Pre-Invexity and Fuzzy Fractional Integral Inequalities via Fuzzy Up and Down Relation

Abstract

The symmetric function class interacts heavily with other types of functions. One of these is the pre-invex function class, which is strongly related to symmetry theory. This paper proposes a novel fuzzy fractional extension of the Hermite-Hadamard, Hermite-Hadamard-Fejér, and Pachpatte type inequalities for up and down pre-invex fuzzy-number-valued mappings. Using fuzzy fractional operators, several generalizations have been developed, where well-known results fit as particular cases. Additionally, some non-trivial examples are included to support the discussion and the applicability of the key findings. The approach appears trustworthy and effective for dealing with various nonlinear problems in science and engineering. The findings are general and may constitute contributions to complex waveform theory.

1. Introduction

In the literature on mathematical inequalities, the Hermite–Hadamard (HH) inequality emerges as an important, and frequently utilized, result [1,2,3,4,5,6,7,8,9,10]. Other classical and significant inequalities include the Beckenbach–Dresher, Gagliardo–Nirenberg, Hardy, Hölder, Ky Fan, Levinson, Lynger, Minkowski, Olsen, Opial, Ostrowski, Young, arithmetic–geometric, and others [11,12,13,14,15,16,17,18,19,20,21,22,23,24,25].

For a convex mapping ${\rm Y}:K\to \mathbb{R}$ on convex set $K$, the HH inequality is written as

$${\rm Y}\left(\frac{\rho +\mu }{2}\right)\le \frac{1}{\mu -\rho }{\int }_{\rho }^{\mu }{\rm Y}\left(\varkappa \right)d\varkappa \le \frac{{\rm Y}\left(\rho \right)+{\rm Y}\left(\mu \right)}{2},$$

(1)

For all $\rho ,\mu \in K,$ with $\rho \le \mu$. If ${\rm Y}$ is concave, then (1) is reversed.

The HH inequality (1), as well as some generalizations, converses, and modifications [26,27,28,29,30,31,32,33,34,35,36,37,38,39], plays a significant role in many scientific disciplines, including statistics, electrical engineering, economics and finance, information sciences, and coding theory.

The notion of convexity has been used in different areas of technology and science. Convex functions and convex sets have been generalized in the scope of numerous fields due to their robustness. The convexity of a function has been demonstrated to exist only when it fulfills an integral inequality (1).

As novel fractional differential and integral operators with exponential kernels emerged as useful tools in several subjects, it became possible to provide novel classes of functional variants for harmonically convex functions, as well as generalizations in the scope of convexity theory.

An important generalization of the HH inequality is the HH–Fejér inequality [40]. Let us consider ${\rm Y}:K\to \mathbb{R}$ a convex mapping on a convex set $K$, and $\rho ,\mu \in K$ with $\rho \le \mu .$ Then, we have

$${\rm Y}\left(\frac{\rho +\mu }{2}\right)\le \frac{1}{{\int }_{\rho }^{\mu }\mathcal{C}\left(\varkappa \right)d\varkappa }{\int }_{\rho }^{\mu }{\rm Y}\left(\varkappa \right)\mathcal{C}\left(\varkappa \right)d\varkappa \le \frac{{\rm Y}\left(\rho \right)+{\rm Y}\left(\mu \right)}{2}.$$

(2)

If $\mathcal{C}\left(\varkappa \right)=1$, then we obtain (1) from (2). For a concave mapping, (2) is reversed. Different inequalities can be derived using distinct symmetric convex mappings, $\mathcal{C}\left(\varkappa \right)$.

Classical calculus underwent enormous development in the last few decades. As an extension of the classical differentiation and integration to non-integer orders, fractional calculus has increased relevance. Indeed, fractional operators can better depict real-world phenomena involving memory and hereditary properties. For instance, Baleanu et al. [41], Miller and Ross [42], and Kilbas et al. [43] discussed numerous applications and extensive methodologies of fractional calculus. The authors of [44,45,46,47,48,49,50,51,52,53,54] provide an overview of many fractional operators.

It has been shown that fractional differential equations capture complex systems’ dynamics more precisely than integer-order ones. Several applications can be found in cosmology, environmental sciences, medicine, biology, materials, image and signal processing, and many other fields.

Fractional integral inequalities are used in the scope of district subjects (please see [55,56,57,58,59,60,61,62,63,64,65,66] and references therein). The calculus of variations in the area of applied sciences has been widely addressed and has become an appealing research-oriented field to address the existence and uniqueness of solutions to fractional equations. For instance, Adil Khan et al. [67] derived the HH inequality for s-convex functions, while Khan et al. [68] considered weighted generalizations of the HH inequality for interval-valued exponential trigonometric functions. For more information, see [69,70,71,72,73,74,75,76,77,78,79,80] and the references therein.

Khan et al. extended the idea of convex interval-valued mappings (I·V·Ms) and convex fuzzy-number-valued mappings ($\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$s) by using fuzzy-order relations, such that convex $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$s include (h1, h2)-convex $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$s [81] and harmonic convex $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$s [82]. Moreover, in order to illustrate inequalities of HH, HH–Fejér, and Pachpatte types, they adopted h-pre-invex $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$s [83], (h1, h2)-pre-invex $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$s [84], and higher-order pre-invex $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$s [85]. In recent research, Khan et al. [86] proposed new forms of HH and HH–Fejér inequalities by using the concept of fuzzy Riemann–Liouville (RL) fractional integrals via up and down (UD) pre-invex $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$s. For up-to-date advancements in fuzzy-interval-valued analysis of integral inequalities, please see [87,88,89,90,91,92,93,94,95,96,97,98] and the references therein.

Based on the aforementioned ideas, in this paper, we establish HH, HH–Fejér, and Pachpatte type integral inequalities for UD pre-invex $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$s using the fuzzy fractional integral operator for inferable functions. Additionally, we show complete agreement between the suggested approach and existing results. The new approach appears reliable and effective for dealing with various nonlinear problems in the scope of science and engineering. The findings are general and may constitute contributions to complex waveform theory.

2. Preliminary Concepts and Definitions

We recall a few definitions that will be relevant in what follows.

Let us consider that ${\mathbb{X}}_o$ is the space of all closed and bounded intervals of $\mathbb{R}$, and that $\mathbb{M}\in {\mathbb{X}}_o$ is given by

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

(3)

If ${\mathbb{M}}_{*}={\mathbb{M}}^{*}$, then we say that $\mathbb{M}$ is degenerate. In the follow-up, all intervals are considered non-degenerate. If ${\mathbb{M}}_{*}\ge 0$, then we say that $\mathbb{M}$ is positive. We denote by ${\mathbb{X}}_o^{+}=\left\{\left[{\mathbb{M}}_{*},{\mathbb{M}}^{*}\right]:\left[{\mathbb{M}}_{*},{\mathbb{M}}^{*}\right]\in {\mathbb{X}}_o\mathrm{a}\mathrm{n}\mathrm{d}{\mathbb{M}}_{*}\ge 0\right\}$ the set of all positive intervals.

Let $\mathfrak{u}\in \mathbb{R}$ and $\mathfrak{u}\cdot \mathbb{M}$ be given by

$$\mathfrak{u}\cdot \mathbb{M}=\left\{\begin{array}{c}\left[\mathfrak{u}{\mathbb{M}}_{*},{\mathfrak{u}\mathbb{M}}^{*}\right]\mathrm{if}\mathfrak{u}>0,\\ \left\{0\right\}\mathrm{if}\mathfrak{u}=0,\\ \left[\mathfrak{u}{\mathbb{M}}^{*}{,\mathfrak{u}\mathbb{M}}_{*}\right]\mathrm{if}\mathfrak{u}<0.\end{array}\right.$$

(4)

We consider the Minkowski sum, $\mathbb{M}+\mathbb{W}$, product, $\mathbb{M}\times \mathbb{W}$, and difference, $\mathbb{W}-\mathbb{M}$, for $\mathbb{M},\mathbb{W}\in {\mathbb{X}}_o$, as

$$\left[{\mathbb{W}}_{*},{\mathbb{W}}^{*}\right]+\left[{\mathbb{M}}_{*},{\mathbb{M}}^{*}\right]=\left[{\mathbb{W}}_{*}+{\mathbb{M}}_{*},{\mathbb{W}}^{*}+{\mathbb{M}}^{*}\right],$$

(5)

$$\begin{array}{l}\left[{\mathbb{W}}_{*},{\mathbb{W}}^{*}\right]\times \left[{\mathbb{M}}_{*},{\mathbb{M}}^{*}\right]\\ =\left[\mathrm{m}\mathrm{i}\mathrm{n}\left\{{\mathbb{W}}_{*}{\mathbb{M}}_{*},{\mathbb{W}}^{*}{\mathbb{M}}_{*},{\mathbb{W}}_{*}{\mathbb{M}}^{*},{\mathbb{W}}^{*}{\mathbb{M}}^{*}\right\},\mathrm{m}\mathrm{a}\mathrm{x}\left\{{\mathbb{W}}_{*}{\mathbb{M}}_{*},{\mathbb{W}}^{*}{\mathbb{M}}_{*},{\mathbb{W}}_{*}{\mathbb{M}}^{*},{\mathbb{W}}^{*}{\mathbb{M}}^{*}\right\}\right]\end{array}$$

(6)

$$\left[{\mathbb{W}}_{*},{\mathbb{W}}^{*}\right]-\left[{\mathbb{M}}_{*},{\mathbb{M}}^{*}\right]=\left[{\mathbb{W}}_{*}-{\mathbb{M}}^{*},{\mathbb{W}}^{*}-{\mathbb{M}}_{*}\right].$$

(7)

Remark 1.

(i) 

For a given $\left[{\mathbb{W}}_{*},{\mathbb{W}}^{*}\right],\left[{\mathbb{M}}_{*},{\mathbb{M}}^{*}\right]\in {\mathbb{X}}_o,$ the relation $“{\supseteq }_I”$, defined on ${\mathbb{X}}_o$ by

$$\left[{\mathbb{M}}_{*},{\mathbb{M}}^{*}\right]{\supseteq }_I\left[{\mathbb{W}}_{*},{\mathbb{W}}^{*}\right]ifandonlyif{\mathbb{M}}_{*}\le {\mathbb{W}}_{*},{\mathbb{W}}^{*}{\le \mathbb{M}}^{*}$$

(8)

for all $\left[{\mathbb{W}}_{*},{\mathbb{W}}^{*}\right],\left[{\mathbb{M}}_{*},{\mathbb{M}}^{*}\right]\in {\mathbb{X}}_o,$ is considered a partial interval inclusion relation. Moreover, $\left[{\mathbb{M}}_{*},{\mathbb{M}}^{*}\right]{\supseteq }_I\left[{\mathbb{W}}_{*},{\mathbb{W}}^{*}\right]$ coincides with $\left[{\mathbb{M}}_{*},{\mathbb{M}}^{*}\right]\supseteq \left[{\mathbb{W}}_{*},{\mathbb{W}}^{*}\right]$ on ${\mathbb{X}}_o.$ The relation $“{\supseteq }_I”$ is of UD order (see [96]).

(ii) 

For a given $\left[{\mathbb{W}}_{*},{\mathbb{W}}^{*}\right],\left[{\mathbb{M}}_{*},{\mathbb{M}}^{*}\right]\in {\mathbb{X}}_o,$ the relation $“{\le }_I”$, defined on ${\mathbb{X}}_o$ by $\left[{\mathbb{W}}_{*},{\mathbb{W}}^{*}\right]{\le }_I\left[{\mathbb{M}}_{*},{\mathbb{M}}^{*}\right]$ if and only if ${\mathbb{W}}_{*}{\le \mathbb{M}}_{*},{\mathbb{W}}^{*}{\le \mathbb{M}}^{*}$ or ${\mathbb{W}}_{*}{\le \mathbb{M}}_{*},{\mathbb{W}}^{*}{<\mathbb{M}}^{*}$, is a partial interval order relation. In addition, we have that $\left[{\mathbb{W}}_{*},{\mathbb{W}}^{*}\right]{\le }_I\left[{\mathbb{M}}_{*},{\mathbb{M}}^{*}\right]$ coincides with $\left[{\mathbb{W}}_{*},{\mathbb{W}}^{*}\right]\le \left[{\mathbb{M}}_{*},{\mathbb{M}}^{*}\right]$ on ${\mathbb{X}}_o.$ The relation $“{\le }_I”$ is of left and right (LR) type [95,96].

Given the intervals $\left[{\mathbb{W}}_{*},{\mathbb{W}}^{*}\right],\left[{\mathbb{M}}_{*},{\mathbb{M}}^{*}\right]\in {\mathbb{X}}_o,$ we consider their Hausdorff–Pompeiu distance as 

$$d_H\left(\left[{\mathbb{W}}_{*},{\mathbb{W}}^{*}\right],\left[{\mathbb{M}}_{*},{\mathbb{M}}^{*}\right]\right)=\mathrm{m}\mathrm{a}\mathrm{x}\left\{\left|{\mathbb{W}}_{*}-{\mathbb{M}}_{*}\right|,\left|{\mathbb{W}}^{*}-{\mathbb{M}}^{*}\right|\right\}.$$

(9)

We have that $\left({\mathbb{X}}_o,d_H\right)$ is a complete metric space [89,93,94].

Definition 1

([88]). A fuzzy subset $L$ of $\mathbb{R}$ is a mapping $\tilde{\mathbb{M}}:\mathbb{R}\to \left[\mathrm{0,1}\right]$, which is the denoted membership mapping of $L$. We adopt the symbol $₤$ to represent the set of all fuzzy subsets of $\mathbb{R}$.

Let us consider $\tilde{\mathbb{M}}\in \mathrm{₤}$. If the following properties hold, then we say that $\tilde{\mathbb{M}}$ is a fuzzy number:

(1)

$\tilde{\mathbb{M}}$ is normal if there exists $\varkappa \in \mathbb{R}$ and $\tilde{\mathbb{M}}\left(\varkappa \right)=1;$

(2)

$\tilde{\mathbb{M}}$ is upper semi-continuous on $\mathbb{R}$ if for $\varkappa \in \mathbb{R}$ there exist $\epsilon >0$ and $\delta >0,$ yielding $\tilde{\mathbb{M}}\left(\varkappa \right)-\tilde{\mathbb{M}}\left(y\right)<\epsilon$ for all $y\in \mathbb{R}$ with $\left|\varkappa -y\right|<\delta ;$

(3)

$\tilde{\mathbb{M}}$ is fuzzy convex, meaning that $\tilde{\mathbb{M}}\left(\left(1-\mathfrak{u}\right)\varkappa +\mathfrak{u}y\right)\ge \mathrm{m}\mathrm{i}\mathrm{n}\left(\tilde{\mathbb{M}}\left(\varkappa \right),\tilde{\mathbb{M}}\left(y\right)\right),$ for all $\varkappa ,y\in \mathbb{R},$ and $\mathfrak{u}\in \left[\mathrm{0,1}\right]$;

(4)

$\tilde{\mathbb{M}}$ is compactly supported, which means that $\mathrm{c}\mathrm{l}\left\{\varkappa \in \mathbb{R}|\tilde{\mathbb{M}}\left(\varkappa \right)>0\right\}$ is compact.

The symbol ${\mathrm{₤}}_o$ will be adopted to designate the set of all fuzzy numbers of $\mathbb{R}$.

Definition 2

([88,89]). For $\tilde{\mathbb{M}}\in {₤}_o$, the $\mathfrak{b}$-level, or $\mathfrak{b}$-cut, sets of $\tilde{\mathbb{M}}$ are ${\left[\tilde{\mathbb{M}}\right]}^{\mathfrak{b}}=\left\{\varkappa \in \mathbb{R}|\tilde{\mathbb{M}}\left(\varkappa \right)>\mathfrak{b}\right\}$ for all $\mathfrak{b}\in [0,1]$, and ${\left[\tilde{\mathbb{M}}\right]}^0=\left\{\varkappa \in \mathbb{R}|\tilde{\mathbb{M}}\left(\varkappa \right)>0\right\}$.

Proposition 1

([90]). Let $\tilde{\mathbb{M}},\tilde{\mathbb{W}}\in {₤}_o$. The relation $“{\le }_{\mathbb{F}}”$, defined on ${₤}_o$ by

$$\tilde{\mathbb{M}}{\le }_{\mathbb{F}}\tilde{\mathbb{W}}whenandonlywhen{{\left[\tilde{\mathbb{M}}\right]}^{\mathfrak{b}}\le }_I{\left[\tilde{\mathbb{W}}\right]}^{\mathfrak{b}},forevery\mathfrak{b}\in [0,1],$$

(10)

is an LR order relation.

Proposition 2

([86]). Let $\tilde{\mathbb{M}},\tilde{\mathbb{W}}\in {₤}_o$. The relation  $“{\supseteq }_{\mathbb{F}}”$, defined on ${₤}_o$ by

$$\tilde{\mathbb{M}}{\supseteq }_{\mathbb{F}}\tilde{\mathbb{W}}\mathrm{when}\mathrm{and}\mathrm{only}\mathrm{when}{\left[\tilde{\mathbb{M}}\right]}^{\mathfrak{b}}{\supseteq }_I{\left[\tilde{\mathbb{W}}\right]}^{\mathfrak{b}},\mathrm{for}\mathrm{every}\mathfrak{b}\in [0,1],$$

(11)

is a UD order relation.

