuzzy interval valued strongly h-convex function.


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## Body

## 1. Introduction

The theory of interval analysis is one of the top areas of research nowadays because of its enormous application in various fields especially in automatic error analysis [1], computer graphics [2], and neural network output optimization [3]. In [4], Moore et al. give the first monograph on interval analysis, and since then, a huge amount of work have been done in interval calculus; for example, in [5], Chalco Cano et al. studied the interval-valued functions using generalized Hukuhara derivative and presented applications of interval valued calculus. An efficient method for solving fuzzy optimization problems using interval valued calculus is presented in [6]. Costa et al. [7] calculated the possible conformations arising from uncertainty in the molecular distance geometry problem using constraint interval analysis. Interval vector spaces and interval optimizations are given in [8], while optimality conditions for generalized differentiable interval valued functions are presented in [9]. Several classical inequalities for interval valued functions have been established in [10].To address the modern problems, the convexity has been generalized in number of ways and some interesting generalizations are stronglyh,s-convex functions [11], strongly generalized convex functions [12], p-convex function [13], and many more. The authors introduce h-convex function in [14] as follows;A functionϕ:I=μ1,μ2⟶ℝ is known as h-convex if
(1)ϕβs+1−βt≤hβϕs+h1−βϕt,for all s,t∈I=μ1,μ2 and β∈0,1.The authors introduce the strongly convex function in [15] as follows:A functionϕ:I=μ1,μ2⟶ℝ with modulus ϑ≥0 is defined as
(2)ϕβs+1−βt≤βϕs+1−βϕt−ϑβ1−βs−t2,for all s,t∈I=μ1,μ2 and β∈0,1 which is called strongly convex.The unification of above two definitions is defined in [16] as follows:A functionϕ:I=μ1,μ2⟶ℝ with modulus ϑ≥0 defined as follows:
(3)ϕβs+1−βt≤hβϕs+h1−βϕt−ϑβ1−βs−t2,for all s,t∈I=μ1,μ2 and β∈0,1 which is known as strongly h-convex.In recent years, mathematical inequalities for interval valued convex and nonconvex function got attention of many mathematician [17]. Bai et al. [10] established Hermite-Hadamard and Jensen-type inequalities for the class of interval valued nonconvex functions. In 2017, Costa [18] presented Jensen-type inequality for the class of fuzzy interval valued functions, and in the same year, Costa and Román–Flores [19] presented some integral inequalities for the same class of functions. Some Opial–type inequalities were studied in [20]. For other remarkable results, we refer [21] and references therein.In this report, we proposed the definition of fuzzy interval valued stronglyh-convex function. We investigated some of its properties and established Hermite-Hadamard and Schur-type inequalities for the proposed definition.The paper is organized as follows: In Section2, we will give some preliminaries and basic definitions, and we will established some basic properties. However, Section 3 is devoted for the establishment of main results like Hermite Hadamard and Schur-type inequalities.


