Harmonically Convex Fuzzy-Interval-Valued Functions and Fuzzy-Interval Riemann–Liouville Fractional Integral Inequalities

Gul Sana; Muhammad Bilal Khan; Muhammad Aslam Noor; Pshtiwan Othman Mohammed; Yu-Ming Chu

doi:10.2991/ijcis.d.210620.001

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Volume 14, Issue 1, 2021, Pages 1809 - 1822

Harmonically Convex Fuzzy-Interval-Valued Functions and Fuzzy-Interval Riemann–Liouville Fractional Integral Inequalities

Authors

Gul Sana¹^,, Muhammad Bilal Khan¹, Muhammad Aslam Noor¹, Pshtiwan Othman Mohammed², Yu-Ming Chu³^{, *}

¹Department of Mathematics, COMSATS University Islamabad, Islamabad, Pakistan

²Department of Mathematics, College of Education, University of Sulaimani, Sulaimani, Iraq

³Department of Mathematics, Huzhou University, Huzhou, P. R. China

^*Corresponding author. Email: chuyuming@zjhu.edu.cn

Corresponding Author

Yu-Ming Chu

Received 5 April 2021, Accepted 14 June 2021, Available Online 28 June 2021.

DOI: 10.2991/ijcis.d.210620.001 How to use a DOI?
Keywords: Harmonically convex fuzzy interval-valued function; Fuzzy interval fractional integral operator; Hermite–Hadamard inequality; Hermite–Hadamard–Fejér inequality
Abstract: It is well known that the concept of convexity establishes strong relationship with integral inequality for single-valued and interval-valued function. The single-valued function and interval-valued function both are special cases of fuzzy interval-valued function. The aim of this paper is to introduce a new class of convex fuzzy interval-valued functions, which is called harmonically convex fuzzy interval-valued functions (harmonically convex fuzzy-IVFs) by means of fuzzy order relation and to investigate this new class via fuzzy-interval Riemann–Liouville fractional operator. With the help of fuzzy order relation and fuzzy-interval Riemann–Liouville fractional, we derive some integrals inequalities of Hermite–Hadamard (H-H) type and Hermite–Hadamard–Fejér (H-H Fejér) type as well as some product inequities for harmonically convex fuzzy-IVFs. Our results represent a significant improvement and refinement of the known results. We hope that these interesting outcomes may open a new direction for fuzzy optimization, modeling and interval-valued function.
Copyright: © 2021 The Authors. Published by Atlantis Press B.V.
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

The Hermite–Hadamard (H-H) inequality was firstly introduced by Hadamard [1] and Hermite [2] for convex functions. This inequality is used as a most useful tool in mathematical analysis and optimization because convex functions establish strong relationship with H-H inequality. Therefore, many authors have discussed the relation of H-H inequality with different kinds of convex and nonconvex functions and many papers have provided refinements, generalizations and extensions, see [3–8]. Besides, fractional integrals have played a critical role in different branches of sciences. It is also a familiar fact that inequalities have become a very popular method using fractional integrals, and that this method has been the driving force behind many studies in recent years. Many forms of inequality have been studied, resulting in the introduction of new trend in inequality theory. Firstly, by using fractional integrals, Sarikaya et al. [9] discovered fractional H-H inequality for classical convex function. After that, many scholars devoted their efforts to present fractional H-H type inequalities for different classes of convex and nonconvex functions see [10–15].

It is well known that interval analysis provides tools to deal with data uncertainty. In general, interval analysis is typically used to deal with the models whose data are composed of inaccuracies that may occur from certain kinds of measurements. In 1966, the concept of interval analysis was firstly introduced by late American mathematician Ramon E. Moore in [16]. Since its inception, various authors in the mathematical community have paid close attention to this area of research. Interval analysis has been found to be useful in global optimization and constraint solution algorithms, according to experts. It has slowly risen in popularity over the last few decades. Scientists and engineers engaged in scientific computation have discovered that interval analysis is useful, especially in terms of accuracy, round-off error affects and automatic validation of results. After the invention of interval analysis, the researchers working in the area of inequalities wants to know whether the inequalities in abovementioned results can be found substituted with inclusions relation. In certain cases, the question is answered correctively. Recently, through interval Riemann integral, interval Riemann–Liouville fractional integrals and fuzzy Riemann integral, several authors presented new versions of various inequalities for interval and fuzzy-interval-valued functions like, as one can see Costa [17], Costa and Roman-Flores [18], Roman-Flores et al. [19,20], and Chalco-Cano et al. [21,22], An et al. [23], Zhao et al. [24], but also to more general set-valued maps by Nikodem et al. [25], Matkowski and Nikodem [26]. In particular, Zhang et al. [27] derived the new version of Jensen's inequalities for set-valued and fuzzy set-valued functions by means of a pseudo order relation and proved that these Jensen's inequalities generalized form of Costa Jensen's inequalities [17]. As a further extension, more and more, H-H type inequalities have been obtained through interval Riemann–Liouville fractional integrals, see for convex-IVFs [28,29], for harmonically convex-IVFs [30]. Moreover, recently, Khan et al. [31] introduced the new class of convex fuzzy mappings is known as h1,h2-convex fuzzy-IVFs by means fuzzy order relation and presented the following new version of H-H-type inequality for h1,h2-convex fuzzy-IVF involving fuzzy-interval Riemann integrals:

Let Q˜:u,ν→F0 be a h1,h2-convex fuzzy-IVF with h1,h2:0,1→ℝ+ and h112h212≠0, whose β-levels define the family of IVFs Qβ:u,ν⊂ℝ→KC+ are given by Qβz=Q∗z,β,Q∗z,β for all z∈u,ν, β∈0,1. If Q˜ is fuzzy-interval Riemann integrable (in sort, FR-integrable), then

12h112h212Q˜u+ν2≼1ν−uFR∫uνQ˜zdz≼Q˜u+˜Q˜ν∫01h1ϱh21−ϱdϱ.(1)

If h1ϱ=ϱ and h2ϱ≡1, then inequality (1) reduces to the following inequality:

Q˜u+ν2≼1ν−uFR∫uνQ˜zdz≼Q˜u+˜Q˜ν2,(2)

where Q˜ is a convex fuzzy-IVF. We urge the readers to [32–42] and the citations therein for further review of related literature on the implementations and characterization of fuzzy-interval, inequalities and generalized convex fuzzy mappings.

This study is organized as follows: Section 2 presents preliminary notions and results in interval space, and fuzzy-interval space. Moreover, Section 2 also discusses new class of convex fuzzy-IVF, which is known as harmonically convex fuzzy-IVF. Section 3 obtains fuzzy-interval H-H inequalities via convex fuzzy-IVFs. In particular, some intriguing examples are provided to support our outcomes. Conclusions and future plans are discussed in Section 4.

