Some New Classes of Preinvex Fuzzy-Interval-Valued Functions and Inequalities

Muhammad Bilal Khan; Muhammad Aslam Noor; Lazim Abdullah; Yu-Ming Chu

doi:10.2991/ijcis.d.210409.001

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Volume 14, Issue 1, 2021, Pages 1403 - 1418

Some New Classes of Preinvex Fuzzy-Interval-Valued Functions and Inequalities

Authors

Muhammad Bilal Khan¹, Muhammad Aslam Noor¹, Lazim Abdullah²^,, Yu-Ming Chu³^,

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

²Management Science Research Group, Faculty of Ocean Engineering Technology and Informatics, University Malaysia Terengganu, Kuala Terengganu, Malaysia

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

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

Corresponding Author

Yu-Ming Chu

Received 24 December 2020, Accepted 18 March 2021, Available Online 19 April 2021.

DOI: 10.2991/ijcis.d.210409.001 How to use a DOI?
Keywords: Fuzzy-interval-valued functions; Fuzzy Riemann integrals; h1,h2-preinvex fuzzy-interval-valued functions; Hermite–Hadamard inequalities
Abstract: It is well known that convexity and nonconvexity develop a strong relationship with different types of integral inequalities. Due to the importance of the concept of nonconvexity and integral inequality, in this paper, we present some new classes of preinvex fuzzy-interval-valued functions involving two arbitrary auxiliary functions is known as h1,h2-preinvex fuzzy-interval-valued functions (h1,h2-preinvex fuzzy-IVFs). With the help of these classes, we derive some new Hermite–Hadamard inequalities (HH-inequalities) by means of fuzzy order relation on fuzzy-interval space and verify with the support of some nontrivial examples. This fuzzy order relation is defined level-wise through Kulisch–Miranker order relation defined on fuzzy-interval space. Moreover, several new and previously known results are also discussed which can be deducted from our main results. These results and different approaches may open new directions for fuzzy optimization problems, modeling, and interval-valued functions.
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

In the development of pure and applied mathematics [1–3] convexity has played a key role. Convex sets and convex functions have been generalized and expanded in many mathematical directions due to their robustness; see [4–10]. In particular, several inequalities can be obtained in literature through convexity theory. In linear programing, combinatory, orthogonal polynomials, quantum theory, number theory, optimization theory, dynamics, and in the theory of relativity, integral inequalities [11–16] have various applications. This subject has gained substantial attention from researchers [17–19] and is therefore considered to be an integrative topic between economics, mathematics, physics, and statistics [1,2,20]. To the best of understanding, the HH-inequality is a familiar, supreme, and broadly useful inequality [2,21–27]. This inequality has fundamental significance [21,22] due to other classical inequalities such as the Oslen and Gagliardo–Nirenberg, Hardy, Oslen, Opial, Young, Linger, Arithmetic’s-Geometric, Ostrowski, levison, Minkowski, Beckenbach-Dresher, Ky-fan and Holer inequality, which are closely linked to the classical HH-inequality. It can be stated as follows:

Let ℱ:K→ℝ be a convex function on a convex set K and u,ϑ∈K with u≤ϑ. Then,

ℱu+ϑ2≤1ϑ−u∫uϑℱxdx≤ℱu+ℱϑ2.(1)

Fejér considered the major generalizations of HH-inequality in [18] which is known as HH-Fejér inequality.

Let ℱ:K→ℝ be a convex function on a convex set K and u,ϑ∈K with u≤ϑ. Then,

ℱu+ϑ2≤1∫uϑΩxdx∫uϑℱxΩxdx≤ℱu+ℱϑ2∫uϑΩ(x))dx(2)

where Ω:u,ϑ→ℝ with Ω(x)≥0, is a symmetric function with respect to u+ϑ2, and ∫uϑΩ(x)dx>0. If Ωx=1 then, we obtain (1) from (2). With the assistance of inequality (1), many classical inequalities can be obtained through special convex function. In addition, these inequalities have a very significant role for convex functions in both pure and applied mathematics.

Hanson [28] initiated to introduce a generalized class of convexity which is known as an invex function. The invex function played a significant role in mathematical programing. A step forward, the invex set and preinvex function were introduced and studied by Israel and Mond [10]. Also, Noor [29] examined the optimality conditions of differentiable preinvex functions and proved that variational-like inequalities would characterize the minimum. Many generalizations and extensions of classical convexity have been investigated by several authors. In 2007, Noor [26] introduced h-preinvex functions and with the help of this concept, presented the following HH-inequality for h-preinvex function:

1h12ℱ2u+∂ϑ,u2≤1∂ϑ,u∫uu+∂ϑ,uℱxdx≤ℱu+ℱϑ∫01h(τ)dτ,(3)

where ∂:u,ϑ×u,ϑ→u,ϑ and ℱ:K→ℝ+ is a preinvex function on the invex set K=u,u+∂ϑ,u with u<u+∂ϑ,u and h:[0,1]→ℝ+ with h12≠0. A step forward, Marian Matloka [30] constructed HH-Fejér inequalities for h-preinvex function and investigated some different properties of differentiable preinvex function. Recently, Noor et al. [31] new classes of preinvex functions involving two arbitrary auxiliary functions and derived some following HH-inequalities for these classes of preinvex functions:

12h112h212ℱ2u+∂ϑ,u2≤1∂ϑ,u∫uu+∂ϑ,uℱxdx≤ℱu+ℱϑ∫01h1τh21−τdτ.(4)

where ℱ:K→ℝ+ is a h1,h2-preinvex function on the invex set K=u,u+∂ϑ,u with u<u+∂ϑ,u and h1,h2:[0,1]→ℝ+ with h112h212≠0.

From the other side, because of the absence of implementations of the theory of interval analysis in other sciences, this theory fell into oblivion for a long time. Moore [32] and Kulish and Miranker [33] suggested and investigated the idea of interval analysis. In numerical analysis, it is first used to evaluate the error bounds of a finite state machine’s numerical solutions. We refer the readers [34–36] and the references therein for basic information and applications.

Recently, Zhao et al. [37] introduced h-convex interval-valued functions (h-convex interval-valued functions (IVFs), in short) and proved the HH-type inequalities and Jensen HH-type inequalities for h-convex IVFs. Besides, An et al. [38] defined the class of h1,h2-convex IVFs and established following interval-valued HH-type inequality for h1,h2-convex IVFs.

Let ℱ:u,ϑ⊂ℝ→KC+ be an (h1,h2)-convex IVF given by ℱx=ℱ*x,ℱ*x for all x∈u,ϑ, with h1,h2:[0,1]→ℝ+ with h112h212≠0, where ℱ*x and ℱ*x both are (h1,h2)-convex function. If ℱ is Riemann integrable (in sort, IR-integrable) then,

12h112h212ℱu+ϑ2⊇1ϑ−uIR∫uϑℱxdx⊇ℱu+ℱϑ∫01h1τh21−τdτ.(5)

For further review of the literature on the applications and properties of generalized convex functions and Hermite–Hadamrd inequalities, see [39–42] and the references therein.

Similarly, the notions of convexity and generalized convexity play a vital role in optimization under fuzzy domain because during characterization of the optimality condition of convexity, we obtain fuzzy variational inequalities so variational inequality theory and fuzzy complementary problem theory established powerful mechanism of the mathematical problems and they have a friendly relationship. Many authors contributed to this fascinating and interesting field. In 1989, Nanda and Kar [43] initiated to introduce convex fuzzy mappings from convex set to the set of fuzzy numbers and characterized the notion of convex fuzzy mapping through the idea of epigraph. A step forward, Furukawa [44] and Syau [45] proposed and examined fuzzy mapping from space ℝn to the set of fuzzy numbers, fuzzy valued Lipschitz continuity, logarithmic convex fuzzy mappings, and quasi-convex fuzzy mappings. Besides, Chang [1] discussed the idea of convex fuzzy mapping and find its optimality condition with the support of fuzzy variational inequality. Generalization and extension of fuzzy convexity play a vital and significant implementation in diverse directions. So let’s note that, one of the most considered classes of nonconvex fuzzy mapping is preinvex fuzzy mapping. Noor [46] introduced this idea and proved some results that distinguish the fuzzy optimality condition of differentiable preinvex fuzzy mappings by fuzzy variational-like inequality. Fuzzy variational inequality theory and complementary problem theory established a strong relationship with mathematical problems. For more useful details about the applications and properties of variational inequalities and generalized convex fuzzy mappings, see [47–52] and the references therein.

