Some New Versions of Hermite–Hadamard Integral Inequalities in Fuzzy Fractional Calculus for Generalized Pre-Invex Functions via Fuzzy-Interval-Valued Settings

Abstract

The purpose of this study is to prove the existence of fractional integral inclusions that are connected to the Hermite–Hadamard and Hermite–Hadamard–Fejér type inequalities for χ-pre-invex fuzzy-interval-valued functions. Some of the related fractional integral inequalities are also proved via Riemann–Liouville fractional integral operator, where integrands are fuzzy-interval-valued functions. To prove the validity of our main results, some of the nontrivial examples are also provided. As specific situations, our findings can provide a variety of new and well-known outcomes which can be viewed as applications of our main results. The results in this paper can be seen as refinements and improvements to previously published findings.

1. Introduction

The Hermite–Hadamard inequality is one of the most well-known inequalities in the theory of convex functions with geometrical meaning, and it has a wide range of applications. This disparity might be seen as a refinement of the idea of convexity. In recent years, the Hermite–Hadamard inequality ($H.H$ inequality) for convex functions has gotten a lot of attention, and there have been some impressive improvements and generalizations, see [1,2].

The relevance of set-valued analysis research from both a theoretical and practical standpoint is well understood. Control theory and dynamical games have fueled several developments in set-valued analysis. Since the dawn of the 1960s, optimal control theory and mathematical programming have been pushing these disciplines. Interval analysis is a special instance that was developed to deal with interval uncertainty that may be found in many mathematical or computer models of deterministic real-world processes. In recent years, a few major inequalities for interval-valued functions, such as $H.H$ and Ostrowski type inequalities have been developed. Chalco-Cano et al. used Hukuhara derivatives for interval-valued functions to develop Ostrowski type inequalities for interval-valued functions in [3,4]. For interval-valued functions, Roman-Flores et al. established Minkowski and Beckenbach’s inequalities in [5]. Readers [6,7,8,9,10,11,12,13] are referred to for further relevant findings.

Novel inequalities for various forms of convexities, pre-invexities, statistical theory, and other fields have lately been developed by a number of authors. Several debates illustrate that inequality theory and convex functions are inextricably linked. For the first time in 1981, Hanson looked at the invex function in the setting of bi-function $\Lambda \left(.,.\right)$ (see [14]). Following in the footsteps of Hanson, Ben-Israel and Mond sought to explore deeper into linked invexity, for the first time presenting the ideas of invex sets and pre-invex functions (see [15]). According to Mohan and Neogy [16], the pre-invex and invex functions in the form of differentiability are equivalent under certain situations. In 2005, Antczak [17] was the first to uncover and study the characteristics of pre-invex functions. Fuzzy mappings (F-Ms) are functions with a fuzzy interval value. Nanda and Kar [18], Syau [19], and Furukawa [20], introduced the notion of convex F-Ms from ${ℝ}^n$ to the set of fuzzy numbers. They also constructed various forms of convex F-Ms, such as logarithmic convex F-Ms and quasi-convex F-Ms, and studied Lipschitz continuity of fuzzy-valued mappings. Based on Goetschel and Voxman’s idea of ordering [21], Yan and Xu [22] presented the concepts of epigraphs and convexity of F-Ms, as well as the features of convex F-Ms and quasi-convex F-Ms,. Noor [23] proposed and investigated the notion of fuzzy pre-invex mapping on the invex set. He also showed how to express the fuzzy optimality conditions of differentiable pre-invex fuzzy mappings using variational inequalities.

Khan et al. [24,25,26,27] extended the class of convex F⋅Ms and defined $h$-convex and $\left(h_1, h_2\right)$-convex F-IV-Fs using fuzzy partial order relation. Moreover, they introduced $H.H$, Hermite–Hadamard–Fejér ($H.H$ Fejér), fractional $H.H$ and $H.H$ Fejér for $h$-convex and $\left(h_1, h_2\right)$-convex F⋅I⋅V⋅Fs via fuzzy Riemannian and fuzzy Riemann–Liouville fractional integrals. We suggest readers to [28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63] and the references therein for more study of literature on the applications and properties of fuzzy-interval, inequalities, and generalized convex F-Ms.

In Section 2, motivated by ongoing research work some preliminary notions, definitions and some new results are introduced. In the Riemann–Liouville interval-valued fractional operator settings, Section 3 develops $H.H$ and $H.H$ Fejér type inequalities and some related inequalities for $\mathit{\chi }$-pre-invex fuzzy-interval-valued functions ($\mathit{\chi }$-pre-invex F-IV-Fs). Moreover, some non-trivial examples are also given to prove the validity of our main results. We end Section 4 with conclusions and future plans.

2. Preliminaries

Let ${\mathcal{K}}_C$ and ${\mathbb{F}}_0$ be the collection of all closed and bounded intervals, and fuzzy intervals of $ℝ$. We use ${\mathcal{K}}_C^{+}$ to represent the set of all positive intervals. The collection of all Riemann integrable real valued functions, Riemann integrable IV-Fs and fuzzy Riemann integrable F-IV-Fs over $\left[t, \upsilon \right]$ is denoted by ${ℛ}_{\left[t, \upsilon \right]}$, $ℐ{ℛ}_{\left[t, \upsilon \right]},$ and $ℱ{ℛ}_{\left(\left[t, \upsilon \right]\right)}$, respectively. For more conceptions on interval-valued functions and fuzzy-interval-valued functions, see [28,34,35]. Moreover, some more Preliminaries notions are as follows:

The inclusion “$\subseteq$” means that

$$\xi \subseteq \eta \mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if},\left[{\xi }_{*}, {\xi }^{*}\right]\subseteq \left[{\eta }_{*}, {\eta }^{*}\right],\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}{\eta }_{*}\le {\xi }_{*},{\xi }^{*}\le {\eta }^{*},$$

(1)

for all $\left[{𝓇}_{*},{𝓇}^{*}\right],\left[{\eta }_{*},{\eta }^{*}\right]\in {\mathcal{K}}_C$.

Remark 1.

[31] The relation “${\le }_I$” defined on ${\mathcal{K}}_C$ by

$$\left[{𝓇}_{*}, {𝓇}^{*}\right]{\le }_I\left[{\eta }_{*}, {\eta }^{*}\right],\mathit{if}\mathit{and}\mathit{only}\mathit{if}{𝓇}_{*}\le {\eta }_{*}, {𝓇}^{*}\le {\eta }^{*},$$

(2)

for all $\left[{𝓇}_{*}, {𝓇}^{*}\right], \left[{\eta }_{*}, {\eta }^{*}\right]\in {\mathcal{K}}_C,$ it is an order relation.

Proposition 1.

[39] If $\xi ,\varpi \in {\mathbb{F}}_0$, then relation “$\preccurlyeq$” defined on ${\mathbb{F}}_0$ by

$$\xi \preccurlyeq \varpi \mathit{if}\mathit{and}\mathit{only}\mathit{if},{\left[\xi \right]}^{\varsigma }{\le }_I{\left[\varpi \right]}^{\varsigma },\mathit{for}\mathit{all}\varsigma \in \left[0, 1\right],$$

(3)

this relation is known as the partial order relation.

Theorem 1.

[29] Let $\tilde{\mathfrak{U}}:\left[t, \upsilon \right]\subset ℝ\to {\mathbb{F}}_0$ be a F-IV-F, whose $\varsigma$-levels define the family of IV-Fs ${\mathfrak{U}}_{\varsigma }:\left[t, \upsilon \right]\subset ℝ\to {\mathcal{K}}_C$ are given by ${\mathfrak{U}}_{\varsigma }\left(\varkappa \right)=\left[{\mathfrak{U}}_{*}\left(\varkappa ,\varsigma \right), {\mathfrak{U}}^{*}\left(\varkappa ,\varsigma \right)\right]$ for all $\varkappa \in \left[t, \upsilon \right]$ and for all $\varsigma \in \left(0, 1\right].$ Then $\tilde{\mathfrak{U}}$ is fuzzy Riemann integrable over $\left[t, \upsilon \right]$ if and only if, ${\mathfrak{U}}_{*}\left(\varkappa ,\varsigma \right)$ and ${\mathfrak{U}}^{*}\left(\varkappa ,\varsigma \right)$ both are Riemann integrable over $\left[t, \upsilon \right]$. Moreover, if $\tilde{\mathfrak{U}}$ is fuzzy Riemann integrable over $\left[t, \upsilon \right],$ then

$${\left[\left(FR\right){\int }_t^{\upsilon }\tilde{\mathfrak{U}}\left(\varkappa \right)d\varkappa \right]}^{\varsigma }=\left[\left(R\right){\int }_t^{\upsilon }{\mathfrak{U}}_{*}\left(\varkappa ,\varsigma \right)d\varkappa , \left(R\right){\int }_t^{\upsilon }{\mathfrak{U}}^{*}\left(\varkappa ,\varsigma \right)d\varkappa \right]=\left(IR\right){\int }_t^{\upsilon }{\mathfrak{U}}_{\varsigma }\left(\varkappa \right)d\varkappa$$

(4)

for all $\varsigma \in \left(0,1\right]$.

The following FI Riemann–Liouville fractional integral operators were introduced by Allahviranloo et al. [33].

Definition 1.

Let $\alpha >0$ and $L\left(\left[t, \upsilon \right],{\mathbb{F}}_0\right)$ be the collection of all Lebesgue measurable F⋅I⋅V⋅Fs on $\left[t,\upsilon \right]$. Then the fuzzy left and right Riemann–Liouville fractional integral of $\tilde{\mathfrak{U}}\in$ $L\left(\left[t, \upsilon \right],{\mathbb{F}}_0\right)$ with order $\alpha >0$ are defined by

$${ℐ}_{t^{+}}^{\alpha } \tilde{\mathfrak{U}}\left(\varkappa \right)=\frac{1}{\Gamma \left(\alpha \right)}{\int }_t^{\varkappa }{\left(\varkappa -s\right)}^{\alpha -1}\tilde{\mathfrak{U}}\left(s\right)ds, \left(\varkappa >t\right),$$

(5)

and

$${ℐ}_{{\upsilon }^{-}}^{\alpha } \tilde{\mathfrak{U}}\left(\varkappa \right)=\frac{1}{\Gamma \left(\alpha \right)}{\int }_{\varkappa }^{\upsilon }{\left(s-\varkappa \right)}^{\alpha -1}\tilde{\mathfrak{U}}\left(s\right)ds, (\varkappa <\upsilon )$$

(6)

respectively, where $\Gamma \left(\varkappa \right)={\int }_0^{\infty }{s}^{\varkappa -1}{e}^{-s}ds$ is the Euler gamma function. The fuzzy left and right Riemann–Liouville fractional integral $\varkappa$ based on left and right end point functions can be defined, that is

$$\begin{gathered} {\left[{ℐ}_{t^{+}}^{\alpha } \tilde{\mathfrak{U}}\left(\varkappa \right)\right]}^{\varsigma }=\frac{1}{\Gamma \left(\alpha \right)}{\int }_t^{\varkappa }{\left(\varkappa -s\right)}^{\alpha -1}{\mathfrak{U}}_{\varsigma }\left(s\right)ds \\ =\frac{1}{\Gamma \left(\alpha \right)}{\int }_t^{\varkappa }{\left(\varkappa -s\right)}^{\alpha -1}\left[{\mathfrak{U}}_{*}\left(s, \varsigma \right),{\mathfrak{U}}^{*}\left(s, \varsigma \right)\right]ds, \left(\varkappa >t\right), \end{gathered}$$

(7)

where

$${ℐ}_{t^{+}}^{\alpha } {\mathfrak{U}}_{*}\left(\varkappa , \varsigma \right)=\frac{1}{\Gamma \left(\alpha \right)}{\int }_t^{\varkappa }{\left(\varkappa -s\right)}^{\alpha -1}{\mathfrak{U}}_{*}\left(s, \varsigma \right)ds, \left(\varkappa >t\right),$$

(8)

and

$${ℐ}_{t^{+}}^{\alpha } {\mathfrak{U}}^{*}\left(\varkappa , \varsigma \right)=\frac{1}{\Gamma \left(\alpha \right)}{\int }_t^{\varkappa }{\left(\varkappa -s\right)}^{\alpha -1}{\mathfrak{U}}^{*}\left(s, \varsigma \right)ds, \left(\varkappa >t\right),$$

(9)

Similarly, the left and right end-point functions can be used to define the right Riemann–Liouville fractional integral $\tilde{\mathfrak{U}}$ of $\varkappa$.

Definition 2.

A non-negative real mapping $\mathfrak{U}:\left[t, \upsilon \right]\to {ℝ}^{+}$ is named as a convex function if

$$\mathfrak{U}\left(s\mu +\left(1-s\right)\varkappa \right)\le s\mathfrak{U}\left(\mu \right)+\left(1-s\right)\mathfrak{U}\left(\varkappa \right),$$

(10)

for all $\mu , \varkappa \in \left[t, \upsilon \right], s\in \left[0, 1\right].$ If (10) is reversed, then $\mathfrak{U}$ is named as a concave.

Definition 3.

[18] The F-IV-F $\tilde{\mathfrak{U}}:\left[t, \upsilon \right]\to {\mathbb{F}}_0$ is named as a convex F-IV-F on $\left[t, \upsilon \right]$ if

$$\tilde{\mathfrak{U}}\left(s\mu +\left(1-s\right)\varkappa \right)\preccurlyeq s\tilde{\mathfrak{U}}\left(\mu \right)\widetilde{+}\left(1-s\right)\tilde{\mathfrak{U}}\left(\varkappa \right),$$

(11)

for all $\mu , \varkappa \in \left[t, \upsilon \right], s\in \left[0, 1\right],$ where $\tilde{\mathfrak{U}}\left(\varkappa \right)\succcurlyeq \tilde{0}$ for all $\varkappa \in \left[t, \upsilon \right].$ If (11) is reversed, then $\tilde{\mathfrak{U}}$ is named as a concave F-IV-F on $\left[t, \upsilon \right]$. $\tilde{\mathfrak{U}}$ is affine if and only if it is both convex and concave F-IV-F.

Remark 2.

If ${\mathfrak{U}}_{*}\left(\varkappa ,\varsigma \right)={\mathfrak{U}}^{*}\left(\varkappa ,\varsigma \right)$ and $\varsigma =1$, then we obtain the classical convex function.

Definition 4.

[23] The F⋅I⋅V⋅F $\tilde{\mathfrak{U}}:\left[t, \upsilon \right]\to {\mathbb{F}}_0$ is named as a pre-invex F⋅I⋅V⋅F on invex interval $\left[t, \upsilon \right]$ if

$$\tilde{\mathfrak{U}}\left(\mu +\left(1-s\right)\Lambda \left(\mu ,\varkappa \right)\right)\preccurlyeq s\tilde{\mathfrak{U}}\left(\mu \right)\widetilde{+}\left(1-s\right)\tilde{\mathfrak{U}}\left(\varkappa \right),$$

(12)

for all $\mu , \varkappa \in \left[t, \upsilon \right], s\in \left[0, 1\right],$ where for all $\tilde{\mathfrak{U}}\left(\mu \right)\succcurlyeq \tilde{0}$ for all $\mu \in \left[t, \upsilon \right].$ If (12) is reversed, then $\tilde{\mathfrak{U}}$ is named as a pre-incave F⋅I⋅V⋅F on $\left[t, \upsilon \right]$. $\tilde{\mathfrak{U}}$ is affine if and only if it is both pre-invex and pre-incave F⋅I⋅V⋅Fs.

Definition 5.

[41] Let ${\mathit{\chi }}_1, {\mathit{\chi }}_2:\left[0, 1]\subseteq [t,\upsilon \right]\to {ℝ}^{+}$ such that ${\mathit{\chi }}_1, {\mathit{\chi }}_2\not\equiv 0$. Then, F-IV-F $\tilde{\mathfrak{U}}:\left[t,\upsilon \right]\to {\mathbb{F}}_0$ is said to be $\left({\mathit{\chi }}_1, {\mathit{\chi }}_2\right)$-pre-invex F-IV-F on $\left[t,\upsilon \right]$ if

$$\tilde{\mathfrak{U}}\left(\mu +\left(1-s\right)\Lambda \left(\mu ,\varkappa \right)\right)\preccurlyeq {\mathit{\chi }}_1\left(s\right){\mathit{\chi }}_2\left(1-s\right)\tilde{\mathfrak{U}}\left(\mu \right)\widetilde{+}{\mathit{\chi }}_1\left(1-s\right){\mathit{\chi }}_2\left(s\right)\tilde{\mathfrak{U}}\left(\varkappa \right),$$

(13)

for all $\mu , \varkappa \in \left[t,\upsilon \right], s\in \left[0, 1\right],$ where $\tilde{\mathfrak{U}}\left(\mu \right)\succcurlyeq \tilde{0}.$ If $\tilde{\mathfrak{U}}$ is $\left({\mathit{\chi }}_1, {\mathit{\chi }}_2\right)$-pre-incave on $\left[t,\upsilon \right]$, then inequality (13) is reversed.

