Generalized Harmonically Convex Fuzzy-Number-Valued Mappings and Fuzzy Riemann–Liouville Fractional Integral Inequalities

Abstract

We propose the concept of up and down harmonically convex mapping for fuzzy-number-valued mapping as our main goal in this work. With the help of up and down harmonically fuzzy-number convexity and the fuzzy fractional integral operator, we also show the results for the Hermite–Hadamard ($H–H$) inequality, the Fejér type inequality, and some other related versions of inequalities. Moreover, some examples are also presented to discuss the validity of the main results. The results from the new technique show how the suggested scheme is accurate, adaptable, efficient, and user-friendly.

1. Introduction

Fractional calculus [1,2,3,4,5,6,7,8,9,10] is crucial in practically every branch of mathematics and other fields of science. Indeed, it is abundantly obvious that fractional operators have appeared in fractional differential equations as well as in every branch of natural research [11,12,13,14,15,16]. It has been particularly useful in the research of liquid waves, sound transmission, gravitational attraction, and string vibrations. For the study of fractional operators, several significant definitions and ideas such as Riemann, Liouville, Caputo, Hadamard, Katugampola, and Atangana–Baleanu operators have been established. Fractional integral differential equations or inclusions have been developed to determine along with several well-known operators if boundary value problems have solutions [17,18,19,20,21].

Fractional integral inequalities are used in the current situation to successfully and methodically investigate a number of significant fractional derivative and integral operators [22,23,24,25]. Variants are recognized to have numerous significant applications in all directions of mathematics as well as numerous fields of natural science. Numerous types of variations, including those of names such as Jensen, Hermite–Hadamard, Hardy, Ostrowski, Minkowski, and others, are particularly noteworthy and have a remarkable impact on important areas of research. In the mathematical sciences, statistics theory, optimization theory, fixed point theory, and various other branches of science and technology, convexity has attracted increased interest, see [26,27]. A remarkable diversity of convexities has been created by altering convex sets and convex functions over time, including Hp and q-convexity [28,29], harmonic convexity [30], strong convexity [31,32], Schur convexity [33,34], quasi-convexity [35], generalized convexity [36], and others. The convexity hypothesis can be used to find several inequalities in the literature [37]. One of the well-known outstanding classical inequalities, the $H-H$ inequality [38] has received a lot of attention recently.

The H–H inequality for convex mapping $\mathfrak{G}:K\to ℝ$ on an interval $K=\left[\mathfrak{d}, 𝔃\right]$

$$\mathfrak{G}\left(\frac{\mathfrak{d}+𝔃}{2}\right)\le \frac{1}{𝔃-\mathfrak{d}}{{\displaystyle \int }}_{\mathfrak{d}}^{𝔃}\mathfrak{G}\left(\varsigma \right)d\varsigma \le \frac{\mathfrak{G}\left(\mathfrak{d}\right)+\mathfrak{G}\left(𝔃\right)}{2},$$

(1)

for all $\mathfrak{d}, 𝔃\in K.$

Both inequalities in (1) hold in the opposite direction if $\mathfrak{G}$ is concave. The $H-H$ inequality (1) has numerous generalizations, modifications, applications, revisions, and variants that can be found in the literature [39].

Fejér derived the weighted version of the $H-H$ inequality (1) that is as follows:

Let $\mathfrak{G}:K\to ℝ$ be a convex mapping on a convex set $K$ and $\mathfrak{d}, 𝔃\in K$ with $\mathfrak{d}\le 𝔃$. Then,

$$\mathfrak{G}\left(\frac{\mathfrak{d}+𝔃}{2}\right)\le \frac{1}{{{\displaystyle \int }}_{\mathfrak{d}}^{𝔃}\mathfrak{V}\left(\varsigma \right)d\varsigma } {{\displaystyle \int }}_{\mathfrak{d}}^{𝔃}\mathfrak{G}\left(\varsigma \right)\mathfrak{V}\left(\varsigma \right)d\varsigma \le \frac{\mathfrak{G}\left(\mathfrak{d}\right)+\mathfrak{G}\left(𝔃\right)}{2}{{\displaystyle \int }}_{\mathfrak{d}}^{𝔃}\mathfrak{V}\left(\varsigma )\right)d\varsigma .$$

(2)

If $\mathfrak{V}\left(\varsigma \right)=1$, then we obtain (1) from (2).

The modifications for double inequality (2) have received extensive research because of the divergence in convexity concepts. We outline a new scheme and a future strategy within the current framework to address the growing tendency of this study topic. By using the Katugampola fractional integral operator, the p-convex function portrays the concept of the interval-valued function in a dynamic manner while also establishing a number of generalizations. For more information related to interval and fuzzy calculus, see [40,41,42,43,44,45,46].

On the other hand, the theory of interval and fuzzy analysis has a long history that dates to Archimedes’ calculation of the circumference of a circle. Interval analysis spent a long time in obscurity due to the lack of applications in various fields. However, research in this field was not intense until the 1950s. Moore, known for pioneering work in the field of interval arithmetic, wrote the first renowned treatise on interval analysis in 1966 to calculate the error bounds of the numerical solutions of a finite state machine. After his exploration, other scholars focused on examining the literature and applications of interval analysis in automatic error analysis, computer graphics, neural network output optimization, robotics, computational physics, and many other well-known areas in science and technology. Since that time, a number of analysts have focused on and extensively researched interval analysis and interval-valued functions in both mathematics and its applications, see [47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76]. Similarly, most of the authors discussed the applications of real-valued functions, interval-valued functions, and fuzzy-number-valued functions in different fields of mathematics. For more information, see [77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105] and the references therein.

This article’s main goal is to introduce the idea of a harmonic convex mapping for fuzzy-number-valued mappings. By harmonic convexity, which correlates with the fuzzy fractional integral operator, we also show the results for the $H-H$ inequality, Fejér type inequality, and some other related versions. Finally, the effects of the technique used show the presentations for various current results. The results from this unique technique show that the proposed scheme is very precise, adaptable, practical, and easy to use.

2. Preliminaries

We recall a few definitions, which can be found in the literature and that will be relevant in the follow-up.

Let us consider that ${\mathbb{X}}_I$ is the space of all closed and bounded intervals of $ℝ$ (set of real numbers), and that $\mathcal{C}\in {\mathbb{X}}_I$ is given by:

$$\mathcal{C}=\left[{\mathcal{C}}_{*}, {\mathcal{C}}^{*}\right]=\left\{\varsigma \in ℝ| {\mathcal{C}}_{*}\le \varsigma \le {\mathcal{C}}^{*}\right\}, \left({\mathcal{C}}_{*}, {\mathcal{C}}^{*}\in ℝ\right).$$

(3)

If ${\mathcal{C}}_{*}={\mathcal{C}}^{*}$, then $\mathcal{C}$ is degenerate. In the follow-up, all of the intervals are considered non-degenerate. If ${\mathcal{C}}_{*}\ge 0$, then $\mathcal{C}$ is positive. We denote by ${\mathbb{X}}_I^{+}=\left\{\left[{\mathcal{C}}_{*}, {\mathcal{C}}^{*}\right]:\left[{\mathcal{C}}_{*}, {\mathcal{C}}^{*}\right]\in {\mathbb{X}}_I \mathrm{and} {\mathcal{C}}_{*}\ge 0\right\}$ the set of all positive intervals.

Let $\mu \in ℝ$ and $\mu \cdot \mathcal{C}$ be given by:

$$\mu \cdot \mathcal{C}=\{\begin{array}{c}\left[\mu {\mathcal{C}}_{*}, \mu {\mathcal{C}}^{*}\right] \mathrm{if} \mu >0,\\ \left\{0\right\} \mathrm{if} \mu =0,\\ \left[\mu {\mathcal{C}}^{*},\mu {\mathcal{C}}_{*}\right] \mathrm{if} \mu <0. \end{array}$$

(4)

We consider the Minkowski sum, $\mathcal{C}+\mathcal{O}$, product, $\mathcal{C}\times \mathcal{O}$, and difference, $\mathcal{O}-\mathcal{C}$, for $\mathcal{C},\mathcal{O}\in {\mathbb{X}}_I$, as:

$$\left[{\mathcal{O}}_{*}, {\mathcal{O}}^{*}\right]+\left[{\mathcal{C}}_{*}, {\mathcal{C}}^{*}\right]=\left[{\mathcal{O}}_{*}+{\mathcal{C}}_{*}, {\mathcal{O}}^{*}+{\mathcal{C}}^{*}\right],$$

(5)

$$\left[{\mathcal{O}}_{*}, {\mathcal{O}}^{*}\right]\times \left[{\mathcal{C}}_{*}, {\mathcal{C}}^{*}\right]=\left[\min \left\{{\mathcal{O}}_{*}{\mathcal{C}}_{*}, {\mathcal{O}}^{*}{\mathcal{C}}_{*}, {\mathcal{O}}_{*}{\mathcal{C}}^{*}, {\mathcal{O}}^{*}{\mathcal{C}}^{*}\right\}, \max \left\{{\mathcal{O}}_{*}{\mathcal{C}}_{*}, {\mathcal{O}}^{*}{\mathcal{C}}_{*}, {\mathcal{O}}_{*}{\mathcal{C}}^{*}, {\mathcal{O}}^{*}{\mathcal{C}}^{*}\right\}\right],$$

(6)

$$\left[{\mathcal{O}}_{*}, {\mathcal{O}}^{*}\right]-\left[{\mathcal{C}}_{*}, {\mathcal{C}}^{*}\right]=\left[{\mathcal{O}}_{*}-{\mathcal{C}}^{*}, {\mathcal{O}}^{*}-{\mathcal{C}}_{*}\right].$$

(7)

Remark 1.

(i) For given $\left[{\mathcal{O}}_{*}, {\mathcal{O}}^{*}\right], \left[{\mathcal{C}}_{*}, {\mathcal{C}}^{*}\right]\in {\mathbb{X}}_I,$ the relation $“{\supseteq }_I”$ defined on ${\mathbb{X}}_I$ by

$$\left[{\mathcal{C}}_{*}, {\mathcal{C}}^{*}\right]{\supseteq }_I\left[{\mathcal{O}}_{*}, {\mathcal{O}}^{*}\right] \mathrm{if} \mathrm{and} \mathrm{only} \mathrm{if} {\mathcal{C}}_{*}\le {\mathcal{O}}_{*}, {\mathcal{O}}^{*}\le {\mathcal{C}}^{*},$$

(8)

for all $\left[{\mathcal{O}}_{*}, {\mathcal{O}}^{*}\right],\left[{\mathcal{C}}_{*}, {\mathcal{C}}^{*}\right]\in {\mathbb{X}}_I,$ is a partial interval inclusion relation. Moreover, $\left[{\mathcal{C}}_{*}, {\mathcal{C}}^{*}\right]{\supseteq }_I\left[{\mathcal{O}}_{*}, {\mathcal{O}}^{*}\right]$ coincids with $\left[{\mathcal{C}}_{*}, {\mathcal{C}}^{*}\right]\supseteq \left[{\mathcal{O}}_{*}, {\mathcal{O}}^{*}\right]$ on ${\mathbb{X}}_I.$ The relation $“{\supseteq }_I”$ is of $U\mathcal{D}$-order [57].

(ii) For given $\left[{\mathcal{O}}_{*}, {\mathcal{O}}^{*}\right],\left[{\mathcal{C}}_{*}, {\mathcal{C}}^{*}\right]\in {\mathbb{X}}_I,$ the relation $“{\le }_I”$, defined on ${\mathbb{X}}_I$ by $\left[{\mathcal{O}}_{*}, {\mathcal{O}}^{*}\right]{\le }_I\left[{\mathcal{C}}_{*}, {\mathcal{C}}^{*}\right]$ if and only if ${\mathcal{O}}_{*}\le {\mathcal{C}}_{*},{\mathcal{O}}^{*}\le {\mathcal{C}}^{*}$ or ${\mathcal{O}}_{*}\le {\mathcal{C}}_{*},{\mathcal{O}}^{*}<{\mathcal{C}}^{*}$, is a partial interval order relation. Plus, we have that $\left[{\mathcal{O}}_{*}, {\mathcal{O}}^{*}\right]{\le }_I\left[{\mathcal{C}}_{*}, {\mathcal{C}}^{*}\right]$ coincides with $\left[{\mathcal{O}}_{*}, {\mathcal{O}}^{*}\right]\le \left[{\mathcal{C}}_{*}, {\mathcal{C}}^{*}\right]$ on ${\mathbb{X}}_I.$ The relation $“{\le }_I”$ is of left and right (LR) type [56,57].

Given the intervals $\left[{\mathcal{O}}_{*}, {\mathcal{O}}^{*}\right], \left[{\mathcal{C}}_{*}, {\mathcal{C}}^{*}\right]\in {\mathbb{X}}_I,$ their Hausdorff–Pompeiu distance is:

$$d_H\left(\left[{\mathcal{O}}_{*}, {\mathcal{O}}^{*}\right], \left[{\mathcal{C}}_{*}, {\mathcal{C}}^{*}\right]\right)=\max \left\{\left|{\mathcal{O}}_{*}-{\mathcal{C}}_{*}\right|, \left|{\mathcal{O}}^{*}-{\mathcal{C}}^{*}\right|\right\}.$$

(9)

We have that $\left({\mathbb{X}}_I,d_H\right)$ is a complete metric space [50,54,55].

Definition 1.

Ref. [49] A fuzzy subset $L$ of $ℝ$ is a mapping $\tilde{\mathcal{C}}:ℝ\to \left[0,1\right]$, denoted membership mapping of $L$. We adopt the symbol $₤$ to represent the set of all fuzzy subsets of $ℝ$.

Let us consider $\tilde{\mathcal{C}}\in ₤$. If the following properties hold, then $\tilde{\mathcal{C}}$ is a fuzzy number:

(1)

$\tilde{\mathcal{C}}$ is normal if there exists $\varsigma \in ℝ$ and $\tilde{\mathcal{C}}\left(\varsigma \right)=1;$

(2)

$\tilde{\mathcal{C}}$ is upper semi-continuous on $ℝ$ if for a $\varsigma \in ℝ$ there exist $\epsilon >0$ and $\delta >0$ yielding $\tilde{\mathcal{C}}\left(\varsigma \right)-\tilde{\mathcal{C}}\left(𝓉\right)<\epsilon$ for all $𝓉\in ℝ$ with $\left|\varsigma -𝓉\right|<\delta ;$

(3)

$\tilde{\mathcal{C}}$ is fuzzy convex, meaning that $\tilde{\mathcal{C}}\left(\left(1-\mu \right)\varsigma +\mu 𝓉\right)\ge \min (\tilde{\mathcal{C}}\left(\varsigma \right), \tilde{\mathcal{C}}\left(𝓉\right)),$ for all $\varsigma ,𝓉\in ℝ,$ and $𝓉\in \left[0, 1\right]$;

(4)

$\tilde{\mathcal{C}}$ is compactly supported, which means that $\mathrm{cl}\left\{\varsigma \in ℝ| \tilde{\mathcal{C}}\left(\varsigma \right)>0\right\}$ is compact.

The symbol ${₤}_I$ will be adopted to designate the set of all fuzzy numbers of $ℝ$.

Definition 2.

Refs. [49,50] For $\tilde{\mathcal{C}}\in {₤}_I$, the $\mathfrak{o}$-level, or $\mathfrak{o}$-cut, sets of $\tilde{\mathcal{C}}$ are ${\left[\tilde{\mathcal{C}}\right]}^{\mathfrak{o}}=\left\{\varsigma \in ℝ| \tilde{\mathcal{C}}\left(\varsigma \right)>\mathfrak{o}\right\}$ for all $\mathfrak{o}\in \left[0, 1\right]$, and ${\left[\tilde{\mathcal{C}}\right]}^0=\left\{\varsigma \in ℝ| \tilde{\mathcal{C}}\left(\varsigma \right)>0\right\}$.

Proposition 1.

Ref. [51] Let $\tilde{\mathcal{C}},\tilde{\mathcal{O}}\in {₤}_I$. The relation $“{\le }_{\mathbb{F}}”$, defined on ${₤}_I$ by

$$\tilde{\mathcal{C}}{\le }_{\mathbb{F}}\tilde{\mathcal{O}} \mathrm{when} \mathrm{and} \mathrm{only} \mathrm{when} {\left[\tilde{\mathcal{C}}\right]}^{\mathfrak{o}}{\le }_I{\left[\tilde{\mathcal{O}}\right]}^{\mathfrak{o}}, \mathrm{for} \mathrm{every} \mathfrak{o}\in \left[0, 1\right],$$

(10)

is a LR order relation.

Proposition 2.

Ref. [47] Let $\tilde{\mathcal{C}},\tilde{\mathcal{O}}\in {₤}_I$. The relation $“{\supseteq }_{\mathbb{F}}”$, defined on ${₤}_I$ by

$$\tilde{\mathcal{C}}{\supseteq }_{\mathbb{F}}\tilde{\mathcal{O}} \mathrm{when} \mathrm{and} \mathrm{only} \mathrm{when} {\left[\tilde{\mathcal{C}}\right]}^{\mathfrak{o}}{\supseteq }_I{\left[\tilde{\mathcal{O}}\right]}^{\mathfrak{o}}, \mathrm{for} \mathrm{every} \mathfrak{o}\in \left[0, 1\right],$$

(11)

is an $U\mathcal{D}$-order relation.

Proof: 

The proof relies on the $U\mathcal{D}$-relation ${\supseteq }_I$ on ${\mathbb{X}}_I$. If $\tilde{\mathcal{C}},\tilde{\mathcal{O}}\in {₤}_I$ and $\mathfrak{o}\in ℝ$, then, for every $\mathfrak{o}\in \left[0, 1\right]$,

$${\left[\tilde{\mathcal{C}}\oplus \tilde{\mathcal{O}}\right]}^{\mathfrak{o}}={\left[\tilde{\mathcal{C}}\right]}^{\mathfrak{o}}+{\left[\tilde{\mathcal{O}}\right]}^{\mathfrak{o}},$$

(12)

$${\left[\tilde{\mathcal{C}}\otimes \tilde{\mathcal{O}}\right]}^{\mathfrak{o}}={\left[\tilde{\mathcal{C}}\right]}^{\mathfrak{o}}\times {\left[ \tilde{\mathcal{O}}\right]}^{\mathfrak{o}},$$

(13)

$${\left[\mu \odot \tilde{\mathcal{C}}\right]}^{\mathfrak{o}}=\mu .{\left[\tilde{\mathcal{C}}\right]}^{\mathfrak{o}},$$

(14)

result from Equations (4)–(6), respectively. □

Theorem 1. 

Ref. [50] For $\tilde{\mathcal{C}}, \tilde{\mathcal{O}}\in {₤}_I$, the supremum metric.

$$d_{\infty }\left(\tilde{\mathcal{C}}, \tilde{\mathcal{O}}\right)=\underset{0\le \mathfrak{o}\le 1}{\sup }{d}_H\left({\left[\tilde{\mathcal{C}}\right]}^{\mathfrak{o}}, {\left[\tilde{\mathcal{O}}\right]}^{\mathfrak{o}}\right).$$

(15)

is a complete metric space, where $H$ stands for the Hausdorff metric on a space of intervals.

Theorem 2.

Refs. [50,51] If $\mathfrak{G}:\left[\mathfrak{d},𝔃\right]\subset ℝ\to {\mathbb{X}}_I$ is an $I\mathcal{V}ℳ$ satisfying $\mathfrak{G}\left(\varsigma \right)=\left[{\mathfrak{G}}_{*}\left(\varsigma \right), {\mathfrak{G}}^{*}\left(\varsigma \right)\right]$, then $\mathfrak{G}$ is Aumann integrable (IA-integrable) over $\left[\mathfrak{d},𝔃\right]$ when and only when ${\mathfrak{G}}_{*}\left(\varsigma \right)$ and ${\mathfrak{G}}^{*}\left(\varsigma \right)$ are integrable over $\left[\mathfrak{d},𝔃\right]$, meaning

$$\left(IA\right){{\displaystyle \int }}_{\mathfrak{d}}^{𝔃}\mathfrak{G}\left(\varsigma \right)d\varsigma =\left[{{\displaystyle \int }}_{\mathfrak{d}}^{\begin{array}{c} \\ z\end{array}}{\mathfrak{G}}_{*}\left(\varsigma \right)d\varsigma , {{\displaystyle \int }}_{\mathfrak{d}}^{𝔃}{\mathfrak{G}}^{*}\left(\varsigma \right)d\varsigma \right]$$

(16)

Definition 3.