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

$${\left[\tilde{\mathbb{M}}\oplus \tilde{\mathbb{W}}\right]}^{\mathfrak{b}}={\left[\tilde{\mathbb{M}}\right]}^{\mathfrak{b}}+{\left[\tilde{\mathbb{W}}\right]}^{\mathfrak{b}},$$

(12)

$${\left[\tilde{\mathbb{M}}\otimes \tilde{\mathbb{W}}\right]}^{\mathfrak{b}}={\left[\tilde{\mathbb{M}}\right]}^{\mathfrak{b}}\times {\left[\tilde{\mathbb{W}}\right]}^{\mathfrak{b}},$$

(13)

$${\left[\mathfrak{u}\odot \tilde{\mathbb{M}}\right]}^{\mathfrak{b}}=\mathfrak{u}.{\left[\tilde{\mathbb{M}}\right]}^{\mathfrak{b}}$$

(14)

These operations follow directly from Equations (4)–(6), respectively.

Theorem 1

([89]). For $\tilde{\mathbb{M}},\tilde{\mathbb{W}}\in {₤}_o$, the supremum metric

$$d_{\infty }\left(\tilde{\mathbb{M}},\tilde{\mathbb{W}}\right)=\underset{0\le \mathfrak{b}\le 1}{\mathit{sup}}{d}_H\left({\left[\tilde{\mathbb{M}}\right]}^{\mathfrak{b}},{\left[\tilde{\mathbb{W}}\right]}^{\mathfrak{b}}\right)$$

(15)

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

Theorem 2

([89,90]). If ${\rm Y}:[\rho ,\mu ]\subset \mathbb{R}\to {\mathbb{X}}_o$ is an I-V·M satisfying ${\rm Y}\left(\varkappa \right)=\left[{{\rm Y}}_{*}\left(\varkappa \right),{{\rm Y}}^{*}\left(\varkappa \right)\right]$, then ${\rm Y}$ is Aumann integrable (IA-integrable) over $[\rho ,\mu ]$ when and only when ${{\rm Y}}_{*}\left(\varkappa \right)$ and ${{\rm Y}}^{*}\left(\varkappa \right)$ are integrable over $\left[\rho ,\mu \right]$, meaning

$$\left(IA\right){\int }_{\rho }^{\mu }{\rm Y}\left(\varkappa \right)d\varkappa =\left[{\int }_{\rho }^{\begin{array}{c}\\ \mu \end{array}}{{\rm Y}}_{*}\left(\varkappa \right)d\varkappa ,{\int }_{\rho }^{\mu }{{\rm Y}}^{*}\left(\varkappa \right)d\varkappa \right].$$

(16)

Definition 3

([95]). Let $\widetilde{{\rm Y}}:\mathbb{I}\subset \mathbb{R}\to {₤}_o$ be an $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$. The family of I-V·Ms, for every $\mathfrak{b}\in \left[0,1\right]$, is ${{\rm Y}}_{\mathfrak{b}}:\mathbb{I}\subset \mathbb{R}\to {\mathbb{X}}_o$ satisfying ${{\rm Y}}_{\mathfrak{b}}\left(\varkappa \right)=\left[{{\rm Y}}_{*}\left(\varkappa ,\mathfrak{b}\right),{{\rm Y}}^{*}\left(\varkappa ,\mathfrak{b}\right)\right]$ for every $\varkappa \in \mathbb{I}.$ For every $\mathfrak{b}\in \left[0,1\right],$ the lower and upper mappings of ${{\rm Y}}_{\mathfrak{b}}$ are the end point real-valued mappings ${{\rm Y}}_{*}\left(·,\mathfrak{b}\right),{{\rm Y}}^{*}\left(·,\mathfrak{b}\right):\mathbb{I}\to \mathbb{R}$.

Definition 4

([95]). Let $\widetilde{{\rm Y}}:\mathbb{I}\subset \mathbb{R}\to {₤}_o$ be an $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$. Then, $\widetilde{{\rm Y}}\left(\varkappa \right)$ is continuous at $\varkappa \in \mathbb{I},$ if for every $\mathfrak{b}\in \left[0,1\right],$ ${{\rm Y}}_{\mathfrak{b}}\left(\varkappa \right)$ is continuous when and only when ${{\rm Y}}_{*}\left(\varkappa ,\mathfrak{b}\right)$ and ${{\rm Y}}^{*}\left(\varkappa ,\mathfrak{b}\right)$ are continuous at $\varkappa \in \mathbb{I}.$

Definition 5

([89]). Let $\widetilde{{\rm Y}}:[\rho ,\mu ]\subset \mathbb{R}\to {₤}_o$ be an $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$. The fuzzy Aumann integral ($FA$-integral) of $\widetilde{{\rm Y}}$ over $\left[\rho ,\mu \right]$ is

$${\left[\left(FA\right){\int }_{\rho }^{\mu }\widetilde{{\rm Y}}\left(\varkappa \right)d\varkappa \right]}^{\begin{array}{c}\\ b\end{array}}=\left(IA\right){\int }_{\rho }^{\mu }{{\rm Y}}_{\mathfrak{b}}\left(\varkappa \right)d\varkappa =\left\{{\int }_{\rho }^{\mu }{\rm Y}\left(\varkappa ,\mathfrak{b}\right)d\varkappa :{\rm Y}\left(\varkappa ,\mathfrak{b}\right)\in S\left({{\rm Y}}_{\mathfrak{b}}\right)\right\},$$

(17)

where $S\left({{\rm Y}}_{\mathfrak{b}}\right)=\left\{{\rm Y}\left(.,\mathfrak{b}\right)\to \mathbb{R}:{\rm Y}\left(.,\mathfrak{b}\right)isintegrable,and{\rm Y}\left(\varkappa ,\mathfrak{b}\right)\in {{\rm Y}}_{\mathfrak{b}}\left(\varkappa \right)\right\}$ for every $\mathfrak{b}\in \left[0,1\right].$ Moreover, $\widetilde{{\rm Y}}$ is $\left(FA\right)$-integrable over $[\rho ,\mu ]$ if $\left(FA\right){\int }_{\rho }^{\mu }\widetilde{{\rm Y}}\left(\varkappa \right)d\varkappa \in {₤}_o.$

Theorem 3

([90]). Let $\widetilde{{\rm Y}}:[\rho ,\mu ]\subset \mathbb{R}\to {₤}_o$ be an $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$ whose $\mathfrak{b}$-levels define the family of I-V·Ms ${{\rm Y}}_{\mathfrak{b}}:[\rho ,\mu ]\subset \mathbb{R}\to {\mathbb{X}}_o$ satisfying ${{\rm Y}}_{\mathfrak{b}}\left(\varkappa \right)=\left[{{\rm Y}}_{*}\left(\varkappa ,\mathfrak{b}\right),{{\rm Y}}^{*}\left(\varkappa ,\mathfrak{b}\right)\right]$ for every $\varkappa \in [\rho ,\mu ]$ and $\mathfrak{b}\in \left[0,1\right].$ $\widetilde{{\rm Y}}$ is $\left(FA\right)$-integrable over $[\rho ,\mu ]$ when and only when ${{\rm Y}}_{*}\left(\varkappa ,\mathfrak{b}\right)$ and ${{\rm Y}}^{*}\left(\varkappa ,\mathfrak{b}\right)$ are integrable over $[\rho ,\mu ]$. Moreover, if $\widetilde{{\rm Y}}$ is $\left(FA\right)$-integrable over $\left[\rho ,\mu \right]$, we have

$${\left[\left(FA\right){\int }_{\rho }^{\mu }\widetilde{{\rm Y}}\left(\varkappa \right)d\varkappa \right]}^{\begin{array}{c}\\ \mathfrak{b}\end{array}}=\left[{\int }_{\rho }^{\mu }{{\rm Y}}_{*}\left(\varkappa ,\mathfrak{b}\right)d\varkappa ,{\int }_{\rho }^{\mu }{{\rm Y}}^{*}\left(\varkappa ,\mathfrak{b}\right)d\varkappa \right]=\left(IA\right){\int }_{\rho }^{\mu }{{\rm Y}}_{\mathfrak{b}}\left(\varkappa \right)d\varkappa$$

(18)

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

Definition 6

([32]). Let $\beta >0$ and $L\left(\left[\rho ,\mu \right],{₤}_o\right)$ be the collection of all Lebesgue measurable fuzzy-number-valued mappings on $[\rho ,\mu ]$. Then, the fuzzy left and right RL fractional integrals of order $\beta >0$ of ${\rm Y}\in$$L\left(\left[\rho ,\mu \right],{₤}_o\right)$ are

$${\mathcal{I}}_{{\rho }^{+}}^{\beta }\widetilde{{\rm Y}}\left(\varkappa \right)=\frac{1}{\Gamma \left(\beta \right)}{\int }_{\rho }^{\varkappa }{\left(\varkappa -m\right)}^{\beta -1}\widetilde{{\rm Y}}\left(m\right)dm,\left(\varkappa >\rho \right),$$

(19)

and

$${\mathcal{I}}_{{\mu }^{-}}^{\beta }\widetilde{{\rm Y}}\left(\varkappa \right)=\frac{1}{\Gamma \left(\beta \right)}{\int }_{\varkappa }^{\mu }{\left(m-\varkappa \right)}^{\beta -1}\widetilde{{\rm Y}}\left(m\right)dm,(\varkappa <\mu )$$

(20)

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

$$\begin{array}{cc}\hfill {\left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }\widetilde{{\rm Y}}\left(\varkappa \right)\right]}^{\mathfrak{b}}& =\frac{1}{\Gamma \left(\beta \right)}{\int }_{\rho }^{\varkappa }{\left(\varkappa -m\right)}^{\beta -1}{{\rm Y}}_{\mathfrak{b}}\left(m\right)dm\hfill \\ & =\frac{1}{\Gamma \left(\beta \right)}{\int }_{\rho }^{\varkappa }{\left(\varkappa -m\right)}^{\beta -1}\left[{{\rm Y}}_{*}\left(m,\mathfrak{b}\right),{{\rm Y}}^{*}\left(m,\mathfrak{b}\right)\right]dm,\left(\varkappa >\rho \right),\hfill \end{array}$$

(21)

where

$${\mathcal{I}}_{{\rho }^{+}}^{\beta }{{\rm Y}}_{*}\left(\varkappa ,\mathfrak{b}\right)=\frac{1}{\Gamma \left(\beta \right)}{\int }_{\rho }^{\varkappa }{\left(\varkappa -m\right)}^{\beta -1}{{\rm Y}}_{*}\left(m,\mathfrak{b}\right)dm,\left(\varkappa >\rho \right),$$

(22)

and

$${\mathcal{I}}_{{\rho }^{+}}^{\beta }{{\rm Y}}^{*}\left(\varkappa ,\mathfrak{b}\right)=\frac{1}{\Gamma \left(\beta \right)}{\int }_{\rho }^{\varkappa }{\left(\varkappa -m\right)}^{\beta -1}{{\rm Y}}^{*}\left(m,\mathfrak{b}\right)dm,\left(\varkappa >\rho \right).$$

(23)

The RL fractional integral $\widetilde{{\rm Y}}$ of $\varkappa$ based on left and right end point mappings can be defined in a similar way.

An interval-valued mapping ${\rm Y}:\mathbb{I}=\left[\rho ,\mu \right]\to {\mathbb{X}}_o$ is a convex interval-valued mapping if

$${\rm Y}\left(\mathfrak{u}\varkappa +\left(1-\mathfrak{u}\right)y\right)\supseteq \mathfrak{u}{\rm Y}\left(\varkappa \right)+(1-\mathfrak{u}){\rm Y}\left(y\right),$$

(24)

For all $\varkappa ,y\in \left[\rho ,\mu \right],\mathfrak{u}\in \left[0,1\right]$, where ${\mathbb{X}}_o$ is the collection of all real-valued intervals. If (24) is reversed, then ${\rm Y}$ is said to be concave (see [91]).

Definition 7

([87]). The $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$$\widetilde{{\rm Y}}:\left[\rho ,\mu \right]\to {₤}_o$ is convex on $\left[\rho ,\mu \right]$ if

$$\widetilde{{\rm Y}}\left(\mathfrak{u}\varkappa +\left(1-\mathfrak{u}\right)y\right){\le }_{\mathbb{F}}\mathfrak{u}\odot \widetilde{{\rm Y}}\left(\varkappa \right)\oplus (1-\mathfrak{u})\odot \widetilde{{\rm Y}}\left(y\right),$$

(25)

for all $\varkappa ,y\in \left[\rho ,\mu \right],\mathfrak{u}\in \left[0,1\right],$ where $\widetilde{{\rm Y}}\left(\varkappa \right){\ge }_{\mathbb{F}}\tilde{0}$ for all $\varkappa \in \left[\rho ,\mu \right].$ If (25) is reversed, then $\widetilde{{\rm Y}}$ is said to be concave. Moreover, $\widetilde{{\rm Y}}$ is named affine if and only if it is convex and concave.

Definition 8

([96]). The $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$ $\widetilde{{\rm Y}}:\left[\rho ,\mu \right]\to {₤}_o$ is UD convex on $\left[\rho ,\mu \right]$ if

$$\widetilde{{\rm Y}}\left(\mathfrak{u}\varkappa +\left(1-\mathfrak{u}\right)y\right){\supseteq }_{\mathbb{F}}\mathfrak{u}\odot \widetilde{{\rm Y}}\left(\varkappa \right)\oplus (1-\mathfrak{u})\odot \widetilde{{\rm Y}}\left(y\right),$$

(26)

for all $\varkappa ,y\in \left[\rho ,\mu \right],\mathfrak{u}\in \left[0,1\right],$ where $\widetilde{{\rm Y}}\left(\varkappa \right){\ge }_{\mathbb{F}}\tilde{0}$ for all $\varkappa \in \left[\rho ,\mathit{\mu }\right].$ If (26) is reversed then, $\widetilde{{\rm Y}}$ is UD concave. Moreover, $\widetilde{{\rm Y}}$ is named UD affine if and only if it is UD convex and concave.

Definition 9

([44]). Let $K$ be an invex set. The $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$ $\widetilde{{\rm Y}}:K\to {₤}_o$ is UD pre-invex on $K$ with respect to $\phi$ if

$$\widetilde{{\rm Y}}\left(\varkappa +\left(1-\mathfrak{u}\right)\phi \left(y,\varkappa \right)\right){\supseteq }_{\mathbb{F}}\mathfrak{u}\odot \widetilde{{\rm Y}}\left(\varkappa \right)\oplus \left(1-\mathfrak{u}\right)\odot \widetilde{{\rm Y}}\left(y\right),$$

(27)

for all $\varkappa ,y\in K,\mathfrak{u}\in \left[0,1\right],$ where $\widetilde{{\rm Y}}\left(\varkappa \right){\ge }_{\mathbb{F}}\tilde{0},$  $\phi :K\times K\to \mathbb{R}.$ The mapping $\widetilde{{\rm Y}}$ is UD pre-incave on $K$ with respect to $\phi$ if (27) is reversed.

Theorem 4

([4]). Let $\widetilde{{\rm Y}}:[\rho ,\mu ]\to {₤}_o$ be an $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$ whose $\mathfrak{b}$-levels describe the family of I-V·Ms ${{\rm Y}}_{\mathfrak{b}}:[\rho ,\mu ]\to {\mathbb{X}}_o^{+}\subset {\mathbb{X}}_o$ as

$${{\rm Y}}_{\mathfrak{b}}\left(\varkappa \right)=\left[{{\rm Y}}_{*}\left(\varkappa ,\mathfrak{b}\right),{{\rm Y}}^{*}\left(\varkappa ,\mathfrak{b}\right)\right],$$

(28)

for all $\varkappa \in [\rho ,\mu ]$ and all $\mathfrak{b}\in \left[0,1\right]$. The $\widetilde{{\rm Y}}$ is UD pre-invex on $[\rho ,\mu ]$ if and only if for all $\mathfrak{b}\in \left[0,1\right],$  ${{\rm Y}}_{*}\left(\varkappa ,\mathfrak{b}\right)$ is pre-invex and ${{\rm Y}}^{*}\left(\varkappa ,\mathfrak{b}\right)$ is pre-incave.

