Some New Estimates of Fuzzy Integral Inequalities for Harmonically Convex Fuzzy-Number-Valued Mappings via up and down Fuzzy Relation

Abstract

In this article, the up and down harmonically convex fuzzy-number-valued mapping which is a novel kind of harmonically convex fuzzy-number-valued mapping is introduced. In addition, it is highlighted that the new idea of up and down harmonically convex fuzzy-number-valued mapping ($\mathcal{U}-\mathfrak{O}-\mathcal{H}$ convex $F-N-V-M$), which is a generalization of the previous class, describes a variety of new and classical classes as special cases by employing some mild restrictions. With the help of fuzzy inclusion relation, the new versions of the Hermite–Hadamard-type (HH-type) inequalities for up and down harmonically convex fuzzy-number-valued mappings are established. Then, we introduce a new version of Hermite–Hadamard Fejér-type inequality via fuzzy inclusion relation by using up and down harmonically convex fuzzy-number-valued mapping. Additionally, several instances are given to illustrate our main findings.

1. Introduction

The convexity of mappings is a potent tool that is primarily used to address a number of difficulties in both pure and practical science. Many academics have recently devoted their time to researching the characteristics related to convexity in various directions; for more information, see [1,2,3,4,5,6,7] and the references therein. Hermite–Hadamard’s inequality [8,9], which is also frequently used in many other parts of practical mathematics, especially in optimization and probability, is one of the most important mathematical inequalities relevant to convex maps. The well-known classical HH-inequality for convex mapping $g:K\to \mathbb{R}$ on an interval $K$ is:

$$g\left(\frac{a+b}{2}\right)\le \frac{1}{b-a}{\int }_a^{b}g\left(\varkappa \right)d\varkappa \le \frac{g\left(a\right)+g\left(b\right)}{2},$$

(1)

for all $a,b\in K$, where $K$ is a convex set.

See [10,11,12,13,14,15,16,17,18,19,20] for a number of intriguing generalizations and extensions of this inequality. The idea of harmonic convexity was presented in 2014 by Iscan [21], who also constructed several Hermite–Hadamard-type inequalities for this class of mappings. These inequalities have been examined in more detail for harmonically convex mappings in [22]. Noor et al. [23] discovered various Hermite–Hadamard-type inequalities and introduced the class of harmonically h-convex mappings in 2015. Readers interested in recent convexity and harmonic convexity research are referred to [24,25,26,27,28,29,30,31,32,33,34,35] and the references therein.

On the other hand, Moore’s renowned book [36] is where interval analysis and interval-valued mappings were first presented in numerical analysis. Interval analysis has been a very important research topic over the past fifty years due to its extensive applicability in numerous domains; see, for example, [37,38,39,40,41,42,43,44,45,46] and the references therein. Numerous classical integral inequalities have recently been expanded to more general set-valued maps by Klarici’c Bakula and Nikodem [47], Matkowski and Nikodem [48], Mitroi et al. [49], and Nikodem et al. [50], in addition to the context of interval-valued mappings by Chalco-Cano et al. [51,52], Román-Flores et al. [53], Flores-Franulič et al. [54], and Costa and Román-Flores [55]. Recently, Khan et al. [56] introduced a new class of harmonic convex functions as well as presented some fractional inequalities over up and down harmonic functions. For more related concepts, see [57,58,59,60] and the references therein.

The findings of Iscan [21] and Noor et al. [23] are the primary sources of inspiration for our investigation. The concept of harmonically h-convexity for interval-valued mappings is introduced first. Then, we demonstrate several new Hermite–Hadamard-type inequalities for the newly introduced class of mappings. The results from [21,23] have fuzzy-number-valued analogs in our inequalities. For more fruitful information, related to fuzzy-valued and interval-valued mappings, see [61,62,63,64,65,66,67,68].

The structure of the essay is as follows. Following Section 2 of the preliminary material, Section 3 introduces the notion of harmonic convexity for fuzzy-number-valued mappings and proves a new version of Hermite–Hadamard-type inequalities. Section 4 presents the conclusions and future work concludes our discussion.

2. Preliminaries

Let ${\mathbb{R}}_I$ be the space of all closed and bounded intervals of $\mathbb{R}$, and $\mathtt{A}\in {\mathbb{R}}_I$ be defined by

$$\mathtt{A}=\left[{\mathtt{A}}_{*},{\mathtt{A}}^{*}\right]=\left\{\varkappa \in \mathbb{R}|{\mathtt{A}}_{*}\le \varkappa \le {\mathtt{A}}^{*}\right\},\left({\mathtt{A}}_{*},{\mathtt{A}}^{*}\in \mathbb{R}\right).$$

(2)

If ${\mathtt{A}}_{*}={\mathtt{A}}^{*}$, then $\mathtt{A}$ is said to be degenerate. In this article, all intervals are nondegenerate intervals. If ${\mathtt{A}}_{*}\ge 0$, then $\left[{\mathtt{A}}_{*},{\mathtt{A}}^{*}\right]$ is named as the positive interval. The set of all positive intervals is denoted by ${\mathbb{R}}_I^{+}$ and defined as ${\mathbb{R}}_I^{+}=\left\{\left[{\mathtt{A}}_{*},{\mathtt{A}}^{*}\right]:\left[{\mathtt{A}}_{*},{\mathtt{A}}^{*}\right]\in {\mathbb{R}}_I \mathrm{a}\mathrm{n}\mathrm{d} {\mathtt{A}}_{*}\ge 0\right\}.$

Let $\mathbb{ϛ}\in \mathbb{R}$ and $\mathtt{ϛ}·\mathtt{A}$ be defined by

$$\mathtt{ϛ}·\mathtt{A}=\left\{\begin{array}{c}\left[ϛ{\mathtt{A}}_{*},{\mathtt{ϛ}\mathtt{A}}^{*}\right]\hspace{1em}\mathrm{if}ϛ>0,\\ \left\{0\right\}\hspace{1em}\hspace{1em}\mathrm{if}ϛ=0,\\ \left[ϛ{\mathtt{A}}^{*}{,\mathtt{ϛ}\mathtt{A}}_{*}\right]\hspace{1em}\mathrm{if}ϛ<0.\end{array}\right.$$

(3)

Then the addition $\mathtt{A}+\mathtt{B}$, multiplication $\mathtt{A}\times \mathtt{B}$, and Minkowski difference $\mathtt{B}-\mathtt{A}$, for $\mathtt{A},\mathtt{B}\in {\mathbb{R}}_I$ are, respectively, defined by

$$\left[{\mathtt{B}}_{*},{\mathtt{B}}^{*}\right]+\left[{\mathtt{A}}_{*},{\mathtt{A}}^{*}\right]=\left[{\mathtt{B}}_{*}+{\mathtt{A}}_{*},{\mathtt{B}}^{*}+{\mathtt{A}}^{*}\right],$$

(4)

$$\left[{\mathtt{B}}_{*},{\mathtt{B}}^{*}\right]\times \left[{\mathtt{A}}_{*},{\mathtt{A}}^{*}\right]=\left[\mathrm{m}\mathrm{i}\mathrm{n}\left\{{\mathtt{B}}_{*}{\mathtt{A}}_{*},{\mathtt{B}}^{*}{\mathtt{A}}_{*},{\mathtt{B}}_{*}{\mathtt{A}}^{*},{\mathtt{B}}^{*}{\mathtt{A}}^{*}\right\},\mathrm{m}\mathrm{a}\mathrm{x}\left\{{\mathtt{B}}_{*}{\mathtt{A}}_{*},{\mathtt{B}}^{*}{\mathtt{A}}_{*},{\mathtt{B}}_{*}{\mathtt{A}}^{*},{\mathtt{B}}^{*}{\mathtt{A}}^{*}\right\}\right],$$

(5)

$$\left[{\mathtt{B}}_{*},{\mathtt{B}}^{*}\right]-\left[{\mathtt{A}}_{*},{\mathtt{A}}^{*}\right]=\left[{\mathtt{B}}_{*}-{\mathtt{A}}^{*},{\mathtt{B}}^{*}-{\mathtt{A}}_{*}\right],$$

(6)

Remark 1.

(i) For given $\left[{\mathtt{B}}_{*},{\mathtt{B}}^{*}\right],\left[{\mathtt{A}}_{*},{\mathtt{A}}^{*}\right]\in {\mathbb{R}}_I,$ the relation $“{\supseteq }_I”$ defined on ${\mathbb{R}}_I$ by

$$\left[{\mathtt{A}}_{*},{\mathtt{A}}^{*}\right]{\supseteq }_I\left[{\mathtt{B}}_{*},{\mathtt{B}}^{*}\right]\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}{\mathtt{A}}_{*}\le {\mathtt{B}}_{*},{\mathtt{B}}^{*}{\le \mathtt{A}}^{*},$$

(7)

it is a partial interval inclusion relation. It can be easily seen that “${\supseteq }_I$” looks like “up and down” on the real line $\mathbb{R},$ so we call “${\supseteq }_I$ ” “up and down” (or “$\mathcal{U}-\mathfrak{O}$” order, in short) [67].

(ii) For given $\left[{\mathtt{B}}_{*},{\mathtt{B}}^{*}\right],\left[{\mathtt{A}}_{*},{\mathtt{A}}^{*}\right]\in {\mathbb{R}}_I,$ we say that $\left[{\mathtt{B}}_{*},{\mathtt{B}}^{*}\right]{\le }_I\left[{\mathtt{A}}_{*},{\mathtt{A}}^{*}\right]$ if and only if ${\mathtt{B}}_{*}{\le \mathtt{A}}_{*},{\mathtt{B}}^{*}{\le \mathtt{A}}^{*}$ , it is a partial interval order relation. It can be easily seen that $“{\le }_I”$ looks like “left and right” on the real line $\mathbb{R},$ so we call $“{\le }_I”$ “left and right” (or “LR” order, in short) [66,67].

For $\left[{\mathtt{B}}_{\mathrm{*}},{\mathtt{B}}^{\mathrm{*}}\right],\left[{\mathtt{A}}_{\mathrm{*}},{\mathtt{A}}^{\mathrm{*}}\right]\in {\mathbb{R}}_I,$ the Hausdorff–Pompeiu distance between intervals $\left[{\mathtt{B}}_{\mathrm{*}},{\mathtt{B}}^{\mathrm{*}}\right]$ and $\left[{\mathtt{A}}_{\mathrm{*}},{\mathtt{A}}^{\mathrm{*}}\right]$ is defined by

$$d_H\left(\left[{\mathtt{B}}_{*},{\mathtt{B}}^{*}\right],\left[{\mathtt{A}}_{*},{\mathtt{A}}^{*}\right]\right)=\mathrm{m}\mathrm{a}\mathrm{x}\left\{\left|{\mathtt{B}}_{*}-{\mathtt{A}}_{*}\right|,\left|{\mathtt{B}}^{*}-{\mathtt{A}}^{*}\right|\right\}.$$

(8)

It is familiar fact that $\left({\mathbb{R}}_I,d_H\right)$ is a complete metric space [60,64,65].

Definition 1

([59,60]). A fuzzy subset $L$ of $\mathbb{R}$ is characterized by a mapping $\tilde{\mathtt{A}}:\mathbb{R}\to [0,1]$ named as the membership mapping of  $L$ . In general, a fuzzy subset A of R is presented as its membership function $\tilde{\mathtt{A}}$ to simplify the notation. In other words, a fuzzy subset $L$ of $\mathbb{R}$ is a mapping $\tilde{\mathtt{A}}:\mathbb{R}\to \left[\mathrm{0,1}\right]$. We appoint $\mathbb{E}$ to denote the set of all fuzzy subsets of $\mathbb{R}$.

Definition 2.

([59,60]). Let $\tilde{\mathtt{A}}\in \mathbb{E}$. Then, $\tilde{\mathtt{A}}$ is known as a fuzzy number or fuzzy interval if the following properties are satisfied by $\tilde{\mathtt{A}}$:

(1)

$\tilde{\mathtt{A}}$ should be normal if there exists $\varkappa \in \mathbb{R}$ and $\tilde{\mathtt{A}}\left(\varkappa \right)=1;$

(2)

$\tilde{\mathtt{A}}$ should be upper semi-continuous on  $\mathbb{R}$ if for given  $\varkappa \in \mathbb{R},$ there exist $\epsilon >0$, there exist $\delta >0$ such that $\tilde{\mathtt{A}}\left(\varkappa \right)-\tilde{\mathtt{A}}\left(y\right)<\epsilon$ for all $y\in \mathbb{R}$ with $\left|\varkappa -y\right|<\delta ;$

(3)

$\tilde{\mathtt{A}}$ should be fuzzy convex that is $\tilde{\mathtt{A}}\left(\left(1-ϛ\right)\varkappa +ϛy\right)\ge \mathrm{m}\mathrm{i}\mathrm{n}\left(\tilde{\mathtt{A}}\left(\varkappa \right),\tilde{\mathtt{A}}\left(y\right)\right),$ for all $\varkappa ,y\in \mathbb{R},$ and $ϛ\in [0,1]$;

(4)

$\tilde{\mathtt{A}}$ should be compactly supported that is $cl\left\{\varkappa \in \mathbb{R}|\tilde{\mathtt{A}}\left(\varkappa \right)>0\right\}$ is compact;

The notion ${\mathbb{E}}_I$ represents the set of all fuzzy numbers of $\mathbb{R}$.

