Generalized Preinvex Interval-Valued Functions and Related Hermite–Hadamard Type Inequalities

Abstract

This study aims to connect the idea of inequalities with Riemann integral operators, which are of interest because of their characteristics and widespread use. We create a new Hermite–Hadamard type integral inequality for an 𝔴-preinvex interval-valued function using an interval integral operator. In the context of pseudo-order relations, we also establish new variations of the Fejér type inequalities and the Pachpatte type inequalities. We further verify the veracity of the conclusions we draw in this study by providing a number of numerical examples. The findings given in this work, in our opinion, are innovative and will help spur more study in this area.

1. Introduction

The Hermite–Hadamard inequality [1,2] has garnered a lot of interest in elementary mathematics since it is the first fundamental conclusion of convex maps with a natural geometric interpretation and wide application. The inequality of the Hermite–Hadamard type, which is defined by:

$$\mathfrak{S}\left(\frac{\varsigma +\upsilon }{2}\right)\le \frac{1}{\upsilon -\varsigma } {\int }_{\varsigma }^{\upsilon }\mathfrak{S}\left(𝓸\right)d𝓸\le \frac{\mathfrak{S}\left(\varsigma \right)+\mathfrak{S}\left(\upsilon \right)}{2}$$

(1)

where $\mathfrak{S}:\mathbb{I}\subset ℝ\to ℝ$, is a convex function on closed bound interval $\mathbb{I}$ of $ℝ$, and $\varsigma ,\upsilon \in \mathbb{I}$ with $\varsigma <\upsilon$, and for applications of the Hermite–Hadamard integral inequality, see [3,4,5,6,7,8,9,10,11,12,13] and the references therein.

Different integral inequalities have been discovered for different integrals operators. For generalizing significant and well-known integral inequalities, these integrals are helpful. The Hermite–Hadamard integral inequality is a particular type of integral inequality. It is frequently used in the literature and outlines the prerequisites and extenuating circumstances for a function to be convex. Using Riemann-Liouville fractional integrals, Sarikaya et al. [14] extended the Hermite–Hadamard inequality. Iscan [15] expanded Sarikaya et al.’s findings to include Hermite–Hadamard–Fejér type inequalities. By utilizing the product of two convex functions, Chen [16] produced fractional Hermite–Hadamard type integral inequalities using the methods of Sarikaya et al. [14]. Convex polytopes and Jensen type inequalities proposals were the subject of Guessab’s [17] study, which also looked at approximation error in convex functions. A sequence of operators need not have an identity limit, according to a Korovkin-type theorem found by Guessab et al. [18]. Additionally, Guessab [19] worked on ideas such as bivariate Hermite interpolation and higher order convexity. In recent years, mathematicians have become increasingly interested in the presentation of a number of well-known integral inequalities using different unique notions of fractional integral operators. The findings in [20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35] might be consulted in this respect.

Set-valued analysis is a subdivision of interval analysis. The value of interval analysis in both basic and practical research cannot be overstated. The error limitations of numerical solutions to finite state machines were one of the earliest applications of interval analysis. However, interval analysis has become more important in recent years as a part of mathematical and computational models for dealing with interval uncertainty. Moore [36], who is credited with being the first to apply intervals in computer mathematics, published the first book on interval analysis in 1966. Following the publication of this book, a number of scientists began examining the theory and uses of interval arithmetic. Due to its widespread use today, interval analysis is a helpful technique in many fields with ambiguous data. Applications may be found in computer graphics, computational and experimental physics, error analysis, robotics, and many other fields.

Convex interval-valued functions have recently been the subject of research on Jensen type inequality and Hermite–Hadamard type inequalities since, as we all know, convex functions and inequalities go hand in hand. It is important to keep in mind that inclusion relations or LU-orders [37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52], which are partial orders, are currently used to generate interval-valued inequalities. The midpoint and radius of the interval were used in 2014 by Bhunia and Samanta [53] to define the cr-order, which is a complete order relation. Rahman [54] developed the cr-convex function and investigated its nonlinearity in 2020. For more information related to interval-valued functions, see [55,56,57,58,59,60,61,62,63,64,65,66,67,68].

To demonstrate new inequalities, we have combined the ideas of an inclusion relation with interval valued analysis. There are still many unanswered questions about integral inequalities involving different kinds of convex functions, despite the fact that there are many studies on the evolution of integral inequalities using convex functions. The main objective of this paper is to develop new Hermite–Hadamard, Pachppate, and Fejér type inequalities for generalized convex interval-valued functions using interval Riemann integral operators.

2. Preliminaries

In this section, we recall some basic preliminary notions, definitions and results. With the help of these results, some new basic definitions and results are also discussed.

We begin by recalling the basic notations and definitions. We define interval as:

$$\left[{\mathfrak{e}}_{*}, {\mathfrak{e}}^{*}\right]=\left\{𝓸\in ℝ:{\mathfrak{e}}_{*}\le 𝓸\le {\mathfrak{e}}^{*} \mathrm{and} {\mathfrak{e}}_{*},{\mathfrak{e}}^{*}\in ℝ \right\},\mathrm{where}{\mathfrak{e}}_{*}\le {\mathfrak{e}}^{*}$$

(2)

We write len $\left[{\mathfrak{e}}_{*}, {\mathfrak{e}}^{*}\right]={\mathfrak{e}}^{*}-{\mathfrak{e}}_{*}$. If len $\left[{\mathfrak{e}}_{*}, {\mathfrak{e}}^{*}\right]=0$ then, $\left[{\mathfrak{e}}_{*}, {\mathfrak{e}}^{*}\right]$ is called degenerate. In this article, all intervals will be non-degenerate intervals. The collection of all closed and bounded intervals of $ℝ$ is denoted and defined as ${ℝ}_{\mathbb{I}}=\left\{\left[{\mathfrak{e}}_{*}, {\mathfrak{e}}^{*}\right]:{\mathfrak{e}}_{*}, {\mathfrak{e}}^{*}\in ℝ \mathrm{and} {\mathfrak{e}}_{*}\le {\mathfrak{e}}^{*}\right\}.$ If ${\mathfrak{e}}_{*}\ge 0$, then $\left[{\mathfrak{e}}_{*}, {\mathfrak{e}}^{*}\right]$ is called a positive interval. The set of all positive intervals is denoted by ${ℝ}_{\mathbb{I}}{}^{+}$ and defined as ${ℝ}_{\mathbb{I}}{}^{+}=\left\{\left[{\mathfrak{e}}_{*}, {\mathfrak{e}}^{*}\right]:\left[{\mathfrak{e}}_{*}, {\mathfrak{e}}^{*}\right]\in {ℝ}_{\mathbb{I}} \mathrm{and}{\mathfrak{e}}_{*}\ge 0\right\}.$

We now consider some of the properties of intervals using arithmetic operations. Let $\left[{\mathfrak{e}}_{*}, {\mathfrak{e}}^{*}\right], \left[{𝓲}_{*}, {𝓲}^{*}\right]\in {ℝ}_{\mathbb{I}}$ and $\rho \in ℝ$, then we have:

$$\left[{\mathfrak{e}}_{*}, {\mathfrak{e}}^{*}\right]+\left[{𝓲}_{*}, {𝓲}^{*}\right]=\left[{\mathfrak{e}}_{*}+{𝓲}_{*}, {\mathfrak{e}}^{*}+{𝓲}^{*}\right],$$

(3)

$$\left[{\mathfrak{e}}_{*}, {\mathfrak{e}}^{*}\right]\times \left[{𝓲}_{*}, {𝓲}^{*}\right]=\left[\begin{array}{c}min\left\{{\mathfrak{e}}_{*}{𝓲}_{*}, {\mathfrak{e}}^{*}{𝓲}_{*}, {\mathfrak{e}}_{*}{𝓲}^{*}, {\mathfrak{e}}^{*}{𝓲}^{*}\right\}, \\ max\left\{{\mathfrak{e}}_{*}{𝓲}_{*}, {\mathfrak{e}}^{*}{𝓲}_{*}, {\mathfrak{e}}_{*}{𝓲}^{*}, {\mathfrak{e}}^{*}{𝓲}^{*}\right\}\end{array}\right]$$

(4)

$$\rho .\left[{\mathfrak{e}}_{*}, {\mathfrak{e}}^{*}\right]=\{\begin{array}{ll}\left[\rho {\mathfrak{e}}_{*}, \rho {\mathfrak{e}}^{*}\right]& \mathrm{if} \rho >0\\ \left\{0\right\}& \mathrm{if} \rho =0\\ \left[\rho {\mathfrak{e}}^{*},\rho {\mathfrak{e}}_{*}\right]& \mathrm{if} \rho <0.\end{array}$$

(5)

Remark 1.

The relation $“\subseteq ”$ is defined on ${ℝ}_{\mathbb{I}}$ by

$$\left[{\mathfrak{e}}_{*}, {\mathfrak{e}}^{*}\right]\subseteq \left[{𝓲}_{*}, {𝓲}^{*}\right],\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if},{𝓲}_{*}\le {\mathfrak{e}}_{*},{\mathfrak{e}}^{*}\le {𝓲}^{*}$$

(6)

for all $\left[{\mathfrak{e}}_{*},{\mathfrak{e}}^{*}\right],\left[{𝓲}_{*},{𝓲}^{*}\right]\in {ℝ}_{\mathbb{I}},$ it is an inclusion relation.

Moore [36] initially proposed the concept of the Riemann integral for IVF, which is defined as follows:

Theorem 1.