Definition 10

([4]). Let $\widetilde{{\rm Y}}:[\rho ,\mu ]\to {₤}_o$ be an $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$ whose $\mathfrak{b}$-levels describe the family of I-V·Ms ${{\rm Y}}_{\mathfrak{b}}:[\rho ,\mu ]\to {\mathcal{C}}_o^{+}\subset {\mathcal{C}}_o$ as

$${{\rm Y}}_{\mathfrak{b}}\left(\varkappa \right)=\left[{{\rm Y}}_{*}\left(\varkappa ,\mathfrak{b}\right),{{\rm Y}}^{*}\left(\varkappa ,\mathfrak{b}\right)\right],$$

for all $\mathit{\varkappa }\in [\mathit{\rho },\mathit{\mu }]$ and all $\mathfrak{b}\in \left[\mathbf{0},\mathbf{1}\right]$. Then, $\widetilde{\mathit{{\rm Y}}}$ is lower UDpre-invex (pre-incave) on $[\mathit{\rho },\mathit{\mu }]$ if and only if for all $\mathfrak{b}\in \left[\mathbf{0},\mathbf{1}\right],$

$${\mathit{{\rm Y}}}_{\mathit{*}}\left(\mathit{\varkappa }\mathbf{+}\left(\mathbf{1}\mathrm{-}\mathfrak{u}\right)\mathit{\phi }\left(\mathit{y},\mathit{\varkappa }\right),\mathfrak{b}\right)\le \left(\ge \right)\mathfrak{u}{\mathit{{\rm Y}}}_{\mathit{*}}\left(\mathit{\varkappa },\mathfrak{b}\right)+\left(\mathbf{1}-\mathfrak{u}\right){\mathit{{\rm Y}}}_{\mathit{*}}\left(\mathit{y},\mathfrak{b}\right),$$

(29)

and

$${\mathit{{\rm Y}}}^{\mathit{*}}\left(\mathit{\varkappa }+\left(\mathbf{1}-\mathfrak{u}\right)\mathit{\phi }\left(\mathit{y},\mathit{\varkappa }\right),\mathfrak{b}\right)=\mathfrak{u}{\mathit{{\rm Y}}}^{\mathit{*}}\left(\mathit{\varkappa },\mathfrak{b}\right)+\left(\mathbf{1}-\mathfrak{u}\right){\mathit{{\rm Y}}}^{\mathit{*}}\left(\mathit{y},\mathfrak{b}\right).$$

(30)

Definition 11

([4]). Let $\widetilde{{\rm Y}}:[\rho ,\mu ]\to {₤}_o$ be an $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$ whose $\mathfrak{b}$-levels describe the family of I-V·Ms ${{\rm Y}}_{\mathfrak{b}}:[\rho ,\mu ]\to {\mathcal{C}}_o^{+}\subset {\mathcal{C}}_o$ as

$${{\rm Y}}_{\mathfrak{b}}\left(\varkappa \right)=\left[{{\rm Y}}_{*}\left(\varkappa ,\mathfrak{b}\right),{{\rm Y}}^{*}\left(\varkappa ,\mathfrak{b}\right)\right],$$

for all $\mathit{\varkappa }\in [\mathit{\rho },\mathit{\mu }]$ and all $\mathfrak{b}\in \left[\mathbf{0},\mathbf{1}\right]$. Then, $\widetilde{\mathit{{\rm Y}}}$ is upper UD pre-invex (pre-incave) on $[\mathit{\rho },\mathit{\mu }]$ if and only if for all $\mathfrak{b}\in \left[\mathbf{0},\mathbf{1}\right],$

$${\mathit{{\rm Y}}}_{\mathit{*}}\left(\mathit{\varkappa }+\left(\mathbf{1}-\mathfrak{u}\right)\mathit{\phi }\left(\mathit{y},\mathit{\varkappa }\right),\mathfrak{b}\right)=\mathfrak{u}{\mathit{{\rm Y}}}_{\mathit{*}}\left(\mathit{\varkappa },\mathfrak{b}\right)+\left(\mathbf{1}-\mathfrak{u}\right){\mathit{{\rm Y}}}_{\mathit{*}}\left(\mathit{y},\mathfrak{b}\right),$$

(31)

and

$${\mathit{{\rm Y}}}^{\mathit{*}}\left(\mathit{\varkappa }+\left(\mathbf{1}-\mathfrak{u}\right)\mathit{\phi }\left(\mathit{y},\mathit{\varkappa }\right),\mathfrak{b}\right)\le \left(\ge \right)\mathfrak{u}{\mathit{{\rm Y}}}^{\mathit{*}}\left(\mathit{\varkappa },\mathfrak{b}\right)+\left(\mathbf{1}-\mathfrak{u}\right){\mathit{{\rm Y}}}^{\mathit{*}}\left(\mathit{y},\mathfrak{b}\right).$$

(32)

Remark 2

([4]). Both concepts UD pre-invex $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$ and pre-invex $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$ concide when $\widetilde{{\rm Y}}$ is a lower UD pre-invex $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$, see [84].

Assumption 1

([7]). Let us assume that $K$ is an invex set with respect to the bi-function $\phi :K\times K\to \mathbb{R}.$ For any $\rho ,\mu \in K$ and $\mathfrak{u}\in \left[0,1\right]$, so it yields

$$\phi \left(\mu ,\rho +\mathfrak{u}\phi (\mu ,\rho )\right)=\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),$$

(33)

$$\phi \left(\rho ,\rho +\mathfrak{u}\phi (\mu ,\rho )\right)=-\mathfrak{u}\phi \left(\mu ,\rho \right).$$

(34)

For $\mathfrak{u}$ = 0, we have $\phi \left(\mu ,\rho \right)$ = 0 if and only if $\mu =\rho$, and for all $\rho ,\mu \in K$.

For more details on Assumption 1, please see references [83,84,85,86].

3. Main Results

We develop new results on the well-known fractional HH and HH–Fejér inequalities in the scope of UD pre-invex $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$s. In the follow-up, we denote by $L\left(\left[\rho ,\rho +\phi \left(\mu ,\rho \right)\right],{₤}_0\right)$ the family of Lebesgue measurable $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$s.

Theorem 5.

Let $\widetilde{{\rm Y}}:\left[\rho ,\rho +\phi \left(\mu ,\rho \right)\right]\to {₤}_0$ be a UD pre-invex $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$ on $\left[\rho ,\rho +\phi \left(\mu ,\rho \right)\right]$ whose $\mathfrak{b}$-levels define the family of I⋅V⋅Ms ${{\rm Y}}_{\mathfrak{b}}:\left[\rho ,\rho +\phi \left(\mu ,\rho \right)\right]\subset \mathbb{R}\to {\mathbb{X}}_o^{+}$ as ${{\rm Y}}_{\mathfrak{b}}\left(\varkappa \right)=\left[{{\rm Y}}_{*}\left(\varkappa ,\mathfrak{b}\right),{{\rm Y}}^{*}\left(\varkappa ,\mathfrak{b}\right)\right]$ for all $\varkappa \in \left[\rho ,\rho +\phi \left(\mu ,\rho \right)\right]$ and for all $\mathfrak{b}\in \left[0,1\right]$. If $\phi$ satisfies Assumption 1 and $\widetilde{{\rm Y}}\in L\left(\left[\rho ,\rho +\phi \left(\mu ,\rho \right)\right],{₤}_0\right)$, then

$$\begin{gathered} \widetilde{{\rm Y}}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2}\right){\supseteq }_{\mathbb{F}}\frac{\Gamma \left(\beta +1\right)}{{2\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\odot \left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }\widetilde{{\rm Y}}\left(\rho +\phi \left(\mu ,\rho \right)\right)\oplus {\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }\widetilde{{\rm Y}}\left(\rho \right)\right] \\ {\supseteq }_{\mathbb{F}}\frac{\widetilde{{\rm Y}}\left(\rho \right)\oplus \widetilde{{\rm Y}}\left(\rho +\phi \left(\mu ,\rho \right)\right)}{2}{\supseteq }_{\mathbb{F}}\frac{\widetilde{{\rm Y}}\left(\rho \right)\oplus \widetilde{{\rm Y}}\left(\mu \right)}{2}. \end{gathered}$$

(35)

If  $\widetilde{{\rm Y}}\left(\varkappa \right)$ is UD pre-incave, then

$$\begin{gathered} \widetilde{{\rm Y}}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2}\right){\subseteq }_{\mathbb{F}}\frac{\Gamma \left(\beta +1\right)}{{2\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\odot \left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }\widetilde{{\rm Y}}\left(\rho +\phi \left(\mu ,\rho \right)\right)\oplus {\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }\widetilde{{\rm Y}}\left(\rho \right)\right] \\ {\subseteq }_{\mathbb{F}}\frac{\widetilde{{\rm Y}}\left(\rho \right)\oplus \widetilde{{\rm Y}}\left(\rho +\phi \left(\mu ,\rho \right)\right)}{2}{\subseteq }_{\mathbb{F}}\frac{\widetilde{{\rm Y}}\left(\rho \right)\oplus \widetilde{{\rm Y}}\left(\mu \right)}{2}. \end{gathered}$$

(36)

Proof. 

Let $\widetilde{{\rm Y}}:\left[\rho ,\rho +\phi \left(\mu ,\rho \right)\right]\to {₤}_0$ be a UD pre-invex $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$. If Assumption 1 is verified, then we have

$$2\odot \widetilde{{\rm Y}}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2}\right){\supseteq }_{\mathbb{F}}\widetilde{{\rm Y}}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right)\right)\oplus \widetilde{{\rm Y}}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right)\right).$$

Therefore, for every $\mathfrak{b}\in [0,1]$, we have

$$\begin{array}{c}2{{\rm Y}}_{*}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2},\mathfrak{b}\right)\le {{\rm Y}}_{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right)+{{\rm Y}}_{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right),\\ 2{{\rm Y}}^{*}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2},\mathfrak{b}\right)\ge {{\rm Y}}^{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right)+{{\rm Y}}^{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right).\end{array}$$

By multiplying both sides by ${\mathfrak{u}}^{\beta -1}$ and integrating the result with respect to $\mathfrak{u}$ over $\left(\mathrm{0,1}\right)$, it yields

$$\begin{array}{c}2{\int }_0^1{\mathfrak{u}}^{\beta -1}{{\rm Y}}_{*}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2},\mathfrak{b}\right)d\mathfrak{u}\\ \le {\int }_0^1{\mathfrak{u}}^{\beta -1}{{\rm Y}}_{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right)d\mathfrak{u}+{\int }_0^1{\mathfrak{u}}^{\beta -1}{{\rm Y}}_{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right)d\mathfrak{u},\\ 2{\int }_0^1{\mathfrak{u}}^{\beta -1}{{\rm Y}}^{*}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2},\mathfrak{b}\right)d\mathfrak{u}\\ \ge {\int }_0^1{\mathfrak{u}}^{\beta -1}{{\rm Y}}^{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right)d\mathfrak{u}+{\int }_0^1{\mathfrak{u}}^{\beta -1}{{\rm Y}}^{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right)d\mathfrak{u}.\end{array}$$

Let $\varkappa =\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right)$ and $y=\rho +\mathfrak{u}\phi \left(\mu ,\rho \right).$ Then, we get

$$\begin{array}{c}\frac{2}{\beta }{{\rm Y}}_{*}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2},\mathfrak{b}\right)\le \frac{1}{{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\underset{\rho }{\overset{\rho +\phi \left(\mu ,\rho \right)}{\int }}{\left(\rho +\phi \left(\mu ,\rho \right)-y\right)}^{\beta -1}{{\rm Y}}_{*}\left(y,\mathfrak{b}\right)dy\\ +\frac{1}{{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\underset{\rho }{\overset{\rho +\phi \left(\mu ,\rho \right)}{\int }}{\left(\varkappa -\rho \right)}^{\beta -1}{{\rm Y}}_{*}\left(\varkappa ,\mathfrak{b}\right)d\varkappa \\ \le \frac{\Gamma \left(\beta \right)}{{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }{{\rm Y}}_{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }{{\rm Y}}_{*}\left(\rho ,\mathfrak{b}\right)\right]\\ \frac{2}{\beta }{{\rm Y}}_{*}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2},\mathfrak{b}\right)\ge \frac{1}{{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\underset{\rho }{\overset{\rho +\phi \left(\mu ,\rho \right)}{\int }}{\left(\rho +\phi \left(\mu ,\rho \right)-y\right)}^{\beta -1}{{\rm Y}}^{*}\left(y,\mathfrak{b}\right)dy\\ +\frac{1}{{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\underset{\rho }{\overset{\rho +\phi \left(\mu ,\rho \right)}{\int }}{\left(\varkappa -\rho \right)}^{\beta -1}{{\rm Y}}^{*}\left(\varkappa ,\mathfrak{b}\right)d\varkappa \\ \ge \frac{\Gamma \left(\beta \right)}{{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }{{\rm Y}}^{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }{{\rm Y}}^{*}\left(\rho ,\mathfrak{b}\right)\right].\end{array}$$

That is,

$$\begin{gathered} \frac{2}{\beta }\left[{{\rm Y}}_{*}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2},\mathfrak{b}\right),{{\rm Y}}^{*}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2},\mathfrak{b}\right)\right] \\ {\supseteq }_I\frac{\Gamma \left(\beta \right)}{{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }{{\rm Y}}_{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }{{\rm Y}}_{*}\left(\rho ,\mathfrak{b}\right),{\mathcal{I}}_{{\rho }^{+}}^{\beta }{{\rm Y}}^{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right) \\ +{\mathcal{I}}_{{\mu }^{-}}^{\beta }{{\rm Y}}^{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\right]. \end{gathered}$$

Thus,

$$\frac{2}{\beta }\odot \widetilde{{\rm Y}}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2}\right){\supseteq }_{\mathbb{F}}\frac{\Gamma \left(\beta \right)}{{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\odot \left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }\widetilde{{\rm Y}}\left(\rho +\phi \left(\mu ,\rho \right)\right)\oplus {\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }\widetilde{{\rm Y}}\left(\rho \right)\right].$$

(37)

Adopting a similar procedure, we have

$$\begin{array}{l}\frac{\Gamma \left(\beta \right)}{{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\\ \odot [{\mathcal{I}}_{{\rho }^{+}}^{\beta }\widetilde{{\rm Y}}\left(\rho +\phi \left(\mu ,\rho \right)\right)\\ \oplus {\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }\widetilde{{\rm Y}}\left(\rho \right)]{\supseteq }_{\mathbb{F}}\frac{\widetilde{{\rm Y}}\left(\rho \right)\oplus \widetilde{{\rm Y}}\left(\rho +\phi \left(\mu ,\rho \right)\right)}{2}{\supseteq }_{\mathbb{F}}\frac{\widetilde{{\rm Y}}\left(\rho \right)\oplus \widetilde{{\rm Y}}\left(\mu \right)}{2}.\end{array}$$

(38)

Combining (37) and (38) yields

$$\begin{array}{l}\widetilde{{\rm Y}}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2}\right){\supseteq }_{\mathbb{F}}\frac{\Gamma \left(\beta +1\right)}{{2\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\\ \odot [{\mathcal{I}}_{{\rho }^{+}}^{\beta }\widetilde{{\rm Y}}\left(\rho +\phi \left(\mu ,\rho \right)\right)\\ \oplus {\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }\widetilde{{\rm Y}}\left(\rho \right)]{\supseteq }_{\mathbb{F}}\frac{\widetilde{{\rm Y}}\left(\rho \right)\oplus \widetilde{{\rm Y}}\left(\rho +\phi \left(\mu ,\rho \right)\right)}{2}{\supseteq }_{\mathbb{F}}\frac{\widetilde{{\rm Y}}\left(\rho \right)\oplus \widetilde{{\rm Y}}\left(\mu \right)}{2}.\end{array}$$

Thus, this concludes the proof. □

Remark 3.

If $\phi \left(\varkappa ,y\right)=\varkappa -y$, then, from Theorem 5, we get [86]

$$\widetilde{{\rm Y}}\left(\frac{\rho +\mu }{2}\right){\supseteq }_{\mathbb{F}}\frac{\Gamma \left(\beta +1\right)}{{2\left(\mu -\rho \right)}^{\beta }}\odot \left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }\widetilde{{\rm Y}}\left(\mu \right)\oplus {\mathcal{I}}_{{\mu }^{-}}^{\beta }\widetilde{{\rm Y}}\left(\rho \right)\right]{\supseteq }_{\mathbb{F}}\frac{\widetilde{{\rm Y}}\left(\rho \right)\oplus \widetilde{{\rm Y}}\left(\mu \right)}{2}$$

(39)

Let us assume $\beta =1$. Then, Theorem 5 yields the result for a pre-invex $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$ [4]

$$\widetilde{{\rm Y}}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2}\right){\supseteq }_{\mathbb{F}}\frac{1}{\phi \left(\mu ,\rho \right)}\odot {\int }_{\rho }^{\rho +\phi \left(\mu ,\rho \right)}\widetilde{{\rm Y}}\left(\varkappa \right)d\varkappa {\supseteq }_{\mathbb{F}}\frac{\widetilde{{\rm Y}}\left(\rho \right)\oplus \widetilde{{\rm Y}}\left(\mu \right)}{2}$$

(40)

Let us assume $\beta =1$ and $\phi \left(\varkappa ,y\right)=\varkappa -y$. Then, Theorem 5 yields the result for convex $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$ [86]

$$\widetilde{{\rm Y}}\left(\frac{\rho +\mu }{2}\right){\supseteq }_{\mathbb{F}}\frac{1}{\mu -\rho }\odot {\int }_{\rho }^{\mu }\widetilde{{\rm Y}}\left(\varkappa \right)d\varkappa {\supseteq }_{\mathbb{F}}\frac{\widetilde{{\rm Y}}\left(\rho \right)\oplus \widetilde{{\rm Y}}\left(\mu \right)}{2}$$

(41)

If $\widetilde{{\rm Y}}$ is a lower UD pre-invex $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$, then, from Theorem 5, we get [83]

$$\begin{gathered} \widetilde{{\rm Y}}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2}\right){\le }_{\mathbb{F}}\frac{\Gamma \left(\beta +1\right)}{{2\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\odot \left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }\widetilde{{\rm Y}}\left(\rho +\phi \left(\mu ,\rho \right)\right)\oplus {\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }\widetilde{{\rm Y}}\left(\rho \right)\right] \\ {\le }_{\mathbb{F}}\frac{\widetilde{{\rm Y}}\left(\rho \right)\oplus \widetilde{{\rm Y}}\left(\rho +\phi \left(\mu ,\rho \right)\right)}{2}{\le }_{\mathbb{F}}\frac{\widetilde{{\rm Y}}\left(\rho \right)\oplus \widetilde{{\rm Y}}\left(\mu \right)}{2}. \end{gathered}$$

(42)

If $\widetilde{{\rm Y}}$ is a lower UD pre-invex $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$ with $\phi \left(\varkappa ,y\right)=\varkappa -y$, then, from Theorem 5, we get (see [83])

$$\widetilde{{\rm Y}}\left(\frac{\rho +\mu }{2}\right){\le }_{\mathbb{F}}\frac{\Gamma \left(\beta +1\right)}{{2\left(\mu -\rho \right)}^{\beta }}\odot \left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }\widetilde{{\rm Y}}\left(\mu \right)\oplus {\mathcal{I}}_{{\mu }^{-}}^{\beta }\widetilde{{\rm Y}}\left(\rho \right)\right]{\le }_{\mathbb{F}}\frac{\widetilde{{\rm Y}}\left(\rho \right)\oplus \widetilde{{\rm Y}}\left(\mu \right)}{2}$$

(43)

Let us consider $\beta =1$ and $\widetilde{{\rm Y}}$ a lower UD pre-invex $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$. Then, Theorem 5 yields the result for pre-invex $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$ (see [84])

$$\widetilde{{\rm Y}}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2}\right){\le }_{\mathbb{F}}\frac{1}{\phi \left(\mu ,\rho \right)}\odot {\int }_{\rho }^{\rho +\phi \left(\mu ,\rho \right)}\widetilde{{\rm Y}}\left(\varkappa \right)d\varkappa {\le }_{\mathbb{F}}\frac{\widetilde{{\rm Y}}\left(\rho \right)\oplus \widetilde{{\rm Y}}\left(\mu \right)}{2}$$

(44)

Let us consider $\beta =1$ and $\widetilde{{\rm Y}}$ a lower UD pre-invex F⋅N⋅V⋅M $\phi \left(\varkappa ,y\right)=\varkappa -y$. Then, Theorem 5 yields the result for a convex $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$ (see [81])

$$\widetilde{{\rm Y}}\left(\frac{\rho +\mu }{2}\right){\le }_{\mathbb{F}}\frac{1}{\mu -\rho }\odot {\int }_{\rho }^{\mu }\widetilde{{\rm Y}}\left(\varkappa \right)d\varkappa {\le }_{\mathbb{F}}\frac{\widetilde{{\rm Y}}\left(\rho \right)\oplus \widetilde{{\rm Y}}\left(\mu \right)}{2}$$

(45)

Let us assume that $\beta =1=\mathfrak{b}$ and ${{\rm Y}}_{*}(\varkappa ,\mathfrak{b})={{\rm Y}}^{*}(\varkappa ,\mathfrak{b})$ with $\phi \left(\varkappa ,y\right)=\varkappa -y$. Then, from Theorem 5, we obtain the classical HH–Fejér type inequality.