2. PRELIMINARIES

Let KC be the space of all closed and bounded intervals of ℝ and η∈KC be defined by

η=η∗,η∗=z∈ℝ|η∗≤z≤η∗,η∗,η∗∈ℝ.(3)

If η∗=η∗ then, η is said to be degenerate. In this article, all intervals will be nondegenerate intervals. If η∗≥0, then η∗,η∗ is called positive interval. The set of all positive interval is denoted by KC+ and defined as KC+=η∗,η∗:η∗,η∗∈KC and η∗≥0.

Let ϱ∈ℝ and ϱη be defined by

ϱ.η=ϱη∗,ϱη∗ if ϱ≥0,ϱη∗,ϱη∗ if ϱ<0.(4)

Then the Minkowski difference ζ−η, addition η+ζ and η×ζ for η,ζ∈KC are defined by

ζ∗,ζ∗−η∗,η∗=ζ∗−η∗,ζ∗−η∗,ζ∗,ζ∗+η∗,η∗=ζ∗+η∗,ζ∗+η∗,(5)

and

ζ∗,ζ∗×η∗,η∗=minζ∗η∗,ζ∗η∗,ζ∗η∗,ζ∗η∗,maxζ∗η∗,ζ∗η∗,ζ∗η∗,ζ∗η∗.(6)

The inclusion “⊆” means that

ζ⊆η If and only if,ζ∗,ζ∗⊆η∗,η∗,if and only ifη∗≤ζ∗,ζ∗≤η∗.(7)

Remark 2.1.

[32] The relation “≤I” defined on KC by

ζ∗,ζ∗≤Iη∗,η∗ if and only if ζ∗≤η∗,ζ∗≤η∗,(8)

for all ζ∗,ζ∗,η∗,η∗∈KC, it is an order relation. For given ζ∗,ζ∗,η∗,η∗∈KC, we say that ζ∗,ζ∗≤Iη∗,η∗ if and only if ζ∗≤η∗,ζ∗≤η∗ or ζ∗≤η∗,ζ∗<η∗.

For ζ∗,ζ∗,η∗,η∗∈ℝI, the Hausdorff–Pompeiu distance between intervals ζ∗,ζ∗ and η∗,η∗ is defined by

dζ∗,ζ∗,η∗,η∗=max|ζ∗−η∗|,|ζ∗−η∗|.(9)

It is familiar fact that ℝI,d is a complete metric space.

Assume ℝ is a set of real numbers. The membership function is a mapping ζ˜:ℝ→0,1 that characterizes a fuzzy subset A of ℝ, for each fuzzy set and β∈(0,1], then β-level sets of ζ˜ is denoted and defined as follows: ζβ=u∈ℝ|ζ˜u≥β. If β=0, thensuppζ˜=z∈ℝ|ζ˜z>0 is called support of ζ˜. By ζ˜0 we define the closure of suppζ˜.

Let Fℝ be the family of all fuzzy sets and ζ˜∈Fℝ denote the family of all nonempty sets. ζ˜∈Fℝ be a fuzzy set. Then we define the following:

ζ˜ is said to be normal if there exists z∈ℝ and ζ˜z=1;
ζ˜ is said to be upper semi continuous on ℝ if for given z∈ℝ, there exist ε>0 there exist δ>0 such that ζ˜z−ζ˜y<ε for all y∈ℝ with |z−y|<δ;
ζ˜ is said to be fuzzy convex if ζβ is convex for every β∈0,1;
ζ˜ is compactly supported if suppζ˜ is compact.

A fuzzy set is called a fuzzy number or fuzzy interval if it has properties (1), (2), (3) and (4). We denote by F0 the family of all interval.

Let ζ˜∈F0 be a fuzzy-interval, if and only if, β levels ζ˜β is a nonempty compact convex set of ℝ. From these definitions, we have

ζ˜β=ζ∗β,ζ∗β,

where

ζ∗β=infz∈ℝ|ζ˜z≥β,ζ∗β=supz∈ℝ|ζ˜z≥β.

Proposition 2.2.

[18] If ζ˜,η˜∈F0 then relation “≼” defined on F0 by

ζ˜≼η˜ if and only if,ζ˜β≤Iη˜β,for all β∈0,1,(10)

this relation is known as partial order relation.

For ζ˜,η˜∈F0 and ϱ∈ℝ, the sum ζ˜+˜η˜, product ζ˜×˜η˜, scalar product ϱ.ζ˜ and sum with scalar are defined by

Then, for all β∈0,1, we have

ζ˜+˜η˜β=ζ˜β+η˜β,(11)

ζ˜×˜η˜β=ζ˜β×η˜β,(12)

ϱ.ζ˜β=ϱ.ζ˜β.(13)

ϱ+˜ζ˜β=ϱ+ζ˜β.(14)

For ζ˜∈F0 such that ξ˜=η˜+˜ζ˜, then by this result we have existence of Hukuhara difference of ξ˜ and η˜, and we say that ζ˜ is the H-difference of ξ˜ and η˜, and denoted by ξ˜−˜η˜. If H-difference exists, then

ζ∗β=ξ−η∗β=ξ∗β−η∗β,ζ∗β=ξ−η∗β=ζ∗β−η∗β.(15)

A partition of u,ν is any finite ordered subset P having the form

P=u=z1<z2<z3<z4<z5……<zk=ν.

The mesh of a partition P is the maximum length of the subintervals containing P, that is,

mashP=maxzj−zj−1:j=1,2,3,……k.

Let Pδ,u,ν be the set of all P∈Pδ,u,ν such that mesh P<δ. For each interval zj−1,zj, where 1≤j≤k, choose an arbitrary point μj and taking the sum

SQ,P,δ=∑j=1kQμjzj−zj−1,

where Q:u,ν→ℝI. We call SQ,P,δ a Riemann sum of Q corresponding to P∈Pδ,u,ν.

Definition 2.3.

[24] A function Q:u,ν→ℝI is called interval Riemann integrable (IR-integrable) on u,ν if there exists B∈ℝI such that, foe each ϵ, there exists δ>0 such that

dSQ,P,δ,B<ϵ,

for every Riemann sum of Q corresponding to P∈Pδ,u,ν and for arbitrary choice of μj∈zj−1,zj for 1≤j≤k. Then we say that B is the IR-integral of Q on u,ν and is denoted by B=IR∫uνQzdz.

Moore [16] firstly proposed the concept of Riemann integral for IVF and it is defined as follows:

Theorem 2.4.

[16] If Q:u,ν⊂ℝ→ℝI is an IVF on such that Qz=Q∗,Q∗. Then Q is Riemann integrable over u,ν if and only if, Q∗ and Q∗ both are Riemann integrable over u,ν such that

IR∫uνQzdz=R∫uνQ∗udz,R∫uνQ∗udz.(16)

Definition 2.5.

[33] A fuzzy map Q˜:K⊂ℝ→F0 is also known as fuzzy-IVF. For each β∈0,1, whose β levels characterize the family of IVFs Qβ:K⊂ℝ→KC are given by Qβz=Q∗z,β,Q∗z,β for all z∈K. Here, for each β∈0,1, the left and right real-valued functions Q∗z,β,Q∗z,β:K→ℝ are also called lower and upper functions of Q˜.

Remark 2.6.

