Riemann–Liouville Fractional Integral Inequalities for Generalized Pre-Invex Functions of Interval-Valued Settings Based upon Pseudo Order Relation

Abstract

The concepts of convex and non-convex functions play a key role in the study of optimization. So, with the help of these ideas, some inequalities can also be established. Moreover, the principles of convexity and symmetry are inextricably linked. In the last two years, convexity and symmetry have emerged as a new field due to considerable association. In this paper, we study a new version of interval-valued functions (I-V·Fs), known as left and right $\chi$-pre-invex interval-valued functions (LR-$\chi$-pre-invex I-V·Fs). For this class of non-convex I-V·Fs, we derive numerous new dynamic inequalities interval Riemann–Liouville fractional integral operators. The applications of these repercussions are taken into account in a unique way. In addition, instructive instances are provided to aid our conclusions. Meanwhile, we’ll discuss a few specific examples that may be extrapolated from our primary findings.

1. Introduction

The Hermite–Hadamard inequality (see [1,2], p. 137) is a well-known inequality in convex function theory, with a geometrical explanation and a wide range of applications. Hermite–Hadamard inequality (H-H inequality) is a development of the concept of convexity, and it logically follows from Jensen’s inequality. In recent years, the H-H inequality for convex functions has sparked a lot of attention, and several refinements and extensions have been investigated; see [3,4,5,6,7,8,9,10,11,12,13,14] and the references therein.

On the other hand, interval analysis is a subset of set-valued analysis and is concerned with the study of intervals in the context of mathematical analysis and topology. It was developed as a means of dealing with interval uncertainty, which is included in many mathematical or computer models of deterministic real-world systems. A historical example of an interval enclosure is Archimedes’ method for calculating the circumference of a circle. In 1966, Moore [15] released the first book on interval analysis. Moore is recognized as being the first usage of intervals in computer mathematics. Following the release of his book, a lot of scientists began to study interval arithmetic’s theory and applications. Because of its universality, interval analysis is currently a useful approach in a range of sectors that are interested in ambiguous data. Moreover, interval analysis has also applications in different fields like in error analysis, computer graphics, error analysis, experimental and computational physics, and many more.

Numerous significant inequalities for I-V·Fs (Hermite–Hadamard, Ostrowski, etc.) have been investigated in recent years. In [16,17], Chalco–Cano et al. constructed Ostrowski type inequalities for I-V·Fs using the Hukuhara derivative for I-V·Fs. Román-Flores et al. developed Minkowski and Beckenbach’s inequality for I-V·Fs in [18]. For more information, see [18,19,20,21,22] and the references therein. Moreover, inequalities can be examined for the more general set-valued mappings for example, Sadowska [23] introduced the H-H inequality general set-valued mappings. Similarly, for generalized inequalities, we refer to the following articles, see [24,25] and the references therein. Recently, Khan et al. extended the interval H-H inequalities in terms of fuzzy interval H-H inequalities using fuzzy Riemannian and fuzzy Riemann–Liouville fractional integral operators such as in [26]. Khan et al. also presented the new class of convex fuzzy mappings known as $\left({\chi }_1, {\chi }_2\right)$-convex fuzzy-interval-valued functions ($\left({\chi }_1, {\chi }_2\right)$-convex F-I-V·F) and obtained the new version of H-H inequalities for $\left({\chi }_1, {\chi }_2\right)$-convex F-I-V·Fs. Moreover, Khan et al. introduced new notions of generalized convex F-I-V·Fs, and derived new fractional H-H type and H-H type inequalities for convex F-I-V·Fs [27,28,29,30,31,32]. For more analysis and applications of F-I-V·Fs, see [33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50] and the references therein.

This study is organized as follows: Section 2 presents preliminary and new concepts and results in interval space, and convex analysis. Section 3 obtains interval H-H inequalities and H-H Fejér inequalities for LR-$\chi$-pre-invex I-V·Fs via interval Riemann–Liouville fractional integral operators. In addition, some interesting examples are also given to verify our results. Section 4 gives conclusions and future plans.

2. Preliminaries

Let ${\mathcal{K}}_C$ stand for the collection of all closed and bounded intervals of $ℝ$. We use ${\mathcal{K}}_C^{+}$ to represent the set of all positive intervals. The collections of all Riemann integrable real valued functions and Riemann integrable I-V-F are denoted by ${ℛ}_{\left[\mu ,\omega \right]}$ and $ℐ{ℛ}_{\left[\mu ,\omega \right]},$ respectively. For more conceptions on I-V·Fs, see [36]. Moreover, we have:

Remark 1.

[35] (i) The relation $“{\le }_p”$ defined on ${\mathcal{K}}_C$ by:

$$\left[{\mathfrak{U}}_{*}, {\mathfrak{U}}^{*}\right]{\le }_p\left[{\mathcal{Z}}_{*}, {\mathcal{Z}}^{*}\right]\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}{\mathfrak{U}}_{*}\le {\mathcal{Z}}_{*}, {\mathfrak{U}}^{*}\le {\mathcal{Z}}^{*},$$

(1)

for all  $\left[{\mathfrak{U}}_{*}, {\mathfrak{U}}^{*}\right], \left[{\mathcal{Z}}_{*}, {\mathcal{Z}}^{*}\right]\in {\mathcal{K}}_C,$ it is a pseudo-order relation. For given  $\left[{\mathfrak{U}}_{*}, {\mathfrak{U}}^{*}\right], \left[{\mathcal{Z}}_{*}, {\mathcal{Z}}^{*}\right]\in {\mathcal{K}}_C,$ we say that  $\left[{\mathfrak{U}}_{*}, {\mathfrak{U}}^{*}\right]{\le }_p\left[{\mathcal{Z}}_{*}, {\mathcal{Z}}^{*}\right]$ if and only if  ${\mathfrak{U}}_{*}\le {\mathcal{Z}}_{*}, {\mathfrak{U}}^{*}\le {\mathcal{Z}}^{*}$ or  ${\mathfrak{U}}_{*}\le {\mathcal{Z}}_{*}, {\mathfrak{U}}^{*}<{\mathcal{Z}}^{*}$. The relation  $\left[{\mathfrak{U}}_{*}, {\mathfrak{U}}^{*}\right]{\le }_p\left[{\mathcal{Z}}_{*}, {\mathcal{Z}}^{*}\right]$ coincident to  $\left[{\mathfrak{U}}_{*}, {\mathfrak{U}}^{*}\right]\le \left[{\mathcal{Z}}_{*}, {\mathcal{Z}}^{*}\right]$ on  ${\mathcal{K}}_C$.

(ii) It can be easily seen that   $“{\le }_p”$ looks like “left and right” on the real line $ℝ,$ so we call $“{\le }_p”$  “left and right” (or “LR” order, in short).

The concept of the Riemann integral for I-V-F first introduced by Moore [15] is defined as follows:

Theorem 1.

[15] If $\mathfrak{S}:\left[\mu ,\omega \right]\subset ℝ\to {\mathcal{K}}_C$ is an I-V·F on $\left[\mu ,\omega \right]$ such that $\mathfrak{S}\left(x\right)=\left[{\mathfrak{S}}_{*}\left(x\right), {\mathfrak{S}}^{*}\left(x\right)\right]$, then $\mathfrak{S}$ is Riemann integrable over $\left[\mu ,\omega \right]$ if and only if, ${\mathfrak{S}}_{*}\left(x\right)$ and ${\mathfrak{S}}^{*}\left(x\right)$ both are Riemann integrable over $\left[\mu ,\omega \right]$ such that:

$$\left(IR\right){{\displaystyle \int }}_{\mu }^{\omega }\mathfrak{S}\left(x\right)dx=\left[\left(R\right){{\displaystyle \int }}_{\mu }^{\omega }{\mathfrak{S}}_{*}\left(x\right)dx, \left(R\right){{\displaystyle \int }}_{\mu }^{\omega }{\mathfrak{S}}^{*}\left(x\right)dx\right]$$

(2)

Lupulescuand Budak et al. [36,37] introduced the following interval Riemann–Liouville fractional integral operators:

Let$\alpha >0$and$L\left(\left[\mu , \omega \right],{\mathcal{K}}_C^{+}\right)$be the collection of all Lebesgue measurableI-V-Fs on$\left[\mu ,\omega \right]$. Then the interval left and right Riemann–Liouville fractional integrals of$\mathfrak{S}\in$ $L\left(\left[\mu , \omega \right],{\mathcal{K}}_C^{+}\right)$with order$\alpha >0$are defined by:

$${ℐ}_{{\mu }^{+}}^{\alpha }\mathfrak{S}\left(x\right)=\frac{1}{\Gamma \left(\alpha \right)}{{\displaystyle \int }}_{\mu }^x{\left(x-\varsigma \right)}^{\alpha -1}\mathfrak{S}\left(\varsigma \right)d\varsigma , \left(x>\mu \right),$$

(3)

and:

$${ℐ}_{{\omega }^{-}}^{\alpha }\mathfrak{S}\left(x\right)=\frac{1}{\Gamma \left(\alpha \right)}{{\displaystyle \int }}_x^{\omega }{\left(\varsigma -x\right)}^{\alpha -1}\mathfrak{S}\left(\varsigma \right)d\varsigma , (x<\omega )$$

(4)

respectively, where  $\Gamma \left(x\right)={{\displaystyle \int }}_0^{\infty }{\varsigma }^{x-1}{e}^{-\varsigma }d\varsigma$ is the Euler gamma function.

Definition 1.

[34] A real-valued function  $\mathfrak{S}:\left[\mu , \omega \right]\to {ℝ}^{+}$ is named as convex function if:

$$\mathfrak{S}\left(\varsigma x+\left(1-\varsigma \right)z\right)\le \varsigma \mathfrak{S}\left(x\right)+\left(1-\varsigma \right)\mathfrak{S}\left(z\right),$$

(5)

for all  $x, z\in \left[\mu , \omega \right], \varsigma \in \left[0, 1\right]$. If (5) is reversed, then $\mathfrak{S}$ is named as concave.

Definition 2.

[40] A real valued function  $\mathfrak{S}:\left[\mu , \omega \right]\to {ℝ}^{+}$ is named as pre-invex function if:

$$\mathfrak{S}\left(x+\left(1-\varsigma \right)\phi \left(z,x\right)\right)\le \varsigma \mathfrak{S}\left(x\right)+\left(1-\varsigma \right)\mathfrak{S}\left(z\right),$$

(6)

for all  $x, z\in \left[\mu , \omega \right], \varsigma \in \left[0, 1\right]$, where $\phi :\left[\mu ,\omega \right]\times \left[\mu ,\omega \right]\to ℝ$. If (6) is reversed, then  $\mathfrak{S}$  is named as pre-incave.

Definition 3.

[35] The I-V·F $\mathfrak{S}:\left[\mu , \omega \right]\to {\mathcal{K}}_C^{+}$ is named as LR-convex I-V·F on$\left[\mu , \omega \right]$ if:

$$\mathfrak{S}\left(\varsigma x+\left(1-\varsigma \right)z\right){\le }_p\varsigma \mathfrak{S}\left(x\right)+\left(1-\varsigma \right)\mathfrak{S}\left(z\right),$$

(7)

for all$x, z\in \left[\mu , \omega \right], \varsigma \in \left[0, 1\right]$. If (7) is reversed, then $\mathfrak{S}$ is named as LR-concave I-V·F on $\left[\mu , \omega \right]$. $\mathfrak{S}$ is affine, if and only if it is both LR-convex and LR-concave I-V·F.

Definition 4.

[41] The I-V·F $\mathfrak{S}:\left[\mu , \omega \right]\to {\mathcal{K}}_C^{+}$ is named as LR-pre-invex I-V·F on invex interval $\left[\mu , \omega \right]$ if:

$$\mathfrak{S}\left(x+\left(1-\varsigma \right)\phi \left(z,x\right)\right){\le }_p\varsigma \mathfrak{S}\left(x\right)+\left(1-\varsigma \right)\mathfrak{S}\left(z\right),$$

(8)

for all$x, z\in \left[\mu , \omega \right], \varsigma \in \left[0, 1\right]$, where $\phi :\left[\mu ,\omega \right]\times \left[\mu ,\omega \right]\to ℝ$. If (8) is reversed then, $\mathfrak{S}$ is named as LR-pre-incave I-V·F on $\left[\mu , \omega \right]$. $\mathfrak{S}$ is a LR-affine if and only if, it is both LR-pre-invex and LR-pre-incave I-V·Fs.

Definition 5.