The fuzzy mappings are fuzzy-IVFs. There are some integrals to deal with fuzzy-IVFs, where the integrands are fuzzy-IVFs. For instance, Oseuna-Gomez et al. [53] and Costa et al. [54] constructed Jensen’s integral inequality for fuzzy-IVFs. By using same approach Costa and Floures also presented Minkowski and Beckenbach’s inequalities, where the integrands are fuzzy-IVFs. Motivated by [41,42,53,54] and especially by [55] because Costa et al established relation between elements of fuzzy-interval space and interval space, and introduced level-wise fuzzy order relation on fuzzy-interval space through Kulisch–Miranker order relation defined on interval space. By using this concept on fuzzy-interval space, we generalize integral inequality (1) by constructing fuzzy-interval integral inequalities for convex fuzzy-IVFs, where the integrands are convex fuzzy-IVFs.

This study is organized as follows: Section 2 presents preliminary and new concepts and results in interval space, the space of fuzzy intervals, and fuzzy convex analysis. Section 3 obtains fuzzy-interval HH-inequalities and HH-Fejér inequalities via h1,h2-preinvex fuzzy-IVFs. In addition, some interesting examples are also given to verify our results. Section 4 gives conclusions and future plans.

2. PRELIMINARIES

In this section, we recall some basic preliminary notions, definitions, and results. With the help of these results, some new basic definitions and results are also discussed.

We begin by recalling basic notations and definitions. We define interval,

ω*,ω*=x∈ℝ:ω*≤x≤ω* and ω*,ω*∈ℝ,

where ω*≤ω*.

We write len ω*,ω*=ω*−ω*, If len ω*,ω*=0 then, ω*,ω* is called degenerate. In this article, all intervals will be nondegenerate intervals. The collection of all closed and bounded intervals of ℝ is denoted defined as KC=ω*,ω*:ω*,ω*∈ℝ and ω*≤ω*. If ω*≥0 then, ω*,ω* is called positive interval. The set of all positive interval is denoted by KC+ and defined as KC+=ω*,ω*:ω*,ω* ∈KC and ω*≥0.

We now discuss some properties of intervals under the arithmetic operations addition, multiplication, and scalar multiplication. If μ*,μ*,ω*,ω*∈KC and ρ∈ℝ then, arithmetic operations are defined by

μ*,μ*+ω*,ω*=μ*+ω*,μ*+ω*,

μ*,μ*×ω*,ω*=minμ*ω*,μ*ω*,μ*ω*,μ*ω*,maxμ*ω*,μ*ω*,μ*ω*,μ*ω*,

ρ.μ*,μ*=ρμ*,ρμ*if ρ≥0,ρμ*,ρμ*if ρ<0.

For μ*,μ*,ω*,ω*∈KC, the inclusion “⊆" is defined by

μ*,μ*⊆ω*,ω*, if and only if ω*≤μ*, μ*≤ω*.

Remark 2.1.

The relation “≤I" defined on KC by

μ*,μ*≤Iω*,ω* if and only if μ*≤ω*,μ*≤ω*,

for all μ*,μ*,ω*,ω*∈KC, it is an order relation, see [33]. For given μ*,μ*,ω*,ω*∈KC, we say that μ*,μ*≤Iω*,ω* if and only if μ*≤ω*,μ*≤ω* or μ*≤ω*,μ*<ω*.

The concept of Riemann integral for IVF first introduced by Moore [32] is defined as follows:

Theorem 2.2

If ℱ:[c,d]⊂ℝ→KC is an IVF such that ℱ*,ℱ* then, ℱ is Riemann integrable over: [c,d] if and only if, ℱ* and ℱ* both are Riemann integrable over: c,d such that

IR∫cdℱxdx=R∫cdℱ*udx,R∫cdℱ*udx

The collection of all Riemann integrable real valued functions and Riemann integrable IVFs is denoted by ℛ[c,d] and ℐℛ[c,d], respectively.

Let ℝ be the set of real numbers. A fuzzy subset set A of ℝ is distinguished by a function φ:ℝ→[0,1] called the membership function. In this study this depiction is approved. Moreover, the collection of all fuzzy subsets of ℝ is denoted by Fℝ.

A real fuzzy-interval φ is a fuzzy set in ℝ with the following properties:

φ is normal, i.e., there exists x∈ℝ such that φx=1;
φ is upper semi continuous, i.e., for given x∈ℝ, there exist ε>0 there exist δ>0 such that φx−φy<ε for all y∈ℝ with |x−y|<δ.
φ is fuzzy convex, i.e., φ1−τx+τy≥min(φ(x), φ(y),∀ x,y∈ℝ and τ∈[0,1];
φ is compactly supported i.e., clx∈ℝ| φx>0 is compact.

The collection of all real fuzzy-intervals is denoted by FCℝ.

Let φ∈FCℝ be real fuzzy-interval, if and only if, γ-levels φγ is a nonempty compact convex set of ℝ. This is represented by

φγ=x∈ℝ|φx≥γ,

from these definitions, we have

φγ=φ*γ,φ*γ,

where

φ*γ=infx∈ℝ|φx≥γ,φ*γ=supx∈ℝ|φx≥γ.

Thus a real fuzzy-interval φ can be identified by a parametrized triples

φ*γ,φ*γ,γ:γ∈0,1.

This leads the following characterization of a real fuzzy-interval in terms of the two end point functions φ*γ and φ*γ.

Theorem 2.3

[9] Suppose that φ*γ:[0,1]→ℝ and φ*γ:[0,1]→ℝ satisfy the following conditions:

φ*γ is a nondecreasing function.
φ*γ is a nonincreasing function.
φ*1≤φ*1.
φ*γ and φ*γ are bounded and left continuous on (0,1] and right continuous at γ=0.

Moreover, If φ:ℝ→[0,1] is a real fuzzy-interval given by φ*γ,φ*γ then, function φ*γ and φ*γ find the conditions (1–4).

Proposition 2.4.

[55] Let φ,ϕ∈FCℝ. Then, fuzzy order relation “≼" given on FCℝ by

φ≼ϕ if and only if, φγ≤Iϕγ for all γ∈[0,1],

it is partial order relation.

We now discuss some properties of real fuzzy-intervals under addition, scalar multiplication, multiplication, and division. If φ,ϕ∈FCℝ and ρ∈ℝ then, arithmetic operations are defined by

φ+˜ϕγ=φγ+ϕγ,(6)

φ×˜ϕγ=φγ×ϕγ,(7)

ρ.φγ=ρ.φγ(8)

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

ψγ=φ−˜ϕγ=φγ−ϕγ,(9)

where ψ*γ=φ−˜ϕ*γ=φ*γ−ϕ*(γ), ψ*γ=φ−˜ϕ*γ=φ*γ−ϕ*γ.

Remark 2.5.

Obviously, FCℝ is closed under addition and nonnegative scaler multiplication and above defined properties on FCℝ are equivalent to those derived from the usual extension principle. Furthermore, for each scalar number ρ∈ℝ,

ρ+˜φγ=ρ+φγ.(10)

Theorem 2.6.