Example 1.

Let $\beta =\frac{1}{2},\varkappa \in \left[\mathrm{2,2}+\phi \left(\mathrm{3,2}\right)\right]$ and the $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$$\widetilde{{\rm Y}}:\left[\rho ,\rho +\phi \left(\mu ,\rho \right)\right]=\left[2,2+\phi \left(\mathrm{3,2}\right)\right]\to {₤}_0$ be defined as

$$\widetilde{{\rm Y}}\left(\varkappa \right)\left(\theta \right)=\left\{\begin{array}{c}\frac{\theta -2+{\varkappa }^{\frac{1}{2}}}{1+{\varkappa }^{\frac{1}{2}}}\theta \in \left[2-{\varkappa }^{\frac{1}{2}},3\right]\\ \frac{2+{\varkappa }^{\frac{1}{2}}-\theta }{{\varkappa }^{\frac{1}{2}}+1}\theta \in (3,2+{\varkappa }^{\frac{1}{2}}]\\ 0otherwise.\end{array}\right.$$

(46)

For each $\mathfrak{b}\in \left[0,1\right],$ we have ${{\rm Y}}_{\mathfrak{b}}\left(\varkappa \right)=\left[\left(1-\mathfrak{b}\right)\left(2-{\varkappa }^{\frac{1}{2}}\right)+3\mathfrak{b},\left(1-\mathfrak{b}\right)\left(2+{\varkappa }^{\frac{1}{2}}\right)+3\mathfrak{b}\right]$. Since the left and right end point functions ${{\rm Y}}_{\mathrm{*}}\left(\varkappa ,\mathfrak{b}\right)=\left(1-\mathfrak{b}\right)\left(2-{\varkappa }^{\frac{1}{2}}\right)+3\mathfrak{b}$ and ${{\rm Y}}^{\mathrm{*}}\left(\varkappa ,\mathfrak{b}\right)=\left(1-\mathfrak{b}\right)\left(2+{\varkappa }^{\frac{1}{2}}\right)+3\mathfrak{b}$ are pre-invex functions with respect to $\phi \left(\mu ,\rho \right)=\mu -\rho$, for each $\mathfrak{b}\in [0,1]$, $\widetilde{{\rm Y}}\left(\varkappa \right)$ is a UD pre-invex $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$. We verify that $\widetilde{{\rm Y}}\in L\left(\left[\rho ,\rho +\phi \left(\mu ,\rho \right)\right],{\mathrm{₤}}_0\right)$ and

$$\begin{gathered} {{\rm Y}}_{*}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2},\mathfrak{b}\right)={{\rm Y}}_{*}\left(\frac{5}{2},\mathfrak{b}\right)=\left(1-\mathfrak{b}\right)\frac{4-\sqrt{10}}{2}+3\mathfrak{b}, \\ {{\rm Y}}^{*}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2},\mathfrak{b}\right)={{\rm Y}}^{*}\left(\frac{5}{2},\mathfrak{b}\right)=\left(1-\mathfrak{b}\right)\frac{4+\sqrt{10}}{2}+3\mathfrak{b}, \\ \frac{{{\rm Y}}_{*}\left(\rho ,\mathfrak{b}\right)+{{\rm Y}}_{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)}{2}=\left(1-\mathfrak{b}\right)\left(\frac{4-\sqrt{2}-\sqrt{3}}{2}\right)+3\mathfrak{b}, \\ \frac{{{\rm Y}}^{*}\left(\rho ,\mathfrak{b}\right)+{{\rm Y}}^{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)}{2}=\left(1-\mathfrak{b}\right)\left(\frac{4+\sqrt{2}+\sqrt{3}}{2}\right)+3\mathfrak{b}. \end{gathered}$$

Note that

$$\begin{gathered} \begin{array}{c}\frac{\Gamma (\beta +1)}{{2\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }{{\rm Y}}_{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }{{\rm Y}}_{*}\left(\rho ,\mathfrak{b}\right)\right]\\ =\frac{\Gamma \left(\frac{3}{2}\right)}{2}\frac{1}{\sqrt{\pi }}{\int }_2^{2+\phi \left(\mathrm{3,2}\right)}{\left(3-\varkappa \right)}^{\frac{-1}{2}}.\left(\left(1-\mathfrak{b}\right)\left(2-{\varkappa }^{\frac{1}{2}}\right)+3\mathfrak{b}\right)d\varkappa \\ +\frac{\Gamma \left(\frac{3}{2}\right)}{2}\frac{1}{\sqrt{\pi }}{\int }_2^{2+\phi \left(\mathrm{3,2}\right)}{\left(\varkappa -2\right)}^{\frac{-1}{2}}.\left(\left(1-\mathfrak{b}\right)\left(2-{\varkappa }^{\frac{1}{2}}\right)+3\mathfrak{b}\right)d\varkappa \\ =\frac{1}{4}\left(1-\mathfrak{b}\right)\left[\frac{7393}{\text{10,000}}+\frac{9501}{\text{10,000}}\right]+3\mathfrak{b}\\ =\left(1-\mathfrak{b}\right)\frac{8447}{\text{20,000}}+3\mathfrak{b},\end{array} \\ \begin{array}{c}\frac{\Gamma (\beta +1)}{{2\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }{{\rm Y}}^{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }{{\rm Y}}^{*}\left(\rho ,\mathfrak{b}\right)\right]\\ =\frac{\Gamma \left(\frac{3}{2}\right)}{2}\frac{1}{\sqrt{\pi }}{\int }_2^{2+\phi \left(\mathrm{3,2}\right)}{\left(3-\varkappa \right)}^{\frac{-1}{2}}.\left(\left(1-\mathfrak{b}\right)\left(2+{\varkappa }^{\frac{1}{2}}\right)+3\mathfrak{b}\right)d\varkappa \\ +\frac{\Gamma \left(\frac{3}{2}\right)}{2}\frac{1}{\sqrt{\pi }}{\int }_2^{2+\phi \left(\mathrm{3,2}\right)}{\left(\varkappa -2\right)}^{\frac{-1}{2}}.\left(\left(1-\mathfrak{b}\right)\left(2+{\varkappa }^{\frac{1}{2}}\right)+3\mathfrak{b}\right)d\varkappa \\ =\frac{1}{4}\left(1-\mathfrak{b}\right)\left[\frac{7260}{1000}+\frac{7049}{1000}\right]+3\mathfrak{b}\\ =\left(1-\mathfrak{b}\right)\frac{\text{14,309}}{4000}+3\mathfrak{b}.\end{array} \end{gathered}$$

Therefore,

$$\begin{array}{l}\left[\left(1-\mathfrak{b}\right)\frac{4-\sqrt{10}}{2}+3\mathfrak{b},\left(1-\mathfrak{b}\right)\frac{4+\sqrt{10}}{2}+3\mathfrak{b}\right]{\supseteq }_I\left[\left(1-\mathfrak{b}\right)\frac{8447}{\mathrm{20,000}}+3\mathfrak{b},\left(1-\mathfrak{b}\right)\frac{\text{14,309}}{4000}+3\mathfrak{b}\right]\\ {\supseteq }_I\left[\left(1-\mathfrak{b}\right)\left(\frac{4-\sqrt{2}-\sqrt{3}}{2}\right)+3\mathfrak{b},\left(1-\mathfrak{b}\right)\left(\frac{4+\sqrt{2}+\sqrt{3}}{2}\right)+3\mathfrak{b}\right],\end{array}$$

and Theorem 5 is verified.

We now introduce some new forms of fuzzy-interval fractional HH type inequalities for the product of UD pre-invex $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$s, which are known as inequalities of Pachpatte type.

Theorem 6.

Let $\widetilde{{\rm Y}},\tilde{\mathcal{J}}:\left[\rho ,\rho +\phi \left(\mu ,\rho \right)\right]\to {₤}_0$ be two UD pre-invex $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$s on $\left[\rho ,\rho +\phi \left(\mu ,\rho \right)\right]$ whose $\mathfrak{b}$-levels ${{\rm Y}}_{\mathfrak{b}},{\mathcal{J}}_{\mathfrak{b}}:\left[\rho ,\rho +\phi \left(\mu ,\rho \right)\right]\subset \mathbb{R}\to {\mathbb{X}}_o^{+}$ are defined by ${{\rm Y}}_{\mathfrak{b}}\left(\varkappa \right)=\left[{{\rm Y}}_{*}\left(\varkappa ,\mathfrak{b}\right),{{\rm Y}}^{*}\left(\varkappa ,\mathfrak{b}\right)\right]$ and ${\mathcal{J}}_{\mathfrak{b}}\left(\varkappa \right)=\left[{\mathcal{J}}_{*}\left(\varkappa ,\mathfrak{b}\right),{\mathcal{J}}^{*}\left(\varkappa ,\mathfrak{b}\right)\right]$ for all $\varkappa \in \left[\rho ,\rho +\phi \left(\mu ,\rho \right)\right]$ and for all $\mathfrak{b}\in \left[0,1\right]$. If $\widetilde{{\rm Y}}\otimes \tilde{\mathcal{J}}\in L\left(\left[\rho ,\rho +\phi \left(\mu ,\rho \right)\right],{₤}_0\right)$ and $\phi$ satisfies Assumption 1, then

$$\begin{gathered} \frac{\Gamma \left(\beta \right)}{{2\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\odot \left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }\widetilde{{\rm Y}}\left(\rho +\phi \left(\mu ,\rho \right)\right)\otimes \tilde{\mathcal{J}}\left(\rho +\phi \left(\mu ,\rho \right)\right)\oplus {\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }\widetilde{{\rm Y}}\left(\rho \right)\otimes \tilde{\mathcal{J}}\left(\rho \right)\right] \\ {\supseteq }_{\mathbb{F}}\left(\frac{1}{2}-\frac{\beta }{(\beta +1)(\beta +2)}\right)\odot \tilde{\mathfrak{B}}\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right)\oplus \left(\frac{\beta }{(\beta +1)(\beta +2)}\right)\odot \tilde{\mathfrak{D}}\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right). \end{gathered}$$

(47)

where  $\tilde{\mathfrak{B}}\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right)=\widetilde{{\rm Y}}\left(\rho \right)\otimes \tilde{\mathcal{J}}\left(\rho \right)\oplus \widetilde{{\rm Y}}\left(\rho +\phi \left(\mu ,\rho \right)\right)\otimes \tilde{\mathcal{J}}\left(\rho +\phi \left(\mu ,\rho \right)\right),$  $\tilde{\mathfrak{D}}\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right)=\widetilde{{\rm Y}}\left(\rho \right)\otimes \tilde{\mathcal{J}}\left(\rho +\phi \left(\mu ,\rho \right)\right)\oplus \widetilde{{\rm Y}}\left(\rho +\phi \left(\mu ,\rho \right)\right)\otimes \tilde{\mathcal{J}}\left(\rho \right),$  ${\mathfrak{B}}_{\mathfrak{b}}\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right)=\left[{\mathfrak{B}}_{*}\left(\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right),\mathfrak{b}\right),{\mathfrak{B}}^{*}\left(\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right),\mathfrak{b}\right)\right],$ and  ${\mathfrak{D}}_{\mathfrak{b}}\left(\rho ,\mu \right)=\left[{\mathfrak{D}}_{*}\left(\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right),\mathfrak{b}\right),{\mathfrak{D}}^{*}\left(\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right),\mathfrak{b}\right)\right].$

Proof. 

Since $\widetilde{{\rm Y}},\tilde{\mathcal{J}}$ are UD pre-invex $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$s and Assumption 1 holds $\mathrm{f}\mathrm{o}\mathrm{r}\phi$, then for each $\mathfrak{b}\in \left[0,1\right]$ we have

$$\begin{array}{c}\\ {{\rm Y}}_{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right)={{\rm Y}}_{*}\left(\rho +\phi \left(\mu ,\rho \right)+\mathfrak{u}\phi \left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right),\mathfrak{b}\right)\\ \le u{{\rm Y}}_{*}\left(\rho ,\mathfrak{b}\right)+\left(1-\mathfrak{u}\right){{\rm Y}}_{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\\ {{\rm Y}}^{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right)={{\rm Y}}^{*}\left(\rho +\phi \left(\mu ,\rho \right)+\mathfrak{u}\phi \left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right),\mathfrak{b}\right)\\ \ge u{{\rm Y}}^{*}\left(\rho ,\mathfrak{b}\right)+\left(1-\mathfrak{u}\right){{\rm Y}}^{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\end{array}$$

and

$$\begin{array}{c}\\ {\mathcal{J}}_{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right)={\mathcal{J}}_{*}\left(\rho +\phi \left(\mu ,\rho \right)+\mathfrak{u}\phi \left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right),\mathfrak{b}\right)\\ \le u{\mathcal{J}}_{*}\left(\rho ,\mathfrak{b}\right)+\left(1-\mathfrak{u}\right){\mathcal{J}}_{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\\ {\mathcal{J}}^{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right)={\mathcal{J}}^{*}\left(\rho +\phi \left(\mu ,\rho \right)+\mathfrak{u}\phi \left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right),\mathfrak{b}\right)\\ \ge u{\mathcal{J}}^{*}\left(\rho ,\mathfrak{b}\right)+\left(1-\mathfrak{u}\right){\mathcal{J}}^{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right).\end{array}$$

From the UD pre-invex $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$ definition, we get $\tilde{0}{\le }_{\mathbb{F}}\widetilde{{\rm Y}}\left(\varkappa \right)$ and $\tilde{0}{\le }_{\mathbb{F}}\mathcal{J}\left(\varkappa \right)$. Therefore,

$$\begin{array}{c}{{\rm Y}}_{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\times {\mathcal{J}}_{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\\ \le \left(u{{\rm Y}}_{*}\left(\rho ,\mathfrak{b}\right)+\left(1-\mathfrak{u}\right){{\rm Y}}_{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\right)\left(u{\mathcal{J}}_{*}\left(\rho ,\mathfrak{b}\right)+\left(1-\mathfrak{u}\right){\mathcal{J}}_{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\right)\\ ={\mathfrak{u}}^2{{\rm Y}}_{*}\left(\rho ,\mathfrak{b}\right)\times {\mathcal{J}}_{*}\left(\rho ,\mathfrak{b}\right)+{\left(1-\mathfrak{u}\right)}^2{{\rm Y}}_{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\times {\mathcal{J}}_{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\\ +\mathfrak{u}\left(1-\mathfrak{u}\right){{\rm Y}}_{*}\left(\rho ,\mathfrak{b}\right)\times {\mathcal{J}}_{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)+\mathfrak{u}\left(1-\mathfrak{u}\right){{\rm Y}}_{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\times {\mathcal{J}}_{*}\left(\rho ,\mathfrak{b}\right),\\ {{\rm Y}}^{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\times {\mathcal{J}}^{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\\ \ge \left(u{{\rm Y}}^{*}\left(\rho ,\mathfrak{b}\right)+\left(1-\mathfrak{u}\right){{\rm Y}}^{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\right)\left(u{\mathcal{J}}^{*}\left(\rho ,\mathfrak{b}\right)+\left(1-\mathfrak{u}\right){\mathcal{J}}^{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\right)\\ ={\mathfrak{u}}^2{{\rm Y}}^{*}\left(\rho ,\mathfrak{b}\right)\times {\mathcal{J}}^{*}\left(\rho ,\mathfrak{b}\right)+{\left(1-\mathfrak{u}\right)}^2{{\rm Y}}^{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right){\times \mathcal{J}}^{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\\ +\mathfrak{u}\left(1-\mathfrak{u}\right){{\rm Y}}^{*}\left(\rho ,\mathfrak{b}\right)\times {\mathcal{J}}^{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)+\mathfrak{u}\left(1-\mathfrak{u}\right){{\rm Y}}^{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\times {\mathcal{J}}^{*}\left(\rho ,\mathfrak{b}\right).\end{array}$$

(48)