Ref. [56] Let $\tilde{\mathfrak{G}}:\mathbb{I}\subset ℝ\to {₤}_I$ be a $ℱ\mathcal{N}\mathcal{V}ℳ$. The family of $I\mathcal{V}ℳ$s, for every $\mathfrak{o}\in \left[0, 1\right]$, is ${\mathfrak{G}}_{\mathfrak{o}}:\mathbb{I}\subset ℝ\to {\mathbb{X}}_I$ satisfying ${\mathfrak{G}}_{\mathfrak{o}}\left(\varsigma \right)=\left[{\mathfrak{G}}_{*}\left(\varsigma ,\mathfrak{o}\right), {\mathfrak{G}}^{*}\left(\varsigma ,\mathfrak{o}\right)\right]$ for every $\varsigma \in \mathbb{I}.$ For every $\mathfrak{o}\in \left[0, 1\right],$ the lower and upper mappings of ${\mathfrak{G}}_{\mathfrak{o}}$ are the endpoint real-valued mappings ${\mathfrak{G}}_{*}\left(·,\mathfrak{o}\right), {\mathfrak{G}}^{*}\left(·,\mathfrak{o}\right):\mathbb{I}\to ℝ$.

Definition 4.

Ref. [56] Let $\tilde{\mathfrak{G}}:\mathbb{I}\subset ℝ\to {₤}_I$ be a $ℱ\mathcal{N}\mathcal{V}ℳ$. Then, $\tilde{\mathfrak{G}}\left(\varsigma \right)$ is continuous at $\varsigma \in \mathbb{I},$ if for every $\mathfrak{o}\in \left[0, 1\right],$ ${\mathfrak{G}}_{\mathfrak{o}}\left(\varsigma \right)$ is continuous when and only when ${\mathfrak{G}}_{*}\left(\varsigma ,\mathfrak{o}\right)$ and ${\mathfrak{G}}^{*}\left(\varsigma ,\mathfrak{o}\right)$ are continuous at $\varsigma \in \mathbb{I}.$

Definition 5.

Ref. [50] Let $\tilde{\mathfrak{G}}:\mathbb{I}=\left[\mathfrak{d}, 𝔃\right]\subset ℝ\to {₤}_I$ be a $ℱ\mathcal{N}\mathcal{V}ℳ$. The fuzzy Aumann integral ($FA$-integral) of $\tilde{\mathfrak{G}}$ over $\left[\mathfrak{d}, 𝔃\right]$ is:

$${\left[\left(FA\right){{\displaystyle \int }}_{\mathfrak{d}}^{𝔃}\tilde{\mathfrak{G}}\left(\varsigma \right)d\varsigma \right]}^{\begin{array}{c} \\ o\end{array}}=\left(IA\right){{\displaystyle \int }}_{\mathfrak{d}}^{𝔃}{\mathfrak{G}}_{\mathfrak{o}}\left(\varsigma \right)d\varsigma =\left\{{{\displaystyle \int }}_{\mathfrak{d}}^{𝔃}\mathfrak{G}\left(\varsigma ,\mathfrak{o}\right)d\varsigma :\mathfrak{G}\left(\varsigma ,\mathfrak{o}\right)\in S\left({\mathfrak{G}}_{\mathfrak{o}}\right)\right\},$$

(17)

where $S\left({\mathfrak{G}}_{\mathfrak{o}}\right)=\left\{\mathfrak{G}\left(.,\mathfrak{o}\right)\to ℝ:\mathfrak{G}\left(.,\mathfrak{o}\right) \mathrm{is} \mathrm{integrable}, \mathrm{and} \mathfrak{G}\left(\varsigma ,\mathfrak{o}\right)\in {\mathfrak{G}}_{\mathfrak{o}}\left(\varsigma \right)\right\},$ for every $\mathfrak{o}\in \left[0, 1\right].$ Moreover, $\tilde{\mathfrak{G}}$ is $\left(FA\right)$-integrable over $\left[\mathfrak{d}, 𝔃\right]$ if $\left(FA\right){{\displaystyle \int }}_{\mathfrak{d}}^{𝔃}\tilde{\mathfrak{G}}\left(\varsigma \right)d\varsigma \in {₤}_I.$

Theorem 3. 

Ref. [51] Let $\tilde{\mathfrak{G}}:\left[\mathfrak{d}, 𝔃\right]\subset ℝ\to {₤}_I$ be a $ℱ\mathcal{N}\mathcal{V}ℳ$, whose $\mathfrak{o}$-levels define the family of $I\mathcal{V}ℳ$s ${\mathfrak{G}}_{\mathfrak{o}}:\left[\mathfrak{d}, 𝔃\right]\subset ℝ\to {\mathbb{X}}_I$ satisfying ${\mathfrak{G}}_{\mathfrak{o}}\left(\varsigma \right)=\left[{\mathfrak{G}}_{*}\left(\varsigma ,\mathfrak{o}\right), {\mathfrak{G}}^{*}\left(\varsigma ,\mathfrak{o}\right)\right]$ for every $\varsigma \in \left[\mathfrak{d}, 𝔃\right]$ and $\mathfrak{o}\in \left[0, 1\right].$ $\tilde{\mathfrak{G}}$ is $\left(FA\right)$-integrable over $\left[\mathfrak{d}, 𝔃\right]$ when and only when ${\mathfrak{G}}_{*}\left(\varsigma ,\mathfrak{o}\right)$ and ${\mathfrak{G}}^{*}\left(\varsigma ,\mathfrak{o}\right)$ are integrable over $\left[\mathfrak{d}, 𝔃\right]$. Moreover, if $\tilde{\mathfrak{G}}$ is $\left(FA\right)$-integrable over $\left[\mathfrak{d}, 𝔃\right]$, then we have:

$${\left[\left(FA\right){{\displaystyle \int }}_{\mathfrak{d}}^{𝔃}\tilde{\mathfrak{G}}\left(\varsigma \right)d\varsigma \right]}^{\begin{array}{c} \\ o\end{array}}=\left[{{\displaystyle \int }}_{\mathfrak{d}}^{𝔃}{\mathfrak{G}}_{*}\left(\varsigma ,\mathfrak{o}\right)d\varsigma , {{\displaystyle \int }}_{\mathfrak{d}}^{𝔃}{\mathfrak{G}}^{*}\left(\varsigma ,\mathfrak{o}\right)d\varsigma \right]=\left(IA\right){{\displaystyle \int }}_{\mathfrak{d}}^{𝔃}{\mathfrak{G}}_{\mathfrak{o}}\left(\varsigma \right)d\varsigma ,$$

(18)

for every $\mathfrak{o}\in \left[0, 1\right].$

Definition 6.

Ref. [63] Let $\beta >0$ and $L\left(\left[\mathfrak{d}, 𝔃\right],{₤}_I\right)$ be the collection of all Lebesgue measurable fuzzy-number-valued mappings on $\left[\mathfrak{d},𝔃\right]$. Then, the fuzzy left and right Riemann–Liouville fractional integrals of order $\beta >0$ of $\mathfrak{G}\in$ $L\left(\left[\mathfrak{d}, 𝔃\right],{₤}_I\right)$ are:

$${ℐ}_{{\mathfrak{d}}^{+}}^{\beta } \tilde{\mathfrak{G}}\left(\varsigma \right)=\frac{1}{\Gamma \left(\beta \right)}{{\displaystyle \int }}_{\mathfrak{d}}^{\varsigma }{\left(\varsigma -q\right)}^{\beta -1}\tilde{\mathfrak{G}}\left(q\right)dq, \left(\varsigma >\mathfrak{d}\right),$$

(19)

and

$${ℐ}_{{𝔃}^{-}}^{\beta }\tilde{\mathfrak{G}}\left(\varsigma \right)=\frac{1}{\Gamma \left(\beta \right)}{{\displaystyle \int }}_{\varsigma }^{𝔃}{\left(q-\varsigma \right)}^{\beta -1}\tilde{\mathfrak{G}}\left(q\right)dq, (\varsigma <𝔃),$$

(20)

respectively, where $\Gamma \left(\varsigma \right)={{\displaystyle \int }}_0^{\infty }{q}^{\varsigma -1}{e}^{-q}dq$ is the Euler gamma function. The fuzzy left and right RL fractional integrals $\varsigma$ based on left and right end point mappings are:

$$\begin{array}{cc}{\left[{ℐ}_{{\mathfrak{d}}^{+}}^{\beta } \tilde{\mathfrak{G}}\left(\varsigma \right)\right]}^{\mathfrak{o}}\hfill & =\frac{1}{\Gamma \left(\beta \right)}{{\displaystyle \int }}_{\mathfrak{d}}^{\varsigma }{\left(\varsigma -q\right)}^{\beta -1}{\mathfrak{G}}_{\mathfrak{o}}\left(q\right)dq\hfill \\ \hfill & =\frac{1}{\Gamma \left(\beta \right)}{{\displaystyle \int }}_{\mathfrak{d}}^{\varsigma }{\left(\varsigma -q\right)}^{\beta -1}\left[{\mathfrak{G}}_{*}\left(q, \mathfrak{o}\right),{\mathfrak{G}}^{*}\left(q, \mathfrak{o}\right)\right]dq, \left(\varsigma >\mathfrak{d}\right),\hfill \end{array}$$

(21)

where:

$${ℐ}_{{\mathfrak{d}}^{+}}^{\beta } {\mathfrak{G}}_{*}\left(\varsigma , \mathfrak{o}\right)=\frac{1}{\Gamma \left(\beta \right)}{{\displaystyle \int }}_{\mathfrak{d}}^{\varsigma }{\left(\varsigma -q\right)}^{\beta -1}{\mathfrak{G}}_{*}\left(q, \mathfrak{o}\right)dq, \left(\varsigma >\mathfrak{d}\right),$$

(22)

and

$${ℐ}_{{\mathfrak{d}}^{+}}^{\beta } {\mathfrak{G}}^{*}\left(\varsigma , \mathfrak{o}\right)=\frac{1}{\Gamma \left(\beta \right)}{{\displaystyle \int }}_{\mathfrak{d}}^{\varsigma }{\left(\varsigma -q\right)}^{\beta -1}{\mathfrak{G}}^{*}\left(q, \mathfrak{o}\right)dq, \left(\varsigma >\mathfrak{d}\right).$$

(23)

The Riemann–Liouville fractional integral $\tilde{\mathfrak{G}}$ of $\varsigma$ based on left and right end point mappings can be defined in a similar way.

Now we recall some basic classical definitions related of convex real-valued, interval-valued and fuzzy-number-valued mappings.

Definition 7.

Ref. [52] An interval valued mapping $\mathfrak{G}:\mathbb{I}=\left[\mathfrak{d}, 𝔃\right]\to {\mathbb{X}}_I$ is a convex inteval valued mapping if:

$$\mathfrak{G}\left(\mu \varsigma +\left(1-\mu \right)𝓉\right)\supseteq \mu \mathfrak{G}\left(\varsigma \right)+\left(1-\mu \right)\mathfrak{G}\left(𝓉\right),$$

(24)

for all $\varsigma , 𝓉\in \left[\mathfrak{d}, 𝔃\right], \mu \in \left[0, 1\right]$, where ${\mathbb{X}}_I$ is the collection of all real valued intervals. If (24) is reversed, then $\mathfrak{G}$ is said to be concave.

Definition 8. 

Ref. [48] The F-I∙V∙M $\tilde{\mathfrak{G}}:\left[\mathfrak{d}, 𝔃\right]\to {₤}_I$ is convex on $\left[\mathfrak{d}, 𝔃\right]$ if:

$$\tilde{\mathfrak{G}}\left(\mu \varsigma +\left(1-\mu \right)𝓉\right){\le }_{\mathbb{F}}\mu \odot \tilde{\mathfrak{G}}\left(\varsigma \right)\oplus \left(1-\mu \right)\odot \tilde{\mathfrak{G}}\left(𝓉\right),$$

(25)

For all $\varsigma , 𝓉\in \left[\mathfrak{d}, 𝔃\right], \mu \in \left[0, 1\right],$ where $\tilde{\mathfrak{G}}\left(\varsigma \right){\ge }_{\mathbb{F}}\tilde{0}$ for all $\varsigma \in \left[\mathfrak{d}, 𝔃\right].$ If (25) is reversed, table. Moreover, $\tilde{\mathfrak{G}}$ is named affine if and only if it is convex and concave.

Definition 9.

Ref. [57] The F-I∙V∙M $\tilde{\mathfrak{G}}:\left[\mathfrak{d}, 𝔃\right]\to {₤}_I$ is $U\mathcal{D}$-convex on $\left[\mathfrak{d}, 𝔃\right]$ if:

$$\tilde{\mathfrak{G}}\left(\mu \varsigma +\left(1-\mu \right)𝓉\right){\supseteq }_{\mathbb{F}}\mu \odot \tilde{\mathfrak{G}}\left(\varsigma \right)\oplus \left(1-\mu \right)\odot \tilde{\mathfrak{G}}\left(𝓉\right),$$

(26)

for all $\varsigma , 𝓉\in \left[\mathfrak{d}, 𝔃\right], \mu \in \left[0, 1\right],$ where $\tilde{\mathfrak{G}}\left(\varsigma \right){\ge }_{\mathbb{F}}\tilde{0}$ for all $\varsigma \in \left[\mathfrak{d}, 𝔃\right].$ If (26) is reversed then, $\tilde{\mathfrak{G}}$ is $U\mathcal{D}$-concave. Moreover, $\tilde{\mathfrak{G}}$ is $U\mathcal{D}$-affine if and only if it is $U\mathcal{D}$-convex and concave.

3. Up and Down Harmonically Convex Fuzzy-Number-Valued Mappings

In the following section, we will introduce new definitions of harmonically convex mappings and related new results with respect to fuzzy-inclusion relation.

Definition 10.

The $ℱ\mathcal{N}\mathcal{V}ℳ$ $\tilde{\mathfrak{G}}:\left[\mathfrak{d}, 𝔃\right]\to {\mathbb{F}}_0$ is named as $U\mathcal{D}$-harmonically convex $ℱ\mathcal{N}\mathcal{V}ℳ$ on $\left[\mathfrak{d}, 𝔃\right]$ if:

$$\tilde{\mathfrak{G}}\left(\frac{\varsigma 𝓉}{\mu \varsigma +\left(1-\mu \right)𝓉}\right){\supseteq }_{\mathbb{F}}\left(1-\mu \right)\odot \tilde{\mathfrak{G}}\left(\varsigma \right)\oplus \mu \odot \tilde{\mathfrak{G}}\left(𝓉\right),$$

(27)

for all $\varsigma , 𝓉\in \left[\mathfrak{d}, 𝔃\right], \mu \in \left[0, 1\right],$ where $\tilde{\mathfrak{G}}\left(\varsigma \right){\ge }_{\mathbb{F}}\tilde{0}$, for all $\varsigma \in \left[\mathfrak{d}, 𝔃\right]$. If (15) is reversed then, $\tilde{\mathfrak{G}}$ is named as $U\mathcal{D}$-harmonically concave $ℱ\mathcal{N}\mathcal{V}ℳ$ on $\left[\mathfrak{d}, 𝔃\right]$. The set of all $U\mathcal{D}$-harmonically convex ($U\mathcal{D}$-harmonically concave) $ℱ\mathcal{N}\mathcal{V}ℳ$ is denoted by:

$$U\mathcal{D}\mathscr{H}ℱ\mathcal{S}\mathcal{X}\left(\left[\mathfrak{d}, 𝔃\right], {\mathbb{F}}_0\right) \left(U\mathcal{D}\mathscr{H}ℱ\mathcal{S}\mathcal{V}\left(\left[\mathfrak{d}, 𝔃\right], {\mathbb{F}}_0\right)\right).$$

Theorem 4.

Let $\left[\mathfrak{d}, 𝔃\right]$ be a harmonically convex set, and let $\tilde{\mathfrak{G}}:\left[\mathfrak{d}, 𝔃\right]\to {\mathbb{F}}_0$ be an $ℱ\mathcal{N}\mathcal{V}ℳ$, whose $\mathfrak{o}$-cuts define the family of $I\mathcal{V}ℳ$s ${\mathfrak{G}}_{\mathfrak{o}}:\left[\mathfrak{d}, 𝔃\right]\subset ℝ\to {\mathbb{X}}_I^{+}\subset {\mathbb{X}}_I$ are provided by:

$${\mathfrak{G}}_{\mathfrak{o}}\left(\varsigma \right)=\left[{\mathfrak{G}}_{*}\left(\varsigma ,\mathfrak{o}\right), {\mathfrak{G}}^{*}\left(\varsigma ,\mathfrak{o}\right)\right], \forall \varsigma \in \left[\mathfrak{d}, 𝔃\right].$$

(28)

for all $\varsigma \in \left[\mathfrak{d}, 𝔃\right]$, $\mathfrak{o}\in \left[0, 1\right]$. Then, $\tilde{\mathfrak{G}}\in U\mathcal{D}\mathscr{H}ℱ\mathcal{S}\mathcal{X}\left(\left[\mathfrak{d}, 𝔃\right], {\mathbb{F}}_0\right),$ if and only if, for all $\in \left[0, 1\right],$ ${\mathfrak{G}}_{*}\left(\varsigma , \mathfrak{o}\right)\in HSX\left(\left[\mathfrak{d}, 𝔃\right], {ℝ}^{+}\right)$ and ${\mathfrak{G}}^{*}\left(\varsigma , \mathfrak{o}\right)\in \left(U\mathcal{D}\mathscr{H}ℱ\mathcal{S}\mathcal{V}\left(\left[\mathfrak{d}, 𝔃\right], {ℝ}^{+}\right)\right)$.

Proof. 

Assume that for each $\mathfrak{o}\in \left[0, 1\right],$ ${\mathfrak{G}}_{*}\left(\varsigma , \mathfrak{o}\right)$ and ${\mathfrak{G}}^{*}\left(\varsigma , \mathfrak{o}\right)$ are harmonically convex on $K.$ Then, from (28), we have:

$${\mathfrak{G}}_{*}\left(\frac{\varsigma 𝓉}{\mu \varsigma +\left(1-\mu \right)𝓉}, \mathfrak{o}\right)\le \left(1-\mu \right){\mathfrak{G}}_{*}\left(\varsigma , \mathfrak{o}\right)+\mu {\mathfrak{G}}_{*}\left(𝓉, \mathfrak{o}\right),$$

and

$${\mathfrak{G}}^{*}\left(\frac{\varsigma 𝓉}{\mu \varsigma +\left(1-\mu \right)𝓉}, \mathfrak{o}\right)\ge \left(1-\mu \right){\mathfrak{G}}^{*}\left(\varsigma , \mathfrak{o}\right)+\mu {\mathfrak{G}}^{*}\left(𝓉, \mathfrak{o}\right).$$

Then, by (28), (12), and (14), we obtain:

$${\mathfrak{G}}_{\mathfrak{o}}\left(\frac{\varsigma 𝓉}{\mu \varsigma +\left(1-\mu \right)𝓉}\right)=\left[{\mathfrak{G}}_{*}\left(\frac{\varsigma 𝓉}{\mu \varsigma +\left(1-\mu \right)𝓉}, \mathfrak{o}\right), {\mathfrak{G}}^{*}\left(\frac{\varsigma 𝓉}{\mu \varsigma +\left(1-\mu \right)𝓉}, \mathfrak{o}\right)\right],{\supseteq }_I\left(1-\mu \right)\left[{\mathfrak{G}}_{*}\left(\varsigma , \mathfrak{o}\right), {\mathfrak{G}}^{*}\left(\varsigma , \mathfrak{o}\right)\right]+\mu \left[{\mathfrak{G}}_{*}\left(𝓉, \mathfrak{o}\right), {\mathfrak{G}}^{*}\left(𝓉, \mathfrak{o}\right)\right],$$

that is:

$$\tilde{\mathfrak{G}}\left(\frac{\varsigma 𝓉}{\mu \varsigma +\left(1-\mu \right)𝓉}\right){\supseteq }_{\mathbb{F}}\left(1-\mu \right)\odot \tilde{\mathfrak{G}}\left(\varsigma \right)\oplus \mu \odot \tilde{\mathfrak{G}}\left(𝓉\right),\forall \varsigma ,𝓉\in K, \mu \in \left[0, 1\right].$$

Hence, $\tilde{\mathfrak{G}}$ is $U\mathcal{D}$-harmonically convex on $K$.