Let$\chi :\left[0, 1]\subseteq [\mu ,\omega \right]\to {ℝ}^{+}$such that$\chi \not\equiv 0$. Then, I-V·F$\mathfrak{S}:\left[\mu ,\omega \right]\to {\mathcal{K}}_C^{+}$is said to be LR-$\chi$-pre-invex I-V·F on$\left[\mu ,\omega \right]$if:

$$\mathfrak{S}\left(x+\left(1-\varsigma \right)\phi \left(x,z\right)\right){\le }_p\chi \left(\varsigma \right)\mathfrak{S}\left(x\right)+\chi \left(1-\varsigma \right)\mathfrak{S}\left(z\right),$$

(9)

for all$x, z\in \left[\mu ,\omega \right], \varsigma \in \left[0, 1\right]$, where$\phi :\left[\mu ,\omega \right]\times \left[\mu ,\omega \right]\to ℝ$. If  $\mathfrak{S}$ is LR-$\chi$-pre-incave on $\left[\mu ,\omega \right]$, then inequality (9) is reversed.

Remark 2.

If$\chi \left(\varsigma \right)=\varsigma ,$then LR-$\chi$-pre-invex I-V·F becomes LR-pre-invex I-V·F. If$\chi \left(\varsigma \right)\equiv 1,$then LR-$\chi$-pre-invex I-V·F becomes LR-$P$I-V·F, that is:

$$\mathfrak{S}\left(x+\left(1-\varsigma \right)\phi \left(x,z\right)\right){\le }_p\mathfrak{S}\left(x\right)+\mathfrak{S}\left(z\right),\forall x, z\in \left[\mu ,\omega \right], \varsigma \in \left[0, 1\right].$$

(10)

Theorem 2.

Let$\chi :\left[0, 1]\subseteq [\mu ,\omega \right]\to ℝ$be a non-negative real valued function such that$\chi \not\equiv 0$and let$\mathfrak{S}:\left[\mu ,\omega \right]\to {\mathcal{K}}_C^{+}$be a I-V·F such that:

$$\mathfrak{S}\left(z\right)=\left[{\mathfrak{S}}_{*}\left(z\right), {\mathfrak{S}}^{*}\left(z\right)\right],$$

(11)

for all  $z\in \left[\mu ,\omega \right]$. Then,  $\mathfrak{S}\left(z\right)$ is LR-$\chi$-pre-invex I-V·F on  $\left[\mu ,\omega \right],$ if and only if,  ${\mathfrak{S}}_{*}\left(z\right)$ and  ${\mathfrak{S}}^{*}\left(z\right)$  both are  $\chi$-pre-invex.

Proof. 

Assume that, ${\mathfrak{S}}_{*}\left(x\right)$ and ${\mathfrak{S}}^{*}\left(x\right)$ are $\chi$-pre-invex on $\left[\mu ,\omega \right]$. Then from (6), we have:

$${\mathfrak{S}}_{*}\left(x+\left(1-\varsigma \right)\phi \left(x,z\right)\right)\le \chi \left(\varsigma \right){\mathfrak{S}}_{*}\left(x\right)+\chi \left(1-\varsigma \right){\mathfrak{S}}_{*}\left(z\right),\forall x,z\in \left[\mu ,\omega \right], \varsigma \in \left[0, 1\right],$$

and:

$${\mathfrak{S}}^{*}\left(x+\left(1-\varsigma \right)\phi \left(x,z\right)\right)\le \chi \left(\varsigma \right){\mathfrak{S}}^{*}\left(x\right)+\chi \left(1-\varsigma \right){\mathfrak{S}}^{*}\left(z\right),\forall x,z\in \left[\mu ,\omega \right], \varsigma \in \left[0, 1\right].$$

Then by (11), we obtain:

$$\begin{array}{ll}\mathfrak{S}\left(x+\left(1-\varsigma \right)\phi \left(x,z\right)\right)& =\left[{\mathfrak{S}}_{*}\left(x+\left(1-\varsigma \right)\phi \left(x,z\right)\right), {\mathfrak{S}}^{*}\left(x+\left(1-\varsigma \right)\phi \left(x,z\right)\right)\right],\\ & \le \left[\chi \left(\varsigma \right){\mathfrak{S}}_{*}\left(x\right), \chi \left(\varsigma \right){\mathfrak{S}}^{*}\left(x\right)\right]+\left[\chi \left(1-\varsigma \right){\mathfrak{S}}_{*}\left(z\right), \chi \left(1-\varsigma \right){\mathfrak{S}}^{*}\left(z\right)\right],\end{array}$$

that is:

$$\mathfrak{S}\left(x+\left(1-\varsigma \right)\phi \left(x,z\right)\right){\le }_p\chi \left(\varsigma \right)\mathfrak{S}\left(x\right)+\chi \left(1-\varsigma \right)\mathfrak{S}\left(z\right),\forall x,z\in \left[\mu ,\omega \right], \varsigma \in \left[0, 1\right].$$

Hence, $\mathfrak{S}$ is LR-$\chi$-pre-invex IVF on $\left[\mu ,\omega \right]$.

Conversely, let $\mathfrak{S}$ be a LR-$\chi$-pre-invex IVF on $\left[\mu ,\omega \right]$. Then for all $x,z\in \left[\mu ,\omega \right]$ and $\varsigma \in \left[0, 1\right],$ we have:

$$\mathfrak{S}\left(x+\left(1-\varsigma \right)\phi \left(x,z\right)\right){\le }_p\chi \left(\varsigma \right)\mathfrak{S}\left(x\right)+\chi \left(1-\varsigma \right)\mathfrak{S}\left(z\right).$$

Therefore, from (11), we have:

$$\mathfrak{S}\left(x+\left(1-\varsigma \right)\phi \left(x,z\right)\right)=\left[{\mathfrak{S}}_{*}\left(x+\left(1-\varsigma \right)\phi \left(x,z\right)\right), {\mathfrak{S}}^{*}\left(x+\left(1-\varsigma \right)\phi \left(x,z\right)\right)\right].$$

Again, from (11), we obtain:

$$\chi \left(\varsigma \right)\mathfrak{S}\left(x\right)+\chi \left(1-\varsigma \right)\mathfrak{S}\left(x\right)=\left[\chi \left(\varsigma \right){\mathfrak{S}}_{*}\left(x\right), \chi \left(\varsigma \right){\mathfrak{S}}^{*}\left(x\right)\right]+\left[\chi \left(1-\varsigma \right){\mathfrak{S}}_{*}\left(z\right), \chi \left(1-\varsigma \right){\mathfrak{S}}^{*}\left(z\right)\right],$$

for all $x,z\in \left[\mu ,\omega \right]$ and $\varsigma \in \left[0, 1\right]$. Then by LR-$\chi$-pre-invexity of $\mathfrak{S}$, we have for all $x,z\in \left[\mu ,\omega \right]$ and $\varsigma \in \left[0, 1\right]$ such that:

$${\mathfrak{S}}_{*}\left(x+\left(1-\varsigma \right)\phi \left(x,z\right)\right)\le \chi \left(\varsigma \right){\mathfrak{S}}_{*}\left(x\right)+\chi \left(1-\varsigma \right){\mathfrak{S}}_{*}\left(z\right),$$

and:

$${\mathfrak{S}}^{*}\left(x+\left(1-\varsigma \right)\phi \left(x,z\right)\right)\le \chi \left(\varsigma \right){\mathfrak{S}}^{*}\left(x\right)+\chi \left(1-\varsigma \right){\mathfrak{S}}^{*}\left(z\right),$$

hence, the result follows. □

Example 1.

We consider$\chi \left(\varsigma \right)=\varsigma ,$for$\varsigma \in \left[0, 1\right]$and the I-V·F$\mathfrak{S}:\left[0, 4\right]\to {\mathcal{K}}_C^{+}$defined by$\mathfrak{S}\left(z\right)=\left[z,2e^{z^2} \right]$. Since end point functions${\mathfrak{S}}_{*}\left(z\right),$ ${\mathfrak{S}}^{*}\left(z\right)$are$\chi$-pre-invex functions with respect to$\phi \left(x,z\right)=x-z$. Hence$\mathfrak{S}\left(z\right)$is LR-$\chi$-pre-invex I-V·F.

Remark 3.

If  $\chi \left(\varsigma \right)\equiv \varsigma$ and  ${\mathfrak{S}}_{*}\left(z\right)={\mathfrak{S}}^{*}\left(z\right)$, then from (8), we obtain the inequality (6).

If $\chi \left(\varsigma \right)\equiv \varsigma$ and ${\mathfrak{S}}_{*}\left(z\right)={\mathfrak{S}}^{*}\left(z\right)$ and $\phi \left(x,z\right)=x-z$, then from (8), we obtain the inequality (5).

We’ll need to make the following assumption about the function $\phi :\left[\mu ,\omega \right]\times \left[\mu ,\omega \right]\to ℝ$, which will be crucial in the major findings.

Condition C. [40]

$$\phi \left(x,z+\varsigma \phi \left(x,z\right)\right)=\left(1-\varsigma \right)\phi \left(x,z\right),$$

$$\phi \left(z,z+\varsigma \phi \left(x,z\right)\right)=-\varsigma \phi \left(x,z\right).$$

Note that $\forall z, x\in \left[\mu ,\omega \right]$ and $\varsigma \in \left[0, 1\right]$, then from condition C we have

$$\phi \left(z+{\varsigma }_2\phi \left(x,z\right),z+{\varsigma }_1\phi \left(x,z\right)\right)=\left({\varsigma }_2-{\varsigma }_1\right)\phi \left(x,z\right)$$

Clearly for $\varsigma =0$, we have $\phi \left(x,z\right)=0$ if and only if $x=z$, for all $z, x\in \left[\mu ,\omega \right]$. For the application of condition C, see [40,41].

3. Interval Fractional Hermite–Hadamard Inequalities

In this section, we will prove some new H-H type inequalities for LR-$\chi$-pre-invex I-V·Fs via Riemann–Liouville fractional integral operators. In the next results, we will denote $L\left(\left[\mu , \mu +\phi \left(\omega ,\mu \right)\right],{\mathcal{K}}_C^{+}\right)$ as the family of Lebesgue measurable I-V·Fs.

Theorem 3.

Let$\mathfrak{S}:\left[\mu , \mu +\phi \left(\omega ,\mu \right)\right]\to {\mathcal{K}}_C^{+}$be a LR-𝜒-pre-invex I-V·F on$\left[\mu , \mu +\phi \left(\omega ,\mu \right)\right]$such that$\mathfrak{S}\left(z\right)=\left[{\mathfrak{S}}_{*}\left(z\right), {\mathfrak{S}}^{*}\left(z\right)\right]$for all$z\in \left[\mu , \mu +\phi \left(\omega ,\mu \right)\right]$. If$\phi$satisfies condition C and$\mathfrak{S}\in L\left(\left[\mu , \mu +\phi \left(\omega ,\mu \right)\right],{\mathcal{K}}_C^{+}\right)$, then:

$$\begin{array}{ll}\frac{1}{\alpha \chi \left(\frac{1}{2}\right)}\mathfrak{S}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right)& {\le }_p\frac{\Gamma \left(\alpha \right)}{{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }}\left[{ℐ}_{{\mu }^{+}}^{\alpha }\mathfrak{S}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha }\mathfrak{S}\left(\mu \right)\right]\\ & {\le }_p\left(\mathfrak{S}\left(\mu \right)+\mathfrak{S}\left(\mu +\phi \left(\omega ,\mu \right)\right)\right){{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}\left[\chi \left(\varsigma \right)-\chi \left(1-\varsigma \right)\right]d\varsigma \\ & {\le }_p\left(\mathfrak{S}\left(\mu \right)+\mathfrak{S}\left(\omega \right)\right){{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}\left[\chi \left(\varsigma \right)-\chi \left(1-\varsigma \right)\right]d\varsigma \end{array}$$

(12)

If  $\mathfrak{S}\left(z\right)$  is pre-incave I-V·F, then:

$$\begin{array}{ll}\frac{1}{\alpha \chi \left(\frac{1}{2}\right)}\mathfrak{S}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right)& {\ge }_p\frac{\Gamma \left(\alpha \right)}{{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }}\left[{ℐ}_{{\mu }^{+}}^{\alpha }\mathfrak{S}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha }\mathfrak{S}\left(\mu \right)\right]\\ & {\ge }_p\left(\mathfrak{S}\left(\mu \right)+\mathfrak{S}\left(\mu +\phi \left(\omega ,\mu \right)\right)\right){{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}\left[\chi \left(\varsigma \right)-\chi \left(1-\varsigma \right)\right]d\varsigma \\ & {\ge }_p\left(\mathfrak{S}\left(\mu \right)+\mathfrak{S}\left(\omega \right)\right){{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}\left[\chi \left(\varsigma \right)-\chi \left(1-\varsigma \right)\right]d\varsigma \end{array}$$

(13)

Proof. 