Definition 3

([59,60]). Given $\tilde{\mathtt{A}}\in {\mathbb{E}}_I$, the level sets are given by ${\left[\tilde{\mathtt{A}}\right]}^{⋌}=\left\{\varkappa \in \mathbb{R}|\tilde{\mathtt{A}}\left(\varkappa \right)\ge ⋌\right\}$ for all $⋌\in \left(\left.0,1\right]\right.$ and by ${\left[\tilde{\mathtt{A}}\right]}^0=cl\left\{\varkappa \in \mathbb{R}|\tilde{\mathtt{A}}\left(\varkappa \right)>0\right\}$ . These sets are known as $⋌$-level sets of $\tilde{\mathtt{A}}$, for all $⋌\in \left[0,1\right]$.

Theorem  1

([68]). If $\tilde{\mathtt{A}}\in {\mathbb{E}}_I$ and ${\left[\tilde{\mathtt{A}}\right]}^{⋌}$ are its $⋌$−level sets, then:

(i)

${\left[\tilde{\mathtt{A}}\right]}^{⋌}$ is a closed interval ${\left[\tilde{\mathtt{A}}\right]}^{⋌}=\left[{{\mathtt{A}}_{*}}_{⋌},{{\mathtt{A}}^{*}}^{⋌}\right]$, for all $⋌\in \left[0,1\right]$.

(ii)

If $0\le {⋌}_1\le {⋌}_2\le 1$, then ${\left[\tilde{\mathtt{A}}\right]}^{{⋌}_2}\subseteq {\left[\tilde{\mathtt{A}}\right]}^{{⋌}_1}$.

(iii)

For any sequence ${\left({⋌}_n\right)}_{n\in N}$ which converges from below to $⋌\in \left(\left.0,1\right]\right.$ , we have ${\bigcap }_{n=1}^{\infty }{\left[\tilde{\mathtt{A}}\right]}^{{⋌}_n}={\left[\tilde{\mathtt{A}}\right]}^{⋌}$.

(iv)

For any sequence ${\left({⋌}_n\right)}_{n\in N}$ which converges from above to 0, we have $cl\left({\bigcup }_{n=1}^{\infty }{\left[\tilde{\mathtt{A}}\right]}^{{⋌}_n}\right)={\left[\tilde{\mathtt{A}}\right]}^0$.

Proposition 1

([55]). Let $\tilde{\mathtt{A}},\tilde{\mathtt{B}}\in {\mathbb{E}}_I$. Then relation $“{\le }_{\mathbb{F}}\mathrm{”}$ given on ${\mathbb{E}}_I$ by

$$\tilde{\mathtt{A}}{\le }_{\mathbb{F}}\tilde{\mathtt{B}}\mathrm{when}\mathrm{and}\mathrm{only}\mathrm{when},{{\left[\tilde{\mathtt{A}}\right]}^{⋌}\le }_I{\left[\tilde{\mathtt{B}}\right]}^{⋌},\mathrm{for}\mathrm{every}⋌\in [0,1],$$

it is left- and right-order relation.

Proposition 2

([61]). Let $\tilde{\mathtt{A}},\tilde{\mathtt{B}}\in {\mathbb{E}}_I$ . Then relation $“{\supseteq }_{\mathbb{F}}\mathrm{”}$ given on ${\mathbb{E}}_I$ by

$$\tilde{\mathtt{A}}{\supseteq }_{\mathbb{F}}\tilde{\mathtt{B}}\mathrm{when}\mathrm{and}\mathrm{only}\mathrm{when},{\left[\tilde{\mathtt{A}}\right]}^{⋌}{\supseteq }_I{\left[\tilde{\mathtt{B}}\right]}^{⋌},\mathrm{for}\mathrm{every}⋌\in [0,1],$$

it is $\mathcal{U}-\mathfrak{O}$ order relation on ${\mathbb{E}}_I$.

If $\tilde{\mathtt{A}},\tilde{\mathtt{B}}\in {\mathbb{E}}_I$ and $\mathbb{ϛ}\in \mathbb{R}$, then, for every $⋌\in \left[0,1\right],$ the arithmetic operations, addition “$\tilde{\mathtt{A}}\oplus \tilde{\mathtt{B}}$”, multiplication “$\tilde{\mathtt{A}}\otimes \tilde{\mathtt{B}}$”, and multiplication by scalar “$ϛ\odot \tilde{\mathtt{A}}$” can be characterized level-wise, respectively, by

$${\left[\tilde{\mathtt{A}}\oplus \tilde{\mathtt{B}}\right]}^{⋌}={\left[\tilde{\mathtt{A}}\right]}^{⋌}+{\left[\tilde{\mathtt{B}}\right]}^{⋌},$$

(9)

$${\left[\tilde{\mathtt{A}}\otimes \tilde{\mathtt{B}}\right]}^{⋌}={\left[\tilde{\mathtt{A}}\right]}^{⋌}\times {\left[\tilde{\mathtt{B}}\right]}^{⋌},$$

(10)

$${\left[ϛ\odot \tilde{\mathtt{A}}\right]}^{⋌}=ϛ·{\left[\tilde{\mathtt{A}}\right]}^{⋌}.$$

(11)

These operations follow directly from Equations (4), (5) and (3), respectively.

Theorem 2

([60]). The space ${\mathbb{E}}_I$ equipped with a supremum metric, i.e., for $\tilde{\mathtt{A}},\tilde{\mathtt{B}}\in {\mathbb{E}}_I$

$$d_{\infty }\left(\tilde{\mathtt{A}},\tilde{\mathtt{B}}\right)=\underset{0\le ⋌\le 1}{\sup }{d}_H\left({\left[\tilde{\mathtt{A}}\right]}^{⋌},{\left[\tilde{\mathtt{B}}\right]}^{⋌}\right),$$

(12)

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

Theorem 3

([60,62]). If $g:[a,b]\subset \mathbb{R}\to {\mathbb{R}}_I$ is an interval-valued mapping (I∙V∙M) satisfying that $g\left(\varkappa \right)=\left[g_{*}\left(\varkappa \right),g^{*}\left(\varkappa \right)\right]$ , then  $g$ is Aumann integrable (IA-integrable) over  $[a,b]$ when and only when,  $g_{*}\left(\varkappa \right)$ and  $g^{*}\left(\varkappa \right)$ both are integrable over $\left[a,b\right]$ such that

$$\left(IA\right){\int }_a^{b}g\left(\varkappa \right)d\varkappa =\left[{\int }_a^{\begin{array}{c}\\ b\end{array}}{g}_{*}\left(\varkappa \right)d\varkappa ,{\int }_a^b{g}^{*}\left(\varkappa \right)d\varkappa \right].$$

(13)

Definition 4

([66]). A mapping $g:\mathbb{I}\subset \mathbb{R}\to {\mathbb{E}}_I$ is named as $F-N-V-M$ . Then, for every $⋌\in \left[0,1\right]$ , as well as $⋌$-levels define the family of $I-V-M$s $g_{⋌}:\mathbb{I}\subset \mathbb{R}\to {\mathbb{R}}_I$ satisfying that $g_{⋌}\left(\varkappa \right)=\left[g_{*}\left(\varkappa ,⋌\right),g^{*}\left(\varkappa ,⋌\right)\right]$ for every $\varkappa \in \mathbb{I}.$ Here, for every $⋌\in \left[0,1\right],$ endpoint real-valued mappings $g_{*}\left(·,⋌\right),g^{*}\left(·,⋌\right):\mathbb{I}\to \mathbb{R}$ are named as lower and upper mappings of $g$.

Definition 5

([66]). A $F-N-V-Mg:\mathbb{I}\subset \mathbb{R}\to {\mathbb{E}}_I$ is said to be continuous at $\varkappa \in \mathbb{I},$ if for every $⋌\in \left[0,1\right],$ both endpoint mappings $g_{*}\left(\varkappa ,⋌\right)$ and $g^{*}\left(\varkappa ,⋌\right)$ are continuous at $\varkappa \in \mathbb{I}.$

Definition 6

([62]). Let $g:[a,b]\subset \mathbb{R}\to {\mathbb{E}}_I$ is $F-N-V-M$ . The fuzzy Aumann integral ($\left(FA\right)$-integral) of $g$ over $\left[a,b\right],$ denoted by $\left(FA\right){\int }_a^{b}g\left(\varkappa \right)d\varkappa$ , is defined level-wise by

$${\left[\left(FA\right){\int }_a^{b}g\left(\varkappa \right)d\varkappa \right]}^{\begin{array}{c}\\ ⋌\end{array}}=\left(IA\right){\int }_a^b{g}_{⋌}\left(\varkappa \right)d\varkappa =\left\{{\int }_a^{b}g\left(\varkappa ,⋌\right)d\varkappa :g\left(\varkappa ,⋌\right)\in S\left(g_{⋌}\right)\right\},$$

(14)

where $S\left(g_{⋌}\right)=\left\{g\left(.,⋌\right)\to \mathbb{R}:g\left(.,⋌\right)is integrable and g\left(\varkappa ,⋌\right)\in g_{⋌}\left(\varkappa \right)\right\},$ for every $⋌\in \left[0,1\right]$. $g$ is $\left(FA\right)$-integrable over $[a,b]$ if $\left(FA\right){\int }_a^{b}g\left(\varkappa \right)d\varkappa \in {\mathbb{E}}_I.$

Theorem 4

([55]). Let $g:[a,b]\subset \mathbb{R}\to {\mathbb{E}}_I$ be an $F-N-V-M$ as well as $⋌$-levels define the family of $I-V-M$s $g_{⋌}:[a,b]\subset \mathbb{R}\to {\mathbb{R}}_I$ satisfying that $g_{⋌}\left(\varkappa \right)=\left[g_{*}\left(\varkappa ,⋌\right),g^{*}\left(\varkappa ,⋌\right)\right]$ for every $\varkappa \in [a,b]$ and for every $⋌\in \left[0,1\right].$ Then $g$ is $\left(FA\right)$-integrable over $[a,b]$ when and only when, $g_{*}\left(\varkappa ,⋌\right)$ and $g^{*}\left(\varkappa ,⋌\right)$ both are integrable over $[a,b]$ . Moreover, if $g$ is $\left(FA\right)$-integrable over $\left[a,b\right],$ then

$${\left[\left(FA\right){\int }_a^{b}g\left(\varkappa \right)d\varkappa \right]}^{\begin{array}{c}\\ ⋌\end{array}}=\left[{\int }_a^b{g}_{*}\left(\varkappa ,⋌\right)d\varkappa ,{\int }_a^b{g}^{*}\left(\varkappa ,⋌\right)d\varkappa \right]=\left(IA\right){\int }_a^b{g}_{⋌}\left(\varkappa \right)d\varkappa ,$$

(15)

for every $⋌\in \left[0,1\right].$

Breckner discussed the coming emerging idea of interval-valued convexity in [63].

An I∙V∙M $g:\mathbb{I}=\left[a,b\right]\to {\mathbb{R}}_I$ is named as convex I∙V∙M if

$$g\left(ϛ\varkappa +\left(1-ϛ\right)c\right)\supseteq ϛg\left(\varkappa \right)+(1-ϛ)g\left(c\right),$$

(16)

for all $\varkappa ,c\in \left[a,b\right],ϛ\in \left[0,1\right]$, where ${\mathbb{R}}_I$ is the collection of real-valued intervals. If (16) is reversed, then $g$ is named as concave.

Definition 7

([58]). The $F-N-V-M$$\tilde{g}:\left[a,b\right]\to {\mathbb{E}}_I$ is named as convex $F-N-V-M$ on$\left[a,b\right]$ if

$$\tilde{g}\left(ϛ\varkappa +\left(1-ϛ\right)c\right){\le }_{\mathbb{F}}ϛ\odot \tilde{g}\left(\varkappa \right)\oplus (1-ϛ)\odot \tilde{g}\left(c\right),$$

(17)

for all $\varkappa ,c\in \left[a,b\right],ϛ\in \left[0,1\right],$ where $\tilde{g}\left(\varkappa \right){\ge }_{\mathbb{F}}\tilde{0}$ for all $\varkappa \in \left[a,b\right].$ If (17) is reversed, then $\tilde{g}$ is named as concave $F-N-V-M$ on $\left[a,b\right]$. $\tilde{g}$ is affine if and only if it is both convex and concave $F-N-V-M$.

Definition 8

([67]). The $F-N-V-M$$\tilde{g}:\left[a,b\right]\to {\mathbb{E}}_I$ is named as $\mathcal{U}-\mathfrak{O}$ convex $F-N-V-M$ on $\left[a,b\right]$ if

$$\tilde{g}\left(ϛ\varkappa +\left(1-ϛ\right)c\right){\supseteq }_{\mathbb{F}}ϛ\odot \tilde{g}\left(\varkappa \right)\oplus (1-ϛ)\odot \tilde{g}\left(c\right),$$

(18)

for all $\varkappa ,c\in \left[a,b\right],ϛ\in \left[0,1\right],$ where $\tilde{g}\left(\varkappa \right){\ge }_{\mathbb{F}}\tilde{0}$ for all $\varkappa \in \left[a,b\right].$ If (18) is reversed, then, $\tilde{g}$ is named as $\mathcal{U}-\mathfrak{O}$ concave $F-N-V-M$ on $\left[a,b\right]$. $\tilde{g}$ is $\mathcal{U}-\mathfrak{O}$ affine $F-N-V-M$ if and only if it is both $\mathcal{U}-\mathfrak{O}$ convex and $\mathcal{U}-\mathfrak{O}$ concave $F-N-V-M$.