Conversely, let $\tilde{\mathfrak{G}}$ be $U\mathcal{D}$-harmonically convex $ℱ\mathcal{N}\mathcal{V}ℳ$ on $K.$ Then, for all $\varsigma ,𝓉\in K, \mu \in \left[0, 1\right]$, we have:

$$\tilde{\mathfrak{G}}\left(\frac{\varsigma 𝓉}{\mu \varsigma +\left(1-\mu \right)𝓉}\right){\supseteq }_{\mathbb{F}}\left(1-\mu \right)\odot \tilde{\mathfrak{G}}\left(\varsigma \right)\oplus \mu \odot \tilde{\mathfrak{G}}\left(𝓉\right).$$

Therefore, from (28), for each $\mathfrak{o}\in \left[0, 1\right]$, left side of above inequality, we have:

$${\mathfrak{G}}_{\mathfrak{o}}\left(\frac{\varsigma 𝓉}{\mu \varsigma +\left(1-\mu \right)𝓉}\right)=\left[{\mathfrak{G}}_{*}\left(\frac{\varsigma 𝓉}{\mu \varsigma +\left(1-\mu \right)𝓉}, \mathfrak{o}\right), {\mathfrak{G}}^{*}\left(\frac{\varsigma 𝓉}{\mu \varsigma +\left(1-\mu \right)𝓉}, \mathfrak{o}\right)\right].$$

Again, from (28), we obtain:

$$\left(1-\mu \right){\mathfrak{G}}_{\mathfrak{o}}\left(\varsigma \right)+\mu {\mathfrak{G}}_{\mathfrak{o}}\left(\varsigma \right)=\left(1-\mu \right)\left[{\mathfrak{G}}_{*}\left(\varsigma , \mathfrak{o}\right), {\mathfrak{G}}^{*}\left(\varsigma , \mathfrak{o}\right)\right]+\mu \left[{\mathfrak{G}}_{*}\left(𝓉, \mathfrak{o}\right), {\mathfrak{G}}^{*}\left(𝓉, \mathfrak{o}\right)\right],$$

for all $\varsigma ,𝓉\in K$, $\mu \in \left[0, 1\right].$ Then, by $U\mathcal{D}$-harmonically convexity of $\tilde{\mathfrak{G}}$, we have for all $\varsigma ,𝓉\in K$, $\mu \in \left[0, 1\right]$ such that:

$${\mathfrak{G}}_{*}\left(\frac{\varsigma 𝓉}{\mu \varsigma +\left(1-\mu \right)𝓉}, \mathfrak{o}\right)\le \left(1-\mu \right){\mathfrak{G}}_{*}\left(\varsigma , \mathfrak{o}\right)+\mu {\mathfrak{G}}_{*}\left(𝓉, \mathfrak{o}\right),$$

and

$${\mathfrak{G}}^{*}\left(\frac{\varsigma 𝓉}{\mu \varsigma +\left(1-\mu \right)𝓉}, \mathfrak{o}\right)\ge \left(1-\mu \right){\mathfrak{G}}^{*}\left(\varsigma , \mathfrak{o}\right)+\mu {\mathfrak{G}}^{*}\left(𝓉, \mathfrak{o}\right),$$

for each $\mathfrak{o}\in \left[0, 1\right].$ Hence, the result follows. □

Example 1.

We consider the $ℱ\mathcal{N}\mathcal{V}ℳ$s $\tilde{\mathfrak{G}}:\left[\frac{1}{2}, 1\right]\to {\mathbb{F}}_0$ defined by:

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

Then, for each $\mathfrak{o}\in \left[0, 1\right],$ we have ${\mathfrak{G}}_{\mathfrak{o}}\left(\varsigma \right)=\left[\left(1-\mathfrak{o}\right){\varsigma }^2+\mathfrak{o},\left(1-\mathfrak{o}\right)\left(5-e^{\varsigma }\right)+\mathfrak{o}\right]$. We can easily see that ${\mathfrak{G}}_{*}\left(\varsigma , \mathfrak{o}\right)\in HSX\left(\left[\mathfrak{d}, 𝔃\right], {ℝ}^{+}\right)$, ${\mathfrak{G}}^{*}\left(\varsigma , \mathfrak{o}\right)\in HSV\left(\left[\mathfrak{d}, 𝔃\right], {ℝ}^{+}\right)$, for each $\mathfrak{o}\in \left[0, 1\right]$. Hence, $\tilde{\mathfrak{G}}\in U\mathcal{D}\mathscr{H}ℱ\mathcal{S}\mathcal{X}\left(\left[\mathfrak{d}, 𝔃\right], {\mathbb{F}}_0\right)$.

Now that we have the new definitions listed below, we can use them to analyze certain classical and contemporary results as subsets of the primary findings.

Definition 11.

Let $\tilde{\mathfrak{G}}:\left[\mathfrak{d},𝔃\right]\to {₤}_I$ be a $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$, whose $\mathfrak{o}$-cuts define the family of $I\mathcal{V}ℳ$s ${\mathfrak{G}}_{\mathfrak{o}}:\left[\mathfrak{d},𝔃\right]\to {\mathbb{X}}_I^{+}\subset {\mathbb{X}}_I$ are provided by:

$${\mathfrak{G}}_{\mathfrak{o}}\left(\varsigma \right)=\left[{\mathfrak{G}}_{*}\left(\varsigma ,\mathfrak{o}\right), {\mathfrak{G}}^{*}\left(\varsigma ,\mathfrak{o}\right)\right],$$

for all $\varsigma \in \left[\mathfrak{d},𝔃\right]$ and for all $\mathfrak{o}\in \left[0, 1\right]$. Then, $\tilde{\mathfrak{G}}$ is lower $U\mathcal{D}$-harmonically convex (concave) $ℱ\mathcal{N}\mathcal{V}ℳ$ on $\left[\mathfrak{d},𝔃\right],$ if and only if:

$${\mathfrak{G}}_{*}\left(\frac{\varsigma 𝓉}{\mu \varsigma +\left(1-\mu \right)𝓉}, \mathfrak{o}\right)\le (\ge )\left(1-\mu \right){\mathfrak{G}}_{*}\left(\varsigma , \mathfrak{o}\right)+\mu {\mathfrak{G}}_{*}\left(𝓉, \mathfrak{o}\right),$$

(29)

and

$${\mathfrak{G}}^{*}\left(\frac{\varsigma 𝓉}{\mu \varsigma +\left(1-\mu \right)𝓉}, \mathfrak{o}\right)=\left(1-\mu \right){\mathfrak{G}}^{*}\left(\varsigma , \mathfrak{o}\right)+\mu {\mathfrak{G}}^{*}\left(𝓉, \mathfrak{o}\right),$$

(30)

for all $\mathfrak{o}\in \left[0, 1\right]$.

Definition 12.

Let $\tilde{\mathfrak{G}}:\left[\mathfrak{d},𝔃\right]\to {₤}_I$ be a $ℱ\mathcal{N}\mathcal{V}ℳ$, whose $\mathfrak{o}$-cuts define the family of $I\mathcal{V}ℳ$s ${\mathfrak{G}}_{\mathfrak{o}}:\left[\mathfrak{d},𝔃\right]\to {\mathbb{X}}_I^{+}\subset {\mathbb{X}}_I$ are provided by:

$${\mathfrak{G}}_{\mathfrak{o}}\left(\varsigma \right)=\left[{\mathfrak{G}}_{*}\left(\varsigma ,\mathfrak{o}\right), {\mathfrak{G}}^{*}\left(\varsigma ,\mathfrak{o}\right)\right],$$

for all $\varsigma \in \left[\mathfrak{d},𝔃\right]$ and for all $\mathfrak{o}\in \left[0, 1\right]$. Then, $\mathfrak{G}$ is upper $U\mathcal{D}$-harmonically convex (concave) $ℱ\mathcal{N}\mathcal{V}ℳ$ on $\left[\mathfrak{d},𝔃\right],$ if and only if:

$${\mathfrak{G}}_{*}\left(\frac{\varsigma 𝓉}{\mu \varsigma +\left(1-\mu \right)𝓉}, \mathfrak{o}\right)=\left(1-\mu \right){\mathfrak{G}}_{*}\left(\varsigma , \mathfrak{o}\right)+\mu {\mathfrak{G}}_{*}\left(𝓉, \mathfrak{o}\right),$$

(31)

and

$${\mathfrak{G}}^{*}\left(\frac{\varsigma 𝓉}{\mu \varsigma +\left(1-\mu \right)𝓉}, \mathfrak{o}\right)\le (\ge )\left(1-\mu \right){\mathfrak{G}}^{*}\left(\varsigma , \mathfrak{o}\right)+\mu {\mathfrak{G}}^{*}\left(𝓉, \mathfrak{o}\right),$$

(32)

for all $\mathfrak{o}\in \left[0, 1\right]$.

Remark 2.

(1) Let ${\mathfrak{G}}_{*}\left(\varsigma , \mathfrak{o}\right)\ne {\mathfrak{G}}^{*}\left(\varsigma , \mathfrak{o}\right)$ with $\mathfrak{o}=1$. Then, $U\mathcal{D}$-harmonically convex (concave) $ℱ\mathcal{N}\mathcal{V}ℳ$ reduces to the interval-valued harmonically convex(concave) mapping, see [31].

(2)

Let $\tilde{\mathfrak{G}}$ be a lower (upper) $U\mathcal{D}$-harmonically convex (concave) $ℱ\mathcal{N}\mathcal{V}ℳ$. Then, we obtain the definition of harmonically convex (concave) $ℱ\mathcal{N}\mathcal{V}ℳ$, see [64].

(3)

If ${\mathfrak{G}}_{*}\left(\varsigma , \mathfrak{o}\right)={\mathfrak{G}}^{*}\left(\varsigma , \mathfrak{o}\right)$ with $\mathfrak{o}=1$, then $U\mathcal{D}$-harmonically convex (concave) $ℱ\mathcal{N}\mathcal{V}ℳ$ reduces to the classical harmonically convex(concave) mapping, see [65].

In the next result, we will develop the relation between $U\mathcal{D}$-harmonically convex $ℱ\mathcal{N}\mathcal{V}ℳ$ and $U\mathcal{D}$-convex $ℱ\mathcal{N}\mathcal{V}ℳ$. This result will be helpful to prove main results.

Theorem 5.

Let $\tilde{\mathfrak{G}}:K\to {₤}_I$ be a $ℱ\mathcal{N}\mathcal{V}ℳ$, where for all $\mathfrak{o}\in \left[0, 1\right]$, whose $\mathfrak{o}$-cuts define the family of $\mathcal{I}\mathcal{V}\mathcal{M}$s ${\mathfrak{G}}_{\mathfrak{o}}:K\subset ℝ\to {\mathcal{K}}_I^{+}\subset {\mathcal{K}}_I$ are provided by ${\mathfrak{G}}_{\mathfrak{o}}\left(\varsigma \right)=\left[{\mathfrak{G}}_{*}\left(\varsigma ,\mathfrak{o}\right), {\mathfrak{G}}^{*}\left(\varsigma ,\mathfrak{o}\right)\right],$ for all $\varsigma \in K$. Then, $\tilde{\mathfrak{G}}\left(\varsigma \right)$ is $U\mathcal{D}$-harmonically convex $ℱ\mathcal{N}\mathcal{V}ℳ$ on $K,$ if and only if, $\tilde{\mathfrak{G}}\left(\frac{1}{\varsigma }\right)$ is $U\mathcal{D}$-convex $ℱ\mathcal{N}\mathcal{V}ℳ$ on $K.$

Proof. 

Since $\tilde{\mathfrak{G}}(\varsigma )$ is a $U\mathcal{D}$-harmonically convex $ℱ\mathcal{N}\mathcal{V}ℳ$ then, for $\varsigma , 𝓉\in \left[\mathfrak{d}, 𝔃\right], \mu \in \left[0, 1\right]$, we have:

$$\tilde{\mathfrak{G}}\left(\frac{\varsigma 𝓉}{\mu \varsigma +\left(1-\mu \right)𝓉}\right){\supseteq }_{\mathbb{F}}\left(1-\mu \right)\odot \tilde{\mathfrak{G}}\left(\varsigma \right)\oplus \mu \odot \tilde{\mathfrak{G}}\left(𝓉\right).$$

Therefore, for each $\mathfrak{o}\in \left[0, 1\right]$, we have:

$$\begin{array}{c}{\mathfrak{G}}_{*}\left(\frac{\varsigma 𝓉}{\mu \varsigma +\left(1-\mu \right)𝓉}, \mathfrak{o}\right)\le \left(1-\mu \right){\mathfrak{G}}_{*}\left(\varsigma , \mathfrak{o}\right)+\mu {\mathfrak{G}}_{*}\left(𝓉, \mathfrak{o}\right), \\ {\mathfrak{G}}^{*}\left(\frac{\varsigma 𝓉}{\mu \varsigma +\left(1-\mu \right)𝓉}, \mathfrak{o}\right)\ge \left(1-\mu \right){\mathfrak{G}}^{*}\left(\varsigma , \mathfrak{o}\right)+\mu {\mathfrak{G}}^{*}\left(𝓉, \mathfrak{o}\right).\end{array}$$

(33)

Consider $\tilde{\phi }\left(\varsigma \right)=\tilde{\mathfrak{G}}\left(\frac{1}{\varsigma }\right)$. Taking $q=\frac{1}{\varsigma }$ and $n=\frac{1}{𝓉}$ to replace $\varsigma$ and $𝓉$, respectively. Then, for each $\mathfrak{o}\in \left[0, 1\right]$, applying (33)

$$\begin{gathered} {\mathfrak{G}}_{*}\left(\frac{\frac{1}{\varsigma 𝓉}}{\mu \frac{1}{\varsigma }+\left(1-\mu \right)\frac{1}{𝓉}}, \mathfrak{o}\right)={\mathfrak{G}}_{*}\left(\frac{1}{\left(1-\mu \right)\varsigma +\mu 𝓉}, \mathfrak{o}\right) \\ ={\phi }_{*}\left(\left(1-\mu \right)\varsigma +\mu 𝓉, \mathfrak{o}\right) \\ \le \mu {\mathfrak{G}}_{*}\left(\frac{1}{𝓉}, \mathfrak{o}\right)+\left(1-\mu \right){\mathfrak{G}}_{*}\left(\frac{1}{\varsigma }, \mathfrak{o}\right) \\ =\mu {\phi }_{*}\left(𝓉, \mathfrak{o}\right)+\left(1-\mu \right){\phi }_{*}\left(\varsigma , \mathfrak{o}\right), \\ {\mathfrak{G}}^{*}\left(\frac{\frac{1}{\varsigma 𝓉}}{\mu \frac{1}{\varsigma }+\left(1-\mu \right)\frac{1}{𝓉}}, \mathfrak{o}\right)={\mathfrak{G}}^{*}\left(\frac{1}{\left(1-\mu \right)\varsigma +\mu 𝓉}, \mathfrak{o}\right) \\ ={\phi }^{*}\left(\left(1-\mu \right)\varsigma +\mu 𝓉, \mathfrak{o}\right) \\ \ge \mu {\mathfrak{G}}^{*}\left(\frac{1}{𝓉}, \mathfrak{o}\right)+\left(1-\mu \right){\mathfrak{G}}^{*}\left(\frac{1}{\varsigma }, \mathfrak{o}\right) \\ =\mu {\phi }^{*}\left(𝓉, \mathfrak{o}\right)+\left(1-\mu \right){\phi }^{*}\left(\varsigma , \mathfrak{o}\right). \end{gathered}$$

It follows that:

$$\begin{gathered} \left[{\mathfrak{G}}_{*}\left(\frac{\frac{1}{\varsigma 𝓉}}{\mu \frac{1}{\varsigma }+\left(1-\mu \right)\frac{1}{𝓉}}, \mathfrak{o}\right), {\mathfrak{G}}^{*}\left(\frac{\frac{1}{\varsigma 𝓉}}{\mu \frac{1}{\varsigma }+\left(1-\mu \right)\frac{1}{𝓉}}, \mathfrak{o}\right)\right] \\ =\left[\begin{array}{c}{\phi }_{*}\left(\left(1-\mu \right)\varsigma +\mu 𝓉, \mathfrak{o}\right), {\phi }^{*}\left(\left(1-\mu \right)\varsigma +\mu 𝓉, \mathfrak{o}\right)\end{array}\right] \\ {\supseteq }_I\mu \left[{\phi }_{*}\left(𝓉, \mathfrak{o}\right), {\phi }^{*}\left(𝓉, \mathfrak{o}\right)\right]+\left(1-\mu \right)\left[{\phi }_{*}\left(\varsigma , \mathfrak{o}\right), {\phi }^{*}\left(\varsigma , \mathfrak{o}\right)\right]. \end{gathered}$$

which implies that:

$${\phi }_{\mathfrak{o}}\left(\left(1-\mu \right)\varsigma +\mu 𝓉\right){\supseteq }_I\mu {\phi }_{\mathfrak{o}}\left(𝓉\right)+\left(1-\mu \right){\phi }_{\mathfrak{o}}\left(\varsigma \right),$$

that is:

$$\tilde{\phi }\left(\left(1-\mu \right)\varsigma +\mu 𝓉 \right){\supseteq }_{\mathbb{F}}\mu \odot \tilde{\phi }\left(𝓉\right)\oplus \left(1-\mu \right)\odot \tilde{\phi }\left(\varsigma \right).$$

This concludes that $\tilde{\phi }\left(\varsigma \right)$ is a convex $ℱ\mathcal{N}\mathcal{V}ℳ$.

Conversely, let $\tilde{\phi }$ is convex $ℱ\mathcal{N}\mathcal{V}ℳ$ on $K.$ Then, for all $\varsigma ,𝓉\in K$, $\mu \in \left[0, 1\right],$ we have:

$$\tilde{\phi }\left(\mu \varsigma +\left(1-\mu \right)𝓉\right){\supseteq }_{\mathbb{F}}\mu \odot \tilde{\phi }\left(\varsigma \right)\oplus \left(1-\mu \right)\odot \tilde{\phi }\left(𝓉\right).$$

By using the same steps as above, for each $\mathfrak{o}\in \left[0, 1\right]$, we have:

$$\begin{gathered} {\phi }_{*}\left(\mu \frac{1}{\varsigma }+\left(1-\mu \right)\frac{1}{𝓉}, \mathfrak{o}\right)={\mathfrak{G}}_{*}\left(\frac{1}{\mu \frac{1}{\varsigma }+\left(1-\mu \right)\frac{1}{𝓉}}, \mathfrak{o}\right) \\ ={\mathfrak{G}}_{*}\left(\frac{\varsigma 𝓉}{\left(1-\mu \right)\varsigma +\mu 𝓉}, \mathfrak{o}\right) \\ \le \mu {\phi }_{*}\left(\frac{1}{\varsigma }, \mathfrak{o}\right)+\left(1-\mu \right){\phi }_{*}\left(\frac{1}{𝓉}, \mathfrak{o}\right) \\ =\mu {\mathfrak{G}}_{*}\left(\varsigma , \mathfrak{o}\right)+\left(1-\mu \right){\mathfrak{G}}_{*}\left(𝓉, \mathfrak{o}\right) \\ {\phi }^{*}\left(\mu \frac{1}{\varsigma }+\left(1-\mu \right)\frac{1}{𝓉}, \mathfrak{o}\right)={\mathfrak{G}}_{*}\left(\frac{1}{\mu \frac{1}{\varsigma }+\left(1-\mu \right)\frac{1}{𝓉}}, \mathfrak{o}\right) \\ ={\mathfrak{G}}_{*}\left(\frac{\varsigma 𝓉}{\left(1-\mu \right)\varsigma +\mu 𝓉}, \mathfrak{o}\right) \\ \ge \mu {\phi }^{*}\left(\frac{1}{\varsigma }, \mathfrak{o}\right)+\left(1-\mu \right){\phi }^{*}\left(\frac{1}{𝓉}, \mathfrak{o}\right) \\ =\mu {\mathfrak{G}}_{*}\left(\varsigma , \mathfrak{o}\right)+\left(1-\mu \right){\mathfrak{G}}_{*}\left(𝓉, \mathfrak{o}\right). \end{gathered}$$

It follows that:

$${\mathfrak{G}}_{\mathfrak{o}}\left(\frac{\varsigma 𝓉}{\mu \varsigma +\left(1-\mu \right)𝓉}\right){\supseteq }_I\left(1-\mu \right){\mathfrak{G}}_{\mathfrak{o}}\left(\varsigma \right)+\mu {\mathfrak{G}}_{\mathfrak{o}}\left(𝓉\right),$$

that is:

$$\tilde{\mathfrak{G}}\left(\frac{\varsigma 𝓉}{\mu \varsigma +\left(1-\mu \right)𝓉}\right){\supseteq }_{\mathbb{F}}\left(1-\mu \right)\odot \tilde{\mathfrak{G}}\left(\varsigma \right)\oplus \mu \odot \tilde{\mathfrak{G}}\left(𝓉\right),$$

the proof the theorem has been completed. □

Remark 3.