Let $\mathfrak{S}:\left[\mu , \mu +\phi \left(\omega ,\mu \right)\right]\to {\mathcal{K}}_C^{+}$ be a LR-𝜒-pre-invex I-V·F. If condition C holds then, by hypothesis, we have:

$$\frac{1}{\chi \left(\frac{1}{2}\right)}\mathfrak{S}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right){\le }_p\mathfrak{S}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)+\mathfrak{S}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right).$$

Therefore, we have:

$$\begin{array}{c}\frac{1}{\chi \left(\frac{1}{2}\right)}{\mathfrak{S}}_{*}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right)\le {\mathfrak{S}}_{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)+{\mathfrak{S}}_{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right), \\ \frac{1}{\chi \left(\frac{1}{2}\right)}{\mathfrak{S}}^{*}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right)\le {\mathfrak{S}}^{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)+{\mathfrak{S}}^{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right).\end{array}$$

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

$$\begin{gathered} \frac{1}{\chi \left(\frac{1}{2}\right)}{{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}{\mathfrak{S}}_{*}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right)d\varsigma \le {{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}{\mathfrak{S}}_{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)d\varsigma +{{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}{\mathfrak{S}}_{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)d\varsigma \\ \frac{1}{\chi \left(\frac{1}{2}\right)}{{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}{\mathfrak{S}}^{*}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right)d\varsigma \le {{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}{\mathfrak{S}}^{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)d\varsigma +{{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}{\mathfrak{S}}^{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)d\varsigma . \end{gathered}$$

Let $x=\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)$ and $x=\mu +\varsigma \phi \left(\omega ,\mu \right)$. Then, we have:

$$\begin{array}{cc}\hfill \frac{1}{\alpha \chi \left(\frac{1}{2}\right)}{\mathfrak{S}}_{*}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right)& \le \frac{1}{{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }} {{\displaystyle \int }}_{\mu }^{\mu +\phi \left(\omega ,\mu \right)}{\left(\mu +\phi \left(\omega ,\mu \right)-x\right)}^{\alpha -1}{\mathfrak{S}}_{*}\left(x\right)dx\hfill \\ & +\frac{1}{{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }}{{\displaystyle \int }}_{\mu }^{\mu +\phi \left(\omega ,\mu \right)}{\left(z-\mu \right)}^{\alpha -1}{\mathfrak{S}}_{*}\left(z\right)dz\hfill \\ \hfill \frac{1}{\alpha \chi \left(\frac{1}{2}\right)}{\mathfrak{S}}^{*}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right)& \le \frac{1}{{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }} {{\displaystyle \int }}_{\mu }^{\mu +\phi \left(\omega ,\mu \right)}{\left(\mu +\phi \left(\omega ,\mu \right)-x\right)}^{\alpha -1}{\mathfrak{S}}^{*}\left(x\right)dx\hfill \\ & +\frac{1}{{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }}{{\displaystyle \int }}_{\mu }^{\mu +\phi \left(\omega ,\mu \right)}{\left(z-\mu \right)}^{\alpha -1}{\mathfrak{S}}^{*}\left(z\right)dz,\hfill \\ & \le \frac{\Gamma \left(\alpha \right)}{{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }}\left[{ℐ}_{{\mu }^{+}}^{\alpha } {\mathfrak{S}}_{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha } {\mathfrak{S}}_{*}\left(\mu \right)\right] \hfill \\ & \le \frac{\Gamma \left(\alpha \right)}{{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }}\left[{ℐ}_{{\mu }^{+}}^{\alpha } {\mathfrak{S}}^{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha } {\mathfrak{S}}^{*}\left(\mu \right)\right],\hfill \end{array}$$

that is:

$$\frac{1}{\alpha \chi \left(\frac{1}{2}\right)}\left[{\mathfrak{S}}_{*}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right), {\mathfrak{S}}^{*}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right)\right]$$

$${\le }_p\frac{\Gamma \left(\alpha \right)}{{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }}\left[{ℐ}_{{\mu }^{+}}^{\alpha } {\mathfrak{S}}_{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha } {\mathfrak{S}}_{*}\left(\mu \right), {ℐ}_{{\mu }^{+}}^{\alpha } {\mathfrak{S}}^{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha } {\mathfrak{S}}^{*}\left(\mu \right)\right]$$

thus,

$$\frac{1}{\alpha \chi \left(\frac{1}{2}\right)}\mathfrak{S}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right){\le }_p\frac{\Gamma \left(\alpha \right)}{{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }}\left[{ℐ}_{{\mu }^{+}}^{\alpha }\mathfrak{S}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha }\mathfrak{S}\left(\mu \right)\right].$$

(14)

In a similar way as above, we have:

$$\begin{gathered} \frac{\Gamma \left(\alpha \right)}{{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }}\left[{ℐ}_{{\mu }^{+}}^{\alpha }\mathfrak{S}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha }\mathfrak{S}\left(\mu \right)\right] \\ {\le }_p\left[\mathfrak{S}\left(\mu \right)+\mathfrak{S}\left(\mu +\phi \left(\omega ,\mu \right)\right)\right]{{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}\left[\chi \left(\varsigma \right)-\chi \left(1-\varsigma \right)\right]d\varsigma . \end{gathered}$$

(15)

Combining (14) and (15), we have:

$$\begin{array}{ll}\frac{1}{\alpha \chi \left(\frac{1}{2}\right)}\mathfrak{S}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right)& {\le }_p\frac{\Gamma \left(\alpha \right)}{{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }}\left[{ℐ}_{{\mu }^{+}}^{\alpha }\mathfrak{S}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha }\mathfrak{S}\left(\mu \right)\right]\\ & {\le }_p\left[\mathfrak{S}\left(\mu \right)+\mathfrak{S}\left(\mu +\phi \left(\omega ,\mu \right)\right)\right]{{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}\left[\chi \left(\varsigma \right)-\chi \left(1-\varsigma \right)\right]d\varsigma \\ & {\le }_p\left[\mathfrak{S}\left(\mu \right)+\mathfrak{S}\left(\omega \right)\right]{{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}\left[\chi \left(\varsigma \right)-\chi \left(1-\varsigma \right)\right]d\varsigma \end{array}$$

hence, the required result. □

Remark 4.

From Theorem 3 we clearly see that:

If$\phi \left(\omega ,\mu \right)=\omega -\mu$, then from Theorem 3, we get the following new result in fractional calculus, see [42].

$$\mathfrak{Q}\left(\frac{\mu +\omega }{2}\right){\le }_p\frac{\Gamma \left(\alpha +1\right)}{2{\left(\omega -\mu \right)}^{\alpha }}\left[{ℐ}_{{\mu }^{+}}^{\alpha }\mathfrak{Q}\left(\omega \right)+{ℐ}_{{\omega }^{-}}^{\alpha }\mathfrak{Q}\left(\mu \right)\right]{\le }_p\frac{\mathfrak{Q}\left(\mu \right)+\mathfrak{Q}\left(\omega \right)}{2}$$

(16)

If  $\alpha =1$, then from Theorem 3, we obtain the following results for LR-𝜒-pre-invex I-V·F, which are also new ones:

$$\begin{gathered} \frac{1}{2\chi \left(\frac{1}{2}\right)}\mathfrak{S}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right){\le }_p\frac{1}{\phi \left(\omega ,\mu \right)} \left(FR\right){{\displaystyle \int }}_{\mu }^{\mu +\phi \left(\omega ,\mu \right)}\mathfrak{S}\left(z\right)dz \\ {\le }_p\left[\mathfrak{S}\left(\mu \right)+\mathfrak{S}\left(\mu +\phi \left(\omega ,\mu \right)\right)\right]{{\displaystyle \int }}_0^1\chi \left(\varsigma \right)d\varsigma . \end{gathered}$$

(17)

If$\chi \left(\varsigma \right)=\varsigma$, then Theorem 3 reduces to the result for LR-pre-invex I-V·F, see [41]:

$$\mathfrak{S}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right){\le }_p\frac{\Gamma \left(\alpha +1\right)}{2{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }}\left[{ℐ}_{{\mu }^{+}}^{\alpha }\mathfrak{S}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha }\mathfrak{S}\left(\mu \right)\right]{\le }_p\frac{\mathfrak{S}\left(\mu \right)+\mathfrak{S}\left(\mu +\phi \left(\omega ,\mu \right)\right)}{2}$$

(18)

Let$\alpha =1$and$\chi \left(\varsigma \right)=\varsigma$. Then, Theorem 3 reduces to the result for LR-pre-invex-I-V·F, see [41]:

$$\mathfrak{S}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right){\le }_p\frac{1}{\phi \left(\omega ,\mu \right)} \left(FR\right){{\displaystyle \int }}_{\mu }^{\mu +\phi \left(\omega ,\mu \right)}\mathfrak{S}\left(z\right)dz{\le }_p\frac{\mathfrak{S}\left(\mu \right)+\mathfrak{S}\left(\omega \right)}{2}$$

(19)

Example 2.

$\alpha =\frac{1}{2}$,$\chi \left(\varsigma \right)=\varsigma$, for all$\varsigma \in \left[0, 1\right]$and the I-V·F$\mathfrak{S}:\left[\mu , \mu +\phi \left(\omega ,\mu \right)\right]=\left[2, 2+\phi \left(3,2\right)\right]\to {\mathcal{K}}_C^{+},$defined by$\mathfrak{S}\left(z\right)=\left[1,2\right]\left(2-z^{\frac{1}{2}}\right)$. Since left and right end-point functions${\mathfrak{S}}_{*}\left(z\right)=\left(2-z^{\frac{1}{2}}\right),$ ${\mathfrak{S}}^{*}\left(z\right)=2\left(2-z^{\frac{1}{2}}\right)$, are LR-$\chi$-pre-invex functions, then$\mathfrak{S}\left(z\right)$is LR-$\chi$-pre-invex I-V·F. We clearly see that$\mathfrak{S}\in L\left(\left[\mu , \mu +\phi \left(\omega ,\mu \right)\right],{\mathcal{K}}_C^{+}\right)$and:

$$\frac{1}{\alpha \chi \left(\frac{1}{2}\right)}{\mathfrak{S}}_{*}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right)={\mathfrak{S}}_{*}\left(\frac{5}{2}\right)=\frac{4-\sqrt{10}}{8}$$

$$\frac{1}{\alpha \chi \left(\frac{1}{2}\right)}{\mathfrak{S}}^{*}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right)={\mathfrak{S}}^{*}\left(\frac{5}{2}\right)=\frac{4-\sqrt{10}}{4},$$

$$\frac{{\mathfrak{S}}_{*}\left(\mu \right)+{\mathfrak{S}}_{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)}{2}{{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}\left[\chi \left(\varsigma \right)-\chi \left(1-\varsigma \right)\right]d\varsigma =\left(4-\sqrt{2}-\sqrt{3}\right)$$

$$\frac{{\mathfrak{S}}^{*}\left(\mu \right)+{\mathfrak{S}}^{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)}{2}{{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}\left[\chi \left(\varsigma \right)-\chi \left(1-\varsigma \right)\right]d\varsigma =2\left(4-\sqrt{2}-\sqrt{3}\right).$$

Note that:

$$\begin{gathered} \frac{\Gamma \left(\alpha \right)}{{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }}\left[{ℐ}_{{\mu }^{+}}^{\alpha } {\mathfrak{S}}_{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha } {\mathfrak{S}}_{*}\left(\mu \right)\right] \\ =\frac{\Gamma \left(\frac{1}{2}\right)}{2}\frac{1}{\sqrt{\pi }}{{\displaystyle \int }}_2^3{\left(3-z\right)}^{\frac{-1}{2}}·\left(2-z^{\frac{1}{2}}\right)dz \\ +\frac{\Gamma \left(\frac{1}{2}\right)}{2}\frac{1}{\sqrt{\pi }}{{\displaystyle \int }}_2^3{\left(z-2\right)}^{\frac{-1}{2}}·\left(2-z^{\frac{1}{2}}\right)dz \\ =\frac{1}{2}\left[\frac{7393}{10,000}+\frac{9501}{10,000}\right] \\ =\frac{8447}{20,000}. \end{gathered}$$

$$\begin{gathered} \frac{\Gamma \left(\alpha \right)}{{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }}\left[{ℐ}_{{\mu }^{+}}^{\alpha } {\mathfrak{S}}^{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha } {\mathfrak{S}}^{*}\left(\mu \right)\right] \\ \frac{\Gamma \left(\alpha \right)}{{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }}\left[{ℐ}_{{\mu }^{+}}^{\alpha } {\mathfrak{S}}^{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha } {\mathfrak{S}}^{*}\left(\mu \right)\right] \\ +\frac{\Gamma \left(\frac{1}{2}\right)}{2}\frac{1}{\sqrt{\pi }}{{\displaystyle \int }}_2^3{\left(z-2\right)}^{\frac{-1}{2}} · 2\left(2-z^{\frac{1}{2}}\right)dz \\ =\left[\frac{7393}{10,000}+\frac{9501}{10,000}\right] \\ =\frac{8447}{10,000}. \end{gathered}$$

Therefore:

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

and Theorem 3 is verified.