Theorem 5

([67]). Let $\tilde{g}:[a,b]\to {\mathbb{E}}_I$ be a $F-N-V-M$ , whose $⋌$-levels define the family of interval-valued mappings $g_{⋌}:[a,b]\to {\mathbb{R}}_I^{+}\subset {\mathbb{R}}_I$ are given by

$$g_{⋌}\left(\varkappa \right)=\left[g_{*}\left(\varkappa ,⋌\right),g^{*}\left(\varkappa ,⋌\right)\right],$$

(19)

for all $\varkappa \in [a,b]$ and for all $⋌\in \left[0,1\right]$. Then, $\tilde{g}$ is $\mathcal{U}-\mathfrak{O}$ convex $F-N-V-M$ on $[a,b],$ if and only if, for all $⋌\in \left[0,1\right],$$g_{*}\left(\varkappa ,⋌\right)$ is a convex mapping and $g^{*}\left(\varkappa ,⋌\right)$ is a concave mapping.

Remark 2.

If $g_{*}\left(\varkappa ,⋌\right)\ne g^{*}\left(\varkappa ,⋌\right)$ and $⋌=1$ , then we obtain the inequality (16).

If $g_{*}\left(\varkappa ,⋌\right)=g^{*}\left(\varkappa ,⋌\right)$ and $⋌=1$, then we obtain the classical definition of convex mappings.

Definition 9

([21]). A set $K=\left[a,b\right]\subset {\mathbb{R}}^{+}=\left(0,\infty \right)$ is said to be convex set, if, for all $\varkappa ,c\in K,ϛ\in \left[0,1\right]$ , we have

$$\frac{\varkappa c}{ϛ\varkappa +\left(1-ϛ\right)c}\in K.$$

(20)

Definition 10

([21]). The $g:\left[a,b\right]\to {\mathbb{R}}^{+}$ is named as harmonically convex ($\mathcal{H}$-convex) mapping on$\left[a,b\right]$ if

$$g\left(\frac{\varkappa c}{ϛ\varkappa +\left(1-ϛ\right)c}\right)\le \left(1-ϛ\right)g\left(\varkappa \right)+ϛg\left(c\right),$$

(21)

for all $\varkappa ,c\in \left[a,b\right],ϛ\in \left[0,1\right],$ where $g\left(\varkappa \right)\ge 0$ for all $\varkappa \in \left[a,b\right].$ If (13) is reversed, then $g$ is named as $\mathcal{H}$-concave $F-N-V-M$ on $\left[a,b\right]$.

Definition 11

([61]). The $F-N-V-M$$\tilde{g}:\left[a,b\right]\to {\mathbb{F}}_0$ is named as $\mathcal{H}$-convex $F-N-V-M$ on $\left[a,b\right]$ if

$$\tilde{g}\left(\frac{\varkappa c}{ϛ\varkappa +\left(1-ϛ\right)c}\right){\le }_{\mathbb{F}}\left(1-ϛ\right)\odot \tilde{g}\left(\varkappa \right)\oplus ϛ\odot \tilde{g}\left(c\right),$$

(22)

for all $\varkappa ,c\in \left[a,b\right],ϛ\in \left[0,1\right],$ where $\tilde{g}\left(\varkappa \right){\ge }_{\mathbb{F}}\tilde{0}$ , for all $\varkappa \in \left[a,b\right].$ If (14) is reversed, then $\tilde{g}$ is named as $\mathcal{H}$-concave $F-N-V-M$ on $\left[a,b\right]$.

3. Fuzzy Hermite–Hadamard Inequalities

In this section, we define $\mathcal{U}-\mathfrak{O}-\mathcal{H}$ convex $F-N-V-M$s. Moreover, we prove H·H Fejér inequalities and weighted symmetric interval-valued mappings for $\mathcal{U}-\mathfrak{O}-\mathcal{H}$ convex interval-valued mappings.

Definition 12.

The $F-N-V-M$$\tilde{g}:\left[a,b\right]\to {\mathbb{F}}_0$ is named as $\mathcal{U}-\mathfrak{O}-\mathcal{H}$ convex $F-N-V-M$ on $\left[a,b\right]$ if

$$\tilde{g}\left(\frac{\varkappa c}{ϛ\varkappa +\left(1-ϛ\right)c}\right){\supseteq }_{\mathbb{F}}\left(1-ϛ\right)\odot \tilde{g}\left(\varkappa \right)\oplus ϛ\odot \tilde{g}\left(c\right),$$

(23)

for all $\varkappa ,c\in \left[a,b\right],ϛ\in \left[0,1\right],$ where $\tilde{g}\left(\varkappa \right){\ge }_{\mathbb{F}}\tilde{0}$ , for all $\varkappa \in \left[a,b\right]$ . If (23) is reversed, then $\tilde{g}$ is named as $\mathcal{U}-\mathfrak{O}-\mathcal{H}$ concave $F-N-V-M$ on $\left[a,b\right]$ . The set of all $\mathcal{U}-\mathfrak{O}-\mathcal{H}$ convex ($\mathcal{U}-\mathfrak{O}-\mathcal{H}$ concave) $F-N-V-M$ is denoted by

$$UDHFSX\left(\left[a,b\right],{\mathbb{F}}_0\right)\left(UDHFSV\left(\left[a,b\right],{\mathbb{F}}_0\right)\right).$$

Theorem 6.

Let $\left[a,b\right]$ be an $\mathcal{H}$-convex set, and let $\tilde{g}:\left[a,b\right]\to {\mathbb{F}}_0$ be an $F-N-V-M$ , whose $⋌$-levels define the family of $I-V-M$s $g_{⋌}:\left[a,b\right]\subset \mathbb{R}\to {\mathbb{R}}_I^{+}\subset {\mathbb{R}}_I$ are given by

$$g_{⋌}\left(\varkappa \right)=\left[g_{*}\left(\varkappa ,⋌\right),g^{*}\left(\varkappa ,⋌\right)\right],\forall \varkappa \in \left[a,b\right].$$

(24)

for all $\varkappa \in \left[a,b\right]$ , $⋌\in \left[0,1\right]$ . Then, $\tilde{g}\in UDHFSX\left(\left[a,b\right],{\mathbb{F}}_0\right),$ if and only if, for all $\in \left[0,1\right],$$g_{*}\left(\varkappa ,⋌\right)\in HSX\left(\left[a,b\right],{\mathbb{R}}^{+}\right)$ and $g^{*}\left(\varkappa ,⋌\right)\in \left(UDHFSV\left(\left[a,b\right],{\mathbb{F}}_0\right)\right)$.

Proof. 

Assume that for each $⋌\in \left[0,1\right],g_{*}\left(\varkappa ,⋌\right)$ and $g^{*}\left(\varkappa ,⋌\right)$ are $\mathcal{H}$-convex on $K.$ Then from (24), we have

$$g_{*}\left(\frac{\varkappa c}{ϛ\varkappa +\left(1-ϛ\right)c},⋌\right)\le (1-ϛ)g_{*}\left(\varkappa ,⋌\right)+ϛg_{*}\left(c,⋌\right),$$

and

$$g^{*}\left(\frac{\varkappa c}{ϛ\varkappa +\left(1-ϛ\right)c},⋌\right)\ge (1-ϛ)g^{*}\left(\varkappa ,⋌\right)+ϛg^{*}\left(c,⋌\right).$$

Then by (24), (19), and (11), we obtain

$$g_{⋌}\left(\frac{\varkappa c}{ϛ\varkappa +\left(1-ϛ\right)c}\right)=\left[g_{*}\left(\frac{\varkappa c}{ϛ\varkappa +\left(1-ϛ\right)c},⋌\right),g^{*}\left(\frac{\varkappa c}{ϛ\varkappa +\left(1-ϛ\right)c},⋌\right)\right],$$

$${\supseteq }_I\left(1-ϛ\right)\left[g_{*}\left(\varkappa ,⋌\right),g^{*}\left(\varkappa ,⋌\right)\right]+ϛ\left[g_{*}\left(c,⋌\right),g^{*}\left(c,⋌\right)\right],$$

that is

$$\tilde{g}\left(\frac{\varkappa c}{ϛ\varkappa +\left(1-ϛ\right)c}\right){\supseteq }_{\mathbb{F}}(1-ϛ)\odot \tilde{g}\left(\varkappa \right)\oplus ϛ\odot \tilde{g}\left(c\right),\forall \varkappa ,c\in K,ϛ\in \left[0,1\right].$$

Hence, $\tilde{g}$ is $\mathcal{U}-\mathfrak{O}-\mathcal{H}$ convex $F-N-V-M$ on $K.$

Conversely, let $\tilde{g}$ be $\mathcal{U}-\mathfrak{O}-\mathcal{H}$ convex $F-N-V-M$ on $K.$ Then for all $\varkappa ,c\in K$, $ϛ\in \left[0,1\right],$ we have

$$\tilde{g}\left(\frac{\varkappa c}{ϛ\varkappa +\left(1-ϛ\right)c}\right){\supseteq }_{\mathbb{F}}(1-ϛ)\odot \tilde{g}\left(\varkappa \right)\oplus ϛ\odot \tilde{g}\left(c\right).$$

Therefore, from (24), for each $⋌\in \left[0,1\right]$, left side of above inequality, we have

$$g_{⋌}\left(\frac{\varkappa c}{ϛ\varkappa +\left(1-ϛ\right)c}\right)=\left[g_{*}\left(\frac{\varkappa c}{ϛ\varkappa +\left(1-ϛ\right)c},⋌\right),g^{*}\left(\frac{\varkappa c}{ϛ\varkappa +\left(1-ϛ\right)c},⋌\right)\right].$$

Again, from (24), we obtain

$$\left(1-ϛ\right)g_{⋌}\left(\varkappa \right)+ϛg_{⋌}\left(\varkappa \right)=(1-ϛ)\left[g_{*}\left(\varkappa ,⋌\right),g^{*}\left(\varkappa ,⋌\right)\right]+ϛ\left[g_{*}\left(c,⋌\right),g^{*}\left(c,⋌\right)\right],$$

for all $\varkappa ,c\in K$, $ϛ\in \left[0,1\right].$ Then, by $\mathcal{U}-\mathfrak{O}-\mathcal{H}$ convexity of $\tilde{g}$, we have for all $\varkappa ,c\in K$, $ϛ\in \left[0,1\right]$ such that

$$g_{*}\left(\frac{\varkappa c}{ϛ\varkappa +\left(1-ϛ\right)c},⋌\right)\le (1-ϛ)g_{*}\left(\varkappa ,⋌\right)+ϛg_{*}\left(c,⋌\right),$$

and

$$g^{*}\left(\frac{\varkappa c}{ϛ\varkappa +\left(1-ϛ\right)c},⋌\right)\ge (1-ϛ)g^{*}\left(\varkappa ,⋌\right)+ϛg^{*}\left(c,⋌\right),$$

for each $⋌\in \left[0,1\right].$ Hence, the result follows. □

Example 1.

We consider the $F-N-V-M$s $\tilde{g}:\left[\frac{1}{2},1\right]\to {\mathbb{E}}_I$ defined by,

$$\tilde{g}\left(\varkappa \right)\left(\partial \right)=\left\{\begin{array}{c}\frac{\partial -{\varkappa }^2}{1-{\varkappa }^2}\hspace{1em}\partial \in \left[{\varkappa }^2,1\right],\\ \frac{5-e^{\varkappa }-\partial }{4-e^{\varkappa }}\hspace{1em}\partial \in \left(1,5-e^{\varkappa }\right],\\ 0\hspace{1em}\mathrm{otherwise}.\end{array}\right.$$

Then, for each $⋌\in \left[0,1\right],$ we have $g_{⋌}\left(\varkappa \right)=\left[\left(1-⋌\right){\varkappa }^2+⋌,\left(1-⋌\right)\left(5-e^{\varkappa }\right)+⋌\right]$ . We can easily see that $g_{*}\left(\varkappa ,⋌\right)\in HSX\left(\left[a,b\right],{\mathbb{R}}^{+}\right)$ , $g^{*}\left(\varkappa ,⋌\right)\in HSV\left(\left[a,b\right],{\mathbb{R}}^{+}\right)$ , for each $⋌\in [0,1]$ . Hence, $\tilde{g}\in UDHFSX\left(\left[a,b\right],{\mathbb{E}}_I\right)$.

We may now utilize the new definitions stated below to study specific classical and modern outcomes as subsets of the primary findings.

Definition 13.

Let $\tilde{g}:[a,b]\to {\mathbb{E}}_I$ be an $F-N-V-M$ , whose $⋌$-levels define the family of $I-V-M$s $g_{⋌}:[a,b]\to {\mathbb{R}}_I^{+}\subset {\mathbb{R}}_I$ are given by

$$g_{⋌}\left(\varkappa \right)=\left[g_{*}\left(\varkappa ,⋌\right),g^{*}\left(\varkappa ,⋌\right)\right],$$

(25)

for all $\varkappa \in [a,b]$and for all $⋌\in \left[0,1\right]$ . Then, $\tilde{g}$ is lower $\mathcal{U}-\mathfrak{O}-\mathcal{H}$ convex (concave) $F-N-V-M$ on $[a,b],$ if and only if,

$$g_{*}\left(\frac{\varkappa c}{ϛ\varkappa +\left(1-ϛ\right)c},⋌\right)\le \left(\ge \right)(1-ϛ)g_{*}\left(\varkappa ,⋌\right)+ϛg_{*}\left(c,⋌\right),$$

and

$$g^{*}\left(\frac{\varkappa c}{ϛ\varkappa +\left(1-ϛ\right)c},⋌\right)=(1-ϛ)g^{*}\left(\varkappa ,⋌\right)+ϛg^{*}\left(c,⋌\right),$$

for all $⋌\in \left[\mathbf{0},\mathbf{1}\right]$.