If $\mathfrak{S}:\left[\varsigma ,\upsilon \right]\subset ℝ\to {ℝ}_{\mathbb{I}}$ is an IVF on such that $\mathfrak{S}\left(𝓸\right)=\left[{\mathfrak{S}}_{*}\left(𝓸\right), {\mathfrak{S}}^{*}\left(𝓸\right)\right].$ Then $\mathfrak{S}$ is Riemann integrable over $\left[\varsigma ,\upsilon \right]$ if and only if,${\mathfrak{S}}_{*}$ and ${\mathfrak{S}}^{*}$ both are Riemann integrable over $\left[\varsigma ,\upsilon \right]$ such that

$$\left(IR\right){\int }_{\varsigma }^{\upsilon }\mathfrak{S}\left(𝓸\right)d𝓸=\left[\left(R\right){{\displaystyle \int }}_{\varsigma }^{\upsilon }{\mathfrak{S}}_{*}\left(𝓸\right)d𝓸, \left(R\right){{\displaystyle \int }}_{\varsigma }^{\upsilon }{\mathfrak{S}}^{*}\left(𝓸\right)d𝓸\right]$$

(7)

Definition 1.

Let$K$be an invex set and$\mathfrak{w}:\left[0, 1\right]\to ℝ$such that$\mathfrak{w}\left(𝓸\right)>0$. Then IVF$\mathfrak{S}:K\to {ℝ}_{\mathbb{I}}^{+}$is said to be $\mathfrak{w}$-preinvex on$K$with respect to$\gamma$if

$$\mathfrak{S}\left(𝓸+\left(1-\rho \right)\gamma \left(y,𝓸\right)\right)\supseteq \mathfrak{w}\left(\rho \right)\mathfrak{S}\left(𝓸\right)+\mathfrak{w}\left(1-\rho \right)\mathfrak{S}\left(y\right),$$

(8)

for all$𝓸, y\in K, \rho \in \left[0, 1\right],$$\gamma :K\times K\to ℝ.$$\mathfrak{S}$is called $\mathcal{h}$-preconcave on $K$ with respect to $\gamma$ if Inequality (8) is reversed. $\mathfrak{S}$ is called affine 𝔴-preinvex on $K$ with respect to $\gamma$ if

$$\mathfrak{S}\left(𝓸+\left(1-\rho \right)\gamma \left(y,𝓸\right) \right)=\mathfrak{w}\left(\rho \right)\mathfrak{S}\left(𝓸\right)+\mathfrak{w}\left(1-\rho \right)\mathfrak{S}\left(y\right),$$

(9)

for all$𝓸, y\in K$.

Remark 2.

The 𝔴-preinvex IVFs have some very nice properties similar to preinvex IVF,

if $\mathfrak{S}$ is 𝔴-preinvex IVF, then ${\rm Y}$ is also 𝔴-preinvex for ${\rm Y}\ge 0$.

if $\mathfrak{S}$ and $\mathfrak{U}$ both are 𝔴-preinvex IVFs, then $\max \left(\mathfrak{S}\left(𝓸\right),\mathfrak{U}\left(𝓸\right)\right)$ is also 𝔴-preinvex IVF.

Now we discuss some new special cases of 𝔴-preinvex IVFs:

(i)

If one takes $\mathfrak{w}\left(\rho \right)={\rho }^s,$ then from (8), one can acquire the following coming inequality, see [37]:

$$\mathfrak{S}\left(𝓸+\left(1-\rho \right)\gamma \left(y,𝓸\right)\right)\supseteq {\rho }^s\mathfrak{S}\left(𝓸\right)+{\left(1-\rho \right)}^s\mathfrak{S}\left(y\right),\forall 𝓸, y\in K, \rho \in \left[0, 1\right].$$

(10)

If one takes $\gamma \left(y,𝓸\right)=y-𝓸,$ then $\mathfrak{S}$ is called $s$-convex IVF.

(ii)

If one takes $\mathfrak{w}\left(\rho \right)=\rho ,$ then from (8), one can acquire the following coming inequality, see [17]:

$$\mathfrak{S}\left(𝓸+\left(1-\rho \right)\gamma \left(y,𝓸\right)\right)\supseteq \rho \mathfrak{S}\left(𝓸\right)+\left(1-\rho \right)\mathfrak{S}\left(y\right),\forall 𝓸, y\in K, \rho \in \left[0, 1\right].$$

(11)

If one takes $\gamma \left(y,𝓸\right)=y-𝓸,$ then $\mathfrak{S}$ is called convex IVF.

(iii)

If one takes $\mathfrak{w}\left(\rho \right)\equiv 1,$ then from (8), one can achieve the following coming inequality:

$$\mathfrak{S}\left(𝓸+\left(1-\rho \right)\gamma \left(y,𝓸\right)\right)\supseteq \mathfrak{S}\left(𝓸\right)+\mathfrak{S}\left(y\right),\forall 𝓸, y\in K, \rho \in \left[0, 1\right].$$

(12)

If one takes $\gamma \left(y,𝓸\right)=y-𝓸,$ then $\mathfrak{S}$ is called $P$-IVF.

Theorem 2.

Let$K$be an invex set and$\mathfrak{w}:\left[0, 1\right]\subseteq K\to {ℝ}^{+}$such that$\mathfrak{w}>0$, and let$\mathfrak{S}:K\to {ℝ}_{\mathbb{I}}^{+}$be a IVF with$\mathfrak{S}\left(𝓸\right){\ge }_{I}0$such that

$$\mathfrak{S}\left(𝓸\right)=\left[{\mathfrak{S}}_{*}\left(𝓸\right), {\mathfrak{S}}^{*}\left(𝓸\right)\right], \forall 𝓸\in K.$$

(13)

for all $𝓸\in K$. Then $\mathfrak{S}$ is 𝔴-preinvex IVF on $K,$ if and only if, ${\mathfrak{S}}_{*}\left(𝓸\right)$ and ${\mathfrak{S}}^{*}\left(𝓸\right)$ both are 𝔴-preinvex functions.

Proof. 

The proof of this result is similar to the proof of Theorem 3.7, see [37].  □

Example 1.

We consider $\mathfrak{w}\left(\rho \right)=\rho ,$ for $\rho \in \left[0, 1\right]$ and the IVF $\mathfrak{S}:{ℝ}^{+}\to {ℝ}_{\mathbb{I}}^{+}$ defined by $\mathfrak{S}\left(𝓸\right)=\left[(e^{𝓸},2e^{𝓸} \right]$. Since ${\mathfrak{S}}_{*}\left(𝓸\right),$ ${\mathfrak{S}}^{*}\left(𝓸\right)$ are 𝔴-preinvex functions $\gamma \left(y,𝓸\right)=y-𝓸$. Hence $\mathfrak{S}\left(𝓸\right)$ is 𝔴-preinvex IVF.

3. Main Results

Now, the application of inequality (2), Definition 1, and Theorems 1, 2 gives the followings results.

Theorem 3.

Let$\mathfrak{S}:\left[\varsigma , \varsigma +\gamma \left(\upsilon , \varsigma \right)\right]\to {ℝ}_{\mathbb{I}}^{+}$be a $\mathfrak{w}$-preinvex IVF with$\mathfrak{w}:\left[0, 1\right]\to {ℝ}^{+}$and$\mathfrak{w}\left(\frac{1}{2}\right)𝓸0$such that$\mathfrak{S}\left(𝓸\right)=\left[{\mathfrak{S}}_{*}\left(𝓸\right), {\mathfrak{S}}^{*}\left(𝓸\right)\right]$for all $𝓸\in \left[\varsigma , \varsigma +\gamma \left(\upsilon , \varsigma \right)\right]$. If $\mathfrak{S}\in ℐ{ℛ}_{\left(\left[\varsigma , \varsigma +\gamma \left(\upsilon , \varsigma \right)\right]\right)}$, then

$$\frac{1}{2\mathfrak{w}\left(\frac{1}{2}\right)}\mathfrak{S}\left(\frac{2\varsigma +\gamma \left(\upsilon , \varsigma \right)}{2}\right)\supseteq \frac{1}{\gamma \left(\upsilon , \varsigma \right)} \left(IR\right){\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}\mathfrak{S}\left(𝓸\right)d𝓸\supseteq \left[\mathfrak{S}\left(\varsigma \right)+\mathfrak{S}\left(\upsilon \right)\right]{\int }_0^1\mathfrak{w}\left(\rho \right)d\rho .$$

(14)

If $\mathfrak{S}$ is 𝔴-preinvex IVF, then (14) is reversed such that

$$\frac{1}{2\mathfrak{w}\left(\frac{1}{2}\right)}\mathfrak{S}\left(\frac{2\varsigma +\gamma \left(\upsilon , \varsigma \right)}{2}\right)\subseteq \frac{1}{\gamma \left(\upsilon , \varsigma \right)} \left(IR\right){\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}\mathfrak{S}\left(𝓸\right)d𝓸\subseteq \left[\mathfrak{S}\left(\varsigma \right)+\mathfrak{S}\left(\upsilon \right)\right]{\int }_0^1\mathfrak{w}\left(\rho \right)d\rho .$$

(15)

Proof. 