Analogously, we have

$$\begin{array}{c}\\ {{\rm Y}}_{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right){\mathcal{J}}_{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\\ \le {\left(1-\mathfrak{u}\right)}^2{{\rm Y}}_{*}\left(\rho ,\mathfrak{b}\right)\times {\mathcal{J}}_{*}\left(\rho ,\mathfrak{b}\right)+{\mathfrak{u}}^2{{\rm Y}}_{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\times {\mathcal{J}}_{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\\ +\mathfrak{u}\left(1-\mathfrak{u}\right){{\rm Y}}_{*}\left(\rho ,\mathfrak{b}\right)\times {\mathcal{J}}_{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)+\mathfrak{u}\left(1-\mathfrak{u}\right){{\rm Y}}_{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\times {\mathcal{J}}_{*}\left(\rho ,\mathfrak{b}\right),\\ {{\rm Y}}^{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\times {\mathcal{J}}^{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\\ \ge {\left(1-\mathfrak{u}\right)}^2{{\rm Y}}^{*}\left(\rho ,\mathfrak{b}\right)\times {\mathcal{J}}^{*}\left(\rho ,\mathfrak{b}\right)+{\mathfrak{u}}^2{{\rm Y}}^{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\times {\mathcal{J}}^{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\\ +u\left(1-\mathfrak{u}\right){{\rm Y}}^{*}\left(\rho ,\mathfrak{b}\right)\times {\mathcal{J}}^{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)+\mathfrak{u}\left(1-\mathfrak{u}\right){{\rm Y}}^{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\times {\mathcal{J}}^{*}\left(\rho ,\mathfrak{b}\right).\end{array}$$

(49)

Adding (48) and (49), we have

$$\begin{array}{c}\\ {{\rm Y}}_{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\times {\mathcal{J}}_{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\\ +{{\rm Y}}_{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\times {\mathcal{J}}_{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\\ \le \left[{{\mathfrak{u}}^2+\left(1-\mathfrak{u}\right)}^2\right]\left[{{\rm Y}}_{*}\left(\rho ,\mathfrak{b}\right)\times {\mathcal{J}}_{*}\left(\rho ,\mathfrak{b}\right)+{{\rm Y}}_{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\times {\mathcal{J}}_{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\right]\\ +2u\left(1-\mathfrak{u}\right)\left[{{\rm Y}}_{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\times {\mathcal{J}}_{*}\left(\rho ,\mathfrak{b}\right)+{{\rm Y}}_{*}\left(\rho ,\mathfrak{b}\right)\times {\mathcal{J}}_{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\right],\\ {{\rm Y}}^{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\times {\mathcal{J}}^{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\\ +{{\rm Y}}^{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\times {\mathcal{J}}^{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\\ \ge \left[{{\mathfrak{u}}^2+\left(1-\mathfrak{u}\right)}^2\right]\left[{{\rm Y}}^{*}\left(\rho ,\mathfrak{b}\right)\times {\mathcal{J}}^{*}\left(\rho ,\mathfrak{b}\right)+{{\rm Y}}^{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\times {\mathcal{J}}^{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\right]\\ +2u\left(1-\mathfrak{u}\right)\left[{{\rm Y}}^{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\times {\mathcal{J}}^{*}\left(\rho ,\mathfrak{b}\right)+{{\rm Y}}^{*}\left(\rho ,\mathfrak{b}\right)\times {\mathcal{J}}^{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\right].\end{array}$$

(50)

Multiplying (50) by ${\mathfrak{u}}^{\beta -1}$ and integrating the result with respect to $\mathfrak{u}$ over (0,1), we obtain

$$\begin{array}{c}\\ {\int }_0^1{\mathfrak{u}}^{\beta -1}{{\rm Y}}_{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\times {\mathcal{J}}_{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\\ +{\mathfrak{u}}^{\beta -1}{{\rm Y}}_{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\times {\mathcal{J}}_{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right)d\mathfrak{u}\\ \le {\mathfrak{B}}_{*}\left(\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right),\mathfrak{b}\right){\int }_0^1{\mathfrak{u}}^{\beta -1}\left[{{\mathfrak{u}}^2+\left(1-\mathfrak{u}\right)}^2\right]d\mathfrak{u}\\ +2{\mathfrak{D}}_{*}\left(\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right),\mathfrak{b}\right){\int }_0^1{\mathfrak{u}}^{\beta -1}\mathfrak{u}\left(1-\mathfrak{u}\right)d\mathfrak{u},\\ {\int }_0^1{\mathfrak{u}}^{\beta -1}{{\rm Y}}^{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\times {\mathcal{J}}^{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\\ +{\mathfrak{u}}^{\beta -1}{{\rm Y}}^{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\times {\mathcal{J}}^{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right)d\mathfrak{u}\\ \ge {\mathfrak{B}}^{*}\left(\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right),\mathfrak{b}\right){\int }_0^1{\mathfrak{u}}^{\beta -1}\left[{{\mathfrak{u}}^2+\left(1-\mathfrak{u}\right)}^2\right]d\mathfrak{u}\\ +2{\mathfrak{D}}^{*}\left(\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right),\mathfrak{b}\right){\int }_0^1{\mathfrak{u}}^{\beta -1}\mathfrak{u}\left(1-\mathfrak{u}\right)d\mathfrak{u}.\end{array}$$

It follows that

$$\begin{array}{c}\\ \frac{\Gamma \left(\beta \right)}{{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }{{\rm Y}}_{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\times {\mathcal{J}}_{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }{{\rm Y}}_{*}\left(\rho ,\mathfrak{b}\right)\times {\mathcal{J}}_{*}\left(\rho ,\mathfrak{b}\right)\right]\\ \le \frac{2}{\beta }\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right){\mathfrak{B}}_{*}\left(\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right),\mathfrak{b}\right)+\frac{2}{\beta }\left(\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right){\mathfrak{D}}_{*}\left(\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right),\mathfrak{b}\right),\\ \\ \frac{\Gamma \left(\beta \right)}{{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }{{\rm Y}}^{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\times {\mathcal{J}}^{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }{{\rm Y}}^{*}\left(\rho ,\mathfrak{b}\right)\times {\mathcal{J}}^{*}\left(\rho ,\mathfrak{b}\right)\right]\\ \ge \frac{2}{\beta }\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right){\mathfrak{B}}^{*}\left(\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right),\mathfrak{b}\right)+\frac{2}{\beta }\left(\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right){\mathfrak{D}}^{*}\left(\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right),\mathfrak{b}\right),\\ \end{array}$$

That is,

$$\begin{gathered} \frac{\Gamma \left(\beta \right)}{{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}[{\mathcal{I}}_{{\rho }^{+}}^{\beta }{{\rm Y}}_{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\times {\mathcal{J}}_{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }{{\rm Y}}_{*}\left(\rho ,\mathfrak{b}\right)\times {\mathcal{J}}_{*}\left(\rho ,\mathfrak{b}\right), \\ {\mathcal{I}}_{{\rho }^{+}}^{\beta }{{\rm Y}}^{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\times {\mathcal{J}}^{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }{{\rm Y}}^{*}\left(\rho ,\mathfrak{b}\right)\times {\mathcal{J}}^{*}\left(\rho ,\mathfrak{b}\right)] \\ {\supseteq }_I\frac{2}{\beta }\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\left[{\mathfrak{B}}_{*}\left(\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right),\mathfrak{b}\right),{\mathfrak{B}}^{*}\left(\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right),\mathfrak{b}\right)\right] \\ +\frac{2}{\beta }\left(\frac{\beta }{(\beta +1)(\beta +2)}\right)\left[{\mathfrak{D}}_{*}\left(\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right),\mathfrak{b}\right),{\mathfrak{D}}^{*}\left(\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right),\mathfrak{b}\right)\right]. \end{gathered}$$

Therefore,

$$\begin{gathered} \frac{\Gamma \left(\beta \right)}{{2\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\odot \left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }\widetilde{{\rm Y}}\left(\rho +\phi \left(\mu ,\rho \right)\right)\otimes \tilde{\mathcal{J}}\left(\rho +\phi \left(\mu ,\rho \right)\right)\oplus {\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }\widetilde{{\rm Y}}\left(\rho \right)\otimes \tilde{\mathcal{J}}\left(\rho \right)\right] \\ {\supseteq }_{\mathbb{F}}\left(\frac{1}{2}-\frac{\beta }{(\beta +1)(\beta +2)}\right)\odot \tilde{\mathfrak{B}}\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right)\oplus \left(\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\odot \tilde{\mathfrak{D}}\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right), \end{gathered}$$

and, thus, the theorem is proven. □

Theorem 7.

Let $\widetilde{{\rm Y}},\tilde{\mathcal{J}}:\left[\rho ,\rho +\phi \left(\mu ,\rho \right)\right]\to {₤}_0$ be two UD pre-invex $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$s whose $\mathfrak{b}$-levels define the family of $I\mathcal{V}\mathcal{M}$s ${{\rm Y}}_{\mathfrak{b}},{\mathcal{J}}_{\mathfrak{b}}:\left[\rho ,\rho +\phi \left(\mu ,\rho \right)\right]\subset \mathbb{R}\to {\mathbb{X}}_o^{+}$ given by ${{\rm Y}}_{\mathfrak{b}}\left(\varkappa \right)=\left[{{\rm Y}}_{*}\left(\varkappa ,\mathfrak{b}\right),{{\rm Y}}^{*}\left(\varkappa ,\mathfrak{b}\right)\right]$ and ${\mathcal{J}}_{\mathfrak{b}}\left(\varkappa \right)=\left[{\mathcal{J}}_{*}\left(\varkappa ,\mathfrak{b}\right),{\mathcal{J}}^{*}\left(\varkappa ,\mathfrak{b}\right)\right]$ for all $\varkappa \in \left[\rho ,\rho +\phi \left(\mu ,\rho \right)\right]$ and for all $\mathfrak{b}\in \left[0,1\right]$. If $\widetilde{{\rm Y}}\otimes \tilde{\mathcal{J}}\in L\left(\left[\rho ,\rho +\phi \left(\mu ,\rho \right)\right],{₤}_0\right)$ and $\phi$ satisfies Assumption 1, then

$$\begin{array}{l}\frac{1}{\beta }\odot \widetilde{{\rm Y}}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2}\right)\otimes \tilde{\mathcal{J}}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2}\right)\\ {\supseteq }_{\mathbb{F}}\frac{\Gamma \left(\beta +1\right)}{{4\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\odot \left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }\widetilde{{\rm Y}}\left(\rho +\phi \left(\mu ,\rho \right)\right)\otimes \tilde{\mathcal{J}}\left(\rho +\phi \left(\mu ,\rho \right)\right)\oplus {\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }\widetilde{{\rm Y}}\left(\rho \right)\otimes \tilde{\mathcal{J}}\left(\rho \right)\right]\\ \oplus \frac{1}{2\beta }\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\odot \tilde{\mathfrak{D}}\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right)\oplus \frac{1}{2\beta }\left(\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\odot \tilde{\mathfrak{B}}\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right),\end{array}$$

(51)

where $\tilde{\mathfrak{B}}\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right)=\widetilde{{\rm Y}}\left(\rho \right)\otimes \tilde{\mathcal{J}}\left(\rho \right)\oplus \widetilde{{\rm Y}}\left(\rho +\phi \left(\mu ,\rho \right)\right)\otimes \tilde{\mathcal{J}}\left(\rho +\phi \left(\mu ,\rho \right)\right),$  $\tilde{\mathfrak{D}}\left(\rho ,\mu \right)=\widetilde{{\rm Y}}\left(\rho \right)\otimes \tilde{\mathcal{J}}\left(\rho +\phi \left(\mu ,\rho \right)\right)\oplus \widetilde{{\rm Y}}\left(\rho +\phi \left(\mu ,\rho \right)\right)\otimes \tilde{\mathcal{J}}\left(\rho \right),$  ${\mathfrak{B}}_{\mathfrak{b}}\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right)=\left[{\mathfrak{B}}_{*}\left(\left(\rho ,\rho +\phi \left(\rho +\phi \left(\mu ,\rho \right)\right)\right),\mathfrak{b}\right),{\mathfrak{B}}^{*}\left(\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right),\mathfrak{b}\right)\right],$ and ${\mathfrak{D}}_{\mathfrak{b}}\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right)=\left[{\mathfrak{D}}_{*}\left(\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right),\mathfrak{b}\right),{\mathfrak{D}}^{*}\left(\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right),\mathfrak{b}\right)\right].$

Proof. 

Consider that $\widetilde{{\rm Y}},\tilde{\mathcal{J}}:\left[\rho ,\rho +\phi \left(\mu ,\rho \right)\right]\to {₤}_0$ are UD pre-invex $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}s$. By hypothesization, for each $\mathfrak{b}\in \left[0,1\right],$ we have

$$\begin{gathered} \begin{array}{c}{{\rm Y}}_{*}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2},\mathfrak{b}\right)\times {\mathcal{J}}_{*}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2},\mathfrak{b}\right)\\ {{\rm Y}}^{*}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2},\mathfrak{b}\right)\times {\mathcal{J}}^{*}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2},\mathfrak{b}\right)\end{array} \\ \begin{array}{c}\le \frac{1}{4}\left[\begin{array}{c}{{\rm Y}}_{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\times {\mathcal{J}}_{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\\ +{{\rm Y}}_{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\times {\mathcal{J}}_{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\end{array}\right]\\ +\frac{1}{4}\left[\begin{array}{c}{{\rm Y}}_{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right){\times \mathcal{J}}_{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\\ +{{\rm Y}}_{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right){\times \mathcal{J}}_{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\end{array}\right]\\ \ge \frac{1}{4}\left[\begin{array}{c}{{\rm Y}}^{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right){\times \mathcal{J}}^{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\\ +{{\rm Y}}^{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right){\times \mathcal{J}}^{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\end{array}\right]\\ +\frac{1}{4}\left[\begin{array}{c}{{\rm Y}}^{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right){\times \mathcal{J}}^{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\\ +{{\rm Y}}^{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right){\times \mathcal{J}}^{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\end{array}\right],\end{array} \\ \begin{array}{c}\le \frac{1}{4}\left[\begin{array}{c}{{\rm Y}}_{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right){\times \mathcal{J}}_{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\\ +{{\rm Y}}_{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right){\times \mathcal{J}}_{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\end{array}\right]\\ +\frac{1}{4}\left[\begin{array}{c}\left(\mathfrak{u}{{\rm Y}}_{*}\left(\rho ,\mathfrak{b}\right)+\left(1-\mathfrak{u}\right){{\rm Y}}_{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\right)\\ \times \left(\left(1-\mathfrak{u}\right){\mathcal{J}}_{*}\left(\rho ,\mathfrak{b}\right)+\mathfrak{u}{\mathcal{J}}_{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\right)\\ +\left({\left(1-\mathfrak{u}\right){\rm Y}}_{*}\left(\rho ,\mathfrak{b}\right)+\mathfrak{u}{{\rm Y}}_{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\right)\\ \times \left(\mathfrak{u}{\mathcal{J}}_{*}\left(\rho ,\mathfrak{b}\right)+\left(1-\mathfrak{u}\right){\mathcal{J}}_{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\right)\end{array}\right]\\ \ge \frac{1}{4}\left[\begin{array}{c}{{\rm Y}}^{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right){\times \mathcal{J}}^{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\\ +{{\rm Y}}^{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right){\times \mathcal{J}}^{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\end{array}\right]\\ +\frac{1}{4}\left[\begin{array}{c}\left(\mathfrak{u}{{\rm Y}}^{*}\left(\rho ,\mathfrak{b}\right)+\left(1-\mathfrak{u}\right){{\rm Y}}^{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\right)\\ \times \left(\left(1-\mathfrak{u}\right){\mathcal{J}}^{*}\left(\rho ,\mathfrak{b}\right)+\mathfrak{u}{\mathcal{J}}^{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\right)\\ +\left(\left(1-\mathfrak{u}\right){{\rm Y}}^{*}\left(\rho ,\mathfrak{b}\right)+\mathfrak{u}{{\rm Y}}^{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\right)\\ \times \left(\mathfrak{u}{\mathcal{J}}^{*}\left(\rho ,\mathfrak{b}\right)+\left(1-\mathfrak{u}\right){\mathcal{J}}^{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\right)\end{array}\right],\end{array} \\ \begin{array}{c}=\frac{1}{4}\left[\begin{array}{c}{{\rm Y}}_{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right){\times \mathcal{J}}_{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\\ +{{\rm Y}}_{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right){\times \mathcal{J}}_{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\end{array}\right]\\ +\frac{1}{4}\left[\begin{array}{c}\left\{{\mathfrak{u}}^2+{\left(1-\mathfrak{u}\right)}^2\right\}{\mathfrak{D}}_{*}\left(\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right),\mathfrak{b}\right)\\ +\left\{\mathfrak{u}\left(1-\mathfrak{u}\right)+\left(1-\mathfrak{u}\right)\mathfrak{u}\right\}{\mathfrak{B}}_{*}\left(\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right),\mathfrak{b}\right)\end{array}\right],\\ =\frac{1}{4}\left[\begin{array}{c}{{\rm Y}}^{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right){\times \mathcal{J}}^{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\\ +{{\rm Y}}^{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right){\times \mathcal{J}}^{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\end{array}\right]\\ +\frac{1}{4}\left[\begin{array}{c}\left\{{\mathfrak{u}}^2+{\left(1-\mathfrak{u}\right)}^2\right\}{\mathfrak{D}}^{*}\left(\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right),\mathfrak{b}\right)\\ +\left\{\mathfrak{u}\left(1-\mathfrak{u}\right)+\left(1-\mathfrak{u}\right)\mathfrak{u}\right\}{\mathfrak{B}}^{*}\left(\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right),\mathfrak{b}\right)\end{array}\right].\end{array} \end{gathered}$$

(52)