If Q˜:K⊂ℝ→F0 is a fuzzy-IVF then, Q˜z is called continuous function at z∈K, if for each β∈0,1, both left and right real-valued functions Q∗z,β and Q∗z,β are continuous at z∈K.

The following conclusion can be drawn from the above literature review, see [16,24,33].

Definition 2.7.

Let Q˜:c,d⊂ℝ→F0 be a fuzzy-IVF. Then, fuzzy Riemann integral of Q˜ over c,d, denoted by FR∫cdQ˜zdz, it is defined level by level

FR∫cdQ˜zdzβ=IR∫cdQβzdz=∫cdQz,βdz:Qz,β∈Rc,d,(17)

for all β∈0,1, where Rc,d contains the family of left and right functions of IVFs. Q˜ is FR-integrable over c,d if FR∫cdQ˜zdz∈F0. Note that, if left and right real-valued functions are Lebesgue-integrable, then Q˜ is fuzzy Aumann-integrable over c,d, denoted by FA∫cdQ˜zdz, see [33].

Theorem 2.8.

Let Q˜:c,d⊂ℝ→F0 be a fuzzy-IVF, whose β levels characterize the collection of IVFs Qβ:c,d⊂ℝ→KC are defined by Qβz=Q∗z,β,Q∗z,β for all z∈c,d and for all β∈0,1. Then Q˜ is FR-integrable over c,d if and only if, Q∗z,β and Q∗z,β both are R-integrable over c,d. Moreover, if Q˜ is FR-integrable over c,d, then

FR∫cdQ˜zdzβ=R∫cdQ∗z,βdz,R∫cdQ∗z,βdz=IR∫cdQβzdz,(18)

for all β∈0,1. For each β∈0,1, QRc,d,β and Rc,d,β denote the collection of all FR-integrable fuzzy-IVFs and, R-integrable left and right functions over c,d.

Allahviranloo et al. [39] introduced the following fuzzy-interval Riemann–Liouville fractional integral operators:

Let α>0 and Lu,ν,F0 be the collection of all Lebesgue measurable fuzzy-IVFs on u,ν. Then the fuzzy-interval left and right Riemann–Liouville fractional integral of Q˜∈Lu,ν,F0 with order α>0 are defined by

Iu+αQ˜z=1Γα∫uzz−ϱα−1Q˜ϱdϱ,z>u,(19)

and

Iν−αQ˜z=1Γα∫zνϱ−zα−1Q˜ϱdϱ,(z<ν),(20)

respectively, where Γz=∫0∞ϱz−1u−ϱdϱ is the Euler gamma function. The fuzzy-interval left and right Riemann–Liouville fractional integral z based on left and right end point functions can be defined, that is,

Iu+αQzβ=1Γα∫uzz−ϱα−1Qβϱdϱ=1Γα∫uzz−ϱα−1Q∗ϱ,β,Q∗ϱ,βdϱ,z>u,(21)

where

Iu+αQ∗z,β=1Γα∫uzz−ϱα−1Q∗ϱ,βdϱ,z>u,(22)

and

Iu+αQ∗z,β=1Γα∫uzz−ϱα−1Q∗ϱ,βdϱ,z>u,(23)

Similarly, we can define right Riemann–Liouville fractional integral Q of z based on left and right end point functions.

Definition 2.9.

A set K=u,v⊂ℝ+=0,∞ is said to be harmonically convex set, if, for all z,y∈K,ϱ∈0,1, we have

zyϱz+1−ϱy∈K.(24)

Definition 2.10.

[3] The fuzzy-IVF Q:u,ν→F0 is called harmonically convex fuzzy-IVF on u,ν if

Qzyϱz+1−ϱy≤1−ϱQz+ϱQy,(25)

for all z,y∈u,ν,ϱ∈0,1, where Qz≥0 for all z∈u,ν. If (25) is reversed then, Q is called harmonically concave fuzzy-IVF on u,ν.

Definition 2.11.

The fuzzy-IVF Q˜:u,ν→F0 is called harmonically convex fuzzy-IVF on u,ν if

Q˜zyϱz+1−ϱy≼1−ϱQ˜z+˜ϱQ˜y,(26)

for all z,y∈u,ν,ϱ∈0,1, where Q˜z≽0˜, for all z∈u,ν. If (26) is reversed then, Q˜ is called concave fuzzy-IVF on u,ν.

Theorem 2.12.

Let K be harmonically convex set, and let Q˜:K→FCℝ be a fuzzy-IVF whose β levels define the family of IVFs Qβ:K⊂ℝ→KC+⊂KC are given by

Qβz=Q∗z,β,Q∗z,β,∀z∈K,(27)

for all z∈K, β∈0,1. Then Q˜ is harmonically convex on K, if and only if, for all β∈0,1,Q∗z,β and Q∗z,β are harmonically convex.

Proof.

Assume that for each β∈0,1,Q∗z,β and Q∗z,β are harmonically convex on K. Then from (25), we have

Q∗zyϱz+1−ϱy,β≤1−ϱQ∗z,β+ϱQ∗y,β,

and

Q∗zyϱz+1−ϱy,β≤1−ϱQ∗z,β+ϱQ∗y,β.

Then by (26), (19) and (21), we obtain

Qβzyϱz+1−ϱy=Q∗ϱz+1−ϱy,β,Q∗ϱz+1−ϱy,β,≤I1−ϱQ∗z,β,Q∗z,β +ϱQ∗y,β,Q∗y,β,

that is,

Q˜zyϱz+1−ϱy≼1−ϱQ˜z+˜ϱQ˜y,

∀z,y∈K,ϱ∈0,1. Hence, Q˜ is harmonically convex fuzzy-IVF on K.

Conversely, let Q˜ be harmonically convex fuzzy-IVF on K. Then for all z,y∈K, ϱ∈0,1, we have

Q˜zyϱz+1−ϱy≼1−ϱQ˜z+˜ϱQ˜y.

Therefore, from (26), for each β∈0,1, left side of above inequality, we have

Qβzyϱz+1−ϱy=Q∗zyϱz+1−ϱy,β,Q∗zyϱz+1−ϱy,β.

Again, from (26), we obtain

1−ϱQβz+ϱQβz=1−ϱQ∗z,β,Q∗z,β+ϱQ∗y,β,Q∗y,β,

for all z,y∈K, ϱ∈0,1. Then by harmonically convexity of Q˜, we have for all z,y∈K, ϱ∈0,1 such that

Q∗zyϱz+1−ϱy,β≤1−ϱQ∗z,β+ϱQ∗y,β,

and

Q∗zyϱz+1−ϱy,β≤1−ϱQ∗z,β+ϱQ∗y,β,

for each β∈0,1. Hence, the result follows.

Remark 2.12.

If Q∗z,β=Q∗z,β and β=1 then from Definition 210, we obtain Definition 2.10.

Example 2.13.

We consider the fuzzy-IVFs Q˜:0,2→FCℝ defined by

Q˜zσ=σzσ∈0,z2−σ2z2σ∈(z,2z]0otherwise.