Remark 3.

[41] If one attempt to take ${\mathit{\chi }}_2\left(s\right)\equiv 1,$ then from $\left({\mathit{\chi }}_1, {\mathit{\chi }}_2\right)$-pre-invex F-IV-F one achieves $\mathit{\chi }$-pre-invex F-IV-F, that is

$$\tilde{\mathfrak{U}}\left(\mu +\left(1-s\right)\Lambda \left(\mu ,\varkappa \right)\right)\preccurlyeq {\mathit{\chi }}_1\left(s\right)\tilde{\mathfrak{U}}\left(\mu \right)\widetilde{+}{\mathit{\chi }}_1\left(1-s\right)\tilde{\mathfrak{U}}\left(\varkappa \right), \forall \mu , \varkappa \in \left[t,\upsilon \right], \in \left[0, 1\right].$$

(14)

If one attempt to take ${\mathit{\chi }}_1\left(s\right)=s,{\mathit{\chi }}_2\left(s\right)\equiv 1,$ then from $\left({\mathit{\chi }}_1, {\mathit{\chi }}_2\right)$-pre-invex F-IV-F one achieves pre-invex F-IV-F, that is

$$\tilde{\mathfrak{U}}\left(\mu +\left(1-s\right)\Lambda \left(\mu ,\varkappa \right)\right)\preccurlyeq s\tilde{\mathfrak{U}}\left(\mu \right)\widetilde{+}\left(1-s\right)\tilde{\mathfrak{U}}\left(\varkappa \right),\forall \mu , \varkappa \in \left[t,\upsilon \right], s\in \left[0, 1\right].$$

(15)

If one attempts to take ${\mathit{\chi }}_1\left(s\right)={\mathit{\chi }}_2\left(s\right)\equiv 1,$ then from $\left({\mathit{\chi }}_1, {\mathit{\chi }}_2\right)$-pre-invex F-IV-F one achieves $P$-pre-invex F-IV-F, that is

$$\tilde{\mathfrak{U}}\left(\mu +\left(1-s\right)\Lambda \left(\mu ,\varkappa \right)\right)\preccurlyeq \tilde{\mathfrak{U}}\left(\mu \right)\widetilde{+}\tilde{\mathfrak{U}}\left(\varkappa \right),\forall \mu , \varkappa \in \left[t,\upsilon \right], s\in \left[0, 1\right].$$

(16)

Theorem 2.

[41] Let $\mathit{\chi }:\left[0, 1]\subseteq [t,\upsilon \right]\to ℝ$ be a non-negative real-valued function such that $\mathit{\chi }\not\equiv 0$ and let $\tilde{\mathfrak{U}}:\left[t,\upsilon \right]\to {\mathbb{F}}_0$ be a F-IV-F, whose $\varsigma$-cuts define the family of IV-Fs ${\mathfrak{U}}_{\varsigma }:\left[t,\upsilon \right]\to {\mathcal{K}}_C{}^{+}\subset {\mathcal{K}}_C$ are given by

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

(17)

for all $\varkappa \in \left[t,\upsilon \right]$ and for all $\varsigma \in \left[0, 1\right]$. Then, $\tilde{\mathfrak{U}}$ is $\mathit{\chi }$-pre-invex F-IV-F on $\left[t,\upsilon \right],$ if and only if, for all $\varsigma \in \left[0, 1\right],$ ${\mathfrak{U}}_{*}\left(\varkappa , \varsigma \right)$ and ${\mathfrak{U}}^{*}\left(\varkappa , \varsigma \right)$ are $\mathit{\chi }$-pre-invex function.

Example 1.

If we attempt to take $\mathit{\chi }\left(s\right)=s,$ for $s\in \left[0, 1\right]$ and the F-IV-F $\tilde{\mathfrak{U}}:\left[0, 4\right]\to {\mathbb{F}}_0$ defined by

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

then, for each $\varsigma \in \left[0, 1\right],$ we have ${\mathfrak{U}}_{\varsigma }\left(\varkappa \right)=\left[2\varsigma e^{{\varkappa }^2},2\left(2-\varsigma \right)e^{{\varkappa }^2}\right]$. Since end point functions ${\mathfrak{U}}_{*}\left(\varkappa ,\varsigma \right),$ ${\mathfrak{U}}^{*}\left(\varkappa ,\varsigma \right)$ are $\mathit{\chi }$-pre-invex functions with respect to $\Lambda \left(\upsilon ,t\right)=\upsilon -t$, for each $\varsigma \in \left[0, 1\right]$. Hence, $\tilde{\mathfrak{U}}\left(\varkappa \right)$ is $\mathit{\chi }$-pre-invex F-IV-F.

3. Main Results

For the purpose of giving improved versions of the $H.H$ inequalities on the fuzzy-interval space, let us first retrospect the generalized $H.H$-type inequality for χ-pre-invex F-IV-Fs. To prove the upcoming results, we need the following assumption regarding the function $\Lambda :\left[t,\upsilon \right]\times \left[t,\upsilon \right]\to ℝ,$ which plays an important role in upcoming results.

Condition C. [16]

$$\begin{gathered} \Lambda \left(\mu ,\varkappa +s\Lambda \left(\mu ,\varkappa \right)\right)=\left(1-s\right)\Lambda \left(\mu ,\varkappa \right), \\ \Lambda \left(\varkappa ,\varkappa +s\Lambda \left(\mu ,\varkappa \right)\right)=-s\Lambda \left(\mu ,\varkappa \right). \end{gathered}$$

Note that $\forall \varkappa , \mu \in \left[t,\upsilon \right]$ and $s\in \left[0, 1\right]$, then from Condition C we have

$$\Lambda \left(\varkappa +s_2\Lambda \left(\mu ,\varkappa \right),\varkappa +s_1\Lambda \left(\mu ,\varkappa \right)\right)=\left(s_2-s_1\right)\Lambda \left(\mu ,\varkappa \right).$$

Clearly for $s$ = 0, we have $\Lambda \left(\mu ,\varkappa \right)$ = 0 if and only if $\mu =\varkappa$, for all $\varkappa , \mu \in \left[t,\upsilon \right]$. For the application of Condition C, see [23,42,44].

Theorem 3.

Let $\tilde{\mathfrak{U}}:\left[t, t+\Lambda \left(\upsilon ,t\right)\right]\to {\mathbb{F}}_0$ be a χ-pre-invex F-IV-F on $\left[t, t+\Lambda \left(\upsilon ,t\right)\right],$ whose $\varsigma$-cuts define the family of IV-Fs ${\mathfrak{U}}_{\varsigma }:\left[t, t+\Lambda \left(\upsilon ,t\right)\right]\subset ℝ\to {\mathcal{K}}_C^{+}$ are given by ${\mathfrak{U}}_{\varsigma }\left(\varkappa \right)=\left[{\mathfrak{U}}_{*}\left(\varkappa ,\varsigma \right), {\mathfrak{U}}^{*}\left(\varkappa ,\varsigma \right)\right]$ for all $\varkappa \in \left[t, t+\Lambda \left(\upsilon ,t\right)\right]$ and for all $\varsigma \in \left[0, 1\right]$. If Λ satisfies Condition C and $\tilde{\mathfrak{U}}\in L\left(\left[t, t+\Lambda \left(\upsilon ,t\right)\right],{\mathbb{F}}_0\right)$, then

$$\begin{gathered} \frac{1}{\alpha \mathit{\chi }\left(\frac{1}{2}\right)}\tilde{\mathfrak{U}}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}\right)\preccurlyeq \frac{\Gamma \left(\alpha \right)}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }}\left[{ℐ}_{t^{+}}^{\alpha } \tilde{\mathfrak{U}}\left(t+\Lambda \left(\upsilon ,t\right)\right)\widetilde{+}{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } \tilde{\mathfrak{U}}\left(t\right)\right] \\ \preccurlyeq \tilde{\mathfrak{U}}\left(t\right)\widetilde{+}\tilde{\mathfrak{U}}\left(t+\Lambda \left(\upsilon ,t\right)\right){\int }_0^1{s}^{\alpha -1}\left[\mathit{\chi }\left(s\right)-\mathit{\chi }\left(1-s\right)\right]ds \\ \preccurlyeq \tilde{\mathfrak{U}}\left(t\right)\widetilde{+}\tilde{\mathfrak{U}}\left(\upsilon \right){\int }_0^1{s}^{\alpha -1}\left[\mathit{\chi }\left(s\right)-\mathit{\chi }\left(1-s\right)\right]ds. \end{gathered}$$

(18)

If $\tilde{\mathfrak{U}}\left(\varkappa \right)$ is pre-incave F-IV-F, then

$$\begin{gathered} \frac{1}{\alpha \mathit{\chi }\left(\frac{1}{2}\right)}\tilde{\mathfrak{U}}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}\right)\succcurlyeq \frac{\Gamma \left(\alpha \right)}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }}\left[{ℐ}_{t^{+}}^{\alpha } \tilde{\mathfrak{U}}\left(t+\Lambda \left(\upsilon ,t\right)\right)\widetilde{+}{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } \tilde{\mathfrak{U}}\left(t\right)\right] \\ \succcurlyeq \tilde{\mathfrak{U}}\left(t\right)\widetilde{+}\tilde{\mathfrak{U}}\left(t+\Lambda \left(\upsilon ,t\right)\right){\int }_0^1{s}^{\alpha -1}\left[\mathit{\chi }\left(s\right)-\mathit{\chi }\left(1-s\right)\right]ds. \\ \succcurlyeq \tilde{\mathfrak{U}}\left(t\right)\widetilde{+}\tilde{\mathfrak{U}}\left(\upsilon \right){\int }_0^1{s}^{\alpha -1}\left[\mathit{\chi }\left(s\right)-\mathit{\chi }\left(1-s\right)\right]ds. \end{gathered}$$

(19)

Proof. 

Let $\tilde{\mathfrak{U}}:\left[t, t+\Lambda \left(\upsilon ,t\right)\right]\to {\mathbb{F}}_0$ be a χ-pre-invex F-IV-F. If Condition C holds then, by hypothesis, we have

$$\frac{1}{\mathit{\chi }\left(\frac{1}{2}\right)}\tilde{\mathfrak{U}}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}\right)\preccurlyeq \tilde{\mathfrak{U}}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right)\right)\widetilde{+}\tilde{\mathfrak{U}}\left(t+s\Lambda \left(\upsilon ,t\right)\right).$$

Therefore, for every $\varsigma \in \left[0, 1\right]$, we have

$$\begin{array}{c}\frac{1}{\mathit{\chi }\left(\frac{1}{2}\right)}{\mathfrak{U}}_{*}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}, \varsigma \right)\le {\mathfrak{U}}_{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)+{\mathfrak{U}}_{*}\left(t+s\Lambda \left(\upsilon ,t\right), \varsigma \right), \\ \frac{1}{\mathit{\chi }\left(\frac{1}{2}\right)}{\mathfrak{U}}^{*}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}, \varsigma \right)\le {\mathfrak{U}}^{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)+{\mathfrak{U}}^{*}\left(t+s\Lambda \left(\upsilon ,t\right),\varsigma \right).\end{array}$$

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

$$\begin{array}{c}\frac{1}{\mathit{\chi }\left(\frac{1}{2}\right)}{\int }_0^1{s}^{\alpha -1}{\mathfrak{U}}_{*}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}, \varsigma \right)ds\\ \le {\int }_0^1{s}^{\alpha -1}{\mathfrak{U}}_{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right),\varsigma \right)ds+{\int }_0^1{s}^{\alpha -1}{\mathfrak{U}}_{*}\left(t+s\Lambda \left(\upsilon ,t\right), \varsigma \right)ds,\\ \frac{1}{\mathit{\chi }\left(\frac{1}{2}\right)}{\int }_0^1{s}^{\alpha -1}{\mathfrak{U}}^{*}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2},\varsigma \right)ds\\ \le {\int }_0^1{s}^{\alpha -1}{\mathfrak{U}}^{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)ds+{\int }_0^1{s}^{\alpha -1}{\mathfrak{U}}^{*}\left(t+s\Lambda \left(\upsilon ,t\right),\varsigma \right)ds.\end{array}$$

(20)

Let $\mu =t+\left(1-s\right)\Lambda \left(\upsilon ,t\right)$ and $\varkappa =t+s\Lambda \left(\upsilon ,t\right).$ Then, we have

$$\begin{array}{c}\frac{1}{\alpha \mathit{\chi }\left(\frac{1}{2}\right)}{\mathfrak{U}}_{*}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}, \varsigma \right) \le \frac{1}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }} {\int }_t^{t+\Lambda \left(\upsilon ,t\right)}{\left(t+\Lambda \left(\upsilon ,t\right)-\mu \right)}^{\alpha -1}{\mathfrak{U}}_{*}\left(\mu ,\varsigma \right)d\mu \\ +\frac{1}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }}{\int }_t^{t+\Lambda \left(\upsilon ,t\right)}{\left(\varkappa -t\right)}^{\alpha -1}{\mathfrak{U}}_{*}\left(\varkappa ,\varsigma \right)d\varkappa \\ \frac{1}{\alpha \mathit{\chi }\left(\frac{1}{2}\right)}{\mathfrak{U}}^{*}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}, \varsigma \right)\le \frac{1}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }} {\int }_t^{t+\Lambda \left(\upsilon ,t\right)}{\left(t+\Lambda \left(\upsilon ,t\right)-\mu \right)}^{\alpha -1}{\mathfrak{U}}^{*}\left(\mu ,\varsigma \right)d\mu \\ +\frac{1}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }}{\int }_t^{t+\Lambda \left(\upsilon ,t\right)}{\left(\varkappa -t\right)}^{\alpha -1}{\mathfrak{U}}^{*}\left(\varkappa ,\varsigma \right)d\varkappa ,\\ \le \frac{\Gamma \left(\alpha \right)}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }}\left[{ℐ}_{t^{+}}^{\alpha } {\mathfrak{U}}_{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } {\mathfrak{U}}_{*}\left(t, \varsigma \right)\right]\\ \le \frac{\Gamma \left(\alpha \right)}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }}\left[{ℐ}_{t^{+}}^{\alpha } {\mathfrak{U}}^{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } {\mathfrak{U}}^{*}\left(t, \varsigma \right)\right],\end{array}$$

(21)

that is

$$\begin{gathered} \frac{1}{\alpha \mathit{\chi }\left(\frac{1}{2}\right)}\left[{\mathfrak{U}}_{*}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}, \varsigma \right), {\mathfrak{U}}^{*}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}, \varsigma \right)\right] \\ {\le }_I\frac{\Gamma \left(\alpha \right)}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }}\left[\left[{ℐ}_{t^{+}}^{\alpha } {\mathfrak{U}}_{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } {\mathfrak{U}}_{*}\left(t, \varsigma \right)\right], \left[{ℐ}_{t^{+}}^{\alpha } {\mathfrak{U}}^{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } {\mathfrak{U}}^{*}\left(t, \varsigma \right)\right]\right], \end{gathered}$$

thus,

$$\frac{1}{\alpha \mathit{\chi }\left(\frac{1}{2}\right)} {\mathfrak{U}}_{\varsigma }\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}\right){\le }_I\frac{\Gamma \left(\alpha \right)}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }}\left[{ℐ}_{t^{+}}^{\alpha } {\mathfrak{U}}_{\varsigma }\left(t+\Lambda \left(\upsilon ,t\right)\right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } {\mathfrak{U}}_{\varsigma }\left(t\right)\right].$$

(22)

In a similar way as above, we have

$$\begin{gathered} \frac{\Gamma \left(\alpha \right)}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }}\left[{ℐ}_{t^{+}}^{\alpha } {\mathfrak{U}}_{\varsigma }\left(t+\Lambda \left(\upsilon ,t\right)\right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } {\mathfrak{U}}_{\varsigma }\left(t\right)\right] \\ {\le }_I\left[{\mathfrak{U}}_{\varsigma }\left(t\right)+{\mathfrak{U}}_{\varsigma }\left(t+\Lambda \left(\upsilon ,t\right)\right)\right]{\int }_0^1{s}^{\alpha -1}\left[\mathit{\chi }\left(s\right)-\mathit{\chi }\left(1-s\right)\right]ds. \end{gathered}$$

(23)

Combining (20) and (21), we have

$$\begin{gathered} \frac{1}{\alpha \mathit{\chi }\left(\frac{1}{2}\right)}{\mathfrak{U}}_{\varsigma }\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}\right){\le }_I\frac{\Gamma \left(\alpha \right)}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }}\left[{ℐ}_{t^{+}}^{\alpha } {\mathfrak{U}}_{\varsigma }\left(t+\Lambda \left(\upsilon ,t\right)\right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } {\mathfrak{U}}_{\varsigma }\left(t\right)\right] \\ {\le }_I\left[{\mathfrak{U}}_{\varsigma }\left(t\right)+{\mathfrak{U}}_{\varsigma }\left(t+\Lambda \left(\upsilon ,t\right)\right)\right]{\int }_0^1{s}^{\alpha -1}\left[\mathit{\chi }\left(s\right)-\mathit{\chi }\left(1-s\right)\right]ds, \end{gathered}$$

that is

$$\begin{gathered} \frac{1}{\alpha \mathit{\chi }\left(\frac{1}{2}\right)}\tilde{\mathfrak{U}}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}\right)\preccurlyeq \frac{\Gamma \left(\alpha \right)}{\left(\Lambda \left(\upsilon ,t\right)\right)}\left[{ℐ}_{t^{+}}^{\alpha } \tilde{\mathfrak{U}}\left(t+\Lambda \left(\upsilon ,t\right)\right)\widetilde{+}{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } \tilde{\mathfrak{U}}\left(t\right)\right] \\ \preccurlyeq \left[\tilde{\mathfrak{U}}\left(t\right)\widetilde{+}\tilde{\mathfrak{U}}\left(t+\Lambda \left(\upsilon ,t\right)\right)\right]{\int }_0^1{s}^{\alpha -1}\left[\mathit{\chi }\left(s\right)-\mathit{\chi }\left(1-s\right)\right]ds \\ \preccurlyeq \tilde{\mathfrak{U}}\left(t\right)\widetilde{+}\tilde{\mathfrak{U}}\left(\upsilon \right){\int }_0^1{s}^{\alpha -1}\left[\mathit{\chi }\left(s\right)-\mathit{\chi }\left(1-s\right)\right]ds. \end{gathered}$$

the theorem has been proved. □

Remark 4.