From Theorems 4 and 5, we obtain some interval fractional integral inequalities related to interval fractional H-H inequalities.

Theorem 4.

Let$\mathfrak{S},\mathscr{H} :\left[\mu , \mu +\phi \left(\omega ,\mu \right)\right]\to {\mathcal{K}}_C^{+}$be LR-${\chi }_1$-pre-invex and LR-${\chi }_2$-pre-invex I-V·Fs on$\left[\mu , \mu +\phi \left(\omega ,\mu \right)\right]$, respectively, such that$\mathfrak{S}\left(z\right)=\left[{\mathfrak{S}}_{*}\left(z\right), {\mathfrak{S}}^{*}\left(z\right)\right]$and$\mathscr{H}\left(z\right)=\left[{\mathscr{H}}_{*}\left(z\right), {\mathscr{H}}^{*}\left(z\right)\right]$for all$z\in \left[\mu , \mu +\phi \left(\omega ,\mu \right)\right]$. If$\phi$satisfies condition C and$\mathfrak{S}\times \mathscr{H}\in L\left(\left[\mu , \mu +\phi \left(\omega ,\mu \right)\right],{\mathcal{K}}_C^{+}\right)$, then:

$$\begin{gathered} \frac{\Gamma \left(\alpha \right)}{{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }}\left[{ℐ}_{{\mu }^{+}}^{\alpha }\mathfrak{S}\left(\mu +\phi \left(\omega ,\mu \right)\right)\times \mathscr{H}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha }\mathfrak{S}\left(\mu \right)\times \mathscr{H}\left(\mu \right)\right] \\ {\le }_p\xi \left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right){{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}\left[{\chi }_1\left(\varsigma \right){\chi }_2\left(\varsigma \right)+{\chi }_1\left(1-\varsigma \right){\chi }_2\left(1-\varsigma \right)\right]d\varsigma \\ +\partial \left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right){{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}\left[{\chi }_1\left(\varsigma \right){\chi }_2\left(1-\varsigma \right)+{\chi }_1\left(1-\varsigma \right){\chi }_2\left(\varsigma \right)\right]d\varsigma . \end{gathered}$$

(20)

where$\xi \left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right)=\mathfrak{S}\left(\mu \right)\times \mathscr{H}\left(\mu \right)+\mathfrak{S}\left(\mu +\phi \left(\omega ,\mu \right)\right)\times \mathscr{H}\left(\mu +\phi \left(\omega ,\mu \right)\right),$ $\partial \left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right)=\mathfrak{S}\left(\mu \right)\times \mathscr{H}\left(\mu +\phi \left(\omega ,\mu \right)\right)+\mathfrak{S}\left(\mu +\phi \left(\omega ,\mu \right)\right)\times \mathscr{H}\left(\mu \right),$and$\xi \left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right)=\left[{\xi }_{*}\left(\left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right)\right), {\xi }^{*}\left(\left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right)\right)\right]$and$\partial \left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right)=\left[{\partial }_{*}\left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right), {\partial }^{*}\left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right)\right]$.

Proof. 

Since $\mathfrak{S},\mathscr{H}$ both are LR-${\chi }_1$-pre-invex and LR-${\chi }_2$-pre-invex I-V·F then, we have:

$$\begin{array}{r}{\mathfrak{S}}_{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)={\mathfrak{S}}_{*}\left(\mu +\phi \left(\omega ,\mu \right)+\varsigma \phi \left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right)\right)\\ \le {\chi }_1\left(\varsigma \right){\mathfrak{S}}_{*}\left(\mu \right)+{\chi }_1\left(1-\varsigma \right){\mathfrak{S}}_{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)\\ {\mathfrak{S}}^{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)={\mathfrak{S}}^{*}\left(\mu +\phi \left(\omega ,\mu \right)+\varsigma \phi \left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right)\right)\\ \le {\chi }_1\left(\varsigma \right){\mathfrak{S}}^{*}\left(\mu \right)+{\chi }_1\left(1-\varsigma \right){\mathfrak{S}}^{*}\left(\mu +\phi \left(\omega ,\mu \right)\right).\end{array}$$

and:

$$\begin{array}{ll}{\mathscr{H}}_{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)=& {\mathscr{H}}_{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\\ & \le {\chi }_2\left(\varsigma \right){\mathscr{H}}_{*}\left(\mu \right)+{\chi }_2\left(1-\varsigma \right){\mathscr{H}}_{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)\\ {\mathscr{H}}^{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)=& {\mathscr{H}}^{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\\ & \le {\chi }_2\left(\varsigma \right){\mathscr{H}}^{*}\left(\mu \right)+{\chi }_2\left(1-\varsigma \right){\mathscr{H}}^{*}\left(\mu +\phi \left(\omega ,\mu \right)\right).\end{array}$$

From the definition of LR-𝜒-pre-invex I-V·F, we have:

$$\begin{array}{l}{\mathfrak{S}}_{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}_{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\\ \hspace{1em}\le {\chi }_1\left(\varsigma \right){\chi }_2\left(\varsigma \right){\mathfrak{S}}_{*}\left(\mu \right)\times {\mathscr{H}}_{*}\left(\mu \right)+{\chi }_1\left(1-\varsigma \right){\chi }_2\left(1-\varsigma \right){\mathfrak{S}}_{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}_{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)\\ \hspace{1em}+{\chi }_1\left(\varsigma \right){\chi }_2\left(1-\varsigma \right){\mathfrak{S}}_{*}\left(\mu \right)\times {\mathscr{H}}_{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{\chi }_1\left(1-\varsigma \right){\chi }_2\left(\varsigma \right){\mathfrak{S}}_{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}_{*}\left(\mu \right)\\ {\mathfrak{S}}^{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}^{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\\ \hspace{1em}\le {\chi }_1\left(\varsigma \right){\chi }_2\left(\varsigma \right){\mathfrak{S}}^{*}\left(\mu \right)\times {\mathscr{H}}^{*}\left(\mu \right)+{\chi }_1\left(1-\varsigma \right){\chi }_2\left(1-\varsigma \right){\mathfrak{S}}^{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}^{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)\\ \hspace{1em}+{\chi }_1\left(\varsigma \right){\chi }_2\left(1-\varsigma \right){\mathfrak{S}}^{*}\left(\mu \right)\times {\mathscr{H}}^{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{\chi }_1\left(1-\varsigma \right){\chi }_2\left(\varsigma \right){\mathfrak{S}}^{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}^{*}\left(\mu \right).\end{array}$$

(21)

Analogously, we have:

$$\begin{array}{l}{\mathfrak{S}}_{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right){\mathscr{H}}_{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\\ \hspace{1em}\le {\chi }_1\left(1-\varsigma \right){\chi }_2\left(1-\varsigma \right){\mathfrak{S}}_{*}\left(\mu \right)\times {\mathscr{H}}_{*}\left(\mu \right)+{\chi }_1\left(\varsigma \right){\chi }_2\left(\varsigma \right){\mathfrak{S}}_{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}_{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)\\ \hspace{1em}+{\chi }_1\left(1-\varsigma \right){\chi }_2\left(\varsigma \right){\mathfrak{S}}_{*}\left(\mu \right)\times {\mathscr{H}}_{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{\chi }_1\left(\varsigma \right){\chi }_2\left(1-\varsigma \right){\mathfrak{S}}_{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}_{*}\left(\mu \right)\\ {\mathfrak{S}}^{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}^{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\\ \hspace{1em}\le {\chi }_1\left(1-\varsigma \right){\chi }_2\left(1-\varsigma \right){\mathfrak{S}}^{*}\left(\mu \right)\times {\mathscr{H}}^{*}\left(\mu \right)+{\chi }_1\left(\varsigma \right){\chi }_2\left(\varsigma \right){\mathfrak{S}}^{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}^{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)\\ \hspace{1em}+{\chi }_1\left(1-\varsigma \right){\chi }_2\left(\varsigma \right){\mathfrak{S}}^{*}\left(\mu \right)\times {\mathscr{H}}^{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{\chi }_1\left(\varsigma \right){\chi }_2\left(1-\varsigma \right){\mathfrak{S}}^{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}^{*}\left(\mu \right).\end{array}$$

(22)

Adding (21) and (22), we have:

$$\begin{array}{l}{\mathfrak{S}}_{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}_{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\\ +{\mathfrak{S}}_{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}_{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\\ \hspace{1em}\le \left[\begin{array}{c}{\chi }_1\left(\varsigma \right){\chi }_2\left(\varsigma \right)\\ +{\chi }_1\left(1-\varsigma \right){\chi }_2\left(1-\varsigma \right)\end{array}\right]\left[{\mathfrak{S}}_{*}\left(\mu \right)\times {\mathscr{H}}_{*}\left(\mu \right)+{\mathfrak{S}}_{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}_{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)\right]\\ +\left[\begin{array}{c}{\chi }_1\left(\varsigma \right){\chi }_2\left(1-\varsigma \right)\\ +{\chi }_1\left(1-\varsigma \right){\chi }_2\left(\varsigma \right)\end{array}\right]\left[{\mathfrak{S}}_{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}_{*}\left(\mu \right)+{\mathfrak{S}}_{*}\left(\mu \right)\times {\mathscr{H}}_{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)\right]\\ {\mathfrak{S}}^{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}^{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\\ +{\mathfrak{S}}^{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}^{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\\ \hspace{1em}\le \left[\begin{array}{c}{\chi }_1\left(\varsigma \right){\chi }_2\left(\varsigma \right)\\ +{\chi }_1\left(1-\varsigma \right){\chi }_2\left(1-\varsigma \right)\end{array}\right]\left[{\mathfrak{S}}^{*}\left(\mu \right)\times {\mathscr{H}}^{*}\left(\mu \right)+{\mathfrak{S}}^{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}^{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)\right]\\ +\left[\begin{array}{c}{\chi }_1\left(\varsigma \right){\chi }_2\left(1-\varsigma \right)\\ +{\chi }_1\left(1-\varsigma \right){\chi }_2\left(\varsigma \right)\end{array}\right]\left[{\mathfrak{S}}^{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}^{*}\left(\mu \right)+{\mathfrak{S}}^{*}\left(\mu \right)\times {\mathscr{H}}^{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)\right].\end{array}$$

(23)

Taking multiplication of (23) with ${\varsigma }^{\alpha -1}$ and integrating the obtained result with respect to $\varsigma$ over (0,1), we have:

$$\begin{array}{l}{{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}{\mathfrak{S}}_{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}_{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\\ +{\varsigma }^{\alpha -1}{\mathfrak{S}}_{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}_{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)d\varsigma \\ \hspace{1em}\le {\xi }_{*}\left(\left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right)\right){{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}\left[{\chi }_1\left(\varsigma \right){\chi }_2\left(\varsigma \right)+{\chi }_1\left(1-\varsigma \right){\chi }_2\left(1-\varsigma \right)\right]d\varsigma \\ +{\partial }_{*}\left(\left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right)\right){{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}\left[{\chi }_1\left(\varsigma \right){\chi }_2\left(1-\varsigma \right)+{\chi }_1\left(1-\varsigma \right){\chi }_2\left(\varsigma \right)\right]d\varsigma \\ {{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}{\mathfrak{S}}^{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}^{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\\ +{\varsigma }^{\alpha -1}{\mathfrak{S}}^{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}^{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)d\varsigma \\ \hspace{1em}\le {\xi }^{*}\left(\left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right)\right){{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}\left[{\chi }_1\left(\varsigma \right){\chi }_2\left(\varsigma \right)+{\chi }_1\left(1-\varsigma \right){\chi }_2\left(1-\varsigma \right)\right]d\varsigma \\ \hspace{1em}+{\partial }^{*}\left(\left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right)\right){{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}\left[{\chi }_1\left(\varsigma \right){\chi }_2\left(1-\varsigma \right)+{\chi }_1\left(1-\varsigma \right){\chi }_2\left(\varsigma \right)\right]d\varsigma .\end{array}$$