## 2. Some Basic Properties of Interval Calculus

LetℝI+ and ℝI be all positive intervals and all intervals. The algebraic operations “+” and “.” are defined as [22] s1,s2+t1,t2=s1+t1,s2+t2 and
(4)λ.s1,s2=λs1,λs2,if0≤λ,λs2,λs1,ifλ<0.A functionF:V⊆ℝ⟶ℝI with Fs=ϕ1s,ϕ2s, where ϕ1,ϕ2:V⟶ℝ are real functions and known as interval valued functions if ϕ1s≤ϕ2s, for all s∈V.For intervalss¯,s¯ and t¯,t¯, the distance ds¯,s¯,t¯,t¯=maxs¯−t¯,s¯−t¯ is the Hausdorff distance. Then ℝI,d is complete.The tagged partitionP of interval s,t is the set of numbers fsi−1,ti,sii=1m if
(5)s=s0<s1<⋯<sm=tand if si−1≤ti≤si, for all i=1,2,⋯,m.Moreover, lettingΔsi=si−si−1 and Δsi≤δ for every i, the partition is said to be δ-fine. The family of the all such δ-fine partitions of s,t is represented as Pδ,s,t [17].LetX be the δ-fine partitions of s,t; then the function ϕ:s,t⟶ℝI is said to be integral sum if
(6)Sϕ,X,δ,s,t=∑i=1mϕtisi−si−1.Proposition 1 (see [17]).
The four arithmetic operators+,−,·,/ for t=t¯,t¯ and s=s¯,s¯ are defined as
(7)t+s=t¯+s¯,t¯+s¯,t−s=t¯−s¯,t¯−s¯,t.s=mint¯.s¯,t¯.s¯,t¯.s¯,t¯.s¯,maxt¯.s¯,t¯.s¯,t¯.s¯,t¯.s¯,ts=mint¯s¯,t¯s¯,t¯s¯,t¯s¯,maxt¯s¯,t¯s¯,t¯s¯,t¯s¯,where 0∉s¯,s¯.Definition 2.
A functionϕ:I=μ1,μ2⟶ℝ is called convex function if
(8)ϕβs+1−βt≤βϕs+1−βϕt,for all s,t∈I=μ1,μ2 and β∈0,1.Definition 3 (see [23]).
Let a nonnegative functionh:a,b⟶ℝ, h≠0 and 0,1⊆μ1,μ2. We say that a function ϕ:μ1,μ2⟶ℝ+I is interval h-convex, if for all s,t∈μ1,μ2 and β∈0,1, we have
(9)hβϕs+h1−βϕt⊆ϕβs+1−βt.
If the set inclusion (9) is reversed, then ϕ is said to be interval h-concave, i.e., ϕ∈SVh,μ1,μ2,ℝ.
We end this section of preliminaries by introducing the new concept of interval valued stronglyh-convex function. Note that for interval s¯,s¯ and t¯,t¯, the inclusion ′′⊆′′ is defined by
(10)s¯,s¯⊆t¯,t¯,ifft¯≤s¯,s¯≤t¯.Definition 4.
LetX be a normed space and D⊆X is convex. h:0,1⟶ℝ is a function. We call a function ϕ:I=μ1,μ2⟶ℝ with modulusϑ≥0 is interval valued strongly h-convex function if
(11)ϕβs+1−βt⊇hβϕs+h1−βϕt−ϑβ1−βs−t2,for all s,t∈I=μ1,μ2 and β∈0,1.The notionSXh;μ1,μ2,ℝ+I is reserved for the class of interval valued strongly h-convex functions.Remark 5.
(1)
By takingϑ=0, then the set inclusion (11) reduces to interval valued h-convex function [23](2)
By takingϕ¯=ϕ¯, then the set inclusion (11) reduces to strongly h-convex function [16](3)
By takingϕ¯=ϕ¯ and ϑ=0, then the set inclusion (11) reduces to h-convex function [14](4)
By takingϕ¯=ϕ¯ and hβ=β, then the set inclusion (11) reduces to strongly-convex function [15](5)
By takingϕ¯=ϕ¯, hβ=β and ϑ=0, then the set inclusion (11) reduces to classical convexity(6)
By takingϕ¯=ϕ¯, hβ=1/β and ϑ=0, then the set inclusion (11) reduces to Godunova- Levin function