[17] The space FCℝ dealing with a supremum metric i.e, for ψ,ϕ∈FCℝ

Dψ,ϕ=sup0≤γ≤1Hψγ,ϕγ,

it is a complete metric space, where H denote the well-known Hausdorff metric on space of intervals.

Definition 2.7.

[55] A fuzzy-interval-valued map ℱ:K⊂ℝ→FCℝ is called fuzzy-IVF. For each γ∈0,1, whose γ-levels define the family of IVFs ℱγ:K⊂ℝ→KC are given by ℱγx=ℱ*x,γ,ℱ*x,γ for all x∈K. Here, for each γ∈0,1, the end point real functions ℱ*.,γ,ℱ*.,γ:K→ℝ are called lower and upper functions of ℱ.

Remark 2.8.

Let ℱ:K⊂ℝ→FCℝ be a fuzzy-IVF. Then, ℱx is said to be continuous at x∈K, if for each γ∈0,1, both end point functions ℱ*x,γ and ℱ*x,γ are continuous at x∈K.

From above literature review, following results can be concluded, see [33,49,50,55]:

Definition 2.9.

Let ℱ:[c,d]⊂ℝ→FCℝ is called fuzzy-IVF. The fuzzy integral of ℱ over c,d, denoted by FR∫cdℱxdx, it is defined level-wise by

FR∫cdℱxdxγ=IR∫cdℱγxdx=∫cdℱx,γdx:ℱx,γ∈ℛ[c,d],(11)

for all γ∈0,1, where ℛ[c,d] is the collection of end point functions of IVFs. ℱ is FR-integrable over [c,d] if FR∫cdℱxdx∈FCℝ. Note that, if both end point functions are Lebesgue-integrable then, ℱ is fuzzy Aumann-integrable, see [33,49,50,55].

Theorem 2.10.

Let ℱ:[c,d]⊂ℝ→FCℝ be a fuzzy-IVF, whose γ-levels define the family of IVFs ℱγ:[c,d]⊂ℝ→KC are given by ℱγx=ℱ*x,γ,ℱ*x,γ for all x∈[c,d] and for all γ∈0,1. Then, ℱ is FR-integrable over [c,d] if and only if, ℱ*x,γ and ℱ*x,γ both are R-integrable over [c,d]. Moreover, if ℱ is FR-integrable over c,d then,

FR∫cdℱxdxγ=R∫cdℱ*x,γdx,R∫cdℱ*x,γdx=IR∫cdℱγxdx,(12)

for all γ∈0,1.

The family of all FR-integrable fuzzy-IVFs and R-integrable functions over [c,d] are denoted by ℱℛc,d,γ and ℛc,d,γ, for all γ∈0,1.

Definition 2.11.

[46] Let K be an invex set. Then, fuzzy-IVF ℱ:K→FCℝ is said to be preinvex on K with respect to ∂ if

ℱx+1−τ∂(y,x)≼τℱx+˜1−τℱy,

for all x,y∈K,τ∈0,1, where ℱx≽0˜, ∂:K×K→ℝ. If ℱ is preconcave fuzzy-IVF then, −ℱ is preinvex fuzzy-IVF.

Definition 2.13.

Let K be an invex set and h1,h2:[0,1]⊆K→ℝ such that h1,h2≢0. Then, fuzzy-IVF ℱ:K→FCℝ is said to be

h1,h2-preinvex on K if
ℱx+1−τ∂y,x≼h1τh21−τℱx+˜h11−τh2τℱy,(13)
for all x,y∈K,τ∈0,1, where ℱx≽0˜ and ∂:K×K→ℝ.
h1,h2-preconcave on K if inequality (13) is reversed.
affine h1,h2-preinvex on K if
ℱx+1−τ∂y,x=h1τh21−τℱx+˜h11−τh2τℱy,(14)

for all x,y∈K,τ∈0,1, where ℱx≽0˜ and ∂:K×K→ℝ.

Remark 2.14.

If h2τ≡1 then, h1,h2-preinvex fuzzy-IVF becomes h1-preinvex fuzzy-IVF, i.e.,

ℱx+1−τ∂y,x≼h1τℱx+˜h11−τℱy,∀ x,y∈K,τ∈0,1.

If h1τ=τs, h2τ≡1 then, h1,h2-preinvex fuzzy-IVF becomes s-preinvex fuzzy-IVF, i.e.,

ℱx+1−τ∂y,x≼τsℱx+˜1−τsℱy,∀x,y∈K,τ∈0,1.

If h1τ=τ,h2τ≡1 then, h1,h2-preinvex fuzzy-IVF becomes preinvex fuzzy-IVF, i.e.,

ℱx+1−τ∂y,x≼ℱx+˜1−τℱy,∀ x,y∈K,τ∈0,1.

If h1τ=h2τ≡1 then, h1,h2-preinvex fuzzy-IVF becomes P-preinvex fuzzy-IVF, i.e.,

ℱx+1−τ∂y,x≼ℱx+˜ℱy,∀ x,y∈K,τ∈0,1.

Theorem 2.15.

Let K be an invex set and non-negative real-valued function h1,h2:[0,1]⊆K→ℝ such that h1,h2≢0. Let ℱ:K→FCℝ be a fuzzy-IVF with ℱx≽0˜, whose γ-levels define the family of IVFs ℱγ:[c,d]⊂ℝ→KC+ are given by

ℱγx=ℱ*x,γ,ℱ*x,γ,∀x∈K.(15)

for all x∈[c,d] and for all γ∈0,1. Then, ℱ is h1,h2-preinvex fuzzy-IVF on K, if and only if, for all γ∈0,1,ℱ*x,γ and ℱ*x,γ both are h1,h2-preinvex functions. (16)

Proof.

Assume that for each γ∈0,1,ℱ*x,γ and ℱ*x,γ are h1,h2-preinvex on K. Then, from (13), we have

ℱ*x+1−τ∂y,x,γ≤h1τh21−τℱ*x,γ+ h11−τh2τℱ*y,γ

for all x,y∈K,τ∈0,1,

And

ℱ*x+1−τ∂y,x,γ≤h1τh21−τℱ*x,γ+h11−τh2τℱ*y,γ,

for all x,y∈K,τ∈0,1.

Then, by (15), (5) and (7), we obtain

ℱγx+1−τ∂y,x=ℱ*x+1−τ∂(y,x),γ,ℱ*x+1−τ∂(y,x),γ,≤h1τh21−τℱ*x,γ,h1τh21−τℱ*x,γ+h11−τh2τℱ*y,γ,h11−τh2τℱ*y,γ,

i.e.,

ℱx+1−τ∂y,x≼h1τh21−τℱx+˜ h11−τh2τℱy, ∀ x,y∈K,τ∈0,1.

Hence, ℱ is h1,h2-preinvex fuzzy-IVF on K.

Conversely, let ℱ is h1,h2-preinvex fuzzy-IVF on K. Then, for all x,y∈K and τ∈0,1, we have ℱx+1−τ∂(y,x)≼h1τh21−τℱx+˜h11−τh2τℱy. Therefore, from (15), we have

ℱγx+1−τ∂(y,x)=ℱ*x+1−τ∂(y,x),γ,ℱ*x+1−τ∂(y,x),γ.

Again, from (15), (5) and (7), we obtain

h1τh21−τℱγx+˜h11−τh2τℱγx=h1τh21−τℱ*x,γ,h1τh21−τℱ*x,γ+h11−τh2τℱ*y,γ,h11−τh2τℱ*y,γ,

for all x,y∈K and τ∈0,1. Then, by h1,h2-preinvexity of ℱ, we have for all x,y∈K and τ∈0,1 such that

ℱ*x+1−τ∂y,x,γ≤h1τh21−τℱ*x,γ+h11−τh2τℱ*y,γ,

and

ℱ*x+1−τ∂y,x,γ≤h1τh21−τℱ*x,γ+h11−τh2τℱ*y,γ,

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

Theorem 2.16.