Let $\mathfrak{S}:\left[\varsigma , \varsigma +\gamma \left(\upsilon , \varsigma \right)\right]\to {ℝ}_{\mathbb{I}}^{+}$ be a 𝔴-preinvex IVF. Then, by hypothesis, we have

$$\frac{1}{\mathfrak{w}\left(\frac{1}{2}\right)}\mathfrak{S}\left(\frac{2\varsigma +\gamma \left(\upsilon , \varsigma \right)}{2}\right)\supseteq \mathfrak{S}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon , \varsigma \right)\right)+\mathfrak{S}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right).$$

Therefore, we have

$$\begin{array}{c}\frac{1}{\mathfrak{w}\left(\frac{1}{2}\right)}{\mathfrak{S}}_{*}\left(\frac{2\varsigma +\gamma \left(\upsilon , \varsigma \right)}{2}\right)\le {\mathfrak{S}}_{*}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon , \varsigma \right)\right)+{\mathfrak{S}}_{*}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right),\\ \frac{1}{\mathfrak{w}\left(\frac{1}{2}\right)}{\mathfrak{S}}^{*}\left(\frac{2\varsigma +\gamma \left(\upsilon , \varsigma \right)}{2}\right)\ge {\mathfrak{S}}^{*}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon , \varsigma \right)\right)+{\mathfrak{S}}^{*}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right).\end{array}$$

Then

$$\begin{array}{c}\frac{1}{\mathfrak{w}\left(\frac{1}{2}\right)}{\int }_0^1{\mathfrak{S}}_{*}\left(\frac{2\varsigma +\gamma \left(\upsilon , \varsigma \right)}{2}\right)d\rho \le {\int }_0^1{\mathfrak{S}}_{*}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon , \varsigma \right)\right)d\rho +{\int }_0^1{\mathfrak{S}}_{*}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)d\rho ,\\ \frac{1}{\mathfrak{w}\left(\frac{1}{2}\right)}{\int }_0^1{\mathfrak{S}}^{*}\left(\frac{2\varsigma +\gamma \left(\upsilon , \varsigma \right)}{2}\right)d\rho \ge {\int }_0^1{\mathfrak{S}}^{*}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon , \varsigma \right)\right)d\rho +{\int }_0^1{\mathfrak{S}}^{*}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)d\rho .\end{array}$$

It follows that

$$\begin{array}{c}\frac{1}{\mathfrak{w}\left(\frac{1}{2}\right)}{\mathfrak{S}}_{*}\left(\frac{2\varsigma +\gamma \left(\upsilon , \varsigma \right)}{2}\right)\le \frac{2}{\gamma \left(\upsilon , \varsigma \right)} {\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}{\mathfrak{S}}_{*}\left(𝓸\right)d𝓸,\\ \frac{1}{\mathfrak{w}\left(\frac{1}{2}\right)}{\mathfrak{S}}^{*}\left(\frac{2\varsigma +\gamma \left(\upsilon , \varsigma \right)}{2}\right)\ge \frac{2}{\gamma \left(\upsilon , \varsigma \right)} {\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}{\mathfrak{S}}^{*}\left(𝓸\right)d𝓸.\end{array}$$

That is

$$\frac{1}{\mathfrak{w}\left(\frac{1}{2}\right)}\left[{\mathfrak{S}}_{*}\left(\frac{2\varsigma +\gamma \left(\upsilon , \varsigma \right)}{2}\right), {\mathfrak{S}}^{*}\left(\frac{2\varsigma +\gamma \left(\upsilon , \varsigma \right)}{2}\right)\right]\supseteq \frac{2}{\gamma \left(\upsilon , \varsigma \right)}\left[{\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}{\mathfrak{S}}_{*}\left(𝓸\right)d𝓸, {\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}{\mathfrak{S}}^{*}\left(𝓸\right)d𝓸\right].$$

Thus,

$$\frac{1}{2\mathfrak{w}\left(\frac{1}{2}\right)}\mathfrak{S}\left(\frac{2\varsigma +\gamma \left(\upsilon , \varsigma \right)}{2}\right)\supseteq \frac{1}{\gamma \left(\upsilon , \varsigma \right)} \left(IR\right){\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}\mathfrak{S}\left(𝓸\right)d𝓸.$$

(16)

In a similar way to above, we have:

$$\frac{1}{\gamma \left(\upsilon , \varsigma \right)} \left(IR\right){\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}\mathfrak{S}\left(𝓸\right)d𝓸\supseteq \left[\mathfrak{S}\left(\varsigma \right)+\mathfrak{S}\left(\upsilon \right)\right]{\int }_0^1\mathfrak{w}\left(\rho \right)d\rho .$$

(17)

Combining (16) and (17), we have

$$\frac{1}{2\mathfrak{w}\left(\frac{1}{2}\right)}\mathfrak{S}\left(\frac{2\varsigma +\gamma \left(\upsilon ,\varsigma \right)}{2}\right)\supseteq \frac{1}{\gamma \left(\upsilon ,\varsigma \right)}\left(IR\right){\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon ,\varsigma \right)}\mathfrak{S}\left(𝓸\right)d𝓸\supseteq \left[\mathfrak{S}\left(\varsigma \right)+\mathfrak{S}\left(\upsilon \right)\right]{\int }_0^1\mathfrak{w}\left(\rho \right)d\rho ,$$

which completes the proof.  □

Note that, inequality (14) is known as fuzzy-interval H-H inequality for 𝔴-preinvex IVF.

Remark 3.

If one takes $\mathfrak{w}\left(\rho \right)={\rho }^s$, then from 14, we achieve the result for $s$-preinvex IVF in the second sense:

$$2^{s-1}\mathfrak{S}\left(\frac{2\varsigma +\gamma \left(\upsilon , \varsigma \right)}{2}\right)\supseteq \frac{1}{\gamma \left(\upsilon , \varsigma \right)} \left(IR\right){\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}\mathfrak{S}\left(𝓸\right)d𝓸\supseteq \frac{1}{s+1}\left[\mathfrak{S}\left(\varsigma \right)+\mathfrak{S}\left(\upsilon \right)\right].$$

(18)

If one takes $\mathfrak{w}\left(\rho \right)=\rho$, then from 14, we obtain the result for preinvex IVF:

$$\mathfrak{S}\left(\frac{2\varsigma +\gamma \left(\upsilon , \varsigma \right)}{2}\right)\supseteq \frac{1}{\gamma \left(\upsilon , \varsigma \right)} \left(IR\right){\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}\mathfrak{S}\left(𝓸\right)d𝓸\supseteq \frac{\mathfrak{S}\left(\varsigma \right)+\mathfrak{S}\left(\upsilon \right)}{2}.$$

(19)

If one takes $\mathfrak{w}\left(\rho \right)\equiv 1$, then we achieve the result for $P$ IVF:

$$\frac{1}{2}\mathfrak{S}\left(\frac{2\varsigma +\gamma \left(\upsilon , \varsigma \right)}{2}\right)\supseteq \frac{1}{\gamma \left(\upsilon , \varsigma \right)} \left(IR\right){\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}\mathfrak{S}\left(𝓸\right)d𝓸\supseteq \mathfrak{S}\left(\varsigma \right)+\mathfrak{S}\left(\upsilon \right).$$

(20)

If one takes ${\mathfrak{S}}_{*}\left(𝓸\right)={\mathfrak{S}}^{*}\left(𝓸\right)$, then we acquire the result for 𝔴-preinvex function, see [37]:

$$\frac{1}{2\mathfrak{w}\left(\frac{1}{2}\right)}\mathfrak{S}\left(\frac{2\varsigma +\gamma \left(\upsilon , \varsigma \right)}{2}\right)\le \frac{1}{\gamma \left(\upsilon , \varsigma \right)} \left(R\right){\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}\mathfrak{S}\left(𝓸\right)d𝓸\le \left[\mathfrak{S}\left(\varsigma \right)+\mathfrak{S}\left(\upsilon \right)\right]{\int }_0^1\mathfrak{w}\left(\rho \right)d\rho .$$

(21)

Note that, if $\gamma \left(\upsilon ,\varsigma \right)=\upsilon -\varsigma ,$ then integral inequalities (18)–(21) reduce to classical ones.

Example 2.

We consider$\mathfrak{w}\left(\rho \right)=\rho ,$for$\rho \in \left[0, 1\right]$, and the IVF$\mathfrak{S}:\left[\varsigma , \varsigma +\gamma \left(\upsilon , \varsigma \right)\right]=\left[0, \gamma \left(2, 0\right)\right]\to {ℝ}_{\mathbb{I}}^{+}$defined by$\mathfrak{S}\left(𝓸\right)=\left[2{𝓸}^2,4𝓸\right]$. Since${\mathfrak{S}}_{*}\left(𝓸\right)=2{𝓸}^2,$ ${\mathfrak{S}}^{*}\left(𝓸\right)=4𝓸$are 𝔴-preinvex functions with respect to $\gamma \left(\upsilon , \varsigma \right)=\upsilon -\varsigma$. Hence $\mathfrak{S}\left(𝓸\right)$ is 𝔴-preinvex IVF with respect to $\gamma \left(\upsilon , \varsigma \right)=\upsilon -\varsigma$. Since ${\mathfrak{S}}_{*}\left(𝓸\right)=2{𝓸}^2$ and ${\mathfrak{S}}^{*}\left(𝓸\right)=4𝓸$then, we compute the following

$$\frac{1}{2\mathfrak{w}\left(\frac{1}{2}\right)} {\mathfrak{S}}_{*}\left(\frac{2\varsigma +\gamma \left(\upsilon , \varsigma \right)}{2}\right)\le \frac{1}{\gamma \left(\upsilon , \varsigma \right)} {\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}{\mathfrak{S}}_{*}\left(𝓸\right)d𝓸\le \left[{\mathfrak{S}}_{*}\left(\varsigma \right)+{\mathfrak{S}}_{*}\left(\upsilon \right)\right]{\int }_0^1\mathfrak{w}\left(\rho \right)d\rho .$$

$$\frac{1}{2\mathfrak{w}\left(\frac{1}{2}\right)} {\mathfrak{S}}_{*}\left(\frac{2\varsigma +\gamma \left(\upsilon , \varsigma \right)}{2}\right)={\mathfrak{S}}_{*}\left(1\right)=2,$$

$$\frac{1}{\gamma \left(\upsilon , \varsigma \right)} {\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}{\mathfrak{S}}_{*}\left(𝓸\right)d𝓸=\frac{1}{2} {\int }_0^{2}2{𝓸}^{2}d𝓸=\frac{8}{3},$$

$$\left[{\mathfrak{S}}_{*}\left(\varsigma \right)+{\mathfrak{S}}_{*}\left(\upsilon \right)\right]{\int }_0^1\mathfrak{w}\left(\rho \right)d\rho =4,$$