## 3. Main Results

In this section, we establish the Hermite-Hadamard- and Schur-type inequalities for the proposed definition.Theorem 6.
LetX be a normed space and D⊆X is convex. h:0,1⟶ℝ is a function with holding condition hβ≥β for each β∈0,1 if g:D⟶ℝ belongs to SXh;μ1,μ2,ℝ+I; then ϕ:D⟶ℝ is defined as
(12)ϕs=gs+ϑs2,which also belongs to SXh;μ1,μ2,ℝ+I.Proof.
Considerg∈SXh;μ1,μ2,ℝ+I; then
(13)gβt+1−βs=ϕβt+1−βs+ϑβt+1−βs2⊇hβgt+h1−βgs+ϑβt+1−βs2⊇hβϕt+h1−βϕs−ϑβt2−ϑ1−βs2+ϑβt+1−βs2⊇hβϕt+h1−βϕs−ϑβt2−ϑ1−βs2+ϑβ2t2+2β1−β<t,s>+1−β2s2⊇hβϕt+h1−βϕs−ϑβ1−βt−s2,which gives the desired result.Theorem 7.
LetX be a normed space and D⊆X is convex. h:0,1⟶ℝ is a function with holding condition hβ≤β for each β∈0,1 if ϕ:D⟶ℝ belongs to SXh;μ1,μ2,ℝ+I; then g:D⟶ℝ is defined as
(14)gs=ϕs+ϑs2,which also belongs to SXh;μ1,μ2,ℝ+I.Proof.
The proof is similar to that of Theorem6.Example 8.
Lethβ=1, β∈0,1. Then ϕ:−1,1⟶ℝ defined by ϕs=1, s∈−1,1 is interval valued strongly h-convex with modulus ϑ=1. Because for every s,t∈−1,1 and β∈0,1, then from (11), we have
(15)ϕ¯βs+1−βt≤hβϕ¯s+h1−βϕ¯t−ϑβ1−βs−t2,(16)ϕ¯βs+1−βt≥hβϕ¯s+h1−βϕ¯t−ϑβ1−βs−t2.
Hence, the inequalities (15) and (16) hold for ϕs=ϕt=1, hβ=1, and ϑ=1. So the set inclusion (11) holds, and we say that g is interval valued strongly h-convex function.
But it is not interval valuedh-convex function. Because by taking β=1/2, hβ=1, s=−1, t=1, and ϑ=1, the inequality (16) is not satisfied. So the set inclusion (9) is holds, and we say that it is not interval valued h-convex function.


### 3.1. Interval Hermite-Hadamard-Type Inequality

Theorem 9.
Let a nonnegative functionh:0,1⟶ℝ. A function ϕ defined over I where I=μ1,μ2 is interval valued strongly h-convex function with modulus ϑ≥0; then
(17)ϕμ1+ϕμ2∫01hβdβ−ϑμ2−μ126⊆1μ2−μ1∫μ1μ2ϕudu⊆12h1/2ϕμ1+μ22+ϑμ2−μ1212.Proof.
Takeu=βμ1+1−βμ2, v=1−βμ1+βμ2, and
(18)ϕμ1+μ22=ϕu+v2,and applying the definition 4, we obtained
(19)ϕμ1+μ22⊇h12ϕu+ϕv−ϑ4u−v2,which implies that
(20)h12ϕ¯u+ϕ¯v−7ϑ4u−v2,h12ϕ¯u+ϕ¯v−ϑ4u−v2⊆ϕ¯μ1+μ22,ϕ¯μ1+μ22,(21)ϕ¯μ1+μ22≤h12ϕ¯u+ϕ¯v−ϑ4u−v2,(22)ϕ¯μ1+μ22≥h12ϕ¯u+ϕ¯v−ϑ4u−v2.
Substituting the values ofu and v in (21), we get
(23)ϕ¯μ1+μ22≤h12ϕ¯βμ1+1−βμ2+ϕ¯1−βμ1+βμ2−ϑ4βμ1+1−βμ2−1−βμ1+βμ22≤h12ϕ¯βμ1+1−βμ2+ϕ¯1−βμ1+βμ2−ϑ42β−1μ1+1−2βμ22,integrating (23) with respect toβover0,1; we obtain
(24)ϕ¯μ1+μ22≤h12∫01ϕ¯βμ1+1−βμ2dβ+∫01ϕ¯1−βμ1+βμ2dβ−ϑ4∫012β−1μ1+1−2βμ22dβ≤h122μ2−μ1∫μ1μ2ϕxdx−ϑ12μ2−μ1.
Similarly substituting the values ofu and v in (22), we get
(25)ϕ¯μ1+μ22≤h122μ2−μ1∫μ1μ2ϕxdx−ϑ12μ2−μ1.
Combining (24) and (25), we obtain
(26)h122μ2−μ1∫μ1μ2ϕxdx−ϑ12μ2−μ1⊆ϕμ1+μ22,which gives the proof of left hand side.
For the proof of right hand side, take(27)1μ2−μ1∫μ1μ2ϕudu=∫01ϕ1−βμ1+βμ2dβ.
Using the definition of interval valued stronglyh-convex function, we obtain
(28)1μ2−μ1∫μ1μ2ϕudu⊇∫01h1−βϕμ1+hβϕμ2−ϑβ1−βμ1−μ22dβ⊇ϕμ1+ϕμ2∫01hβdβ−ϑμ2−μ126,which gives right hand side.Combining (26) and (28), we obtain desire result.Remark 10.
(1)
By takingϑ=0, then (17) becomes Hermite Hadamard-type inequality for interval valued h-convex function [23](2)
By substitutingϕ¯=ϕ¯, then (17) becomes Hermite Hadamard-type inequality for strongly h-convex function [16](3)
By substitutingϕ¯=ϕ¯ and ht=t, then (17) becomes Hermite Hadamard-type inequality for strongly convex function [15](4)
Forϕ¯=ϕ¯, ϑ=0 and ht=t, then (17) becomes classical Hermite Hadamard-type inequalityExample 11.
Considerht=t for t∈0,1, μ1,μ2=−1,1, and ϕ:μ1,μ2⟶ℝ+I be defined by ϕt=t2,4−et and ϑ=1; then, we have
(29)12h1/2ϕμ1+μ22=ϕ0=13,103,(30)1μ2−μ1∫μ1μ2ϕtdt=13,4−e−e−12.
Also,(31)ϕμ1+ϕμ2∫01hxdx−ϑμ2−μ126=13,22−3e+e−16.
Returning to (29), (30), and (31), we deduce
(32)13,103⊇13,4−e−e−12⊇13,22−3e+e−16.Consequently, Theorem9 is verified.