Let K be an invex set and non-negative real-valued function h1,h2:[0,1]⊆K→ℝ such that h1,h2≢0. Let ℱ:K→FCℝ be a fuzzy-IVF with ℱx≽0˜, whose γ-levels define the family of IVFs ℱγ:[c,d]⊂ℝ→KC+ are given by

ℱγx=ℱ*x,γ,ℱ*x,γ,∀x∈K.(17)

for all x∈[c,d] and for all γ∈0,1. Then, ℱ is h1,h2-preconcave fuzzy-INF on K, if and only if, for all γ∈0,1, ℱ*x,γ and ℱ*x,γ both are h1,h2-preconcave function. (18)

Proof.

The demonstration is analogous to the demonstration of Theorem 2.15.

3. MAIN RESULTS

In this section, we put forward some HH-inequalities for h1,h2-preinvex fuzzy-IVFs via fuzzy Riemann integrals.

Theorem 3.1.

Let ℱ:u,u+∂(ϑ,u)→FCℝ be a h1,h2-preinvex fuzzy-IVF with non-negative real-valued functions h1,h2:[0,1]→ℝ and h112h212≠0, whose γ-levels define the family of IVFs ℱγ:u,u+∂(ϑ,u)⊂ℝ→KC+ are given by ℱγx=ℱ*x,γ,ℱ*x,γ for all x∈u,u+∂(ϑ,u) and for all γ∈0,1. If ℱ∈ℱℛu,u+∂(ϑ,u),γ then,

12h112h212ℱ2u+∂ϑ,u2≼1∂ϑ,uFR∫uu+∂ϑ,uℱxdx≼ℱu+˜ℱϑ∫01h1τh21−τdτ.(19)

Proof.

Let ℱ:u,u+∂ϑ,u→FCℝ, h1,h2-preinvex fuzzy-IVF. Then, by hypothesis, we have

1h112h212ℱ2u+∂ϑ,u2≼ℱu+1−τ∂ϑ,u+˜ℱu+τ∂ϑ,u.

Therefore, for every γ∈[0,1], we have

1h112h212ℱ*2u+∂ϑ,u2,γ≤ℱ*u+1−τ∂ϑ,u,γ+ℱ*u+τ∂ϑ,u,γ,1h112h212ℱ*2u+∂ϑ,u2,γ≤ℱ*u+1−τ∂ϑ,u,γ+ℱ*u+τ∂ϑ,u,γ.

Then,

1h112h212∫01ℱ*2u+∂ϑ,u2,γdτ≤∫01ℱ*u+1−τ∂ϑ,u,γdτ+∫01ℱ*u+τ∂ϑ,u,γdτ,1h112h212∫01ℱ*2u+∂ϑ,u2,γdτ≤∫01ℱ*u+1−τ∂ϑ,u,γdτ+∫01ℱ*u+τ∂ϑ,u,γdτ.

It follows that

1h112h212ℱ*2u+∂ϑ,u2,γ≤2∂ϑ,u∫uu+∂ϑ,uℱ*x,γdx,1h112h212ℱ*2u+∂ϑ,u2,γ≤2∂ϑ,u∫uu+∂ϑ,uℱ*x,γdx.

That is,

1h112h212ℱ*2u+∂ϑ,u2,γ,ℱ*2u+∂ϑ,u2,γ≤I2∂ϑ,u∫uu+∂ϑ,uℱ*x,γdx,∫uu+∂ϑ,uℱ*x,γdx.

Thus,

12h112h212ℱ2u+∂ϑ,u2≼1∂ϑ,uFR∫uu+∂ϑ,uℱxdx.(20)

In a similar way as above, we have

1∂ϑ,uFR∫uu+∂ϑ,uℱxdx≼ℱu+˜ℱϑ∫01h1τh21−τdτ.(21)

Combining (20) and (21), we have

12h112h212ℱ2u+∂ϑ,u2≼1∂ϑ,uFR∫uu+∂ϑ,uℱxdx≼ℱu+˜ℱϑ∫01h1τh21−τdτ.

This completes the proof.

Note that, if ℱ(x) is h1,h2-preconcave fuzzy-IVF then, integral inequality (19) is reversed

Remark 3.2.

If h2τ≡1 then, Theorem 3.1 reduces to the result for h1-preinvex fuzzy-IVF:

12h112ℱ2u+∂ϑ,u2≼1∂ϑ,uFR∫uu+∂ϑ,uℱxdx≼ℱu+˜ℱϑ∫01h1τdτ.

If h1τ=τs and h2τ≡1 then, Theorem 3.1 reduces to the result for s-preinvex fuzzy-IVF:

2s−1ℱ2u+∂ϑ,u2≼1∂ϑ,uFR∫uu+∂ϑ,uℱxdx≼1s+1ℱu+˜ℱϑ.

If h1τ=τ and h2τ≡1 then, Theorem 3.1 reduces to the result for preinvex fuzzy-IVF:

ℱ2u+∂ϑ,u2≼1∂ϑ,uFR∫uu+∂ϑ,uℱxdx≼ℱu+˜ℱϑ2.

If h1τ=h2τ≡1 then, Theorem 3.1 reduces to the result for P-preinvex fuzzy-IVF:

12ℱ2u+∂ϑ,u2≼1∂ϑ,uFR∫uu+∂ϑ,uℱxdx≼ℱu+˜ℱϑ.

If ℱ*u,γ=ℱ*ϑ,γ with γ=1 then, we obtain (4) from (19).

If ℱ*u,γ=ℱ*ϑ,γ with γ=1 and h2τ≡1 then, we obtain (3) from (19).

Note that, if ∂ϑ,u=ϑ−u then, above integral inequalities reduce to new ones.

Example 3.3.

We consider h1τ=τ,h2τ≡1, for τ∈0,1 and the fuzzy-IVF ℱ:u,u+∂ϑ,u=0,∂2,0→FCℝ defined by,

ℱxσ=σ2x2σ∈0,2x24x2−σ2x2σ∈(2x2,4x2]0otherwise,

Then, for each γ∈0,1, we have ℱγx=2γx2,(4−2γ)x2. Since end point functions ℱ*x,γ=2γx2, ℱ*x,γ=(4−2γ)x2 are h1,h2-preinvex functions with respect to ∂ϑ,u=ϑ−u for each γ∈[0,1] then, ℱx is h1,h2-preinvex fuzzy-IVF with respect to ∂ϑ,u=ϑ−u. Now we compute the following:

12h112h212ℱ*2u+∂ϑ,u2,γ≤1∂ϑ,u∫uu+∂ϑ,uℱ*x,γdx≤ℱ*u,γ+ℱ*ϑ,γ∫01h1τh21−τdτ.

Now

12h112h212ℱ*2u+∂ϑ,u2,γ=ℱ*1,γ=2γ,

1∂ϑ,u∫uu+∂ϑ,uℱ*x,γdx=12∫022γx2dx=8γ3,

ℱ*u,γ+ℱ*ϑ,γ∫01h1τh21−τdτ=4γ,

for all γ∈0,1. That means

2γ≤8γ3≤4γ.

Similarly, it can be easily show that

12h112h212ℱ*2u+∂ϑ,u2,γ≤1∂ϑ,u∫uu+∂ϑ,uℱ*x,γdx≤ℱ*u,γ+ℱ*ϑ,γ∫01h1τh21−τdτ.

for all γ∈0,1, such that

12h112h212ℱ*2u+∂ϑ,u2,γ=ℱ*1,γ=2(2−γ),

1∂ϑ,u∫uu+∂ϑ,uℱ*x,γdx=12∫02(4−2γ)x2dx=8(2−γ)3,

ℱ*u,γ+ℱ*ϑ,γ∫01h1τh21−τdτ=4(2−γ),

From which, it follows that

4−2γ≤44−2γ3≤24−2γ,

i.e.,

2γ,22−γ≤I8γ3,82−γ3≤I4γ,42−1γ, for all γ∈0,1.