Then, for each β∈0,1, we have Qβz=βz,2−βz. Since end point functions Q∗z,β,Q∗z,β are harmonically convex functions for each β∈0,1. Hence Q˜z is harmonically convex fuzzy-IVF.

In next result, we will establish a relation between convex fuzzy-IVF and harmonically convex fuzzy-IVF.

Theorem 2.14.

Let Q˜:K→FCℝ be a fuzzy-IVF, where for all β∈0,1, whose β levels define the family of IVFs Qβ:K⊂ℝ→KC+⊂KC are given by Qβz=Q∗z,β,Q∗z,β, for all z∈K. Then Q˜z is harmonically convex fuzzy-IVF on K, if and only if, Q˜1z is convex fuzzy-IVF on K.

Proof.

Since Q˜z is a harmonically convex fuzzy-IVF then, for z,y∈u,ν,ϱ∈0,1, we have

Q˜zyϱz+1−ϱy≼1−ϱQ˜z+˜ϱQ˜y.

Therefore, for each β∈0,1, we have

Q∗zyϱz+1−ϱy,β≤1−ϱQ∗z,β+ϱQ∗y,β,Q∗zyϱz+1−ϱy,β≤1−ϱQ∗z,β+ϱQ∗y,β.(28)

Consider φ˜z=Q˜1z. Taking m=1z and n=1y to replace z and y, respectively. Then for each β∈0,1, applying (28)

Q∗1zyϱ1z+1−ϱ1y,β=Q∗11−ϱz+ϱy,β=φ∗1−ϱz+ϱy,β≤ϱQ∗1y,β+1−ϱQ∗1z,β=ϱφ∗y,β+1−ϱφ∗z,β,Q∗1zyϱ1z+1−ϱ1y,β=Q∗11−ϱz+ϱy,β=φ∗1−ϱz+ϱy,β≤ϱQ∗1y,β+1−ϱQ∗1z,β=ϱφ∗y,β+1−ϱφ∗z,β.

It follows that

Q∗1zyϱ1z+1−ϱ1y,β,Q∗1zyϱ1z+1−ϱ1y,β=φ∗1−ϱz+ϱy,β,φ∗1−ϱz+ϱy,β≤ϱφ∗y,β,φ∗y,β+1−ϱφ∗z,β,φ∗z,β.

which implies that

φβ1−ϱz+ϱy≤Iϱφβy+1−ϱφβz,

that is,

φ˜1−ϱz+ϱy≼ϱφ˜y+˜1−ϱφ˜z.

This concludes that φ˜z is a convex fuzzy-IVF.

Conversely, let φ˜ is convex fuzzy-IVF on K. Then, for all z,y∈K, ϱ∈0,1, we have

φ˜ϱz+1−ϱy≼ϱφ˜z+˜1−ϱφ˜y.

By using same steps as above, for each β∈0,1, we have

φ∗ϱ1z+1−ϱ1y,β=Q∗1ϱ1z+1−ϱ1y,β=Q∗zy1−ϱz+ϱy,β≤ϱφ∗1z,β+1−ϱφ∗1y,β=ϱQ∗z,β+1−ϱQ∗y,β

φ∗ϱ1z+1−ϱ1y,β=Q∗1ϱ1z+1−ϱ1y,β=Q∗zy1−ϱz+ϱy,β≤ϱφ∗1z,β+1−ϱφ∗1y,β=ϱQ∗z,β+1−ϱQ∗y,β.

It follows that

Qβzyϱz+1−ϱy≤I1−ϱQβz+ϱQβy,

that is,

Q˜zyϱz+1−ϱy≼1−ϱQ˜z+˜ϱQ˜y,

the proof the theorem has been completed.

Remark 2.15.

If Q∗z,β=Q∗z,β and β=1 then from Theorem 2.14, we obtain Lemma 2.1 of [13].

3. FUZZY-INTERVAL FRACTIONAL HERMITE–HADAMARD INEQUALITIES

In this section, we shall continue with the following the fractional H−H inequality for harmonically convex fuzzy-IVFs and we also give fractional H−H Fejér inequality for harmonically convex fuzzy-IVF through fuzzy order relation. In what follows, we denote by Lu,ν,F0 the family of Lebesgue measureable fuzzy-IVFs.

Theorem 3.1.

Let Q˜:u,ν→F0 be a harmonically convex fuzzy-IVF on u,ν, whose β levels define the family of IVFs Qβ:u,ν⊂ℝ→KC+ are given by Qβz=Q∗z,β,Q∗z,β for all z∈u,ν, β∈0,1. If Q˜∈Lu,ν,F0, then

Q˜2uνu+ν≼Γα+12ν−uαI1u−αQ˜ ∘ ψ1ν+˜I1ν+αQ˜ ∘ ψ1u ≼Q˜u+˜Q˜ν2.(29)

If Q˜z is concave fuzzy-IVF then

Q˜2uνu+ν≽Γα+12ν−uαI1u−αQ˜ ∘ ψ1ν+˜I1ν+αQ˜ ∘ ψ1u ≽Q˜u+˜Q˜ν2.(30)

where ψz=1z.

Proof.

Let Q˜:u,ν→F0 be harmonically convex fuzzy-IVF. Then, by hypothesis, we have

2Q˜2uνu+ν≼Q˜uνϱu+1−ϱν+˜Q˜uν1−ϱu+ϱν.

Therefore, for each β∈0,1, we have

2Q∗2uνu+ν,β≤Q∗uνϱu+1−ϱν,β+Q∗uν1−ϱu+ϱν,β,2Q∗2uνu+ν,β≤Q∗uνϱu+1−ϱν,β+Q∗uν1−ϱu+ϱν,β.

Consider φ˜z=Q˜1z. By Theorem 2.14 we have φ˜z is convex fuzzy-IVF then for each β∈0,1, above inequality, we have

2φ∗u+ν2uν,β≤φ∗ϱu+1−ϱνuν,β+φ∗1−ϱu+ϱνuν,β.

Multiplying both sides by ϱα−1 and integrating the obtained result with respect to ϱ over 0,1, we have

2∫01ϱα−1φ∗u+ν2uν,βdϱ≤∫01ϱα−1φ∗ϱu+1−ϱνuν,βdϱ+∫01ϱα−1φ∗1−ϱu+ϱνuν,βdϱ.

Let z=1−ϱu+ϱνuν and y=ϱu+1−ϱνuν. Then we have

2αφ∗u+ν2uν,β≤uνν−uα∫1ν1u1u−yα−1φ∗y,βdy +uνν−uα∫1ν1uz−1να−1φ∗z,βdz=Γαuνν−uαI1u−αφ∗1ν,β+I1ν+αφ∗1u,β.

Similarly, for Q∗z,γ, we have

2αφ∗u+ν2uν,β≤Γαuνν−uαI1u−αφ∗1ν,β+I1ν+αφ∗1u,β.

It follows that

2φ∗u+ν2uν,β,φ∗u+ν2uν,β≤IΓα+1uνν−uαI1u−αφ∗1ν,β+I1ν+α φ∗1u,β,I1u−αφ∗1ν,β+I1ν+αφ∗1u,β.