### 3.2. Interval Schur-Type Inequality

Theorem 12.
Considerg:I=u1,u2⟶ℝ be the interval valued strongly h-convex function and a nonnegative super multiplicative function h:J⟶ℝ for all s1,s2,s3∈I=u1,u2 and s1<s2<s3 such that s3−s1,s3−s2,s2−s1∈J; then the following inequality holds:
(33)hs3−s1gs2⊇hs3−s2gs1+hs2−s1gs3−ϑs3−s2s2−s1,where 0,1⊆J iff g is interval valued strongly h-convex function.Proof.
Lets1,s2,s3∈I be numbers which satisfy assumptions of the proposition. Then s3−s2/s3−s1∈0,1⊆J, s3−s2/s3−s1∈0,1⊆J, and
(34)s3−s2s3−s1+s2−s1s3−s1=1.
Also,(35)hs3−s2=hs3−s2s3−s1.s3−s1≥hs3−s2s3−s1hs3−s1,(36)hs2−s1≥hs2−s1s3−s1hs3−s1.
Assume thaths3−s1≥0and g is interval valued strongly h−convex function with modulus ϑ≥0. Let s1,s2,s3∈I=u1,u2; then substituting β=s3−s2/s3−s1, s=s1, and t=s3 in (11) yields that
(37)gs3−s2s3−s1s1+1−s3−s2s3−s1s3⊇hs3−s2s3−s1gs1+h1−s3−s2s3−s1gs3−ϑs3−s2s3−s11−s3−s2s3−s1s1−s32,implying
(38)gs2⊇hs3−s2s3−s1gs1+hs2−s1s3−s1gs3−ϑs3−s2s2−s1.
Now, we write the above set inclusion as(39)g¯s2≤hs3−s2s3−s1g¯s1+hs2−s1s3−s1g¯s3−ϑs3−s2s2−s1,g¯s2≥hs3−s2s3−s1g¯s1+hs2−s1s3−s1g¯s3−ϑs3−s2s2−s1.
Using the condition (35), we obtain
(40)g¯s2≤hs3−s2hs3−s1g¯s1+hs2−s1hs3−s1g¯s3−ϑs3−s2s2−s1,g¯s2≥hs3−s2hs3−s1g¯s1+hs2−s1hs3−s1g¯s3−ϑs3−s2s2−s1.
Now, we write the above inequalities in set inclusion form(41)hs3−s1gs2⊇hs3−s2gs1+hs2−s1gs3−ϑs3−s2s2−s1hs3−s1,as required. Conversely suppose that (33) holds, and we insert s1=u, s2=βu+1−βv, and s3=v in (33); we get
(42)hv−ugβu+1−βv⊇hβv−ugu+hv−u1−βgv−ϑβ1−βhv−uv−u2.
Use condition (35), and simplify the above inclusion set yields that g is interval valued strongly h-convex, which completes the proof.Remark 13.
If we takeϑ=0 and g¯=g¯ in Theorem 12, then, we obtain Schur-type inequality for h-convex function [24].