Hence, the Theorem 3.1 is verified.

Theorem 3.4.

Let ℱ,J:u,u+∂(ϑ,u)→FCℝ be two h1,h2-preinvex fuzzy-IVFs with nonnegative real-valued functions h1,h2:[0,1]→ℝ, whose γ-levels define the family of IVFs ℱγ,Jγ:u,u+∂(ϑ,u)⊂ℝ→KC+ are given by ℱγx=ℱ*x,γ,ℱ*x,γ and Jγx=J*x,γ,J*x,γ for all x∈u,u+∂(ϑ,u) and for all γ∈0,1. If ℱ,J and ℱJ∈ℐℛu,u+∂(ϑ,u),γ then,

1∂ϑ,uFR∫uu+∂ϑ,uℱx×˜Jxdx≼ℳu,ϑ∫01h1τh21−τ2dτ+˜Nu,ϑ∫01h1τh2τh11−τh21−τdτ,

where ℳu,ϑ=ℱu×˜Ju+˜ℱϑ×˜Jϑ, Nu,ϑ=ℱu×˜Jϑ+˜ℱϑ×˜Ju with ℳγu,ϑ=ℳ*u,ϑ,γ,ℳ*u,ϑ,γ and Nγu,ϑ=N*u,ϑ,γ,N*u,ϑ,γ.

Proof.

Since ℱ and J both are h1,h2-preinvex fuzzy-IVFs on u,u+∂(ϑ,u) then, for each γ∈0,1 we have

ℱ*u+1−τ∂ϑ,u,γ≤h1τh21−τℱ*u,γ+h11−τh2τℱ*ϑ,γ,,ℱ*u+1−τ∂ϑ,u,γ≤h1τh21−τℱ*u,γ+h11−τh2τℱ*ϑ,γ.

And

J*u+1−τ∂ϑ,u,γ≤h1τh21−τJ*u,γ+h11−τh2τJ*ϑ,γ,,J*u+1−τ∂ϑ,u,γ≤h1τh21−τJ*u,γ+h11−τh2τJ*ϑ,γ..

From the definition of h1,h2-preinvex fuzzy-IVFs it follows that ℱx≽0˜ and Jx≽0˜, so

ℱ*u+1−τ∂ϑ,u,γ×J*u+1−τ∂ϑ,u,γ≤h1τh21−τℱ*u,γ+h11−τh2τℱ*ϑ,γh1τh21−τJ*u,γ+h11−τh2τJ*ϑ,γ=ℱ*u,γ×J*u,γh1τh21−τ2+ℱ*ϑ,γ×J*ϑ,γh1τh21−τ2+ℱ*u,γ×J*ϑ,γh1τh2τh11−τh21−τ+ℱ*ϑ,γ×J*u,γh1τh2τh11−τh21−τ,

Integrating both sides of above inequality over [0, 1] we get

∫01ℱ*u+1−τ∂ϑ,u,γ×J*u+1−τ∂ϑ,uγ=1∂ϑ,u∫uu+∂ϑ,uℱ*x,γ×J*x,γdx≤ℱ*u,γ×J*u,γ+ℱ*ϑ,γ×J*ϑ,γ∫01h1τh21−τ2dτ+ℱ*u,γ×J*ϑ,γ+ℱ*ϑ,γ×J*u,γ.∫01h1τh2τh11−τh21−τdτ,

It follows that,

1∂ϑ,u∫uu+∂ϑ,uℱ*x,γ×J*x,γdx≤ℳ*u,ϑ,γ∫01h1τh21−τ2dτ+N*u,ϑ,γ∫01h1τh2τh11−τh21−τdτ,1∂ϑ,u∫uu+∂ϑ,uℱ*x,γ×J*x,γdx

≤ℳ*u,ϑ,γ∫01h1τh21−τ2dτ+N*u,ϑ,γ∫01h1τh2τh11−τh21−τdτ,

i.e.,

1∂ϑ,u∫uu+∂ϑ,uℱ*x,γ×J*x,γdx,∫uu+∂ϑ,uℱ*x,γ×J*x,γdx≤lℳ*u,ϑ,γ,ℳ*u,ϑ,γ∫01h1τh21−τ2dτ+N*u,ϑ,γ,N*u,ϑ,γ∫01h1τh2τh11−τh21−τdτ.

Thus,

1∂ϑ,uFR∫uu+∂ϑ,uℱx×˜Jxdx≼ℳu,ϑ∫01h1τh21−τ2dτ+˜Nu,ϑ∫01h1τh2τh11−τh21−τdτ,

and the theorem has been established.

Following assumption is required to prove next result regarding the bi-function ∂:K×K→ℝ which is known as

Condition C. Let K be an invex set with respect to ∂. For any u,ϑ∈K and τ∈0,1,

∂ϑ,u+τ∂(ϑ,u)=1−τ∂ϑ,u,

∂u,u+τξ(ϑ,u)=−τ∂ϑ,u.

Clearly for τ=0, we have ∂ϑ,u=0 if and only if, ϑ=u, for all u,ϑ∈K. For the applications of Condition C, see [12,26,31,46,51].

Theorem 3.5.

Let ℱ,J:u,u+∂(ϑ,u)→FCℝ be two h1,h2-preinvex fuzzy-IVFs with nonnegative real-valued functions h1,h2:[0,1]→ℝ and h112h212≠0, whose γ-levels define the family of IVFs ℱγ,Jγ:u,u+∂(ϑ,u)⊂ℝ→KC+ are given by ℱγx=ℱ*x,γ,ℱ*x,γ and Jγx=J*x,γ,J*x,γ for all x∈u,u+∂(ϑ,u) and for all γ∈0,1. If ℱ,J and ℱJ∈ℐℛu,u+∂(ϑ,u),γ, and Condition C hold for ∂ then,

12h112h2122ℱ2u+∂ϑ,u2×˜J2u+∂ϑ,u2≼1∂ϑ,uFR∫uu+∂ϑ,uℱx×˜Jxdx+˜ℳu,ϑ∫01h1τh2τh11−τh21−τdτ+˜Nu,ϑ∫01h1τh21−τ2dτ,

where ℳu,ϑ=ℱu×˜Ju+˜ℱϑ×˜Jϑ, Nu,ϑ= ℱu×˜Jϑ+˜ℱϑ×˜Ju, and ℳγu,ϑ=ℳ*u,ϑ,γ, ℳ*u,ϑ,γ and Nγu,ϑ=N*u,ϑ,γ,N*u,ϑ,γ.

Proof.

Using Condition C, we can write

u+12∂ϑ,u=u+τ∂ϑ,u+12∂u+1−τ∂ϑ,u,u+τ∂ϑ,u.