Multiplying (52) with ${\mathfrak{u}}^{\beta -1}$ and integrating over $(0,1),$ we get

$$\begin{array}{c}\frac{1}{\beta }{{\rm Y}}_{*}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2},\mathfrak{b}\right){\times \mathcal{J}}_{*}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2},\mathfrak{b}\right)\\ \le \frac{1}{{4\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\left[\begin{array}{c}{\int }_{\rho }^{\rho +\phi \left(\mu ,\rho \right)}{\left(\rho +\phi \left(\mu ,\rho \right)-\varkappa \right)}^{\beta -1}{{\rm Y}}_{*}\left(\varkappa ,\mathfrak{b}\right){\times \mathcal{J}}_{*}\left(\varkappa ,\mathfrak{b}\right)d\varkappa \\ +{\int }_{\rho }^{\rho +\phi \left(\mu ,\rho \right)}{\left(y-\rho \right)}^{\beta -1}{{\rm Y}}_{*}\left(y,\mathfrak{b}\right){\times \mathcal{J}}_{*}\left(y,\mathfrak{b}\right)dy\end{array}\right]\\ +\frac{1}{2\beta }\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right){\mathfrak{D}}_{*}\left(\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right),\mathfrak{b}\right)+\frac{1}{2\beta }\left(\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right){\mathfrak{B}}_{*}\left(\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right),\mathfrak{b}\right)\\ =\frac{\Gamma \left(\beta +1\right)}{{4\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }{{\rm Y}}_{*}\left(\rho +\phi \left(\mu ,\rho \right)\right){\times \mathcal{J}}_{*}\left(\rho +\phi \left(\mu ,\rho \right)\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }{{\rm Y}}_{*}\left(\rho \right){\times \mathcal{J}}_{*}\left(\rho \right)\right]\\ +\frac{1}{2\beta }\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right){\mathfrak{D}}_{*}\left(\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right),\mathfrak{b}\right)+\frac{1}{2\beta }\left(\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right){\mathfrak{B}}_{*}\left(\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right),\mathfrak{b}\right),\\ \frac{1}{\beta }{{\rm Y}}^{*}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2},\mathfrak{b}\right)\times {\mathcal{J}}^{*}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2},\mathfrak{b}\right)\\ \ge \frac{1}{{4\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\left[\begin{array}{c}{\int }_{\rho }^{\rho +\phi \left(\mu ,\rho \right)}{\left(\rho +\phi \left(\mu ,\rho \right)-\varkappa \right)}^{\beta -1}{{\rm Y}}^{*}\left(\varkappa ,\mathfrak{b}\right)\times {\mathcal{J}}^{*}\left(\varkappa ,\mathfrak{b}\right)d\varkappa \\ +{\int }_{\rho }^{\rho +\phi \left(\mu ,\rho \right)}{\left(y-\rho \right)}^{\beta -1}{{\rm Y}}^{*}\left(y,\mathfrak{b}\right)\times {\mathcal{J}}^{*}\left(y,\mathfrak{b}\right)dy\end{array}\right]\\ +\frac{1}{2\beta }\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right){\mathfrak{D}}^{*}\left(\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right),\mathfrak{b}\right)+\frac{1}{2\beta }\left(\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right){\mathfrak{B}}^{*}\left(\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right),\mathfrak{b}\right)\\ =\frac{\Gamma \left(\beta +1\right)}{{4\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }{{\rm Y}}^{*}\left(\rho +\phi \left(\mu ,\rho \right)\right)\times {\mathcal{J}}^{*}\left(\rho +\phi \left(\mu ,\rho \right)\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }{{\rm Y}}^{*}\left(\rho \right)\times {\mathcal{J}}^{*}\left(\rho \right)\right]\\ +\frac{1}{2\beta }\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right){\mathfrak{D}}^{*}\left(\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right),\mathfrak{b}\right)+\frac{1}{2\beta }\left(\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right){\mathfrak{B}}^{*}\left(\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right),\mathfrak{b}\right),\end{array}$$

That is,

$$\begin{gathered} \frac{1}{\beta }\odot \widetilde{{\rm Y}}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2}\right)\tilde{\times }\tilde{\mathcal{J}}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2}\right){\supseteq }_{\mathbb{F}}\frac{\Gamma \left(\beta +1\right)}{{4\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\odot \\ \left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }\widetilde{{\rm Y}}\left(\rho +\phi \left(\mu ,\rho \right)\right)\otimes \tilde{\mathcal{J}}\left(\rho +\phi \left(\mu ,\rho \right)\right)\oplus {\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }\widetilde{{\rm Y}}\left(\rho \right)\otimes \tilde{\mathcal{J}}\left(\rho \right)\right]\oplus \frac{1}{2\beta }\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\odot \\ \tilde{\mathfrak{D}}\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right)\oplus \frac{1}{2\beta }\left(\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\odot \tilde{\mathfrak{B}}\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right). \end{gathered}$$

Thus, this concludes the proof. □

Example 2.

Let $\left[\rho ,\rho +\phi \left(\mu ,\rho \right)\right]=[0,\phi \left(\mathrm{2,0}\right)]$, $\beta =\frac{1}{2},\widetilde{{\rm Y}}\left(\varkappa \right)=\left[\varkappa ,2\varkappa \right]$, and $\tilde{\mathcal{J}}\left(\varkappa \right)=\left[\varkappa ,3\varkappa \right],$ as

$$\widetilde{{\rm Y}}\left(\varkappa \right)\left(\theta \right)=\left\{\begin{array}{c}\frac{\theta }{\varkappa }\theta \in \left[0,\varkappa \right]\\ \frac{2\varkappa -\theta }{\varkappa }\theta \in (\varkappa ,2\varkappa ]\\ 0otherwise,\end{array}\right.$$

(53)

$$\tilde{\mathcal{J}}\left(\varkappa \right)\left(\theta \right)=\left\{\begin{array}{c}\frac{\theta }{2\varkappa }\theta \in \left[\varkappa ,2\right]\\ \frac{4\varkappa -\theta }{2\varkappa }\theta \in (2,(2,8-e^{\varkappa }]]\\ 0otherwise.\end{array}\right.$$

(54)

Then, for each $\mathfrak{b}\in \left[0,1\right],$ we have ${{\rm Y}}_{\mathfrak{b}}\left(\varkappa \right)=\left[\mathfrak{b}\varkappa ,(2-\mathfrak{b})\varkappa \right]$ and ${\mathcal{J}}_{\mathfrak{b}}\left(\varkappa \right)=\left[\left(1-\mathfrak{b}\right)\varkappa +2\mathfrak{b},\left(1-\mathfrak{b}\right)\left(8-e^{\varkappa }\right)+2\mathfrak{b}\right].$ Since the left and right end point functions ${{\rm Y}}_{\mathrm{*}}\left(\varkappa ,\mathfrak{b}\right)=\mathfrak{b}\varkappa ,$ ${{\rm Y}}^{\mathrm{*}}\left(\varkappa ,\mathfrak{b}\right)=(2-\mathfrak{b})\varkappa$, ${\mathcal{J}}_{\mathrm{*}}\left(\varkappa ,\mathfrak{b}\right)=\left(1-\mathfrak{b}\right)\varkappa +2\mathfrak{b},$ and ${\mathcal{J}}^{\mathrm{*}}\left(\varkappa ,\mathfrak{b}\right)=\left(1-\mathfrak{b}\right)\left(8-e^{\varkappa }\right)+2\mathfrak{b}$ are pre-invex functions with respect to $\phi \left(\mu ,\rho \right)=\mu -\rho$ and for each $\mathfrak{b}\in [0,1]$, then $\widetilde{{\rm Y}}\left(\varkappa \right)$ and $\tilde{\mathcal{J}}\left(\varkappa \right)$ are UD pre-invex $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$s. We see that $\widetilde{{\rm Y}}\left(\varkappa \right)\otimes \tilde{\mathcal{J}}\left(\varkappa \right)\in L\left(\left[\rho ,\rho +\phi \left(\mu ,\rho \right)\right],{\mathrm{₤}}_0\right)$ and

$$\begin{gathered} \frac{\Gamma \left(1+\beta \right)}{2{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }{{\rm Y}}_{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\times {\mathcal{J}}_{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }{{\rm Y}}_{*}\left(\rho ,\mathfrak{b}\right)\times {\mathcal{J}}_{*}\left(\rho ,\mathfrak{b}\right)\right] \\ =\frac{\Gamma \left(\frac{3}{2}\right)}{2\sqrt{2}}\frac{1}{\sqrt{\pi }}{\int }_0^{\phi \left(\mathrm{2,0}\right)}{\left(2-\varkappa \right)}^{\frac{-1}{2}}\left(\mathfrak{b}\left(1-\mathfrak{b}\right){\varkappa }^2+2{\mathfrak{b}}^2\varkappa \right)d\varkappa +\frac{\Gamma \left(\frac{3}{2}\right)}{2\sqrt{2}}\frac{1}{\sqrt{\pi }}{\int }_0^{\phi \left(\mathrm{2,0}\right)}{\left(\varkappa \right)}^{\frac{-1}{2}}\left(\mathfrak{b}\left(1-\mathfrak{b}\right){\varkappa }^2+2{\mathfrak{b}}^2\varkappa \right)d\varkappa =\frac{2}{15}\mathfrak{b}\left(4\mathfrak{b}+11\right), \\ \frac{\Gamma \left(1+\beta \right)}{2{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }{{\rm Y}}^{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\times {\mathcal{J}}^{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }{{\rm Y}}^{*}\left(\rho ,\mathfrak{b}\right)\times {\mathcal{J}}^{*}\left(\rho ,\mathfrak{b}\right)\right] \\ =\frac{\Gamma \left(\frac{3}{2}\right)}{2\sqrt{2}}\frac{1}{\sqrt{\pi }}{\int }_0^{\phi \left(\mathrm{2,0}\right)}{\left(2-\varkappa \right)}^{\frac{-1}{2}}.\left[\left(1-\mathfrak{b}\right)\left(2-\mathfrak{b}\right)\varkappa \left(8-e^{\varkappa }\right)+2\mathfrak{b}\left(2-\mathfrak{b}\right)\varkappa \right]d\varkappa \\ +\frac{\Gamma \left(\frac{3}{2}\right)}{2\sqrt{2}}\frac{1}{\sqrt{\pi }}{\int }_0^{\phi \left(\mathrm{2,0}\right)}{\left(\varkappa \right)}^{\frac{-1}{2}}.\left[\left(1-\mathfrak{b}\right)\left(2-\mathfrak{b}\right)\varkappa \left(8-e^{\varkappa }\right)+2\mathfrak{b}\left(2-\mathfrak{b}\right)\varkappa \right]d\varkappa =\frac{2}{5}\left(2-\mathfrak{b}\right)\left(8-3\mathfrak{b}\right). \end{gathered}$$

Note that

$$\begin{gathered} \left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right){\mathfrak{B}}_{*}\left(\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right),\mathfrak{b}\right)=[{{\rm Y}}_{*}\left(\rho ,\mathfrak{b}\right)\times {\mathcal{J}}_{*}\left(\rho ,\mathfrak{b}\right)+ \\ {{\rm Y}}_{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\times {\mathcal{J}}_{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)]=\frac{22}{15}\mathfrak{b}, \\ \left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right){\mathfrak{B}}^{*}\left(\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right),\mathfrak{b}\right) \\ =\left[{{\rm Y}}^{*}\left(\rho ,\mathfrak{b}\right)\times {\mathcal{J}}^{*}\left(\rho ,\mathfrak{b}\right)+{{\rm Y}}^{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\times {\mathcal{J}}^{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\right]= \\ \frac{11}{15}.\left(2-\mathfrak{b}\right)\left[\left(1-\mathfrak{b}\right)\left(8-e^2\right)+2\mathfrak{b}\right], \\ \left(\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right){\mathfrak{D}}_{*}\left(\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right),\mathfrak{b}\right) \\ =\left[{{\rm Y}}_{*}\left(\rho ,\mathfrak{b}\right)\times {\mathcal{J}}_{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)+{{\rm Y}}_{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\times {\mathcal{J}}_{*}\left(\rho ,\mathfrak{b}\right)\right] \\ =\frac{8}{15}{\mathfrak{b}}^2, \\ \left(\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right){\mathfrak{D}}_{*}\left(\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right),\mathfrak{b}\right) \\ =\left[{{\rm Y}}^{*}\left(\rho ,\mathfrak{b}\right)\times {\mathcal{J}}^{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)+{{\rm Y}}^{*}\left(\rho +\phi \left(\mu ,\rho \right)\right)\times {\mathcal{J}}^{*}\left(\rho ,\mathfrak{b}\right)\right] \\ =\frac{4}{15}\left(2-\mathfrak{b}\right)\left(7-5\mathfrak{b}\right). \end{gathered}$$

Therefore, we have

$$\begin{array}{c}\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right){\mathfrak{B}}_{\mathfrak{b}}\left(\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right),\mathfrak{b}\right)+\left(\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right){\mathfrak{D}}_{\mathfrak{b}}\left(\left(\rho ,\rho +\phi \left(\mu ,\rho \right)\right),\mathfrak{b}\right)\\ =\frac{1}{15}\left[2\mathfrak{b}\left(11+4\mathfrak{b}\right),\left(2-\mathfrak{b}\right)\left[11\left(1-\mathfrak{b}\right)\left(8-e^2\right)+2\mathfrak{b}+28\right]\right].\end{array}$$

It follows that

$$\begin{gathered} \frac{1}{5}[\frac{2}{3}\mathfrak{b}\left(11+4\mathfrak{b}\right),2\left(2-\mathfrak{b}\right)\left(8-3\mathfrak{b}\right)] \\ {\supseteq }_I\frac{1}{15}\left[2\mathfrak{b}\left(11+4\mathfrak{b}\right),\left(2-\mathfrak{b}\right)\left[11\left(1-\mathfrak{b}\right)\left(8-e^2\right)+2\mathfrak{b}+28\right]\right] \end{gathered}$$

and Theorem 7 has been illustrated.

We now obtain second and first fuzzy fractional HH–Fejér type inequalities for UD pre-invex $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$.

Theorem 8.

Let $\widetilde{{\rm Y}}:\left[\rho ,\rho +\phi \left(\mu ,\rho \right)\right]\to {₤}_0$ be a UD pre-invex $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$ with $\rho <+\phi \left(\mu ,\rho \right)$ whose $\mathfrak{b}$-levels define the family of $I\mathcal{V}\mathcal{M}$${{\rm Y}}_{\mathfrak{b}}:\left[\rho ,\rho +\phi \left(\mu ,\rho \right)\right]\subset \mathbb{R}\to {\mathbb{X}}_o^{+}$ given by ${{\rm Y}}_{\mathfrak{b}}\left(\varkappa \right)=\left[{{\rm Y}}_{*}\left(\varkappa ,\mathfrak{b}\right),{{\rm Y}}^{*}\left(\varkappa ,\mathfrak{b}\right)\right]$ for all $\varkappa \in \left[\rho ,\rho +\phi \left(\mu ,\rho \right)\right]$ and for all $\mathfrak{b}\in \left[0,1\right]$. Let $\widetilde{{\rm Y}}\in L\left(\left[\rho ,\rho +\phi \left(\mu ,\rho \right)\right],{₤}_0\right)$ and $\mathcal{C}:\left[\rho ,\rho +\phi \left(\mu ,\rho \right)\right]\to \mathbb{R},\mathcal{C}\left(\varkappa \right)\ge 0$ be symmetric with respect to $\frac{2\rho +\phi \left(\mu ,\rho \right)}{2}.$ If $\phi$ satisfies Assumption 1, then

$$\begin{gathered} \left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }\widetilde{{\rm Y}}\mathcal{C}\left(\rho +\phi \left(\mu ,\rho \right)\right)\oplus {\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }\widetilde{{\rm Y}}\mathcal{C}\left(\rho \right)\right] \\ {\supseteq }_{\mathbb{F}}\frac{\widetilde{{\rm Y}}\left(\rho \right)\oplus \widetilde{{\rm Y}}\left(\rho +\phi \left(\mu ,\rho \right)\right)}{2}\odot \left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }\mathcal{C}\left(\rho +\phi \left(\mu ,\rho \right)\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }\mathcal{C}\left(\rho \right)\right] \\ {\supseteq }_{\mathbb{F}}\frac{\widetilde{{\rm Y}}\left(\rho \right)\oplus \widetilde{{\rm Y}}\left(\mu \right)}{2}\odot \left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }\mathcal{C}\left(\rho +\phi \left(\mu ,\rho \right)\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }\mathcal{C}\left(\rho \right)\right]. \end{gathered}$$

(55)

If $\widetilde{{\rm Y}}$ is UD pre-incave, then inequality (55) is reversed.

Proof. 