That is,

2φ˜u+ν2uν≼Γα+1uνν−uαI1u−αφ˜1ν+˜I1ν+αφ˜1u.(31)

In a similar way as above, we have

Γαuνν−uαI1u−αφ˜1ν+˜I1ν+αφ˜1u≼φ˜1u+˜φ˜1να.(32)

Combining (31) and (32), we have

φ˜u+ν2uν≼Γα+1uνν−uα2I1u−αφ˜1ν+˜I1ν+αφ˜1u ≼φ˜1u+˜φ˜1ν2.

That is,

Q˜2uνu+ν≼Γα+12ν−uαI1u−αQ˜ ∘ ψ1ν+˜I1ν+αQ˜ ∘ ψ1u ≼Q˜u+˜Q˜ν2.

Hence, the required result.

Remark 3.2.

If α=1, then inequality (29) reduces to the following inequality which is also new one:

Q˜2uνu+ν≼uνν−u∫uνQ˜zz2dz≼Q˜u+˜Q˜ν2.(33)

If Q∗z,β=Q∗z,β with β=1 then, we obtain classical fractional H-H inequality for harmonically convex function which is given in [13]:

Q2uνu+ν≤Γα+12ν−uαI1u−αQ ∘ ψ1ν+I1ν+αQ ∘ ψ1u ≤Qu+Qν2.(34)

If Q∗z,β=Q∗z,β with β=1 and α=1 then, we obtain classical H-H inequality for harmonically convex function which is given in [3].

Q2uνu+ν≤uνν−u∫uνQzz2dz≤Qu+Qν2.(35)

Theorem 3.3.

(Second fuzzy fractional H−H Fejér inequality) Let Q˜:u,ν→F0 be a harmonically convex fuzzy-IVF with u<ν, whose β levels define the family of IVFs Qβ:u,ν⊂ℝ→KC+ are given by Qβz=Q∗z,β,Q∗z,β for all z∈u,ν, β∈0,1. If Q˜∈Lu,ν,F0 and Ω:u,ν→ℝ,Ω11u+1ν−1z=Ωz≥0, then

Iu+αQ˜Ω ∘ ψν+˜Iν−αQ˜Ω ∘ ψu≼Q˜u+˜Q˜ν2I1ν+αΩ ∘ ψ1u+I1u−αΩ ∘ ψ1ν.(36)

If Q˜ is concave fuzzy-IVF then, inequality (36) is reversed.

Proof.

Let Q˜ be a harmonically convex fuzzy-IVF and ϱα−1Ωuνϱu+1−ϱν≥0. Then, for each β∈0,1, we have

ϱα−1Q∗uνϱu+1−ϱν,βΩuνϱu+1−ϱν≤ϱα−11−ϱQ∗u,β+ϱQ∗ν,βΩuνϱu+1−ϱν,(37)

and

ϱα−1Q∗uν1−ϱu+ϱν,βΩuνϱu+1−ϱν≤ϱα−1ϱQ∗u,β+1−ϱQ∗ν,βΩuνϱu+1−ϱν.(38)

After adding (37) and (38), and integrating over 0,1, we get

∫01ϱα−1Q∗uνϱu+1−ϱν,βΩuνϱu+1−ϱνdϱ +∫01ϱα−1Q∗uν1−ϱu+ϱν,βΩuνϱu+1−ϱνdϱ≤∫01ϱα−1Q∗u,βϱ+1−ϱΩuνϱu+1−ϱν+ϱα−1Q∗ν,β1−ϱ+ϱΩuνϱu+1−ϱνdϱ,=Q∗u,β∫01ϱα−1Ωuνϱu+1−ϱνdϱ +Q∗ν,β∫01ϱα−1Ωuνϱu+1−ϱνdϱ,

Similarly, for Q∗z,γ, we have

∫01ϱα−1Q∗uνϱu+1−ϱν,βΩuνϱu+1−ϱνdϱ +∫01ϱα−1Q∗uν1−ϱu+ϱν,βΩuνϱu+1−ϱνdϱ=Q∗u,β∫01ϱα−1Ωuνϱu+1−ϱνdϱ +Q∗ν,β∫01ϱα−1Ωuνϱu+1−ϱνdϱ.

From which, we have

Γαuνν−uαIu+αQβΩ ∘ ψν+Iν−αQβΩ ∘ ψu≤IΓαuνν−uαQβu+Qβν2I1ν+αΩ ∘ ψ1u+I1u−αΩ ∘ ψ1ν,

that is,

Iu+αQ˜Ω ∘ ψν+˜Iν−αQ˜Ω ∘ ψu≼Q˜u+˜Q˜ν2I1ν+αΩ ∘ ψ1u+I1u−αΩ ∘ ψ1ν.(39)

Theorem 3.4.

(First fuzzy fractional H−H Fejér inequality) Let Q˜:u,ν→F0 be a harmonically convex fuzzy-IVF with u<ν, whose β levels define the family of IVFs Qβ:u,ν⊂ℝ→KC+ are given by Qβz=Q∗z,β,Q∗z,β for all z∈u,ν, β∈0,1. If Q˜∈Lu,ν,F0 and Ω:u,ν→ℝ,Ω11u+1ν−1z=Ωz≥0, then

Q˜2uνu+νI1ν+αΩ ∘ ψ1u+I1u−αΩ ∘ ψ1ν≼I1ν+αQ˜Ω ∘ ψ1u+˜I1u−αQ˜Ω ∘ ψ1ν≼Q˜u+˜Q˜ν2I1ν+αΩ ∘ ψ1u+I1u−αΩ ∘ ψ1ν.(40)

If Q˜ is concave fuzzy-IVF then, inequality (40) is reversed.

Proof.

Since Q˜ is a harmonically convex fuzzy-IVF, then for β∈0,1, we have

Q∗2uνu+ν,β≤12Q∗uνϱu+1−ϱν,β+Q∗uν1−ϱu+ϱν,β.(41)

Multiplying both sides by (41) by ϱα−1Ωuν1−ϱu+ϱν and then integrating the resultant with respect to ϱ over 0,1, we obtain

Q∗2uνu+ν,β∫01ϱα−1Ωuν1−ϱu+ϱνdϱ≤12∫01ϱα−1Q∗uνϱu+1−ϱν,βΩuν1−ϱu+ϱνdϱ+∫01ϱα−1Q∗uν1−ϱu+ϱν,βΩuν1−ϱu+ϱνdϱ.(42)

Let z=uνϱu+1−ϱν. Then, we have

2uνν−uαQ∗2uνu+ν,β∫1ν1uz−1να−1Ω1z,βdz≤uνν−uα∫1ν1uz−1να−1Q∗11u+1ν−1z,βΩ1zdz +uνν−uα∫u1uz−1να−1Q∗1z,βΩ1zdz=uνν−uα∫1ν1u1u−zα−1Q∗z,βΩ11u+1ν−1zdz +uνν−uα∫1ν1uz−1να−1Q∗1z,βΩ1zdz=Γαuνν−uαℐ1ν+αQ∗Ω1u+ℐ1u−αQ∗Ω1ν,(43)