By hypothesis, for each γ∈0,1, we have

ℱ*2u+∂ϑ,u2,γ×J*2u+∂ϑ,u2,γ

ℱ*2u+∂ϑ,u2,γ×J*2u+∂ϑ,u2,γ=ℱ*u+τ∂ϑ,u+12∂u+1−τ∂ϑ,u,u+τ∂ϑ,u,γ× J*u+τ∂ϑ,u+12∂u+1−τ∂ϑ,u,u+τ∂ϑ,u,γ,=ℱ*u+τ∂ϑ,u+12∂u+1−τ∂ϑ,u,u+τ∂ϑ,u,γ× J*u+τ∂ϑ,u+12∂u+1−τ∂ϑ,u,u+τ∂ϑ,u,γ,

≤h112h2122ℱ*u+1−τ∂ϑ,u,γ×J*u+1−τ∂ϑ,u,γ+ℱ*u+1−τ∂ϑ,u,γ×J*u+τ∂ϑ,u,γ+h112h2122ℱ*u+τ∂ϑ,u,γ×J*u+1−τ∂ϑ,u,γ+ℱ*u+τ∂ϑ,u,γ×J*u+τ∂ϑ,u,γ,≤h112h2122ℱ*u+1−τ∂ϑ,u,γ×J*u+1−τ∂ϑ,u,γ+ℱ*u+1−τ∂ϑ,u,γ×J*u+τ∂ϑ,u,γ+h112h2122ℱ*u+τ∂ϑ,u,γ×J*u+1−τ∂ϑ,u,γ+ℱ*u+τ∂ϑ,u,γ×J*u+τ∂ϑ,u,γ,

≤h112h2122ℱ*u+1−τ∂ϑ,u,γ×J*u+1−τ∂ϑ,u,γ+ℱ*u+τ∂ϑ,u,γ×J*u+τ∂(ϑ,u),γ+h112h2122h1τh21−τℱ*u,γ+h11−τh2τℱ*ϑ,γ×h11−τh2τJ*u,γ+h1τh21−τJ*ϑ,γ+h11−τh2τℱ*u,γ+h1τh21−τℱ*ϑ,γ×h1τh21−τJ*u,γ+h11−τh2τJ*ϑ,γ,

≤h112h2122ℱ*u+1−τ∂ϑ,u,γ×J*u+1−τ∂ϑ,u,γ+ℱ*u+τ∂ϑ,u,γ×J*u+τ∂ϑ,u,γ+h112h2122h1τh21−τℱ*u,γ+h11−τh2τℱ*ϑ,γ×h11−τh2τJ*u,γ+h1τh21−τJ*ϑ,γ+h11−τh2τℱ*u,γ+h1τh21−τℱ*ϑ,γ×h1τh21−τJ*u,γ+h11−τh2τJ*ϑ,γ,

=h112h2122ℱ*u+1−τ∂ϑ,u,γ×J*u+1−τ∂ϑ,u,γ+ℱ*u+τ∂ϑ,u,γ×J*u+τ∂ϑ,u,γ+2h112h2122h1τh2τh11−τh21−τ.ℳ*u,ϑ,γ+h1τh21−τ2N*u,ϑ,γ,=h112h2122ℱ*u+1−τ∂ϑ,u,γ×J*u+1−τ∂ϑ,u,γ+ℱ*u+τ∂ϑ,u,γ×J*u+τ∂ϑ,u,γ+2h112h2122h1τh2τh11−τh21−τ.ℳ*u,ϑ,γ+h1τh21−τ2N*u,ϑ,γ,

integrating over 0,1, we have

12h112h2122ℱ*2u+∂ϑ,u2,γ×J*2u+∂ϑ,u2,γ≤1∂ϑ,u∫0u+∂ϑ,uℱ*x,γ×J*x,γdx+ℳ*u,ϑ,γ∫01h1τh2τh11−τh21−τdτ+N*u,ϑ,γ∫01h1τh21−τ2dτ,12h112h2122ℱ*2u+∂ϑ,u2,γ×J*2u+∂ϑ,u2,γ≤1∂ϑ,u∫uu+∂ϑ,uℱ*x,γ×J*x,γdx+ℳ*u,ϑ,γ∫01h1τh2τh11−τh21−τdτ+N*u,ϑ,γ∫01h1τh21−τ2dτ,

from which, we have

12h112h2122ℱ*2u+∂ϑ,u2,γ×J*2u+∂ϑ,u2,γ,ℱ*2u+∂ϑ,u2,γ×J*2u+∂ϑ,u2,γ≤l1∂ϑ,u∫uu+∂ϑ,uℱ*x,γ×J*x,γdx,∫uu+∂ϑ,uℱ*x,γ×J*x,γdx+∫01h1τh2τh11−τh21−τdτ ℳ*u,ϑ,γ,ℳ*u,ϑ,γ+N*u,ϑ,γ,N*u,ϑ,γ∫01h1τh21−τ2dτ,

i.e.,

12h112h2122ℱ2u+∂ϑ,u2×˜J2u+∂ϑ,u2≼1∂ϑ,uFR∫uu+∂ϑ,uℱx×˜Jxdx+˜ℳu,ϑ∫01h1τh2τh11−τh21−τdτ+˜Nu,ϑ∫01h1τh21−τ2dτ,

hence, the required result.

Remark 3.6.

If h2τ≡1, τ∈0,1 then, above theorems reduces for h1-preinvex fuzzy-IVFs.

If h1τ=τ and h2τ≡1, τ∈0,1 then, above theorems reduces for preinvex fuzzy-IVFs.

If in the above theorem ℱ*u,γ=ℱ*u,γ with γ=1 then, we obtain the appropriate theorems for h1,h2-preinvex functions, see [31].

If in the above theorem ℱ*u,γ=ℱ*u,γ with γ=1 and h2τ≡1, τ∈0,1 then, we obtain the appropriate theorems for h1-preinvex functions, see [30].

If in the above theorems ℱ*u+1−τ∂ϑ,u,γ=ℱ*u+1−τ∂ϑ,u,γ with γ=1, ∂ϑ,u=ϑ−u, h1τ=τ and h2τ≡1, τ∈0,1 then, we obtain the appropriate theorems for convex functions, see [56].

If in the above theorems ℱ*u+1−τ∂ϑ,u,γ=ℱ*u+1−τ∂ϑ,u,γ with γ=1, ∂ϑ,u=ϑ−u, h1τ=τs and h2τ≡1, τ∈0,1, s∈0,1 then, we obtain the appropriate theorems for s-convex functions in the second sense, see [57].

Example 3.7.

We consider h1τ=τ,h2τ≡1, for τ∈0,1, and the fuzzy-IVFs ℱ,J:u,u+∂(ϑ,u)=[0,∂(1,0)]→FCℝ defined by,

ℱxσ=σ2x2,σ∈0,2x2,4x2−σ2x2,σ∈2x2,4x2,0,otherwise,

Jxσ=σxσ∈0,x2x−σxσ∈(x,2x]0otherwise,

Then, for each γ∈0,1, we have ℱγx=2γx2,(4−2γ)x2 and Jγx=γx,(2−γ)x. Since end point functions ℱ*x,γ=2γx2 and ℱ*x,γ=(4−2γ)x2 both are h1,h2-preinvex functions, and J*x,γ=γx, and J*x,γ=(2−γ)x both are also h1,h2 preinvex functions with respect to same ∂ϑ,u=ϑ−u, for each γ∈[0,1] then, ℱ and J both are h1,h2-preinvex fuzzy-IVFs, respectively. Since ℱ*x,γ=2γx2 and ℱ*x,γ=(4−2γ)x2, and J*x,γ=γx, and J*x,γ=(2−γ)x then,

1∂ϑ,u∫uu+∂ϑ,uℱ*x,γ×J*x,γdx=∫012γx2γxdx=γ22,1∂ϑ,u∫uu+∂ϑ,uℱ*x,γ×J*x,γdx=∫01(4−2γ)u2(2−γ)udx=(2−γ)22,

ℳ*u,ϑ,γ∫01h1τh21−τ2dτ=2γ23,ℳ*u,ϑ,γ∫01h1τh21−τ2dτ=2(2−γ)23,

N*u,ϑ,γ∫01h1τh21−τh11−τh2τdτ=0N*u,ϑ,γ∫01h1τh21−τh11−τh2τdτ=0,

for each γ∈0,1, that means

γ22≤2γ23+0=2γ23,(2−γ)22≤2(2−γ)23+0=2(2−γ)23,

Hence, Theorem 3.4 is verified.