Definition 14.

Let $\tilde{g}:[a,b]\to {\mathbb{E}}_I$ be a $F-N-V-M$ , whose $⋌$-levels define the family of $I-V-M$s $g_{⋌}:[a,b]\to {\mathbb{R}}_I^{+}\subset {\mathbb{R}}_I$ are given by

$$g_{⋌}\left(\varkappa \right)=\left[g_{*}\left(\varkappa ,⋌\right),g^{*}\left(\varkappa ,⋌\right)\right],$$

(26)

for all $\varkappa \in [a,b]$ and for all $⋌\in \left[0,1\right]$ . Then, $g$ is upper $\mathcal{U}-\mathfrak{O}-\mathcal{H}$ convex (concave) $F-N-V-M$ on $[a,b],$ if and only if,

$$g_{*}\left(\frac{\varkappa c}{ϛ\varkappa +\left(1-ϛ\right)c},⋌\right)=(1-ϛ)g_{*}\left(\varkappa ,⋌\right)+ϛg_{*}\left(c,⋌\right),$$

and

$$g^{*}\left(\frac{\varkappa c}{ϛ\varkappa +\left(1-ϛ\right)c},⋌\right)\le \left(\ge \right)(1-ϛ)g^{*}\left(\varkappa ,⋌\right)+ϛg^{*}\left(c,⋌\right),$$

for all $⋌\in \left[0,1\right]$.

Remark 3.

Let $g_{*}\left(\varkappa ,⋌\right)\ne g^{*}\left(\varkappa ,⋌\right)$ with $⋌=1$ . Then, $\mathcal{U}-\mathfrak{O}-\mathcal{H}$ convex (concave) $F-N-V-M$ reduces to the classical interval-valued $\mathcal{H}$-convex(concave) mapping.

Let $\tilde{g}$ be a lower $\mathcal{U}-\mathfrak{O}-\mathcal{H}$ convex (concave) $F-N-V-M$ . Then, we obtain the definition of $\mathcal{H}$-convex (concave) $F-N-V-M$ , see [61].

If $g_{*}\left(\varkappa ,⋌\right)=g^{*}\left(\varkappa ,⋌\right)$ with $⋌=1$ , then $\mathcal{H}$-convex (concave) $F-N-V-M$ reduces to the classical $\mathcal{H}$-convex(concave) mapping, see [21].

In our next main result, we prove H·H-type inequalities for $\mathcal{U}-\mathfrak{O}-\mathcal{H}$ convex (concave) $F-N-V-M$ . Firstly, we prove H·H-type inequality for $\mathcal{U}-\mathfrak{O}-\mathcal{H}$ convex (concave) $F-N-V-M$.

Theorem 7.

Let $\tilde{g}\in UDHFSX\left(\left[a,b\right],{\mathbb{F}}_0\right)$ , whose $⋌$-levels define the family of $I-V-M$s $g_{⋌}:\left[a,b\right]\subset \mathbb{R}\to {\mathbb{R}}_I^{+}$ are given by $g_{⋌}\left(\varkappa \right)=\left[g_{*}\left(\varkappa ,⋌\right),g^{*}\left(\varkappa ,⋌\right)\right]$ for all $\varkappa \in \left[a,b\right]$ , $⋌\in \left[0,1\right]$ . If $\tilde{g}\in {\mathcal{F}\mathcal{A}}_{\left(\left[a,b\right],⋌\right)}$ , then

$$\tilde{g}\left(\frac{2ab}{a+b}\right){\supseteq }_{\mathbb{F}}\frac{\begin{array}{c}\\ ab\end{array}}{b-a}\odot \left(FA\right){\int }_a^b\frac{\tilde{g}\left(\varkappa \right)}{{\varkappa }^2}d\varkappa {\supseteq }_{\mathbb{F}}\frac{\tilde{g}\left(a\right)\oplus \tilde{g}\left(b\right)}{2}.$$

(27)

If $\tilde{g}\in UDHFSV\left(\left[a,b\right],{\mathbb{F}}_0\right)$, then

$$\tilde{g}\left(\frac{2ab}{a+b}\right){\subseteq }_{\mathbb{F}}\frac{\begin{array}{c}\\ ab\end{array}}{b-a}\odot {\int }_a^b\frac{\tilde{g}\left(\varkappa \right)}{{\varkappa }^2}d\varkappa {\subseteq }_{\mathbb{F}}\frac{\tilde{g}\left(a\right)\oplus \tilde{g}\left(b\right)}{2}.$$

(28)

Proof. 

Let $\tilde{g}\in UDHFSX\left(\left[a,b\right],{\mathbb{F}}_0\right)$. Then, by hypothesis, we have

$$2\odot \tilde{g}\left(\frac{2ab}{a+b}\right){\supseteq }_{\mathbb{F}}\tilde{g}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b}\right)\oplus \tilde{g}\left(\frac{ab}{\left(1-ϛ\right)a+ϛb}\right).$$

Therefore, for each $⋌\in [0,1]$, we have

$$\begin{array}{c}2g_{*}\left(\frac{2ab}{a+b},⋌\right)\le g_{*}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b},⋌\right)+g_{*}\left(\frac{ab}{\left(1-ϛ\right)a+ϛb},⋌\right),\\ 2g^{*}\left(\frac{2ab}{a+b},⋌\right)\ge g^{*}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b},⋌\right)+g^{*}\left(\frac{ab}{\left(1-ϛ\right)a+ϛb},⋌\right).\end{array}$$

Then

$$\begin{array}{c}2{\int }_0^1{g}_{*}\left(\frac{2ab}{a+b},⋌\right)dϛ\le {\int }_0^1{g}_{*}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b},⋌\right)dϛ+{\int }_0^1{g}_{*}\left(\frac{ab}{\left(1-ϛ\right)a+ϛb},⋌\right)dϛ,\\ 2{\int }_0^1{g}^{*}\left(\frac{2ab}{a+b},⋌\right)dϛ\ge {\int }_0^1{g}^{*}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b},⋌\right)dϛ+{\int }_0^1{g}^{*}\left(\frac{ab}{\left(1-ϛ\right)a+ϛb},⋌\right)dϛ.\end{array}$$

It follows that

$$\begin{array}{c}{g}_{*}\left(\frac{2ab}{a+b},⋌\right)\le \frac{\begin{array}{c}\\ ab\end{array}}{b-a}{\int }_a^b\frac{g_{*}\left(\varkappa ,⋌\right)}{{\varkappa }^2}d\varkappa ,\\ g^{*}\left(\frac{2ab}{a+b},⋌\right)\ge \frac{\begin{array}{c}\\ ab\end{array}}{b-a}{\int }_a^b\frac{g^{*}\left(\varkappa ,⋌\right)}{{\varkappa }^2}d\varkappa .\end{array}$$

That is

$$\left[g_{*}\left(\frac{2ab}{a+b},⋌\right),g^{*}\left(\frac{2ab}{a+b},⋌\right)\right]{\supseteq }_I\frac{\begin{array}{c}\\ ab\end{array}}{b-a}\left[{\int }_a^b\frac{g_{*}\left(\varkappa ,⋌\right)}{{\varkappa }^2}d\varkappa ,{\int }_a^b\frac{g^{*}\left(\varkappa ,⋌\right)}{{\varkappa }^2}d\varkappa \right].$$

Thus,

$$\tilde{g}\left(\frac{2ab}{a+b}\right){\supseteq }_{\mathbb{F}}\frac{\begin{array}{c}\\ ab\end{array}}{b-a}\odot \left(FA\right){\int }_a^b\frac{\tilde{g}\left(\varkappa \right)}{{\varkappa }^2}d\varkappa .$$

(29)

In a similar way as above, we have

$$\frac{\begin{array}{c}\\ ab\end{array}}{b-a}\odot \left(FA\right){\int }_a^b\frac{\tilde{g}\left(\varkappa \right)}{{\varkappa }^2}d\varkappa {\supseteq }_{\mathbb{F}}\frac{\tilde{g}\left(a\right)\oplus \tilde{g}\left(b\right)}{2}.$$

(30)

Combining (29) and (30), we have

$$\tilde{g}\left(\frac{2ab}{a+b}\right){\supseteq }_{\mathbb{F}}\frac{\begin{array}{c}\\ ab\end{array}}{b-a}\odot \left(FA\right){\int }_a^b\frac{\tilde{g}\left(\varkappa \right)}{{\varkappa }^2}d\varkappa {\supseteq }_{\mathbb{F}}\frac{\tilde{g}\left(a\right)\oplus \tilde{g}\left(b\right)}{2}.$$

Hence, the required result. □

Remark 4.

Let $\tilde{g}$ be a lower $\mathcal{U}-\mathfrak{O}-\mathcal{H}$ convex $F-N-V-M$. Then, we obtain the result for definition of $\mathcal{H}$-convex $F-N-V-M$, see [23].

If $g_{*}\left(\varkappa ,⋌\right)\ne g^{*}\left(\varkappa ,⋌\right)$ with $⋌=1$, then we obtain the result for classical definition of $\mathcal{H}$-convex I∙V∙M, see [23]:

$$g\left(\frac{2ab}{a+b}\right)\supseteq \frac{\begin{array}{c}\\ ab\end{array}}{b-a}\left(IA\right){\int }_a^b\frac{g\left(\varkappa \right)}{{\varkappa }^2}d\varkappa \supseteq \frac{g\left(a\right)+g\left(b\right)}{2}.$$

(31)

If $g_{*}\left(\varkappa ,⋌\right)=g^{*}\left(\varkappa ,⋌\right)$ with $⋌=1$, then Theorem 7 reduces to the result for classical $\mathcal{H}$-convex mapping, see [21]:

$$\left(\frac{2ab}{a+b}\right)\le \frac{\begin{array}{c}\\ ab\end{array}}{b-a}\left(R\right){\int }_a^b\frac{g\left(\varkappa \right)}{{\varkappa }^2}d\varkappa \le \frac{g\left(a\right)+g\left(b\right)}{2}.$$

(32)

Example 2.

We consider the $F-N-V-M$s $\tilde{g}:\left[\frac{1}{2},1\right]\to {\mathbb{E}}_I,$ as in Example 1. Then, for each $⋌\in \left[0,1\right],$ we have $g_{⋌}\left(\varkappa \right)=\left[\left(1-⋌\right){\varkappa }^2+⋌,\left(1-⋌\right)\left(5-e^{\varkappa }\right)+⋌\right]$ is $\mathcal{U}-\mathfrak{O}-\mathcal{H}$ convex $F-N-V-M$. Since, $g_{*}\left(\varkappa ,⋌\right)=\left(1-⋌\right){\varkappa }^2+⋌,$$g^{*}\left(\varkappa ,⋌\right)=\left(1-⋌\right)\left(5-e^{\varkappa }\right)+⋌$ . We now compute the following:

$$g_{*}\left(\frac{2ab}{a+b},⋌\right)\le \frac{\begin{array}{c}\\ ab\end{array}}{b-a}{\int }_a^b\frac{g_{*}\left(\varkappa ,⋌\right)}{{\varkappa }^2}d\varkappa \le \frac{g_{*}\left(a,⋌\right)+g_{*}\left(b,⋌\right)}{2},$$

$$g_{*}\left(\frac{2ab}{a+b},⋌\right)=g_{*}\left(\frac{2}{3},⋌\right)=\frac{4}{9}\left(1-⋌\right)+⋌,$$

$$\frac{\begin{array}{c}\\ ab\end{array}}{b-a}{\int }_a^b\frac{g_{*}\left(\varkappa ,⋌\right)}{{\varkappa }^2}d\varkappa =\frac{1}{2}{\int }_0^2\frac{\left(1-⋌\right){\varkappa }^2+⋌}{{\varkappa }^2}d\varkappa =\frac{1}{2}\left(1+⋌\right),$$

$$\frac{g_{*}\left(a,⋌\right)+g_{*}\left(b,⋌\right)}{2}=\frac{5+3⋌}{4},$$

for all $⋌\in \left[0,1\right].$ That means

$$\frac{4}{9}\left(1-⋌\right)+⋌\le \frac{1}{2}\left(1+⋌\right)\le \frac{5+3⋌}{4}.$$

Similarly, it can be easily shown that

$$g^{*}\left(\frac{2ab}{a+b},⋌\right)\ge \frac{\begin{array}{c}\\ ab\end{array}}{b-a}{\int }_a^b\frac{g^{*}\left(\varkappa ,⋌\right)}{{\varkappa }^2}d\varkappa \ge \frac{g^{*}\left(a,⋌\right)+g^{*}\left(b,⋌\right)}{2}.$$

for all $⋌\in \left[0,1\right],$ such that

$$g^{*}\left(\frac{2ab}{a+b},⋌\right)=g_{*}\left(\frac{2}{3},⋌\right)=\left(1-⋌\right)\left(5-e^{\frac{2}{3}}\right)+⋌,$$

$$\frac{\begin{array}{c}\\ ab\end{array}}{b-a}{\int }_a^b\frac{g^{*}\left(\varkappa ,⋌\right)}{{\varkappa }^2}d\varkappa ={\int }_0^2\frac{\left(1-⋌\right)\left(5-e^{\varkappa }\right)+⋌}{{\varkappa }^2}d\varkappa \approx 3-2⋌,$$

$$\frac{g^{*}\left(a,⋌\right)+g^{*}\left(b,⋌\right)}{2}=\frac{\left(5-e\right)\left(1-⋌\right)+\left(5-e^{\frac{1}{2}}\right)\left(1-⋌\right)+2⋌}{2}.$$

From which, we have

$$\left(1-⋌\right)\left(5-e^{\frac{2}{3}}\right)+⋌\ge 3-2⋌\ge \frac{\left(5-e\right)\left(1-⋌\right)+\left(5-e^{\frac{1}{2}}\right)\left(1-⋌\right)+2⋌}{2},$$

that is

$$\left[\frac{4}{9}\left(1-⋌\right)+⋌,\left(1-⋌\right)\left(5-e^{\frac{2}{3}}\right)+⋌\right]{\supseteq }_I\left[\frac{1}{2}\left(1+⋌\right),3-2⋌\right]{\supseteq }_I\left[\frac{5+3⋌}{4},\frac{\left(5-e\right)\left(1-⋌\right)+\left(5-e^{\frac{1}{2}}\right)\left(1-⋌\right)+2⋌}{2}\right],$$

for all $⋌\in \left[0,1\right].$

Hence,

$$\tilde{g}\left(\frac{2ab}{a+b}\right){\supseteq }_{\mathbb{F}}\frac{\begin{array}{c}\\ ab\end{array}}{b-a}\odot {\int }_a^b\frac{\tilde{g}\left(\varkappa \right)}{{\varkappa }^2}d\varkappa {\supseteq }_{\mathbb{F}}\frac{\tilde{g}\left(a\right)\oplus \tilde{g}\left(b\right)}{2}.$$

Theorem 8.