It follows that:

$$\begin{array}{l}\frac{\Gamma \left(\alpha \right)}{{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }}\left[{ℐ}_{{\mu }^{+}}^{\alpha } {\mathfrak{S}}_{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}_{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha } {\mathfrak{S}}_{*}\left(\mu \right)\times {\mathscr{H}}_{*}\left(\mu \right)\right]\\ \hspace{1em}\le {\xi }_{*}\left(\left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right)\right){{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}\left[{\chi }_1\left(\varsigma \right){\chi }_2\left(\varsigma \right)+{\chi }_1\left(1-\varsigma \right){\chi }_2\left(1-\varsigma \right)\right]d\varsigma \\ \hspace{1em}+{\partial }_{*}\left(\left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right)\right){{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}\left[{\chi }_1\left(\varsigma \right){\chi }_2\left(1-\varsigma \right)+{\chi }_1\left(1-\varsigma \right){\chi }_2\left(\varsigma \right)\right]d\varsigma \\ \frac{\Gamma \left(\alpha \right)}{{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }}\left[{ℐ}_{{\mu }^{+}}^{\alpha } {\mathfrak{S}}^{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}^{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha } {\mathfrak{S}}^{*}\left(\mu \right)\times {\mathscr{H}}^{*}\left(\mu \right)\right]\\ \hspace{1em}\le {\xi }^{*}\left(\left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right)\right){{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}\left[{\chi }_1\left(\varsigma \right){\chi }_2\left(\varsigma \right)+{\chi }_1\left(1-\varsigma \right){\chi }_2\left(1-\varsigma \right)\right]d\varsigma \\ \hspace{1em}+{\partial }^{*}\left(\left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right)\right){{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}\left[{\chi }_1\left(\varsigma \right){\chi }_2\left(1-\varsigma \right)+{\chi }_1\left(1-\varsigma \right){\chi }_2\left(\varsigma \right)\right]d\varsigma .\end{array}$$

It results that:

$$\begin{gathered} \frac{\Gamma \left(\alpha \right)}{{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }}[{ℐ}_{{\mu }^{+}}^{\alpha } {\mathfrak{S}}_{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}_{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha } {\mathfrak{S}}_{*}\left(\mu \right)\times {\mathscr{H}}_{*}\left(\mu \right), \\ {ℐ}_{{\mu }^{+}}^{\alpha } {\mathfrak{S}}^{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}^{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha } {\mathfrak{S}}^{*}\left(\mu \right)\times {\mathscr{H}}^{*}\left(\mu \right)] \\ {\le }_p\left[{\xi }_{*}\left(\left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right)\right), {\xi }^{*}\left(\left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right)\right)\right]{{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}\left[{\chi }_1\left(\varsigma \right){\chi }_2\left(\varsigma \right)+{\chi }_1\left(1-\varsigma \right){\chi }_2\left(1-\varsigma \right)\right]d\varsigma \\ +\left[{\partial }_{*}\left(\left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right)\right), {\partial }^{*}\left(\left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right)\right)\right]{{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}\left[{\chi }_1\left(\varsigma \right){\chi }_2\left(1-\varsigma \right)+{\chi }_1\left(1-\varsigma \right){\chi }_2\left(\varsigma \right)\right]d\varsigma \end{gathered}$$

that is:

$$\begin{array}{l}\frac{\Gamma \left(\alpha \right)}{{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }}\left[{ℐ}_{{\mu }^{+}}^{\alpha }\mathfrak{S}\left(\mu +\phi \left(\omega ,\mu \right)\right)\times \mathscr{H}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha }\mathfrak{S}\left(\mu \right)\times \mathscr{H}\left(\mu \right)\right]\\ \hspace{1em}{\le }_p\xi \left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right){{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}\left[{\chi }_1\left(\varsigma \right){\chi }_2\left(\varsigma \right)+{\chi }_1\left(1-\varsigma \right){\chi }_2\left(1-\varsigma \right)\right]d\varsigma \\ \hspace{1em}+\partial \left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right){{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}\left[{\chi }_1\left(\varsigma \right){\chi }_2\left(1-\varsigma \right)+{\chi }_1\left(1-\varsigma \right){\chi }_2\left(\varsigma \right)\right]d\varsigma \end{array}$$

and the theorem has been established. □

Theorem 5.

Let$\mathfrak{S}, \mathscr{H} :\left[\mu , \mu +\phi \left(\omega ,\mu \right)\right]\to {\mathcal{K}}_C^{+}$be two LR-${\chi }_1$-pre-invex and LR-${\chi }_2$-pre-invex I-V·Fs, respectively, such that$\mathfrak{S}\left(z\right)=\left[{\mathfrak{S}}_{*}\left(z\right), {\mathfrak{S}}^{*}\left(z\right)\right]$and$\mathscr{H}\left(z\right)=\left[{\mathscr{H}}_{*}\left(z\right), {\mathscr{H}}^{*}\left(z\right)\right]$for all$z\in \left[\mu , \mu +\phi \left(\omega ,\mu \right)\right]$. If$\phi$satisfies condition C and$\mathfrak{S}\times \mathscr{H}\in L\left(\left[\mu , \mu +\phi \left(\omega ,\mu \right)\right],{\mathcal{K}}_C^{+}\right)$, then:

$$\begin{gathered} \frac{1}{\alpha {\chi }_1\left(\frac{1}{2}\right){\chi }_2\left(\frac{1}{2}\right)}\mathfrak{S}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right)\times \mathscr{H}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right) \\ {\le }_p\frac{\Gamma \left(\alpha \right)}{{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }}\left[{ℐ}_{{\mu }^{+}}^{\alpha }\mathfrak{S}\left(\mu +\phi \left(\omega ,\mu \right)\right)\times \mathscr{H}\left(\omega \right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha }\mathfrak{S}\left(\mu \right)\times \mathscr{H}\left(\mu \right)\right] \\ +\partial \left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right){{\displaystyle \int }}_0^1\left[{\varsigma }^{\alpha -1}+{\left(1-\varsigma \right)}^{\alpha -1}\right]{\chi }_1\left(\varsigma \right){\chi }_2\left(1-\varsigma \right)d\varsigma \\ +\xi \left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right){{\displaystyle \int }}_0^1\left[{\varsigma }^{\alpha -1}+{\left(1-\varsigma \right)}^{\alpha -1}\right]{\chi }_1\left(1-\varsigma \right){\chi }_2\left(1-\varsigma \right)d\varsigma , \end{gathered}$$

(24)

where$\xi \left(u,u+\phi \left(\nu ,u\right)\right)=\mathfrak{S}\left(u\right)\times \mathscr{H}\left(u\right)+\mathfrak{S}\left(\mu +\phi \left(\omega ,\mu \right)\right)\times \mathscr{H}\left(\mu +\phi \left(\omega ,\mu \right)\right),$ $\partial \left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right)=\mathfrak{S}\left(\mu \right)\times \mathscr{H}\left(\mu +\phi \left(\omega ,\mu \right)\right)+\mathfrak{S}\left(\mu +\phi \left(\omega ,\mu \right)\right)\times \mathscr{H}\left(\mu \right),$and$\xi \left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right)=\left[{\xi }_{*}\left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right), {\xi }^{*}\left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right)\right]$and$\partial \left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right)=\left[{\partial }_{*}\left(\left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right)\right), {\partial }^{*}\left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right)\right]$.

Proof. 

Consider $\mathfrak{S},\mathscr{H} :\left[\mu , \mu +\phi \left(\omega ,\mu \right)\right]\to {\mathcal{K}}_C^{+}$. are LR-${\chi }_1$-pre-invex and LR-${\chi }_2$-pre-invex I-V·Fs. Then, by hypothesis, we have:

$$\begin{gathered} \begin{array}{c}{\mathfrak{S}}_{*}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2} \right)\times {\mathscr{H}}_{*}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right) \\ {\mathfrak{S}}^{*}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right)\times {\mathscr{H}}^{*}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right)\end{array} \\ \begin{array}{c}\le {\chi }_1\left(\frac{1}{2}\right){\chi }_2\left(\frac{1}{2}\right)\left[\begin{array}{c}{\mathfrak{S}}_{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}_{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\\ +{\mathfrak{S}}_{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}_{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\end{array}\right]\\ + {\chi }_1\left(\frac{1}{2}\right){\chi }_2\left(\frac{1}{2}\right)\left[\begin{array}{c}{\mathfrak{S}}_{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}_{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\\ +{\mathfrak{S}}_{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}_{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\end{array}\right]\\ \le {\chi }_1\left(\frac{1}{2}\right){\chi }_2\left(\frac{1}{2}\right)\left[\begin{array}{c}{\mathfrak{S}}^{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}^{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\\ +{\mathfrak{S}}^{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}^{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\end{array}\right]\\ + {\chi }_1\left(\frac{1}{2}\right){\chi }_2\left(\frac{1}{2}\right)\left[\begin{array}{c}{\mathfrak{S}}^{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}^{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\\ +{\mathfrak{S}}^{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}^{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\end{array}\right],\end{array} \\ \begin{array}{l}\le {\chi }_1\left(\frac{1}{2}\right){\chi }_2\left(\frac{1}{2}\right)\left[\begin{array}{c}{\mathfrak{S}}_{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}_{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\\ +{\mathfrak{S}}_{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}_{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\end{array}\right]\\ + {\chi }_1\left(\frac{1}{2}\right){\chi }_2\left(\frac{1}{2}\right)\left[\begin{array}{c}\left({\chi }_1\left(\varsigma \right){\mathfrak{S}}_{*}\left(\mu \right)+{\chi }_1\left(1-\varsigma \right){\mathfrak{S}}_{*}\left(\mu +\phi \left(\omega ,\mu \right),\right)\right)\\ \times \left({\chi }_2\left(1-\varsigma \right){\mathscr{H}}_{*}\left(\mu \right)+{\chi }_2\left(\varsigma \right){\mathscr{H}}_{*}\left(\mu +\phi \left(\omega ,\mu \right),\right)\right)\\ +\left({\chi }_1\left(1-\varsigma \right){\mathfrak{S}}_{*}\left(\mu \right)+{\chi }_1\left(\varsigma \right){\mathfrak{S}}_{*}\left(\mu +\phi \left(\omega ,\mu \right),\right)\right)\\ \times \left({\chi }_2\left(\varsigma \right){\mathscr{H}}_{*}\left(\mu \right)+{\chi }_2\left(1-\varsigma \right){\mathscr{H}}_{*}\left(\mu +\phi \left(\omega ,\mu \right),\right)\right)\end{array}\right]\\ \le {\chi }_1\left(\frac{1}{2}\right){\chi }_2\left(\frac{1}{2}\right)\left[\begin{array}{c}{\mathfrak{S}}^{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}^{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\\ +{\mathfrak{S}}^{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}^{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\end{array}\right]\\ + {\chi }_1\left(\frac{1}{2}\right){\chi }_2\left(\frac{1}{2}\right)\left[\begin{array}{c}\left({\chi }_1\left(\varsigma \right){\mathfrak{S}}^{*}\left(\mu \right)+{\chi }_1\left(1-\varsigma \right){\mathfrak{S}}^{*}\left(\mu +\phi \left(\omega ,\mu \right),\right)\right)\\ \times \left({\chi }_2\left(1-\varsigma \right){\mathscr{H}}^{*}\left(\mu \right)+{\chi }_2\left(\varsigma \right){\mathscr{H}}^{*}\left(\mu +\phi \left(\omega ,\mu \right),\right)\right)\\ +\left({\chi }_1\left(1-\varsigma \right){\mathfrak{S}}^{*}\left(\mu \right)+{\chi }_1\left(\varsigma \right){\mathfrak{S}}^{*}\left(\mu +\phi \left(\omega ,\mu \right),\right)\right)\\ \times \left({\chi }_2\left(\varsigma \right){\mathscr{H}}^{*}\left(\mu \right)+{\chi }_2\left(1-\varsigma \right){\mathscr{H}}^{*}\left(\mu +\phi \left(\omega ,\mu \right),\right)\right)\end{array}\right],\end{array} \\ \begin{array}{l}={\chi }_1\left(\frac{1}{2}\right){\chi }_2\left(\frac{1}{2}\right)\left[\begin{array}{c}{\mathfrak{S}}_{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}_{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\\ +{\mathfrak{S}}_{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}_{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\end{array}\right]\\ + {\chi }_1\left(\frac{1}{2}\right){\chi }_2\left(\frac{1}{2}\right)\left[\begin{array}{c}\left\{{\chi }_1\left(\varsigma \right){\chi }_2\left(1-\varsigma \right)+{\chi }_1\left(1-\varsigma \right){\chi }_2\left(\varsigma \right)\right\}{\partial }_{*}\left(\left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right)\right)\\ +\left\{{\chi }_1\left(\varsigma \right){\chi }_2\left(\varsigma \right)+{\chi }_1\left(1-\varsigma \right){\chi }_2\left(1-\varsigma \right)\right\}{\xi }_{*}\left(\left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right)\right)\end{array}\right]\\ ={\chi }_1\left(\frac{1}{2}\right){\chi }_2\left(\frac{1}{2}\right)\left[\begin{array}{c}{\mathfrak{S}}^{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}^{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\\ +{\mathfrak{S}}^{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}^{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\end{array}\right]\\ + {\chi }_1\left(\frac{1}{2}\right){\chi }_2\left(\frac{1}{2}\right)\left[\begin{array}{c}\left\{{\chi }_1\left(\varsigma \right){\chi }_2\left(1-\varsigma \right)+{\chi }_1\left(1-\varsigma \right){\chi }_2\left(\varsigma \right)\right\}{\partial }^{*}\left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right)\\ +\left\{{\chi }_1\left(\varsigma \right){\chi }_2\left(\varsigma \right)+{\chi }_1\left(1-\varsigma \right){\chi }_2\left(1-\varsigma \right)\right\}{\xi }^{*}\left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right)\end{array}\right].\end{array} \end{gathered}$$