If ${\mathfrak{G}}_{*}\left(\varsigma ,\mathfrak{o}\right)={\mathfrak{G}}^{*}\left(\varsigma ,\mathfrak{o}\right)$ and $\mathfrak{o}=1$ then from Theorem 5, we obtain Lemma 2.1 of [66].

4. Fuzzy-Interval Fractional Hermite–Hadamard Inequalities

In this section, we will propose new versions of Hermite–Hadamard-type Inequalities for $U\mathcal{D}$-harmonically convex ($U\mathcal{D}$-harmonically concave) $ℱ\mathcal{N}\mathcal{V}ℳ$. Moreover, some exceptional new and classical cases are also discussed.

Theorem 6.

Let $\tilde{\mathfrak{G}}:\left[\mathfrak{d}, 𝔃\right]\to {₤}_I$ be a $U\mathcal{D}$-harmonically convex $ℱ\mathcal{N}\mathcal{V}ℳ$ on $\left[\mathfrak{d}, 𝔃\right],$ whose $\mathfrak{o}$-cuts define the family of $I\mathcal{V}ℳ$s ${\mathfrak{G}}_{\mathfrak{o}}:\left[\mathfrak{d}, 𝔃\right]\subset ℝ\to {\mathcal{K}}_I^{+}$ are provided by ${\mathfrak{G}}_{\mathfrak{o}}\left(\varsigma \right)=\left[{\mathfrak{G}}_{*}\left(\varsigma ,\mathfrak{o}\right), {\mathfrak{G}}^{*}\left(\varsigma ,\mathfrak{o}\right)\right]$ for all $\varsigma \in \left[\mathfrak{d}, 𝔃\right]$, $\mathfrak{o}\in \left[0, 1\right]$. If $\tilde{\mathfrak{G}}\in L\left(\left[\mathfrak{d}, 𝔃\right],{𝔃}_I\right)$, then

$$\tilde{\mathfrak{G}}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃}\right){\supseteq }_{\mathbb{F}}\frac{\Gamma \left(\beta +1\right)}{2{\left(𝔃-\mathfrak{d}\right)}^{\beta }}\odot \left[{ℐ}_{{\frac{1}{\mathfrak{d}}}^{-}}^{\beta }\left(\tilde{\mathfrak{G}}\circ \psi \right)\left(\frac{1}{𝔃}\right)\oplus {ℐ}_{{\frac{1}{𝔃}}^{+}}^{\beta } \left(\tilde{\mathfrak{G}}\circ \psi \right)\left(\frac{1}{\mathfrak{d}}\right)\right]{\supseteq }_{\mathbb{F}}\frac{\tilde{\mathfrak{G}}\left(\mathfrak{d}\right)\oplus \tilde{\mathfrak{G}}\left(𝔃\right)}{2}.$$

(34)

If $\tilde{\mathfrak{G}}\left(\varsigma \right)$ is concave $ℱ\mathcal{N}\mathcal{V}ℳ$, then

$$\tilde{\mathfrak{G}}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃}\right){\subseteq }_{\mathbb{F}}\frac{\Gamma \left(\beta +1\right)}{2{\left(𝔃-\mathfrak{d}\right)}^{\beta }}\odot \left[{ℐ}_{{\frac{1}{\mathfrak{d}}}^{-}}^{\beta } \left(\tilde{\mathfrak{G}}\circ \psi \right)\left(\frac{1}{𝔃}\right)\oplus {ℐ}_{{\frac{1}{𝔃}}^{+}}^{\beta } \left(\tilde{\mathfrak{G}}\circ \psi \right)\left(\frac{1}{\mathfrak{d}}\right)\right]{\subseteq }_{\mathbb{F}}\frac{\tilde{\mathfrak{G}}\left(\mathfrak{d}\right)\oplus \tilde{\mathfrak{G}}\left(𝔃\right)}{2}.$$

(35)

where $\psi \left(\varsigma \right)=\frac{1}{\varsigma }$.

Proof. 

Let $\tilde{\mathfrak{G}}:\left[\mathfrak{d}, 𝔃\right]\to {₤}_I$ be $U\mathcal{D}$-harmonically convex $ℱ\mathcal{N}\mathcal{V}ℳ$. Then, by hypothesis, we have:

$$2\odot \tilde{\mathfrak{G}}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃}\right){\supseteq }_{\mathbb{F}}\tilde{\mathfrak{G}}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}\right)\oplus \tilde{\mathfrak{G}}\left(\frac{\mathfrak{d}𝔃}{\left(1-\mu \right)\mathfrak{d}+\mu 𝔃}\right).$$

Therefore, for each $\mathfrak{o}\in \left[0, 1\right]$, we have:

$$\begin{array}{c}2{\mathfrak{G}}_{*}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃}, \mathfrak{o}\right)\le {\mathfrak{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}, \mathfrak{o}\right)+{\mathfrak{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\left(1-\mu \right)\mathfrak{d}+\mu 𝔃}, \mathfrak{o}\right), \\ 2{\mathfrak{G}}^{*}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃}, \mathfrak{o}\right)\ge {\mathfrak{G}}^{*}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}, \mathfrak{o}\right)+{\mathfrak{G}}^{*}\left(\frac{\mathfrak{d}𝔃}{\left(1-\mu \right)\mathfrak{d}+\mu 𝔃},\mathfrak{o}\right).\end{array}$$

Consider $\tilde{\phi }\left(\varsigma \right)=\tilde{\mathfrak{G}}\left(\frac{1}{\varsigma }\right).$ By Theorem 5, we have $\tilde{\phi }\left(\varsigma \right)$ which is convex $ℱ\mathcal{N}\mathcal{V}ℳ$, then for each $\mathfrak{o}\in \left[0, 1\right]$, the above inequality, we have:

$$\begin{array}{c}2{\phi }_{*}\left(\frac{\mathfrak{d}+𝔃}{2\mathfrak{d}𝔃}, \mathfrak{o}\right)\le {\phi }_{*}\left(\frac{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}{\mathfrak{d}𝔃}, \mathfrak{o}\right)+{\phi }_{*}\left(\frac{\left(1-\mu \right)\mathfrak{d}+\mu 𝔃}{\mathfrak{d}𝔃}, \mathfrak{o}\right). \end{array}$$

Multiplying both sides by ${\mu }^{\beta -1}$ and integrating the obtained result with respect to $\mu$ over $\left(0,1\right)$, we have:

$$\begin{array}{c}2{{\displaystyle \int }}_0^1{\mu }^{\beta -1}{\phi }_{*}\left(\frac{\mathfrak{d}+𝔃}{2\mathfrak{d}𝔃}, \mathfrak{o}\right)d\mu \\ \le {{\displaystyle \int }}_0^1{\mu }^{\beta -1}{\phi }_{*}\left(\frac{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}{\mathfrak{d}𝔃},\mathfrak{o}\right)d\mu +{{\displaystyle \int }}_0^1{\mu }^{\beta -1}{\phi }_{*}\left(\frac{\left(1-\mu \right)\mathfrak{d}+\mu 𝔃}{\mathfrak{d}𝔃}, \mathfrak{o}\right)d\mu .\end{array}$$

Let $\varsigma =\frac{\left(1-\mu \right)\mathfrak{d}+\mu 𝔃}{\mathfrak{d}𝔃}$ and $𝓉=\frac{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}{\mathfrak{d}𝔃}.$ Then, we have:

$$\begin{gathered} \begin{array}{c}\frac{2}{\beta }{\phi }_{*}\left(\frac{\mathfrak{d}+𝔃}{2\mathfrak{d}𝔃}, \mathfrak{o}\right)\le {\left(\frac{\mathfrak{d}𝔃}{𝔃-\mathfrak{d}}\right)}^{\beta }{{\displaystyle \int }}_{\frac{1}{𝔃}}^{\frac{1}{\mathfrak{d}}}{\left(\frac{1}{\mathfrak{d}}-𝓉\right)}^{\beta -1}{\phi }_{*}\left(𝓉,\mathfrak{o}\right)d𝓉+{\left(\frac{\mathfrak{d}𝔃}{𝔃-\mathfrak{d}}\right)}^{\beta }{{\displaystyle \int }}_{\frac{1}{𝔃}}^{\frac{1}{\mathfrak{d}}}{\left(\varsigma -\frac{1}{𝔃}\right)}^{\beta -1}{\phi }_{*}\left(\varsigma ,\mathfrak{o}\right)d\varsigma \end{array} \\ \begin{array}{c}=\Gamma \left(\beta \right){\left(\frac{\mathfrak{d}𝔃}{𝔃-\mathfrak{d}}\right)}^{\beta }\left[{ℐ}_{{\left(\frac{1}{\mathfrak{d}}\right)}^{-}}^{\beta } {\phi }_{*}\left(\frac{1}{𝔃}, \mathfrak{o}\right)+{ℐ}_{{\left(\frac{1}{𝔃}\right)}^{+}}^{\beta } {\phi }_{*}\left(\frac{1}{\mathfrak{d}}, \mathfrak{o}\right)\right]. \end{array} \end{gathered}$$

Similarly, for ${\mathfrak{G}}^{*}\left(\varsigma ,\mathfrak{o}\right)$, we have:

$$\frac{2}{\beta }{\phi }^{*}\left(\frac{\mathfrak{d}+𝔃}{2\mathfrak{d}𝔃}, \mathfrak{o}\right)\ge \Gamma \left(\beta \right){\left(\frac{\mathfrak{d}𝔃}{𝔃-\mathfrak{d}}\right)}^{\beta }\left[{ℐ}_{{\left(\frac{1}{\mathfrak{d}}\right)}^{-}}^{\beta } {\phi }^{*}\left(\frac{1}{𝔃}, \mathfrak{o}\right)+{ℐ}_{{\left(\frac{1}{𝔃}\right)}^{+}}^{\beta } {\phi }^{*}\left(\frac{1}{\mathfrak{d}}, \mathfrak{o}\right)\right].$$

It follows that:

$$2\left[{\phi }_{*}\left(\frac{\mathfrak{d}+𝔃}{2\mathfrak{d}𝔃}, \mathfrak{o}\right), {\phi }^{*}\left(\frac{\mathfrak{d}+𝔃}{2\mathfrak{d}𝔃}, \mathfrak{o}\right)\right] {\supseteq }_I\Gamma \left(\beta +1\right){\left(\frac{\mathfrak{d}𝔃}{𝔃-\mathfrak{d}}\right)}^{\beta }\left[{ℐ}_{{\left(\frac{1}{\mathfrak{d}}\right)}^{-}}^{\beta } {\phi }_{*}\left(\frac{1}{𝔃}, \mathfrak{o}\right)+{ℐ}_{{\left(\frac{1}{𝔃}\right)}^{+}}^{\beta } {\phi }_{*}\left(\frac{1}{\mathfrak{d}}, \mathfrak{o}\right), {ℐ}_{{\left(\frac{1}{\mathfrak{d}}\right)}^{-}}^{\beta } {\phi }^{*}\left(\frac{1}{𝔃}, \mathfrak{o}\right)+{ℐ}_{{\left(\frac{1}{𝔃}\right)}^{+}}^{\beta } {\phi }^{*}\left(\frac{1}{\mathfrak{d}}, \mathfrak{o}\right)\right].$$

That is,

$$2\odot \tilde{\phi }\left(\frac{\mathfrak{d}+𝔃}{2\mathfrak{d}𝔃}\right){\supseteq }_{\mathbb{F}}\Gamma \left(\beta +1\right){\left(\frac{\mathfrak{d}𝔃}{𝔃-\mathfrak{d}}\right)}^{\beta }\odot \left[{ℐ}_{{\left(\frac{1}{\mathfrak{d}}\right)}^{-}}^{\beta }\tilde{\phi }\left(\frac{1}{𝔃}\right)\oplus {ℐ}_{{\left(\frac{1}{𝔃}\right)}^{+}}^{\beta }\tilde{\phi }\left(\frac{1}{\mathfrak{d}}\right)\right].$$

(36)

In a similar way as above, we have:

$$\Gamma \left(\beta \right){\left(\frac{\mathfrak{d}𝔃}{𝔃-\mathfrak{d}}\right)}^{\beta }\odot \left[{ℐ}_{{\left(\frac{1}{\mathfrak{d}}\right)}^{-}}^{\beta } \tilde{\phi }\left(\frac{1}{𝔃}\right)\oplus {ℐ}_{{\left(\frac{1}{𝔃}\right)}^{+}}^{\beta } \tilde{\phi }\left(\frac{1}{\mathfrak{d}}\right)\right]{\supseteq }_{\mathbb{F}}\frac{\tilde{\phi }\left(\frac{1}{\mathfrak{d}}\right)\oplus \tilde{\phi }\left(\frac{1}{𝔃}\right)}{\beta }.$$

(37)

Combining (36) and (37), we have:

$$\tilde{\phi }\left(\frac{\mathfrak{d}+𝔃}{2\mathfrak{d}𝔃}\right){\supseteq }_{\mathbb{F}}\frac{\Gamma \left(\beta +1\right) {\left(\frac{\mathfrak{d}𝔃}{𝔃-\mathfrak{d}}\right)}^{\beta }}{2}\odot \left[{ℐ}_{{\left(\frac{1}{\mathfrak{d}}\right)}^{-}}^{\beta } \tilde{\phi }\left(\frac{1}{𝔃}\right)\oplus {ℐ}_{{\left(\frac{1}{𝔃}\right)}^{+}}^{\beta } \tilde{\phi }\left(\frac{1}{\mathfrak{d}}\right)\right]{\supseteq }_{\mathbb{F}}\frac{\tilde{\phi }\left(\frac{1}{\mathfrak{d}}\right)\oplus \tilde{\phi }\left(\frac{1}{𝔃}\right)}{2}.$$

that is:

$$\tilde{\mathfrak{G}}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃}\right){\supseteq }_{\mathbb{F}}\frac{\Gamma \left(\beta +1\right)}{2{\left(𝔃-\mathfrak{d}\right)}^{\beta }}\odot \left[{ℐ}_{{\frac{1}{\mathfrak{d}}}^{-}}^{\beta }\left(\tilde{\mathfrak{G}}\circ \psi \right)\left(\frac{1}{𝔃}\right)\oplus {ℐ}_{{\frac{1}{𝔃}}^{+}}^{\beta } \left(\tilde{\mathfrak{G}}\circ \psi \right)\left(\frac{1}{\mathfrak{d}}\right)\right]{\supseteq }_{\mathbb{F}}\frac{\tilde{\mathfrak{G}}\left(\mathfrak{d}\right)\oplus \tilde{\mathfrak{G}}\left(𝔃\right)}{2}$$

Hence, the required result. □

Remark 4.

(1) If $\beta =1$, then inequality (34) reduces to the following inequality which is also new one:

$$\tilde{\mathfrak{G}}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃}\right){\supseteq }_{\mathbb{F}}\frac{\begin{array}{c} \\ dz\end{array}}{𝔃-\mathfrak{d}}\odot {{\displaystyle \int }}_{\mathfrak{d}}^{𝔃}\frac{\tilde{\mathfrak{G}}\left(\varsigma \right)}{{\varsigma }^2}d\varsigma {\supseteq }_{\mathbb{F}}\frac{\begin{array}{c} \\ \tilde{\mathfrak{G}}\left(\mathfrak{d}\right)\oplus \tilde{\mathfrak{G}}\left(𝔃\right)\end{array}}{2}.$$

(38)

(2)

If $\beta =1$, then inequality (34) reduces to the following inequality which is also new one:

$$\tilde{\mathfrak{G}}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃}\right){\supseteq }_{\mathbb{F}}\frac{\begin{array}{c} \\ dz\end{array}}{𝔃-\mathfrak{d}}\odot {{\displaystyle \int }}_{\mathfrak{d}}^{𝔃}\frac{\tilde{\mathfrak{G}}\left(\varsigma \right)}{{\varsigma }^2}d\varsigma {\supseteq }_{\mathbb{F}}\frac{\begin{array}{c} \\ \tilde{\mathfrak{G}}\left(\mathfrak{d}\right)\oplus \tilde{\mathfrak{G}}\left(𝔃\right)\end{array}}{2}.$$

(39)

(3)

If $\tilde{\mathfrak{G}}$ is lower $U\mathcal{D}$-harmonically convex $ℱ\mathcal{N}\mathcal{V}ℳ$, then inequality (34) reduces to the following inequality, see [64]:

$$\tilde{\mathfrak{G}}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃}\right){\le }_{\mathbb{F}}\frac{\Gamma \left(\beta +1\right)}{2{\left(𝔃-\mathfrak{d}\right)}^{\beta }}\left[{ℐ}_{{\frac{1}{\mathfrak{d}}}^{-}}^{\beta }\left(\tilde{\mathfrak{G}}\circ \psi \right)\left(\frac{1}{𝔃}\right)+{ℐ}_{{\frac{1}{𝔃}}^{+}}^{\beta } \left(\tilde{\mathfrak{G}}\circ \psi \right)\left(\frac{1}{\mathfrak{d}}\right)\right]{\le }_{\mathbb{F}}\frac{\tilde{\mathfrak{G}}\left(\mathfrak{d}\right)+\tilde{\mathfrak{G}}\left(𝔃\right)}{2}.$$

(40)

(4)

If $\tilde{\mathfrak{G}}$ is lower $U\mathcal{D}$-harmonically convex $ℱ\mathcal{N}\mathcal{V}ℳ$ and $\beta =1$, then inequality (34) reduces to the following inequality, see [64]:

$$\tilde{\mathfrak{G}}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃}\right){\le }_{\mathbb{F}}\frac{\begin{array}{c} \\ dz\end{array}}{𝔃-\mathfrak{d}}\odot {{\displaystyle \int }}_{\mathfrak{d}}^{𝔃}\frac{\tilde{\mathfrak{G}}\left(\varsigma \right)}{{\varsigma }^2}d\varsigma {\le }_{\mathbb{F}}\frac{\begin{array}{c} \\ \tilde{\mathfrak{G}}\left(\mathfrak{d}\right)\oplus \tilde{\mathfrak{G}}\left(𝔃\right)\end{array}}{2}.$$

(41)

(5)

If $\tilde{\mathfrak{G}}$ is lower $U\mathcal{D}$-harmonically convex $ℱ\mathcal{N}\mathcal{V}ℳ$ and $\beta =1$, then inequality (34) reduces to the following inequality, see [64]:

$$\tilde{\mathfrak{G}}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃}\right){\le }_{\mathbb{F}}\frac{\begin{array}{c} \\ dz\end{array}}{𝔃-\mathfrak{d}}\odot {{\displaystyle \int }}_{\mathfrak{d}}^{𝔃}\frac{\tilde{\mathfrak{G}}\left(\varsigma \right)}{{\varsigma }^2}d\varsigma {\le }_{\mathbb{F}}\frac{\begin{array}{c} \\ \tilde{\mathfrak{G}}\left(\mathfrak{d}\right)\oplus \tilde{\mathfrak{G}}\left(𝔃\right)\end{array}}{2}.$$

(42)

(6)

If ${\mathfrak{G}}_{*}\left(\varsigma ,\mathfrak{o}\right)={\mathfrak{G}}^{*}\left(\varsigma ,\mathfrak{o}\right)$ with $\mathfrak{o}=1$ then, we obtain classical fractional 𝐻–𝐻 inequality for harmonically convex function which is given in [66]:

$$\mathfrak{G}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃}\right)\le \frac{\Gamma \left(\beta +1\right)}{2{\left(𝔃-\mathfrak{d}\right)}^{\beta }}\left[{ℐ}_{{\frac{1}{\mathfrak{d}}}^{-}}^{\beta }\left(\mathfrak{G}\circ \psi \right)\left(\frac{1}{𝔃}\right)+{ℐ}_{{\frac{1}{𝔃}}^{+}}^{\beta } \left(\mathfrak{G}\circ \psi \right)\left(\frac{1}{\mathfrak{d}}\right)\right]\le \frac{\mathfrak{G}\left(\mathfrak{d}\right)+\mathfrak{G}\left(𝔃\right)}{2}.$$

(43)

(7)

If ${\mathfrak{G}}_{*}\left(\varsigma ,\mathfrak{o}\right)={\mathfrak{G}}^{*}\left(\varsigma ,\mathfrak{o}\right)$ with $\mathfrak{o}=1$ and $\beta =1$, then we obtain classical 𝐻–𝐻 inequality for harmonically convex function which is given in [65].

$$\mathfrak{G}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃}\right)\le \frac{\mathfrak{d}𝔃}{𝔃-\mathfrak{d}} {{\displaystyle \int }}_{\mathfrak{d}}^{𝔃}\frac{\mathfrak{G}\left(\varsigma \right)}{{\varsigma }^2}d\varsigma \le \frac{\mathfrak{G}\left(\mathfrak{d}\right)+\mathfrak{G}\left(𝔃\right)}{2}.$$

(44)

Example 2.