Let $\tilde{g}\in UDHFSX\left(\left[a,b\right],{\mathbb{F}}_0\right)$ , whose $⋌$-levels define the family of $I-V-M$s $g_{⋌}:\left[a,b\right]\subset \mathbb{R}\to {\mathbb{R}}_I^{+}$ are given by $g_{⋌}\left(\varkappa \right)=\left[g_{*}\left(\varkappa ,⋌\right),g^{*}\left(\varkappa ,⋌\right)\right]$ for all $\varkappa \in \left[a,b\right]$ , $⋌\in \left[0,1\right]$ . If $\tilde{g}\in {\mathcal{F}\mathcal{A}}_{\left(\left[a,b\right],⋌\right)}$ , then

$$\tilde{g}\left(\frac{2ab}{a+b}\right){\supseteq }_{\mathbb{F}}{\mathcal{V}}_2{\supseteq }_{\mathbb{F}}\frac{\begin{array}{c}\\ ab\end{array}}{b-a}\odot \left(FA\right){\int }_a^b\frac{\tilde{g}\left(\varkappa \right)}{{\varkappa }^2}d\varkappa {\supseteq }_{\mathbb{F}}{\mathcal{V}}_1{\supseteq }_{\mathbb{F}}\frac{\tilde{g}\left(a\right)\oplus \tilde{g}\left(b\right)}{2},$$

(33)

where

$${\mathcal{V}}_1=\frac{1}{2}\odot \left[\frac{\tilde{g}\left(a\right)\oplus \tilde{g}\left(b\right)}{2}\oplus \tilde{g}\left(\frac{2ab}{a+b}\right)\right],$$

$${\mathcal{V}}_2=\frac{1}{2}\odot \left[\tilde{g}\left(\frac{4ab}{a+3b}\right)\oplus \tilde{g}\left(\frac{4ab}{3a+b}\right)\right],$$

and ${\mathcal{V}}_1=\left[{{\mathcal{V}}_1}_{*},{{\mathcal{V}}_1}^{*}\right]$ , ${\mathcal{V}}_2=\left[{{\mathcal{V}}_2}_{*},{{\mathcal{V}}_2}^{*}\right].$

If $\tilde{g}\in UDHFSV\left(\left[a,b\right],{\mathbb{F}}_0\right)$ , then inequality (33) is reversed.

Proof. 

Take $\left[a,\frac{2ab}{a+b}\right],$ we have

$$2\odot \tilde{g}\left(\frac{a\frac{4ab}{a+b}}{ϛa+\left(1-ϛ\right)\frac{2ab}{a+b}}+\frac{a\frac{4ab}{a+b}}{(1-ϛ)a+ϛ\frac{2ab}{a+b}}\right){\supseteq }_{\mathbb{F}}\tilde{g}\left(\frac{a\frac{2ab}{a+b}}{ϛa+\left(1-ϛ\right)\frac{2ab}{a+b}}\right)\oplus \tilde{g}\left(\frac{a\frac{2ab}{a+b}}{(1-ϛ)a+ϛ\frac{2ab}{a+b}}\right).$$

Therefore, for every $⋌\in [0,1]$, we have

$$\begin{array}{c}2g_{*}\left(\frac{a\frac{4ab}{a+b}}{ϛa+\left(1-ϛ\right)\frac{2ab}{a+b}}+\frac{a\frac{4ab}{a+b}}{(1-ϛ)a+ϛ\frac{2ab}{a+b}},⋌\right)\le g_{*}\left(\frac{a\frac{2ab}{a+b}}{ϛa+\left(1-ϛ\right)\frac{2ab}{a+b}},⋌\right)+g_{*}\left(\frac{a\frac{2ab}{a+b}}{(1-ϛ)a+ϛ\frac{2ab}{a+b}},⋌\right),\\ 2g^{*}\left(\frac{a\frac{4ab}{a+b}}{ϛa+\left(1-ϛ\right)\frac{2ab}{a+b}}+\frac{a\frac{4ab}{a+b}}{(1-ϛ)a+ϛ\frac{2ab}{a+b}},⋌\right)\ge g^{*}\left(\frac{a\frac{2ab}{a+b}}{ϛa+\left(1-ϛ\right)\frac{2ab}{a+b}},⋌\right)+g^{*}\left(\frac{a\frac{2ab}{a+b}}{(1-ϛ)a+ϛ\frac{2ab}{a+b}},⋌\right).\end{array}$$

In consequence, we obtain

$$\begin{array}{c}\frac{1}{2}{g}_{*}\left(\frac{4ab}{a+3b},⋌\right)\le \frac{\begin{array}{c}\\ ab\end{array}}{b-a}{\int }_a^{\frac{2ab}{a+b}}\frac{g_{*}\left(\varkappa ,⋌\right)}{{\varkappa }^2}d\varkappa ,\\ \frac{1}{2}{g}^{*}\left(\frac{4ab}{a+3b},⋌\right)\ge \frac{\begin{array}{c}\\ ab\end{array}}{b-a}{\int }_a^{\frac{2ab}{a+b}}\frac{g^{*}\left(\varkappa ,⋌\right)}{{\varkappa }^2}d\varkappa .\end{array}$$

That is

$$\frac{1}{2}\left[g_{*}\left(\frac{4ab}{a+3b},⋌\right),g^{*}\left(\frac{4ab}{a+3b},⋌\right)\right]{\supseteq }_I\frac{\begin{array}{c}\\ ab\end{array}}{b-a}\left[{\int }_a^{\frac{2ab}{a+b}}\frac{g_{*}\left(\varkappa ,⋌\right)}{{\varkappa }^2}d\varkappa ,{\int }_a^{\frac{2ab}{a+b}}\frac{g^{*}\left(\varkappa ,⋌\right)}{{\varkappa }^2}d\varkappa \right].$$

It follows that

$$\frac{1}{2}\odot \tilde{g}\left(\frac{4ab}{a+3b}\right){\supseteq }_{\mathbb{F}}\frac{\begin{array}{c}\\ ab\end{array}}{b-a}\odot {\int }_a^{\frac{2ab}{a+b}}\frac{\tilde{g}\left(\varkappa \right)}{{\varkappa }^2}d\varkappa .$$

(34)

In a similar way as above, we have

$$\frac{1}{2}\odot \tilde{g}\left(\frac{4ab}{3a+b}\right){\supseteq }_{\mathbb{F}}\frac{\begin{array}{c}\\ ab\end{array}}{b-a}\odot {\int }_{\frac{2ab}{a+b}}^b\frac{\tilde{g}\left(\varkappa \right)}{{\varkappa }^2}d\varkappa .$$

(35)

Combining (34) and (35), we have

$$\frac{1}{2}\odot \left[\tilde{g}\left(\frac{4ab}{a+3b}\right)\oplus \tilde{g}\left(\frac{4ab}{3a+b}\right)\right]{\supseteq }_{\mathbb{F}}\frac{\begin{array}{c}\\ ab\end{array}}{b-a}\odot {\int }_a^b\frac{\tilde{g}\left(\varkappa \right)}{{\varkappa }^2}d\varkappa .$$

(36)

Therefore, for every $⋌\in [0,1]$, by using Theorem 7, we have

$$\begin{array}{c}{g}_{*}\left(\frac{2ab}{a+b},⋌\right)\le \frac{1}{2}\left[g_{*}\left(\frac{4ab}{a+3b},⋌\right)+g_{*}\left(\frac{4ab}{3a+b},⋌\right)\right]={{\mathcal{V}}_2}_{*}\le \frac{\begin{array}{c}\\ ab\end{array}}{b-a}{\int }_a^{\begin{array}{c}\\ b\end{array}}\frac{g_{*}\left(\varkappa ,⋌\right)}{{\varkappa }^2}d\varkappa \le \frac{1}{2}\left[\frac{g_{*}\left(a,⋌\right)+g_{*}\left(b,⋌\right)}{2}+g_{*}\left(\frac{2ab}{a+b},⋌\right)\right],\\ g^{*}\left(\frac{2ab}{a+b},⋌\right)\ge \frac{1}{2}\left[g^{*}\left(\frac{4ab}{a+3b},⋌\right)+g^{*}\left(\frac{4ab}{3a+b},⋌\right)\right]={{\mathcal{V}}_2}^{*}\ge \frac{\begin{array}{c}\\ ab\end{array}}{b-a}{\int }_a^b\frac{g^{*}\left(\varkappa ,⋌\right)}{{\varkappa }^2}d\varkappa \ge \frac{1}{2}\left[\frac{g^{*}\left(a,⋌\right)+g^{*}\left(b,⋌\right)}{2}+g^{*}\left(\frac{2ab}{a+b},⋌\right)\right],\end{array}$$

$$\begin{array}{c}={{\mathcal{V}}_1}_{*}\le \frac{1}{2}\left[\frac{g_{*}\left(a,⋌\right)+g_{*}\left(b,⋌\right)}{2}+\frac{1}{2}\left(g_{*}\left(a,⋌\right)+g_{*}\left(b,⋌\right)\right)\right]=\frac{1}{2}\left[g_{*}\left(a,⋌\right)+g_{*}\left(b,⋌\right)\right],\\ ={{\mathcal{V}}_1}^{*}\ge \frac{1}{2}\left[\frac{g^{*}\left(a,⋌\right)+g^{*}\left(b,⋌\right)}{2}+\frac{1}{2}\left(g^{*}\left(a,⋌\right)+g^{*}\left(b,⋌\right)\right)\right]=\frac{1}{2}\left[g^{*}\left(a,⋌\right)+g^{*}\left(b,⋌\right)\right],\end{array}$$

that is

$$\tilde{g}\left(\frac{2ab}{a+b}\right){\supseteq }_{\mathbb{F}}{\mathcal{V}}_2{\supseteq }_{\mathbb{F}}\frac{\begin{array}{c}\\ ab\end{array}}{b-a}\odot \left(FA\right){\int }_a^b\frac{\tilde{g}\left(\varkappa \right)}{{\varkappa }^2}d\varkappa {\supseteq }_{\mathbb{F}}{\mathcal{V}}_1{\supseteq }_{\mathbb{F}}\frac{1}{2}\odot \left[\tilde{g}\left(a\right)\oplus \tilde{g}\left(b\right)\right].$$

□

Theorem 9.

Let $\tilde{g}\in UDHFSX\left(\left[a,b\right],{\mathbb{F}}_0\right)$ and $\tilde{\mathfrak{T}}\in UDHFSX\left(\left[a,b\right],{\mathbb{F}}_0\right)$ , whose $⋌$-levels $g_{⋌},{\mathfrak{T}}_{⋌}:\left[a,b\right]\subset \mathbb{R}\to {\mathbb{R}}_I^{+}$ are defined by $g_{⋌}\left(\varkappa \right)=\left[g_{*}\left(\varkappa ,⋌\right),g^{*}\left(\varkappa ,⋌\right)\right]$ and ${\mathfrak{T}}_{⋌}\left(\varkappa \right)=\left[{\mathfrak{T}}_{*}\left(\varkappa ,⋌\right),{\mathfrak{T}}^{*}\left(\varkappa ,⋌\right)\right]$ for all $\varkappa \in \left[a,b\right]$ , $⋌\in \left[0,1\right]$ , respectively. If $\tilde{g}\otimes \tilde{\mathfrak{T}}\in {\mathcal{F}\mathcal{A}}_{\left(\left[a,b\right],⋌\right)}$ , then

$$\frac{\begin{array}{c}\\ ab\end{array}}{b-a}\odot \left(FA\right){\int }_a^b\frac{\tilde{g}\left(\varkappa \right)\otimes \tilde{\mathfrak{T}}\left(\varkappa \right)}{{\varkappa }^2}d\varkappa {\supseteq }_{\mathbb{F}}\frac{\tilde{P}\left(a,b\right)}{3}\oplus \frac{\tilde{Q}\left(a,b\right)}{6},$$

(37)

where $\tilde{P}\left(a,b\right)=\tilde{g}\left(a\right)\otimes \tilde{\mathfrak{T}}\left(a\right)\oplus \tilde{g}\left(b\right)\otimes \tilde{\mathfrak{T}}\left(b\right),$$\tilde{Q}\left(a,b\right)=\tilde{g}\left(a\right)\otimes \tilde{\mathfrak{T}}\left(b\right)\oplus \tilde{g}\left(b\right)\otimes \tilde{\mathfrak{T}}\left(a\right),$ and $P_{⋌}\left(a,b\right)=\left[P_{*}\left(\left(a,b\right),⋌\right),P^{*}\left(\left(a,b\right),⋌\right)\right]$ and $Q_{⋌}\left(a,b\right)=\left[Q_{*}\left(\left(a,b\right),⋌\right),Q^{*}\left(\left(a,b\right),⋌\right)\right].$

Proof. 