## 3.1. Interval Hermite-Hadamard-Type Inequality

Theorem 9.
Let a nonnegative functionh:0,1⟶ℝ. A function ϕ defined over I where I=μ1,μ2 is interval valued strongly h-convex function with modulus ϑ≥0; then
(17)ϕμ1+ϕμ2∫01hβdβ−ϑμ2−μ126⊆1μ2−μ1∫μ1μ2ϕudu⊆12h1/2ϕμ1+μ22+ϑμ2−μ1212.Proof.
Takeu=βμ1+1−βμ2, v=1−βμ1+βμ2, and
(18)ϕμ1+μ22=ϕu+v2,and applying the definition 4, we obtained
(19)ϕμ1+μ22⊇h12ϕu+ϕv−ϑ4u−v2,which implies that
(20)h12ϕ¯u+ϕ¯v−7ϑ4u−v2,h12ϕ¯u+ϕ¯v−ϑ4u−v2⊆ϕ¯μ1+μ22,ϕ¯μ1+μ22,(21)ϕ¯μ1+μ22≤h12ϕ¯u+ϕ¯v−ϑ4u−v2,(22)ϕ¯μ1+μ22≥h12ϕ¯u+ϕ¯v−ϑ4u−v2.
Substituting the values ofu and v in (21), we get
(23)ϕ¯μ1+μ22≤h12ϕ¯βμ1+1−βμ2+ϕ¯1−βμ1+βμ2−ϑ4βμ1+1−βμ2−1−βμ1+βμ22≤h12ϕ¯βμ1+1−βμ2+ϕ¯1−βμ1+βμ2−ϑ42β−1μ1+1−2βμ22,integrating (23) with respect toβover0,1; we obtain
(24)ϕ¯μ1+μ22≤h12∫01ϕ¯βμ1+1−βμ2dβ+∫01ϕ¯1−βμ1+βμ2dβ−ϑ4∫012β−1μ1+1−2βμ22dβ≤h122μ2−μ1∫μ1μ2ϕxdx−ϑ12μ2−μ1.
Similarly substituting the values ofu and v in (22), we get
(25)ϕ¯μ1+μ22≤h122μ2−μ1∫μ1μ2ϕxdx−ϑ12μ2−μ1.
Combining (24) and (25), we obtain
(26)h122μ2−μ1∫μ1μ2ϕxdx−ϑ12μ2−μ1⊆ϕμ1+μ22,which gives the proof of left hand side.
For the proof of right hand side, take(27)1μ2−μ1∫μ1μ2ϕudu=∫01ϕ1−βμ1+βμ2dβ.
Using the definition of interval valued stronglyh-convex function, we obtain
(28)1μ2−μ1∫μ1μ2ϕudu⊇∫01h1−βϕμ1+hβϕμ2−ϑβ1−βμ1−μ22dβ⊇ϕμ1+ϕμ2∫01hβdβ−ϑμ2−μ126,which gives right hand side.Combining (26) and (28), we obtain desire result.Remark 10.
(1)
By takingϑ=0, then (17) becomes Hermite Hadamard-type inequality for interval valued h-convex function [23](2)
By substitutingϕ¯=ϕ¯, then (17) becomes Hermite Hadamard-type inequality for strongly h-convex function [16](3)
By substitutingϕ¯=ϕ¯ and ht=t, then (17) becomes Hermite Hadamard-type inequality for strongly convex function [15](4)
Forϕ¯=ϕ¯, ϑ=0 and ht=t, then (17) becomes classical Hermite Hadamard-type inequalityExample 11.
Considerht=t for t∈0,1, μ1,μ2=−1,1, and ϕ:μ1,μ2⟶ℝ+I be defined by ϕt=t2,4−et and ϑ=1; then, we have
(29)12h1/2ϕμ1+μ22=ϕ0=13,103,(30)1μ2−μ1∫μ1μ2ϕtdt=13,4−e−e−12.
Also,(31)ϕμ1+ϕμ2∫01hxdx−ϑμ2−μ126=13,22−3e+e−16.
Returning to (29), (30), and (31), we deduce
(32)13,103⊇13,4−e−e−12⊇13,22−3e+e−16.Consequently, Theorem9 is verified.