that means

$$2\le \frac{8}{3}\le 4.$$

Similarly, it can be easily show that

$$\frac{1}{2\mathfrak{w}\left(\frac{1}{2}\right)} {\mathfrak{S}}^{*}\left(\frac{2\varsigma +\gamma \left(\upsilon , \varsigma \right)}{2}\right)\ge \frac{1}{\gamma \left(\upsilon , \varsigma \right)} {\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}{\mathfrak{S}}^{*}\left(𝓸\right)d𝓸\ge \left[{\mathfrak{S}}^{*}\left(\varsigma \right)+{\mathfrak{S}}^{*}\left(\upsilon \right)\right]{\int }_0^1\mathfrak{w}\left(\rho \right)d\rho .$$

such that

$$\frac{1}{2\mathfrak{w}\left(\frac{1}{2}\right)} {\mathfrak{S}}^{*}\left(\frac{2\varsigma +\gamma \left(\upsilon , \varsigma \right)}{2}\right)={\mathfrak{S}}_{*}\left(1\right)=4,$$

$$\frac{1}{\gamma \left(\upsilon , \varsigma \right)} {\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}{\mathfrak{S}}^{*}\left(𝓸\right)d𝓸=\frac{1}{2} {\int }_0^{2}4{𝓸}^{2}d𝓸=4,$$

$$\left[{\mathfrak{S}}^{*}\left(\varsigma \right)+{\mathfrak{S}}^{*}\left(\upsilon \right)\right]{\int }_0^1\mathfrak{w}\left(\rho \right)d\rho =4.$$

From which, it follows that

$$4\ge 4\ge 4,$$

that is

$$\left[2, 4\right]\supseteq \left[\frac{8}{3}, 4\right]\supseteq \left[4, 4\right]$$

Hence,

$$\frac{1}{2\mathfrak{w}\left(\frac{1}{2}\right)}\mathfrak{S}\left(\frac{2\varsigma +\gamma \left(\upsilon , \varsigma \right)}{2}\right)\supseteq \frac{1}{\gamma \left(\upsilon , \varsigma \right)} \left(IR\right){\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}\mathfrak{S}\left(𝓸\right)d𝓸\supseteq \left[\mathfrak{S}\left(\varsigma \right)+\mathfrak{S}\left(\upsilon \right)\right]{\int }_0^1\mathfrak{w}\left(\rho \right)d\rho ,$$

and the Theorem 3 is verified.

Theorem 4.

Let $\mathfrak{S},\mathfrak{U}:\left[\mathsf{\varsigma },\mathsf{\varsigma }+\mathsf{\gamma }\left(\mathsf{\upsilon },\mathsf{\varsigma }\right)\right]\to {ℝ}_{\mathbb{I}}^{+}$ be two${\mathfrak{w}}_1$and${\mathfrak{w}}_2$-preinvex IVFs with ${\mathfrak{w}}_1,{\mathfrak{w}}_2:\left[0,1\right]\to {ℝ}^{+}$ such that $\mathfrak{S}\left(𝓸\right)=\left[{\mathfrak{S}}_{*}\left(𝓸\right),{\mathfrak{S}}^{*}\left(𝓸\right)\right]$ and $\mathfrak{U}\left(𝓸\right)=\left[{\mathfrak{U}}_{*}\left(𝓸\right),{\mathfrak{U}}^{*}\left(𝓸\right)\right]$ for all $𝓸\in \left[\mathsf{\varsigma },\mathsf{\varsigma }+\mathsf{\gamma }\left(\mathsf{\upsilon },\mathsf{\varsigma }\right)\right]$. If $\mathfrak{S}\mathfrak{U}\in ℐ{ℛ}_{\left(\left[\mathsf{\varsigma },\mathsf{\varsigma }+\mathsf{\gamma }\left(\mathsf{\upsilon },\mathsf{\varsigma }\right)\right]\right)}$, then

$$\begin{array}{l}\frac{1}{\gamma \left(\upsilon , \varsigma \right)} \left(IR\right){\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}\mathfrak{S}\left(𝓸\right)\times \mathfrak{U}\left(𝓸\right)d𝓸\\ \supseteq \mathcal{K}\left(\varsigma ,\upsilon \right){\int }_0^1{\mathfrak{w}}_1\left(\rho \right){\mathfrak{w}}_2\left(\rho \right)d\rho +ℒ\left(\varsigma ,\upsilon \right){\int }_0^1{\mathfrak{w}}_1\left(\rho \right){\mathfrak{w}}_2\left(1-\rho \right)d\rho ,\end{array}$$

where $\mathcal{K}\left(\varsigma ,\upsilon \right)=\mathfrak{S}\left(\varsigma \right)\times \mathfrak{U}\left(\varsigma \right)+\mathfrak{S}\left(\upsilon \right)\times \mathfrak{U}\left(\upsilon \right),$ $ℒ\left(\varsigma ,\upsilon \right)=\mathfrak{S}\left(\varsigma \right)\times \mathfrak{U}\left(\upsilon \right)+\mathfrak{S}\left(\upsilon \right)\times \mathfrak{U}\left(\varsigma \right)$ with $\mathcal{K}\left(\varsigma ,\upsilon \right)=\left[{\mathcal{K}}_{*}\left(\varsigma ,\upsilon \right),{\mathcal{K}}^{*}\left(\varsigma ,\upsilon \right)\right]$ and $ℒ\left(\varsigma ,\upsilon \right)=\left[{ℒ}_{*}\left(\varsigma ,\upsilon \right),{ℒ}^{*}\left(\varsigma ,\upsilon \right)\right].$

Example 3.

We consider ${\mathfrak{w}}_1\left(\rho \right)=\rho , {\mathfrak{w}}_2\left(\rho \right)\equiv \rho ,$ for $\rho \in \left[0, 1\right]$, and the IVFs $\mathfrak{S},\mathfrak{U}:\left[\varsigma , \varsigma +\gamma \left(\upsilon , \varsigma \right)\right]=\left[0, \gamma \left(1, 0\right)\right]\to {ℝ}_{\mathbb{I}}^{+}$ defined by $\mathfrak{S}\left(𝓸\right)=\left[2{𝓸}^2,4𝓸\right]$ and $\mathfrak{U}\left(𝓸\right)=\left[𝓸,2𝓸\right].$ Since ${\mathfrak{S}}_{*}\left(𝓸\right)=2{𝓸}^2$ and ${\mathfrak{S}}^{*}\left(𝓸\right)=4𝓸$ both are ${\mathfrak{w}}_1$-preinvex functions, and ${\mathfrak{U}}_{*}\left(𝓸\right)=𝓸$, and ${\mathfrak{U}}^{*}\left(𝓸\right)=2𝓸$ both are also ${\mathfrak{w}}_2$-preinvex functions with respect to same $\gamma \left(\upsilon , \varsigma \right)=\upsilon -\varsigma$ then, $\mathfrak{S}$ and $\mathfrak{U}$ both are ${\mathfrak{w}}_1$ and ${\mathfrak{w}}_2$-preinvex IVFs, respectively. Since ${\mathfrak{S}}_{*}\left(𝓸\right)=2{𝓸}^2$ and ${\mathfrak{S}}^{*}\left(𝓸\right)=4𝓸$, and ${\mathfrak{U}}_{*}\left(𝓸\right)=𝓸$, and ${\mathfrak{U}}^{*}\left(𝓸\right)=2𝓸$, then we compute the following:

$$\begin{array}{c}\frac{1}{\gamma \left(\upsilon , \varsigma \right)} {\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}{\mathfrak{S}}_{*}\left(𝓸\right)\times {\mathfrak{U}}_{*}\left(𝓸\right)d𝓸={\int }_0^1\left(2{𝓸}^2\right)\left(𝓸\right)d𝓸=\frac{1}{2} \\ \frac{1}{\gamma \left(\upsilon , \varsigma \right)} {\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}{\mathfrak{S}}^{*}\left(𝓸\right)\times {\mathfrak{U}}^{*}\left(𝓸\right)d𝓸={\int }_0^1\left(4𝓸\right)\left(2𝓸\right)d𝓸=\frac{8}{3},\end{array}$$

$$\begin{array}{c}{\mathcal{K}}_{*}\left(\varsigma ,\upsilon \right){\int }_0^1{\mathfrak{w}}_1\left(\rho \right){\mathfrak{w}}_2\left(\rho \right)d\rho =\frac{2}{3},\\ {\mathcal{K}}^{*}\left(\varsigma ,\upsilon \right){\int }_0^1{\mathfrak{w}}_1\left(\rho \right){\mathfrak{w}}_2\left(\rho \right)d\rho =\frac{8}{3},\end{array}$$

$$\begin{array}{c}{ℒ}_{*}\left(\varsigma ,\upsilon \right){\int }_0^1{\mathfrak{w}}_1\left(\rho \right){\mathfrak{w}}_2\left(1-\rho \right)d\rho =0\\ {ℒ}^{*}\left(\varsigma ,\upsilon \right){\int }_0^1{\mathfrak{w}}_1\left(\rho \right){\mathfrak{w}}_2\left(1-\rho \right)d\rho =0,\end{array}$$

that means

$$\begin{array}{c}\frac{1}{2}\le \frac{2}{3}+0=\frac{2}{3},\\ \frac{8}{3}\ge \frac{8}{3}+0=\frac{8}{3},\end{array}$$

hence, Theorem 4 is verified.

The following assumption is required to prove the next result regarding the bi-function $\gamma :K\times K\to ℝ$ which is known as:

Condition C.

