# Hermite-Hadamard- and Jensen-Type Inequalities for Intervalh1,h2 Nonconvex Function

Authors: Hongxin Bai; Muhammad Shoaib Saleem; Waqas Nazeer; Muhammad Sajid Zahoor; Taiyin Zhao
Journal: Journal of Mathematics
 (2020)

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## Abstract

In the present study, we will introduce the definition of intervalh1,h2 nonconvex function. We will investigate some properties of interval h1,h2 nonconvex function. Moreover, we will develop Hermite-Hadamard- and Jensen-type inequalities for interval h1,h2 nonconvex function.


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## Body

## 1. Introduction

Since its inception five decades ago, the theory of fuzzy sets has advanced in a variety of ways. Application of the theory of fuzzy sets covered many areas like artificial intelligence, decision theory, computer science, logic operational research, and robotics [1–3]. Initially good books like probability theory by Dubois and Prade in 1988, Behavioral and social science by Smithsons in 1987, and Fuzzy Control by sugeno 1985 and pedrycz 1989 and others have been published. We refer [4, 5] recent developments in this field.For some other results and application of interval analysis theory, we refer the readers [4, 6–13]. Due to vast application of fuzzy sets, many integral inequalities have been derived by different authors [1–3, 14–17]Costa [18] there is a new fuzzy version of Jensen-type integral inequality for fuzzy interval valued function. Also in [19], Zhao et al. develop new Harmite-Hadamard-type inequality for h-convex interval valued function.For more about Hermite-Hadamard inequalities, refer [20–24]. We will introduce the interval h1,h2 non-convex function. The second objective of this article is to develop Hermite-Hadamard- and Jensen-type inequality for above said generalization.