If one attempts to take $\alpha =1$, then from inequality (18) one achieves the result for χ-pre-invex F-IV-F, see [41]:

$$\frac{1}{2\mathit{\chi }\left(\frac{1}{2}\right)} \tilde{\mathfrak{U}}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}\right)\preccurlyeq \frac{1}{\Lambda \left(\upsilon ,t\right)} \left(FR\right){\int }_t^{t+\Lambda \left(\upsilon ,t\right)}\tilde{\mathfrak{U}}\left(\varkappa \right)d\varkappa \preccurlyeq \left[\tilde{\mathfrak{U}}\left(t\right) \widetilde{+} \tilde{\mathfrak{U}}\left(t+\Lambda \left(\upsilon ,t\right)\right)\right]{\int }_0^1\mathit{\chi }\left(s\right)ds.$$

(24)

If one attempt to take $\mathit{\chi }\left(s\right)=s$, then from inequality (18) one achieves the result for pre-invex F-IV-F, see [24]:

$$\tilde{\mathfrak{U}}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}\right)\preccurlyeq \frac{\Gamma \left(\alpha +1\right)}{2{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }}\left[{ℐ}_{t^{+}}^{\alpha } \tilde{\mathfrak{U}}\left(t+\Lambda \left(\upsilon ,t\right)\right)\widetilde{+}{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } \tilde{\mathfrak{U}}\left(t\right)\right]\preccurlyeq \frac{\tilde{\mathfrak{U}}\left(t\right)\widetilde{+}\tilde{\mathfrak{U}}\left(t+\Lambda \left(\upsilon ,t\right)\right)}{2}.$$

(25)

Let one attempt to take $\alpha =1$ and $\mathit{\chi }\left(s\right)=s$. Then, from inequality (18) one acquires the result for pre-invex- IV-F given in [41]:

$$\tilde{\mathfrak{U}}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}\right)\preccurlyeq \frac{1}{\Lambda \left(\upsilon ,t\right)} \left(FR\right){\int }_t^{t+\Lambda \left(\upsilon ,t\right)}\tilde{\mathfrak{U}}\left(\varkappa \right)d\varkappa \preccurlyeq \frac{\tilde{\mathfrak{U}}\left(t\right)\widetilde{+}\tilde{\mathfrak{U}}\left(\upsilon \right)}{2}.$$

(26)

If one attempt to take ${\mathfrak{U}}_{*}\left(\varkappa , \varsigma \right)={\mathfrak{U}}^{*}\left(\varkappa , \varsigma \right)$ and $\varsigma =1,$ then, from inequality (18) one acquires coming inequality given in [42]:

$$\begin{gathered} \frac{1}{\alpha \mathit{\chi }\left(\frac{1}{2}\right)}\mathfrak{U}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}\right)\le \frac{\Gamma \left(\alpha \right)}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }}\left[{ℐ}_{t^{+}}^{\alpha }\mathfrak{U}\left(t+\Lambda \left(\upsilon ,t\right)\right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha }\mathfrak{U}\left(t\right)\right] \\ \le \left[\mathfrak{U}\left(t\right)+\mathfrak{U}\left(t+\Lambda \left(\upsilon ,t\right)\right)\right]{\int }_0^1{s}^{\alpha -1}\left[\mathit{\chi }\left(s\right)-\mathit{\chi }\left(1-s\right)\right]ds. \end{gathered}$$

(27)

Let one attempt to take $\alpha =1=\varsigma$ and ${\mathfrak{U}}_{*}\left(\varkappa , \varsigma \right)={\mathfrak{U}}^{*}\left(\varkappa , \varsigma \right)$. Then, from inequality (18) one acquires coming inequality given in [43]:

$$\frac{1}{2\mathit{\chi }\left(\frac{1}{2}\right)}\mathfrak{U}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}\right)\le \frac{1}{\Lambda \left(\upsilon ,t\right)} \left(R\right){\int }_t^{t+\Lambda \left(\upsilon ,t\right)}\mathfrak{U}\left(\varkappa \right)d\varkappa \le \left[\mathfrak{U}\left(t\right)+\mathfrak{U}\left(t+\Lambda \left(\upsilon ,t\right)\right)\right]{\int }_0^1\mathit{\chi }\left(s\right)ds.$$

(28)

Example 2.

If we attempt to take $=\frac{1}{2}$, $\mathit{\chi }\left(s\right)=s$, for all $s\in \left[0, 1\right]$ and the F-IV-F $\tilde{\mathfrak{U}}:\left[t, t+\Lambda \left(\upsilon ,t\right)\right]=\left[2, 2+\Lambda \left(3,2\right)\right]\to {\mathbb{F}}_0,$ defined by

$$\tilde{\mathfrak{U}}\left(\varkappa \right)\left(\theta \right)=\{\begin{array}{c}\frac{\theta }{2-{\varkappa }^{\frac{1}{2}}}, \theta \in \left[0, 2-{\varkappa }^{\frac{1}{2}}\right]\\ \frac{ 2\left(2-{\varkappa }^{\frac{1}{2}}\right)-\theta }{2-{\varkappa }^{\frac{1}{2}}}, \theta \in \left(2-{\varkappa }^{\frac{1}{2}}, 2\left(2-{\varkappa }^{\frac{1}{2}}\right)\right]\\ 0, \mathrm{otherwise}.\end{array}$$

(29)

Then, for each $\varsigma \in \left[0, 1\right],$ we have ${\mathfrak{U}}_{\varsigma }\left(\varkappa \right)=\left[\varsigma \left(2-{\varkappa }^{\frac{1}{2}}\right),\left(2-\varsigma \right)\left(2-{\varkappa }^{\frac{1}{2}}\right)\right]$. Since left and right end-point functions ${\mathfrak{U}}_{*}\left(\varkappa ,\varsigma \right)=\varsigma \left(2-{\varkappa }^{\frac{1}{2}}\right),$ ${\mathfrak{U}}^{*}\left(\varkappa , \varsigma \right)=\left(2-\varsigma \right)\left(2-{\varkappa }^{\frac{1}{2}}\right)$, are $\mathit{\chi }$-pre-invex functions for each $\varsigma \in \left[0, 1\right]$, then $\tilde{\mathfrak{U}}\left(\varkappa \right)$ is $\mathit{\chi }$-pre-invex F-IV-F. We clearly see that $\tilde{\mathfrak{U}}\in L\left(\left[t, t+\Lambda \left(\upsilon ,t\right)\right],{\mathbb{F}}_0\right)$ and

$$\begin{gathered} \frac{1}{\alpha \mathit{\chi }\left(\frac{1}{2}\right)}{\mathfrak{U}}_{*}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}, \varsigma \right)={\mathfrak{U}}_{*}\left(\frac{5}{2}, \varsigma \right)=\varsigma \frac{4-\sqrt{10}}{8} \\ \frac{1}{\alpha \mathit{\chi }\left(\frac{1}{2}\right)}{\mathfrak{U}}^{*}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}, \varsigma \right)={\mathfrak{U}}^{*}\left(\frac{5}{2}, \varsigma \right)=\left(2-\varsigma \right)\frac{4-\sqrt{10}}{8}, \\ \frac{{\mathfrak{U}}_{*}\left(t,\varsigma \right)+{\mathfrak{U}}_{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)}{2}{\int }_0^1{s}^{\alpha -1}\left[\mathit{\chi }\left(s\right)-\mathit{\chi }\left(1-s\right)\right]ds=\varsigma \left(4-\sqrt{2}-\sqrt{3}\right) \\ \frac{{\mathfrak{U}}^{*}\left(t,\varsigma \right)+{\mathfrak{U}}^{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)}{2}{\int }_0^1{s}^{\alpha -1}\left[\mathit{\chi }\left(s\right)-\mathit{\chi }\left(1-s\right)\right]ds=\left(2-\varsigma \right)\left(4-\sqrt{2}-\sqrt{3}\right). \end{gathered}$$

(30)

Note that

$$\begin{gathered} \frac{\Gamma \left(\alpha \right)}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }}\left[{ℐ}_{t^{+}}^{\alpha } {\mathfrak{U}}_{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } {\mathfrak{U}}_{*}\left(t,\varsigma \right)\right] \\ =\frac{\Gamma \left(\frac{1}{2}\right)}{2}\frac{1}{\sqrt{\pi }}{\int }_2^3{\left(3-\varkappa \right)}^{\frac{-1}{2}} . \varsigma \left(2-{\varkappa }^{\frac{1}{2}}\right)d\varkappa \\ +\frac{\Gamma \left(\frac{1}{2}\right)}{2}\frac{1}{\sqrt{\pi }}{\int }_2^3{\left(\varkappa -2\right)}^{\frac{-1}{2}} . \varsigma \left(2-{\varkappa }^{\frac{1}{2}}\right)d\varkappa \\ =\frac{1}{2}\varsigma \left[\frac{7393}{\mathrm{10,000}}+\frac{9501}{\mathrm{10,000}}\right] \\ =\varsigma \frac{8447}{\mathrm{20,000}}. \\ \frac{\Gamma \left(\alpha \right)}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }}\left[{ℐ}_{t^{+}}^{\alpha } {\mathfrak{U}}^{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } {\mathfrak{U}}^{*}\left(t,\varsigma \right)\right] \\ =\frac{\Gamma \left(\frac{1}{2}\right)}{2}\frac{1}{\sqrt{\pi }}{\int }_2^3{\left(3-\varkappa \right)}^{\frac{-1}{2}} . \left(2-\varsigma \right)\left(2-{\varkappa }^{\frac{1}{2}}\right)d\varkappa \\ +\frac{\Gamma \left(\frac{1}{2}\right)}{2}\frac{1}{\sqrt{\pi }}{\int }_2^3{\left(\varkappa -2\right)}^{\frac{-1}{2}} . \left(2-\varsigma \right)\left(2-{\varkappa }^{\frac{1}{2}}\right)d\varkappa \\ =\frac{1}{2}\left(2-\varsigma \right)\left[\frac{7393}{\mathrm{10,000}}+\frac{9501}{\mathrm{10,000}}\right] \\ =\left(2-\varsigma \right)\frac{8447}{\mathrm{20,000}}. \end{gathered}$$

Therefore

$$\left[\varsigma \frac{4-\sqrt{10}}{8},\left(2-\varsigma \right)\frac{4-\sqrt{10}}{8}\right]{\le }_I\left[\varsigma \frac{8447}{\mathrm{20,000}},\left(2-\varsigma \right)\frac{8447}{\mathrm{20,000}}\right]{\le }_I\left[\varsigma \left(4-\sqrt{2}-\sqrt{3}\right),\left(2-\varsigma \right)\left(4-\sqrt{2}-\sqrt{3}\right)\right],$$

and Theorem 3 is verified.

We get various fuzzy-interval-fractional-integral inequalities connected to fuzzy-interval-fractional-H⋅H inequalities from Theorems 4 and 5 via products of two $\mathit{\chi }$-pre-invex F-IV-Fs.

Theorem 4.

Let $\tilde{\mathfrak{U}},\tilde{\mathscr{H}} :\left[t, t+\Lambda \left(\upsilon ,t\right)\right]\to {\mathbb{F}}_0$ be ${\mathit{\chi }}_1$-pre-invex and ${\mathit{\chi }}_2$-pre-invex F-IV-Fs on $\left[t, t+\Lambda \left(\upsilon ,t\right)\right]$, respectively, whose $\varsigma$-cuts ${\mathfrak{U}}_{\varsigma }, {\mathscr{H}}_{\varsigma }:\left[t, t+\Lambda \left(\upsilon ,t\right)\right]\subset ℝ\to {\mathcal{K}}_C^{+}$ are defined by ${\mathfrak{U}}_{\varsigma }\left(\varkappa \right)=\left[{\mathfrak{U}}_{*}\left(\varkappa ,\varsigma \right), {\mathfrak{U}}^{*}\left(\varkappa ,\varsigma \right)\right]$ and ${\mathscr{H}}_{\varsigma }\left(\varkappa \right)=\left[{\mathscr{H}}_{*}\left(\varkappa ,\varsigma \right), {\mathscr{H}}^{*}\left(\varkappa ,\varsigma \right)\right]$ for all $\varkappa \in \left[t, t+\Lambda \left(\upsilon ,t\right)\right]$ and for all $\varsigma \in \left[0, 1\right]$. If $\Lambda$ satisfies Condition C and $\tilde{\mathfrak{U}}\tilde{\times }\tilde{\mathscr{H}}\in L\left(\left[t, t+\Lambda \left(\upsilon ,t\right)\right],{\mathbb{F}}_0\right)$, then

$$\begin{gathered} \frac{\Gamma \left(\alpha \right)}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }}\left[{ℐ}_{t^{+}}^{\alpha } \tilde{\mathfrak{U}}\left(t+\Lambda \left(\upsilon ,t\right)\right)\tilde{\times }\tilde{\mathscr{H}}\left(t+\Lambda \left(\upsilon ,t\right)\right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha }\tilde{\mathfrak{U}}\left(t\right)\tilde{\times }\tilde{\mathscr{H}}\left(t\right)\right] \\ \preccurlyeq \tilde{\mathsf{\Delta }}\left(t,t+\Lambda \left(\upsilon ,t\right)\right){\int }_0^1{s}^{\alpha -1}\left[{\mathit{\chi }}_1\left(s\right){\mathit{\chi }}_2\left(s\right)+{\mathit{\chi }}_1\left(1-s\right){\mathit{\chi }}_2\left(1-s\right)\right]ds \\ +\tilde{\mathit{\nabla }}\left(t,t+\Lambda \left(\upsilon ,t\right)\right){\int }_0^1{s}^{\alpha -1}\left[{\mathit{\chi }}_1\left(s\right){\mathit{\chi }}_2\left(1-s\right)+{\mathit{\chi }}_1\left(1-s\right){\mathit{\chi }}_2\left(s\right)\right]ds. \end{gathered}$$

(31)

where $\tilde{\mathsf{\Delta }}\left(t,t+\Lambda \left(\upsilon ,t\right)\right)=\tilde{\mathfrak{U}}\left(t\right)\tilde{\times }\tilde{\mathscr{H}}\left(t\right) \widetilde{+} \tilde{\mathfrak{U}}\left(t+\Lambda \left(\upsilon ,t\right)\right)\tilde{\times }\tilde{\mathscr{H}}\left(t+\Lambda \left(\upsilon ,t\right)\right),$ $\tilde{\mathit{\nabla }}\left(t,t+\Lambda \left(\upsilon ,t\right)\right)=\tilde{\mathfrak{U}}\left(t\right)\tilde{\times }\tilde{\mathscr{H}}\left(t+\Lambda \left(\upsilon ,t\right)\right) \widetilde{+} \tilde{\mathfrak{U}}\left(t+\Lambda \left(\upsilon ,t\right)\right)\tilde{\times }\tilde{\mathscr{H}}\left(t\right),$ and ${\mathsf{\Delta }}_{\varsigma }\left(t,t+\Lambda \left(\upsilon ,t\right)\right)=\left[{\mathsf{\Delta }}_{*}\left(\left(t,t+\Lambda \left(\upsilon ,t\right)\right), \varsigma \right), {\mathsf{\Delta }}^{*}\left(\left(t,t+\Lambda \left(\upsilon ,t\right)\right), \varsigma \right)\right]$ and ${\mathit{\nabla }}_{\varsigma }\left(t,t+\Lambda \left(\upsilon ,t\right)\right)=\left[{\mathit{\nabla }}_{*}\left(\left(t,t+\Lambda \left(\upsilon ,t\right)\right), \varsigma \right), {\mathit{\nabla }}^{*}\left(\left(t,t+\Lambda \left(\upsilon ,t\right)\right), \varsigma \right)\right]$.