Similarly, for Q∗z,γ, we have

2uνν−uαQ∗2uνu+ν,β∫1ν1uz−1να−1Ω1z,βdz≤Γαuνν−uαI1ν+αQ∗Ω1u+I1u−αQ∗Ω1ν.(44)

From (43) and (44), we have

Γαuνν−uαQ∗2uνu+ν,β,Q∗2uνu+ν,β .I1ν+αΩ1u+I1u−αΩ1ν≤IΓαuνν−uαI1ν+αQ∗Ω1u+I1u−αQ∗Ω1ν,I1ν+αQ∗Ω1u+I1u−αQ∗Ω1ν,

that is,

Q˜2uνu+νI1ν+αΩ ∘ ψ1u+I1u−αΩ ∘ ψ1ν≼I1ν+αQ˜Ω ∘ ψ1u+˜I1u−αQ˜Ω ∘ ψ1ν.(45)

Similarly, if Q˜ be a harmonically convex fuzzy-IVF and ϱα−1Ωuνϱu+1−ϱν≥0, then, for each β∈0,1, we have

ϱα−1Q∗uνϱu+1−ϱν,βΩuνϱu+1−ϱν≤ϱα−11−ϱQ∗u,β+ϱQ∗ν,βΩuνϱu+1−ϱν(46)

and

ϱα−1Q∗uν1−ϱu+ϱν,βΩuνϱu+1−ϱν≤ϱα−1ϱQ∗u,β+1−ϱQ∗ν,βΩuνϱu+1−ϱν.(47)

After adding (46) and (47), and integrating the resultant over 0,1, we get

∫01ϱα−1Q∗uνϱu+1−ϱν,βΩuνϱu+1−ϱνdϱ +∫01ϱα−1Q∗uν1−ϱu+ϱν,βΩuνϱu+1−ϱνdϱ≤∫01ϱα−1Q∗u,βϱ+1−ϱuνϱu+1−ϱν+ϱα−1Q∗ν,β1−ϱ+ϱuνϱu+1−ϱνdϱ,=Q∗u,β∫01ϱα−1Ωuνϱu+1−ϱνdϱ +Q∗ν,β∫01ϱα−1Ωuνϱu+1−ϱνdϱ.

Similarly, for Q∗z,γ, we have

∫01ϱα−1Q∗uνϱu+1−ϱν,βΩuνϱu+1−ϱνdϱ +∫01ϱα−1Q∗uν1−ϱu+ϱν,βΩuνϱu+1−ϱνdϱ=Q∗u,β∫01ϱα−1Ωuνϱu+1−ϱνdϱ +Q∗ν,β∫01ϱα−1Ωuνϱu+1−ϱνdϱ.

From which, we have

Γαuνν−uαI1ν+αQβΩ ∘ ψν+I1u−αQβΩ ∘ ψ1ν≤IΓαuνν−uαQβu+Qβν2 I1ν+αΩ ∘ ψ1u+I1uαΩ ∘ ψ1ν,

that is,

I1ν+αQ˜Ω ∘ ψ1u+˜I1u−αQ˜Ω ∘ ψ1ν≼Q˜u+˜Q˜ν2I1ν+αΩ ∘ ψ1u+I1u−αΩ ∘ ψ1ν.(48)

By combining (45) and (48), we obtain the required inequality (40).

Remark 3.5.

Let α=1. Then from Theorems 3.3 and 3.4, we get following H-H inequality for harmonically convex fuzzy-IVF which is also new one:

Q˜2uνu+ν∫uνΩzz2dz≼∫uνQ˜zz2Ωzdz ≼Q˜u+˜Q˜ν2∫uνΩzz2dz.

Let Ωz=1. Then from Theorems 3.3 and 3.4, we obtain inequality (29).

Let Ωz=1 and α=1, then from Theorems 3.3 and 3.4, we get H-H inequality for harmonically convex fuzzy-IVF:

Q˜2uνu+ν≼uνν−u∫uνQ˜zz2dz≼Q˜u+˜Q˜ν2.

If Q∗z,β=Q∗z,β with β=1 then from Theorems 3.3 and 3.4, we obtain classical fractional H-H Fejér inequality for harmonically convex function, given in [10].

Let Q∗z,β=Q∗z,β with β=1 and α=1. Then, from Theorems 3.3 and 3.4, we obtain classical H-H-Fejér inequality for harmonically convex function, given in [4].

If Q∗z,β=Q∗z,β with Ωz=β=1 then from Theorems 3.3 and 3.4, we obtain classical fractional H-H inequality for harmonically convex function.

If Q∗z,β=Q∗z,β and Ωz=β=α=1 then from Theorems 3.3 and 3.4, we obtain classical H-H inequality for harmonically convex function.

Now in next results, we will establish some H-H type inequalities for the products of two harmonically convex fuzzy-IVFs involving fuzzy-interval Riemann–Liouville fractional integral. These inequalities about harmonically convex fuzzy-IVFs are analogous generalization for some classical results provided by Noor [7], and Chen [6,13] for convex and generalized harmonically convex functions.

Theorem 3.6.

Let Q˜,P˜:u,ν→F0 be two harmonically convex fuzzy-IVFs on u,ν, whose β levels Qβ,Pβ:u,ν⊂ℝ→KC+ are defined by Qβz=Q∗z,β,Q∗z,β and Pβz=P∗z,β,P∗z,β for all z∈u,ν, β∈0,1. If Q˜ ×˜ P˜∈Lu,ν,F0, then

Γα+12uνν−uαI1ν+αQ˜ ∘ ψ1u×˜P˜ ∘ ψ1u+I1u−αQ˜ ∘ ψ1ν×˜P˜ ∘ ψ1ν≼12−αα+1α+2M˜u,ν+αα+1α+2N˜u,ν.

where M˜u,ν=Q˜u×˜ P˜u+˜ Q˜ν×˜ P˜ν,N˜u,ν=Q˜u×˜ P˜ν+˜ Q˜ν×˜ P˜u, and Mβu,ν=M∗u,ν,β,M∗u,ν,β and Nβu,ν=N∗u,ν,β,N∗u,ν,β.

Proof.

Since Q˜,P˜ both are harmonically convex fuzzy-IVFs then, for each β∈0,1 we have

Q∗uνϱu+1−ϱν,β≤1−ϱQ∗u,β+ϱQ∗ν,β

and

P∗uνϱu+1−ϱν,β≤1−ϱP∗u,β+ϱP∗ν,β.