Since $\tilde{g},\tilde{\mathfrak{T}}$ are $\mathcal{U}-\mathfrak{O}-\mathcal{H}$ convex $F-N-V-M$s, then for each $⋌\in \left[0,1\right]$ we have

$$\begin{array}{c}\\ g_{*}\left(\frac{ab}{\left(1-ϛ\right)a+ϛb},⋌\right)\le ϛg_{*}\left(a,⋌\right)+\left(1-ϛ\right)g_{*}\left(b,⋌\right),\\ g^{*}\left(\frac{ab}{\left(1-ϛ\right)a+ϛb},⋌\right)\ge ϛg^{*}\left(a,⋌\right)+\left(1-ϛ\right)g^{*}\left(b,⋌\right).\end{array}$$

and

$$\begin{array}{c}\\ {\mathfrak{T}}_{*}\left(\frac{ab}{\left(1-ϛ\right)a+ϛb},⋌\right)\le ϛ{\mathfrak{T}}_{*}\left(a,⋌\right)+\left(1-ϛ\right){\mathfrak{T}}_{*}\left(b,⋌\right),\\ {\mathfrak{T}}^{*}\left(\frac{ab}{\left(1-ϛ\right)a+ϛb},⋌\right)\ge ϛ{\mathfrak{T}}^{*}\left(a,⋌\right)+\left(1-ϛ\right){\mathfrak{T}}^{*}\left(b,⋌\right).\end{array}$$

From the definition of $\mathcal{U}-\mathfrak{O}-\mathcal{H}$ convexity of $F-N-V-M$s, it follows that $\tilde{g}\left(\varkappa \right){\ge }_{\mathbb{F}}\tilde{0}$ and $\tilde{\mathfrak{T}}\left(\varkappa \right){\ge }_{\mathbb{F}}\tilde{0}$, so

$$\begin{array}{c}\\ g_{*}\left(\frac{ab}{\left(1-ϛ\right)a+ϛb},⋌\right)\times {\mathfrak{T}}_{*}\left(\frac{ab}{\left(1-ϛ\right)a+ϛb},⋌\right)\le \left(ϛg_{*}\left(a,⋌\right)+\left(1-ϛ\right)g_{*}\left(b,⋌\right)\right)\left(ϛ{\mathfrak{T}}_{*}\left(a,⋌\right)+\left(1-ϛ\right){\mathfrak{T}}_{*}\left(b,⋌\right)\right)\\ =g_{*}\left(a,⋌\right){\times \mathfrak{T}}_{*}\left(a,⋌\right)\left[\left(ϛ\right)\left(ϛ\right)\right]+g_{*}\left(b,⋌\right)\times {\mathfrak{T}}_{*}\left(b,⋌\right)\left[\left(1-ϛ\right)\left(1-ϛ\right)\right]+g_{*}\left(a,⋌\right){\mathfrak{T}}_{*}\left(b,⋌\right)ϛ\left(1-ϛ\right)+g_{*}\left(b,⋌\right)\times {\mathfrak{T}}_{*}\left(a,⋌\right)ϛ\left(1-ϛ\right),\\ g^{*}\left(\frac{ab}{\left(1-ϛ\right)a+ϛb},⋌\right)\times {\mathfrak{T}}^{*}\left(\frac{ab}{\left(1-ϛ\right)a+ϛb},⋌\right)\ge \left(ϛg^{*}\left(a,⋌\right)+\left(1-ϛ\right)g^{*}\left(b,⋌\right)\right)\left(ϛ{\mathfrak{T}}^{*}\left(a,⋌\right)+\left(1-ϛ\right){\mathfrak{T}}^{*}\left(b,⋌\right)\right)\\ =g^{*}\left(a,⋌\right)\times {\mathfrak{T}}^{*}\left(a,⋌\right)\left[\left(ϛ\right)\left(ϛ\right)\right]+g^{*}\left(b,⋌\right)\times {\mathfrak{T}}^{*}\left(b,⋌\right)\left[\left(1-ϛ\right)\left(1-ϛ\right)\right]+g^{*}\left(a,⋌\right){\times \mathfrak{T}}^{*}\left(b,⋌\right)ϛ\left(1-ϛ\right)+g^{*}\left(b,⋌\right)\times {\mathfrak{T}}^{*}\left(a,⋌\right)ϛ\left(1-ϛ\right).\end{array}$$

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

$$\begin{array}{c}\\ {\int }_0^1{g}_{*}\left(\frac{ab}{\left(1-ϛ\right)a+ϛb},⋌\right)\times {\mathfrak{T}}_{*}\left(\frac{ab}{\left(1-ϛ\right)a+ϛb},⋌\right)=\frac{\begin{array}{c}\\ ab\end{array}}{b-a}{\int }_a^b\frac{g_{*}\left(\varkappa ,⋌\right)\times {\mathfrak{T}}_{*}\left(\varkappa ,⋌\right)}{{\varkappa }^2}d\varkappa \le \left(g_{*}\left(a,⋌\right)\times {\mathfrak{T}}_{*}\left(a,⋌\right)+g_{*}\left(b,⋌\right){\times \mathfrak{T}}_{*}\left(b,⋌\right)\right){\int }_0^1\left(ϛ\right)\left(ϛ\right)dϛ\\ +\left(g_{*}\left(a,⋌\right)\times {\mathfrak{T}}_{*}\left(b,⋌\right)+g_{*}\left(b,⋌\right)\times {\mathfrak{T}}_{*}\left(a,⋌\right)\right){\int }_0^1ϛ\left(1-ϛ\right)dϛ,\\ {\int }_0^1{g}^{*}\left(\frac{ab}{\left(1-ϛ\right)a+ϛb},⋌\right)\times {\mathfrak{T}}^{*}\left(\frac{ab}{\left(1-ϛ\right)a+ϛb},⋌\right)=\frac{\begin{array}{c}\\ ab\end{array}}{b-a}{\int }_a^b\frac{g^{*}\left(\varkappa ,⋌\right)\times {\mathfrak{T}}^{*}\left(\varkappa ,⋌\right)}{{\varkappa }^2}d\varkappa \ge \left(g^{*}\left(a,⋌\right)\times {\mathfrak{T}}^{*}\left(a,⋌\right)+g^{*}\left(b,⋌\right)\times {\mathfrak{T}}^{*}\left(b,⋌\right)\right){\int }_0^1\left(ϛ\right)\left(ϛ\right)dϛ\\ +\left(g^{*}\left(a,⋌\right)\times {\mathfrak{T}}^{*}\left(b,⋌\right)+g^{*}\left(b,⋌\right)\times {\mathfrak{T}}^{*}\left(a,⋌\right)\right){\int }_0^1ϛ\left(1-ϛ\right)dϛ.\end{array}$$

It follows that,

$$\begin{array}{c}\\ \frac{\begin{array}{c}\\ ab\end{array}}{b-a}{\int }_a^b{g}_{*}\left(\varkappa ,⋌\right)\times {\mathfrak{T}}_{*}\left(\varkappa ,⋌\right)d\varkappa \le \frac{P_{*}\left(\left(a,b\right),⋌\right)}{3}+\frac{Q_{*}\left(\left(a,b\right),⋌\right)}{6}\\ \frac{\begin{array}{c}\\ ab\end{array}}{b-a}{\int }_a^b{g}^{*}\left(\varkappa ,⋌\right)\times {\mathfrak{T}}^{*}\left(\varkappa ,⋌\right)d\varkappa \ge \frac{P^{*}\left(\left(a,b\right),⋌\right)}{3}+\frac{Q^{*}\left(\left(a,b\right),⋌\right)}{6},\end{array}$$

that is

$$\frac{\begin{array}{c}\\ ab\end{array}}{b-a}\left[{\int }_a^b{g}_{*}\left(\varkappa ,⋌\right)\times {\mathfrak{T}}_{*}\left(\varkappa ,⋌\right)d\varkappa ,{\int }_a^b{g}^{*}\left(\varkappa ,⋌\right)\times {\mathfrak{T}}^{*}\left(\varkappa ,⋌\right)d\varkappa \right]{\supseteq }_I\frac{1}{3}\left[P_{*}\left(\left(a,b\right),⋌\right),P^{*}\left(\left(a,b\right),⋌\right)\right]+\frac{1}{6}\left[Q_{*}\left(\left(a,b\right),⋌\right),Q^{*}\left(\left(a,b\right),⋌\right)\right].$$

Thus,

$$\frac{\begin{array}{c}\\ ab\end{array}}{b-a}\odot \left(FA\right){\int }_a^b\frac{\tilde{g}\left(\varkappa \right)\otimes \tilde{\mathfrak{T}}\left(\varkappa \right)}{{\varkappa }^2}d\varkappa {\supseteq }_{\mathbb{F}}\frac{\tilde{P}\left(a,b\right)}{3}\oplus \frac{\tilde{Q}\left(a,b\right)}{6}.$$

□

Theorem 10.

Let $\tilde{g},\tilde{\mathfrak{T}}\in UDHFSX\left(\left[a,b\right],{\mathbb{F}}_0\right)$ , whose $⋌$-levels $g_{⋌},{\mathfrak{T}}_{⋌}:\left[a,b\right]\subset \mathbb{R}\to {\mathbb{R}}_I^{+}$ are defined by $g_{⋌}\left(\varkappa \right)=\left[g_{*}\left(\varkappa ,⋌\right),g^{*}\left(\varkappa ,⋌\right)\right]$ and ${\mathfrak{T}}_{⋌}\left(\varkappa \right)=\left[{\mathfrak{T}}_{*}\left(\varkappa ,⋌\right),{\mathfrak{T}}^{*}\left(\varkappa ,⋌\right)\right]$ for all $\varkappa \in \left[a,b\right]$ , $⋌\in \left[0,1\right]$ , respectively. If $\tilde{g}\otimes \tilde{\mathfrak{T}}\in {\mathcal{F}\mathcal{A}}_{\left(\left[a,b\right],⋌\right)}$ , then

$$2\odot \tilde{g}\left(\frac{2ab}{a+b}\right)\otimes \tilde{\mathfrak{T}}\left(\frac{2ab}{a+b}\right){\supseteq }_{\mathbb{F}}\frac{\begin{array}{c}\\ ab\end{array}}{b-a}\odot \left(FA\right){\int }_a^b\frac{\tilde{g}\left(\varkappa \right)\otimes \tilde{\mathfrak{T}}\left(\varkappa \right)}{{\varkappa }^2}d\varkappa \oplus \frac{\tilde{P}\left(a,b\right)}{6}\oplus \frac{\tilde{Q}\left(a,b\right)}{3},$$

(38)

where $\tilde{P}\left(a,b\right)=\tilde{g}\left(a\right)\otimes \tilde{\mathfrak{T}}\left(a\right)\oplus \tilde{g}\left(b\right)\otimes \tilde{\mathfrak{T}}\left(b\right),$$\tilde{Q}\left(a,b\right)=\tilde{g}\left(a\right)\otimes \tilde{\mathfrak{T}}\left(b\right)\oplus \tilde{g}\left(b\right)\otimes \tilde{\mathfrak{T}}\left(a\right),$ and $P_{⋌}\left(a,b\right)=\left[P_{*}\left(\left(a,b\right),⋌\right),P^{*}\left(\left(a,b\right),⋌\right)\right]$ and $Q_{⋌}\left(a,b\right)=\left[Q_{*}\left(\left(a,b\right),⋌\right),Q^{*}\left(\left(a,b\right),⋌\right)\right].$

Proof. 