Proof. 

Since $\tilde{\mathfrak{U}}, \tilde{\mathscr{H}}$ both are ${\mathit{\chi }}_1$-pre-invex and ${\mathit{\chi }}_2$-pre-invex F-IV-F then, for each $\varsigma \in \left[0, 1\right]$ we have

$$\begin{array}{c}{\mathfrak{U}}_{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)={\mathfrak{U}}_{*}\left(t+\Lambda \left(\upsilon ,t\right)+s\Lambda \left(t,t+\Lambda \left(\upsilon ,t\right)\right), \varsigma \right) \\ \le {\mathit{\chi }}_1\left(s\right){\mathfrak{U}}_{*}\left(t,\varsigma \right)+{\mathit{\chi }}_1\left(1-s\right){\mathfrak{U}}_{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)\\ {\mathfrak{U}}^{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)={\mathfrak{U}}^{*}\left(t+\Lambda \left(\upsilon ,t\right)+s\Lambda \left(t,t+\Lambda \left(\upsilon ,t\right)\right), \varsigma \right)\\ \le {\mathit{\chi }}_1\left(s\right){\mathfrak{U}}^{*}\left(t,\varsigma \right)+{\mathit{\chi }}_1\left(1-s\right){\mathfrak{U}}^{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right).\end{array}$$

and

$$\begin{array}{c}{\mathscr{H}}_{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)={\mathscr{H}}_{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)\\ \le {\mathit{\chi }}_2\left(s\right){\mathscr{H}}_{*}\left(t,\varsigma \right)+{\mathit{\chi }}_2\left(1-s\right){\mathscr{H}}_{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)\\ {\mathscr{H}}^{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)={\mathscr{H}}^{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)\\ \le {\mathit{\chi }}_2\left(s\right){\mathscr{H}}^{*}\left(t,\varsigma \right)+{\mathit{\chi }}_2\left(1-s\right){\mathscr{H}}^{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right).\end{array}$$

(32)

From the definition of χ-pre-invex F-IV-Fs it follows that $\tilde{0}\preccurlyeq \tilde{\mathfrak{U}}\left(\varkappa \right)$ and $\tilde{0}\preccurlyeq \tilde{\mathscr{H}}\left(\varkappa \right)$, so

$$\begin{array}{c}{\mathfrak{U}}_{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)\times {\mathscr{H}}_{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)\\ \le {\mathit{\chi }}_1\left(s\right){\mathit{\chi }}_2\left(s\right){\mathfrak{U}}_{*}\left(t,\varsigma \right)\times {\mathscr{H}}_{*}\left(t,\varsigma \right)+{\mathit{\chi }}_1\left(1-s\right){\mathit{\chi }}_2\left(1-s\right){\mathfrak{U}}_{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)\times {\mathscr{H}}_{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)\\ +{\mathit{\chi }}_1\left(s\right){\mathit{\chi }}_2\left(1-s\right){\mathfrak{U}}_{*}\left(t,\varsigma \right)\times {\mathscr{H}}_{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)+{\mathit{\chi }}_1\left(1-s\right){\mathit{\chi }}_2\left(s\right){\mathfrak{U}}_{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)\times {\mathscr{H}}_{*}\left(t,\varsigma \right)\\ {\mathfrak{U}}^{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)\times {\mathscr{H}}^{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right) \\ \le {\mathit{\chi }}_1\left(s\right){\mathit{\chi }}_2\left(s\right){\mathfrak{U}}^{*}\left(t,\varsigma \right)\times {\mathscr{H}}^{*}\left(t,\varsigma \right)+{\mathit{\chi }}_1\left(1-s\right){\mathit{\chi }}_2\left(1-s\right){\mathfrak{U}}^{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)\times {\mathscr{H}}^{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)\\ +{\mathit{\chi }}_1\left(s\right){\mathit{\chi }}_2\left(1-s\right){\mathfrak{U}}^{*}\left(t,\varsigma \right)\times {\mathscr{H}}^{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)+{\mathit{\chi }}_1\left(1-s\right){\mathit{\chi }}_2\left(s\right){\mathfrak{U}}^{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)\times {\mathscr{H}}^{*}\left(t,\varsigma \right).\end{array}$$

(33)

Analogously, we have

$$\begin{array}{c}{\mathfrak{U}}_{*}\left(t+s\Lambda \left(\upsilon ,t\right), \varsigma \right){\mathscr{H}}_{*}\left(t+s\Lambda \left(\upsilon ,t\right), \varsigma \right) \\ \le {\mathit{\chi }}_1\left(1-s\right){\mathit{\chi }}_2\left(1-s\right){\mathfrak{U}}_{*}\left(t,\varsigma \right)\times {\mathscr{H}}_{*}\left(t,\varsigma \right)+{\mathit{\chi }}_1\left(s\right){\mathit{\chi }}_2\left(s\right){\mathfrak{U}}_{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)\times {\mathscr{H}}_{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)\\ +{\mathit{\chi }}_1\left(1-s\right){\mathit{\chi }}_2\left(s\right){\mathfrak{U}}_{*}\left(t,\varsigma \right)\times {\mathscr{H}}_{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)+{\mathit{\chi }}_1\left(s\right){\mathit{\chi }}_2\left(1-s\right){\mathfrak{U}}_{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)\times {\mathscr{H}}_{*}\left(t,\varsigma \right)\\ {\mathfrak{U}}^{*}\left(t+s\Lambda \left(\upsilon ,t\right), \varsigma \right)\times {\mathscr{H}}^{*}\left(t+s\Lambda \left(\upsilon ,t\right), \varsigma \right) \\ \le {\mathit{\chi }}_1\left(1-s\right){\mathit{\chi }}_2\left(1-s\right){\mathfrak{U}}^{*}\left(t,\varsigma \right)\times {\mathscr{H}}^{*}\left(t,\varsigma \right)+{\mathit{\chi }}_1\left(s\right){\mathit{\chi }}_2\left(s\right){\mathfrak{U}}^{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)\times {\mathscr{H}}^{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)\\ +{\mathit{\chi }}_1\left(1-s\right){\mathit{\chi }}_2\left(s\right){\mathfrak{U}}^{*}\left(t,\varsigma \right)\times {\mathscr{H}}^{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)+{\mathit{\chi }}_1\left(s\right){\mathit{\chi }}_2\left(1-s\right){\mathfrak{U}}^{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)\times {\mathscr{H}}^{*}\left(t,\varsigma \right).\end{array}$$

(34)

Adding (27) and (28), we have

$$\begin{array}{c}{\mathfrak{U}}_{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)\times {\mathscr{H}}_{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right) \\ +{\mathfrak{U}}_{*}\left(t+s\Lambda \left(\upsilon ,t\right), \varsigma \right)\times {\mathscr{H}}_{*}\left(t+s\Lambda \left(\upsilon ,t\right), \varsigma \right) \\ \le \left[{\mathit{\chi }}_1\left(s\right){\mathit{\chi }}_2\left(s\right)+{\mathit{\chi }}_1\left(1-s\right){\mathit{\chi }}_2\left(1-s\right)\right]\left[{\mathfrak{U}}_{*}\left(t,\varsigma \right)\times {\mathscr{H}}_{*}\left(t,\varsigma \right)+{\mathfrak{U}}_{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)\times {\mathscr{H}}_{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)\right]\\ +\left[{\mathit{\chi }}_1\left(s\right){\mathit{\chi }}_2\left(1-s\right)+{\mathit{\chi }}_1\left(1-s\right){\mathit{\chi }}_2\left(s\right)\right]\left[{\mathfrak{U}}_{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)\times {\mathscr{H}}_{*}\left(t,\varsigma \right)+{\mathfrak{U}}_{*}\left(t,\varsigma \right)\times {\mathscr{H}}_{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)\right]\\ {\mathfrak{U}}^{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)\times {\mathscr{H}}^{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right) \\ +{\mathfrak{U}}^{*}\left(t+s\Lambda \left(\upsilon ,t\right), \varsigma \right)\times {\mathscr{H}}^{*}\left(t+s\Lambda \left(\upsilon ,t\right), \varsigma \right)\\ \le \left[{\mathit{\chi }}_1\left(s\right){\mathit{\chi }}_2\left(s\right)+{\mathit{\chi }}_1\left(1-s\right){\mathit{\chi }}_2\left(1-s\right)\right]\left[{\mathfrak{U}}^{*}\left(t,\varsigma \right)\times {\mathscr{H}}^{*}\left(t,\varsigma \right)+{\mathfrak{U}}^{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)\times {\mathscr{H}}^{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)\right]\\ +\left[{\mathit{\chi }}_1\left(s\right){\mathit{\chi }}_2\left(1-s\right)+{\mathit{\chi }}_1\left(1-s\right){\mathit{\chi }}_2\left(s\right)\right]\left[{\mathfrak{U}}^{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)\times {\mathscr{H}}^{*}\left(t,\varsigma \right)+{\mathfrak{U}}^{*}\left(t,\varsigma \right)\times {\mathscr{H}}^{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)\right].\end{array}$$

(35)

Taking multiplication of (29) with $s^{\alpha -1}$ and integrating the obtained result with respect to $s$ over (0, 1), we have

$$\begin{array}{c}{\int }_0^1{s}^{\alpha -1}{\mathfrak{U}}_{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)\times {\mathscr{H}}_{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right) \\ +s^{\alpha -1}{\mathfrak{U}}_{*}\left(t+s\Lambda \left(\upsilon ,t\right), \varsigma \right)\times {\mathscr{H}}_{*}\left(t+s\Lambda \left(\upsilon ,t\right), \varsigma \right)ds\\ \le {\mathsf{\Delta }}_{*}\left(\left(t,t+\Lambda \left(\upsilon ,t\right)\right), \varsigma \right){\int }_0^1{s}^{\alpha -1}\left[{\mathit{\chi }}_1\left(s\right){\mathit{\chi }}_2\left(s\right)+{\mathit{\chi }}_1\left(1-s\right){\mathit{\chi }}_2\left(1-s\right)\right]ds\\ +{\mathit{\nabla }}_{*}\left(\left(t,t+\Lambda \left(\upsilon ,t\right)\right), \varsigma \right){\int }_0^1{s}^{\alpha -1}\left[{\mathit{\chi }}_1\left(s\right){\mathit{\chi }}_2\left(1-s\right)+{\mathit{\chi }}_1\left(1-s\right){\mathit{\chi }}_2\left(s\right)\right]ds \\ {\int }_0^1{s}^{\alpha -1}{\mathfrak{U}}^{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)\times {\mathscr{H}}^{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)\\ +s^{\alpha -1}{\mathfrak{U}}^{*}\left(t+s\Lambda \left(\upsilon ,t\right), \varsigma \right)\times {\mathscr{H}}^{*}\left(t+s\Lambda \left(\upsilon ,t\right), \varsigma \right)ds \\ \le {\mathsf{\Delta }}^{*}\left(\left(t,t+\Lambda \left(\upsilon ,t\right)\right), \varsigma \right){\int }_0^1{s}^{\alpha -1}\left[{\mathit{\chi }}_1\left(s\right){\mathit{\chi }}_2\left(s\right)+{\mathit{\chi }}_1\left(1-s\right){\mathit{\chi }}_2\left(1-s\right)\right]ds\\ +{\mathit{\nabla }}^{*}\left(\left(t,t+\Lambda \left(\upsilon ,t\right)\right), \varsigma \right){\int }_0^1{s}^{\alpha -1}\left[{\mathit{\chi }}_1\left(s\right){\mathit{\chi }}_2\left(1-s\right)+{\mathit{\chi }}_1\left(1-s\right){\mathit{\chi }}_2\left(s\right)\right]ds.\end{array}$$

It follows that,

$$\begin{array}{c}\frac{\Gamma \left(\alpha \right)}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }}\left[{ℐ}_{t^{+}}^{\alpha } {\mathfrak{U}}_{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)\times {\mathscr{H}}_{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } {\mathfrak{U}}_{*}\left(t,\varsigma \right)\times {\mathscr{H}}_{*}\left(t,\varsigma \right)\right]\\ \le {\mathsf{\Delta }}_{*}\left(\left(t,t+\Lambda \left(\upsilon ,t\right)\right), \varsigma \right){\int }_0^1{s}^{\alpha -1}\left[{\mathit{\chi }}_1\left(s\right){\mathit{\chi }}_2\left(s\right)+{\mathit{\chi }}_1\left(1-s\right){\mathit{\chi }}_2\left(1-s\right)\right]ds\\ +{\mathit{\nabla }}_{*}\left(\left(t,t+\Lambda \left(\upsilon ,t\right)\right), \varsigma \right){\int }_0^1{s}^{\alpha -1}\left[{\mathit{\chi }}_1\left(s\right){\mathit{\chi }}_2\left(1-s\right)+{\mathit{\chi }}_1\left(1-s\right){\mathit{\chi }}_2\left(s\right)\right]ds\\ \frac{\Gamma \left(\alpha \right)}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }}\left[{ℐ}_{t^{+}}^{\alpha } {\mathfrak{U}}^{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)\times {\mathscr{H}}^{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } {\mathfrak{U}}^{*}\left(t,\varsigma \right)\times {\mathscr{H}}^{*}\left(t,\varsigma \right)\right]\\ \le {\mathsf{\Delta }}^{*}\left(\left(t,t+\Lambda \left(\upsilon ,t\right)\right), \varsigma \right){\int }_0^1{s}^{\alpha -1}\left[{\mathit{\chi }}_1\left(s\right){\mathit{\chi }}_2\left(s\right)+{\mathit{\chi }}_1\left(1-s\right){\mathit{\chi }}_2\left(1-s\right)\right]ds\\ +{\mathit{\nabla }}^{*}\left(\left(t,t+\Lambda \left(\upsilon ,t\right)\right), \varsigma \right){\int }_0^1{s}^{\alpha -1}\left[{\mathit{\chi }}_1\left(s\right){\mathit{\chi }}_2\left(1-s\right)+{\mathit{\chi }}_1\left(1-s\right){\mathit{\chi }}_2\left(s\right)\right]ds.\end{array}$$

It follows that

$$\begin{gathered} \frac{\Gamma \left(\alpha \right)}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }}[{ℐ}_{t^{+}}^{\alpha } {\mathfrak{U}}_{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)\times {\mathscr{H}}_{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } {\mathfrak{U}}_{*}\left(t,\varsigma \right)\times {\mathscr{H}}_{*}\left(t,\varsigma \right), {ℐ}_{t^{+}}^{\alpha } {\mathfrak{U}}^{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)\times \\ {\mathscr{H}}^{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } {\mathfrak{U}}^{*}\left(t,\varsigma \right)\times {\mathscr{H}}^{*}\left(t, \varsigma \right)] \\ {\le }_I\left[{\mathsf{\Delta }}_{*}\left(\left(t,t+\Lambda \left(\upsilon ,t\right)\right), \varsigma \right), {\mathsf{\Delta }}^{*}\left(\left(t,t+\Lambda \left(\upsilon ,t\right)\right), \varsigma \right)\right]{\int }_0^1{s}^{\alpha -1}\left[{\mathit{\chi }}_1\left(s\right){\mathit{\chi }}_2\left(s\right)+{\mathit{\chi }}_1\left(1-s\right){\mathit{\chi }}_2\left(1-s\right)\right]ds \\ +\left[{\mathit{\nabla }}_{*}\left(\left(t,t+\Lambda \left(\upsilon ,t\right)\right), \varsigma \right), {\mathit{\nabla }}^{*}\left(\left(t,t+\Lambda \left(\upsilon ,t\right)\right), \varsigma \right)\right]{\int }_0^1{s}^{\alpha -1}\left[{\mathit{\chi }}_1\left(s\right){\mathit{\chi }}_2\left(1-s\right)+{\mathit{\chi }}_1\left(1-s\right){\mathit{\chi }}_2\left(s\right)\right]ds, \end{gathered}$$

that is

$$\begin{gathered} \frac{\Gamma \left(\alpha \right)}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }}\left[{ℐ}_{t^{+}}^{\alpha } {\mathfrak{U}}_{\varsigma }\left(t+\Lambda \left(\upsilon ,t\right)\right)\times {\mathscr{H}}_{\varsigma }\left(t+\Lambda \left(\upsilon ,t\right)\right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha }{\mathfrak{U}}_{\varsigma }\left(t\right)\times {\mathscr{H}}_{\varsigma }\left(t\right)\right] \\ {\le }_I{\mathsf{\Delta }}_{\varsigma }\left(t,t+\Lambda \left(\upsilon ,t\right)\right){\int }_0^1{s}^{\alpha -1}\left[{\mathit{\chi }}_1\left(s\right){\mathit{\chi }}_2\left(s\right)+{\mathit{\chi }}_1\left(1-s\right){\mathit{\chi }}_2\left(1-s\right)\right]ds \\ +{\mathit{\nabla }}_{\varsigma }\left(t,t+\Lambda \left(\upsilon ,t\right)\right){\int }_0^1{s}^{\alpha -1}\left[{\mathit{\chi }}_1\left(s\right){\mathit{\chi }}_2\left(1-s\right)+{\mathit{\chi }}_1\left(1-s\right){\mathit{\chi }}_2\left(s\right)\right]ds. \end{gathered}$$

Thus,

$$\begin{gathered} \frac{\Gamma \left(\alpha \right)}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }}\left[{ℐ}_{t^{+}}^{\alpha } \tilde{\mathfrak{U}}\left(t+\Lambda \left(\upsilon ,t\right)\right)\tilde{\times }\tilde{\mathscr{H}}\left(t+\Lambda \left(\upsilon ,t\right)\right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha }\tilde{\mathfrak{U}}\left(t\right)\tilde{\times }\tilde{\mathscr{H}}\left(t\right)\right] \\ \preccurlyeq \tilde{\mathsf{\Delta }}\left(t,t+\Lambda \left(\upsilon ,t\right)\right){\int }_0^1{s}^{\alpha -1}\left[{\mathit{\chi }}_1\left(s\right){\mathit{\chi }}_2\left(s\right)+{\mathit{\chi }}_1\left(1-s\right){\mathit{\chi }}_2\left(1-s\right)\right]ds \\ +\tilde{\mathit{\nabla }}\left(t,t+\Lambda \left(\upsilon ,t\right)\right){\int }_0^1{s}^{\alpha -1}\left[{\mathit{\chi }}_1\left(s\right){\mathit{\chi }}_2\left(1-s\right)+{\mathit{\chi }}_1\left(1-s\right){\mathit{\chi }}_2\left(s\right)\right]ds. \end{gathered}$$

and the theorem has been established. □

Theorem 5.