We consider $\beta =1$, and the $ℱ\mathcal{N}\mathcal{V}ℳ$s $\tilde{\mathfrak{G}}:\left[\frac{1}{2}, 1\right]\to {\mathbb{F}}_0,$ as in Example 1. Then, for each $\mathsf{ɷ}\in \left[0, 1\right],$ we have ${\mathfrak{G}}_{\mathsf{ɷ}}\left(\varsigma \right)=\left[\left(1-\mathsf{ɷ}\right){\varsigma }^2+\mathsf{ɷ},\left(1-\mathsf{ɷ}\right)\left(5-e^{\varsigma }\right)+\mathsf{ɷ}\right]$ is UD-harmonically convex $ℱ\mathcal{N}\mathcal{V}ℳ$. Since ${\mathfrak{G}}_{*}\left(\varsigma ,\mathsf{ɷ}\right)=\left(1-\mathsf{ɷ}\right){\varsigma }^2+\mathsf{ɷ},$ ${\mathfrak{G}}^{*}\left(\varsigma , \mathsf{ɷ}\right)=\left(1-\mathsf{ɷ}\right)\left(5-e^{\varsigma }\right)+\mathsf{ɷ}$. We now compute the following:

$${\mathfrak{G}}_{*}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃}, \mathsf{ɷ}\right)\le \frac{\Gamma \left(\beta +1\right)}{2{\left(𝔃-\mathfrak{d}\right)}^{\beta }}\left[{ℐ}_{{\frac{1}{\mathfrak{d}}}^{-}}^{\beta }\left(\left({\mathfrak{G}}_{*}\circ \psi \right)\left(\frac{1}{𝔃}\right), \mathsf{ɷ}\right)+{ℐ}_{{\frac{1}{𝔃}}^{+}}^{\beta }\left(\left({\mathfrak{G}}_{*}\circ \psi \right)\left(\frac{1}{\mathfrak{d}}\right), \mathsf{ɷ}\right)\right]\le \frac{{\mathfrak{G}}_{*}\left(\mathfrak{d}, \mathsf{ɷ}\right)+{\mathfrak{G}}_{*}\left(𝔃, \mathsf{ɷ}\right)}{2}.$$

$${\mathfrak{G}}_{*}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃}, \mathsf{ɷ}\right)={\mathfrak{G}}_{*}\left(\frac{2}{3}, \mathsf{ɷ}\right)=\frac{4}{9}\left(1-\mathsf{ɷ}\right)+\mathsf{ɷ},$$

$$\frac{\Gamma \left(\beta +1\right)}{2{\left(𝔃-\mathfrak{d}\right)}^{\beta }}\left[{ℐ}_{{\frac{1}{\mathfrak{d}}}^{-}}^{\beta }\left(\left({\mathfrak{G}}_{*}\circ \psi \right)\left(\frac{1}{𝔃}\right), \mathsf{ɷ}\right)+{ℐ}_{{\frac{1}{𝔃}}^{+}}^{\beta }\left(\left({\mathfrak{G}}_{*}\circ \psi \right)\left(\frac{1}{\mathfrak{d}}\right), \mathsf{ɷ}\right)\right]=\frac{1}{2}\left(1+\mathsf{ɷ}\right),$$

$$\frac{{\mathfrak{G}}_{*}\left(\mathfrak{d}, \mathsf{ɷ}\right)+{\mathfrak{G}}_{*}\left(𝔃, \mathsf{ɷ}\right)}{2}=\frac{5+3\mathsf{ɷ}}{4},$$

for all $\mathsf{ɷ}\in \left[0, 1\right].$ That means:

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

Similarly, it can be easily show that:

$${\mathfrak{G}}^{*}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃}, \mathsf{ɷ}\right)\ge \frac{\Gamma \left(\beta +1\right)}{2{\left(𝔃-\mathfrak{d}\right)}^{\beta }}\left[{ℐ}_{{\frac{1}{\mathfrak{d}}}^{-}}^{\beta }\left(\left({\mathfrak{G}}^{*}\circ \psi \right)\left(\frac{1}{𝔃}\right), \mathsf{ɷ}\right)+{ℐ}_{{\frac{1}{𝔃}}^{+}}^{\beta }\left(\left({\mathfrak{G}}^{*}\circ \psi \right)\left(\frac{1}{\mathfrak{d}}\right), \mathsf{ɷ}\right)\right]\ge \frac{{\mathfrak{G}}^{*}\left(\mathfrak{d}, \mathsf{ɷ}\right)+{\mathfrak{G}}^{*}\left(𝔃, \mathsf{ɷ}\right)}{2}$$

for all $\mathsf{ɷ}\in \left[0, 1\right],$ such that:

$${\mathfrak{G}}^{*}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃}, \mathsf{ɷ}\right)={\mathfrak{G}}_{*}\left(\frac{2}{3}, \mathsf{ɷ}\right)=\left(1-\mathsf{ɷ}\right)\left(5-e^{\frac{2}{3}}\right)+\mathsf{ɷ},$$

$$\frac{\Gamma \left(\beta +1\right)}{2{\left(𝔃-\mathfrak{d}\right)}^{\beta }}\left[{ℐ}_{{\frac{1}{\mathfrak{d}}}^{-}}^{\beta }\left(\left({\mathfrak{G}}^{*}\circ \psi \right)\left(\frac{1}{𝔃}\right), \mathsf{ɷ}\right)+{ℐ}_{{\frac{1}{𝔃}}^{+}}^{\beta }\left(\left({\mathfrak{G}}^{*}\circ \psi \right)\left(\frac{1}{\mathfrak{d}}\right), \mathsf{ɷ}\right)\right]\approx 3-2\mathsf{ɷ},$$

$$\frac{{\mathfrak{G}}^{*}\left(\mathfrak{d}, \mathsf{ɷ}\right)+{\mathfrak{G}}^{*}\left(𝔃, \mathsf{ɷ}\right)}{2}=\frac{\left(5-e\right)\left(1-\mathsf{ɷ}\right)+\left(5-e^{\frac{1}{2}}\right)\left(1-\mathsf{ɷ}\right)+2\mathsf{ɷ}}{2}.$$

From which, we have:

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

that is:

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

for all $\mathsf{ɷ}\in \left[0, 1\right].$

Hence,

$$\tilde{\mathfrak{G}}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃}\right){\supseteq }_{\mathbb{F}}\frac{\Gamma \left(\beta +1\right)}{2{\left(𝔃-\mathfrak{d}\right)}^{\beta }}\odot \left[{ℐ}_{{\frac{1}{\mathfrak{d}}}^{-}}^{\beta }\left(\tilde{\mathfrak{G}}\circ \psi \right)\left(\frac{1}{𝔃}\right)\oplus {ℐ}_{{\frac{1}{𝔃}}^{+}}^{\beta }\left(\tilde{\mathfrak{G}}\circ \psi \right)\left(\frac{1}{\mathfrak{d}}\right)\right]{\supseteq }_{\mathbb{F}}\frac{\tilde{\mathfrak{G}}\left(\mathfrak{d}\right)\oplus \tilde{\mathfrak{G}}\left(𝔃\right)}{2}.$$

In next outcome, we will present new fuzzy fractional $H-H$ Fejér inequality for $U\mathcal{D}$-harmonically convex $ℱ\mathcal{N}\mathcal{V}ℳ$.

Theorem 7.

Let $\tilde{\mathfrak{G}}:\left[\mathfrak{d}, 𝔃\right]\to {₤}_I$ be a $U\mathcal{D}$-harmonically convex $ℱ\mathcal{N}\mathcal{V}ℳ$ with $\mathfrak{d}<𝔃$, whose $\mathfrak{o}$-cuts define the family of $I\mathcal{V}ℳ$s ${\mathfrak{G}}_{\mathfrak{o}}:\left[\mathfrak{d}, 𝔃\right]\subset ℝ\to {\mathcal{K}}_I^{+}$ are provided by ${\mathfrak{G}}_{\mathfrak{o}}\left(\varsigma \right)=\left[{\mathfrak{G}}_{*}\left(\varsigma ,\mathfrak{o}\right), {\mathfrak{G}}^{*}\left(\varsigma ,\mathfrak{o}\right)\right]$ for all $\varsigma \in \left[\mathfrak{d}, 𝔃\right]$, $\mathfrak{o}\in \left[0, 1\right]$. If $\tilde{\mathfrak{G}}\in L\left(\left[\mathfrak{d}, 𝔃\right],{₤}_I\right)$ and $\mathfrak{V}:\left[\mathfrak{d}, 𝔃\right]\to ℝ, \mathfrak{V}\left(\frac{1}{\frac{1}{\mathfrak{d}}+\frac{1}{𝔃}-\frac{1}{\varsigma }}\right)=\mathfrak{V}\left(\varsigma \right)\ge 0,$ then:

$$\begin{array}{cc}\tilde{\mathfrak{G}}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃}\right)\odot \left[{ℐ}_{{\left(\frac{1}{𝔃}\right)}^{+}}^{\beta }\left(\mathfrak{V}\circ \psi \right)\left(\frac{1}{\mathfrak{d}}\right)+{ℐ}_{{\left(\frac{1}{\mathfrak{d}}\right)}^{-}}^{\beta }\left(\mathfrak{V}\circ \psi \right)\left(\frac{1}{𝔃}\right)\right]\hfill & {\supseteq }_{\mathbb{F}}\left[{ℐ}_{{\left(\frac{1}{𝔃}\right)}^{+}}^{\beta }\left(\tilde{\mathfrak{G}}\mathfrak{V}\circ \psi \right)\left(\frac{1}{\mathfrak{d}}\right)\oplus {ℐ}_{{\left(\frac{1}{\mathfrak{d}}\right)}^{-}}^{\beta }\left(\tilde{\mathfrak{G}}\mathfrak{V}\circ \psi \right)\left(\frac{1}{𝔃}\right)\right]\hfill \\ & {\supseteq }_{\mathbb{F}}\frac{\tilde{\mathfrak{G}}\left(\mathfrak{d}\right)\oplus \tilde{\mathfrak{G}}\left(𝔃\right)}{2}\odot \left[{ℐ}_{{\frac{1}{𝔃}}^{+}}^{\beta } \left(\mathfrak{V}\circ \psi \right)\left(\frac{1}{\mathfrak{d}}\right)+{ℐ}_{{\frac{1}{\mathfrak{d}}}^{-}}^{\beta }\left(\mathfrak{V}\circ \psi \right)\left(\frac{1}{𝔃}\right)\right].\hfill \end{array}$$

(45)

If $\tilde{\mathfrak{G}}$ is concave $ℱ\mathcal{N}\mathcal{V}ℳ$, then inequality (45) is reversed.

Proof. 

Since $\tilde{\mathfrak{G}}$ is a $U\mathcal{D}$-harmonically convex $ℱ\mathcal{N}\mathcal{V}ℳ$, then for $\mathfrak{o}\in \left[0, 1\right],$ we have:

$$\begin{array}{c} \\ {\mathfrak{G}}_{*}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃}, \mathfrak{o}\right) \\ \le \frac{1}{2}\left({\mathfrak{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}, \mathfrak{o}\right)+{\mathfrak{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\left(1-\mu \right)\mathfrak{d}+\mu 𝔃}, \mathfrak{o}\right)\right).\end{array}$$

(46)

Multiplying both sides by (46) by ${\mu }^{\beta -1}\mathfrak{V}\left(\frac{\mathfrak{d}𝔃}{\left(1-\mu \right)\mathfrak{d}+\mu 𝔃}\right)$ and then integrating the resultant with respect to $\mu$ over $\left[0, 1\right],$ we obtain:

$$\begin{array}{c}{\mathfrak{G}}_{*}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃}, \mathfrak{o}\right){{\displaystyle \int }}_0^1{\mu }^{\beta -1}\mathfrak{V}\left(\frac{\mathfrak{d}𝔃}{\left(1-\mu \right)\mathfrak{d}+\mu 𝔃}\right)d\mu \hfill \\ \le \frac{1}{2}\left(\begin{array}{c}{{\displaystyle \int }}_0^1{\mu }^{\beta -1}{\mathfrak{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}, \mathfrak{o}\right)\mathfrak{V}\left(\frac{\mathfrak{d}𝔃}{\left(1-\mu \right)\mathfrak{d}+\mu 𝔃}\right)d\mu \\ +{{\displaystyle \int }}_0^1{\mu }^{\beta -1}{\mathfrak{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\left(1-\mu \right)\mathfrak{d}+\mu 𝔃}, \mathfrak{o}\right)\mathfrak{V}\left(\frac{\mathfrak{d}𝔃}{\left(1-\mu \right)\mathfrak{d}+\mu 𝔃}\right)d\mu \end{array}\right).\hfill \end{array}$$

(47)

Let $\varsigma =\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}$. Then, we have:

$$\begin{gathered} 2{\left(\frac{\mathfrak{d}𝔃}{𝔃-\mathfrak{d}}\right)}^{\beta }{\mathfrak{G}}_{*}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃}, \mathfrak{o}\right)\underset{\frac{1}{𝔃}}{\overset{\frac{1}{\mathfrak{d}}}{{\displaystyle \int }}}{\left(\varsigma -\frac{1}{𝔃}\right)}^{\beta -1}\mathfrak{V}\left(\frac{1}{\varsigma },\mathfrak{o}\right)d\varsigma \\ \le {\left(\frac{\mathfrak{d}𝔃}{𝔃-\mathfrak{d}}\right)}^{\beta } {{\displaystyle \int }}_{\frac{1}{𝔃}}^{\frac{1}{\mathfrak{d}}}{\left(\varsigma -\frac{1}{𝔃}\right)}^{\beta -1}{\mathfrak{G}}_{*}\left(\frac{1}{\frac{1}{\mathfrak{d}}+\frac{1}{𝔃}-\frac{1}{\varsigma }},\mathfrak{o}\right)\mathfrak{V}\left(\frac{1}{\varsigma }\right)d\varsigma \\ +{\left(\frac{\mathfrak{d}𝔃}{𝔃-\mathfrak{d}}\right)}^{\beta } {{\displaystyle \int }}_{\mathfrak{d}}^{\frac{1}{\mathfrak{d}}}{\left(\varsigma -\frac{1}{𝔃}\right)}^{\beta -1}{\mathfrak{G}}_{*}\left(\frac{1}{\varsigma },\mathfrak{o}\right)\mathfrak{V}\left(\frac{1}{\varsigma }\right)d\varsigma \\ ={\left(\frac{\mathfrak{d}𝔃}{𝔃-\mathfrak{d}}\right)}^{\beta } {{\displaystyle \int }}_{\frac{1}{𝔃}}^{\frac{1}{\mathfrak{d}}}{\left(\frac{1}{\mathfrak{d}}-\varsigma \right)}^{\beta -1}{\mathfrak{G}}_{*}\left(\varsigma ,\mathfrak{o}\right)\mathfrak{V}\left(\frac{1}{\frac{1}{\mathfrak{d}}+\frac{1}{𝔃}-\frac{1}{\varsigma }}\right)d\varsigma \\ +{\left(\frac{\mathfrak{d}𝔃}{𝔃-\mathfrak{d}}\right)}^{\beta } {{\displaystyle \int }}_{\frac{1}{𝔃}}^{\frac{1}{\mathfrak{d}}}{\left(\varsigma -\frac{1}{𝔃}\right)}^{\beta -1}{\mathfrak{G}}_{*}\left(\frac{1}{\varsigma },\mathfrak{o}\right)\mathfrak{V}\left(\frac{1}{\varsigma }\right)d\varsigma \\ =\Gamma \left(\beta \right){\left(\frac{\mathfrak{d}𝔃}{𝔃-\mathfrak{d}}\right)}^{\beta }\left[{ℐ}_{{\left(\frac{1}{𝔃}\right)}^{+}}^{\beta } {\mathfrak{G}}_{*}\mathfrak{V}\left(\frac{1}{\mathfrak{d}}\right)+{ℐ}_{{\left(\frac{1}{\mathfrak{d}}\right)}^{-}}^{\beta } {\mathfrak{G}}_{*}\mathfrak{V}\left(\frac{1}{𝔃}\right)\right], \end{gathered}$$

(48)

Similarly, for ${\mathfrak{G}}^{*}\left(\varsigma ,\mathfrak{o}\right)$, we have:

$$2{\left(\frac{\mathfrak{d}𝔃}{𝔃-\mathfrak{d}}\right)}^{\beta }{\mathfrak{G}}^{*}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃}, \mathfrak{o}\right){{\displaystyle \int }}_{\frac{1}{𝔃}}^{\frac{1}{\mathfrak{d}}}{\left(\varsigma -\frac{1}{𝔃}\right)}^{\beta -1}\mathfrak{V}\left(\frac{1}{\varsigma },\mathfrak{o}\right)d\varsigma \ge \Gamma \left(\beta \right){\left(\frac{\mathfrak{d}𝔃}{𝔃-\mathfrak{d}}\right)}^{\beta }\left[{ℐ}_{{\left(\frac{1}{𝔃}\right)}^{+}}^{\beta } {\mathfrak{G}}^{*}\mathfrak{V}\left(\frac{1}{\mathfrak{d}}\right)+{ℐ}_{{\left(\frac{1}{\mathfrak{d}}\right)}^{-}}^{\beta } {\mathfrak{G}}^{*}\mathfrak{V}\left(\frac{1}{𝔃}\right)\right].$$

(49)

From (48) and (49), we have:

$$\begin{array}{c}\Gamma \left(\beta \right){\left(\frac{\mathfrak{d}𝔃}{𝔃-\mathfrak{d}}\right)}^{\beta }\left[{\mathfrak{G}}_{*}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃}, \mathfrak{o}\right), {\mathfrak{G}}^{*}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃}, \mathfrak{o}\right)\right].\left[{ℐ}_{{\left(\frac{1}{𝔃}\right)}^{+}}^{\beta } \mathfrak{V}\left(\frac{1}{\mathfrak{d}}\right)+{ℐ}_{{\left(\frac{1}{\mathfrak{d}}\right)}^{-}}^{\beta } \mathfrak{V}\left(\frac{1}{𝔃}\right)\right]\hfill \\ {\supseteq }_I\Gamma \left(\beta \right){\left(\frac{\mathfrak{d}𝔃}{𝔃-\mathfrak{d}}\right)}^{\beta }\left[\begin{array}{c} {ℐ}_{{\left(\frac{1}{𝔃}\right)}^{+}}^{\beta } {\mathfrak{G}}_{*}\mathfrak{V}\left(\frac{1}{\mathfrak{d}}\right)+{ℐ}_{{\left(\frac{1}{\mathfrak{d}}\right)}^{-}}^{\beta } {\mathfrak{G}}_{*}\mathfrak{V}\left(\frac{1}{𝔃}\right), {ℐ}_{{\left(\frac{1}{𝔃}\right)}^{+}}^{\beta } {\mathfrak{G}}^{*}\mathfrak{V}\left(\frac{1}{\mathfrak{d}}\right)+{ℐ}_{{\left(\frac{1}{\mathfrak{d}}\right)}^{-}}^{\beta } {\mathfrak{G}}^{*}\mathfrak{V}\left(\frac{1}{𝔃}\right)\end{array}\right],\hfill \end{array}$$

that is:

$$\begin{gathered} \tilde{\mathfrak{G}}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃}\right)\odot \left[{ℐ}_{{\left(\frac{1}{𝔃}\right)}^{+}}^{\beta } \left(\mathfrak{V}\circ \psi \right)\left(\frac{1}{\mathfrak{d}}\right)\oplus {ℐ}_{{\left(\frac{1}{\mathfrak{d}}\right)}^{-}}^{\beta }\left(\mathfrak{V}\circ \psi \right)\left(\frac{1}{𝔃}\right)\right] \\ {\supseteq }_{\mathbb{F}}\left[{ℐ}_{{\left(\frac{1}{𝔃}\right)}^{+}}^{\beta } \left(\tilde{\mathfrak{G}}\mathfrak{V}\circ \psi \right)\left(\frac{1}{\mathfrak{d}}\right)\oplus {ℐ}_{{\left(\frac{1}{\mathfrak{d}}\right)}^{-}}^{\beta } \left(\tilde{\mathfrak{G}}\mathfrak{V}\circ \psi \right)\left(\frac{1}{𝔃}\right)\right]. \end{gathered}$$