(25)

Taking multiplication of (25) with ${\varsigma }^{\alpha -1}$ and integrating over $\left(0, 1\right),$ we get:

$$\begin{array}{l}\frac{1}{\alpha {\chi }_1\left(\frac{1}{2}\right){\chi }_2\left(\frac{1}{2}\right)}{\mathfrak{S}}_{*}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right)\times {\mathscr{H}}_{*}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\le \frac{\Gamma \left(\alpha \right)}{{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }}\left[{ℐ}_{{\mu }^{+}}^{\alpha } {\mathfrak{S}}_{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}_{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha } {\mathfrak{S}}_{*}\left(\mu \right)\times {\mathscr{H}}_{*}\left(\mu \right)\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\partial }_{*}\left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right){{\displaystyle \int }}_0^1\left[{\varsigma }^{\alpha -1}+{\left(1-\varsigma \right)}^{\alpha -1}\right]{\chi }_1\left(\varsigma \right){\chi }_2\left(1-\varsigma \right)d\varsigma b\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\xi }_{*}\left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right){{\displaystyle \int }}_0^1\left[{\varsigma }^{\alpha -1}+{\left(1-\varsigma \right)}^{\alpha -1}\right]{\chi }_1\left(1-\varsigma \right){\chi }_2\left(1-\varsigma \right)d\varsigma \\ \frac{1}{\alpha {\chi }_1\left(\frac{1}{2}\right){\chi }_2\left(\frac{1}{2}\right)}{\mathfrak{S}}^{*}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right)\times {\mathscr{H}}^{*}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\le \frac{\Gamma \left(\alpha \right)}{{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }}\left[{ℐ}_{{\mu }^{+}}^{\alpha } {\mathfrak{S}}^{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)\times {\mathscr{H}}^{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha } {\mathfrak{S}}^{*}\left(\mu \right)\times {\mathscr{H}}^{*}\left(\mu \right)\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\partial }^{*}\left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right){{\displaystyle \int }}_0^1\left[{\varsigma }^{\alpha -1}+{\left(1-\varsigma \right)}^{\alpha -1}\right]{\chi }_1\left(\varsigma \right){\chi }_2\left(1-\varsigma \right)d\varsigma \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\xi }^{*}\left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right){{\displaystyle \int }}_0^1\left[{\varsigma }^{\alpha -1}+{\left(1-\varsigma \right)}^{\alpha -1}\right]{\chi }_1\left(1-\varsigma \right){\chi }_2\left(1-\varsigma \right)d\varsigma .\end{array}$$

It follows that:

$$\begin{array}{c}\frac{1}{\alpha {\chi }_1\left(\frac{1}{2}\right){\chi }_2\left(\frac{1}{2}\right)}\mathfrak{S}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right)\times \mathscr{H}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right)\\ {\le }_p\frac{\Gamma \left(\alpha \right)}{{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }}[{ℐ}_{{\mu }^{+}}^{\alpha }\mathfrak{S}\left(\mu +\phi \left(\omega ,\mu \right)\right)\times \mathscr{H}\left(\mu +\phi \left(\omega ,\mu \right)\right)\\ +{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha }\mathfrak{S}\left(\mu \right)\times \mathscr{H}\left(\mu \right)]\\ +\partial \left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right){{\displaystyle \int }}_0^1\left[{\varsigma }^{\alpha -1}+{\left(1-\varsigma \right)}^{\alpha -1}\right]{\chi }_1\left(\varsigma \right){\chi }_2\left(1-\varsigma \right)d\varsigma \\ +\xi \left(\mu ,\mu +\phi \left(\omega ,\mu \right)\right){{\displaystyle \int }}_0^1\left[{\varsigma }^{\alpha -1}+{\left(1-\varsigma \right)}^{\alpha -1}\right]{\chi }_1\left(1-\varsigma \right){\chi }_2\left(1-\varsigma \right)d\varsigma \end{array}$$

Hence, the required result. □

Now, we present the successful reformative version of the generalized version of interval H-H inequality on invex set for LR-$\chi$-pre-invex I-V·F via interval Riemann–Liouville fractional integral.

Theorem 6.

(Second fractionalH-HFejér inequality) Let$\mathfrak{S}:\left[\mu , \mu +\phi \left(\omega ,\mu \right)\right]\to {\mathcal{K}}_C^{+}$be a LR-$\chi$-pre-invex I-V·F with$\mu <\mu +\phi \left(\omega ,\mu \right)$, such that$\mathfrak{S}\left(z\right)=\left[{\mathfrak{S}}_{*}\left(z\right), {\mathfrak{S}}^{*}\left(z\right)\right]$for all$z\in \left[\mu , \mu +\phi \left(\omega ,\mu \right)\right]$. If$\mathfrak{S}\in L\left(\left[\mu , \mu +\phi \left(\omega ,\mu \right)\right],{\mathcal{K}}_C^{+}\right)$and$\mathcal{D}:\left[\mu , \mu +\phi \left(\omega ,\mu \right)\right]\to ℝ, \mathcal{D}\left(z\right)\ge 0,$symmetric with respect to$\frac{2\mu +\phi \left(\omega ,\mu \right)}{2},$then:

$$\begin{gathered} \frac{\Gamma \left(\alpha \right)}{{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }}\left[{ℐ}_{{\mu }^{+}}^{\alpha }\mathfrak{S}\mathcal{D}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha }\mathfrak{S}\mathcal{D}\left(\mu \right)\right] \\ {\le }_p\left(\mathfrak{S}\left(\mu \right)+\mathfrak{S}\left(\mu +\phi \left(\omega ,\mu \right)\right)\right){{\displaystyle \int }}_0^1\begin{array}{c}{\varsigma }^{\alpha -1}\left[\chi \left(\varsigma \right)+\chi \left(1-\varsigma \right)\right]D\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)d\varsigma \end{array}. \end{gathered}$$

(26)

If  $\mathfrak{S}$ is pre-incave I-V·F, then inequality (26) is reversed.

Proof. 

Let $\mathfrak{S}$ be a LR-$\chi$-pre-invex I-V·F and ${\varsigma }^{\alpha -1}\mathcal{D}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\ge 0$. Then, we have:

$$\begin{array}{l}{\varsigma }^{\alpha -1}{\mathfrak{S}}_{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\mathcal{D}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\le {\varsigma }^{\alpha -1}\left(\chi \left(\varsigma \right){\mathfrak{S}}_{*}\left(\mu \right)+\chi \left(1-\varsigma \right){\mathfrak{S}}_{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)\right)\mathcal{D}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\\ {\varsigma }^{\alpha -1}{\mathfrak{S}}^{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\mathcal{D}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\le {\varsigma }^{\alpha -1}\left(\chi \left(\varsigma \right){\mathfrak{S}}^{*}\left(\mu \right)+\chi \left(1-\varsigma \right){\mathfrak{S}}^{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)\right)\mathcal{D}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right),\end{array}$$

(27)

and:

$$\begin{array}{l}{\varsigma }^{\alpha -1}{\mathfrak{S}}_{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\mathcal{D}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\le {\varsigma }^{\alpha -1}\left(\chi \left(1-\varsigma \right){\mathfrak{S}}_{*}\left(\mu \right)+\chi \left(\varsigma \right){\mathfrak{S}}_{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)\right)\mathcal{D}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\\ {\varsigma }^{\alpha -1}{\mathfrak{S}}^{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\mathcal{D}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\le {\varsigma }^{\alpha -1}\left(\chi \left(1-\varsigma \right){\mathfrak{S}}^{*}\left(\mu \right)+\chi \left(\varsigma \right){\mathfrak{S}}^{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)\right)\mathcal{D}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right).\end{array}$$

(28)

After adding (27) and (28), and integrating over $\left[0, 1\right],$ we get:

$$\begin{array}{l}{{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}{\mathfrak{S}}_{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\mathcal{D}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)d\varsigma \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}{\mathfrak{S}}_{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\mathcal{D}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)d\varsigma \\ \le {{\displaystyle \int }}_0^1\left[\begin{array}{c}{\varsigma }^{\alpha -1}{\mathfrak{S}}_{*}\left(\mu \right)\left\{\chi \left(\varsigma \right)\mathcal{D}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)+\chi \left(1-\varsigma \right)\mathcal{D}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\right\}\\ +{\varsigma }^{\alpha -1}{\mathfrak{S}}_{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)\left\{\chi \left(1-\varsigma \right)\mathcal{D}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)+\chi \left(\varsigma \right)\mathcal{D}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\right\}\end{array}\right]d\varsigma ,\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\mathfrak{S}}_{*}\left(\mu \right){{\displaystyle \int }}_0^1\begin{array}{c}{\varsigma }^{\alpha -1}\left[\chi \left(\varsigma \right)+\chi \left(1-\varsigma \right)\right]D\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\end{array}d\varsigma \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\mathfrak{S}}_{*}\left(\mu +\phi \left(\omega ,\mu \right)\right){{\displaystyle \int }}_0^1\begin{array}{c}{\varsigma }^{\alpha -1}\left[\chi \left(\varsigma \right)+\chi \left(1-\varsigma \right)\right]D\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\end{array}d\varsigma ,\\ {{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}{\mathfrak{S}}^{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\mathcal{D}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)d\varsigma \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}{\mathfrak{S}}^{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\mathcal{D}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)d\varsigma \\ \le {{\displaystyle \int }}_0^1\left[\begin{array}{c}{\varsigma }^{\alpha -1}{\mathfrak{S}}^{*}\left(\mu \right)\left\{\chi \left(\varsigma \right)\mathcal{D}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)+\chi \left(1-\varsigma \right)\mathcal{D}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\right\}\\ +{\varsigma }^{\alpha -1}{\mathfrak{S}}^{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)\left\{\chi \left(1-\varsigma \right)\mathcal{D}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)+\chi \left(\varsigma \right)\mathcal{D}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\right\}\end{array}\right]d\varsigma ,\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\mathfrak{S}}^{*}\left(\mu \right){{\displaystyle \int }}_0^1\begin{array}{c}{\varsigma }^{\alpha -1}\left[\chi \left(\varsigma \right)+\chi \left(1-\varsigma \right)\right]D\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\end{array}d\varsigma \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\mathfrak{S}}^{*}\left(\mu +\phi \left(\omega ,\mu \right)\right){{\displaystyle \int }}_0^1\begin{array}{c}{\varsigma }^{\alpha -1}\left[\chi \left(\varsigma \right)+\chi \left(1-\varsigma \right)\right]D\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\end{array}d\varsigma .\end{array}$$

(29)