Let $\tilde{\mathfrak{U}},\tilde{\mathscr{H}} :\left[t, t+\Lambda \left(\upsilon ,t\right)\right]\to {\mathbb{F}}_0$ be two ${\mathit{\chi }}_1$-pre-invex and ${\mathit{\chi }}_2$-pre-invex F-IV-Fs, respectively, whose $\varsigma$-cuts define the family of IV-Fs ${\mathfrak{U}}_{\varsigma }, {\mathscr{H}}_{\varsigma }:\left[t, t+\Lambda \left(\upsilon ,t\right)\right]\subset ℝ\to {\mathcal{K}}_C^{+}$ are given by ${\mathfrak{U}}_{\varsigma }\left(\varkappa \right)=\left[{\mathfrak{U}}_{*}\left(\varkappa ,\varsigma \right), {\mathfrak{U}}^{*}\left(\varkappa ,\varsigma \right)\right]$ and ${\mathscr{H}}_{\varsigma }\left(\varkappa \right)=\left[{\mathscr{H}}_{*}\left(\varkappa ,\varsigma \right), {\mathscr{H}}^{*}\left(\varkappa ,\varsigma \right)\right]$ for all $\varkappa \in \left[t, t+\Lambda \left(\upsilon ,t\right)\right]$ and for all $\varsigma \in \left[0, 1\right]$. If $\Lambda$ satisfies Condition C and $\tilde{\mathfrak{U}}\tilde{\times }\tilde{\mathscr{H}}\in L\left(\left[t, t+\Lambda \left(\upsilon ,t\right)\right],{\mathbb{F}}_0\right)$, then

$$\begin{gathered} \frac{1}{\alpha {\mathit{\chi }}_1\left(\frac{1}{2}\right){\mathit{\chi }}_2\left(\frac{1}{2}\right)} \tilde{\mathfrak{U}}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}\right)\tilde{\times }\tilde{\mathscr{H}}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}\right)\preccurlyeq \frac{\Gamma \left(\alpha \right)}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }}\left[{ℐ}_{t^{+}}^{\alpha } \tilde{\mathfrak{U}}\left(t+\Lambda \left(\upsilon ,t\right)\right)\tilde{\times }\tilde{\mathscr{H}}\left(\upsilon \right)\widetilde{+}{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } \tilde{\mathfrak{U}}\left(t\right)\tilde{\times }\tilde{\mathscr{H}}\left(t\right)\right] \\ \widetilde{+}\tilde{\mathit{\nabla }}\left(t,t+\Lambda \left(\upsilon ,t\right)\right){\int }_0^1\left[s^{\alpha -1}+{\left(1-s\right)}^{\alpha -1}\right]{\mathit{\chi }}_1\left(s\right){\mathit{\chi }}_2\left(1-s\right)ds \\ \widetilde{+}\tilde{\mathsf{\Delta }}\left(t,t+\Lambda \left(\upsilon ,t\right)\right){\int }_0^1\left[s^{\alpha -1}+{\left(1-s\right)}^{\alpha -1}\right]{\mathit{\chi }}_1\left(1-s\right){\mathit{\chi }}_2\left(1-s\right)ds. \end{gathered}$$

(36)

where $\tilde{\mathsf{\Delta }}\left(t,t+\Lambda \left(\upsilon ,t\right)\right)=\tilde{\mathfrak{U}}\left(t\right)\tilde{\times }\tilde{\mathscr{H}}\left(t\right) \widetilde{+} \tilde{\mathfrak{U}}\left(\upsilon \right)\tilde{\times }\tilde{\mathscr{H}}\left(\upsilon \right),$ $\tilde{\mathit{\nabla }}\left(t,t+\Lambda \left(\upsilon ,t\right)\right)=\tilde{\mathfrak{U}}\left(t\right)\tilde{\times }\tilde{\mathscr{H}}\left(\upsilon \right) \widetilde{+} \tilde{\mathfrak{U}}\left(\upsilon \right)\tilde{\times }\tilde{\mathscr{H}}\left(t\right),$ and ${\mathsf{\Delta }}_{\varsigma }\left(t,t+\Lambda \left(\upsilon ,t\right)\right)=\left[{\mathsf{\Delta }}_{*}\left(\left(t,t+\Lambda \left(\upsilon ,t\right)\right), \varsigma \right), {\mathsf{\Delta }}^{*}\left(\left(t,t+\Lambda \left(\upsilon ,t\right)\right), \varsigma \right)\right]$ and ${\mathit{\nabla }}_{\varsigma }\left(t,t+\Lambda \left(\upsilon ,t\right)\right)=\left[{\mathit{\nabla }}_{*}\left(\left(t,t+\Lambda \left(\upsilon ,t\right)\right), \varsigma \right), {\mathit{\nabla }}^{*}\left(\left(t,t+\Lambda \left(\upsilon ,t\right)\right), \varsigma \right)\right]$.

Proof. 

Consider $\tilde{\mathfrak{U}},\tilde{\mathscr{H}} :\left[t, t+\Lambda \left(\upsilon ,t\right)\right]\to {\mathbb{F}}_0$ are ${\mathit{\chi }}_1$-pre-invex and ${\mathit{\chi }}_2$-pre-invex F-IV-Fs. Then, by hypothesis, for each $\varsigma \in \left[0, 1\right],$ we have

$$\begin{array}{c}\begin{array}{c}{\mathfrak{U}}_{*}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2} ,\varsigma \right)\times {\mathscr{H}}_{*}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2},\varsigma \right) \\ {\mathfrak{U}}^{*}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2},\varsigma \right)\times {\mathscr{H}}^{*}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2},\varsigma \right)\end{array}\\ \le {\mathit{\chi }}_1\left(\frac{1}{2}\right){\mathit{\chi }}_2\left(\frac{1}{2}\right)\left[\begin{array}{c}{\mathfrak{U}}_{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)\times {\mathscr{H}}_{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)\\ +{\mathfrak{U}}_{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)\times {\mathscr{H}}_{*}\left(t+s\Lambda \left(\upsilon ,t\right), \varsigma \right)\end{array}\right]\\ + {\mathit{\chi }}_1\left(\frac{1}{2}\right){\mathit{\chi }}_2\left(\frac{1}{2}\right)\left[\begin{array}{c}{\mathfrak{U}}_{*}\left(t+s\Lambda \left(\upsilon ,t\right), \varsigma \right)\times {\mathscr{H}}_{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)\\ +{\mathfrak{U}}_{*}\left(t+s\Lambda \left(\upsilon ,t\right), \varsigma \right)\times {\mathscr{H}}_{*}\left(t+s\Lambda \left(\upsilon ,t\right), \varsigma \right)\end{array}\right]\\ \le {\mathit{\chi }}_1\left(\frac{1}{2}\right){\mathit{\chi }}_2\left(\frac{1}{2}\right)\left[\begin{array}{c}{\mathfrak{U}}^{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)\times {\mathscr{H}}^{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)\\ +{\mathfrak{U}}^{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)\times {\mathscr{H}}^{*}\left(t+s\Lambda \left(\upsilon ,t\right), \varsigma \right)\end{array}\right]\\ + {\mathit{\chi }}_1\left(\frac{1}{2}\right){\mathit{\chi }}_2\left(\frac{1}{2}\right)\left[\begin{array}{c}{\mathfrak{U}}^{*}\left(t+s\Lambda \left(\upsilon ,t\right), \varsigma \right)\times {\mathscr{H}}^{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)\\ +{\mathfrak{U}}^{*}\left(t+s\Lambda \left(\upsilon ,t\right), \varsigma \right)\times {\mathscr{H}}^{*}\left(t+s\Lambda \left(\upsilon ,t\right), \varsigma \right)\end{array}\right],\\ \le {\mathit{\chi }}_1\left(\frac{1}{2}\right){\mathit{\chi }}_2\left(\frac{1}{2}\right)\left[\begin{array}{c}{\mathfrak{U}}_{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)\times {\mathscr{H}}_{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)\\ +{\mathfrak{U}}_{*}\left(t+s\Lambda \left(\upsilon ,t\right), \varsigma \right)\times {\mathscr{H}}_{*}\left(t+s\Lambda \left(\upsilon ,t\right), \varsigma \right)\end{array}\right]\\ + {\mathit{\chi }}_1\left(\frac{1}{2}\right){\mathit{\chi }}_2\left(\frac{1}{2}\right)\left[\begin{array}{c}\left({\mathit{\chi }}_1\left(s\right){\mathfrak{U}}_{*}\left(t, \varsigma \right)+{\mathit{\chi }}_1\left(1-s\right){\mathfrak{U}}_{*}\left(t+\Lambda \left(\upsilon ,t\right),, \varsigma \right)\right)\\ \times \left({\mathit{\chi }}_2\left(1-s\right){\mathscr{H}}_{*}\left(t, \varsigma \right)+{\mathit{\chi }}_2\left(s\right){\mathscr{H}}_{*}\left(t+\Lambda \left(\upsilon ,t\right),, \varsigma \right)\right)\\ +\left({\mathit{\chi }}_1\left(1-s\right){\mathfrak{U}}_{*}\left(t, \varsigma \right)+{\mathit{\chi }}_1\left(s\right){\mathfrak{U}}_{*}\left(t+\Lambda \left(\upsilon ,t\right),, \varsigma \right)\right)\\ \times \left({\mathit{\chi }}_2\left(s\right){\mathscr{H}}_{*}\left(t, \varsigma \right)+{\mathit{\chi }}_2\left(1-s\right){\mathscr{H}}_{*}\left(t+\Lambda \left(\upsilon ,t\right),, \varsigma \right)\right)\end{array}\right]\\ \le {\mathit{\chi }}_1\left(\frac{1}{2}\right){\mathit{\chi }}_2\left(\frac{1}{2}\right)\left[\begin{array}{c}{\mathfrak{U}}^{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)\times {\mathscr{H}}^{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)\\ +{\mathfrak{U}}^{*}\left(t+s\Lambda \left(\upsilon ,t\right), \varsigma \right)\times {\mathscr{H}}^{*}\left(t+s\Lambda \left(\upsilon ,t\right), \varsigma \right)\end{array}\right]\\ + {\mathit{\chi }}_1\left(\frac{1}{2}\right){\mathit{\chi }}_2\left(\frac{1}{2}\right)\left[\begin{array}{c}\left({\mathit{\chi }}_1\left(s\right){\mathfrak{U}}^{*}\left(t, \varsigma \right)+{\mathit{\chi }}_1\left(1-s\right){\mathfrak{U}}^{*}\left(t+\Lambda \left(\upsilon ,t\right),, \varsigma \right)\right)\\ \times \left({\mathit{\chi }}_2\left(1-s\right){\mathscr{H}}^{*}\left(t, \varsigma \right)+{\mathit{\chi }}_2\left(s\right){\mathscr{H}}^{*}\left(t+\Lambda \left(\upsilon ,t\right),, \varsigma \right)\right)\\ +\left({\mathit{\chi }}_1\left(1-s\right){\mathfrak{U}}^{*}\left(t, \varsigma \right)+{\mathit{\chi }}_1\left(s\right){\mathfrak{U}}^{*}\left(t+\Lambda \left(\upsilon ,t\right),, \varsigma \right)\right)\\ \times \left({\mathit{\chi }}_2\left(s\right){\mathscr{H}}^{*}\left(t, \varsigma \right)+{\mathit{\chi }}_2\left(1-s\right){\mathscr{H}}^{*}\left(t+\Lambda \left(\upsilon ,t\right),, \varsigma \right)\right)\end{array}\right],\\ ={\mathit{\chi }}_1\left(\frac{1}{2}\right){\mathit{\chi }}_2\left(\frac{1}{2}\right)\left[\begin{array}{c}{\mathfrak{U}}_{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)\times {\mathscr{H}}_{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)\\ +{\mathfrak{U}}_{*}\left(t+s\Lambda \left(\upsilon ,t\right), \varsigma \right)\times {\mathscr{H}}_{*}\left(t+s\Lambda \left(\upsilon ,t\right), \varsigma \right)\end{array}\right]\\ + {\mathit{\chi }}_1\left(\frac{1}{2}\right){\mathit{\chi }}_2\left(\frac{1}{2}\right)\left[\begin{array}{c}\left\{{\mathit{\chi }}_1\left(s\right){\mathit{\chi }}_2\left(1-s\right)+{\mathit{\chi }}_1\left(1-s\right){\mathit{\chi }}_2\left(s\right)\right\}{\mathit{\nabla }}_{*}\left(\left(t,t+\Lambda \left(\upsilon ,t\right)\right), \varsigma \right)\\ +\left\{{\mathit{\chi }}_1\left(s\right){\mathit{\chi }}_2\left(s\right)+{\mathit{\chi }}_1\left(1-s\right){\mathit{\chi }}_2\left(1-s\right)\right\}{\mathsf{\Delta }}_{*}\left(\left(t,t+\Lambda \left(\upsilon ,t\right)\right), \varsigma \right)\end{array}\right]\\ ={\mathit{\chi }}_1\left(\frac{1}{2}\right){\mathit{\chi }}_2\left(\frac{1}{2}\right)\left[\begin{array}{c}{\mathfrak{U}}^{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)\times {\mathscr{H}}^{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)\\ +{\mathfrak{U}}^{*}\left(t+s\Lambda \left(\upsilon ,t\right), \varsigma \right)\times {\mathscr{H}}^{*}\left(t+s\Lambda \left(\upsilon ,t\right), \varsigma \right)\end{array}\right] \\ + {\mathit{\chi }}_1\left(\frac{1}{2}\right){\mathit{\chi }}_2\left(\frac{1}{2}\right)\left[\begin{array}{c}\left\{{\mathit{\chi }}_1\left(s\right){\mathit{\chi }}_2\left(1-s\right)+{\mathit{\chi }}_1\left(1-s\right){\mathit{\chi }}_2\left(s\right)\right\}{\mathit{\nabla }}^{*}\left(\left(t,t+\Lambda \left(\upsilon ,t\right)\right), \varsigma \right)\\ +\left\{{\mathit{\chi }}_1\left(s\right){\mathit{\chi }}_2\left(s\right)+{\mathit{\chi }}_1\left(1-s\right){\mathit{\chi }}_2\left(1-s\right)\right\}{\mathsf{\Delta }}^{*}\left(\left(t,t+\Lambda \left(\upsilon ,t\right)\right), \varsigma \right)\end{array}\right].\end{array}$$

(37)