(50)

Similarly, if $\tilde{\mathfrak{G}}$ is a $U\mathcal{D}$-harmonically convex $ℱ\mathcal{N}\mathcal{V}ℳ$, and ${\mu }^{\beta -1}\mathfrak{V}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}\right)\ge 0$, then, for each $\mathfrak{o}\in [0,1]$ we have:

$$\begin{array}{c} \\ {\mu }^{\beta -1}{\mathfrak{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}, \mathfrak{o}\right)\mathfrak{V}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}\right) \\ \le {\mu }^{\beta -1}\left(\left(1-\mu \right){\mathfrak{G}}_{*}\left(\mathfrak{d}, \mathfrak{o}\right)+\mu {\mathfrak{G}}_{*}\left(𝔃, \mathfrak{o}\right)\right)\mathfrak{V}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}\right),\end{array}$$

(51)

and

$$\begin{array}{c} \\ {\mu }^{\beta -1}{\mathfrak{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\left(1-\mu \right)\mathfrak{d}+\mu 𝔃}, \mathfrak{o}\right)\mathfrak{V}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}\right) \\ \le {\mu }^{\beta -1}\left(\mu {\mathfrak{G}}_{*}\left(\mathfrak{d}, \mathfrak{o}\right)+\left(1-\mu \right){\mathfrak{G}}_{*}\left(𝔃, \mathfrak{o}\right)\right)\mathfrak{V}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}\right).\end{array}$$

(52)

After adding (51) and (52), and integrating the resultant over $\left[0, 1\right],$ we obtain:

$$\begin{array}{c} {{\displaystyle \int }}_0^1{\mu }^{\beta -1}{\mathfrak{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}, \mathfrak{o}\right)\mathfrak{V}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}\right)d\mu \\ +{{\displaystyle \int }}_0^1{\mu }^{\beta -1}{\mathfrak{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\left(1-\mu \right)\mathfrak{d}+\mu 𝔃}, \mathfrak{o}\right)\mathfrak{V}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}\right)d\mu \\ \le {{\displaystyle \int }}_0^1\left[\begin{array}{c}{\mu }^{\beta -1}{\mathfrak{G}}_{*}\left(\mathfrak{d}, \mathfrak{o}\right)\left\{\mu +\left(1-\mu \right)\right\}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}\right)\\ +{\mu }^{\beta -1}{\mathfrak{G}}_{*}\left(𝔃, \mathfrak{o}\right)\left\{\left(1-\mu \right)+\mu \right\}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}\right)\end{array}\right]d\mu ,\\ ={\mathfrak{G}}_{*}\left(\mathfrak{d}, \mathfrak{o}\right){{\displaystyle \int }}_0^1\begin{array}{c}{\mu }^{\beta -1}V\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}\right)\end{array}d\mu \\ +{\mathfrak{G}}_{*}\left(𝔃, \mathfrak{o}\right){{\displaystyle \int }}_0^1\begin{array}{c}{\mu }^{\beta -1}V\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}\right)\end{array}d\mu .\end{array}$$

Similarly, for ${\mathfrak{G}}^{*}\left(\varsigma ,\mathfrak{o}\right)$, we have:

$$\begin{array}{c} {{\displaystyle \int }}_0^1{\mu }^{\beta -1}{\mathfrak{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}, \mathfrak{o}\right)\mathfrak{V}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}\right)d\mu \\ +{{\displaystyle \int }}_0^1{\mu }^{\beta -1}{\mathfrak{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\left(1-\mu \right)\mathfrak{d}+\mu 𝔃}, \mathfrak{o}\right)\mathfrak{V}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}\right)d\mu \\ \ge {\mathfrak{G}}_{*}\left(\mathfrak{d}, \mathfrak{o}\right){{\displaystyle \int }}_0^1\begin{array}{c}{\mu }^{\beta -1}V\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}\right)\end{array}d\mu \\ +{\mathfrak{G}}_{*}\left(𝔃, \mathfrak{o}\right){{\displaystyle \int }}_0^1\begin{array}{c}{\mu }^{\beta -1}V\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}\right)\end{array}d\mu .\end{array}$$

From which, we have:

$$\begin{array}{cc}\Gamma \left(\beta \right){\left(\frac{\mathfrak{d}𝔃}{𝔃-\mathfrak{d}}\right)}^{\beta }\hfill & \left[{ℐ}_{{\frac{1}{𝔃}}^{+}}^{\beta } {\mathfrak{G}}_{\mathfrak{o}}\mathfrak{V}\circ \psi \left(𝔃\right)+{ℐ}_{{\left(\frac{1}{\mathfrak{d}}\right)}^{-}}^{\beta } {\mathfrak{G}}_{\mathfrak{o}}\mathfrak{V}\circ \psi \left(\frac{1}{𝔃}\right)\right]\hfill \\ & {\supseteq }_I\Gamma \left(\beta \right){\left(\frac{\mathfrak{d}𝔃}{𝔃-\mathfrak{d}}\right)}^{\beta }\frac{{\mathfrak{G}}_{\mathfrak{o}}\left(\mathfrak{d}\right)+{\mathfrak{G}}_{\mathfrak{o}}\left(𝔃\right)}{2}\left[{ℐ}_{{\frac{1}{𝔃}}^{+}}^{\beta } \left(\mathfrak{V}\circ \psi \right)\left(\frac{1}{\mathfrak{d}}\right)+{ℐ}_{\left(\frac{1}{\mathfrak{d}}\right)}^{\beta }\left(\mathfrak{V}\circ \psi \right)\left(\frac{1}{𝔃}\right)\right],\hfill \end{array}$$

that is:

$$\left[{ℐ}_{{\left(\frac{1}{𝔃}\right)}^{+}}^{\beta }\tilde{\mathfrak{G}}\mathfrak{V}\circ \psi \left(\frac{1}{\mathfrak{d}}\right)\oplus {ℐ}_{{\left(\frac{1}{\mathfrak{d}}\right)}^{-}}^{\beta }\tilde{\mathfrak{G}}\mathfrak{V}\circ \psi \left(\frac{1}{𝔃}\right)\right]{\supseteq }_{\mathbb{F}}\frac{\tilde{\mathfrak{G}}\left(\mathfrak{d}\right)\oplus \tilde{\mathfrak{G}}\left(𝔃\right)}{2}\left[{ℐ}_{{\left(\frac{1}{𝔃}\right)}^{+}}^{\beta }\left(\mathfrak{V}\circ \psi \right)\left(\frac{1}{\mathfrak{d}}\right)+{ℐ}_{{\left(\frac{1}{\mathfrak{d}}\right)}^{-}}^{\beta }\left(\mathfrak{V}\circ \psi \right)\left(\frac{1}{𝔃}\right)\right].$$

(53)

By combining (52) and (53), we obtain the required inequality (45). □

Remark 5.

(1) Let $\beta =1$. Then, from Theorem 7, we acquire 𝐻–𝐻 inequality for $U\mathcal{D}$-harmonically convex $ℱ\mathcal{N}\mathcal{V}ℳ$, which is also a new one:

$$\tilde{\mathfrak{G}}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃}\right)\odot {{\displaystyle \int }}_{\mathfrak{d}}^{𝔃}\frac{\mathfrak{V}\left(\varsigma \right)}{{\varsigma }^2}d\varsigma {\supseteq }_{\mathbb{F}} {{\displaystyle \int }}_{\mathfrak{d}}^{𝔃}\frac{\tilde{\mathfrak{G}}\left(\varsigma \right)}{{\varsigma }^2}\mathfrak{V}\left(\varsigma \right)d\varsigma {\supseteq }_{\mathbb{F}}\frac{\tilde{\mathfrak{G}}\left(\mathfrak{d}\right) \oplus \tilde{\mathfrak{G}}\left(𝔃\right)}{2}\odot {{\displaystyle \int }}_{\mathfrak{d}}^{𝔃}\frac{\mathfrak{V}\left(\varsigma \right)}{{\varsigma }^2}d\varsigma$$

(54)

(2)

Let $\mathfrak{V}\left(\varsigma \right)=1$. Then, from Theorem 7, we achieve inequality (34)

(3)

Let $\mathfrak{V}\left(\varsigma \right)=1$ and $\beta =1$, then from Theorem 7, we obtain 𝐻–𝐻 inequality for $U\mathcal{D}$-harmonically convex $ℱ\mathcal{N}\mathcal{V}ℳ$:

$$\tilde{\mathfrak{G}}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃}\right){\supseteq }_{\mathbb{F}}\frac{\mathfrak{d}𝔃}{𝔃-\mathfrak{d}}\odot {{\displaystyle \int }}_{\mathfrak{d}}^{𝔃}\frac{\tilde{\mathfrak{G}}\left(\varsigma \right)}{{\varsigma }^2}d\varsigma {\supseteq }_{\mathbb{F}}\frac{\tilde{\mathfrak{G}}\left(\mathfrak{d}\right)\oplus \tilde{\mathfrak{G}}\left(𝔃\right)}{2}$$

(55)

(4)

Let $\tilde{\mathfrak{G}}$ be lower $U\mathcal{D}$-harmonically convex $ℱ\mathcal{N}\mathcal{V}ℳ$. Then, we obtain the following inequality, see [64]:

$$\begin{gathered} \tilde{\mathfrak{G}}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃}\right)\odot \left[{ℐ}_{{\left(\frac{1}{𝔃}\right)}^{+}}^{\beta } \left(\mathfrak{V}\circ \psi \right)\left(\frac{1}{\mathfrak{d}}\right)+{ℐ}_{{\left(\frac{1}{\mathfrak{d}}\right)}^{-}}^{\beta }\left(\mathfrak{V}\circ \psi \right)\left(\frac{1}{𝔃}\right)\right]{\le }_{\mathbb{F}}\left[{ℐ}_{{\left(\frac{1}{𝔃}\right)}^{+}}^{\beta } \left(\tilde{\mathfrak{G}}\mathfrak{V}\circ \psi \right)\left(\frac{1}{\mathfrak{d}}\right)\oplus {ℐ}_{{\left(\frac{1}{\mathfrak{d}}\right)}^{-}}^{\beta } \left(\tilde{\mathfrak{G}}\mathfrak{V}\circ \psi \right)\left(\frac{1}{𝔃}\right)\right] \\ {\le }_{\mathbb{F}}\frac{\tilde{\mathfrak{G}}\left(\mathfrak{d}\right)\oplus \tilde{\mathfrak{G}}\left(𝔃\right)}{2}\odot \left[{ℐ}_{{\frac{1}{𝔃}}^{+}}^{\beta } \left(\mathfrak{V}\circ \psi \right)\left(\frac{1}{\mathfrak{d}}\right)+{ℐ}_{{\frac{1}{\mathfrak{d}}}^{-}}^{\beta }\left(\mathfrak{V}\circ \psi \right)\left(\frac{1}{𝔃}\right)\right]. \end{gathered}$$

(56)

(5)

Let lower $U\mathcal{D}$-harmonically convex $ℱ\mathcal{N}\mathcal{V}ℳ$ and $\beta =1$. Then, from Theorem 7, we acquire 𝐻–𝐻 inequality for harmonically convex $ℱ\mathcal{N}\mathcal{V}ℳ$, see [64]:

$$\tilde{\mathfrak{G}}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃}\right)\odot {{\displaystyle \int }}_{\mathfrak{d}}^{𝔃}\frac{\mathfrak{V}\left(\varsigma \right)}{{\varsigma }^2}d\varsigma {\le }_{\mathbb{F}} {{\displaystyle \int }}_{\mathfrak{d}}^{𝔃}\frac{\tilde{\mathfrak{G}}\left(\varsigma \right)}{{\varsigma }^2}\mathfrak{V}\left(\varsigma \right)d\varsigma {\le }_{\mathbb{F}}\frac{\tilde{\mathfrak{G}}\left(\mathfrak{d}\right)\oplus \tilde{\mathfrak{G}}\left(𝔃\right)}{2}\odot {{\displaystyle \int }}_{\mathfrak{d}}^{𝔃}\frac{\mathfrak{V}\left(\varsigma \right)}{{\varsigma }^2}d\varsigma .$$

(57)

(6)

Let lower $U\mathcal{D}$-harmonically convex $ℱ\mathcal{N}\mathcal{V}ℳ$ and, $\mathfrak{V}\left(\varsigma \right)=1$, $\beta =1$, then, from Theorem 7, we acquire 𝐻–𝐻 inequality for harmonically convex $ℱ\mathcal{N}\mathcal{V}ℳ$ see [64]:

$$\tilde{\mathfrak{G}}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃}\right){\le }_{\mathbb{F}}\frac{\mathfrak{d}𝔃}{𝔃-\mathfrak{d}} \odot {{\displaystyle \int }}_{\mathfrak{d}}^{𝔃}\frac{\tilde{\mathfrak{G}}\left(\varsigma \right)}{{\varsigma }^2}d\varsigma {\le }_{\mathbb{F}}\frac{\tilde{\mathfrak{G}}\left(\mathfrak{d}\right)\oplus \tilde{\mathfrak{G}}\left(𝔃\right)}{2}.$$

(58)

(7)

If ${\mathfrak{G}}_{*}\left(\varsigma ,\mathfrak{o}\right)={\mathfrak{G}}^{*}\left(\varsigma ,\mathfrak{o}\right)$ with $\mathfrak{o}=1$ then from Theorem 7, we achieve classical fractional 𝐻–𝐻–Fejér inequality for harmonically convex function, given in [66].

(8)

Let ${\mathfrak{G}}_{*}\left(\varsigma ,\mathfrak{o}\right)={\mathfrak{G}}^{*}\left(\varsigma ,\mathfrak{o}\right)$ with $\mathfrak{o}=1$ and $\beta =1$. Then, from Theorem 7, in our case classical 𝐻–𝐻–Fejér inequality for harmonically convex function, given in [66].

(9)

If ${\mathfrak{G}}_{*}\left(\varsigma ,\mathfrak{o}\right)={\mathfrak{G}}^{*}\left(\varsigma ,\mathfrak{o}\right)$ with $\mathfrak{V}\left(\varsigma \right)=\mathfrak{o}=1$ then from Theorem 7, in our case classical fractional 𝐻–𝐻 inequality for harmonically convex function, see [65].

(10)

If ${\mathfrak{G}}_{*}\left(\varsigma ,\mathfrak{o}\right)={\mathfrak{G}}^{*}\left(\varsigma ,\mathfrak{o}\right)$ and $\mathfrak{V}\left(\varsigma \right)=\mathfrak{o}=\beta =1$ then from Theorem 7, in our case classical 𝐻–𝐻 inequality for harmonically convex function, [65].

In next two outcomes, we will prove fuzzy fractional $H-H$ inequality for the product of two $U\mathcal{D}$-harmonically convex $ℱ\mathcal{N}\mathcal{V}ℳ$. These types of inequalities are known as Pachpatte-type inequalities.

Theorem 8.

Let $\tilde{\mathfrak{G}},\tilde{\mathcal{G}} :\left[\mathfrak{d}, 𝔃\right]\to {₤}_I$ be two $U\mathcal{D}$-harmonically convex $ℱ\mathcal{N}\mathcal{V}ℳ$s on $\left[\mathfrak{d}, 𝔃\right]$ $,$ whose $\mathfrak{o}$-cuts ${\mathfrak{G}}_{\mathfrak{o}}, {\mathcal{G}}_{\mathfrak{o}}:\left[\mathfrak{d}, 𝔃\right]\subset ℝ\to {\mathcal{K}}_I^{+}$ are characterized by ${\mathfrak{G}}_{\mathfrak{o}}\left(\varsigma \right)=\left[{\mathfrak{G}}_{*}\left(\varsigma ,\mathfrak{o}\right), {\mathfrak{G}}^{*}\left(\varsigma ,\mathfrak{o}\right)\right]$ and ${\mathcal{G}}_{\mathfrak{o}}\left(\varsigma \right)=\left[{\mathcal{G}}_{*}\left(\varsigma ,\mathfrak{o}\right), {\mathcal{G}}^{*}\left(\varsigma ,\mathfrak{o}\right)\right]$ for all $\varsigma \in \left[\mathfrak{d}, 𝔃\right]$, $\mathfrak{o}\in \left[0, 1\right]$. If $\tilde{\mathfrak{G}}\otimes \tilde{\mathcal{G}}\in L\left(\left[\mathfrak{d}, 𝔃\right],{₤}_I\right)$, then:

$$\begin{gathered} \frac{\Gamma \left(\beta +1\right)}{2}{\left(\frac{\mathfrak{d}𝔃}{𝔃-\mathfrak{d}}\right)}^{\beta }\odot \left[\begin{array}{c}{ℐ}_{{\left(\frac{1}{𝔃}\right)}^{+}}^{\beta } \tilde{\mathfrak{G}}\circ \psi \left(\frac{1}{\mathfrak{d}}\right)\otimes \tilde{\mathcal{G}}\circ \psi \left(\frac{1}{\mathfrak{d}}\right)\oplus {ℐ}_{{\left(\frac{1}{\mathfrak{d}}\right)}^{-}}^{\beta }\tilde{\mathfrak{G}}\circ \psi \left(\frac{1}{𝔃}\right)\otimes \tilde{\mathcal{G}}\circ \psi \left(\frac{1}{𝔃}\right)\end{array}\right] \\ {\supseteq }_{\mathbb{F}}\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\odot \tilde{\mathfrak{P}}\left(\mathfrak{d},𝔃\right)\oplus \left(\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\odot \tilde{\mathfrak{F}}\left(\mathfrak{d},𝔃\right). \end{gathered}$$

(59)

where $\tilde{\mathfrak{P}}\left(\mathfrak{d},𝔃\right)=\tilde{\mathfrak{G}}\left(\mathfrak{d}\right)\otimes \tilde{\mathcal{G}}\left(\mathfrak{d}\right)\oplus \tilde{\mathfrak{G}}\left(𝔃\right)\otimes \tilde{\mathcal{G}}\left(𝔃\right),$ $\tilde{\mathfrak{F}}\left(\mathfrak{d},𝔃\right)=\tilde{\mathfrak{G}}\left(\mathfrak{d}\right)\otimes \tilde{\mathcal{G}}\left(𝔃\right)\oplus \tilde{\mathfrak{G}}\left(𝔃\right)\otimes \tilde{\mathcal{G}}\left(\mathfrak{d}\right),$ and ${\mathfrak{P}}_{\mathfrak{o}}\left(\mathfrak{d},𝔃\right)=\left[{\mathfrak{P}}_{*}\left(\left(\mathfrak{d},𝔃\right), \mathfrak{o}\right), {\mathfrak{P}}^{*}\left(\left(\mathfrak{d},𝔃\right), \mathfrak{o}\right)\right]$ and ${\mathfrak{F}}_{\mathfrak{o}}\left(\mathfrak{d},𝔃\right)=\left[{\mathfrak{F}}_{*}\left(\left(\mathfrak{d},𝔃\right), \mathfrak{o}\right), {\mathfrak{F}}^{*}\left(\left(\mathfrak{d},𝔃\right), \mathfrak{o}\right)\right].$

Proof. 