From the definition of harmonically convex fuzzy-IVFs it follows that 0˜≼Q˜z and 0˜≼P˜z, so

Q∗uνϱu+1−ϱν,β×P∗uνϱu+1−ϱν,β≤1−ϱQ∗u,β+ϱQ∗ν,β1−ϱP∗u,β+ϱP∗ν,β=1−ϱ2Q∗u,β×P∗u,β+ϱ2Q∗ν,β×P∗ν,β + ϱ1−ϱQ∗u,β×P∗ν,β + ϱ1−ϱQ∗ν,β×P∗u,β(49)

Analogously, we have

Q∗uν1−ϱu+ϱν,βP∗uν1−ϱu+ϱν,β≤ϱ2Q∗u,β×P∗u,β +1−ϱ2Q∗ν,β×P∗ν,β +ϱ1−ϱQ∗u,β×P∗ν,β +ϱ1−ϱQ∗ν,β×P∗u,β(50)

Adding (49) and (50), we have

Q∗uνϱu+1−ϱν,β×P∗uνϱu+1−ϱν,β +Q∗uν1−ϱu+ϱν,β×P∗uν1−ϱu+ϱν,β≤ϱ2+1−ϱ2Q∗u,β×P∗u,β+Q∗ν,β×P∗ν,β +2ϱ1−ϱQ∗ν,β×P∗u,β+Q∗u,β×P∗ν,β(51)

Taking multiplication of (51) by ϱα−1 and integrating the obtained result with respect to ϱ over (0, 1), we have

∫01ϱα−1Q∗uνϱu+1−ϱν,β×P∗uνϱu+1−ϱν,βdϱ +∫01ϱα−1Q∗uν1−ϱu+ϱν,β×P∗uν1−ϱu+ϱν,βdϱ≤M∗u,ν,β∫01ϱα−1ϱ2+1−ϱ2dϱ +2N∗u,ν,β∫01ϱα−1ϱ1−ϱdϱ.

It follows that,

Γαuνν−uαI1ν+αQ∗1u,β×P∗1u,β+I1u−αQ∗1ν,β×P∗1ν,β≤2α12−αα+1α+2M∗u,ν,β +2ααα+1α+2N∗u,ν,β

Similarly, for Q∗z,γ, we have

Γαuνν−uαI1ν+αQ∗1u,β×P∗1u,β+I1u−αQ∗1ν,β×P∗1ν,β≤2α12−αα+1α+2M∗u,ν,β +2ααα+1α+2N∗u,ν,β,

that is,

Γαuνν−uαI1ν+αQ∗1u,β×P∗1u,β+I1u−αQ∗1ν,β×P∗1ν,β,I1ν+αQ∗1u,β× P∗1u,β+I1u−αQ∗1ν,β×P∗1ν,β≤I2α12−αα+1α+2M∗u,ν,β,M∗u,ν,β +2ααα+1α+2N∗u,ν,β,N∗u,ν,β.

Thus,

Γα+12uνν−uαI1ν+αQ˜ ∘ ψ1u×˜P˜ ∘ ψ1u+I1u−αQ˜ ∘ ψ1ν×˜P˜ ∘ ψ1ν≼12−αα+1α+2M˜u,ν+αα+1α+2N˜u,ν.

and the theorem has been established.

Theorem 3.7.

Let Q˜,P˜:u,ν→F0 be two harmonically convex fuzzy-IVFs, whose β levels define the family of IVFs Qβ,Pβ:u,ν⊂ℝ→KC+ are given by Qβz=Q∗z,β,Q∗z,β and Pβz=P∗z,β,P∗z,β for all z∈u,ν, β∈0,1. If Q˜ ×˜ P˜∈Lu,ν,F0, then

Q˜2uνu+ν ×˜P˜2uνu+ν≼Γα+14uνν−uαI1ν+αQ˜1u×˜ P˜1u+I1u−αQ˜1ν×˜ P˜1ν +1212−αα+1α+2N˜u,ν +12αα+1α+2M˜u,ν.

where M˜u,ν=Q˜u×˜ P˜u+˜ Q˜ν×˜ P˜ν,N˜u,ν=Q˜u×˜ P˜ν+˜ Q˜ν×˜ P˜u, and Mβu,ν=M∗u,ν,β,M∗u,ν,β and Nβu,ν=N∗u,ν,β,N∗u,ν,β.

Proof.

Consider Q˜,P˜:u,ν→F0 are harmonically convex fuzzy-IVFs. Then by hypothesis, for each β∈0,1, we have

Q∗2uνu+ν,β×P∗2uνu+ν,β≤14Q∗uνϱu+1−ϱν,β×P∗uνϱu+1−ϱν,β+Q∗uνϱu+1−ϱν,β×P∗uν1−ϱu+ϱν,β +14Q∗uν1−ϱu+ϱν,β×P∗uνϱu+1−ϱν,β+Q∗uν1−ϱu+ϱν,β×P∗uν1−ϱu+ϱν,β,≤14Q∗uνϱu+1−ϱν,β×P∗uνϱu+1−ϱν,β+Q∗uν1−ϱu+ϱν,β×P∗uν1−ϱu+ϱν,β +14ϱQ∗u,β+1−ϱQ∗ν,β×1−ϱP∗u,β+ϱP∗ν,β+1−ϱQ∗u,β+ϱQ∗ν,β×ϱP∗u,β+1−ϱP∗ν,β,=14Q∗uνϱu+1−ϱν,β×P∗uνϱu+1−ϱν,β+Q∗uν1−ϱu+ϱν,β×P∗uν1−ϱu+ϱν,β +14ϱ2+1−ϱ2N∗u,ν,β+ϱ1−ϱ+1−ϱϱM∗u,ν,β(52)

Multiplying inequality (52) by ϱα−1 and integrating over 0,1,

Q∗2uνu+ν,β×P∗2uνu+ν,β≤14∫01ϱα−1Q∗uνϱu+1−ϱν,β× P∗uνϱu+1−ϱν,βdϱ+∫01ϱα−1Q∗uν1−ϱu+ϱν,β× P∗uν1−ϱu+ϱν,βdϱ +14N∗u,ν,β∫01ϱα−1ϱ2+1−ϱ2dϱ+ 2M∗u,ν,β∫01ϱα−1ϱ1−ϱdϱ

Taking z=uνϱu+1−ϱν and y=uν1−ϱu+ϱν, then we get

1αQ∗2uνu+ν,β×P∗2uνu+ν,β≤Γα4uνν−uαI1ν+αQ∗ ∘ ψ1u×P∗ ∘ ψ1u+I1u−αQ∗ ∘ ψ1ν×P∗ ∘ ψ1ν +12α12−αα+1α+2N∗u,ν,β +12ααα+1α+2M∗u,ν,β,

1αQ∗2uνu+ν,β×P∗2uνu+ν,β≤Γα4uνν−uαI1ν+αQ∗ ∘ ψ1u×P∗ ∘ ψ1u++I1u−αQ∗ ∘ ψ1ν,β×P∗ ∘ ψ1ν,β +12α12−αα+1α+2N∗u,ν,β +12ααα+1α+2M∗u,ν,β,

Similarly, for Q∗z,γ, we have

1αQ∗2uνu+ν,β×P∗2uνu+ν,β≤Γα4uνν−uαI1ν+αQ∗ ∘ ψ1u×P∗ ∘ ψ1u+I1u−αQ∗ ∘ ψ1ν,β×P∗ ∘ ψ1ν,β +12α12−αα+1α+2N∗u,ν,β +12ααα+1α+2M∗u,ν,β,

that is,

Q˜2uνu+ν×˜P˜2uνu+ν≼Γα+14uνν−uαI1ν+αQ˜1u×˜P˜1u+I1u−αQ˜1ν×˜P˜1ν +1212−αα+1α+2N˜u,ν +12αα+1α+2M˜u,ν.