Taking the right hand side of inequality (29), we have:

$$\begin{array}{l}{{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}{\mathfrak{S}}_{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\mathcal{D}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)d\varsigma \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}{\mathfrak{S}}_{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\mathcal{D}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)d\varsigma \\ \hspace{1em}\hspace{1em}\hspace{1em}=\frac{1}{{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }} {{\displaystyle \int }}_{\mu }^{\mu +\phi \left(\omega ,\mu \right)}{\left(z-\mu \right)}^{\alpha -1}{\mathfrak{S}}_{*}\left(2\mu +\phi \left(\omega ,\mu \right)-z\right)\mathcal{D}\left(z\right)dz\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{1}{{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }} {{\displaystyle \int }}_{\mu }^{\mu +\phi \left(\omega ,\mu \right)}{\left(z-\mu \right)}^{\alpha -1}{\mathfrak{S}}_{*}\left(z\right)\mathcal{D}\left(z\right)dz\\ \hspace{1em}\hspace{1em}\hspace{1em}=\frac{1}{{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }} {{\displaystyle \int }}_{\mu }^{\mu +\phi \left(\omega ,\mu \right)}{\left(\mu +\phi \left(\omega ,\mu \right)-z\right)}^{\alpha -1}{\mathfrak{S}}_{*}\left(z\right)\mathcal{D}\left(2\mu +\phi \left(\omega ,\mu \right)-z\right)dz\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{1}{{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }} {{\displaystyle \int }}_{\mu }^{\mu +\phi \left(\omega ,\mu \right)}{\left(z-\mu \right)}^{\alpha -1}{\mathfrak{S}}_{*}\left(z\right)\mathcal{D}\left(z\right)dz\\ \hspace{1em}\hspace{1em}\hspace{1em}=\frac{\Gamma \left(\alpha \right)}{{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }}\left[{ℐ}_{{\mu }^{+}}^{\alpha } {\mathfrak{S}}_{*}\mathcal{D}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha } {\mathfrak{S}}_{*}\mathcal{D}\left(\mu \right)\right],\\ {{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}{\mathfrak{S}}^{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\mathcal{D}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)d\varsigma \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}{\mathfrak{S}}^{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\mathcal{D}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)d\varsigma \\ \hspace{1em}\hspace{1em}\hspace{1em}=\frac{\Gamma \left(\alpha \right)}{{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }}\left[{ℐ}_{{\mu }^{+}}^{\alpha } {\mathfrak{S}}^{*}\mathcal{D}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha } {\mathfrak{S}}^{*}\mathcal{D}\left(\mu \right)\right]\end{array}$$

(30)

From (30), we have:

$$\begin{array}{l}\frac{\Gamma \left(\alpha \right)}{{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }}\left[{ℐ}_{{\mu }^{+}}^{\alpha } {\mathfrak{S}}_{*}\mathcal{D}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha } {\mathfrak{S}}_{*}\mathcal{D}\left(\mu \right)\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le \left({\mathfrak{S}}_{*}\left(\mu \right)+{\mathfrak{S}}_{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)\right){{\displaystyle \int }}_0^1\begin{array}{c}{\varsigma }^{\alpha -1}\left[\begin{array}{c}\chi \left(\varsigma \right)\\ +\chi \left(1-\varsigma \right)\end{array}\right]D\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\end{array}\\ \frac{\Gamma \left(\alpha \right)}{{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }}\left[{ℐ}_{{\mu }^{+}}^{\alpha } {\mathfrak{S}}^{*}\mathcal{D}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha } {\mathfrak{S}}^{*}\mathcal{D}\left(\mu \right)\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le \left({\mathfrak{S}}^{*}\left(\mu \right)+{\mathfrak{S}}^{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)\right){{\displaystyle \int }}_0^1\begin{array}{c}{\varsigma }^{\alpha -1}\left[\begin{array}{c}\chi \left(\varsigma \right)\\ +\chi \left(1-\varsigma \right)\end{array}\right]D\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\end{array},\end{array}$$

that is:

$$\begin{gathered} \frac{\Gamma \left(\alpha \right)}{{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }}\left[{ℐ}_{{\mu }^{+}}^{\alpha } {\mathfrak{S}}_{*}\mathcal{D}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha } {\mathfrak{S}}_{*}\mathcal{D}\left(\mu \right), {ℐ}_{{\mu }^{+}}^{\alpha } {\mathfrak{S}}^{*}\mathcal{D}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha } {\mathfrak{S}}^{*}\mathcal{D}\left(\mu \right)\right] \\ {\le }_p\left[{\mathfrak{S}}_{*}\left(\mu \right)+{\mathfrak{S}}_{*}\left(\mu +\phi \left(\omega ,\mu \right)\right), {\mathfrak{S}}^{*}\left(\mu \right)+{\mathfrak{S}}^{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)\right]{{\displaystyle \int }}_0^1\begin{array}{c}{\varsigma }^{\alpha -1}\left[\chi \left(\varsigma \right)+\chi \left(1-\varsigma \right)\right]D\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)d\varsigma \end{array} \end{gathered}$$

Hence,

$$\begin{gathered} \frac{\Gamma \left(\alpha \right)}{{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }}\left[{ℐ}_{{\mu }^{+}}^{\alpha }\mathfrak{S}\mathcal{D}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha }\mathfrak{S}\mathcal{D}\left(\mu \right)\right] \\ {\le }_p\left(\mathfrak{S}\left(\mu \right)+\mathfrak{S}\left(\omega \right)\right){{\displaystyle \int }}_0^1\begin{array}{c}{\varsigma }^{\alpha -1}\left[\chi \left(\varsigma \right)+\chi \left(1-\varsigma \right)\right]D\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)d\varsigma \end{array} \end{gathered}$$

□

Now, we propose the first H-H Fejér inequality for LR-𝜒-pre-invex I-V·F using the interval Riemann–Liouville fractional integral. Then we will prove the validity of Theorem 6 and Theorem 7 with a nontrivial Example 3.

Theorem 7.

(First fractionalH-H Fejér inequality) Let$\mathfrak{S}:\left[\mu , \mu +\phi \left(\omega ,\mu \right)\right]\to {\mathcal{K}}_C^{+}$be a LR-𝜒-pre-invex I-V·F such that$\mathfrak{S}\left(z\right)=\left[{\mathfrak{S}}_{*}\left(z\right), {\mathfrak{S}}^{*}\left(z\right)\right]$for all$z\in \left[\mu , \mu +\phi \left(\omega ,\mu \right)\right]$. Let$\mathfrak{S}\in L\left(\left[\mu , \mu +\phi \left(\omega ,\mu \right)\right],{\mathcal{K}}_C^{+}\right)$and$\mathcal{D}:\left[\mu , \mu +\phi \left(\omega ,\mu \right)\right]\to ℝ, \mathcal{D}\left(z\right)\ge 0,$symmetric with respect to$\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}$. If$\phi$satisfies condition C, then:

$$\begin{gathered} \frac{1}{2\chi \left(\frac{1}{2}\right)}\mathfrak{S}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right)\left[{ℐ}_{{\mu }^{+}}^{\alpha } \mathcal{D}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha } \mathcal{D}\left(\mu \right)\right] \\ {\le }_p\left[{ℐ}_{{\mu }^{+}}^{\alpha }\mathfrak{S}\mathcal{D}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha }\mathfrak{S}\mathcal{D}\left(\mu \right)\right] \end{gathered}$$

(31)

If  $\mathfrak{S}$ is pre-incave I-V·F, then inequality (31) is reversed.

Proof. 

Since $\mathfrak{S}$ is a LR-$\chi$-pre-invex I-V·F then, we have:

$$\begin{array}{c} \\ {\mathfrak{S}}_{*}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right)\le \chi \left(\frac{1}{2}\right)\left({\mathfrak{S}}_{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)+{\mathfrak{S}}_{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\right)\\ {\mathfrak{S}}^{*}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right)\le \chi \left(\frac{1}{2}\right)\left({\mathfrak{S}}^{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)+{\mathfrak{S}}^{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\right).\end{array}$$

(32)

Since $\mathcal{D}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)=\mathcal{D}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)$, then by multiplying (32) by ${\varsigma }^{\alpha -1}\mathcal{D}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)$ and integrate it with respect to $\varsigma$ over $\left[0, 1\right],$ we obtain:

$$\begin{array}{l}{\mathfrak{S}}_{*}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right){{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}\mathcal{D}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)d\varsigma \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le \chi \left(\frac{1}{2}\right)\left(\begin{array}{c}{{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}{\mathfrak{S}}_{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\mathcal{D}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)d\varsigma \\ +{{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}{\mathfrak{S}}_{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\mathcal{D}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)d\varsigma \end{array}\right),\\ {\mathfrak{S}}^{*}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right){{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}\mathcal{D}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)d\varsigma \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le \chi \left(\frac{1}{2}\right)\left(\begin{array}{c}{{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}{\mathfrak{S}}^{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\mathcal{D}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)d\varsigma \\ +{{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}{\mathfrak{S}}^{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\mathcal{D}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)d\varsigma \end{array}\right).\end{array}$$

(33)

Let $x=\mu +\varsigma \phi \left(\omega ,\mu \right)$. Then, on the right hand side of inequality (33), we have:

$$\begin{array}{l}{{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}{\mathfrak{S}}_{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\mathcal{D}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)d\varsigma \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}{\mathfrak{S}}_{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\mathcal{D}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)d\varsigma \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\frac{1}{{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }} {{\displaystyle \int }}_{\mu }^{\mu +\phi \left(\omega ,\mu \right)}{\left(z-\mu \right)}^{\alpha -1}{\mathfrak{S}}_{*}\left(2\mu +\phi \left(\omega ,\mu \right)-z\right)\mathcal{D}\left(z\right)dz\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{1}{{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }} {{\displaystyle \int }}_{\mu }^{\mu +\phi \left(\omega ,\mu \right)}{\left(z-\mu \right)}^{\alpha -1}{\mathfrak{S}}_{*}\left(z\right)\mathcal{D}\left(z\right)dz\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\frac{1}{{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }} {{\displaystyle \int }}_{\mu }^{\mu +\phi \left(\omega ,\mu \right)}{\left(z-\mu \right)}^{\alpha -1}{\mathfrak{S}}_{*}\left(z\right)\mathcal{D}\left(\mu -\omega -z\right)dz\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{1}{{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }} {{\displaystyle \int }}_{\mu }^{\mu +\phi \left(\omega ,\mu \right)}{\left(z-\mu \right)}^{\alpha -1}{\mathfrak{S}}_{*}\left(z\right)\mathcal{D}\left(z\right)dz\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\frac{\Gamma \left(\alpha \right)}{{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }}\left[{ℐ}_{{\mu }^{+}}^{\alpha } {\mathfrak{S}}_{*}\mathcal{D}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha } {\mathfrak{S}}_{*}\mathcal{D}\left(\mu \right)\right],\\ {{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}{\mathfrak{S}}^{*}\left(\mu +\left(1-\varsigma \right)\phi \left(\omega ,\mu \right)\right)\mathcal{D}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)d\varsigma \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{{\displaystyle \int }}_0^1{\varsigma }^{\alpha -1}{\mathfrak{S}}^{*}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\mathcal{D}\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)d\varsigma \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\frac{\Gamma \left(\alpha \right)}{{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }}\left[{ℐ}_{{\mu }^{+}}^{\alpha } {\mathfrak{S}}^{*}\mathcal{D}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha } {\mathfrak{S}}^{*}\mathcal{D}\left(\mu \right)\right].\end{array}$$

(34)

Then from (34), we have:

$$\begin{array}{l}\frac{1}{2\chi \left(\frac{1}{2}\right)}{\mathfrak{S}}_{*}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right)\left[{ℐ}_{{\mu }^{+}}^{\alpha } \mathcal{D}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha } \mathcal{D}\left(\mu \right)\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le \left[{ℐ}_{{\mu }^{+}}^{\alpha } {\mathfrak{S}}_{*}\mathcal{D}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha } {\mathfrak{S}}_{*}\mathcal{D}\left(\mu \right)\right]\\ \frac{1}{2\chi \left(\frac{1}{2}\right)}{\mathfrak{S}}^{*}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right)\left[{ℐ}_{{\mu }^{+}}^{\alpha } \mathcal{D}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha } \mathcal{D}\left(\mu \right)\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le \left[{ℐ}_{{\mu }^{+}}^{\alpha } {\mathfrak{S}}^{*}\mathcal{D}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha } {\mathfrak{S}}^{*}\mathcal{D}\left(\mu \right)\right],\end{array}$$

from which, we have:

$$\begin{array}{l}\frac{1}{2\chi \left(\frac{1}{2}\right)}\left[{\mathfrak{S}}_{*}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right), {\mathfrak{S}}^{*}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right)\right]\left[{ℐ}_{{\mu }^{+}}^{\alpha } \mathcal{D}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha } \mathcal{D}\left(\mu \right)\right]\\ \hspace{1em}{\begin{array}{c}\begin{array}{c}\le \end{array}\end{array}}_p\left[ {ℐ}_{{\mu }^{+}}^{\alpha } {\mathfrak{S}}_{*}\mathcal{D}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha } {\mathfrak{S}}_{*}\mathcal{D}\left(\mu \right), {ℐ}_{{\mu }^{+}}^{\alpha } {\mathfrak{S}}^{*}\mathcal{D}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha } {\mathfrak{S}}^{*}\mathcal{D}\left(\mu \right)\right],\end{array}$$

and it follows that:

$$\begin{gathered} \frac{1}{2\chi \left(\frac{1}{2}\right)}\mathfrak{S}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right)\left[{ℐ}_{{\mu }^{+}}^{\alpha } \mathcal{D}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha } \mathcal{D}\left(\mu \right)\right] \\ {\begin{array}{c}\begin{array}{c}\le \end{array}\end{array}}_p\left[{ℐ}_{{\mu }^{+}}^{\alpha }\mathfrak{S}\mathcal{D}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha }\mathfrak{S}\mathcal{D}\left(\mu \right)\right] \end{gathered}$$

This completes the proof. □

Remark 5.