Let $\widetilde{{\rm Y}}$ be a UD pre-invex $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$ and ${\mathfrak{u}}^{\beta -1}\mathcal{C}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right)\right)\ge 0$. Then, for each $\mathfrak{b}\in \left[0,1\right],$ we have

$$\begin{array}{c}\\ {\mathfrak{u}}^{\beta -1}{{\rm Y}}_{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right)C\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right)\right)\\ \le {\mathfrak{u}}^{\beta -1}\left(\mathfrak{u}{{\rm Y}}_{*}\left(\rho ,\mathfrak{b}\right)+\left(1-\mathfrak{u}\right){{\rm Y}}_{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\right)C\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\rho +\phi \left(\mu ,\rho \right),\rho \right)\right)\\ {\mathfrak{u}}^{\beta -1}{{\rm Y}}^{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right)C\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right)\right)\\ \ge {\mathfrak{u}}^{\beta -1}\left(\mathfrak{u}{{\rm Y}}^{*}\left(\rho ,\mathfrak{b}\right)+\left(1-\mathfrak{u}\right){{\rm Y}}^{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\right)C\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\rho +\phi \left(\mu ,\rho \right),\rho \right)\right),\end{array}$$

(56)

and

$$\begin{array}{c}\\ {\mathfrak{u}}^{\beta -1}{{\rm Y}}_{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right)C\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right)\right)\\ \le {\mathfrak{u}}^{\beta -1}\left(\left(1-\mathfrak{u}\right){{\rm Y}}_{*}\left(\rho ,\mathfrak{b}\right)+\mathfrak{u}{{\rm Y}}_{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\right)C\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right)\right)\\ {\mathfrak{u}}^{\beta -1}{{\rm Y}}^{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right)C\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right)\right)\\ \ge {\mathfrak{u}}^{\beta -1}\left(\left(1-\mathfrak{u}\right){{\rm Y}}^{*}\left(\rho ,\mathfrak{b}\right)+\mathfrak{u}{{\rm Y}}^{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\right)C\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right)\right).\end{array}$$

(57)

Adding (56) and (57), and integrating over $\left[0,1\right],$ we get

$$\begin{gathered} \begin{array}{c}\\ {\int }_0^1{\mathfrak{u}}^{\beta -1}{{\rm Y}}_{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\mathcal{C}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right)\right)d\mathfrak{u}\\ +{\int }_0^1{\mathfrak{u}}^{\beta -1}{{\rm Y}}_{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\mathcal{C}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right)\right)d\mathfrak{u}\\ \le {\int }_0^1\left[\begin{array}{c}{\mathfrak{u}}^{\beta -1}{{\rm Y}}_{*}\left(\rho ,\mathfrak{b}\right)\left\{\mathfrak{u}\mathcal{C}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right)\right)+\left(1-\mathfrak{u}\right)\mathcal{C}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right)\right)\right\}\\ +{\mathfrak{u}}^{\beta -1}{{\rm Y}}_{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\left\{\left(1-\mathfrak{u}\right)\mathcal{C}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right)\right)+\mathfrak{u}\mathcal{C}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right)\right)\right\}\end{array}\right]d\mathfrak{u},\\ {\int }_0^1{\mathfrak{u}}^{\beta -1}{{\rm Y}}^{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\mathcal{C}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right)\right)d\mathfrak{u}\\ +{\int }_0^1{\mathfrak{u}}^{\beta -1}{{\rm Y}}^{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\mathcal{C}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right)\right)d\mathfrak{u}\\ \ge {\int }_0^1\left[\begin{array}{c}{\mathfrak{u}}^{\beta -1}{{\rm Y}}^{*}\left(\rho ,\mathfrak{b}\right)\left\{\mathfrak{u}\mathcal{C}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right)\right)+\left(1-\mathfrak{u}\right)\mathcal{C}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right)\right)\right\}\\ +{\mathfrak{u}}^{\beta -1}{{\rm Y}}^{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\left\{\left(1-\mathfrak{u}\right)\mathcal{C}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right)\right)+\mathfrak{u}\mathcal{C}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right)\right)\right\}\end{array}\right]d\mathfrak{u},\end{array} \\ \begin{array}{c}\\ ={{\rm Y}}_{*}\left(\rho ,\mathfrak{b}\right){\int }_0^1{\mathfrak{u}}^{\beta -1}C\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right)\right)d\mathfrak{u}+{{\rm Y}}_{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right){\int }_0^1\begin{array}{c}{\mathfrak{u}}^{\beta -1}C\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right)\right)\end{array}d\mathfrak{u},\\ ={{\rm Y}}^{*}\left(\rho ,\mathfrak{b}\right){\int }_0^1{\mathfrak{u}}^{\beta -1}C\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right)\right)d\mathfrak{u}+{{\rm Y}}^{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right){\int }_0^1\begin{array}{c}{\mathfrak{u}}^{\beta -1}C\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right)\right)\end{array}d\mathfrak{u}.\end{array} \end{gathered}$$

Since $\mathcal{C}$ is symmetric, then

$$\begin{gathered} \begin{array}{c}\\ =\left[{{\rm Y}}_{*}\left(\rho ,\mathfrak{b}\right)+{{\rm Y}}_{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\right]{\int }_0^1\begin{array}{c}{\mathfrak{u}}^{\beta -1}C\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right)\right)\end{array}d\mathfrak{u},\\ =\left[{{\rm Y}}^{*}\left(\rho ,\mathfrak{b}\right)+{{\rm Y}}^{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\right]{\int }_0^1\begin{array}{c}{\mathfrak{u}}^{\beta -1}C\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right)\right)\end{array}d\mathfrak{u}.\end{array} \\ \begin{array}{c}\\ =\frac{{{\rm Y}}_{*}\left(\rho ,\mathfrak{b}\right)+{{\rm Y}}_{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)}{2}\frac{\Gamma \left(\beta \right)}{{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }\mathcal{C}\left(\rho +\phi \left(\mu ,\rho \right)\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }\mathcal{C}\left(\rho \right)\right],\\ =\frac{{{\rm Y}}^{*}\left(\rho ,\mathfrak{b}\right)+{{\rm Y}}^{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)}{2}\frac{\Gamma \left(\beta \right)}{{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }\mathcal{C}\left(\rho +\phi \left(\mu ,\rho \right)\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }\mathcal{C}\left(\rho \right)\right].\end{array} \end{gathered}$$

(58)

Since

$$\begin{array}{c}\\ {\int }_0^1{\mathfrak{u}}^{\beta -1}{{\rm Y}}_{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\mathcal{C}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right)\right)d\mathfrak{u}\\ +{\int }_0^1{\mathfrak{u}}^{\beta -1}{{\rm Y}}_{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\mathcal{C}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right)\right)d\mathfrak{u}\\ =\frac{1}{{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}{\int }_{\rho }^{\rho +\phi \left(\mu ,\rho \right)}{\left(\varkappa -\rho \right)}^{\beta -1}{{\rm Y}}_{*}\left(2\rho +\phi \left(\mu ,\rho \right)-\varkappa ,\mathfrak{b}\right)\mathcal{C}\left(\varkappa \right)d\varkappa \\ +\frac{1}{{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}{\int }_{\rho }^{\rho +\phi \left(\mu ,\rho \right)}{\left(\varkappa -\rho \right)}^{\beta -1}{{\rm Y}}_{*}\left(\varkappa ,\mathfrak{b}\right)\mathcal{C}\left(\varkappa \right)d\varkappa \\ =\frac{1}{{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}{\int }_{\rho }^{\rho +\phi \left(\mu ,\rho \right)}{\left(\varkappa -\rho \right)}^{\beta -1}{{\rm Y}}_{*}\left(\varkappa ,\mathfrak{b}\right)\mathcal{C}\left(2\rho +\phi \left(\mu ,\rho \right)-\varkappa \right)d\varkappa \\ +\frac{1}{{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}{\int }_{\rho }^{\rho +\phi \left(\mu ,\rho \right)}{\left(\varkappa -\rho \right)}^{\beta -1}{{\rm Y}}_{*}\left(\varkappa ,\mathfrak{b}\right)\mathcal{C}\left(\varkappa \right)d\varkappa \\ =\frac{\Gamma \left(\beta \right)}{{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }{{\rm Y}}_{*}\mathcal{C}\left(\mu \right)+{\mathcal{I}}_{{\mu }^{-}}^{\beta }{{\rm Y}}_{*}\mathcal{C}\left(\rho \right)\right],\\ {\int }_0^1{\mathfrak{u}}^{\beta -1}{{\rm Y}}^{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\mathcal{C}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right)\right)d\mathfrak{u}\\ +{\int }_0^1{\mathfrak{u}}^{\beta -1}{{\rm Y}}^{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\mathcal{C}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right)\right)d\mathfrak{u}\\ =\frac{\Gamma \left(\beta \right)}{{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }{{\rm Y}}^{*}\mathcal{C}\left(\rho +\phi \left(\mu ,\rho \right)\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }{{\rm Y}}^{*}\mathcal{C}\left(\rho \right)\right].\\ \end{array}$$

(59)

Then, from (58), we have

$$\begin{array}{c}\\ \frac{\Gamma \left(\beta \right)}{{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }{{\rm Y}}_{*}\mathcal{C}\left(\rho +\phi \left(\mu ,\rho \right)\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }{{\rm Y}}_{*}\mathcal{C}\left(\rho \right)\right]\\ \le \frac{{{\rm Y}}_{*}\left(\rho ,\mathfrak{b}\right)+{{\rm Y}}_{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)}{2}\frac{\Gamma \left(\beta \right)}{{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }\mathcal{C}\left(\rho +\phi \left(\mu ,\rho \right)\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }\mathcal{C}\left(\rho \right)\right]\\ \le \frac{{{\rm Y}}_{*}\left(\rho ,\mathfrak{b}\right)+{{\rm Y}}_{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)}{2}\frac{\Gamma \left(\beta \right)}{{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }\mathcal{C}\left(\rho +\phi \left(\mu ,\rho \right)\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }\mathcal{C}\left(\rho \right)\right],\\ \\ \frac{\Gamma \left(\beta \right)}{{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }{{\rm Y}}^{*}\mathcal{C}\left(\rho +\phi \left(\mu ,\rho \right)\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }{{\rm Y}}^{*}\mathcal{C}\left(\rho \right)\right]\\ \ge \frac{{{\rm Y}}^{*}\left(\rho ,\mathfrak{b}\right)+{{\rm Y}}^{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)}{2}\frac{\Gamma \left(\beta \right)}{{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }\mathcal{C}\left(\rho +\phi \left(\mu ,\rho \right)\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }\mathcal{C}\left(\rho \right)\right]\\ \ge \frac{{{\rm Y}}^{*}\left(\rho ,\mathfrak{b}\right)+{{\rm Y}}^{*}\left(\rho +\phi \left(\mu \right),\mathfrak{b}\right)}{2}\frac{\Gamma \left(\beta \right)}{{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }\mathcal{C}\left(\rho +\phi \left(\mu ,\rho \right)\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }\mathcal{C}\left(\rho \right)\right],\end{array}$$

That is,

$$\begin{gathered} \frac{\Gamma \left(\beta \right)}{{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }{{\rm Y}}_{*}\mathcal{C}\left(\rho +\phi \left(\mu ,\rho \right)\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }{{\rm Y}}_{*}\mathcal{C}\left(\rho \right),{\mathcal{I}}_{{\rho }^{+}}^{\beta }{{\rm Y}}^{*}\mathcal{C}\left(\rho +\phi \left(\mu ,\rho \right)\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }{{\rm Y}}^{*}\mathcal{C}\left(\rho \right)\right] \\ {\supseteq }_I\frac{\Gamma \left(\beta \right)}{{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\left[\frac{{{\rm Y}}_{*}\left(\rho ,\mathfrak{b}\right)+{{\rm Y}}_{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)}{2},\frac{{{\rm Y}}^{*}\left(\rho ,\mathfrak{b}\right)+{{\rm Y}}^{*}\left(\rho +\phi \left(\mu ,\rho \right),\mathfrak{b}\right)}{2}\right]\left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }\mathcal{C}\left(\rho +\phi \left(\mu ,\rho \right)\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }\mathcal{C}\left(\rho \right)\right] \\ {\supseteq }_I\frac{\Gamma \left(\beta \right)}{{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\left[\frac{{{\rm Y}}_{*}\left(\rho ,\mathfrak{b}\right)+{{\rm Y}}_{*}\left(\mu ,\mathfrak{b}\right)}{2},\frac{{{\rm Y}}^{*}\left(\rho ,\mathfrak{b}\right)+{{\rm Y}}^{*}\left(\mu ,\mathfrak{b}\right)}{2}\right]\left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }\mathcal{C}\left(\rho +\phi \left(\mu ,\rho \right)\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }\mathcal{C}\left(\rho \right)\right]. \end{gathered}$$

Hence,

$$\begin{gathered} \left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }\widetilde{{\rm Y}}\mathcal{C}\left(\rho +\phi \left(\mu ,\rho \right)\right)\oplus {\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }\widetilde{{\rm Y}}\mathcal{C}\left(\rho \right)\right] \\ {\supseteq }_{\mathbb{F}}\frac{\widetilde{{\rm Y}}\left(\rho \right)\oplus \widetilde{{\rm Y}}\left(\rho +\phi \left(\mu ,\rho \right)\right)}{2}\odot \left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }\mathcal{C}\left(\rho +\phi \left(\mu ,\rho \right)\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }\mathcal{C}\left(\rho \right)\right] \\ {\supseteq }_{\mathbb{F}}\frac{\widetilde{{\rm Y}}\left(\rho \right)\oplus \widetilde{{\rm Y}}\left(\mu \right)}{2}\odot \left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }\mathcal{C}\left(\rho +\phi \left(\mu ,\rho \right)\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }\mathcal{C}\left(\rho \right)\right]. \end{gathered}$$

 □

Theorem 9.

Let $\widetilde{{\rm Y}}:\left[\rho ,\rho +\phi \left(\mu ,\rho \right)\right]\to {₤}_0$ be a UD pre-invex $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$ with $\rho <\mu$ whose $\mathfrak{b}$-levels define the family of $I\mathcal{V}\mathcal{M}$s ${{\rm Y}}_{\mathfrak{b}}:\left[\rho ,\rho +\phi \left(\mu ,\rho \right)\right]\subset \mathbb{R}\to {\mathbb{X}}_o^{+}$ given by ${{\rm Y}}_{\mathfrak{b}}\left(\varkappa \right)=\left[{{\rm Y}}_{*}\left(\varkappa ,\mathfrak{b}\right),{{\rm Y}}^{*}\left(\varkappa ,\mathfrak{b}\right)\right]$ for all $\varkappa \in \left[\rho ,\rho +\phi \left(\mu ,\rho \right)\right]$ and for all $\mathfrak{b}\in \left[0,1\right]$. Let $\widetilde{{\rm Y}}\in L\left(\left[\rho ,\rho +\phi \left(\mu ,\rho \right)\right],{₤}_0\right)$ and $\mathcal{C}:\left[\rho ,\rho +\phi \left(\mu ,\rho \right)\right]\to \mathbb{R},\mathcal{C}\left(\varkappa \right)\ge 0$ be symmetric with respect to $\frac{2\rho +\phi \left(\mu ,\rho \right)}{2}$. If $\phi$ satisfies Assumption 1, then

$$\begin{gathered} \widetilde{{\rm Y}}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2}\right)\odot \left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }\mathcal{C}\left(\rho +\phi \left(\mu ,\rho \right)\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }\mathcal{C}\left(\rho \right)\right] \\ {\supseteq }_{\mathbb{F}}\left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }\widetilde{{\rm Y}}\mathcal{C}\left(\rho +\phi \left(\mu ,\rho \right)\right)\oplus {\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }\widetilde{{\rm Y}}\mathcal{C}\left(\rho \right)\right]. \end{gathered}$$

(60)

If $\widetilde{{\rm Y}}$ is UD pre-incave, then the inequality (60) is reversed.

Proof. 

Since $\widetilde{{\rm Y}}$ is a ρ pre-invex $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$, then for $\mathfrak{b}\in \left[0,1\right],$ we have

$$\begin{array}{c}\\ {{\rm Y}}_{*}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2},\mathfrak{b}\right)\le \frac{1}{2}\left({{\rm Y}}_{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right)+{{\rm Y}}_{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\right)\\ {{\rm Y}}^{*}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2},\mathfrak{b}\right)\ge \frac{1}{2}\left({{\rm Y}}^{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right)+{{\rm Y}}^{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\right).\end{array}$$

(61)

Since $\mathcal{C}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right)\right)=\mathcal{C}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right)\right)$, then by multiplying (61) by ${\mathfrak{u}}^{\beta -1}\mathcal{C}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right)\right)$ and integrating with respect to $\mathfrak{u}$ over $\left[0,1\right],$ we obtain

$$\begin{array}{l}\\ {{\rm Y}}_{*}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2},\mathfrak{b}\right){\int }_0^1{\mathfrak{u}}^{\beta -1}\mathcal{C}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right)\right)d\mathfrak{u}\\ \le \frac{1}{2}\left(\begin{array}{c}{\int }_0^1{\mathfrak{u}}^{\beta -1}{{\rm Y}}_{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\mathcal{C}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right)\right)d\mathfrak{u}\\ +{\int }_0^1{\mathfrak{u}}^{\beta -1}{{\rm Y}}_{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\mathcal{C}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right)\right)d\mathfrak{u}\end{array}\right),\\ {{\rm Y}}^{*}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2},\mathfrak{b}\right){\int }_0^1\mathcal{C}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right)\right)d\mathfrak{u}\\ \ge \frac{1}{2}\left(\begin{array}{c}{\int }_0^1{\mathfrak{u}}^{\beta -1}{{\rm Y}}^{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\mathcal{C}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right)\right)d\mathfrak{u}\\ +{\int }_0^1{\mathfrak{u}}^{\beta -1}{{\rm Y}}^{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\mathcal{C}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right)\right)d\mathfrak{u}\end{array}\right).\end{array}$$

(62)