## 3.2. Interval Schur-Type Inequality

Theorem 12.
Considerg:I=u1,u2⟶ℝ be the interval valued strongly h-convex function and a nonnegative super multiplicative function h:J⟶ℝ for all s1,s2,s3∈I=u1,u2 and s1<s2<s3 such that s3−s1,s3−s2,s2−s1∈J; then the following inequality holds:
(33)hs3−s1gs2⊇hs3−s2gs1+hs2−s1gs3−ϑs3−s2s2−s1,where 0,1⊆J iff g is interval valued strongly h-convex function.Proof.
Lets1,s2,s3∈I be numbers which satisfy assumptions of the proposition. Then s3−s2/s3−s1∈0,1⊆J, s3−s2/s3−s1∈0,1⊆J, and
(34)s3−s2s3−s1+s2−s1s3−s1=1.
Also,(35)hs3−s2=hs3−s2s3−s1.s3−s1≥hs3−s2s3−s1hs3−s1,(36)hs2−s1≥hs2−s1s3−s1hs3−s1.
Assume thaths3−s1≥0and g is interval valued strongly h−convex function with modulus ϑ≥0. Let s1,s2,s3∈I=u1,u2; then substituting β=s3−s2/s3−s1, s=s1, and t=s3 in (11) yields that
(37)gs3−s2s3−s1s1+1−s3−s2s3−s1s3⊇hs3−s2s3−s1gs1+h1−s3−s2s3−s1gs3−ϑs3−s2s3−s11−s3−s2s3−s1s1−s32,implying
(38)gs2⊇hs3−s2s3−s1gs1+hs2−s1s3−s1gs3−ϑs3−s2s2−s1.
Now, we write the above set inclusion as(39)g¯s2≤hs3−s2s3−s1g¯s1+hs2−s1s3−s1g¯s3−ϑs3−s2s2−s1,g¯s2≥hs3−s2s3−s1g¯s1+hs2−s1s3−s1g¯s3−ϑs3−s2s2−s1.
Using the condition (35), we obtain
(40)g¯s2≤hs3−s2hs3−s1g¯s1+hs2−s1hs3−s1g¯s3−ϑs3−s2s2−s1,g¯s2≥hs3−s2hs3−s1g¯s1+hs2−s1hs3−s1g¯s3−ϑs3−s2s2−s1.
Now, we write the above inequalities in set inclusion form(41)hs3−s1gs2⊇hs3−s2gs1+hs2−s1gs3−ϑs3−s2s2−s1hs3−s1,as required. Conversely suppose that (33) holds, and we insert s1=u, s2=βu+1−βv, and s3=v in (33); we get
(42)hv−ugβu+1−βv⊇hβv−ugu+hv−u1−βgv−ϑβ1−βhv−uv−u2.
Use condition (35), and simplify the above inclusion set yields that g is interval valued strongly h-convex, which completes the proof.Remark 13.
If we takeϑ=0 and g¯=g¯ in Theorem 12, then, we obtain Schur-type inequality for h-convex function [24].


## 4. Concluding Remarks

In this report, we introduced the fuzzy interval valued stronglyh-convex function. We also investigated some of its properties. Moreover, we established Hermite-Hadamard and Schur-type inequalities for the proposed definition. Our results are more general than many existing results. It is interesting for the researcher to find other types of inequalities for the proposed class of functions in the setting of various fractional integrals.



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