For Theorem 3.5, we have

12h112h212ℱ*2u+∂ϑ,u2,γ×J*2u+∂ϑ,u2,γ=γ22,12h112h212ℱ*2u+∂ϑ,u2,γ×J*2u+∂ϑ,u2,γ=(2−γ)22,

ℳ*u,ϑ,γ∫01h1τh21−τh11−τh2τdτ=γ23,ℳ*u,ϑ,γ∫01h1τh21−τh11−τh2τdτ=(2−γ)23,

N*u,ϑ,γ∫01h1τh21−τ2dτ=0,N*u,ϑ,γ∫01h1τh21−τ2dτ=0,

for each γ∈0,1, that means

γ22≤γ22+0+γ23=5γ26,(2−γ)22≤2−γ22+0+(2−γ)23=5(2−γ)26,

hence, Theorem 3.5 is demonstrated.

Theorem 3.8.

(The second HH-Fejér inequality for h1,h2-preinvex fuzzy-IVFs). Let ℱ:u,u+∂(ϑ,u)→FCℝ be a h1,h2-preinvex fuzzy-IVF with u<u+∂(ϑ,u) and nonnegative real-valued functions h1,h2:[0,1]→ℝ, whose γ-levels define the family of IVFs ℱγ:u,u+∂(ϑ,u)⊂ℝ→KC+ are given by ℱγx=ℱ*x,γ,ℱ*x,γ for all x∈u,u+∂(ϑ,u) and for all γ∈0,1. If ℱ∈ℱℛu,u+∂(ϑ,u),γ and Ω:u,u+∂(ϑ,u)→ℝ,Ω(x)≥0, symmetric with respect to u+12∂ϑ,u then,

1∂ϑ,uFR∫uu+∂ϑ,uℱxΩxdx≼Fu+˜Fϑ∫01h1τh21−τΩu+τ∂(ϑ,u)dτ.(22)

Proof.

Let ℱ be a (h1,h2)-preinvex fuzzy-IVF. Then, for each γ∈0,1, we have

ℱ*u+1−τ∂ϑ,u,γΩu+1−τ∂ϑ,u≤h1τh21−τℱ*u,γ+h11−τh2τℱ*ϑ,γΩu+1−τ∂(ϑ,u),ℱ*u+1−τ∂ϑ,u,γΩu+1−τ∂ϑ,u≤h1τh21−τℱ*u,γ+h11−τh2τℱ*ϑ,γΩu+1−τ∂(ϑ,u).(23)

And

ℱ*u+τ∂ϑ,u,γΩu+τ∂ϑ,u≤h11−τh2τℱ*u,γ+h1τh21−τℱ*ϑ,γΩu+τ∂(ϑ,u),ℱ*u+τ∂ϑ,u,γΩu+τ∂ϑ,u≤h11−τh2τℱ*u,γ+h1τh21−τℱ*ϑ,γΩu+τ∂(ϑ,u).(24)

After adding (23) and (24), and integrating over 0,1, we get

∫01ℱ*u+1−τ∂ϑ,u,γ.Ωu+1−τ∂ϑ,udτ+∫01ℱ*u+τ∂ϑ,u,γΩu+τ∂ϑ,udτ≤∫01ℱ*u,γh1τh21−τ.Ωu+1−τ∂ϑ,u+h11−τh2τ.Ωu+τ∂ϑ,u+ℱ*ϑ,γh11−τh2τ.Ωu+1−τ∂ϑ,u+h1τh21−τ.Ωu+τ∂ϑ,udt,

∫01ℱ*u+τ∂ϑ,u,γ.Ωu+τ∂ϑ,udτ+∫01ℱ*u+1−τ∂ϑ,u,γ.Ωu+1−τ∂ϑ,udτ≤∫01ℱ*u,γh1τh21−τ.Ωu+1−τ∂ϑ,u+h11−τh2τ.Ωu+τ∂ϑ,u+ℱ*ϑ,γh11−τh2τ.Ωu+1−τ∂ϑ,u+h1τh21−τ.Ωu+τ∂ϑ,udt.

=2ℱ*u,γ∫01h1τh21−τΩu+1−τ∂ϑ,udt+2ℱ*ϑ,γ∫01h1τh21−τΩu+τ∂(ϑ,u)dt,=2ℱ*u,γ∫01h1τh21−τΩu+1−τ∂ϑ,udt+2ℱ*ϑ,γ∫01h1τh21−τΩu+τ∂(ϑ,u)dt.

Since Ω is symmetric then,

=2ℱ*u,γ+ℱ*ϑ,γ∫01h1τh21−τΩu+τ∂(ϑ,u)dt,=2ℱ*u,γ+ℱ*ϑ,γ∫01h1τh21−τΩu+τ∂(ϑ,u)dt.(25)

Since

∫01ℱ*u+1−τ∂ϑ,u,γΩu+1−τ∂ϑ,udτ=∫01ℱ*u+τ∂ϑ,u,γΩu+τ∂ϑ,udτ=1∂ϑ,u∫uu+∂ϑ,uℱ*x,γΩxdx,∫01ℱ*u+τ∂ϑ,u,γΩu+τ∂ϑ,udτ=∫01ℱ*u+1−τ∂ϑ,u,γΩu+1−τ∂ϑ,u.dτ=1∂ϑ,u∫uu+∂ϑ,uℱ*x,γΩxdx.(26)

From (25) and (26), we have

1∂ϑ,u∫uu+∂ϑ,uℱ*x,γΩxdx≤ℱ*u,γ+ℱ*ϑ,γ∫01h1τh21−τΩu+τ∂ϑ,udt,1∂ϑ,u∫uu+∂ϑ,uℱ*x,γΩxdx≤ℱ*u,γ+ℱ*ϑ,γ∫01h1τh21−τΩu+τ∂ϑ,udt,

i.e.,

1∂ϑ,u∫uu+∂ϑ,uℱ*x,γ.Ωxdx,1∂ϑ,u∫uu+∂ϑ,uℱ*x,γ.Ωxdx≤lℱ*u,γ+ℱ*ϑ,γ,ℱ*u,γ+ℱ*ϑ,γ.∫01h1τh21−τΩu+τ∂ϑ,udt,

hence

1∂ϑ,uFR∫uu+∂ϑ,uℱxΩxdx≼ℱu+˜ℱϑ∫01h1τh21−τ.Ωu+τ∂ϑ,udτ,

then, we complete the proof.

Theorem 3.9.

(The first HH-Fejér inequality for h1,h2-preinvex fuzzy-IVFs). Let ℱ:u,u+∂(ϑ,u)→FCℝ be a h1,h2-preinvex fuzzy-IVF with u<u+∂(ϑ,u) and nonnegative real-valued functions h1,h2:[0,1]→ℝ, whose γ-levels define the family of IVFs ℱγ:u,u+∂(ϑ,u)⊂ℝ→KC+ are given by ℱγx=ℱ*x,γ,ℱ*x,γ for all x∈u,u+∂(ϑ,u) and for all γ∈0,1. If ℱ∈ℱℛu,u+∂(ϑ,u),γ and Ω:u,u+∂(ϑ,u)→ℝ,Ω(x)≥0, symmetric with respect to u+12∂ϑ,u, and ∫uu+∂(ϑ,u)Ω(x)dx>0, and Condition C holds for ∂ then,

ℱu+12∂ϑ,u≼2h112h212∫uu+∂(ϑ,u)Ω(x)dxFR∫uu+∂(ϑ,u)ℱxΩ(x)dx.(27)

Proof.

Using Condition C, we can write

u+12∂ϑ,u=u+τ∂ϑ,u+12∂u+1−τ∂ϑ,u,u+τ∂ϑ,u.