(see [60]) Let $K$ be an invex set with respect to $\gamma .$ For any $\varsigma , \upsilon \in K$ and $\rho \in \left[0, 1\right]$,

$$\gamma \left(\upsilon ,\varsigma +\rho \gamma \left(\upsilon ,\varsigma \right)\right)=\left(1-\rho \right)\gamma \left(\upsilon ,\varsigma \right),$$

$$\gamma \left(\varsigma ,\varsigma +\rho \gamma \left(\upsilon ,\varsigma \right)\right)=-\rho \gamma \left(\upsilon ,\varsigma \right).$$

Clearly for $\rho$ = 0, we have $\gamma \left(\upsilon ,\varsigma \right)$ = 0 if and only if, $\upsilon =\varsigma$, for all $\varsigma , \upsilon \in K$. For the applications of Condition C, see [55,56,58,59,60].

Theorem 5.

Let $\mathfrak{S},\mathfrak{U}:\left[\varsigma , \varsigma +\gamma \left(\upsilon , \varsigma \right)\right]\to {ℝ}_{\mathbb{I}}^{+}$ be two ${\mathfrak{w}}_1$ and ${\mathfrak{w}}_2$-preinvex IVFs with ${\mathfrak{w}}_1, {\mathfrak{w}}_2:\left[0, 1\right]\to {ℝ}^{+}$ given by $\mathfrak{S}\left(𝓸\right)=\left[{\mathfrak{S}}_{*}\left(𝓸\right), {\mathfrak{S}}^{*}\left(𝓸\right)\right]$ and $\mathfrak{U}\left(𝓸\right)=\left[{\mathfrak{U}}_{*}\left(𝓸\right), {\mathfrak{U}}^{*}\left(𝓸\right)\right]$ for all $𝓸\in \left[\varsigma , \varsigma +\gamma \left(\upsilon , \varsigma \right)\right]$. If $\mathfrak{S}\mathfrak{U}\in ℐ{ℛ}_{\left(\left[\varsigma , \varsigma +\gamma \left(\upsilon , \varsigma \right)\right]\right)}$ and condition C hold for $\gamma$, then

$$\begin{array}{l}\frac{1}{2{\mathfrak{w}}_1\left(\frac{1}{2}\right){\mathfrak{w}}_2\left(\frac{1}{2}\right)}\mathfrak{S}\left(\frac{2\varsigma +\gamma \left(\upsilon , \varsigma \right)}{2}\right)\times \mathfrak{U}\left(\frac{2\varsigma +\gamma \left(\upsilon , \varsigma \right)}{2}\right)\supseteq \frac{1}{\gamma \left(\upsilon , \varsigma \right)} \left(IR\right){\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}\mathfrak{S}\left(𝓸\right)\times \mathfrak{U}\left(𝓸\right)d𝓸\\ +\mathcal{K}\left(\varsigma ,\upsilon \right){\int }_0^1{\mathfrak{w}}_1\left(\rho \right)w_2\left(1-\rho \right)d\rho +ℒ\left(\varsigma ,\upsilon \right){\int }_0^1{\mathfrak{w}}_1\left(\rho \right){\mathfrak{w}}_2\left(\rho \right)d\rho ,\end{array}$$

where $\mathcal{K}\left(\varsigma ,\upsilon \right)=\mathfrak{S}\left(\varsigma \right)\times \mathfrak{U}\left(\varsigma \right)+\mathfrak{S}\left(\upsilon \right)\times \mathfrak{U}\left(\upsilon \right),$ $ℒ\left(\varsigma ,\upsilon \right)=\mathfrak{S}\left(\varsigma \right)\times \mathfrak{U}\left(\upsilon \right)+\mathfrak{S}\left(\upsilon \right)\times \mathfrak{U}\left(\varsigma \right),$ and $\mathcal{K}\left(\varsigma ,\upsilon \right)=\left[{\mathcal{K}}_{*}\left(\varsigma ,\upsilon \right), {\mathcal{K}}^{*}\left(\varsigma ,\upsilon \right)\right]$ and $ℒ\left(\varsigma ,\upsilon \right)=\left[{ℒ}_{*}\left(\varsigma ,\upsilon \right), {ℒ}^{*}\left(\varsigma ,\upsilon \right)\right].$

Proof. 

Using condition C, we can write

$$\varsigma +\frac{1}{2}\gamma \left(\upsilon , \varsigma \right)=\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)+\frac{1}{2}\gamma \left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon ,\varsigma \right), \varsigma +\rho \gamma \left(\upsilon ,\varsigma \right)\right).$$

By hypothesis, we have

$$\begin{array}{c}{\mathfrak{S}}_{*}\left(\frac{2\varsigma +\gamma \left(\upsilon , \varsigma \right)}{2}\right)\times {\mathfrak{U}}_{*}\left(\frac{2\varsigma +\gamma \left(\upsilon , \varsigma \right)}{2}\right)\\ {\mathfrak{S}}^{*}\left(\frac{2\varsigma +\gamma \left(\upsilon , \varsigma \right)}{2}\right)\times {\mathfrak{U}}^{*}\left(\frac{2\varsigma +\gamma \left(\upsilon , \varsigma \right)}{2}\right)\\ ={\mathfrak{S}}_{*}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)+\frac{1}{2}\gamma \left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon ,\varsigma \right), \varsigma +\rho \gamma \left(\upsilon ,\varsigma \right)\right)\right)\\ \times {\mathfrak{U}}_{*}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)+\frac{1}{2}\gamma \left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon ,\varsigma \right), \varsigma +\rho \gamma \left(\upsilon ,\varsigma \right)\right)\right), \\ ={\mathfrak{S}}^{*}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)+\frac{1}{2}\gamma \left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon ,\varsigma \right), \varsigma +\rho \gamma \left(\upsilon ,\varsigma \right)\right)\right)\\ \times {\mathfrak{U}}^{*}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)+\frac{1}{2}\gamma \left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon ,\varsigma \right), \varsigma +\rho \gamma \left(\upsilon ,\varsigma \right)\right)\right),\\ \le {\mathfrak{w}}_1\left(\frac{1}{2}\right){\mathfrak{w}}_2\left(\frac{1}{2}\right)\left[\begin{array}{c}{\mathfrak{S}}_{*}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon , \varsigma \right)\right)\times {\mathfrak{U}}_{*}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon , \varsigma \right)\right)\\ +{\mathfrak{S}}_{*}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon , \varsigma \right)\right)\times {\mathfrak{U}}_{*}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)\end{array}\right]\\ + {\mathfrak{w}}_1\left(\frac{1}{2}\right){\mathfrak{w}}_2\left(\frac{1}{2}\right)\left[\begin{array}{c}{\mathfrak{S}}_{*}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)\times {\mathfrak{U}}_{*}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon , \varsigma \right)\right)\\ +{\mathfrak{S}}_{*}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)\times {\mathfrak{U}}_{*}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)\end{array}\right],\\ \ge {\mathfrak{w}}_1\left(\frac{1}{2}\right){\mathfrak{w}}_2\left(\frac{1}{2}\right)\left[\begin{array}{c}{\mathfrak{S}}^{*}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon , \varsigma \right)\right)\times {\mathfrak{U}}^{*}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon , \varsigma \right)\right)\\ +{\mathfrak{S}}^{*}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon , \varsigma \right)\right)\times {\mathfrak{U}}^{*}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)\end{array}\right]\\ +{\mathfrak{w}}_1\left(\frac{1}{2}\right){\mathfrak{w}}_2\left(\frac{1}{2}\right)\left[\begin{array}{c}{\mathfrak{S}}^{*}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)\times {\mathfrak{U}}^{*}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon , \varsigma \right)\right)\\ +{\mathfrak{S}}^{*}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)\times {\mathfrak{U}}^{*}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)\end{array}\right],\\ \le {\mathfrak{w}}_1\left(\frac{1}{2}\right){\mathfrak{w}}_2\left(\frac{1}{2}\right)\left[\begin{array}{c}{\mathfrak{S}}_{*}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon , \varsigma \right)\right)\times {\mathfrak{U}}_{*}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon , \varsigma \right)\right)\\ +{\mathfrak{S}}_{*}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)\times {\mathfrak{U}}_{*}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)\end{array}\right]\\ +{\mathfrak{w}}_1\left(\frac{1}{2}\right){\mathfrak{w}}_2\left(\frac{1}{2}\right)\left[\begin{array}{c}\left({\mathfrak{w}}_1\left(\rho \right){\mathfrak{S}}_{*}\left(\varsigma \right)+{\mathfrak{w}}_1\left(1-\rho \right){\mathfrak{S}}_{*}\left(\upsilon \right)\right)\\ \times \left({\mathfrak{w}}_2\left(1-\rho \right){\mathfrak{U}}_{*}\left(\varsigma \right)+{\mathfrak{w}}_2\left(\rho \right){\mathfrak{U}}_{*}\left(\upsilon \right)\right)\\ +\left({\mathfrak{w}}_1\left(1-\rho \right){\mathfrak{S}}_{*}\left(\varsigma \right)+{\mathfrak{w}}_1\left(\rho \right){\mathfrak{S}}_{*}\left(\upsilon \right)\right)\\ \times \left({\mathfrak{w}}_2\left(\rho \right){\mathfrak{U}}_{*}\left(\varsigma \right)+{\mathfrak{w}}_2\left(1-\rho \right){\mathfrak{U}}_{*}\left(\upsilon \right)\right)\end{array}\right],\\ \ge {\mathfrak{w}}_1\left(\frac{1}{2}\right){\mathfrak{w}}_2\left(\frac{1}{2}\right)\left[\begin{array}{c}{\mathfrak{S}}^{*}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon , \varsigma \right)\right)\times {\mathfrak{U}}^{*}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon , \varsigma \right)\right)\\ +{\mathfrak{S}}^{*}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)\times {\mathfrak{U}}^{*}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)\end{array}\right]\\ + {\mathfrak{w}}_1\left(\frac{1}{2}\right){\mathfrak{w}}_2\left(\frac{1}{2}\right)\left[\begin{array}{c}\left({\mathfrak{w}}_1\left(\rho \right){\mathfrak{S}}^{*}\left(\varsigma \right)+{\mathfrak{w}}_1\left(1-\rho \right){\mathfrak{S}}^{*}\left(\upsilon \right)\right)\\ \times \left({\mathfrak{w}}_2\left(1-\rho \right){\mathfrak{U}}^{*}\left(\varsigma \right)+{\mathfrak{w}}_2\left(\rho \right){\mathfrak{U}}^{*}\left(\upsilon \right)\right)\\ +\left({\mathfrak{w}}_1\left(1-\rho \right){\mathfrak{S}}^{*}\left(\varsigma \right)+{\mathfrak{w}}_1\left(\rho \right){\mathfrak{S}}^{*}\left(\upsilon \right)\right)\\ \times \left({\mathfrak{w}}_2\left(\rho \right){\mathfrak{U}}^{*}\left(\varsigma \right)+{\mathfrak{w}}_2\left(1-\rho \right){\mathfrak{U}}^{*}\left(\upsilon \right)\right)\end{array}\right],\\ ={\mathfrak{w}}_1\left(\frac{1}{2}\right){\mathfrak{w}}_2\left(\frac{1}{2}\right)\left[\begin{array}{c}{\mathfrak{S}}_{*}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon , \varsigma \right)\right)\times {\mathfrak{U}}_{*}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon , \varsigma \right)\right)\\ +{\mathfrak{S}}_{*}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)\times {\mathfrak{U}}_{*}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)\end{array}\right]\\ +2{\mathfrak{w}}_1\left(\frac{1}{2}\right){\mathfrak{w}}_2\left(\frac{1}{2}\right)\left[\begin{array}{c}\left\{{\mathfrak{w}}_1\left(\rho \right){\mathfrak{w}}_2\left(\rho \right)+{\mathfrak{w}}_1\left(1-\rho \right){\mathfrak{w}}_2\left(1-\rho \right)\right\}{ℒ}_{*}\left(\varsigma ,\upsilon \right)\\ +\left\{{\mathfrak{w}}_1\left(\rho \right){\mathfrak{w}}_2\left(1-\rho \right)+{\mathfrak{w}}_1\left(1-\rho \right){\mathfrak{w}}_2\left(\rho \right)\right\}{\mathcal{K}}_{*}\left(\varsigma ,\upsilon \right)\end{array}\right],\\ ={\mathfrak{w}}_1\left(\frac{1}{2}\right){\mathfrak{w}}_2\left(\frac{1}{2}\right)\left[\begin{array}{c}{\mathfrak{S}}^{*}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon , \varsigma \right)\right)\times {\mathfrak{U}}^{*}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon , \varsigma \right)\right)\\ +{\mathfrak{S}}^{*}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)\times {\mathfrak{U}}^{*}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)\end{array}\right]\\ +2{\mathfrak{w}}_1\left(\frac{1}{2}\right){\mathfrak{w}}_2\left(\frac{1}{2}\right)\left[\begin{array}{c}\left\{{\mathfrak{w}}_1\left(\rho \right){\mathfrak{w}}_2\left(\rho \right)+{\mathfrak{w}}_1\left(1-\rho \right){\mathfrak{w}}_2\left(1-\rho \right)\right\}{ℒ}^{*}\left(\varsigma ,\upsilon \right)\\ +\left\{{\mathfrak{w}}_1\left(\rho \right){\mathfrak{w}}_2\left(1-\rho \right)+{\mathfrak{w}}_1\left(1-\rho \right){\mathfrak{w}}_2\left(\rho \right)\right\}{\mathcal{K}}^{*}\left(\varsigma ,\upsilon \right)\end{array}\right],\end{array}$$