Taking multiplication of (30) with $s^{\alpha -1}$ and integrating over $\left(0, 1\right),$ we get

$$\begin{array}{c}\frac{1}{\alpha {\mathit{\chi }}_1\left(\frac{1}{2}\right){\mathit{\chi }}_2\left(\frac{1}{2}\right)}{\mathfrak{U}}_{*}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2},\varsigma \right)\times {\mathscr{H}}_{*}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2},\varsigma \right) \\ \le \frac{\Gamma \left(\alpha \right)}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }}\left[{ℐ}_{t^{+}}^{\alpha } {\mathfrak{U}}_{*}\left(t+\Lambda \left(\upsilon ,t\right)\right)\times {\mathscr{H}}_{*}\left(t+\Lambda \left(\upsilon ,t\right)\right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } {\mathfrak{U}}_{*}\left(t\right)\times {\mathscr{H}}_{*}\left(t\right)\right]\\ +{\mathit{\nabla }}_{*}\left(\left(t,t+\Lambda \left(\upsilon ,t\right)\right), \varsigma \right){\int }_0^1\left[s^{\alpha -1}+{\left(1-s\right)}^{\alpha -1}\right]{\mathit{\chi }}_1\left(s\right){\mathit{\chi }}_2\left(1-s\right)ds\\ +{\mathsf{\Delta }}_{*}\left(\left(t,t+\Lambda \left(\upsilon ,t\right)\right), \varsigma \right){\int }_0^1\left[s^{\alpha -1}+{\left(1-s\right)}^{\alpha -1}\right]{\mathit{\chi }}_1\left(1-s\right){\mathit{\chi }}_2\left(1-s\right)ds\\ \frac{1}{\alpha {\mathit{\chi }}_1\left(\frac{1}{2}\right){\mathit{\chi }}_2\left(\frac{1}{2}\right)}{\mathfrak{U}}^{*}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2},\varsigma \right)\times {\mathscr{H}}^{*}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2},\varsigma \right) \\ \le \frac{\Gamma \left(\alpha \right)}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }}\left[{ℐ}_{t^{+}}^{\alpha } {\mathfrak{U}}^{*}\left(t+\Lambda \left(\upsilon ,t\right)\right)\times {\mathscr{H}}^{*}\left(t+\Lambda \left(\upsilon ,t\right)\right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } {\mathfrak{U}}^{*}\left(t\right)\times {\mathscr{H}}^{*}\left(t\right)\right]\\ +{\mathit{\nabla }}^{*}\left(\left(t,t+\Lambda \left(\upsilon ,t\right)\right), \varsigma \right){\int }_0^1\left[s^{\alpha -1}+{\left(1-s\right)}^{\alpha -1}\right]{\mathit{\chi }}_1\left(s\right){\mathit{\chi }}_2\left(1-s\right)ds\\ +{\mathsf{\Delta }}^{*}\left(\left(t,t+\Lambda \left(\upsilon ,t\right)\right), \varsigma \right){\int }_0^1\left[s^{\alpha -1}+{\left(1-s\right)}^{\alpha -1}\right]{\mathit{\chi }}_1\left(1-s\right){\mathit{\chi }}_2\left(1-s\right)ds,\end{array}$$

(38)

It follows that

$$\begin{gathered} \frac{1}{\alpha {\mathit{\chi }}_1\left(\frac{1}{2}\right){\mathit{\chi }}_2\left(\frac{1}{2}\right)} {\mathfrak{U}}_{\varsigma }\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}\right)\times {\mathscr{H}}_{\varsigma }\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}\right) \\ {\le }_I\frac{\Gamma \left(\alpha \right)}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }}\left[{ℐ}_{t^{+}}^{\alpha } {\mathfrak{U}}_{\varsigma }\left(t+\Lambda \left(\upsilon ,t\right)\right)\times {\mathscr{H}}_{\varsigma }\left(t+\Lambda \left(\upsilon ,t\right)\right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } {\mathfrak{U}}_{\varsigma }\left(t\right)\times {\mathscr{H}}_{\varsigma }\left(t\right)\right] \\ +{\mathit{\nabla }}_{\varsigma }\left(t,t+\Lambda \left(\upsilon ,t\right)\right){\int }_0^1\left[s^{\alpha -1}+{\left(1-s\right)}^{\alpha -1}\right]{\mathit{\chi }}_1\left(1-s\right){\mathit{\chi }}_2\left(1-s\right)ds \\ +{\mathsf{\Delta }}_{\varsigma }\left(t,t+\Lambda \left(\upsilon ,t\right)\right){\int }_0^1\left[s^{\alpha -1}+{\left(1-s\right)}^{\alpha -1}\right]{\mathit{\chi }}_1\left(1-s\right){\mathit{\chi }}_2\left(1-s\right)ds, \end{gathered}$$

that is

$$\begin{gathered} \frac{1}{\alpha {\mathit{\chi }}_1\left(\frac{1}{2}\right){\mathit{\chi }}_2\left(\frac{1}{2}\right)} \tilde{\mathfrak{U}}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}\right)\tilde{\times }\tilde{\mathscr{H}}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}\right) \\ \preccurlyeq \frac{\Gamma \left(\alpha \right)}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }}\left[{ℐ}_{t^{+}}^{\alpha } \tilde{\mathfrak{U}}\left(t+\Lambda \left(\upsilon ,t\right)\right)\tilde{\times }\tilde{\mathscr{H}}\left(t+\Lambda \left(\upsilon ,t\right)\right)\widetilde{+}{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } \tilde{\mathfrak{U}}\left(t\right)\tilde{\times }\tilde{\mathscr{H}}\left(t\right)\right] \\ \widetilde{+}\tilde{\mathit{\nabla }}\left(t,t+\Lambda \left(\upsilon ,t\right)\right){\int }_0^1\left[s^{\alpha -1}+{\left(1-s\right)}^{\alpha -1}\right]{\mathit{\chi }}_1\left(s\right){\mathit{\chi }}_2\left(1-s\right)ds \\ \widetilde{+}\tilde{\mathsf{\Delta }}\left(t,t+\Lambda \left(\upsilon ,t\right)\right){\int }_0^1\left[s^{\alpha -1}+{\left(1-s\right)}^{\alpha -1}\right]{\mathit{\chi }}_1\left(1-s\right){\mathit{\chi }}_2\left(1-s\right)ds. \end{gathered}$$

Hence, the required result. □

Theorems 6 and 7 are, respectively, linked to the right and left halves of the standard H⋅H-Fejér inequality. The left and right component of the conventional H⋅H-Fejér inequality are obtained using a fuzzy-Riemann–Liouville-fractional integral inequalities, and these are known as the first and second fuzzy-fractional-H⋅H-Fejér inequalities, respectively. Firstly, we obtain second H⋅H-Fejér inequality.

Theorem 6.

Let $\tilde{\mathfrak{U}}:\left[t, t+\Lambda \left(\upsilon ,t\right)\right]\to {\mathbb{F}}_0$ be a $\mathit{\chi }$-pre-invex F-IV-F with $t<\upsilon$, whose $\varsigma$-cuts define the family of IV-Fs ${\mathfrak{U}}_{\varsigma }:\left[t, t+\Lambda \left(\upsilon ,t\right)\right]\subset ℝ\to {\mathcal{K}}_C^{+}$ are given by ${\mathfrak{U}}_{\varsigma }\left(\varkappa \right)=\left[{\mathfrak{U}}_{*}\left(\varkappa ,\varsigma \right), {\mathfrak{U}}^{*}\left(\varkappa ,\varsigma \right)\right]$ for all $\varkappa \in \left[t, t+\Lambda \left(\upsilon ,t\right)\right]$ and for all $\varsigma \in \left[0, 1\right]$. If $\tilde{\mathfrak{U}}\in L\left(\left[t, t+\Lambda \left(\upsilon ,t\right)\right],{\mathbb{F}}_0\right)$ and $\mathsf{\Omega }:\left[t, t+\Lambda \left(\upsilon ,t\right)\right]\to ℝ, \mathsf{\Omega }\left(\varkappa \right)\ge 0,$ and symmetric with respect to $\frac{2t+\Lambda \left(\upsilon ,t\right)}{2},$ then

$$\begin{gathered} \frac{\Gamma \left(\alpha \right)}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }}\left[{ℐ}_{t^{+}}^{\alpha } \tilde{\mathfrak{U}}\mathsf{\Omega }\left(t+\Lambda \left(\upsilon ,t\right)\right)\widetilde{+}{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } \tilde{\mathfrak{U}}\mathsf{\Omega }\left(t\right)\right] \\ \preccurlyeq \left(\tilde{\mathfrak{U}}\left(t\right)\widetilde{+}\tilde{\mathfrak{U}}\left(t+\Lambda \left(\upsilon ,t\right)\right)\right){\int }_0^1\begin{array}{c}{s}^{\alpha -1}\left[\mathit{\chi }\left(s\right)+\mathit{\chi }\left(1-s\right)\right]\mathsf{\Omega }\left(t+s\Lambda \left(\upsilon ,t\right)\right)ds\end{array} \\ \preccurlyeq \left(\tilde{\mathfrak{U}}\left(t\right)\widetilde{+}\tilde{\mathfrak{U}}\left(\upsilon \right)\right){\int }_0^1{s}^{\alpha -1}\left[\mathit{\chi }\left(s\right)+\mathit{\chi }\left(1-s\right)\right]\mathsf{\Omega }\left(t+s\Lambda \left(\upsilon ,t\right)\right)ds. \end{gathered}$$

(39)

If $\tilde{\mathfrak{U}}$ is pre-incave F-IV-F, then inequality (31) is reversed.

Proof. 

Let $\tilde{\mathfrak{U}}$ be a χ-pre-invex F-IV-F and $s^{\alpha -1}\mathsf{\Omega }\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right)\right)\ge 0$. Then, for each $\varsigma \in \left[0, 1\right],$ we have

$$\begin{array}{c}{s}^{\alpha -1}{\mathfrak{U}}_{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)\mathsf{\Omega }\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right)\right)\\ \le s^{\alpha -1}\left(\mathit{\chi }\left(s\right){\mathfrak{U}}_{*}\left(t, \varsigma \right)+\mathit{\chi }\left(1-s\right){\mathfrak{U}}_{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)\right)\mathsf{\Omega }\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right)\right)\\ s^{\alpha -1}{\mathfrak{U}}^{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)\mathsf{\Omega }\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right)\right)\\ \le s^{\alpha -1}\left(\mathit{\chi }\left(s\right){\mathfrak{U}}^{*}\left(t, \varsigma \right)+\mathit{\chi }\left(1-s\right){\mathfrak{U}}^{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)\right)\mathsf{\Omega }\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right)\right),\end{array}$$

(40)

and

$$\begin{array}{c}{s}^{\alpha -1}{\mathfrak{U}}_{*}\left(t+s\Lambda \left(\upsilon ,t\right), \varsigma \right)\mathsf{\Omega }\left(t+s\Lambda \left(\upsilon ,t\right)\right)\\ \le s^{\alpha -1}\left(\mathit{\chi }\left(1-s\right){\mathfrak{U}}_{*}\left(t, \varsigma \right)+\mathit{\chi }\left(s\right){\mathfrak{U}}_{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)\right)\mathsf{\Omega }\left(t+s\Lambda \left(\upsilon ,t\right)\right)\\ s^{\alpha -1}{\mathfrak{U}}^{*}\left(t+s\Lambda \left(\upsilon ,t\right), \varsigma \right)\mathsf{\Omega }\left(t+s\Lambda \left(\upsilon ,t\right)\right)\\ \le s^{\alpha -1}\left(\mathit{\chi }\left(1-s\right){\mathfrak{U}}^{*}\left(t, \varsigma \right)+\mathit{\chi }\left(s\right){\mathfrak{U}}^{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)\right)\mathsf{\Omega }\left(t+s\Lambda \left(\upsilon ,t\right)\right).\end{array}$$

(41)

After adding (32) and (33), and integrating over $\left[0, 1\right],$ we get

$$\begin{array}{c}{\int }_0^1{s}^{\alpha -1}{\mathfrak{U}}_{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)\mathsf{\Omega }\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right)\right)ds \\ +{\int }_0^1{s}^{\alpha -1}{\mathfrak{U}}_{*}\left(t+s\Lambda \left(\upsilon ,t\right), \varsigma \right)\mathsf{\Omega }\left(t+s\Lambda \left(\upsilon ,t\right)\right)ds \\ \le {\int }_0^1\left[\begin{array}{c}{s}^{\alpha -1}{\mathfrak{U}}_{*}\left(t, \varsigma \right)\left\{\mathit{\chi }\left(s\right)\mathsf{\Omega }\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right)\right)+\mathit{\chi }\left(1-s\right)\mathsf{\Omega }\left(t+s\Lambda \left(\upsilon ,t\right)\right)\right\}\\ +s^{\alpha -1}{\mathfrak{U}}_{*}\left(t+\Lambda \left(\upsilon ,t\right),\varsigma \right)\left\{\mathit{\chi }\left(1-s\right)\mathsf{\Omega }\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right)\right)+\mathit{\chi }\left(s\right)\mathsf{\Omega }\left(t+s\Lambda \left(\upsilon ,t\right)\right)\right\}\end{array}\right]ds,\\ ={\mathfrak{U}}_{*}\left(t, \varsigma \right){\int }_0^1\begin{array}{c}{s}^{\alpha -1}\left[\mathit{\chi }\left(s\right)+\mathit{\chi }\left(1-s\right)\right]\mathsf{\Omega }\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right)\right)\end{array}ds\\ +{\mathfrak{U}}_{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right){\int }_0^1\begin{array}{c}{s}^{\alpha -1}\left[\mathit{\chi }\left(s\right)+\mathit{\chi }\left(1-s\right)\right]\mathsf{\Omega }\left(t+s\Lambda \left(\upsilon ,t\right)\right)\end{array}ds,\\ {\int }_0^1{s}^{\alpha -1}{\mathfrak{U}}^{*}\left(t+s\Lambda \left(\upsilon ,t\right), \varsigma \right)\mathsf{\Omega }\left(t+s\Lambda \left(\upsilon ,t\right)\right)ds \\ +{\int }_0^1{s}^{\alpha -1}{\mathfrak{U}}^{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)\mathsf{\Omega }\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right)\right)ds \\ \le {\int }_0^1\left[\begin{array}{c}{s}^{\alpha -1}{\mathfrak{U}}^{*}\left(t, \varsigma \right)\left\{\mathit{\chi }\left(s\right)\mathsf{\Omega }\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right)\right)+\mathit{\chi }\left(1-s\right)\mathsf{\Omega }\left(t+s\Lambda \left(\upsilon ,t\right)\right)\right\}\\ +s^{\alpha -1}{\mathfrak{U}}^{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)\left\{\mathit{\chi }\left(1-s\right)\mathsf{\Omega }\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right)\right)+\mathit{\chi }\left(s\right)\mathsf{\Omega }\left(t+s\Lambda \left(\upsilon ,t\right)\right)\right\}\end{array}\right]ds,\\ ={\mathfrak{U}}^{*}\left(t, \varsigma \right){\int }_0^1\begin{array}{c}{s}^{\alpha -1}\left[\mathit{\chi }\left(s\right)+\mathit{\chi }\left(1-s\right)\right]\mathsf{\Omega }\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right)\right)\end{array}ds\\ +{\mathfrak{U}}^{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right){\int }_0^1\begin{array}{c}{s}^{\alpha -1}\left[\mathit{\chi }\left(s\right)+\mathit{\chi }\left(1-s\right)\right]\mathsf{\Omega }\left(t+s\Lambda \left(\upsilon ,t\right)\right)\end{array}ds.\end{array}$$

(42)

Taking the right-hand side of inequality (34), we have

$$\begin{array}{c}{\int }_0^1{s}^{\alpha -1}{\mathfrak{U}}_{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)\mathsf{\Omega }\left(t+s\Lambda \left(\upsilon ,t\right)\right)ds \\ +{\int }_0^1{s}^{\alpha -1}{\mathfrak{U}}_{*}\left(t+s\Lambda \left(\upsilon ,t\right), \varsigma \right)\mathsf{\Omega }\left(t+s\Lambda \left(\upsilon ,t\right)\right)ds\\ =\frac{1}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }} {\int }_t^{t+\Lambda \left(\upsilon ,t\right)}{\left(\varkappa -t\right)}^{\alpha -1}{\mathfrak{U}}_{*}\left(2t+\Lambda \left(\upsilon ,t\right)-\varkappa ,\varsigma \right)\mathsf{\Omega }\left(\varkappa \right)d\varkappa \\ +\frac{1}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }} {\int }_t^{t+\Lambda \left(\upsilon ,t\right)}{\left(\varkappa -t\right)}^{\alpha -1}{\mathfrak{U}}_{*}\left(\varkappa ,\varsigma \right)\mathsf{\Omega }\left(\varkappa \right)d\varkappa \\ =\frac{1}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }} {\int }_t^{t+\Lambda \left(\upsilon ,t\right)}{\left(t+\Lambda \left(\upsilon ,t\right)-\varkappa \right)}^{\alpha -1}{\mathfrak{U}}_{*}\left(\varkappa ,\varsigma \right)\mathsf{\Omega }\left(2t+\Lambda \left(\upsilon ,t\right)-\varkappa \right)d\varkappa \\ +\frac{1}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }} {\int }_t^{t+\Lambda \left(\upsilon ,t\right)}{\left(\varkappa -t\right)}^{\alpha -1}{\mathfrak{U}}_{*}\left(\varkappa ,\varsigma \right)\mathsf{\Omega }\left(\varkappa \right)d\varkappa \\ =\frac{\Gamma \left(\alpha \right)}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }}\left[{ℐ}_{t^{+}}^{\alpha } {\mathfrak{U}}_{*}\mathsf{\Omega }\left(t+\Lambda \left(\upsilon ,t\right)\right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } {\mathfrak{U}}_{*}\mathsf{\Omega }\left(t\right)\right],\\ {\int }_0^1{s}^{\alpha -1}{\mathfrak{U}}^{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)\mathsf{\Omega }\left(t+s\Lambda \left(\upsilon ,t\right)\right)ds \\ +{\int }_0^1{s}^{\alpha -1}{\mathfrak{U}}^{*}\left(t+s\Lambda \left(\upsilon ,t\right), \varsigma \right)\mathsf{\Omega }\left(t+s\Lambda \left(\upsilon ,t\right)\right)ds\\ =\frac{\Gamma \left(\alpha \right)}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }}\left[{ℐ}_{t^{+}}^{\alpha } {\mathfrak{U}}^{*}\mathsf{\Omega }\left(t+\Lambda \left(\upsilon ,t\right)\right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } {\mathfrak{U}}^{*}\mathsf{\Omega }\left(t\right)\right].\end{array}$$