If$\mathcal{D}\left(z\right)=1$, then from (26) and (31), we get Theorem 3.

If$\chi \left(\varsigma \right)=\varsigma$, then from (26) and (31), we achieve the following coming inequality, see [42]:

$$\begin{gathered} \mathfrak{S}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right)\left[{ℐ}_{{\mu }^{+}}^{\alpha } \mathcal{D}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha } \mathcal{D}\left(\mu \right)\right] \\ {\le }_p\left[{ℐ}_{{\mu }^{+}}^{\alpha }\mathfrak{S}\mathcal{D}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha }\mathfrak{S}\mathcal{D}\left(\mu \right)\right] \\ {\le }_p\frac{\mathfrak{S}\left(\mu \right)+\mathfrak{S}\left(\mu +\phi \left(\omega ,\mu \right)\right)}{2}\left[{ℐ}_{{\mu }^{+}}^{\alpha } \mathcal{D}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha } \mathcal{D}\left(\mu \right)\right] \\ {\le }_p\frac{\mathfrak{S}\left(\mu \right)+\mathfrak{S}\left(\omega \right)}{2}\left[{ℐ}_{{\mu }^{+}}^{\alpha } \mathcal{D}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha } \mathcal{D}\left(\mu \right)\right] \end{gathered}$$

(35)

Let  $\chi \left(\varsigma \right)=\varsigma$ and  $\alpha =1$. Then, from (26) and (31), we achieve the following coming inequality:

$$\mathfrak{S}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right){\le }_p\frac{1}{{{\displaystyle \int }}_{\mu }^{\mu +\phi \left(\omega ,\mu \right)}\mathcal{D}\left(z\right)dz} \left(FR\right){{\displaystyle \int }}_{\mu }^{\mu +\phi \left(\omega ,\mu \right)}\mathfrak{S}\left(z\right)\mathcal{D}\left(z\right)dz{\le }_p\frac{\mathfrak{S}\left(\mu \right)+\mathfrak{S}\left(\omega \right)}{2}$$

(36)

If  ${\mathfrak{S}}_{*}\left(z\right)={\mathfrak{S}}^{*}\left(z\right)$ and  $\chi \left(\varsigma \right)=\varsigma$, then from (26) and (31), we achieve the following coming inequality:

$$\begin{gathered} \mathfrak{S}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right)\left[{ℐ}_{{\mu }^{+}}^{\alpha } \mathcal{D}\left(\omega \right)+{ℐ}_{{\omega }^{-}}^{\alpha } \mathcal{D}\left(\mu \right)\right]\le \left[{ℐ}_{{\mu }^{+}}^{\alpha }\mathfrak{S}\mathcal{D}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha }\mathfrak{S}\mathcal{D}\left(\mu \right)\right] \\ \le \frac{\mathfrak{S}\left(\mu \right)+\mathfrak{S}\left(\mu +\phi \left(\omega ,\mu \right)\right)}{2}\left[{ℐ}_{{\mu }^{+}}^{\alpha } \mathcal{D}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha } \mathcal{D}\left(\mu \right)\right] \\ \le \frac{\mathfrak{S}\left(\mu \right)+\mathfrak{S}\left(\omega \right)}{2}\left[{ℐ}_{{\mu }^{+}}^{\alpha } \mathcal{D}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha } \mathcal{D}\left(\mu \right)\right] \end{gathered}$$

(37)

If  ${\mathfrak{S}}_{*}\left(z\right)={\mathfrak{S}}^{*}\left(z\right)$ and  $\alpha =1$ and  $\chi \left(\varsigma \right)=\varsigma$, then from (26) and (31), we acquire the classical H-H Fejér inequality.

If  ${\mathfrak{S}}_{*}\left(z\right)={\mathfrak{S}}^{*}\left(z\right)$ and  $\mathcal{D}\left(z\right)=\alpha =1$ and  $\chi \left(\varsigma \right)=\varsigma$, then from (26) and (31), we acquire the classical H-H inequality.

Example 3.

We consider the I-V·F$\mathfrak{S}:\left[0, \phi \left(2,0\right)\right]\to {\mathcal{K}}_C^{+}$defined by,$\mathfrak{S}\left(z\right)=\left[1,2\right]\left(2-\sqrt{z}\right)$. Since end-point functions${\mathfrak{S}}_{*}\left(z\right),$ ${\mathfrak{S}}^{*}\left(z\right)$are LR-$\chi$-pre-invex functions, then$\mathfrak{S}\left(z\right)$is LR-$\chi$-pre-invex I-V·F.

If:

$$\mathcal{D}\left(z\right)=\{\begin{array}{c}\sqrt{z}, \sigma \in \left[0,1\right],\\ \sqrt{2-z}, \sigma \in \left(1, 2\right], \end{array}$$

then $\mathcal{D}\left(2-z\right)=\mathcal{D}\left(z\right)\ge 0$, for all $z\in \left[0, 2\right]$. Since ${\mathfrak{S}}_{*}\left(z\right)=\left(2-\sqrt{z}\right)$ and ${\mathfrak{S}}^{*}\left(z\right)=2\left(2-\sqrt{z}\right)$. If $\chi \left(\varsigma \right)=\varsigma$ and $\alpha =\frac{1}{2}$, then we compute the following:

$$\begin{array}{c}\frac{{\mathfrak{S}}_{*}\left(\mu \right)+{\mathfrak{S}}_{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)}{2}{{\displaystyle \int }}_0^1\begin{array}{c}{\varsigma }^{\alpha -1}\left[\chi \left(\varsigma \right)+\chi \left(1-\varsigma \right)\right]D\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\end{array}=\frac{\pi }{\sqrt{2}}\left(\frac{4-\sqrt{2}}{2}\right),\\ \frac{{\mathfrak{S}}^{*}\left(\mu \right)+{\mathfrak{S}}^{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)}{2}{{\displaystyle \int }}_0^1\begin{array}{c}{\varsigma }^{\alpha -1}\left[\chi \left(\varsigma \right)+\chi \left(1-\varsigma \right)\right]D\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\end{array}=\frac{\pi }{\sqrt{2}}\left(4-\sqrt{2}\right), \end{array}$$

(38)

$$\begin{gathered} \begin{array}{c}\frac{{\mathfrak{S}}_{*}\left(\mu \right)+{\mathfrak{S}}_{*}\left(\mu +\phi \left(\mu +\phi \left(\omega ,\mu \right),\mu \right)\right)}{2}{{\displaystyle \int }}_0^1\begin{array}{c}{\varsigma }^{\alpha -1}\left[\chi \left(\varsigma \right)+\chi \left(1-\varsigma \right)\right]D\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\end{array}=\frac{\pi }{\sqrt{2}}\left(\frac{4-\sqrt{2}}{2}\right),\\ \frac{{\mathfrak{S}}^{*}\left(\mu \right)+{\mathfrak{S}}^{*}\left(\mu +\phi \left(\omega ,\mu \right)\right)}{2}{{\displaystyle \int }}_0^1\begin{array}{c}{\varsigma }^{\alpha -1}\left[\chi \left(\varsigma \right)+\chi \left(1-\varsigma \right)\right]D\left(\mu +\varsigma \phi \left(\omega ,\mu \right)\right)\end{array}=\frac{\pi }{\sqrt{2}}\left(4-\sqrt{2}\right), \end{array} \\ \begin{array}{c}\frac{\Gamma \left(\alpha \right)}{{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }}\left[{ℐ}_{{\mu }^{+}}^{\alpha } {\mathfrak{S}}_{*}\mathcal{D}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha } {\mathfrak{S}}_{*}\mathcal{D}\left(\mu \right)\right]=\frac{1}{\sqrt{\pi }}\left(2\pi +\frac{4-8\sqrt{2}}{3}\right),\\ \frac{\Gamma \left(\alpha \right)}{{\left(\phi \left(\omega ,\mu \right)\right)}^{\alpha }}\left[{ℐ}_{{\mu }^{+}}^{\alpha } {\mathfrak{S}}^{*}\mathcal{D}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha } {\mathfrak{S}}^{*}\mathcal{D}\left(\mu \right)\right]=\frac{2}{\sqrt{\pi }}\left(2\pi +\frac{4-8\sqrt{2}}{3}\right). \end{array} \end{gathered}$$

(39)

From (38) and (39), we have:

$$\frac{1}{\sqrt{\pi }}\left[\left(2\pi +\frac{4-8\sqrt{2}}{3}\right), 2\left(2\pi +\frac{4-8\sqrt{2}}{3}\right)\right]{\begin{array}{c} \begin{array}{c}\le \end{array}\end{array}}_p\frac{\pi }{\sqrt{2}}\left[\frac{4-\sqrt{2}}{2}, 4-\sqrt{2}\right].$$

Hence, Theorem 6 is verified.

For Theorem 7, we have:

$$\begin{array}{l}{ℐ}_{{\mu }^{+}}^{\alpha } {\mathfrak{S}}_{*}\mathcal{D}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha } {\mathfrak{S}}_{*}\mathcal{D}\left(\mu \right)\\ \hspace{1em}\hspace{1em}=\frac{1}{\sqrt{\pi }}{{\displaystyle \int }}_0^2{\left(2-z\right)}^{\frac{-1}{2}} \mathcal{D}\left(z\right)\left(2-\sqrt{z}\right)dz+\frac{1}{\sqrt{\pi }}{{\displaystyle \int }}_0^2{\left(z\right)}^{\frac{-1}{2}} \mathcal{D}\left(z\right)\left(2-\sqrt{z}\right)dz\\ =\frac{1}{\sqrt{\pi }}\left(\pi +\frac{8-8\sqrt{2}}{3}\right)+\frac{1}{\sqrt{\pi }}\left(\pi -\frac{4}{3}\right)=\frac{1}{\sqrt{\pi }}\left(2\pi +\frac{4-8\sqrt{2}}{3}\right)\\ {ℐ}_{{\mu }^{+}}^{\alpha } {\mathfrak{S}}^{*}\mathcal{D}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha } {\mathfrak{S}}^{*}\mathcal{D}\left(\mu \right)\\ \hspace{1em}\hspace{1em}=\frac{2}{\sqrt{\pi }}{{\displaystyle \int }}_0^2{\left(2-z\right)}^{\frac{-1}{2}} \mathcal{D}\left(z\right)\left(2-\sqrt{z}\right)dz+\frac{2}{\sqrt{\pi }}{{\displaystyle \int }}_0^2{\left(z\right)}^{\frac{-1}{2}} \mathcal{D}\left(z\right)\left(2-\sqrt{z}\right)dz\\ \hspace{1em}\hspace{1em}=\frac{2}{\sqrt{\pi }}\left(\pi +\frac{8-8\sqrt{2}}{3}\right)+\frac{2}{\sqrt{\pi }}\left(\pi -\frac{4}{3}\right)=\frac{2}{\sqrt{\pi }}\left(2\pi +\frac{4-8\sqrt{2}}{3}\right).\end{array}$$

(40)

$$\begin{array}{c}\frac{1}{2\chi \left(\frac{1}{2}\right)}{\mathfrak{S}}_{*}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right)\left[{ℐ}_{{\mu }^{+}}^{\alpha } \mathcal{D}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha } \mathcal{D}\left(\mu \right)\right]=\sqrt{\pi }, \\ \frac{1}{2\chi \left(\frac{1}{2}\right)}{\mathfrak{S}}^{*}\left(\frac{2\mu +\phi \left(\omega ,\mu \right)}{2}\right)\left[{ℐ}_{{\mu }^{+}}^{\alpha } \mathcal{D}\left(\mu +\phi \left(\omega ,\mu \right)\right)+{ℐ}_{\mu +\phi {\left(\omega ,\mu \right)}^{-}}^{\alpha } \mathcal{D}\left(\mu \right)\right]=2\sqrt{\pi }. \end{array}$$

(41)

4. Conclusions

We have proposed the class of LR-$\chi$-pre-invexity for I-V·Fs. By using this class, we have presented several interval H-H inequalities and interval H-H Fejér inequalities using interval Riemann–Liouville fractional integral operators. Useful examples that illustrate the applicability of theory developed in this study are also presented. In future, we intend to discuss generalized LR-$\chi$-pre-invex I-V·Fs. We hope that this concept will be helpful for other authors to play their roles in different fields of sciences.