Since $\tilde{\mathfrak{G}}, \tilde{\mathcal{G}}$ both are $U\mathcal{D}$-harmonically convex $ℱ\mathcal{N}\mathcal{V}ℳ$s, then for each $\mathfrak{o}\in \left[0, 1\right]$, we have:

$$\begin{array}{c} \\ {\mathfrak{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}, \mathfrak{o}\right)\le \left(1-\mu \right){\mathfrak{G}}_{*}\left(\mathfrak{d},\mathfrak{o}\right)+\mu {\mathfrak{G}}_{*}\left(𝔃, \mathfrak{o}\right),\end{array}$$

and

$$\begin{array}{c} \\ {\mathcal{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}, \mathfrak{o}\right)\le \left(1-\mu \right){\mathcal{G}}_{*}\left(\mathfrak{d},\mathfrak{o}\right)+\mu {\mathcal{G}}_{*}\left(𝔃, \mathfrak{o}\right).\end{array}$$

From the definition of $U\mathcal{D}$-harmonically convex $ℱ\mathcal{N}\mathcal{V}ℳ$s, it follows that $\tilde{0}{\le }_{\mathbb{F}}\tilde{\mathfrak{G}}\left(\varsigma \right)$ and $\tilde{0}{\le }_{\mathbb{F}}\tilde{\mathcal{G}}\left(\varsigma \right)$, so:

$$\begin{gathered} {\mathfrak{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}, \mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}, \mathfrak{o}\right) \\ \le \left(\begin{array}{c}\left(1-\mu \right){\mathfrak{G}}_{*}\left(\mathfrak{d},\mathfrak{o}\right)+\mu {\mathfrak{G}}_{*}\left(𝔃, \mathfrak{o}\right)\end{array}\right)\left(\begin{array}{c}\left(1-\mu \right){\mathcal{G}}_{*}\left(\mathfrak{d},\mathfrak{o}\right)\\ +\mu {\mathcal{G}}_{*}\left(𝔃, \mathfrak{o}\right)\end{array}\right) \\ ={\left(1-\mu \right)}^2{\mathfrak{G}}_{*}\left(\mathfrak{d},\mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(\mathfrak{d},\mathfrak{o}\right)+{\mu }^2{\mathfrak{G}}_{*}\left(𝔃, \mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(𝔃, \mathfrak{o}\right) \\ +\mu \left(1-\mu \right){\mathfrak{G}}_{*}\left(\mathfrak{d},\mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(𝔃, \mathfrak{o}\right) \\ +\mu \left(1-\mu \right){\mathfrak{G}}_{*}\left(𝔃, \mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(\mathfrak{d},\mathfrak{o}\right). \end{gathered}$$

(60)

Analogously, we have:

$$\begin{array}{c}{\mathfrak{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\left(1-\mu \right)\mathfrak{d}+\mu 𝔃}, \mathfrak{o}\right){\mathcal{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\left(1-\mu \right)\mathfrak{d}+\mu 𝔃}, \mathfrak{o}\right) \\ \le {\mu }^2{\mathfrak{G}}_{*}\left(\mathfrak{d},\mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(\mathfrak{d},\mathfrak{o}\right)\\ +{\left(1-\mu \right)}^2{\mathfrak{G}}_{*}\left(𝔃, \mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(𝔃, \mathfrak{o}\right)\\ +\mu \left(1-\mu \right){\mathfrak{G}}_{*}\left(\mathfrak{d},\mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(𝔃, \mathfrak{o}\right)\\ +\mu \left(1-\mu \right){\mathfrak{G}}_{*}\left(𝔃, \mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(\mathfrak{d},\mathfrak{o}\right).\end{array}$$

(61)

Adding (60) and (61), we have:

$$\begin{array}{c}{\mathfrak{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}, \mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}, \mathfrak{o}\right) \\ +{\mathfrak{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\left(1-\mu \right)\mathfrak{d}+\mu 𝔃}, \mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\left(1-\mu \right)\mathfrak{d}+\mu 𝔃}, \mathfrak{o}\right) \\ \le \left[{\mu }^2+{\left(1-\mu \right)}^2\right]\left[\begin{array}{c}{\mathfrak{G}}_{*}\left(\mathfrak{d},\mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(\mathfrak{d},\mathfrak{o}\right)\\ +{\mathfrak{G}}_{*}\left(𝔃, \mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(𝔃, \mathfrak{o}\right)\end{array}\right]\\ +2\mu \left(1-\mu \right)\left[\begin{array}{c}{\mathfrak{G}}_{*}\left(𝔃, \mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(\mathfrak{d},\mathfrak{o}\right)\\ +{\mathfrak{G}}_{*}\left(\mathfrak{d},\mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(𝔃, \mathfrak{o}\right)\end{array}\right].\end{array}$$

(62)

Taking multiplication of (62) by ${\mu }^{\beta -1}$ and integrating the obtained result with respect to $\mu$ over (0, 1), we have:

$$\begin{array}{c}{{\displaystyle \int }}_0^1{\mu }^{\beta -1}{\mathfrak{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}, \mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}, \mathfrak{o}\right)d\mu \\ +{{\displaystyle \int }}_0^1{\mu }^{\beta -1}{\mathfrak{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\left(1-\mu \right)\mathfrak{d}+\mu 𝔃}, \mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\left(1-\mu \right)\mathfrak{d}+\mu 𝔃}, \mathfrak{o}\right)d\mu \\ \le {\mathfrak{P}}_{*}\left(\left(\mathfrak{d},𝔃\right), \mathfrak{o}\right){{\displaystyle \int }}_0^1{\mu }^{\beta -1}\left[{\mu }^2+{\left(1-\mu \right)}^2\right]d\mu \\ +2{\mathfrak{F}}_{*}\left(\left(\mathfrak{d},𝔃\right), \mathfrak{o}\right){{\displaystyle \int }}_0^1{\mu }^{\beta -1}\mu \left(1-\mu \right)d\mu .\end{array}$$

It follows that:

$$\begin{array}{c} \\ \Gamma \left(\beta \right){\left(\frac{\mathfrak{d}𝔃}{𝔃-\mathfrak{d}}\right)}^{\beta }\left[\begin{array}{c}{ℐ}_{{\left(\frac{1}{𝔃}\right)}^{+}}^{\beta } {\mathfrak{G}}_{*}\left(\frac{1}{\mathfrak{d}}, \mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(\frac{1}{\mathfrak{d}}, \mathfrak{o}\right)+{ℐ}_{{\left(\frac{1}{\mathfrak{d}}\right)}^{-}}^{\beta } {\mathfrak{G}}_{*}\left(\frac{1}{𝔃},\mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(\frac{1}{𝔃},\mathfrak{o}\right)\end{array}\right] \\ \le \frac{2}{\beta }\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right){\mathfrak{P}}_{*}\left(\left(\mathfrak{d},𝔃\right), \mathfrak{o}\right)+\frac{2}{\beta }\left(\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right){\mathfrak{F}}_{*}\left(\left(\mathfrak{d},𝔃\right), \mathfrak{o}\right).\end{array}$$

Similarly, for ${\mathfrak{G}}^{*}\left(\varsigma ,\mathfrak{o}\right)$, we have:

$$\begin{gathered} \Gamma \left(\beta \right){\left(\frac{\mathfrak{d}𝔃}{𝔃-\mathfrak{d}}\right)}^{\beta }\left[\begin{array}{c}{ℐ}_{{\left(\frac{1}{𝔃}\right)}^{+}}^{\beta }{\mathfrak{G}}^{*}\left(\frac{1}{\mathfrak{d}}, \mathfrak{o}\right)\times {\mathcal{G}}^{*}\left(\frac{1}{\mathfrak{d}}, \mathfrak{o}\right)+{ℐ}_{{\left(\frac{1}{\mathfrak{d}}\right)}^{-}}^{\beta }{\mathfrak{G}}^{*}\left(\frac{1}{𝔃},\mathfrak{o}\right)\times {\mathcal{G}}^{*}\left(\frac{1}{𝔃},\mathfrak{o}\right)\end{array}\right] \\ \ge \frac{2}{\beta }\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right){\mathfrak{P}}^{*}\left(\left(\mathfrak{d},𝔃\right), \mathfrak{o}\right) \\ +\frac{2}{\beta }\left(\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right){\mathfrak{F}}^{*}\left(\left(\mathfrak{d},𝔃\right), \mathfrak{o}\right), \end{gathered}$$

that is:

$$\begin{gathered} \Gamma \left(\beta \right){\left(\frac{\mathfrak{d}𝔃}{𝔃-\mathfrak{d}}\right)}^{\beta }\left[{ℐ}_{{\left(\frac{1}{𝔃}\right)}^{+}}^{\beta }{\mathfrak{G}}_{*}\left(\frac{1}{\mathfrak{d}}, \mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(\frac{1}{\mathfrak{d}}, \mathfrak{o}\right)+{ℐ}_{{\left(\frac{1}{\mathfrak{d}}\right)}^{-}}^{\beta }{\mathfrak{G}}_{*}\left(\frac{1}{𝔃},\mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(\frac{1}{𝔃},\mathfrak{o}\right), {ℐ}_{{\left(\frac{1}{𝔃}\right)}^{+}}^{\beta }{\mathfrak{G}}^{*}\left(\frac{1}{\mathfrak{d}}, \mathfrak{o}\right)\times {\mathcal{G}}^{*}\left(\frac{1}{\mathfrak{d}}, \mathfrak{o}\right)+{ℐ}_{{\left(\frac{1}{\mathfrak{d}}\right)}^{-}}^{\beta }{\mathfrak{G}}^{*}\left(\frac{1}{𝔃},\mathfrak{o}\right)\times {\mathcal{G}}^{*}\left(\frac{1}{𝔃}, \mathfrak{o}\right)\right] \\ {\supseteq }_I\frac{2}{\beta }\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\left[{\mathfrak{P}}_{*}\left(\left(\mathfrak{d},𝔃\right), \mathfrak{o}\right), {\mathfrak{P}}^{*}\left(\left(\mathfrak{d},𝔃\right), \mathfrak{o}\right)\right]+\frac{2}{\beta }\left(\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\left[{\mathfrak{F}}_{*}\left(\left(\mathfrak{d},𝔃\right), \mathfrak{o}\right), {\mathfrak{F}}^{*}\left(\left(\mathfrak{d},𝔃\right), \mathfrak{o}\right)\right]. \end{gathered}$$

Thus,

$$\begin{gathered} \frac{\Gamma \left(\beta +1\right)}{2}{\left(\frac{\mathfrak{d}𝔃}{𝔃-\mathfrak{d}}\right)}^{\beta }\odot \left[\begin{array}{c}{ℐ}_{{\left(\frac{1}{𝔃}\right)}^{+}}^{\beta } \tilde{\mathfrak{G}}\circ \psi \left(\frac{1}{\mathfrak{d}}\right)\otimes \tilde{\mathcal{G}}\circ \psi \left(\frac{1}{\mathfrak{d}}\right)\oplus {ℐ}_{{\left(\frac{1}{\mathfrak{d}}\right)}^{-}}^{\beta }\tilde{\mathfrak{G}}\circ \psi \left(\frac{1}{𝔃}\right)\otimes \tilde{\mathcal{G}}\circ \psi \left(\frac{1}{𝔃}\right)\end{array}\right] \\ {\supseteq }_{\mathbb{F}}\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\odot \tilde{\mathfrak{P}}\left(\mathfrak{d},𝔃\right)\oplus \left(\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\odot \tilde{\mathfrak{F}}\left(\mathfrak{d},𝔃\right). \end{gathered}$$

and the theorem has been established. □

Theorem 9.

Let $\tilde{\mathfrak{G}}, \tilde{\mathcal{G}} :\left[\mathfrak{d}, 𝔃\right]\to {₤}_I$ be two $U\mathcal{D}$-harmonically convex $ℱ\mathcal{N}\mathcal{V}ℳ$s, whose $\mathfrak{o}$-cuts define the family of $I\mathcal{V}ℳ$s ${\mathfrak{G}}_{\mathfrak{o}}, {\mathcal{G}}_{\mathfrak{o}}:\left[\mathfrak{d}, 𝔃\right]\subset ℝ\to {\mathcal{K}}_I^{+}$ are provided by ${\mathfrak{G}}_{\mathfrak{o}}\left(\varsigma \right)=\left[{\mathfrak{G}}_{*}\left(\varsigma ,\mathfrak{o}\right), {\mathfrak{G}}^{*}\left(\varsigma ,\mathfrak{o}\right)\right]$ and ${\mathcal{G}}_{\mathfrak{o}}\left(\varsigma \right)=\left[{\mathcal{G}}_{*}\left(\varsigma ,\mathfrak{o}\right), {\mathcal{G}}^{*}\left(\varsigma ,\mathfrak{o}\right)\right]$ for all $\varsigma \in \left[\mathfrak{d}, 𝔃\right]$, $\mathfrak{o}\in \left[0, 1\right]$. If $\tilde{\mathfrak{G}}\tilde{\times }\tilde{\mathcal{G}}\in L\left(\left[\mathfrak{d}, 𝔃\right],{₤}_I\right)$, then:

$$\begin{gathered} \tilde{\mathfrak{G}}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃}\right)\otimes \tilde{\mathcal{G}}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃}\right){\supseteq }_{\mathbb{F}}\frac{\Gamma \left(\beta +1\right)}{4}{\left(\frac{\mathfrak{d}𝔃}{𝔃-\mathfrak{d}}\right)}^{\beta }\odot \left[\begin{array}{c}{ℐ}_{{\left(\frac{1}{𝔃}\right)}^{+}}^{\beta } \tilde{\mathfrak{G}}\left(\frac{1}{\mathfrak{d}}\right)\otimes \tilde{\mathcal{G}}\left(\frac{1}{\mathfrak{d}}\right)\oplus {ℐ}_{{\left(\frac{1}{\mathfrak{d}}\right)}^{-}}^{\beta } \tilde{\mathfrak{G}}\left(\frac{1}{𝔃}\right)\otimes \tilde{\mathcal{G}}\left(\frac{1}{𝔃}\right)\end{array}\right] \\ \oplus \frac{1}{2}\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\odot \tilde{\mathfrak{F}}\left(\mathfrak{d},𝔃\right)\oplus \frac{1}{2}\left(\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\odot \tilde{\mathfrak{P}}\left(\mathfrak{d},𝔃\right). \end{gathered}$$

(63)

where $\tilde{\mathfrak{P}}\left(\mathfrak{d},𝔃\right)=\tilde{\mathfrak{G}}\left(\mathfrak{d}\right)\otimes \tilde{\mathcal{G}}\left(\mathfrak{d}\right)\oplus \tilde{\mathfrak{G}}\left(𝔃\right)\otimes \tilde{\mathcal{G}}\left(𝔃\right),$ $\tilde{\mathfrak{F}}\left(\mathfrak{d},𝔃\right)=\tilde{\mathfrak{G}}\left(\mathfrak{d}\right)\otimes \tilde{\mathcal{G}}\left(𝔃\right)\oplus \tilde{\mathfrak{G}}\left(𝔃\right)\otimes \tilde{\mathcal{G}}\left(\mathfrak{d}\right),$ and ${\mathfrak{P}}_{\mathfrak{o}}\left(\mathfrak{d},𝔃\right)=\left[{\mathfrak{P}}_{*}\left(\left(\mathfrak{d},𝔃\right), \mathfrak{o}\right), {\mathfrak{P}}^{*}\left(\left(\mathfrak{d},𝔃\right), \mathfrak{o}\right)\right]$ and ${\mathfrak{F}}_{\mathfrak{o}}\left(\mathfrak{d},𝔃\right)=\left[{\mathfrak{F}}_{*}\left(\left(\mathfrak{d},𝔃\right), \mathfrak{o}\right), {\mathfrak{F}}^{*}\left(\left(\mathfrak{d},𝔃\right), \mathfrak{o}\right)\right].$

Proof. 

Consider $\tilde{\mathfrak{G}},\tilde{\mathcal{G}} :\left[\mathfrak{d}, 𝔃\right]\to {₤}_I$ are $U\mathcal{D}$-harmonically convex $ℱ\mathcal{N}\mathcal{V}ℳ$s. Then by hypothesis, for each $\mathfrak{o}\in \left[0, 1\right],$ we have:

$$\begin{array}{cc}{\mathfrak{G}}_{*}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃},\mathfrak{o}\right)\times {\mathcal{G}}_{*}\hfill & \left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃},\mathfrak{o}\right) \hfill \\ \hfill & \begin{array}{c}\le \frac{1}{4}\left[\begin{array}{c}{\mathfrak{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}, \mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}, \mathfrak{o}\right)\\ +{\mathfrak{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}, \mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\left(1-\mu \right)\mathfrak{d}+\mu 𝔃}, \mathfrak{o}\right)\end{array}\right] \\ +\frac{1}{4}\left[\begin{array}{c}{\mathfrak{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\left(1-\mu \right)\mathfrak{d}+\mu 𝔃}, \mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}, \mathfrak{o}\right)\\ +{\mathfrak{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\left(1-\mu \right)\mathfrak{d}+\mu 𝔃}, \mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\left(1-\mu \right)\mathfrak{d}+\mu 𝔃}, \mathfrak{o}\right)\end{array}\right],\end{array}\hfill \\ \hfill & \begin{array}{c}\le \frac{1}{4}\left[\begin{array}{c}{\mathfrak{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}, \mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}, \mathfrak{o}\right)\\ +{\mathfrak{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\left(1-\mu \right)\mathfrak{d}+\mu 𝔃}, \mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\left(1-\mu \right)\mathfrak{d}+\mu 𝔃}, \mathfrak{o}\right)\end{array}\right]\\ +\frac{1}{4}\left[\begin{array}{c}\left(\mu {\mathfrak{G}}_{*}\left(\mathfrak{d}, \mathfrak{o}\right)+\left(1-\mu \right){\mathfrak{G}}_{*}\left(𝔃, \mathfrak{o}\right)\right)\\ \times \left(\left(1-\mu \right){\mathcal{G}}_{*}\left(\mathfrak{d}, \mathfrak{o}\right)+\mu {\mathcal{G}}_{*}\left(𝔃, \mathfrak{o}\right)\right)\\ +\left(\left(1-\mu \right){\mathfrak{G}}_{*}\left(\mathfrak{d}, \mathfrak{o}\right)+\mu {\mathfrak{G}}_{*}\left(𝔃, \mathfrak{o}\right)\right)\\ \times \left(\mu {\mathcal{G}}_{*}\left(\mathfrak{d}, \mathfrak{o}\right)+\left(1-\mu \right){\mathcal{G}}_{*}\left(𝔃, \mathfrak{o}\right)\right)\end{array}\right], \end{array}\hfill \\ \hfill & \begin{array}{c}=\frac{1}{4}\left[\begin{array}{c}{\mathfrak{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}, \mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}, \mathfrak{o}\right)\\ +{\mathfrak{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\left(1-\mu \right)\mathfrak{d}+\mu 𝔃}, \mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\left(1-\mu \right)\mathfrak{d}+\mu 𝔃}, \mathfrak{o}\right)\end{array}\right] \\ +\frac{1}{4}\left[\begin{array}{c}\left\{{\mu }^2+{\left(1-\mu \right)}^2\right\}{\mathfrak{F}}_{*}\left(\left(\mathfrak{d},𝔃\right), \mathfrak{o}\right)\\ +\left\{\mu \left(1-\mu \right)+\left(1-\mu \right)\mu \right\}{\mathfrak{P}}_{*}\left(\left(\mathfrak{d},𝔃\right), \mathfrak{o}\right)\end{array}\right]. \end{array}\hfill \end{array}$$

(64)

Multiplying inequality (64) by ${\mu }^{\beta -1}$ and integrating over $\left(0, 1\right)$:

$$\begin{gathered} {\mathfrak{G}}_{*}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃},\mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃},\mathfrak{o}\right) \\ \le \frac{1}{4}\left[\begin{array}{c}{{\displaystyle \int }}_0^1{\mu }^{\beta -1}{\mathfrak{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}, \mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}, \mathfrak{o}\right)d\mu \\ +{{\displaystyle \int }}_0^1{\mu }^{\beta -1}{\mathfrak{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\left(1-\mu \right)\mathfrak{d}+\mu 𝔃}, \mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\left(1-\mu \right)\mathfrak{d}+\mu 𝔃}, \mathfrak{o}\right)\end{array}\right]d\mu \\ +\left[\begin{array}{c}\frac{1}{4}{\mathfrak{F}}_{*}\left(\left(\mathfrak{d},𝔃\right), \mathfrak{o}\right){{\displaystyle \int }}_0^1{\mu }^{\beta -1}\left[{\mu }^2+{\left(1-\mu \right)}^2\right]d\mu \\ +2{\mathfrak{P}}_{*}\left(\left(\mathfrak{d},𝔃\right), \mathfrak{o}\right){{\displaystyle \int }}_0^1{\mu }^{\beta -1}\mu \left(1-\mu \right)d\mu \end{array}\right]. \end{gathered}$$