(43)

From (35), we have

$$\begin{array}{c}\frac{\Gamma \left(\alpha \right)}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }}\left[{ℐ}_{t^{+}}^{\alpha } {\mathfrak{U}}_{*}\mathsf{\Omega }\left(t+\Lambda \left(\upsilon ,t\right)\right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } {\mathfrak{U}}_{*}\mathsf{\Omega }\left(t\right)\right]\\ \le \frac{{\mathfrak{U}}_{*}\left(t, \varsigma \right)+{\mathfrak{U}}_{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)}{2}{\int }_0^1\begin{array}{c}{s}^{\alpha -1}\left[\begin{array}{c}\mathit{\chi }\left(s\right)\\ +\mathit{\chi }\left(1-s\right)\end{array}\right]\mathsf{\Omega }\left(t+s\Lambda \left(\upsilon ,t\right)\right)\end{array}\\ \frac{\Gamma \left(\alpha \right)}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }}\left[{ℐ}_{t^{+}}^{\alpha } {\mathfrak{U}}^{*}\mathsf{\Omega }\left(t+\Lambda \left(\upsilon ,t\right)\right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } {\mathfrak{U}}^{*}\mathsf{\Omega }\left(t\right)\right]\\ \le \frac{{\mathfrak{U}}^{*}\left(t, \varsigma \right)+{\mathfrak{U}}^{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)}{2}{\int }_0^1\begin{array}{c}{s}^{\alpha -1}\left[\begin{array}{c}\mathit{\chi }\left(s\right)\\ +\mathit{\chi }\left(1-s\right)\end{array}\right]\mathsf{\Omega }\left(t+s\Lambda \left(\upsilon ,t\right)\right)\end{array},\end{array}$$

(44)

that is

$$\begin{gathered} \frac{\Gamma \left(\alpha \right)}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }}\left[{ℐ}_{t^{+}}^{\alpha } {\mathfrak{U}}_{*}\mathsf{\Omega }\left(t+\Lambda \left(\upsilon ,t\right)\right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } {\mathfrak{U}}_{*}\mathsf{\Omega }\left(t\right), {ℐ}_{t^{+}}^{\alpha } {\mathfrak{U}}^{*}\mathsf{\Omega }\left(t+\Lambda \left(\upsilon ,t\right)\right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } {\mathfrak{U}}^{*}\mathsf{\Omega }\left(t\right)\right] \\ {\le }_I\left[\frac{{\mathfrak{U}}_{*}\left(t, \varsigma \right)+{\mathfrak{U}}_{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)}{2}, \frac{{\mathfrak{U}}^{*}\left(t, \varsigma \right)+{\mathfrak{U}}^{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)}{2}\right]{\int }_0^1{s}^{\alpha -1}\left[\mathit{\chi }\left(s\right)+\mathit{\chi }\left(1-s\right)\right]\mathsf{\Omega }\left(t+s\Lambda \left(\upsilon ,t\right)\right)ds, \end{gathered}$$

Hence

$$\begin{gathered} \frac{\Gamma \left(\alpha \right)}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }}\left[{ℐ}_{t^{+}}^{\alpha } \tilde{\mathfrak{U}}\mathsf{\Omega }\left(t+\Lambda \left(\upsilon ,t\right)\right)\widetilde{+}{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } \tilde{\mathfrak{U}}\mathsf{\Omega }\left(t\right)\right] \\ \preccurlyeq \frac{\tilde{\mathfrak{U}}\left(t\right)\widetilde{+}\tilde{\mathfrak{U}}\left(t+\Lambda \left(\upsilon ,t\right)\right)}{2}{\int }_0^1{s}^{\alpha -1}\left[\mathit{\chi }\left(s\right)+\mathit{\chi }\left(1-s\right)\right]\mathsf{\Omega }\left(t+s\Lambda \left(\upsilon ,t\right)\right)ds. \\ \preccurlyeq \frac{\tilde{\mathfrak{U}}\left(t\right)\widetilde{+}\tilde{\mathfrak{U}}\left(\upsilon \right)}{2}{\int }_0^1{s}^{\alpha -1}\left[\mathit{\chi }\left(s\right)+\mathit{\chi }\left(1-s\right)\right]\mathsf{\Omega }\left(t+s\Lambda \left(\upsilon ,t\right)\right)ds. \end{gathered}$$

(45)

 □

Theorem 7.

Let $\tilde{\mathfrak{U}}:\left[t, t+\Lambda \left(\upsilon ,t\right)\right]\to {\mathbb{F}}_0$ be a χ-pre-invex F-IV-F with $t<t+\Lambda \left(\upsilon ,t\right),$ whose $\varsigma$-cuts define the family of IV-Fs ${\mathfrak{U}}_{\varsigma }:\left[t, t+\Lambda \left(\upsilon ,t\right)\right]\subset ℝ\to {\mathcal{K}}_C^{+}$ are given by ${\mathfrak{U}}_{\varsigma }\left(\varkappa \right)=\left[{\mathfrak{U}}_{*}\left(\varkappa ,\varsigma \right), {\mathfrak{U}}^{*}\left(\varkappa ,\varsigma \right)\right]$ for all $\varkappa \in \left[t, t+\Lambda \left(\upsilon ,t\right)\right]$ and for all $\varsigma \in \left[0, 1\right]$. Let $\tilde{\mathfrak{U}}\in L\left(\left[t, t+\Lambda \left(\upsilon ,t\right)\right],{\mathbb{F}}_0\right)$ and $\mathsf{\Omega }:\left[t, t+\Lambda \left(\upsilon ,t\right)\right]\to ℝ, \mathsf{\Omega }\left(\varkappa \right)\ge 0,$ symmetric with respect to $\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}$. If $\Lambda$ satisfies Condition C, then

$$\frac{1}{2\mathit{\chi }\left(\frac{1}{2}\right)}\tilde{\mathfrak{U}}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}\right)\left[{ℐ}_{t^{+}}^{\alpha } \mathsf{\Omega }\left(t+\Lambda \left(\upsilon ,t\right)\right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } \mathsf{\Omega }\left(t\right)\right]\preccurlyeq \left[{ℐ}_{t^{+}}^{\alpha } \tilde{\mathfrak{U}}\mathsf{\Omega }\left(t+\Lambda \left(\upsilon ,t\right)\right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } \tilde{\mathfrak{U}}\mathsf{\Omega }\left(t\right)\right].$$

(46)

If $\tilde{\mathfrak{U}}$ is pre-incave F-IV-F, then inequality (36) is reversed.

Proof. 

Since $\tilde{\mathfrak{U}}$ is a χ-pre-invex F-IV-F, then for $\varsigma \in \left[0, 1\right],$ we have

$$\begin{array}{c}{\mathfrak{U}}_{*}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}, \varsigma \right)\le \mathit{\chi }\left(\frac{1}{2}\right)\left({\mathfrak{U}}_{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)+{\mathfrak{U}}_{*}\left(t+s\Lambda \left(\upsilon ,t\right), \varsigma \right)\right)\\ {\mathfrak{U}}^{*}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}, \varsigma \right)\le \mathit{\chi }\left(\frac{1}{2}\right)\left({\mathfrak{U}}^{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)+{\mathfrak{U}}^{*}\left(t+s\Lambda \left(\upsilon ,t\right), \varsigma \right)\right).\end{array}$$

(47)

Since $\mathsf{\Omega }\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right)\right)=\mathsf{\Omega }\left(t+s\Lambda \left(\upsilon ,t\right)\right)$, then by multiplying (37) by $s^{\alpha -1}\mathsf{\Omega }\left(t+s\Lambda \left(\upsilon ,t\right)\right)$ and integrate it with respect to $s$ over $\left[0, 1\right],$ we obtain

$$\begin{array}{c}{\mathfrak{U}}_{*}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}, \varsigma \right){\int }_0^1{s}^{\alpha -1}\mathsf{\Omega }\left(t+s\Lambda \left(\upsilon ,t\right)\right)ds\\ \le \mathit{\chi }\left(\frac{1}{2}\right)\left(\begin{array}{c}{\int }_0^1{s}^{\alpha -1}{\mathfrak{U}}_{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)\mathsf{\Omega }\left(t+s\Lambda \left(\upsilon ,t\right)\right)ds\\ +{\int }_0^1{s}^{\alpha -1}{\mathfrak{U}}_{*}\left(t+s\Lambda \left(\upsilon ,t\right), \varsigma \right)\mathsf{\Omega }\left(t+s\Lambda \left(\upsilon ,t\right)\right)ds\end{array}\right),\\ {\mathfrak{U}}^{*}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}, \varsigma \right){\int }_0^1{s}^{\alpha -1}\mathsf{\Omega }\left(t+s\Lambda \left(\upsilon ,t\right)\right)ds\\ \le \mathit{\chi }\left(\frac{1}{2}\right)\left(\begin{array}{c}{\int }_0^1{s}^{\alpha -1}{\mathfrak{U}}^{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)\mathsf{\Omega }\left(t+s\Lambda \left(\upsilon ,t\right)\right)ds\\ +{\int }_0^1{s}^{\alpha -1}{\mathfrak{U}}^{*}\left(t+s\Lambda \left(\upsilon ,t\right), \varsigma \right)\mathsf{\Omega }\left(t+s\Lambda \left(\upsilon ,t\right)\right)ds\end{array}\right).\end{array}$$

(48)

Let $\varkappa =t+s\Lambda \left(\upsilon ,t\right)$. Then, right-hand side of inequality (38), we have

$$\begin{array}{c}{\int }_0^1{s}^{\alpha -1}{\mathfrak{U}}_{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)\mathsf{\Omega }\left(t+s\Lambda \left(\upsilon ,t\right)\right)ds\\ +{\int }_0^1{s}^{\alpha -1}{\mathfrak{U}}_{*}\left(t+s\Lambda \left(\upsilon ,t\right), \varsigma \right)\mathsf{\Omega }\left(t+s\Lambda \left(\upsilon ,t\right)\right)ds\\ =\frac{1}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }} {\int }_t^{t+\Lambda \left(\upsilon ,t\right)}{\left(\varkappa -t\right)}^{\alpha -1}{\mathfrak{U}}_{*}\left(2t+\Lambda \left(\upsilon ,t\right)-\varkappa ,\varsigma \right)\mathsf{\Omega }\left(\varkappa \right)d\varkappa \\ +\frac{1}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }} {\int }_t^{t+\Lambda \left(\upsilon ,t\right)}{\left(\varkappa -t\right)}^{\alpha -1}{\mathfrak{U}}_{*}\left(\varkappa ,\varsigma \right)\mathsf{\Omega }\left(\varkappa \right)d\varkappa \\ =\frac{1}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }} {\int }_t^{t+\Lambda \left(\upsilon ,t\right)}{\left(\varkappa -t\right)}^{\alpha -1}{\mathfrak{U}}_{*}\left(\varkappa ,\varsigma \right)\mathsf{\Omega }\left(2t+\Lambda \left(\upsilon ,t\right)-\varkappa \right)d\varkappa \\ +\frac{1}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }} {\int }_t^{t+\Lambda \left(\upsilon ,t\right)}{\left(\varkappa -t\right)}^{\alpha -1}{\mathfrak{U}}_{*}\left(\varkappa ,\varsigma \right)\mathsf{\Omega }\left(\varkappa \right)d\varkappa \\ =\frac{\Gamma \left(\alpha \right)}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }}\left[{ℐ}_{t^{+}}^{\alpha } {\mathfrak{U}}_{*}\mathsf{\Omega }\left(t+\Lambda \left(\upsilon ,t\right)\right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } {\mathfrak{U}}_{*}\mathsf{\Omega }\left(t\right)\right],\\ {\int }_0^1{s}^{\alpha -1}{\mathfrak{U}}^{*}\left(t+\left(1-s\right)\Lambda \left(\upsilon ,t\right), \varsigma \right)\mathsf{\Omega }\left(t+s\Lambda \left(\upsilon ,t\right)\right)ds \\ +{\int }_0^1{s}^{\alpha -1}{\mathfrak{U}}^{*}\left(t+s\Lambda \left(\upsilon ,t\right), \varsigma \right)\mathsf{\Omega }\left(t+s\Lambda \left(\upsilon ,t\right)\right)ds\\ =\frac{\Gamma \left(\alpha \right)}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }}\left[{ℐ}_{t^{+}}^{\alpha } {\mathfrak{U}}^{*}\mathsf{\Omega }\left(t+\Lambda \left(\upsilon ,t\right)\right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } {\mathfrak{U}}^{*}\mathsf{\Omega }\left(t\right)\right].\end{array}$$

(49)

Then from (39), we have

$$\begin{array}{c}\frac{1}{2\mathit{\chi }\left(\frac{1}{2}\right)}{\mathfrak{U}}_{*}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}, \varsigma \right)\left[{ℐ}_{t^{+}}^{\alpha } \mathsf{\Omega }\left(t+\Lambda \left(\upsilon ,t\right)\right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } \mathsf{\Omega }\left(t\right)\right]\le \left[{ℐ}_{t^{+}}^{\alpha } {\mathfrak{U}}_{*}\mathsf{\Omega }\left(t+\Lambda \left(\upsilon ,t\right)\right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } {\mathfrak{U}}_{*}\mathsf{\Omega }\left(t\right)\right]\\ \frac{1}{2\mathit{\chi }\left(\frac{1}{2}\right)}{\mathfrak{U}}^{*}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}, \varsigma \right)\left[{ℐ}_{t^{+}}^{\alpha } \mathsf{\Omega }\left(t+\Lambda \left(\upsilon ,t\right)\right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } \mathsf{\Omega }\left(t\right)\right]\le \left[{ℐ}_{t^{+}}^{\alpha } {\mathfrak{U}}^{*}\mathsf{\Omega }\left(t+\Lambda \left(\upsilon ,t\right)\right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } {\mathfrak{U}}^{*}\mathsf{\Omega }\left(t\right)\right],\end{array}$$

from which, we have

$$\begin{array}{c}\frac{1}{2\mathit{\chi }\left(\frac{1}{2}\right)}\left[{\mathfrak{U}}_{*}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}, \varsigma \right), {\mathfrak{U}}^{*}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}, \varsigma \right)\right]\left[{ℐ}_{t^{+}}^{\alpha } \mathsf{\Omega }\left(t+\Lambda \left(\upsilon ,t\right)\right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } \mathsf{\Omega }\left(t\right)\right]\\ {\le }_I\left[{ℐ}_{t^{+}}^{\alpha } {\mathfrak{U}}_{*}\mathsf{\Omega }\left(t+\Lambda \left(\upsilon ,t\right)\right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } {\mathfrak{U}}_{*}\mathsf{\Omega }\left(t\right), {ℐ}_{t^{+}}^{\alpha } {\mathfrak{U}}^{*}\mathsf{\Omega }\left(t+\Lambda \left(\upsilon ,t\right)\right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } {\mathfrak{U}}^{*}\mathsf{\Omega }\left(t\right)\right],\end{array}$$

it follows that

$$\frac{1}{2\mathit{\chi }\left(\frac{1}{2}\right)}{\mathfrak{U}}_{\varsigma }\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}\right)\left[{ℐ}_{t^{+}}^{\alpha } \mathsf{\Omega }\left(t+\Lambda \left(\upsilon ,t\right)\right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } \mathsf{\Omega }\left(t\right)\right]{\le }_I\left[{ℐ}_{t^{+}}^{\alpha } {\mathfrak{U}}_{\varsigma }\mathsf{\Omega }\left(t+\Lambda \left(\upsilon ,t\right)\right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } {\mathfrak{U}}_{\varsigma }\mathsf{\Omega }\left(t\right)\right],$$

that is

$$\frac{1}{2\mathit{\chi }\left(\frac{1}{2}\right)}\tilde{\mathfrak{U}}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}\right)\left[{ℐ}_{t^{+}}^{\alpha } \mathsf{\Omega }\left(t+\Lambda \left(\upsilon ,t\right)\right)\widetilde{+}{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } \mathsf{\Omega }\left(t\right)\right]\preccurlyeq \left[{ℐ}_{t^{+}}^{\alpha } \tilde{\mathfrak{U}}\mathsf{\Omega }\left(t+\Lambda \left(\upsilon ,t\right)\right)\widetilde{+}{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } \tilde{\mathfrak{U}}\mathsf{\Omega }\left(t\right)\right].$$

(50)

This completes the proof. □

Remark 5.

If one attempt to take $\mathsf{\Omega }\left(\varkappa \right)=1$, then from (31) and (36) one achieves Theorem 3.