## 2. Preliminaries

In this section, we define some basic definitions, properties, results, and notations on interval analysis, which are used throughout the paper [17, 25]. Here, ℝI and ℝI+ denote the family of all intervals and positive interval and it is equipped with the algebraic operations “+” and “.” given, respectively, by x1,x2+y1,y2=x1+y1,x2+y2 and(1)α⋅x1,x2=αx1,αx2,if0≤α,αx2,αx1,ifα<0.A functionF:V⊆ℝ⟶ℝI with Fx=f1x,f2x, where f1,f2:V⟶ℝ are real functions with f1x≤f2x for all x∈V, it is called an interval-valued function.For intervalsx¯,x¯and y¯,y¯, the Hausdorff distance is defined by dx¯,x¯,y¯,y¯=maxx¯−y¯,x¯−y¯.ThenℝI,d is complete.A set of numbersfxi−1,yi,xii=1m is said to be a tagged partition P of x,y if(2)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 if for each i, then we say that the partition is δ-fine. The family of all δ-fine partitions of x,y is denoted by Pδ,x,y.GivenX∈Pδ,x,y, we define a integral sum of f:x,y⟶ℝI as follows:(3)Sf,X,δ,x,y=∑i=1mfyixi−xi−1.Throughout the paper,IR-integrable means interval Riemann integrable. The concept of IR-integrable is given in [19], Definition 2.2, is equivalent to IR integral given in [10], Definition 9.1.Definition 1.
Letf:x,y⟶ℝI. f is called IR-integrable on x,y with IR-integral A=IR∫xyftdt, if there exists an A∈RI such that for any ϵ>0 there exists a δ>0 such that(4)dSf,X,δ,x,y,A<ϵ,for each X∈Pd,x,y. Let IRx,y denote the set of all IR-integrable functions on x,y.Definition 2.
(Seep-convex set, [26]).
The real intervalI is known as p-convex set if for all x,y∈I and α∈0,1, implies that(5)αxp+1−αyp1/p∈I,where p=2k+1 or p=n1/n2, n1=2r+1, n2=2t+1, and k,r,t∈N.Definition 3.
(Seep-convex function, [26]). For a p-convex set I, the mapping f:I⟶ℝ:(6)fλp+1−λyp1/p≤λfx+1−λfy,is called p-convex function, for all x,y∈I, and λ∈0,1.Definition 4.
(Seeh-convex function, [27]). The nonnegative function f:a,b⟶ℝ is h-convex if(7)fλx+1−λy≤hλfx+h1−λfy.
∀
x
,
y
∈
a
,
b and λ∈0,1andh≠0, h is nonnegative real-valued function or f belongs to the class SXh;a,b.
If the inequality (7) is reversed, then f is said to be h-concave, i.e., f∈SVh,I.Definition 5.
h
1
,
h
2-convex function [28]. Let h1,h2:0,1⊆J⟶ℝ+ be a nonnegative function, h1,h2≠0. The nonnegative function f:J⟶ℝ is an h1,h2-convex if(8)fλx+1−λy≤h1λh21−λfx+h11−λh2λfy,for all x,y∈J and λ∈0,1 or that f belongs to the class SXh1,h2;J,ℝ. If the inequality (8) is reversed, then f is said to be h1,h2-concave, i.e., f∈SVh1,h2,J,ℝI+.Definition 6.
Intervalh1,h2-convex function [29]. Let h1,h2:0,1⊆J⟶ℝ+ be a nonnegative function, h1,h2≠0. We say that f:J⟶ℝI+ is an interval h1,h2-convex function or that f belongs to the class SXh1,h2;J,ℝI+, if f is nonnegative and for all x,y∈J and λ∈0,1, we have(9)h1λh21−λfx+h11−λh2λfy⊆fλx+1−λy.
If the inequality (9) is reversed, then f is said to be interval h1,h2-concave, i.e., f∈SVh1,h2,J,ℝ.Remark 1.
(1)
Ifh2=1, then Definition 6 becomes interval h-convex in [19].
(2)
Ifh1=h2=1, then Definition 6 becomes P-function in [30].
(3)
Ifh1t=ts, h2=1, then Definition 6 becomes to s-convex fuzzy process in [31].
We wind up the current section by introducing the new concept of intervalh1,h2 nonconvexity. This idea is inspired by An et al. [29]. Throughout the paper, for interval x¯,x¯ and y¯,y¯,(10)x¯,x¯⊆y¯,y¯⇒y¯≤x¯andx¯≤y¯.Definition 7.
Intervalh1,h2 nonconvex function. Let h1,h2:0,1⊆J⟶ℝ+ be a nonnegative function, h1,h2≠0. For a nonconvex function, f:J⟶ℝI+ is an interval h1,h2 nonconvex function if(11)h1λh21−λfx+h11−λh2λfy⊆fλxp+1−λyp1/p.
∀
x
,
y
∈
J, and λ∈0,1
For convenience, in this paper, the class of interval non-convex function is denoted bySXh1,h2,p;J,ℝI+. If the inequality (11) is reversed, then f is said to be interval h1,h2 nonconcave, i.e., f∈SVh1,h2,p,J,ℝ.
Throughout the paperHx,y=h1xh2y for. x,y∈0,1.Theorem 1.
Letf:r,s⟶ℝI+ be an interval-valued function such that fλ=f¯λ,f¯λ. Then f∈SXh1,h2,p,r,s,ℝI+ if and only if f¯∈SXh1,h2,p,r,s,ℝI+ and f¯∈SVh1,h2,p,r,s,ℝI+.Proof.
Letf be a interval h1,h2 non-convex function and suppose that x,y∈r,s, λ∈0,1, then(12)h1λh21−λfx+h11−λh2λfy⊆fλxp+1−λyp1/p,that is,(13)h1λh21−λf¯x+h11−λh2λf¯y×h1λh21−λf¯x+h11−λh2λf¯y⊆f¯λxp+1−λyp1/p,f¯λxp+1−λyp1/p.
It follows that(14)h1λh21−λf¯x+h11−λh2λf¯y≥f¯λxp+1−λyp1/ph1λh21−λf¯x+h11−λh2λf¯y≤f¯λxp+1−λyp1/p.
This shows thatf¯∈SXh1,h2,p,r,s,ℝI+ and. f¯∈SVh1,h2,p,r,s,ℝI+.
Conversely suppose that
f
¯
∈
S
X
h
1
,
h
2
,
p
,
r
,
s
,
ℝ
I
+ and f¯∈SVh1,h2,p,r,s,ℝI+ then from above definition and inclusion set (13), it follows that f∈SXh1,h2,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∈SVh1,h2,p,r,s,ℝI+ if and only if f¯∈SVh1,h2,p,r,s,ℝI+ and f¯∈SXh1,h2,p,r,s,ℝI+.Proof.
The proof is similar to that of Theorem1.