By hypothesis, for each $⋌\in \left[0,1\right],$ we have

$$\begin{array}{c}{g}_{*}\left(\frac{2ab}{a+b},⋌\right)\times {\mathfrak{T}}_{*}\left(\frac{2ab}{a+b},⋌\right)\le \frac{1}{4}\left[\begin{array}{c}{g}_{*}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b},⋌\right)\times {\mathfrak{T}}_{*}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b},⋌\right)+g_{*}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b},⋌\right)\times {\mathfrak{T}}_{*}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b},⋌\right)\end{array}\right]\\ +\frac{1}{4}\left[\begin{array}{c}{g}_{*}\left(\frac{ab}{\left(1-ϛ\right)a+ϛb},⋌\right)\times {\mathfrak{T}}_{*}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b},⋌\right)+g_{*}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b},⋌\right)\times {\mathfrak{T}}_{*}\left(\frac{ab}{\left(1-ϛ\right)a+ϛb},⋌\right)\end{array}\right],\\ g^{*}\left(\frac{2ab}{a+b},⋌\right)\times {\mathfrak{T}}^{*}\left(\frac{2ab}{a+b},⋌\right)\ge \frac{1}{4}\left[\begin{array}{c}{g}^{*}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b},⋌\right)\times {\mathfrak{T}}^{*}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b},⋌\right)+g^{*}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b},⋌\right)\times {\mathfrak{T}}^{*}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b},⋌\right)\end{array}\right]\\ +\frac{1}{4}\left[\begin{array}{c}{g}^{*}\left(\frac{ab}{\left(1-ϛ\right)a+ϛb},⋌\right)\times {\mathfrak{T}}^{*}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b},⋌\right)+g^{*}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b},⋌\right)\times {\mathfrak{T}}^{*}\left(\frac{ab}{\left(1-ϛ\right)a+ϛb},⋌\right)\end{array}\right],\end{array}$$

$$\begin{array}{c}\le \frac{1}{4}\left[\begin{array}{c}{g}_{*}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b},⋌\right){\times \mathfrak{T}}_{*}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b},⋌\right)+g_{*}\left(\frac{ab}{\left(1-ϛ\right)a+ϛb},⋌\right)\times {\mathfrak{T}}_{*}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b},⋌\right)\end{array}\right]\\ +\frac{1}{4}\left[\begin{array}{c}\left(ϛg_{*}\left(a,⋌\right)+\left(1-ϛ\right)g_{*}\left(b,⋌\right)\right)\times \left(\left(1-ϛ\right){\mathfrak{T}}_{*}\left(a,⋌\right)+ϛ{\mathfrak{T}}_{*}\left(b,⋌\right)\right)\\ +\left({\left(1-ϛ\right)g}_{*}\left(a,⋌\right)+ϛg_{*}\left(b,⋌\right)\right)\times \left(ϛ{\mathfrak{T}}_{*}\left(a,⋌\right)+\left(1-ϛ\right){\mathfrak{T}}_{*}\left(b,⋌\right)\right)\end{array}\right],\\ \ge \frac{1}{4}\left[\begin{array}{c}{g}^{*}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b},⋌\right)\times {\mathfrak{T}}^{*}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b},⋌\right)+g^{*}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b}⋌\right)\times {\mathfrak{T}}^{*}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b},⋌\right)\end{array}\right]\\ +\frac{1}{4}\left[\begin{array}{c}\left(ϛg^{*}\left(a,⋌\right)+\left(1-ϛ\right)g^{*}\left(b,⋌\right)\right)\times \left(\left(1-ϛ\right){\mathfrak{T}}^{*}\left(a,⋌\right)+ϛ{\mathfrak{T}}^{*}\left(b,⋌\right)\right)\\ +\left(\left(1-ϛ\right)g^{*}\left(a,⋌\right)+ϛg^{*}\left(b,⋌\right)\right)\times \left(ϛ{\mathfrak{T}}^{*}\left(a,⋌\right)+\left(1-ϛ\right){\mathfrak{T}}^{*}\left(b,⋌\right)\right)\end{array}\right],\end{array}$$

$$\begin{array}{c}=\frac{1}{4}\left[\begin{array}{c}{g}_{*}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b},⋌\right)\times {\mathfrak{T}}_{*}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b},⋌\right)+g_{*}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b},⋌\right){\times \mathfrak{T}}_{*}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b},⋌\right)\end{array}\right]\\ +\frac{1}{4}\left[\begin{array}{c}\left\{\left(ϛ\right)\left(ϛ\right)+\left(1-ϛ\right)\left(1-ϛ\right)\right\}{Q}_{*}\left(\left(a,b\right),⋌\right)+\left\{ϛ\left(1-ϛ\right)+ϛ\left(1-ϛ\right)\right\}{P}_{*}\left(\left(a,b\right),⋌\right)\end{array}\right],\\ =\frac{1}{4}\left[\begin{array}{c}{g}^{*}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b},⋌\right)\times {\mathfrak{T}}^{*}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b},⋌\right)+g^{*}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b},⋌\right)\times {\mathfrak{T}}^{*}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b},⋌\right)\end{array}\right]\\ +\frac{1}{4}\left[\begin{array}{c}\left\{\left(ϛ\right)\left(ϛ\right)+\left(1-ϛ\right)\left(1-ϛ\right)\right\}{Q}^{*}\left(\left(a,b\right),⋌\right)+\left\{ϛ\left(1-ϛ\right)+ϛ\left(1-ϛ\right)\right\}{P}^{*}\left(\left(a,b\right),⋌\right)\end{array}\right].\end{array}$$

Integrating over $\left[0,1\right],$ we have

$$\begin{array}{c}\\ 2g_{*}\left(\frac{2ab}{a+b},⋌\right)\times {\mathfrak{T}}_{*}\left(\frac{2ab}{a+b},⋌\right)\le \frac{1}{b-a}{\int }_a^b{g}_{*}\left(\varkappa ,⋌\right)\times {\mathfrak{T}}_{*}\left(\varkappa ,⋌\right)d\varkappa +P_{*}\left(\left(a,b\right),⋌\right){\int }_0^1ϛ\left(1-ϛ\right)dϛ+Q_{*}\left(\left(a,b\right),⋌\right){\int }_0^1\left(ϛ\right)\left(ϛ\right)dϛ,\\ 2g^{*}\left(\frac{2ab}{a+b},⋌\right)\times {\mathfrak{T}}^{*}\left(\frac{2ab}{a+b},⋌\right)\ge \frac{1}{b-a}{\int }_a^b{g}^{*}\left(\varkappa ,⋌\right){\times \mathfrak{T}}^{*}\left(\varkappa ,⋌\right)d\varkappa +P^{*}\left(\left(a,b\right),⋌\right){\int }_0^1ϛ\left(1-ϛ\right)dϛ+Q^{*}\left(\left(a,b\right),⋌\right){\int }_0^1\left(ϛ\right)\left(ϛ\right)dϛ,\end{array}$$

that is

$$2\odot \tilde{g}\left(\frac{2ab}{a+b}\right)\otimes \tilde{\mathfrak{T}}\left(\frac{2ab}{a+b}\right){\supseteq }_{\mathbb{F}}\frac{\begin{array}{c}\\ ab\end{array}}{b-a}\odot \left(FA\right){\int }_a^b\frac{\tilde{g}\left(\varkappa \right)\otimes \tilde{\mathfrak{T}}\left(\varkappa \right)}{{\varkappa }^2}d\varkappa \oplus \frac{\tilde{P}\left(a,b\right)}{6}\oplus \frac{\tilde{Q}\left(a,b\right)}{3}.$$

Hence, the theorem has been proved. □

3.1. Second Fuzzy Hermite–Hadamard Fejér-Type Inequality

Theorem 11.

Let$\tilde{g}\in UDHFSX\left(\left[a,b\right],{\mathbb{F}}_0\right)$ , whose $⋌$-levels define the family of $I-V-M$s  $g_{⋌}:\left[a,b\right]\subset \mathbb{R}\to {\mathbb{R}}_I^{+}$ are given by $g_{⋌}\left(\varkappa \right)=\left[g_{*}\left(\varkappa ,⋌\right),g^{*}\left(\varkappa ,⋌\right)\right]$ for all $\varkappa \in \left[a,b\right]$ ,$⋌\in \left[0,1\right]$ . If $\tilde{g}\in {\mathcal{F}\mathcal{A}}_{\left(\left[a,b\right],⋌\right)}$ and $M:\left[a,b\right]\to \mathbb{R},M\left(\frac{1}{\frac{1}{a}+\frac{1}{b}-\frac{1}{\varkappa }}\right)=M\left(\varkappa \right)\ge 0,$ then

$$\left(FA\right){\int }_a^b\frac{\tilde{g}\left(\varkappa \right)}{{\varkappa }^2}M\left(\varkappa \right)d\varkappa {\supseteq }_{\mathbb{F}}\frac{\tilde{g}\left(a\right)\oplus \tilde{g}\left(b\right)}{2}\odot {\int }_0^1\frac{M\left(\varkappa \right)}{{\varkappa }^2}d\varkappa .$$

(39)

If  $\tilde{g}\in UDHFSV\left(\left[a,b\right],{\mathbb{F}}_0\right)$ , then inequality (39) is reversed.

Proof. 

Let $\tilde{g}$ be a $\mathcal{U}-\mathfrak{O}-\mathcal{H}$ convex $F-N-V-M$. Then, for each $⋌\in \left[0,1\right],$ we have

$$\begin{array}{c}\\ g_{*}\left(\frac{ab}{\left(1-ϛ\right)a+ϛb},⋌\right)M\left(\frac{ab}{\left(1-ϛ\right)a+ϛb}\right)\le \left(ϛg_{*}\left(a,⋌\right)+\left(1-ϛ\right)g_{*}\left(b,⋌\right)\right)M\left(\frac{ab}{\left(1-ϛ\right)a+ϛb}\right),\\ g^{*}\left(\frac{ab}{\left(1-ϛ\right)a+ϛb},⋌\right)M\left(\frac{ab}{\left(1-ϛ\right)a+ϛb}\right)\ge \left(ϛg^{*}\left(a,⋌\right)+\left(1-ϛ\right)g^{*}\left(b,⋌\right)\right)M\left(\frac{ab}{\left(1-ϛ\right)a+ϛb}\right),\end{array}$$

(40)

and

$$\begin{array}{c}\\ g_{*}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b},⋌\right)M\left(\frac{ab}{ϛa+\left(1-ϛ\right)b}\right)\le \left(\left(1-ϛ\right)g_{*}\left(a,⋌\right)+ϛg_{*}\left(b,⋌\right)\right)M\left(\frac{ab}{ϛa+\left(1-ϛ\right)b}\right),\\ g^{*}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b},⋌\right)M\left(\frac{ab}{ϛa+\left(1-ϛ\right)b}\right)\ge \left(\left(1-ϛ\right)g^{*}\left(a,⋌\right)+ϛg^{*}\left(b,⋌\right)\right)M\left(\frac{ab}{ϛa+\left(1-ϛ\right)b}\right).\end{array}$$

(41)

After adding (40) and (41), and integrating over $\left[0,1\right],$ we obtain

$$\begin{array}{c}\\ {\int }_0^1{g}_{*}\left(\frac{ab}{\left(1-ϛ\right)a+ϛb},⋌\right)M\left(\frac{ab}{\left(1-ϛ\right)a+ϛb}\right)dϛ+{\int }_0^1{g}_{*}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b},⋌\right)M\left(\frac{ab}{ϛa+\left(1-ϛ\right)b}\right)dϛ\\ \le {\int }_0^1\left[\begin{array}{c}{g}_{*}\left(a,⋌\right)\left\{\begin{array}{c}ϛM\left(\frac{ab}{\left(1-ϛ\right)a+ϛb}\right)+\left(1-ϛ\right)M\left(\frac{ab}{ϛa+\left(1-ϛ\right)b}\right)\end{array}\right\}+g_{*}\left(b,⋌\right)\left\{\begin{array}{c}\left(1-ϛ\right)M\left(\frac{ab}{\left(1-ϛ\right)a+ϛb}\right)+ϛM\left(\frac{ab}{ϛa+\left(1-ϛ\right)b}\right)\end{array}\right\}\end{array}\right]dϛ\\ =2g_{*}\left(a,⋌\right){\int }_0^1\begin{array}{c}ϛM\left(\frac{ab}{\left(1-ϛ\right)a+ϛb}\right)\end{array}dϛ+2g_{*}\left(b,⋌\right){\int }_0^1\begin{array}{c}ϛM\left(\frac{ab}{ϛa+\left(1-ϛ\right)b}\right)\end{array}dϛ,\\ {\int }_0^1{g}^{*}\left(\frac{ab}{\left(1-ϛ\right)a+ϛb},⋌\right)M\left(\frac{ab}{\left(1-ϛ\right)a+ϛb}\right)dϛ+{\int }_0^1{g}^{*}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b},⋌\right)M\left(\frac{ab}{ϛa+\left(1-ϛ\right)b}\right)dϛ\\ \ge {\int }_0^1\left[\begin{array}{c}{g}^{*}\left(a,⋌\right)\left\{\begin{array}{c}ϛM\left(\frac{ab}{\left(1-ϛ\right)a+ϛb}\right)+\left(1-ϛ\right)M\left(\frac{ab}{ϛa+\left(1-ϛ\right)b}\right)\end{array}\right\}+g^{*}\left(b,⋌\right)\left\{\begin{array}{c}\left(1-ϛ\right)M\left(\frac{ab}{\left(1-ϛ\right)a+ϛb}\right)+ϛM\left(\frac{ab}{ϛa+\left(1-ϛ\right)b}\right)\end{array}\right\}\end{array}\right]dϛ\\ =2g^{*}\left(a,⋌\right){\int }_0^1\begin{array}{c}ϛM\left(\frac{ab}{\left(1-ϛ\right)a+ϛb}\right)\end{array}dϛ+2g^{*}\left(b,⋌\right){\int }_0^1\begin{array}{c}ϛM\left(\frac{ab}{ϛa+\left(1-ϛ\right)b}\right)\end{array}dϛ.\end{array}$$