Since ℱ is a h1,h2-preinvex then, for γ∈0,1, we have

ℱ*u+12∂ϑ,u,γ=ℱ*u+τ∂ϑ,u+12∂u+1−τ∂ϑ,u,u+τ∂ϑ,u,γ≤h112h212ℱ*u+1−τ∂ϑ,u,γ+ℱ*u+τ∂ϑ,u,γ,ℱ*u+12∂ϑ,u,γ=ℱ*u+τ∂ϑ,u+12∂u+1−τ∂ϑ,u,u+τ∂ϑ,u,γ≤h112h212ℱ*u+1−τ∂ϑ,u,γ+ℱ*u+τ∂ϑ,u,γ,(28)

By multiplying (28) by Ωu+1−τ∂ϑ,u=Ωu+τ∂ϑ,u and integrate it by τ over 0,1, we obtain

ℱ*u+12∂ϑ,u,γ∫01Ωu+τ∂ϑ,udτ≤h112h212∫01ℱ*u+1−τ∂ϑ,u,γΩu+1−τ∂ϑ,udτ+∫01ℱ*u+τ∂ϑ,u,γΩu+τ∂ϑ,udτ,ℱ*u+12∂ϑ,u,γ∫01Ωu+τ∂ϑ,udτ≤h112h212∫01ℱ*u+1−τ∂ϑ,u,γΩu+1−τ∂ϑ,udτ+∫01ℱ*u+τ∂ϑ,u,γ.Ωu+τ∂ϑ,udτ,(29)

Since

∫01ℱ*u+1−τ∂ϑ,u,γΩu+1−τ∂ϑ,udτ=∫01ℱ*u+τ∂ϑ,u,γΩu+τ∂ϑ,udτ=1∂ϑ,u∫uu+∂ϑ,uℱ*x,γΩxdx,∫01ℱ*u+τ∂ϑ,u,γΩu+τ∂ϑ,udτ=∫01ℱ*u+1−τ∂ϑ,u,γ.Ωu+1−τ∂ϑ,udτ=1∂ϑ,u∫uu+∂ϑ,uℱ*x,γΩxdx(30)

From (29) and (30), we have

ℱ*u+12∂ϑ,u,γ≤2h112h212∫uu+∂ϑ,uΩxdx∫uu+∂ϑ,uℱ*x,γΩxdx,ℱ*u+12∂ϑ,u,γ≤2h112h212∫uu+∂ϑ,uΩxdx∫uu+∂ϑ,uℱ*x,γΩxdx,

From which, we have

ℱ*u+12∂ϑ,u,γ,ℱ*u+12∂ϑ,u,γ≤l2h112h212∫uu+∂ϑ,uΩxdx∫uu+∂ϑ,uℱ*x,γ.Ωxdx,∫uu+∂ϑ,uℱ*x,γ.Ωxdx,

i.e.,

ℱu+12∂ϑ,u≼2h112h212∫uu+∂ϑ,uΩxdxFR∫uu+∂ϑ,uℱxΩxdx,

this completes the proof.

Remark 3.10.

If h2τ≡1, τ∈0,1 then, inequalities in Theorems 3.8 and 3.9 reduce for h1-preinvex fuzzy-IVFs which are also new one.

If h1τ=τ and h2τ≡1, τ∈0,1 then, inequalities in Theorems 3.8 and 3.9 reduce for preinvex fuzzy-IVFs which are also new one.

If in the Theorems 3.8 and 3.9 h2τ≡1 and ∂ϑ,u=ϑ−u then, we obtain the appropriate theorems for h1-convex fuzzy-IVFs which are also new one.

If in the Theorems 3.8 and 3.9 h1τ=τ, h2τ≡1 and ∂ϑ,u=ϑ−u then, we obtain the appropriate theorems for convex fuzzy-IVFs which are also new one.

If ℱ*u,γ=ℱ*u,γ with γ=1 and h2τ≡1 then, Theorems 3.8 and 3.9 reduces to classical first and second HH-Fejér inequality for h-preinvex function, see [30].

If in the Theorems 3.8 and 3.9 ℱ*u,γ=ℱ*u,γ with γ=1, h2τ≡1 and ∂ϑ,u=ϑ−u then, we obtain the appropriate theorems for h-convex function, see [39].

If Ωx=1 then, combining Theorems 3.8 and 3.9, we get Theorem 3.1.

Example 3.11.

We consider h1τ=τ, h2τ=1 for τ∈0,1 and the fuzzy-IVF ℱ:1,1+∂(5,1)→FCℝ defined by,

ℱxσ=σ−exex,σ∈ex,2ex,4ex−σ2ex,σ∈2ex,4ex,0,otherwise,

Then, for each γ∈0,1, we have ℱγx=(1+γ)ex,2(2−γ)ex. Since end point functions ℱ*x,γ, ℱ*x,γ are h1,h2-preinvex functions ∂y,x=y−x for each γ∈[0,1] then, ℱx is h1,h2-preinvex fuzzy-IVF. If

Ωx=x−1,σ∈1,52,4−x,σ∈52,4,

then, we have

1∂4,1∫11+∂5,1ℱ*x,γΩxdx=13∫14ℱ*x,γΩxdx=13∫152ℱ*x,γΩxdx+13∫524ℱ*x,γΩxdx,1∂4,1∫11+∂5,1ℱ*x,γΩxdx=13∫14ℱ*x,γΩxdx=13∫152ℱ*x,γΩxdx+13∫524ℱ*x,γΩxdx,

=131+γ∫152exx−1dx+131+γ∫524ex4−xdx≈111+γ,=232−γ∫152exx−1dx+232−γ∫524ex4−xdx≈212−γ,(31)

and

ℱ*u,γ+ℱ*ϑ,γ∫01h1τh21−τ.Ωu+τ∂ϑ,udτℱ*u,γ+ℱ*ϑ,γ∫01h1τh21−τ.Ωu+τ∂ϑ,udτ

=1+γe+e4∫0123τ2dx+∫121τ3−3τdτ≈4321+γ,=22−γe+e4∫0123τ2dx+∫121τ3−3τdτ≈432−γ,(32)

From (31) and (32), we have

111+γ,212−γ≤l4321+γ,432−γ,

for each γ∈0,1. Hence, Theorem 3.8 is verified.

For Theorem 3.9, we have

ℱ*u+12∂ϑ,u,γ≈6451+γ,ℱ*u+12∂ϑ,u,γ≈12252−γ,(33)

∫uu+∂ϑ,uΩxdx=∫152x−1dx+∫uu+∂ϑ,u4−xdx=94,

2h112h212∫uu+∂ϑ,uΩxdx∫uu+∂ϑ,uℱ*x,γΩxdx ≈7351+γ2h112h212∫uu+∂ϑ,uΩxdx∫uu+∂ϑ,uℱ*x,γΩxdx ≈293102−γ(34)

From (33) and (34), we have

6451+γ,12252−γ≤l7351+γ,293102−γ.

Hence, Theorem 3.9 is verified.

4. CONCLUSION AND FUTURE STUDY

In this article, we introduced new class of nonconvex functions is known as h1,h2-preinvex fuzzy-IVFs. With the help of this class, we derived some new HH-inequalities by means of fuzzy order relation. To strengthen our result, we provided some examples to illustrate the validation of our results. Moreover, several new and previously known results also obtained. In future, we will try to explore these concepts for fuzzy fractional integral operators.

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: 1403 - 1418
Publication Date: 2021/04/19
ISSN (Online): 1875-6883
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TY  - JOUR
AU  - Muhammad Bilal Khan
AU  - Muhammad Aslam Noor
AU  - Lazim Abdullah
AU  - Yu-Ming Chu
PY  - 2021
DA  - 2021/04/19
TI  - Some New Classes of Preinvex Fuzzy-Interval-Valued Functions and Inequalities
JO  - International Journal of Computational Intelligence Systems
SP  - 1403
EP  - 1418
VL  - 14
IS  - 1
SN  - 1875-6883
UR  - https://doi.org/10.2991/ijcis.d.210409.001
DO  - 10.2991/ijcis.d.210409.001
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