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

$$\begin{array}{cc}\hfill \frac{1}{2{\mathfrak{w}}_1\left(\frac{1}{2}\right){\mathfrak{w}}_2\left(\frac{1}{2}\right)} {\mathfrak{S}}_{*}& \left(\frac{2\varsigma +\gamma \left(\upsilon , \varsigma \right)}{2}\right)\times {\mathfrak{U}}_{*}\left(\frac{2\varsigma +\gamma \left(\upsilon , \varsigma \right)}{2}\right)\le \frac{1}{\gamma \left(\upsilon , \varsigma \right)} {\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}{\mathfrak{S}}_{*}\left(𝓸\right)\times {\mathfrak{U}}_{*}\left(𝓸\right)d𝓸\hfill \\ & +{\mathcal{K}}_{*}\left(\varsigma ,\upsilon \right){\int }_0^1{\mathfrak{w}}_1\left(\rho \right){\mathfrak{w}}_2\left(1-\rho \right)d\rho +{ℒ}_{*}\left(\varsigma ,\upsilon \right){\int }_0^1{\mathfrak{w}}_1\left(\rho \right){\mathfrak{w}}_2\left(\rho \right)d\rho , \hfill \\ \hfill \frac{1}{2{\mathfrak{w}}_1\left(\frac{1}{2}\right){\mathfrak{w}}_2\left(\frac{1}{2}\right)} {\mathfrak{S}}^{*}& \left(\frac{2\varsigma +\gamma \left(\upsilon , \varsigma \right)}{2}\right)\times {\mathfrak{U}}^{*}\left(\frac{2\varsigma +\gamma \left(\upsilon , \varsigma \right)}{2}\right)\ge \frac{1}{\gamma \left(\upsilon , \varsigma \right)} {\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}{\mathfrak{S}}^{*}\left(𝓸\right)\times {\mathfrak{U}}^{*}\left(𝓸\right)d𝓸\hfill \\ & + {\mathcal{K}}^{*}\left(\varsigma ,\upsilon \right){\int }_0^1{\mathfrak{w}}_1\left(\rho \right){\mathfrak{w}}_2\left(1-\rho \right)d\rho +{ℒ}^{*}\left(\varsigma ,\upsilon \right){\int }_0^1{\mathfrak{w}}_1\left(\rho \right){\mathfrak{w}}_2\left(\rho \right)d\rho ,\hfill \end{array}$$

from which, we have

$$\begin{array}{c}\frac{1}{2{\mathfrak{w}}_1\left(\frac{1}{2}\right){\mathfrak{w}}_2\left(\frac{1}{2}\right)}\left[{\mathfrak{S}}_{*}\left(\frac{2\varsigma +\gamma \left(\upsilon , \varsigma \right)}{2}\right)\times {\mathfrak{U}}_{*}\left(\frac{2\varsigma +\gamma \left(\upsilon , \varsigma \right)}{2}\right), {\mathfrak{S}}^{*}\left(\frac{2\varsigma +\gamma \left(\upsilon , \varsigma \right)}{2}\right)\times {\mathfrak{U}}^{*}\left(\frac{2\varsigma +\gamma \left(\upsilon , \varsigma \right)}{2}\right)\right]\\ \supseteq \frac{1}{\gamma \left(\upsilon , \varsigma \right)}\left[{{\displaystyle \int }}_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}{\mathfrak{S}}_{*}\left(𝓸\right)\times {\mathfrak{U}}_{*}\left(𝓸\right)d𝓸 , {{\displaystyle \int }}_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}{\mathfrak{S}}^{*}\left(𝓸\right)\times {\mathfrak{U}}^{*}\left(𝓸\right)d𝓸\right]\\ +{\int }_0^1{\mathfrak{w}}_1\left(\rho \right){\mathfrak{w}}_2\left(1-\rho \right)d\rho \left[{\mathcal{K}}_{*}\left(\varsigma ,\upsilon \right), {\mathcal{K}}^{*}\left(\varsigma ,\upsilon \right)\right]+\left[{ℒ}_{*}\left(\varsigma ,\upsilon \right), {ℒ}^{*}\left(\varsigma ,\upsilon \right)\right]{\int }_0^1{\mathfrak{w}}_1\left(\rho \right){\mathfrak{w}}_2\left(\rho \right)d\rho ,\end{array}$$

that is

$$\begin{array}{c}\frac{1}{2{\mathfrak{w}}_1\left(\frac{1}{2}\right){\mathfrak{w}}_2\left(\frac{1}{2}\right)}\mathfrak{S}\left(\frac{2\varsigma +\gamma \left(\upsilon , \varsigma \right)}{2}\right)\times \mathfrak{U}\left(\frac{2\varsigma +\gamma \left(\upsilon , \varsigma \right)}{2}\right)\supseteq \frac{1}{\gamma \left(\upsilon , \varsigma \right)} \left(IR\right){\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}\mathfrak{S}\left(𝓸\right)\times \mathfrak{U}\left(𝓸\right)d𝓸\\ +\mathcal{K}\left(\varsigma ,\upsilon \right){\int }_0^1{\mathfrak{w}}_1\left(\rho \right){\mathfrak{w}}_2\left(1-\rho \right)d\rho +ℒ\left(\varsigma ,\upsilon \right){\int }_0^1{\mathfrak{w}}_1\left(\rho \right){\mathfrak{w}}_2\left(\rho \right)d\rho ,\end{array}$$

this completes the result.  □

Example 4.