## 3. Main Result

Theorem 3.
Letf:r,s⟶ℝI+, h1,h2:0,1ℝ+ and H1/2,1/2≠0 if f∈SXh1,h2,p,r,s,ℝI+ and f∈IRr,s then(15)12H1/2,1/2frp+sp21/p⊇psp−rp∫rsλp−1fλdλ⊇fr+fs∫01Hx,1−xdx.Proof.
By assumption, we have(16)H12,12f¯xrp+1−xsp1/p+H12,12f¯1−xrp+xsp1/p≥f¯rp+sp21/pH12,12f¯xrp+1−xsp1/p+H12,12f¯1−xrp+xsp1/p≤f¯rp+sp21/p.
Integrating above w.r.t. “x” on [0, 1], we get(17)∫01f¯xrp+1−xsp1/pdx+∫01f¯1−xrp+xsp1/pdx≥1H1/2,1/2f¯rp+sp21/p,∫01f¯xrp+1−xsp1/pdx+∫01f¯1−xrp+xsp1/pdx≤1H1/2,1/2f¯rp+sp21/p,it follows that(18)2psP−rp∫01λp−1f¯λdλ≥1H1/2,1/2f¯rp+sp21/p2psP−rp∫01λp−1f¯λdλ≤1H1/2,1/2f¯rp+sp21/p.
This implies that(19)psP−rp∫01λp−1fλdλ⊆12H1/2,1/2frp+sp21/p.
Now by def. of interval nonconvex function, we have(20)h1xh21−xfr+h11−xh2xfs⊆fxrp+1−xsp1/p,integrated with respect to “x” on [0, 1], we get(21)fr∫01h1xh21−xdx+fs∫01h11−xh2xdx⊆∫01fxrp+1−xsp1/pdxfr+fs×∫01h1xh21−xdx⊆∫01fxrp+1−xsp1/p,it follows that(22)fr+fs⋅∫01h1xh21−xdx⊆psP−rp∫01λp−1fλdλ.
Combining (19) and (22) we get (27).Remark 2.
(1)
IfHx,y=h1x and f¯=f¯ then Theorem 3 becomes ([32], Theorem 5).
(2)
IfHx,y=h1x and p=1 then Theorem 3 becomes ([19], Theorem 4.1).
(3)
Ifh1x=xs, h2x=1 and p=1 then Theorem 3 becomes ([31], Theorem 4).
(4)
Iff¯=f¯ then Theorem 3 becomes Hermite-Hadamard inequality for h1,h2-convex function.Example 1.
Considerh1t=t,h2t=1 for t∈0,1, r,s=−1,1 and f:r,s⟶ℝI+ be defind by fλ=λp,4−eλp and p be an odd number, then we have(23)12H1/2,1/2frp+sp21/p=f0=0,3,and(24)psp−rp∫rsλp−1fλdλ=p2∫−11λp−1λpdλ,∫−11λp−14−eλpdλ=p2∫−11λ2p−1dλ,8p−∫−11λp−1eλpdλ=0,4−e−e−12.
Puttingz=λp in (24) and simplifying, we get(25)fr+fs∫01Hx,1−xdx=0,4−e+e−12.
Combining (23)–(25), we get(26)0,3⊇0,4−e−e−12⊇0,4−e+e−12.
Consequently, verify the Theorem3.Theorem 4.
Letf:r,s⟶ℝI+, h1,h2:0,1ℝ+, and H1/2,1/2≠0 if f∈SXh1,h2,p,r,s,ℝI+ and f∈IRr,s then(27)14H1/2,1/22frp+sp21/p⊇Δ1⊇psp−rp∫rsλp−1fλdλ⊇Δ2⊇fr+fs12+H12,12×∫01Hx,1−xdx,where(28)Δ1=14H1/2,1/2f3rp+sp41/p+frp+3sp41/p,Δ2=12fr+fs+2frp+sp21/p∫01Hx,1−xdx.Proof.
Forrp,rp+sp/2, one has(29)H12,12fxrp+1−xrp+sp21/p+H12,12f1−xrp+xrp+sp21/p⊆f3rp+sp41/p.
Consequently, we get(30)4psp−rp∫rrp+sp/21/pλp−1fλdλ⊆1H1/2,1/2f3rp+sp41/p.
Similarly forrp+sp/2,sp(31)4psp−rp∫rp+sp/21/psλp−1fλdλ⊆1H1/2,1/2frp+3sp41/p.
Adding (30) and (31), we get(32)psp−rp∫rsλp−1fλdλ⊆14H1/2,1/2f3rp+sp41/p+frp+3sp41/ppsp−rp∫rsλp−1fλdλ⊆Δ1,and from Theorem 3, we have(33)14H1/2,1/22frp+sp21/p=14H1/2,1/22f123rp+sp4+12rp+3sp41/p⊇14H1/2,1/2f3rp+sp4+rp+3sp4⊇Δ1⊇psp−rp∫rsλp−1fλdλ⊇12fr+fs+2frp+sp21/p∫01Hx,1−xdx=Δ2⊇fr+fs12+H12,12∫01Hx,1−xdx,and the result follows.Theorem 5 (Jensen-type inequality).
Letw1,w2,…,wn∈ℝ+ with n≥2. If h1,h2 is nonnegative super multiplicative function and if f∈SXh1,h2,p,u,v,ℝI+ and x1,x2,…,xn∈I. Then, we have following inequality:(34)f1Wn∑i=1nwixip1/p⊇∑i=1nHwiWn,Wn−1Wnfxi,where Wn=∑inwiProof.
Whenn = 2 inequality (33) is true. Consider the inequality (33) is true for n−1, then(35)f1Wn∑i=1nwixip1/p=fwnWnxnp+∑i=1n−1wiWnxip1/p⊇h1wnWnh2Wn−1Wnfxn+h1Wn−1Wnh2wnWnf∑i=1n−1wiWnxip⊇h1wnWnh2Wn−1Wnfxn+h1Wn−1Wnh2wnWn∑i=1n−1HwiWn,Wn−2Wn−1fxi⊇∑i=1nHwiWn,Wn−1Wnfxi.
Thus, by mathematical induction the result is proved.


## 4. Conclusions

Hermite-Hadamard- and Jensen-type inequalities are true for this new concept of interval value convexity. Moreover by suitable substitution, the obtained results reduce to the results of [19, 31, 32].



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