Let $\varkappa =\rho +\mathfrak{u}\phi \left(\mu ,\rho \right)$. Then, we have

$$\begin{array}{c}\\ {\int }_0^1{\mathfrak{u}}^{\beta -1}{{\rm Y}}_{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\mathcal{C}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right)\right)d\mathfrak{u}\\ +{\int }_0^1{\mathfrak{u}}^{\beta -1}{{\rm Y}}_{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\mathcal{C}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right)\right)d\mathfrak{u}\\ =\frac{1}{{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}{\int }_{\rho }^{\rho +\phi \left(\mu ,\rho \right)}{\left(\varkappa -\rho \right)}^{\beta -1}{{\rm Y}}_{*}\left(2\rho +\phi \left(\mu ,\rho \right)-\varkappa ,\mathfrak{b}\right)\mathcal{C}\left(\varkappa \right)d\varkappa \\ +\frac{1}{{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}{\int }_{\rho }^{\rho +\phi \left(\mu ,\rho \right)}{\left(\varkappa -\rho \right)}^{\beta -1}{{\rm Y}}_{*}\left(\varkappa ,\mathfrak{b}\right)\mathcal{C}\left(\varkappa \right)d\varkappa \\ =\frac{1}{{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}{\int }_{\rho }^{\rho +\phi \left(\mu ,\rho \right)}{\left(\varkappa -\rho \right)}^{\beta -1}{{\rm Y}}_{*}\left(\varkappa ,\mathfrak{b}\right)\mathcal{C}\left(2\rho +\phi \left(\mu ,\rho \right)-\varkappa \right)d\varkappa \\ +\frac{1}{{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}{\int }_{\rho }^{\rho +\phi \left(\mu ,\rho \right)}{\left(\varkappa -\rho \right)}^{\beta -1}{{\rm Y}}_{*}\left(\varkappa ,\mathfrak{b}\right)\mathcal{C}\left(\varkappa \right)d\varkappa \\ =\frac{\Gamma \left(\beta \right)}{{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }{{\rm Y}}_{*}\mathcal{C}\left(\rho +\phi \left(\mu ,\rho \right)\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }{{\rm Y}}_{*}\mathcal{C}\left(\rho \right)\right],\\ {\int }_0^1{\mathfrak{u}}^{\beta -1}{{\rm Y}}^{*}\left(\rho +\left(1-\mathfrak{u}\right)\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\mathcal{C}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right)\right)d\mathfrak{u}\\ +{\int }_0^1{\mathfrak{u}}^{\beta -1}{{\rm Y}}^{*}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right),\mathfrak{b}\right)\mathcal{C}\left(\rho +\mathfrak{u}\phi \left(\mu ,\rho \right)\right)d\mathfrak{u}\\ =\frac{\Gamma \left(\beta \right)}{{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }{{\rm Y}}^{*}\mathcal{C}\left(\rho +\phi \left(\mu ,\rho \right)\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }{{\rm Y}}^{*}\mathcal{C}\left(\rho \right)\right].\end{array}$$

(63)

Then, from (63), we have

$$\begin{array}{c}\\ \frac{\Gamma \left(\beta \right)}{{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}{{\rm Y}}_{*}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2},\mathfrak{b}\right)\left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }\mathcal{C}\left(\rho +\phi \left(\mu ,\rho \right)\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }\mathcal{C}\left(\rho \right)\right]\\ \le \frac{\Gamma \left(\beta \right)}{{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }{{\rm Y}}_{*}\mathcal{C}\left(\rho +\phi \left(\mu ,\rho \right)\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }{{\rm Y}}_{*}\mathcal{C}\left(\rho \right)\right],\\ \frac{\Gamma \left(\beta \right)}{{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}{{\rm Y}}^{*}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2},\mathfrak{b}\right)\left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }\mathcal{C}\left(\rho +\phi \left(\mu ,\rho \right)\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }\mathcal{C}\left(\rho \right)\right]\\ \ge \frac{\Gamma \left(\beta \right)}{{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }{{\rm Y}}^{*}\mathcal{C}\left(\rho +\phi \left(\mu ,\rho \right)\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }{{\rm Y}}^{*}\mathcal{C}\left(\rho \right)\right],\end{array}$$

from which we have

$$\begin{array}{c}\\ \frac{\Gamma \left(\beta \right)}{{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\left[{{\rm Y}}_{*}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2},\mathfrak{b}\right),{{\rm Y}}^{*}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2},\mathfrak{b}\right)\right]\left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }\mathcal{C}\left(\rho +\phi \left(\mu ,\rho \right)\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }\mathcal{C}\left(\rho \right)\right]\\ {\begin{array}{c}\begin{array}{c}\supseteq \end{array}\end{array}}_I\frac{\Gamma \left(\beta \right)}{{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }{{\rm Y}}_{*}\mathcal{C}\left(\rho +\phi \left(\mu ,\rho \right)\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }{{\rm Y}}_{*}\mathcal{C}\left(\rho \right),{\mathcal{I}}_{{\rho }^{+}}^{\beta }{{\rm Y}}^{*}\mathcal{C}\left(\rho +\phi \left(\mu ,\rho \right)\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }{{\rm Y}}^{*}\mathcal{C}\left(\rho \right)\right].\\ \end{array}$$

That is,

$$\begin{gathered} \frac{\Gamma \left(\beta \right)}{{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\odot \widetilde{{\rm Y}}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2}\right)\odot \left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }\mathcal{C}\left(\rho +\phi \left(\mu ,\rho \right)\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }\mathcal{C}\left(\rho \right)\right] \\ {\supseteq }_{\mathbb{F}}\frac{\Gamma \left(\beta \right)}{{\left(\phi \left(\mu ,\rho \right)\right)}^{\beta }}\odot \left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }\widetilde{{\rm Y}}\mathcal{C}\left(\rho +\phi \left(\mu ,\rho \right)\right)\oplus {\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }\widetilde{{\rm Y}}\mathcal{C}\left(\rho \right)\right]. \end{gathered}$$

This finishes the proof. □

Example 3.

Let us consider the $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$ $\widetilde{{\rm Y}}:\left[0,2\right]\to {₤}_0$ as

$$\widetilde{{\rm Y}}\left(\varkappa \right)\left(m\right)=\left\{\begin{array}{c}\frac{\begin{array}{c}m\end{array}}{\frac{1}{2}+\sqrt{\varkappa }},m\in \left[2-\sqrt{\varkappa },\frac{3}{2}\right],\\ \frac{2+\sqrt{\varkappa }-m}{\sqrt{\varkappa }-\frac{1}{2}},m\in \left(\frac{3}{2},2+\sqrt{\varkappa }\right],\\ 0,otherwise.\end{array}\right.$$

(64)

Then, for each $\mathfrak{b}\in \left[0,1\right],$ we get ${{\rm Y}}_{\mathfrak{b}}\left(\varkappa \right)=\left[\left(1-\mathfrak{b}\right)\left(2-\sqrt{\varkappa }\right)+\frac{3}{2}\mathfrak{b},\left(1-\mathfrak{b}\right)\left(2+\sqrt{\varkappa }\right)+\frac{3}{2}\mathfrak{b})\right]$. Since the end point functions ${{\rm Y}}_{\mathrm{*}}\left(\varkappa ,\mathfrak{b}\right),$ ${{\rm Y}}^{\mathrm{*}}\left(\varkappa ,\mathfrak{b}\right)$ are pre-invex with respect to $\phi \left(\mu ,\rho \right)=\mu -\rho$ for each $\mathfrak{b}\in [0,1]$, then $\widetilde{{\rm Y}}\left(\varkappa \right)$ is ρ pre-invex $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$. If

$$\mathcal{C}\left(\varkappa \right)=\left\{\begin{array}{c}\sqrt{\varkappa },m\in \left[\mathrm{0,1}\right],\\ \sqrt{2-\varkappa },m\in \left(1,2\right],\end{array}\right.$$

(65)

then $\mathcal{C}\left(2-\varkappa \right)=\mathcal{C}\left(\varkappa \right)\ge 0$ for all $\varkappa \in \left[0,2\right]$. Since ${{\rm Y}}_{\mathrm{*}}\left(\varkappa ,\mathfrak{b}\right)=\left(1-\mathfrak{b}\right)\left(2-\sqrt{\varkappa }\right)+\frac{3}{2}\mathfrak{b}$ and ${{\rm Y}}^{\mathrm{*}}\left(\varkappa ,\mathfrak{b}\right)=\left(1-\mathfrak{b}\right)\left(2+\sqrt{\varkappa }\right)+\frac{3}{2}\mathfrak{b}$, if $\beta =\frac{1}{2}$, then we get

$$\begin{gathered} \left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }\widetilde{{\rm Y}}\mathcal{C}\left(\rho +\phi \left(\mu ,\rho \right)\right)\oplus {\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }\widetilde{{\rm Y}}\mathcal{C}\left(\rho \right)\right] \\ {\supseteq }_{\mathbb{F}}\frac{\widetilde{{\rm Y}}\left(\rho \right)\oplus \widetilde{{\rm Y}}\left(\rho +\phi \left(\mu ,\rho \right)\right)}{2}\odot \left[\begin{array}{c}{\mathcal{I}}_{{\rho }^{+}}^{\beta }C\left(\rho +\phi \left(\mu ,\rho \right)\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }C\left(\rho \right)\end{array}\right] \\ \begin{array}{c}\frac{{{\rm Y}}_{*}\left(\rho \right)+{{\rm Y}}_{*}\left(\rho +\phi \left(\mu ,\rho \right)\right)}{2}\left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }\mathcal{C}\left(\rho +\phi \left(\mu ,\rho \right)\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }\mathcal{C}\left(\rho \right)\right]=\left(1-\mathfrak{b}\right)\sqrt{\pi }\left(\frac{4-\sqrt{2}}{2}\right)+\frac{3}{2}\sqrt{\pi }b\\ \frac{{{\rm Y}}^{*}\left(\rho \right)+{{\rm Y}}^{*}\left(\rho +\phi \left(\mu ,\rho \right)\right)}{2}\left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }\mathcal{C}\left(\rho +\phi \left(\mu ,\rho \right)\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }\mathcal{C}\left(\rho \right)\right]=\left(1-\mathfrak{b}\right)\sqrt{\pi }\left(\frac{4+\sqrt{2}}{2}\right)+\frac{3}{2}\sqrt{\pi }b,\end{array} \end{gathered}$$

(66)

$$\begin{array}{c}\frac{{{\rm Y}}_{*}\left(\rho \right)+{{\rm Y}}_{*}\left(\mu \right)}{2}\left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }\mathcal{C}\left(\rho +\phi \left(\mu ,\rho \right)\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }\mathcal{C}\left(\rho \right)\right]=\left(1-\mathfrak{b}\right)\sqrt{\pi }\left(\frac{4-\sqrt{2}}{2}\right)+\frac{3}{2}\sqrt{\pi }b\\ \frac{{{\rm Y}}^{*}\left(\rho \right)\widetilde{+}{{\rm Y}}^{*}\left(\mu \right)}{2}\left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }\mathcal{C}\left(\rho +\phi \left(\mu ,\rho \right)\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }\mathcal{C}\left(\rho \right)\right]=\left(1-\mathfrak{b}\right)\sqrt{\pi }\left(\frac{4+\sqrt{2}}{2}\right)+\frac{3}{2}\sqrt{\pi }b,\end{array}$$

(67)

$$\begin{array}{c}\left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }{{\rm Y}}_{*}\mathcal{C}\left(\rho +\phi \left(\mu ,\rho \right)\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }{{\rm Y}}_{*}\mathcal{C}\left(\rho \right)\right]=\frac{1}{\sqrt{\pi }}\left(1-\mathfrak{b}\right)\left(2\pi +\frac{4-8\sqrt{2}}{3}\right)+\frac{3}{2}\sqrt{\pi }b,\\ \left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }{{\rm Y}}^{*}\mathcal{C}\left(\rho +\phi \left(\mu ,\rho \right)\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }{{\rm Y}}^{*}\mathcal{C}\left(\rho \right)\right]=\frac{1}{\sqrt{\pi }}\left(1-\mathfrak{b}\right)\left(2\pi +\frac{8\sqrt{2}-4}{3}\right)+\frac{3}{2}\sqrt{\pi }b.\end{array}$$

(68)

From (66)–(68), we have

$$\begin{gathered} \left[\frac{\left(1-\mathfrak{b}\right)}{\sqrt{\pi }}\left(2\pi +\frac{4-8\sqrt{2}}{3}\right)+\frac{3}{2}\sqrt{\pi }\mathfrak{b},\frac{\left(1-\mathfrak{b}\right)}{\sqrt{\pi }}\left(2\pi +\frac{8\sqrt{2}-4}{3}\right)+\frac{3}{2}\sqrt{\pi }\mathfrak{b}\right] \\ {\supseteq }_I\sqrt{\pi }\left[\left(1-\mathfrak{b}\right)\left(\frac{4-\sqrt{2}}{2}\right)+\frac{3}{2}\mathfrak{b},(1-\mathfrak{b})\left(\frac{4+\sqrt{2}}{2}\right)+\frac{3}{2}\mathfrak{b}\right], \\ =\sqrt{\pi }\left[\left(1-\mathfrak{b}\right)\left(\frac{4-\sqrt{2}}{2}\right)+\frac{3}{2}\mathfrak{b},(1-\mathfrak{b})\left(\frac{4+\sqrt{2}}{2}\right)+\frac{3}{2}\mathfrak{b}\right], \end{gathered}$$

for each $\mathfrak{b}\in \left[0,1\right].$ Therefore, we have verified Theorem 8.

For Theorem 9, it yields

$$\begin{array}{c}{{\rm Y}}_{\mathrm{*}}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2},\mathfrak{b}\right)\left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }\mathcal{C}\left(\rho +\phi \left(\mu ,\rho \right)\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }\mathcal{C}\left(\rho \right)\right]=\left(1-\mathfrak{b}\right)\sqrt{\pi }+\frac{3}{2}\sqrt{\pi }b,\\ {{\rm Y}}^{\mathrm{*}}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2},\mathfrak{b}\right)\left[{\mathcal{I}}_{{\rho }^{+}}^{\beta }\mathcal{C}\left(\rho +\phi \left(\mu ,\rho \right)\right)+{\mathcal{I}}_{{\rho +\phi \left(\mu ,\rho \right)}^{-}}^{\beta }\mathcal{C}\left(\rho \right)\right]=3\left(1-\mathfrak{b}\right)\sqrt{\pi }+\frac{3}{2}\sqrt{\pi }b.\end{array}$$

(69)

From (68) and (69), we have

$$\begin{gathered} \sqrt{\pi }\left[\left(1-\mathfrak{b}\right)+\frac{3}{2}\mathfrak{b},3\left(1-\mathfrak{b}\right)+\frac{3}{2}\mathfrak{b}\right]{\supseteq }_I[\frac{\left(1-\mathfrak{b}\right)}{\sqrt{\pi }}\left(2\pi +\frac{4-8\sqrt{2}}{3}\right) \\ +\frac{3}{2}\sqrt{\pi }\mathfrak{b},\frac{\left(1-\mathfrak{b}\right)}{\sqrt{\pi }}\left(2\pi +\frac{8\sqrt{2}-4}{3}\right)+\frac{3}{2}\sqrt{\pi }\mathfrak{b}] \end{gathered}$$

for each $\mathfrak{b}\in \left[0,1\right].$

Remark 4.

If $\mathcal{C}\left(\varkappa \right)=1$, then, from Theorems 8 and 9, we get Theorem 5.

Let $\beta =1$. Then, we obtain the HH–Fejér type inequality for ρ pre-invex $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$ [4]

$$\begin{array}{ll}\widetilde{{\rm Y}}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2}\right)& {\supseteq }_{\mathbb{F}}\frac{1}{{\int }_{\rho }^{\rho +\phi \left(\mu ,\rho \right)}\mathcal{C}\left(\varkappa \right)d\varkappa }\\ & \odot (FR){\int }_{\rho }^{\rho +\phi \left(\mu ,\rho \right)}\widetilde{{\rm Y}}\left(\varkappa \right)\odot \mathcal{C}\left(\varkappa \right)d\varkappa {\supseteq }_{\mathbb{F}}\frac{\widetilde{{\rm Y}}\left(\rho \right)\oplus \widetilde{{\rm Y}}\left(\mu \right)}{2}\end{array}$$

(70)

Let $\beta =1$ and $\widetilde{{\rm Y}}$ be a lower ρ pre-invex $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$. Then, we obtain the HH–Fejér type inequality for lower ρ pre-invex $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$ [84]

$$\begin{array}{ll}\widetilde{{\rm Y}}\left(\frac{2\rho +\phi \left(\mu ,\rho \right)}{2}\right)& {\le }_{\mathbb{F}}\frac{1}{{\int }_{\rho }^{\rho +\phi \left(\mu ,\rho \right)}\mathcal{C}\left(\varkappa \right)d\varkappa }\\ & \odot (FR){\int }_{\rho }^{\rho +\phi \left(\mu ,\rho \right)}\widetilde{{\rm Y}}\left(\varkappa \right)\odot \mathcal{C}\left(\varkappa \right)d\varkappa {\le }_{\mathbb{F}}\frac{\widetilde{{\rm Y}}\left(\rho \right)\oplus \widetilde{{\rm Y}}\left(\mu \right)}{2}\end{array}$$

(71)

If ${{\rm Y}}_{*}(\varkappa ,\mathfrak{b})={{\rm Y}}^{*}(\varkappa ,\mathfrak{b})$ with $\phi \left(\varkappa ,y\right)=\varkappa -y$ and $\mathcal{C}\left(\varkappa \right)=\beta =1=\mathfrak{b}$, then, by means of Theorems 8 and 9, we get the classical HH inequality [97].

If ${{\rm Y}}_{*}(\varkappa ,\mathfrak{b})={{\rm Y}}^{*}(\varkappa ,\mathfrak{b})$ with $\phi \left(\varkappa ,y\right)=\varkappa -y$ and $\beta =1$, then, by using Theorems 8 and 9, we obtain the classical HH–Fejér inequality (2).

4. Conclusions and Future Work

In this study, the HH, HH–Fejér, and Pachpatte type integral inequalities were successfully derived using the fuzzy fractional integral operator and by employing functions that have UD fuzzy-number pre-invexity properties. A number of generalizations for the pre-invexity theory could be derived. The results appeared trustworthy and effective for dealing with various nonlinear issues that arise in science and engineering. The findings have a general nature and may constitute contributions to complex waveform theory. Further research is required to confirm such a probable connection. In the future, we will focus this concept on the field of quantum calculus.