We consider ${\mathfrak{w}}_1\left(\rho \right)=\rho , {\mathfrak{w}}_2\left(\rho \right)=\rho ,$ for $\rho \in \left[0, 1\right]$, and the IVFs $\mathfrak{S},\mathfrak{U}:\left[\varsigma , \varsigma +\gamma \left(\upsilon , \varsigma \right)\right]=\left[0, \gamma \left(1, 0\right)\right]\to {ℝ}_{\mathbb{I}}^{+}$ defined by $\mathfrak{S}\left(𝓸\right)=\left[2{𝓸}^2,4𝓸\right]$ and $\mathfrak{U}\left(𝓸\right)=\left[𝓸,2𝓸\right],$ as in Example 3, and $\mathfrak{S}\left(𝓸\right), \mathfrak{U}\left(𝓸\right)$ both are ${\mathfrak{w}}_1$ and ${\mathfrak{w}}_2$-preinvex IVFs with respect to $\gamma \left(\upsilon , \varsigma \right)=\upsilon -\varsigma$, respectively. Since ${\mathfrak{S}}_{*}\left(𝓸\right)=2{𝓸}^2,$ ${\mathfrak{S}}^{*}\left(𝓸\right)=4𝓸$ and ${\mathfrak{U}}_{*}\left(𝓸\right)=𝓸$, ${\mathfrak{U}}^{*}\left(𝓸\right)=2𝓸$ then, we have

$$\begin{array}{c}\frac{1}{2{\mathfrak{w}}_1\left(\frac{1}{2}\right){\mathfrak{w}}_2\left(\frac{1}{2}\right)} {\mathfrak{S}}_{*}\left(\frac{2\varsigma +\gamma \left(\upsilon , \varsigma \right)}{2}\right)\times {\mathfrak{U}}_{*}\left(\frac{2\varsigma +\gamma \left(\upsilon , \varsigma \right)}{2}\right)=\frac{1}{2}, \\ \frac{1}{2{\mathfrak{w}}_1\left(\frac{1}{2}\right){\mathfrak{w}}_2\left(\frac{1}{2}\right)} {\mathfrak{S}}^{*}\left(\frac{2\varsigma +\gamma \left(\upsilon , \varsigma \right)}{2}\right)\times {\mathfrak{U}}^{*}\left(\frac{2\varsigma +\gamma \left(\upsilon , \varsigma \right)}{2}\right)=4,\end{array}$$

$$\begin{array}{c}\frac{1}{\gamma \left(\upsilon , \varsigma \right)} {\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}{\mathfrak{S}}_{*}\left(𝓸\right)\times {\mathfrak{U}}_{*}\left(𝓸\right)d𝓸=\frac{1}{2} \\ \frac{1}{\gamma \left(\upsilon , \varsigma \right)} {\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}{\mathfrak{S}}^{*}\left(𝓸\right)\times {\mathfrak{U}}^{*}\left(𝓸\right)d𝓸=4,\end{array}$$

$$\begin{array}{c}{\mathcal{K}}_{*}\left(\varsigma ,\upsilon \right){\int }_0^1{\mathfrak{w}}_1\left(\rho \right){\mathfrak{w}}_2\left(1-\rho \right)d\rho =\frac{1}{3}, \\ {\mathcal{K}}^{*}\left(\varsigma ,\upsilon \right){\int }_0^1{\mathfrak{w}}_1\left(\rho \right){\mathfrak{w}}_2\left(1-\rho \right)d\rho =\frac{4}{3},\end{array}$$

$$\begin{array}{c}{ℒ}_{*}\left(\varsigma ,\upsilon \right){\int }_0^1{\mathfrak{w}}_1\left(\rho \right){\mathfrak{w}}_2\left(\rho \right)d\rho =0,\\ {ℒ}^{*}\left(\varsigma ,\upsilon \right){\int }_0^1{\mathfrak{w}}_1\left(\rho \right){\mathfrak{w}}_2\left(\rho \right)d\rho =0,\end{array}$$

that means

$$\begin{array}{c}\frac{1}{2}\le \frac{1}{2}+0+\frac{1}{3}=\frac{5}{6}, \\ 4\ge 4+0+\frac{4}{3}=4,\end{array}$$

hence, Theorem 5 is demonstrated.

We now give H–H Fejér inequalities for 𝔴-preinvex IVFs. Firstly, we obtain the second H–H Fejér inequality for 𝔴-preinvex IVF.

Theorem 6.

Let $\mathfrak{S}:\left[\varsigma , \varsigma +\gamma \left(\upsilon , \varsigma \right)\right]\to {ℝ}_{\mathbb{I}}^{+}$ be a 𝔴-preinvex IVF with $\varsigma <\varsigma +\gamma \left(\upsilon , \varsigma \right)$ and $\mathfrak{w}:\left[0, 1\right]\to {ℝ}^{+}$given by $\mathfrak{S}\left(𝓸\right)=\left[{\mathfrak{S}}_{*}\left(𝓸\right), {\mathfrak{S}}^{*}\left(𝓸\right)\right]$for all $𝓸\in \left[\varsigma , \varsigma +\gamma \left(\upsilon , \varsigma \right)\right]$. If $\mathfrak{S}\in ℐ{ℛ}_{\left(\left[\varsigma , \varsigma +\gamma \left(\upsilon , \varsigma \right)\right]\right)}$ and $\mathfrak{X}:\left[\varsigma , \varsigma +\gamma \left(\upsilon , \varsigma \right)\right]\to ℝ, \mathfrak{X}\left(𝓸\right)\ge 0,$ symmetric with respect to $\varsigma +\frac{1}{2}\gamma \left(\upsilon , \varsigma \right),$ then

$$\frac{1}{\gamma \left(\upsilon , \varsigma \right)} \left(IR\right){\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}\mathfrak{S}\left(𝓸\right)\mathfrak{X}\left(𝓸\right)d𝓸\supseteq \left[\mathfrak{S}\left(\varsigma \right)+\mathfrak{S}\left(\upsilon \right)\right]{\int }_0^1\mathfrak{w}\left(\rho \right)\mathfrak{X}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)d\rho .$$

(22)

Proof. 

Let $\mathfrak{S}$ be a 𝔴-preinvex IVF. Then, we have

$$\begin{array}{l}{\mathfrak{S}}_{*}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon ,\varsigma \right)\right)\mathfrak{X}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon ,\varsigma \right)\right)\\ \le \left(\mathfrak{w}\left(\rho \right){\mathfrak{S}}_{*}\left(\varsigma \right)+\mathfrak{w}\left(1-\rho \right){\mathfrak{S}}_{*}\left(\upsilon \right)\right)\mathfrak{X}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon ,\varsigma \right)\right),\\ {\mathfrak{S}}^{*}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon ,\varsigma \right)\right)\mathfrak{X}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon ,\varsigma \right)\right)\\ \ge \left(\mathfrak{w}\left(\rho \right){\mathfrak{S}}^{*}\left(\varsigma \right)+\mathfrak{w}\left(1-\rho \right){\mathfrak{S}}^{*}\left(\upsilon \right)\right)\mathfrak{X}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon ,\varsigma \right)\right).\end{array}$$

(23)

And

$$\begin{array}{c}{\mathfrak{S}}_{*}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)X\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)\le \left(\mathfrak{w}\left(1-\rho \right){\mathfrak{S}}_{*}\left(\varsigma \right)+\mathfrak{w}\left(\rho \right){\mathfrak{S}}_{*}\left(\upsilon \right)\right)X\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right),\\ {\mathfrak{S}}^{*}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)X\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)\ge \left(\mathfrak{w}\left(1-\rho \right){\mathfrak{S}}^{*}\left(\varsigma \right)+\mathfrak{w}\left(\rho \right){\mathfrak{S}}^{*}\left(\upsilon \right)\right)X\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right).\end{array}$$

(24)

After adding (23) and (24), and integrating over $\left[0, 1\right],$ we obtain

$$\begin{array}{l}{\int }_0^1{\mathfrak{S}}_{*}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon ,\varsigma \right)\right)\mathfrak{X}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon ,\varsigma \right)\right)d\rho +{\int }_0^1{\mathfrak{S}}_{*}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)\mathfrak{X}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)d\rho \\ \le {\int }_0^1\left[\begin{array}{c}{\mathfrak{S}}_{*}\left(\varsigma \right)\left\{\mathfrak{w}\left(\rho \right)\mathfrak{X}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon ,\varsigma \right)\right)+\mathfrak{w}\left(1-\rho \right)\mathfrak{X}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)\right\}\\ +{\mathfrak{S}}_{*}\left(\upsilon \right)\left\{\mathfrak{w}\left(1-\rho \right)\mathfrak{X}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon ,\varsigma \right)\right)+\mathfrak{w}\left(\rho \right)\mathfrak{X}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)\right\}\end{array}\right]d\rho ,\\ {\int }_0^1{\mathfrak{S}}^{*}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)\mathfrak{X}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)d\rho +{\int }_0^1{\mathfrak{S}}^{*}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon ,\varsigma \right)\right)\mathfrak{X}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon ,\varsigma \right)\right)d\rho \\ \ge {\int }_0^1\left[\begin{array}{c}{\mathfrak{S}}^{*}\left(\varsigma \right)\left\{\mathfrak{w}\left(\rho \right)\mathfrak{X}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon ,\varsigma \right)\right)+\mathfrak{w}\left(1-\rho \right)\mathfrak{X}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)\right\}\\ +{\mathfrak{S}}^{*}\left(\upsilon \right)\left\{\mathfrak{w}\left(1-\rho \right)\mathfrak{X}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon ,\varsigma \right)\right)+\mathfrak{w}\left(\rho \right)\mathfrak{X}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)\right\}\end{array}\right]d\rho .\end{array}$$

$$\begin{array}{c}=2{\mathfrak{S}}_{*}\left(\varsigma \right){\int }_0^1\begin{array}{c}\mathfrak{w}\left(\rho \right)\mathfrak{X}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon ,\varsigma \right)\right)\end{array}d\rho +2{\mathfrak{S}}_{*}\left(\upsilon \right){\int }_0^1\begin{array}{c}\mathfrak{w}\left(\rho \right)X\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)\end{array}d\rho ,\\ =2{\mathfrak{S}}^{*}\left(\varsigma \right){\int }_0^1\begin{array}{c}\mathfrak{w}\left(\rho \right)\mathfrak{X}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon ,\varsigma \right)\right)\end{array}d\rho +2{\mathfrak{S}}^{*}\left(\upsilon \right){\int }_0^1\begin{array}{c}\mathfrak{w}\left(\rho \right)X\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)\end{array}d\rho .\end{array}$$