Taking $\varsigma =\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}$ and $𝓉=\frac{\mathfrak{d}𝔃}{\left(1-\mu \right)\mathfrak{d}+\mu 𝔃}$, we then obtain:

$$\begin{gathered} \frac{1}{\beta } {\mathfrak{G}}_{*}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃},\mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃},\mathfrak{o}\right) \\ \le \frac{\Gamma \left(\beta \right)}{4}{\left(\frac{\mathfrak{d}𝔃}{𝔃-\mathfrak{d}}\right)}^{\beta }\left[\begin{array}{c}{ℐ}_{{\left(\frac{1}{𝔃}\right)}^{+}}^{\beta } {\mathfrak{G}}_{*}\circ \psi \left(\frac{1}{\mathfrak{d}}\right)\times {\mathcal{G}}_{*}\circ \psi \left(\frac{1}{\mathfrak{d}}\right)\\ +{ℐ}_{{\left(\frac{1}{\mathfrak{d}}\right)}^{-}}^{\beta } {\mathfrak{G}}_{*}\circ \psi \left(\frac{1}{𝔃}\right)\times {\mathcal{G}}_{*}\circ \psi \left(\frac{1}{𝔃}\right)\end{array}\right] \\ +\frac{1}{2\beta }\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right){\mathfrak{F}}_{*}\left(\left(\mathfrak{d},𝔃\right), \mathfrak{o}\right) \\ +\frac{1}{2\beta }\left(\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right){\mathfrak{P}}_{*}\left(\left(\mathfrak{d},𝔃\right), \mathfrak{o}\right). \end{gathered}$$

Similarly, for ${\mathfrak{G}}^{*}\left(\varsigma ,\mathfrak{o}\right)$, we have:

$$\begin{gathered} \frac{1}{\beta } {\mathfrak{G}}^{*}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃},\mathfrak{o}\right)\times {\mathcal{G}}^{*}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃},\mathfrak{o}\right) \\ \ge \frac{\Gamma \left(\beta \right)}{4}{\left(\frac{\mathfrak{d}𝔃}{𝔃-\mathfrak{d}}\right)}^{\beta }\left[\begin{array}{c}{ℐ}_{{\left(\frac{1}{𝔃}\right)}^{+}}^{\beta } {\mathfrak{G}}^{*}\circ \psi \left(\frac{1}{\mathfrak{d}}\right)\times {\mathcal{G}}^{*}\circ \psi \left(\frac{1}{\mathfrak{d}}\right)++{ℐ}_{{\left(\frac{1}{\mathfrak{d}}\right)}^{-}}^{\beta } {\mathfrak{G}}^{*}\circ \psi \left(\frac{1}{𝔃},\mathfrak{o}\right)\times {\mathcal{G}}^{*}\circ \psi \left(\frac{1}{𝔃},\mathfrak{o}\right)\end{array}\right] \\ +\frac{1}{2\beta }\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right){\mathfrak{F}}^{*}\left(\left(\mathfrak{d},𝔃\right), \mathfrak{o}\right)+\frac{1}{2\beta }\left(\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right){\mathfrak{P}}^{*}\left(\left(\mathfrak{d},𝔃\right), \mathfrak{o}\right), \end{gathered}$$

that is:

$$\begin{gathered} \tilde{\mathfrak{G}}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃}\right)\otimes \tilde{\mathcal{G}}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃}\right){\supseteq }_{\mathbb{F}}\frac{\Gamma \left(\beta +1\right)}{4}{\left(\frac{\mathfrak{d}𝔃}{𝔃-\mathfrak{d}}\right)}^{\beta }\odot \left[\begin{array}{c}{ℐ}_{\left(\frac{1}{𝔃}\right)}^{\beta }+ \tilde{\mathfrak{G}}\left(\frac{1}{\mathfrak{d}}\right)\otimes \tilde{\mathcal{G}}\left(\frac{1}{\mathfrak{d}}\right)\oplus {ℐ}_{\left(\frac{1}{\mathfrak{d}}\right)}^{\beta }- \tilde{\mathfrak{G}}\left(\frac{1}{𝔃}\right)\otimes \tilde{\mathcal{G}}\left(\frac{1}{𝔃}\right)\end{array}\right] \\ \oplus \frac{1}{2}\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\odot \tilde{\mathfrak{F}}\left(\mathfrak{d},𝔃\right)\oplus \frac{1}{2}\left(\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\odot \tilde{\mathfrak{P}}\left(\mathfrak{d},𝔃\right). \end{gathered}$$

Hence, the required result. □

Theorem 10.

Let $\tilde{\mathfrak{G}},\tilde{\mathcal{G}} :\left[\mathfrak{d}, 𝔃\right]\to {₤}_I$ be two $U\mathcal{D}$-harmonically convex $ℱ\mathcal{N}\mathcal{V}ℳ$s, whose $\mathfrak{o}$-cuts define the family of $I\mathcal{V}ℳ$s ${\mathfrak{G}}_{\mathfrak{o}}, {\mathcal{G}}_{\mathfrak{o}}:\left[\mathfrak{d}, 𝔃\right]\subset ℝ\to {\mathcal{K}}_I^{+}$ are provided by ${\mathfrak{G}}_{\mathfrak{o}}\left(\varsigma \right)=\left[{\mathfrak{G}}_{*}\left(\varsigma ,\mathfrak{o}\right), {\mathfrak{G}}^{*}\left(\varsigma ,\mathfrak{o}\right)\right]$ and ${\mathcal{G}}_{\mathfrak{o}}\left(\varsigma \right)=\left[{\mathcal{G}}_{*}\left(\varsigma ,\mathfrak{o}\right), {\mathcal{G}}^{*}\left(\varsigma ,\mathfrak{o}\right)\right]$ for all $\varsigma \in \left[\mathfrak{d}, 𝔃\right]$, $\mathfrak{o}\in \left[0, 1\right]$. If $\tilde{\mathfrak{G}}\otimes \tilde{\mathcal{G}}\in L\left(\left[\mathfrak{d}, 𝔃\right],{₤}_I\right)$, then:

$$\begin{array}{c}2\odot \tilde{\mathfrak{G}}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃}\right)\otimes \tilde{\mathcal{G}}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃}\right)\hfill \\ {\supseteq }_{\mathbb{F}}\frac{\Gamma \left(\beta +1\right)}{2^{1-\beta }}{\left(\frac{\mathfrak{d}𝔃}{𝔃-\mathfrak{d}}\right)}^{\beta }\odot \left[\begin{array}{c}{ℐ}_{{\left(\frac{\mathfrak{d}+𝔃}{2\mathfrak{d}𝔃}\right)}^{+}}^{\beta }\tilde{\mathfrak{G}}\circ \psi \left(\frac{1}{\mathfrak{d}}\right)\otimes \tilde{\mathcal{G}}\circ \psi \left(\frac{1}{\mathfrak{d}}\right)\oplus {ℐ}_{{\left(\frac{\mathfrak{d}+𝔃}{2\mathfrak{d}𝔃}\right)}^{-}}^{\beta }\tilde{\mathfrak{G}}\circ \psi \left(\frac{1}{𝔃}\right)\otimes \tilde{\mathcal{G}}\circ \psi \left(\frac{1}{𝔃}\right)\end{array}\right]\hfill \\ \oplus \left(\frac{1}{2}-\frac{{\beta }^2+3\beta }{4\left(\beta +1\right)\left(\beta +2\right)}\right)\odot \tilde{\mathfrak{F}}\left(\mathfrak{d},𝔃\right)\oplus \frac{{\beta }^2+3\beta }{4\left(\beta +1\right)\left(\beta +2\right)}\odot \tilde{\mathfrak{P}}\left(\mathfrak{d},𝔃\right).\hfill \end{array}$$

(65)

where $\tilde{\mathfrak{P}}\left(\mathfrak{d},𝔃\right)=\tilde{\mathfrak{G}}\left(\mathfrak{d}\right)\otimes \tilde{\mathcal{G}}\left(\mathfrak{d}\right)\oplus \tilde{\mathfrak{G}}\left(𝔃\right)\otimes \tilde{\mathcal{G}}\left(𝔃\right),$ $\tilde{\mathfrak{F}}\left(\mathfrak{d},𝔃\right)=\tilde{\mathfrak{G}}\left(\mathfrak{d}\right)\otimes \tilde{\mathcal{G}}\left(𝔃\right)\oplus \tilde{\mathfrak{G}}\left(𝔃\right)\otimes \tilde{\mathcal{G}}\left(\mathfrak{d}\right),$ and ${\mathfrak{P}}_{\mathfrak{o}}\left(\mathfrak{d},𝔃\right)=\left[{\mathfrak{P}}_{*}\left(\left(\mathfrak{d},𝔃\right), \mathfrak{o}\right), {\mathfrak{P}}^{*}\left(\left(\mathfrak{d},𝔃\right), \mathfrak{o}\right)\right]$ and ${\mathfrak{F}}_{\mathfrak{o}}\left(\mathfrak{d},𝔃\right)=\left[{\mathfrak{F}}_{*}\left(\left(\mathfrak{d},𝔃\right), \mathfrak{o}\right), {\mathfrak{F}}^{*}\left(\left(\mathfrak{d},𝔃\right), \mathfrak{o}\right)\right].$

Proof. 

Consider $\tilde{\mathfrak{G}},\tilde{\mathcal{G}} :\left[\mathfrak{d}, 𝔃\right]\to {\mathbb{F}}_0$ are $U\mathcal{D}$-harmonically convex $ℱ\mathcal{N}\mathcal{V}ℳ$s. Then by hypothesis, for each $\mathfrak{o}\in \left[0, 1\right],$ we have:

$$\begin{array}{cc}{\mathfrak{G}}_{*}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃},\mathfrak{o}\right)\times {\mathcal{G}}_{*}\hfill & \left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃},\mathfrak{o}\right) \hfill \\ \hfill & \begin{array}{c}\le \frac{1}{4}\left[\begin{array}{c}{\mathfrak{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}, \mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}, \mathfrak{o}\right)\\ +{\mathfrak{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}, \mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\left(1-\mu \right)\mathfrak{d}+\mu 𝔃}, \mathfrak{o}\right)\end{array}\right] \\ +\frac{1}{4}\left[\begin{array}{c}{\mathfrak{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\left(1-\mu \right)\mathfrak{d}+\mu 𝔃}, \mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}, \mathfrak{o}\right)\\ +{\mathfrak{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\left(1-\mu \right)\mathfrak{d}+\mu 𝔃}, \mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\left(1-\mu \right)\mathfrak{d}+\mu 𝔃}, \mathfrak{o}\right)\end{array}\right],\end{array}\hfill \\ \hfill & \begin{array}{c}\le \frac{1}{4}\left[\begin{array}{c}{\mathfrak{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}, \mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}, \mathfrak{o}\right)\\ +{\mathfrak{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\left(1-\mu \right)\mathfrak{d}+\mu 𝔃}, \mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\left(1-\mu \right)\mathfrak{d}+\mu 𝔃}, \mathfrak{o}\right)\end{array}\right]\\ +\frac{1}{4}\left[\begin{array}{c}\left(\mu {\mathfrak{G}}_{*}\left(\mathfrak{d}, \mathfrak{o}\right)+\left(1-\mu \right){\mathfrak{G}}_{*}\left(𝔃, \mathfrak{o}\right)\right)\\ \times \left(\left(1-\mu \right){\mathcal{G}}_{*}\left(\mathfrak{d}, \mathfrak{o}\right)+\mu {\mathcal{G}}_{*}\left(𝔃, \mathfrak{o}\right)\right)\\ +\left(\left(1-\mu \right){\mathfrak{G}}_{*}\left(\mathfrak{d}, \mathfrak{o}\right)+\mu {\mathfrak{G}}_{*}\left(𝔃, \mathfrak{o}\right)\right)\\ \times \left(\mu {\mathcal{G}}_{*}\left(\mathfrak{d}, \mathfrak{o}\right)+\left(1-\mu \right){\mathcal{G}}_{*}\left(𝔃, \mathfrak{o}\right)\right)\end{array}\right], \end{array}\hfill \\ \hfill & \begin{array}{c}=\frac{1}{4}\left[\begin{array}{c}{\mathfrak{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}, \mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}, \mathfrak{o}\right)\\ +{\mathfrak{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\left(1-\mu \right)\mathfrak{d}+\mu 𝔃}, \mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\left(1-\mu \right)\mathfrak{d}+\mu 𝔃}, \mathfrak{o}\right)\end{array}\right] \\ +\frac{1}{4}\left[\begin{array}{c}\left\{{\mu }^2+{\left(1-\mu \right)}^2\right\}{\mathfrak{F}}_{*}\left(\left(\mathfrak{d},𝔃\right), \mathfrak{o}\right)\\ +2\mu \left(1-\mu \right){\mathfrak{P}}_{*}\left(\left(\mathfrak{d},𝔃\right), \mathfrak{o}\right)\end{array}\right]. \end{array}\hfill \end{array}$$

(66)

Multiplying inequality (66) by $2^{1+\beta }\beta {\mu }^{\beta -1}$ and then integrating the obtain outcome over $\left[0, \frac{1}{2}\right]$:

$$\begin{gathered} {\mathfrak{G}}_{*}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃},\mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃},\mathfrak{o}\right) \\ \le \frac{1}{4}{{\displaystyle \int }}_0^{\frac{1}{2}}\begin{array}{c}{2}^{1+\beta }\beta {\mu }^{\beta -1}\left[\begin{array}{c}{\mathfrak{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}, \mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}, \mathfrak{o}\right)\\ +{\mathfrak{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\left(1-\mu \right)\mathfrak{d}+\mu 𝔃}, \mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(\frac{\mathfrak{d}𝔃}{\left(1-\mu \right)\mathfrak{d}+\mu 𝔃}, \mathfrak{o}\right)\end{array}\right]d\mu \end{array} \\ +\frac{1}{4}\left[\begin{array}{c}{\mathfrak{F}}_{*}\left(\left(\mathfrak{d},𝔃\right), \mathfrak{o}\right){{\displaystyle \int }}_0^{\frac{1}{2}}{2}^{1+\beta }\beta {\mu }^{\beta -1}\left[{\mu }^2+{\left(1-\mu \right)}^2\right]d\mu \\ +2{\mathfrak{P}}_{*}\left(\left(\mathfrak{d},𝔃\right), \mathfrak{o}\right){{\displaystyle \int }}_0^{\frac{1}{2}}{2}^{1+\beta }\beta {\mu }^{\beta -1}\mu \left(1-\mu \right)d\mu \end{array}\right]. \end{gathered}$$

Taking $\varsigma =\frac{\mathfrak{d}𝔃}{\mu \mathfrak{d}+\left(1-\mu \right)𝔃}$ and $𝔃=\frac{\mathfrak{d}𝔃}{\left(1-\mu \right)\mathfrak{d}+\mu 𝔃}$, we then obtain:

$$\begin{array}{c}2{\mathfrak{G}}_{*}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃},\mathfrak{o}\right)\times {\mathcal{G}}_{*}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃},\mathfrak{o}\right) \\ \le \frac{\Gamma \left(\beta +1\right)}{2^{1-\beta }}{\left(\frac{\mathfrak{d}𝔃}{𝔃-\mathfrak{d}}\right)}^{\beta }\left[\begin{array}{c}{\mathcal{I}}_{{\left(\frac{1}{𝔃}\right)}^{+}}^{\beta } {\mathfrak{G}}_{*}\circ \psi \left(\frac{1}{\mathfrak{d}}\right)\times {\mathcal{G}}_{*}\circ \psi \left(\frac{1}{\mathfrak{d}}\right)+{\mathcal{I}}_{{\left(\frac{1}{\mathfrak{d}}\right)}^{-}}^{\beta } {\mathfrak{G}}_{*}\circ \psi \left(\frac{1}{𝔃}\right)\times {\mathcal{G}}_{*}\circ \psi \left(\frac{1}{𝔃}\right)\end{array}\right] \\ +\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right){\mathfrak{F}}_{*}\left(\left(\mathfrak{d},𝔃\right), \mathfrak{o}\right)+\left(\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right){\mathfrak{P}}_{*}\left(\left(\mathfrak{d},𝔃\right), \mathfrak{o}\right).\end{array}$$

Similarly, for ${\mathfrak{G}}^{*}\left(\varsigma ,\mathfrak{o}\right)$, we have:

$$\begin{gathered} 2 {\mathfrak{G}}^{*}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃},\mathfrak{o}\right)\times {\mathcal{G}}^{*}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃},\mathfrak{o}\right) \\ \ge \frac{\Gamma \left(\beta +1\right)}{2^{1-\beta }}{\left(\frac{\mathfrak{d}𝔃}{𝔃-\mathfrak{d}}\right)}^{\beta }\left[\begin{array}{c}{ℐ}_{{\left(\frac{1}{𝔃}\right)}^{+}}^{\beta } {\mathfrak{G}}^{*}\circ \psi \left(\frac{1}{\mathfrak{d}}\right)\times {\mathcal{G}}^{*}\circ \psi \left(\frac{1}{\mathfrak{d}}\right)\\ +{ℐ}_{{\left(\frac{1}{\mathfrak{d}}\right)}^{-}}^{\beta } {\mathfrak{G}}^{*}\circ \psi \left(\frac{1}{𝔃}\right)\times {\mathcal{G}}^{*}\circ \psi \left(\frac{1}{𝔃},\mathfrak{o}\right)\end{array}\right] \\ +\left(\frac{1}{2}-\frac{{\beta }^2+3\beta }{4\left(\beta +1\right)\left(\beta +2\right)}\right){\mathfrak{F}}^{*}\left(\left(\mathfrak{d},𝔃\right), \mathfrak{o}\right) \\ +\frac{{\beta }^2+3\beta }{4\left(\beta +1\right)\left(\beta +2\right)}{\mathfrak{P}}^{*}\left(\left(\mathfrak{d},𝔃\right), \mathfrak{o}\right), \end{gathered}$$

that is:

$$\begin{array}{c}2\odot \tilde{\mathfrak{G}}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃}\right)\otimes \tilde{\mathcal{G}}\left(\frac{2\mathfrak{d}𝔃}{\mathfrak{d}+𝔃}\right)\hfill \\ {\supseteq }_{\mathbb{F}}\frac{\Gamma \left(\beta +1\right)}{2^{1-\beta }}{\left(\frac{\mathfrak{d}𝔃}{𝔃-\mathfrak{d}}\right)}^{\beta }\odot \left[\begin{array}{c}{ℐ}_{{\left(\frac{\mathfrak{d}+𝔃}{2\mathfrak{d}𝔃}\right)}^{+}}^{\beta }\tilde{\mathfrak{G}}\circ \psi \left(\frac{1}{\mathfrak{d}}\right)\otimes \tilde{\mathcal{G}}\circ \psi \left(\frac{1}{\mathfrak{d}}\right)\oplus {ℐ}_{{\left(\frac{\mathfrak{d}+𝔃}{2\mathfrak{d}𝔃}\right)}^{-}}^{\beta }\tilde{\mathfrak{G}}\circ \psi \left(\frac{1}{𝔃}\right)\otimes \tilde{\mathcal{G}}\circ \psi \left(\frac{1}{𝔃}\right)\end{array}\right]\hfill \\ \oplus \left(\frac{1}{2}-\frac{{\beta }^2+3\beta }{4\left(\beta +1\right)\left(\beta +2\right)}\right)\odot \tilde{\mathfrak{F}}\left(\mathfrak{d},𝔃\right)\oplus \frac{{\beta }^2+3\beta }{4\left(\beta +1\right)\left(\beta +2\right)}\odot \tilde{\mathfrak{P}}\left(\mathfrak{d},𝔃\right).\end{array}$$

□

5. Conclusions

Our study of interval integral operator-type integral inequalities will broaden their applications. We provide an up and down harmonically concept for fuzzy-number-valued settings in this paper. The H–H-type inequalities were developed utilizing this idea. This study expands on several recent findings made by Kunt, and İşcan et al. [65,66], and the researchers who came after them, Refs. [61,64]. Furthermore, some non-trivial cases are provided to verify the accuracy of our primary conclusions. In the future, it will be fascinating to look into how analogous inequalities are established for other convexity types and by employing various integral operators. Fuzzy convex optimization theory may take a new turn as a result of this idea. Other researchers working in a range of scientific subjects may find the idea useful. In the future, we will try to explore this concept in a fuzzy environment.