Since $M$ is symmetric, then

$$\begin{array}{c}\\ {\int }_0^1{g}_{*}\left(\frac{ab}{\left(1-ϛ\right)a+ϛb},⋌\right)M\left(\frac{ab}{\left(1-ϛ\right)a+ϛb}\right)dϛ+{\int }_0^1{g}_{*}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b},⋌\right)M\left(\frac{ab}{ϛa+\left(1-ϛ\right)b}\right)dϛ\le 2\left[g_{*}\left(a,⋌\right)+g_{*}\left(b,⋌\right)\right]{\int }_0^1\begin{array}{c}ϛM\left(\frac{ab}{ϛa+\left(1-ϛ\right)b}\right)\end{array}dϛ,\\ {\int }_0^1{g}^{*}\left(\frac{ab}{\left(1-ϛ\right)a+ϛb},⋌\right)M\left(\frac{ab}{\left(1-ϛ\right)a+ϛb}\right)dϛ+{\int }_0^1{g}^{*}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b},⋌\right)M\left(\frac{ab}{ϛa+\left(1-ϛ\right)b}\right)dϛ\ge 2\left[g^{*}\left(a,⋌\right)+g^{*}\left(b,⋌\right)\right]{\int }_0^1\begin{array}{c}ϛM\left(\frac{ab}{ϛa+\left(1-ϛ\right)b}\right)\end{array}dϛ.\end{array}$$

(42)

Since

$$\begin{array}{c}\\ {\int }_0^1{g}_{*}\left(ϛa+\left(1-ϛ\right)b,⋌\right)M\left(\frac{ab}{\left(1-ϛ\right)a+ϛb}\right)dϛ={\int }_0^1{g}_{*}\left(\left(1-ϛ\right)a+ϛb,⋌\right)M\left(\frac{ab}{ϛa+\left(1-ϛ\right)b}\right)dϛ=\frac{ab}{b-a}{\int }_a^b{g}_{*}\left(\varkappa ,⋌\right)M\left(\varkappa \right)d\varkappa \\ \\ {\int }_0^1{g}^{*}\left(\left(1-ϛ\right)a+ϛb,⋌\right)M\left(\frac{ab}{ϛa+\left(1-ϛ\right)b}\right)dϛ={\int }_0^1{g}^{*}\left(ϛa+\left(1-ϛ\right)b,⋌\right)M\left(\frac{ab}{ϛa+\left(1-ϛ\right)b}\right)dϛ=\frac{ab}{b-a}{\int }_a^b{g}^{*}\left(\varkappa ,⋌\right)M\left(\varkappa \right)d\varkappa .\end{array}$$

(43)

From (42) and (43), we have

$$\begin{array}{c}\frac{ab}{b-a}{\int }_a^b{g}_{*}\left(\varkappa ,⋌\right)M\left(\varkappa \right)d\varkappa \le \left[g_{*}\left(a,⋌\right)+g_{*}\left(b,⋌\right)\right]{\int }_0^1\begin{array}{c}ϛM\left(\frac{ab}{ϛa+\left(1-ϛ\right)b}\right)\end{array}dϛ,\\ \frac{ab}{b-a}{\int }_a^b{g}^{*}\left(\varkappa ,⋌\right)M\left(\varkappa \right)d\varkappa \ge \left[g^{*}\left(a,⋌\right)+g^{*}\left(b,⋌\right)\right]{\int }_0^1\begin{array}{c}ϛM\left(\frac{ab}{ϛa+\left(1-ϛ\right)b}\right)\end{array}dϛ,\\ \end{array}$$

that is

$$\left[\frac{ab}{b-a}{\int }_a^b{g}_{*}\left(\varkappa ,⋌\right)M\left(\varkappa \right)d\varkappa ,\frac{ab}{b-a}{\int }_a^b{g}^{*}\left(\varkappa ,⋌\right)M\left(\varkappa \right)d\varkappa \right]{\supseteq }_I\left[g_{*}\left(a,⋌\right)+g_{*}\left(b,⋌\right),g^{*}\left(a,⋌\right)+g^{*}\left(b,⋌\right)\right]{\int }_0^1\begin{array}{c}ϛM\left(\frac{ab}{ϛa+\left(1-ϛ\right)b}\right)\end{array}dϛ,$$

hence

$$\begin{gathered} \frac{ab}{b-a}\odot \left(FA\right){\int }_a^b\frac{\tilde{g}\left(\varkappa \right)}{{\varkappa }^2}M\left(\varkappa \right)d\varkappa {\supseteq }_{\mathbb{F}}\left[\tilde{g}\left(a\right)\oplus \tilde{g}\left(b\right)\right] \\ \odot {\int }_0^1ϛM\left(\frac{ab}{ϛa+\left(1-ϛ\right)b}\right)dϛ, \end{gathered}$$

and this concludes the proof. □

3.2. First fuzzy Hermite–Hadamard Fejér-type inequality for $\mathcal{U}-\mathfrak{O}-\mathcal{H}$ convex $F-N-V-M$

Theorem 12.

Let $\tilde{g}\in UDHFSX\left(\left[a,b\right],{\mathbb{F}}_0\right)$ , whose $⋌$-levels define the family of $I-V-M$s  $g_{⋌}:\left[a,b\right]\subset \mathbb{R}\to {\mathbb{R}}_I^{+}$ are given by $g_{⋌}\left(\varkappa \right)=\left[g_{*}\left(\varkappa ,⋌\right),g^{*}\left(\varkappa ,⋌\right)\right]$ for all  $\varkappa \in \left[a,b\right]$,  $⋌\in \left[0,1\right]$. If  $\tilde{g}\in {\mathcal{F}\mathcal{A}}_{\left(\left[a,b\right],⋌\right)}$ and  $M:\left[a,b\right]\to \mathbb{R},M\left(\frac{1}{\frac{1}{a}+\frac{1}{b}-\frac{1}{\varkappa }}\right)=M\left(\varkappa \right)\ge 0,$ then

$$\tilde{g}\left(\frac{2ab}{a+b}\right)\odot {\int }_a^b\frac{\tilde{g}\left(\varkappa \right)}{{\varkappa }^2}d\varkappa {\supseteq }_{\mathbb{F}}\left(FA\right){\int }_a^b\frac{\tilde{g}\left(\varkappa \right)}{{\varkappa }^2}\odot M\left(\varkappa \right)d\varkappa .$$

(44)

If  $\tilde{g}\in UDHFSV\left(\left[a,b\right],{\mathbb{F}}_0\right)$, then inequality (44) is reversed.

Proof. 

Since $\tilde{g}$ is an $\mathcal{U}-\mathfrak{O}-\mathcal{H}$ convex, then for $⋌\in \left[0,1\right],$ we have

$$\begin{array}{c}\\ g_{*}\left(\frac{2ab}{a+b},⋌\right)\le \frac{1}{2}\left(\begin{array}{c}{g}_{*}\left(\frac{ab}{\left(1-ϛ\right)a+ϛb},⋌\right)+g_{*}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b},⋌\right)\end{array}\right)\\ \\ g^{*}\left(\frac{2ab}{a+b},⋌\right)\ge \frac{1}{2}\left(\begin{array}{c}{g}^{*}\left(\frac{ab}{\left(1-ϛ\right)a+ϛb},⋌\right)+g^{*}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b},⋌\right)\end{array}\right).\\ \end{array}$$

(45)

By multiplying (45) by $M\left(\frac{ab}{\left(1-ϛ\right)a+ϛb}\right)=M\left(\frac{ab}{ϛa+\left(1-ϛ\right)b}\right)$ and integrating it by $ϛ$ over $\left[0,1\right],$ we obtain

$$\begin{array}{c}\\ g_{*}\left(\frac{2ab}{a+b},⋌\right){\int }_0^{1}M\left(\frac{ab}{ϛa+\left(1-ϛ\right)b}\right)dϛ\le \frac{1}{2}\left(\begin{array}{c}{\int }_0^1{g}_{*}\left(\frac{ab}{\left(1-ϛ\right)a+ϛb},⋌\right)M\left(\frac{ab}{ϛa+\left(1-ϛ\right)b}\right)dϛ+{\int }_0^1{g}_{*}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b},⋌\right)M\left(\frac{ab}{ϛa+\left(1-ϛ\right)b}\right)dϛ\end{array}\right)\\ g^{*}\left(\frac{2ab}{a+b},⋌\right){\int }_0^{1}M\left(\frac{ab}{ϛa+\left(1-ϛ\right)b}\right)dϛ\ge \frac{1}{2}\left(\begin{array}{c}{\int }_0^1{g}^{*}\left(\frac{ab}{\left(1-ϛ\right)a+ϛb},⋌\right)M\left(\frac{ab}{ϛa+\left(1-ϛ\right)b}\right)dϛ+{\int }_0^1{g}^{*}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b},⋌\right)M\left(\frac{ab}{ϛa+\left(1-ϛ\right)b}\right)dϛ\end{array}\right).\\ \end{array}$$

(46)

Since

$$\begin{array}{c}\\ {\int }_0^1{g}_{*}\left(\frac{ab}{\left(1-ϛ\right)a+ϛb},⋌\right)M\left(\frac{ab}{\left(1-ϛ\right)a+ϛb}\right)dϛ={\int }_0^1{g}_{*}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b},⋌\right)M\left(\frac{ab}{ϛa+\left(1-ϛ\right)b}\right)dϛ=\frac{ab}{b-a}{\int }_a^b\frac{g_{*}\left(\varkappa ,⋌\right)}{{\varkappa }^2}M\left(\varkappa \right)d\varkappa ,\\ {\int }_0^1{g}^{*}\left(\frac{ab}{ϛa+\left(1-ϛ\right)b},⋌\right)M\left(\frac{ab}{ϛa+\left(1-ϛ\right)b}\right)dϛ={\int }_0^1{g}^{*}\left(\frac{ab}{\left(1-ϛ\right)a+ϛb},⋌\right)M\left(\frac{ab}{\left(1-ϛ\right)a+ϛb}\right)dϛ=\frac{ab}{b-a}{\int }_a^b\frac{g^{*}\left(\varkappa ,⋌\right)}{{\varkappa }^2}M\left(\varkappa \right)d\varkappa .\end{array}$$

(47)

From (47), (46) we have

$$\begin{array}{c}\\ g_{*}\left(\frac{2ab}{a+b},⋌\right)\le \frac{1}{{\int }_a^b\frac{M\left(\varkappa \right)}{{\varkappa }^2}d\varkappa }{\int }_a^b\frac{g_{*}\left(\varkappa ,⋌\right)}{{\varkappa }^2}M\left(\varkappa \right)d\varkappa ,\\ g^{*}\left(\frac{2ab}{a+b},⋌\right)\ge \frac{1}{{\int }_a^b\frac{M\left(\varkappa \right)}{{\varkappa }^2}d\varkappa }{\int }_a^b\frac{g^{*}\left(\varkappa ,⋌\right)}{{\varkappa }^2}M\left(\varkappa \right)d\varkappa .\\ \end{array}$$

From which, we have

$$\begin{array}{c}\\ \left[g_{*}\left(\frac{2ab}{a+b},⋌\right),g^{*}\left(\frac{2ab}{a+b},⋌\right)\right]{\supseteq }_I\frac{1}{{\int }_a^b\frac{M\left(\varkappa \right)}{{\varkappa }^2}d\varkappa }\left[{\int }_a^b\frac{g_{*}\left(\varkappa ,⋌\right)}{{\varkappa }^2}M\left(\varkappa \right)d\varkappa ,{\int }_a^b\frac{g^{*}\left(\varkappa ,⋌\right)}{{\varkappa }^2}M\left(\varkappa \right)d\varkappa \right],\\ \end{array}$$

that is

$$\tilde{g}\left(\frac{2ab}{a+b}\right)\odot {\int }_a^b\frac{M\left(\varkappa \right)}{{\varkappa }^2}d\varkappa {\supseteq }_{\mathbb{F}}\left(FA\right){\int }_a^b\frac{\tilde{g}\left(\varkappa \right)}{{\varkappa }^2}\odot M\left(\varkappa \right)d\varkappa ,$$

and the proof has been completed. □

Remark 4.

If  $M\left(\varkappa \right)=1$ , then from Theorems 11 and 12, we obtain inequality (17).

If  $g_{*}\left(\varkappa ,⋌\right)\ne g^{*}\left(\varkappa ,⋌\right)$ with  $⋌=1$ , then we obtain the result for definition of  $\mathcal{H}$-convex I∙V∙M from Theorems 11 and 12, see [23].

If  $g_{*}\left(\varkappa ,⋌\right)\ne g^{*}\left(\varkappa ,⋌\right)$ with  $⋌=1$ , then Theorems 11 and 12 reduce to classical first and second classical H·H Fejér inequality for classical  $\mathcal{H}$-convex mapping, see [21].

4. Conclusions and Future Plan

In this study, by using the definition of $\mathcal{U}-\mathfrak{O}$ fuzzy relation, we developed a novel definition of harmonically convex mappings. Additionally, utilizing the definition of $\mathcal{U}-\mathfrak{O}-\mathcal{H}$ fuzzy convexity (concavity), Hermite–Hadamard Fejér-type inequalities for $\mathcal{U}-\mathfrak{O}-\mathcal{H}$ convex (concave) $F-N-V-M$s were derived. A few nontrivial examples were then given to illustrate our auxiliary findings. The findings may also open up new avenues for mathematical sciences research and serve as an example for both inexperienced and seasoned academics working in the area of fuzzy fractional integral inequalities.