If one attempt to take $\mathit{\chi }\left(s\right)=s$, then from (31) and (36) one achieves coming inequality, see [24]:

$$\begin{gathered} \tilde{\mathfrak{U}}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}\right)\left[{ℐ}_{t^{+}}^{\alpha } \mathsf{\Omega }\left(t+\Lambda \left(\upsilon ,t\right)\right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } \mathsf{\Omega }\left(t\right)\right]\preccurlyeq \left[{ℐ}_{t^{+}}^{\alpha } \tilde{\mathfrak{U}}\mathsf{\Omega }\left(t+\Lambda \left(\upsilon ,t\right)\right)\widetilde{+}{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } \tilde{\mathfrak{U}}\mathsf{\Omega }\left(t\right)\right] \\ \preccurlyeq \frac{\tilde{\mathfrak{U}}\left(t\right)\widetilde{+}\tilde{\mathfrak{U}}\left(t+\Lambda \left(\upsilon ,t\right)\right)}{2}\left[{ℐ}_{t^{+}}^{\alpha } \mathsf{\Omega }\left(t+\Lambda \left(\upsilon ,t\right)\right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } \mathsf{\Omega }\left(t\right)\right]. \\ \preccurlyeq \frac{\tilde{\mathfrak{U}}\left(t\right)\widetilde{+}\tilde{\mathfrak{U}}\left(\upsilon \right)}{2}\left[{ℐ}_{t^{+}}^{\alpha } \mathsf{\Omega }\left(t+\Lambda \left(\upsilon ,t\right)\right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } \mathsf{\Omega }\left(t\right)\right]. \end{gathered}$$

(51)

Let one attempt to take $\mathit{\chi }\left(s\right)=s$ and $\alpha =1$. Then, from (31) and (36) one achieves coming inequality for pre-invex F-IV-F, see [41].

$$\tilde{\mathfrak{U}}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}\right)\preccurlyeq \frac{1}{{\int }_t^{t+\Lambda \left(\upsilon ,t\right)}\mathsf{\Omega }\left(\varkappa \right)d\varkappa } \left(FR\right){\int }_t^{t+\Lambda \left(\upsilon ,t\right)}\tilde{\mathfrak{U}}\left(\varkappa \right)\mathsf{\Omega }\left(\varkappa \right)d\varkappa \preccurlyeq \frac{\tilde{\mathfrak{U}}\left(t\right)\widetilde{+}\tilde{\mathfrak{U}}\left(\upsilon \right)}{2}.$$

(52)

Let one attempt to take $\mathit{\chi }\left(s\right)=s$ and $\alpha =1=\mathsf{\Omega }\left(\varkappa \right)$. Then, from (31) and (36) one achieves coming inequality for pre-invex F-IV-F given in [28]:

$$\tilde{\mathfrak{U}}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}\right)\preccurlyeq \left(FR\right){\int }_t^{t+\Lambda \left(\upsilon ,t\right)}\tilde{\mathfrak{U}}\left(\varkappa \right)d\varkappa \preccurlyeq \frac{\tilde{\mathfrak{U}}\left(t\right)\widetilde{+}\tilde{\mathfrak{U}}\left(\upsilon \right)}{2}.$$

(53)

If one attempts to take ${\mathfrak{U}}_{*}\left(\varkappa , \varsigma \right)={\mathfrak{U}}^{*}\left(\varkappa , \varsigma \right)$ and $1=\varsigma$ and $\mathit{\chi }\left(s\right)=s$, then from (31) and (36) one achieves coming inequality given in [27]:

$$\begin{gathered} \mathfrak{U}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}\right)\left[{ℐ}_{t^{+}}^{\alpha } \mathsf{\Omega }\left(\upsilon \right)+{ℐ}_{{\upsilon }^{-}}^{\alpha } \mathsf{\Omega }\left(t\right)\right]\le \left[{ℐ}_{t^{+}}^{\alpha }\mathfrak{U}\mathsf{\Omega }\left(t+\Lambda \left(\upsilon ,t\right)\right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha }\mathfrak{U}\mathsf{\Omega }\left(t\right)\right] \\ \le \frac{\mathfrak{U}\left(t\right)+\mathfrak{U}\left(t+\Lambda \left(\upsilon ,t\right)\right)}{2}\left[{ℐ}_{t^{+}}^{\alpha } \mathsf{\Omega }\left(t+\Lambda \left(\upsilon ,t\right)\right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } \mathsf{\Omega }\left(t\right)\right] \\ \le \frac{\mathfrak{U}\left(t\right)+\mathfrak{U}\left(\upsilon \right)}{2}\left[{ℐ}_{t^{+}}^{\alpha } \mathsf{\Omega }\left(t+\Lambda \left(\upsilon ,t\right)\right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } \mathsf{\Omega }\left(t\right)\right]. \end{gathered}$$

(54)

If one attempt to take ${\mathfrak{U}}_{*}\left(\varkappa , \varsigma \right)={\mathfrak{U}}^{*}\left(\varkappa , \varsigma \right)$ and $\alpha =1=\varsigma$ and $\mathit{\chi }\left(s\right)=s$, then from (31) and (36) one achieves the classical $H$-Fejér inequality, see [38].

If one attempt to take ${\mathfrak{U}}_{*}\left(\varkappa , \varsigma \right)={\mathfrak{U}}^{*}\left(\varkappa , \varsigma \right)$ and $\mathsf{\Omega }\left(\varkappa \right)=\alpha =1=\varsigma$ and $\mathit{\chi }\left(s\right)=s$, then from (31) and (36) one achieves the classical $HH$-inequality.

Example 3.

If we attempt to take F-IV-F $\tilde{\mathfrak{U}}:\left[0, 2\right]\to {\mathbb{F}}_0$ defined by,

$$\tilde{\mathfrak{U}}\left(\varkappa \right)\left(\theta \right)=\{\begin{array}{c}\frac{\theta }{2-\sqrt{\varkappa }}, \theta \in \left[0, 2-\sqrt{\varkappa }\right],\\ \frac{2\left(2-\sqrt{\varkappa }\right)-\theta }{2-\sqrt{\varkappa }}, \theta \in \left(2-\sqrt{\varkappa }, 2\left(2-\sqrt{\varkappa }\right)\right], \\ 0, \mathrm{otherwise}.\end{array}$$

Then, for each $\varsigma \in \left[0, 1\right],$ we have ${\mathfrak{U}}_{\varsigma }\left(\varkappa \right)=\left[\varsigma \left(2-\sqrt{\varkappa }\right),\left(2-\varsigma \right)\left(2-\sqrt{\varkappa }\right)\right]$. Since end point functions ${\mathfrak{U}}_{*}\left(\varkappa ,\varsigma \right),$ ${\mathfrak{U}}^{*}\left(\varkappa ,\varsigma \right)$ are $\mathit{\chi }$-pre-invex functions for each $\varsigma \in \left[0, 1\right]$, then $\tilde{\mathfrak{U}}\left(\varkappa \right)$ is $\mathit{\chi }$-pre-invex F-IV-F. If

$$\mathsf{\Omega }\left(\varkappa \right)=\{\begin{array}{c}\sqrt{\varkappa }, \theta \in \left[0,1\right],\\ \sqrt{2-\varkappa }, \theta \in \left(1, 2\right],\end{array}$$

then $\mathsf{\Omega }\left(2-\varkappa \right)=\mathsf{\Omega }\left(\varkappa \right)\ge 0$, for all $\varkappa \in \left[0, 2\right]$. Since ${\mathfrak{U}}_{*}\left(\varkappa ,\varsigma \right)=\varsigma \left(2-\sqrt{\varkappa }\right)$ and ${\mathfrak{U}}^{*}\left(\varkappa ,\varsigma \right)=\left(2-\varsigma \right)\left(2-\sqrt{\varkappa }\right)$. If $\mathit{\chi }\left(s\right)=s$ and $\alpha =\frac{1}{2}$, then we compute the following:

$$\begin{array}{c}\frac{{\mathfrak{U}}_{*}\left(t, \varsigma \right)+{\mathfrak{U}}_{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)}{2}{\int }_0^1\begin{array}{c}{s}^{\alpha -1}\left[\mathit{\chi }\left(s\right)+\mathit{\chi }\left(1-s\right)\right]\mathsf{\Omega }\left(t+s\Lambda \left(\upsilon ,t\right)\right)\end{array}=\frac{\pi }{\sqrt{2}}\varsigma \left(\frac{4-\sqrt{2}}{2}\right),\\ \frac{{\mathfrak{U}}^{*}\left(t, \varsigma \right)+{\mathfrak{U}}^{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)}{2}{\int }_0^1{s}^{\alpha -1}\left[\mathit{\chi }\left(s\right)+\mathit{\chi }\left(1-s\right)\right]\mathsf{\Omega }\left(t+s\Lambda \left(\upsilon ,t\right)\right)=\frac{\pi }{\sqrt{2}}\left(2-\varsigma \right)\left(\frac{4-\sqrt{2}}{2}\right),\end{array}$$

(55)

$$\begin{array}{c}\frac{{\mathfrak{U}}_{*}\left(t, \varsigma \right)+{\mathfrak{U}}_{*}\left(t+\Lambda \left(t+\Lambda \left(\upsilon ,t\right),t\right), \varsigma \right)}{2}{\int }_0^1{s}^{\alpha -1}\left[\mathit{\chi }\left(s\right)+\mathit{\chi }\left(1-s\right)\right]\mathsf{\Omega }\left(t+s\Lambda \left(\upsilon ,t\right)\right)=\frac{\pi }{\sqrt{2}}\varsigma \left(\frac{4-\sqrt{2}}{2}\right),\\ \frac{{\mathfrak{U}}^{*}\left(t, \varsigma \right)+{\mathfrak{U}}^{*}\left(t+\Lambda \left(\upsilon ,t\right), \varsigma \right)}{2}{\int }_0^1{s}^{\alpha -1}\left[\mathit{\chi }\left(s\right)+\mathit{\chi }\left(1-s\right)\right]\mathsf{\Omega }\left(t+s\Lambda \left(\upsilon ,t\right)\right)=\frac{\pi }{\sqrt{2}}\left(2-\varsigma \right)\left(\frac{4-\sqrt{2}}{2}\right),\end{array}$$

$$\begin{array}{c}\frac{\Gamma \left(\alpha \right)}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }}\left[{ℐ}_{t^{+}}^{\alpha } {\mathfrak{U}}_{*}\mathsf{\Omega }\left(t+\Lambda \left(\upsilon ,t\right)\right)\widetilde{+}{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } {\mathfrak{U}}_{*}\mathsf{\Omega }\left(t\right)\right]=\frac{1}{\sqrt{\pi }}\varsigma \left(2\pi +\frac{4-8\sqrt{2}}{3}\right),\\ \frac{\Gamma \left(\alpha \right)}{{\left(\Lambda \left(\upsilon ,t\right)\right)}^{\alpha }}\left[{ℐ}_{t^{+}}^{\alpha } {\mathfrak{U}}^{*}\mathsf{\Omega }\left(t+\Lambda \left(\upsilon ,t\right)\right)\widetilde{+}{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } {\mathfrak{U}}^{*}\mathsf{\Omega }\left(t\right)\right]=\frac{1}{\sqrt{\pi }}\left(2-\varsigma \right)\left(2\pi +\frac{4-8\sqrt{2}}{3}\right).\end{array}$$

(56)

From (44) and (45), we have $\frac{1}{\sqrt{\pi }}\left[\varsigma \left(2\pi +\frac{4-8\sqrt{2}}{3}\right), \left(2-\varsigma \right)\left(2\pi +\frac{4-8\sqrt{2}}{3}\right)\right]{\le }_I\frac{\pi }{\sqrt{2}}\left[\varsigma \left(\frac{4-\sqrt{2}}{2}\right), \left(2-\varsigma \right)\left(\frac{4-\sqrt{2}}{2}\right)\right],$ for each $\varsigma \in \left[0, 1\right]$.

Hence, (31) is verified.

For (36), we have

$$\begin{array}{c}{ℐ}_{t^{+}}^{\alpha } {\mathfrak{U}}_{*}\mathsf{\Omega }\left(t+\Lambda \left(\upsilon ,t\right)\right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } {\mathfrak{U}}_{*}\mathsf{\Omega }\left(t\right) \\ =\frac{1}{\sqrt{\pi }}\underset{0}{\overset{2}{{\displaystyle \int }}}{\left(2-\varkappa \right)}^{\frac{-1}{2}} \mathsf{\Omega }\left(\varkappa \right)\left(\varsigma \left(2-\sqrt{\varkappa }\right)\right)d\varkappa +\frac{1}{\sqrt{\pi }}\underset{0}{\overset{2}{{\displaystyle \int }}}{\left(\varkappa \right)}^{\frac{-1}{2}} \mathsf{\Omega }\left(\varkappa \right)\left(\varsigma \left(2-\sqrt{\varkappa }\right)\right)d\varkappa \\ =\frac{1}{\sqrt{\pi }}\varsigma \left(\pi +\frac{8-8\sqrt{2}}{3}\right)+\frac{1}{\sqrt{\pi }}\varsigma \left(\pi -\frac{4}{3}\right)=\frac{1}{\sqrt{\pi }}\varsigma \left(2\pi +\frac{4-8\sqrt{2}}{3}\right) \\ {ℐ}_{t^{+}}^{\alpha } {\mathfrak{U}}^{*}\mathsf{\Omega }\left(t+\Lambda \left(\upsilon ,t\right)\right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } {\mathfrak{U}}^{*}\mathsf{\Omega }\left(t\right) \\ =\frac{1}{\sqrt{\pi }}\underset{0}{\overset{2}{{\displaystyle \int }}}{\left(2-\varkappa \right)}^{\frac{-1}{2}} \mathsf{\Omega }\left(\varkappa \right)\left(\left(2-\varsigma \right)\left(2-\sqrt{\varkappa }\right)\right)d\varkappa +\frac{1}{\sqrt{\pi }}\underset{0}{\overset{2}{{\displaystyle \int }}}{\left(\varkappa \right)}^{\frac{-1}{2}} \mathsf{\Omega }\left(\varkappa \right)\left(\left(2-\varsigma \right)\left(2-\sqrt{\varkappa }\right)\right)d\varkappa \\ =\frac{1}{\sqrt{\pi }}\left(2-\varsigma \right)\left(\pi +\frac{8-8\sqrt{2}}{3}\right)+\frac{1}{\sqrt{\pi }}\left(2-\varsigma \right)\left(\pi -\frac{4}{3}\right)=\frac{1}{\sqrt{\pi }}\left(2-\varsigma \right)\left(2\pi +\frac{4-8\sqrt{2}}{3}\right).\end{array}$$

(57)

$$\begin{array}{c}\frac{1}{2\mathit{\chi }\left(\frac{1}{2}\right)}{\mathfrak{U}}_{*}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}, \varsigma \right)\left[{ℐ}_{t^{+}}^{\alpha } \mathsf{\Omega }\left(t+\Lambda \left(\upsilon ,t\right)\right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } \mathsf{\Omega }\left(t\right)\right]=\varsigma \sqrt{\pi },\\ \frac{1}{2\mathit{\chi }\left(\frac{1}{2}\right)}{\mathfrak{U}}^{*}\left(\frac{2t+\Lambda \left(\upsilon ,t\right)}{2}, \varsigma \right)\left[{ℐ}_{t^{+}}^{\alpha } \mathsf{\Omega }\left(t+\Lambda \left(\upsilon ,t\right)\right)+{ℐ}_{t+\Lambda {\left(\upsilon ,t\right)}^{-}}^{\alpha } \mathsf{\Omega }\left(t\right)\right]=\left(2-\varsigma \right)\sqrt{\pi }.\end{array}$$

(58)

From (46) and (47), we have $\sqrt{\pi }\left[\varsigma , \left(2-\varsigma \right)\right]{\begin{array}{c} \begin{array}{c}\le \end{array}\end{array}}_I\frac{1}{\sqrt{\pi }}\left[\varsigma \left(2\pi +\frac{4-8\sqrt{2}}{3}\right), \left(2-\varsigma \right)\left(2\pi +\frac{4-8\sqrt{2}}{3}\right)\right],$ for each $\varsigma \in \left[0, 1\right]$.

4. Conclusions and Future Plan

We use fuzzy-interval-valued-fractional-integral operators with $\mathit{\chi }$-pre-invex F-IV-F to infer various inclusions in the Hermite–Hadamard and Hermite–Hadamard–Fejér type inequalities in this paper. We show the relationships between the examined results and previously published ones to show their generic properties. In addition, three instances are given to demonstrate the accuracy of the results derived in the study. The point we wish to make here is that interval-valued analyses are commonly used in practical mathematics, particularly in the field of optimality analysis (see [10,26,29]). This important field in interval-valued analysis using fractional integral operators merits more investigation. In future, we will try to explore in this concept for $p$-pre-invex F-IV-Fs and generalized $p$-pre-invex F-IV-Fs by using fuzzy-interval-Katugampola-fractional operators.