Hence, the required result.

Theorem 3.8.

Let Q˜,P˜:u,ν→F0 be two harmonically convex fuzzy-IVFs, whose β levels define the family of IVFs Qβ,Pβ:u,ν⊂ℝ→KC+ are given by Qβz=Q∗z,β,Q∗z,β and Pβz=P∗z,β,P∗z,β for all z∈u,ν, β∈0,1. If Q˜ ×˜ P˜∈Lu,ν,F0, then

2Q˜2uνu+ν ×˜ P˜2uνu+ν≼Γα+121−αuνν−uαIu+ν2uν+αQ˜ ∘ ψ1u×˜P˜ ∘ ψ1u+Iu+ν2uν−αQ˜ ∘ ψ1ν×˜P˜ ∘ ψ1ν +12−α2+3α4α+1α+2N˜u,ν +α2+3α4α+1α+2M˜u,ν.

where M˜u,ν=Q˜u×˜ P˜u+˜ Q˜ν×˜ P˜ν,N˜u,ν=Q˜u×˜ P˜ν+˜ Q˜ν×˜ P˜u, and Mβu,ν=M∗u,ν,β,M∗u,ν,β and Nβu,ν=N∗u,ν,β,N∗u,ν,β.

Proof.

Consider Q˜,P˜:u,ν→F0 are harmonically convex fuzzy-IVFs. Then by hypothesis, for each β∈0,1, we have

Q∗2uνu+ν,β×P∗2uνu+ν,β≤14Q∗uνϱu+1−ϱν,β×P∗uνϱu+1−ϱν,β+Q∗uνϱu+1−ϱν,β×P∗uν1−ϱu+ϱν,β +14Q∗uν1−ϱu+ϱν,β×P∗uνϱu+1−ϱν,β+Q∗uν1−ϱu+ϱν,β×P∗uν1−ϱu+ϱν,β,≤14Q∗uνϱu+1−ϱν,β×P∗uνϱu+1−ϱν,β+Q∗uν1−ϱu+ϱν,β×P∗uν1−ϱu+ϱν,β +14ϱQ∗u,β+1−ϱQ∗ν,β×1−ϱP∗u,β+ϱP∗ν,β+1−ϱQ∗u,β+ϱQ∗ν,β×ϱP∗u,β+1−ϱP∗ν,β,=14Q∗uνϱu+1−ϱν,β×P∗uνϱu+1−ϱν,β+Q∗uν1−ϱu+ϱν,β×P∗uν1−ϱu+ϱν,β +14ϱ2+1−ϱ2N∗u,ν,β+2ϱ1−ϱM∗u,ν,β(53)

Multiplying inequality (53) by 21+ααϱα−1 and then integrating the obtain outcome over 0,12,

Q∗2uνu+ν,β×P∗2uνu+ν,β≤14∫01221+ααϱα−1Q∗uνϱu+1−ϱν,β× P∗uνϱu+1−ϱν,β+ Q∗uν1−ϱu+ϱν,β× P∗uν1−ϱu+ϱν,βdϱ. +14N∗u,ν,β∫01221+ααϱα−1ϱ2+1−ϱ2dϱ+2M∗u,ν,β∫01221+ααϱα−1ϱ1−ϱdϱ

Taking z=uνϱu+1−ϱν and y=uν1−ϱu+ϱν, then we get

2Q∗2uνu+ν,β×P∗2uνu+ν,β≤Γα+121−αuνν−uαI1ν+αQ∗ ∘ ψ1u×P∗ ∘ ψ1u+I1u−αQ∗ ∘ ψ1ν×P∗ ∘ ψ1ν +12−αα+1α+2N∗u,ν,β +αα+1α+2M∗u,ν,β

Similarly, for Q∗z,γ, we have

2Q∗2uνu+ν,β×P∗2uνu+ν,β≤Γα+121−αuνν−uαI1ν+αQ∗ ∘ ψ1u×P∗ ∘ ψ1u+I1u−αQ∗ ∘ ψ1ν×P∗ ∘ ψ1ν,β +12−α2+3α4α+1α+2N∗u,ν,β +α2+3α4α+1α+2M∗u,ν,β,

that is,

2Q˜2uνu+ν×˜ P˜2uνu+ν≼Γα+121−αuνν−uαIu+ν2uν+αQ˜ ∘ ψ1u×˜ P˜ ∘ ψ1u+˜Iu+ν2uν−αQ˜ ∘ ψ1ν×˜ P˜ ∘ ψ1ν+12−α2+3α4α+1α+2N˜u,ν+α2+3α4α+1α+2M˜u,ν.

4. CONCLUSION AND FUTURE STUDY

In this study, firstly we introduced the class of harmonically convex fuzzy-IVFs by means of fuzzy-order relation. Then we established H-H and H-H Fejér type inequalities for convex fuzzy-IVFs involving fuzzy Riemann–Liouville fractional integrals, and H-H inequalities are true for this concept of harmonically convex fuzzy-IVFs. As a future research, we try to explore this concept for generalized harmonically convex fuzzy-IVFs and some applications in fuzzy-interval nonlinear programing. By using this concept, the new direction of study can be found in optimization theory and convex analysis. We hope that this concept will be helpful for other authors to pay their roles in different fields of sciences.

AVAILABILITY OF DATA AND MATERIALS

Not applicable.

CONFLICTS OF INTEREST

The authors declare that they have no competing interests.

AUTHORS' CONTRIBUTIONS

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Funding Statement

The research is supported by the National Natural Science Foundation of China (Grant No. 61673169).

ACKNOWLEDGMENTS

The authors would like to thank the Rector, COMSATS University Islamabad, Islamabad, Pakistan, for providing excellent research and academic environments.

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Journal: International Journal of Computational Intelligence Systems
Volume-Issue: 14 - 1
Pages: 1809 - 1822
Publication Date: 2021/06/28
ISSN (Online): 1875-6883
ISSN (Print): 1875-6891
DOI: 10.2991/ijcis.d.210620.001 How to use a DOI?
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TY  - JOUR
AU  - Gul Sana
AU  - Muhammad Bilal Khan
AU  - Muhammad Aslam Noor
AU  - Pshtiwan Othman Mohammed
AU  - Yu-Ming Chu
PY  - 2021
DA  - 2021/06/28
TI  - Harmonically Convex Fuzzy-Interval-Valued Functions and Fuzzy-Interval Riemann–Liouville Fractional Integral Inequalities
JO  - International Journal of Computational Intelligence Systems
SP  - 1809
EP  - 1822
VL  - 14
IS  - 1
SN  - 1875-6883
UR  - https://doi.org/10.2991/ijcis.d.210620.001
DO  - 10.2991/ijcis.d.210620.001
ID  - Sana2021
ER  -

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