Since $\mathfrak{X}$ is symmetric, then

$$\begin{array}{c}=2\left[{\mathfrak{S}}_{*}\left(\varsigma \right)+{\mathfrak{S}}_{*}\left(\upsilon \right)\right]{\int }_0^1\begin{array}{c}\mathfrak{w}\left(\rho \right)X\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)\end{array}d\rho ,\\ =2\left[{\mathfrak{S}}^{*}\left(\varsigma \right)+{\mathfrak{S}}^{*}\left(\upsilon \right)\right]{\int }_0^1\begin{array}{c}\mathfrak{w}\left(\rho \right)X\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)\end{array}d\rho .\end{array}$$

(25)

Since

$$\begin{array}{l}{\int }_0^1{\mathfrak{S}}_{*}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon ,\varsigma \right)\right)\mathfrak{X}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon ,\varsigma \right)\right)d\rho \\ ={\int }_0^1{\mathfrak{S}}_{*}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)\mathfrak{X}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)d\rho \\ =\frac{1}{\gamma \left(\upsilon , \varsigma \right)} {\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}{\mathfrak{S}}_{*}\left(𝓸\right)\mathfrak{X}\left(𝓸\right)d𝓸 \\ {\int }_0^1{\mathfrak{S}}^{*}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon ,\varsigma \right)\right)\mathfrak{X}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon ,\varsigma \right)\right)d\rho \\ ={\int }_0^1{\mathfrak{S}}^{*}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)\mathfrak{X}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)d\rho \\ =\frac{1}{\gamma \left(\upsilon , \varsigma \right)} {\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}{\mathfrak{S}}^{*}\left(𝓸\right)\mathfrak{X}\left(𝓸\right)d𝓸.\end{array}$$

(26)

From (25) and (26), we have

$$\begin{array}{c}\frac{1}{\gamma \left(\upsilon , \varsigma \right)} {\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}{\mathfrak{S}}_{*}\left(𝓸\right)\mathfrak{X}\left(𝓸\right)d𝓸\le \left[{\mathfrak{S}}_{*}\left(\varsigma \right)+{\mathfrak{S}}_{*}\left(\upsilon \right)\right]{\int }_0^1\begin{array}{c}\mathfrak{w}\left(\rho \right)X\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)\end{array}d\rho ,\\ \frac{1}{\gamma \left(\upsilon , \varsigma \right)} {\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}{\mathfrak{S}}^{*}\left(𝓸\right)\mathfrak{X}\left(𝓸\right)d𝓸\ge \left[{\mathfrak{S}}^{*}\left(\varsigma \right)+{\mathfrak{S}}^{*}\left(\upsilon \right)\right]{\int }_0^1\begin{array}{c}\mathfrak{w}\left(\rho \right)X\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)\end{array}d\rho ,\end{array}$$

that is

$$\begin{array}{l}\left[\frac{1}{\gamma \left(\upsilon , \varsigma \right)} {\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}{\mathfrak{S}}_{*}\left(𝓸\right)\mathfrak{X}\left(𝓸\right)d𝓸, \frac{1}{\gamma \left(\upsilon , \varsigma \right)} {\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}{\mathfrak{S}}^{*}\left(𝓸\right)\mathfrak{X}\left(𝓸\right)d𝓸\right]\\ \supseteq \left[{\mathfrak{S}}_{*}\left(\varsigma \right)+{\mathfrak{S}}_{*}\left(\upsilon \right), {\mathfrak{S}}^{*}\left(\varsigma \right)+{\mathfrak{S}}^{*}\left(\upsilon \right)\right]{\int }_0^1\begin{array}{c}\mathfrak{w}\left(\rho \right)X\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)\end{array}d\rho ,\end{array}$$

hence

$$\frac{1}{\gamma \left(\upsilon , \varsigma \right)} \left(IR\right){\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}\mathfrak{S}\left(𝓸\right)\mathfrak{X}\left(𝓸\right)d𝓸\supseteq \left[\mathfrak{S}\left(\varsigma \right) \widetilde{+}\mathfrak{S}\left(\upsilon \right)\right]{\int }_0^1\mathfrak{w}\left(\rho \right)\mathfrak{X}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)d\rho .$$

This completes the proof.  □

Next, we construct the first H–H Fejér inequality for 𝔴-preinvex IVF, which generalizes the first H–H Fejér inequality for the 𝔴-preinvex function, see [58,59].

Theorem 7.

Let $\mathfrak{S}:\left[\varsigma , \varsigma +\gamma \left(\upsilon , \varsigma \right)\right]\to {ℝ}_{\mathbb{I}}^{+}$ be a 𝔴-preinvex IVF with $\varsigma <\varsigma +\gamma \left(\upsilon , \varsigma \right)$ and $\mathfrak{w}:\left[0, 1\right]\to {ℝ}^{+}$, such that $\mathfrak{S}\left(𝓸\right)=\left[{\mathfrak{S}}_{*}\left(𝓸\right), {\mathfrak{S}}^{*}\left(𝓸\right)\right]$ for all $𝓸\in \left[\varsigma , \varsigma +\gamma \left(\upsilon , \varsigma \right)\right]$. If $\mathfrak{S}\in ℐ{ℛ}_{\left(\left[\varsigma , \varsigma +\gamma \left(\upsilon , \varsigma \right)\right]\right)}$ and $\mathfrak{X}:\left[\varsigma , \varsigma +\gamma \left(\upsilon , \varsigma \right)\right]\to ℝ, \mathfrak{X}\left(𝓸\right)\ge 0,$ symmetric with respect to $\varsigma +\frac{1}{2}\gamma \left(\upsilon , \varsigma \right),$ and ${\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}\mathfrak{X}\left(𝓸\right)d𝓸>0$, and Condition C for $\gamma$, then

$$\mathfrak{S}\left(\varsigma +\frac{1}{2}\gamma \left(\upsilon , \varsigma \right)\right)\supseteq \frac{2\mathfrak{w}\left(\frac{1}{2}\right)}{{\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}\mathfrak{X}\left(𝓸\right)d𝓸} \left(IR\right){\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}\mathfrak{S}\left(𝓸\right)\mathfrak{X}\left(𝓸\right)d𝓸.$$

(27)

Proof. 

Using condition C, we can write

$$\varsigma +\frac{1}{2}\gamma \left(\upsilon , \varsigma \right)=\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)+\frac{1}{2}\gamma \left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon ,\varsigma \right), \varsigma +\rho \gamma \left(\upsilon ,\varsigma \right)\right).$$

Since $\mathfrak{S}$ is a 𝔴-preinvex, we have

$$\begin{array}{cc}\hfill {\mathfrak{S}}_{*}\left(\varsigma +\frac{1}{2}\gamma \left(\upsilon , \varsigma \right)\right)=& {\mathfrak{S}}_{*}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)+\frac{1}{2}\gamma \left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon ,\varsigma \right), \varsigma +\rho \gamma \left(\upsilon ,\varsigma \right)\right)\right),\hfill \\ & \le w\left(\frac{1}{2}\right)\left({\mathfrak{S}}_{*}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon ,\varsigma \right)\right)+{\mathfrak{S}}_{*}\left(\varsigma +\rho \gamma \left(\upsilon ,\varsigma \right)\right)\right),\hfill \\ \hfill {\mathfrak{S}}^{*}\left(\varsigma +\frac{1}{2}\gamma \left(\upsilon , \varsigma \right)\right)=& {\mathfrak{S}}^{*}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)+\frac{1}{2}\gamma \left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon ,\varsigma \right), \varsigma +\rho \gamma \left(\upsilon ,\varsigma \right)\right)\right),\hfill \\ & \ge w\left(\frac{1}{2}\right)\left({\mathfrak{S}}^{*}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon ,\varsigma \right)\right)+{\mathfrak{S}}^{*}\left(\varsigma +\rho \gamma \left(\upsilon ,\varsigma \right)\right)\right),\hfill \end{array}$$

(28)

By multiplying (28) by $\mathfrak{X}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon ,\varsigma \right)\right)=\mathfrak{X}\left(\varsigma +\rho \gamma \left(\upsilon ,\varsigma \right)\right)$ and integrating it by $\rho$ over $\left[0, 1\right],$ we obtain:

$$\begin{array}{l}{\mathfrak{S}}_{*}\left(\varsigma +\frac{1}{2}\gamma \left(\upsilon , \varsigma \right)\right){\int }_0^1\mathfrak{X}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)d\rho \\ \le \mathfrak{w}\left(\frac{1}{2}\right)\left(\begin{array}{c}{\int }_0^1{\mathfrak{S}}_{*}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon ,\varsigma \right)\right)\mathfrak{X}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon , \varsigma \right)\right)d\rho \\ +{\int }_0^1{\mathfrak{S}}_{*}\left(\varsigma +\rho \gamma \left(\upsilon ,\varsigma \right)\right)\mathfrak{X}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)d\rho \end{array}\right),\\ {\mathfrak{S}}^{*}\left(\varsigma +\frac{1}{2}\gamma \left(\upsilon , \varsigma \right)\right){\int }_0^1\mathfrak{X}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)d\rho \\ \ge \mathfrak{w}\left(\frac{1}{2}\right)\left(\begin{array}{c}{\int }_0^1{\mathfrak{S}}^{*}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon ,\varsigma \right)\right)\mathfrak{X}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon , \varsigma \right)\right)d\rho \\ +{\int }_0^1{\mathfrak{S}}^{*}\left(\varsigma +\rho \gamma \left(\upsilon ,\varsigma \right)\right)\mathfrak{X}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)d\rho \end{array}\right),\end{array}$$

(29)

Since:

$$\begin{array}{l}{\int }_0^1{\mathfrak{S}}_{*}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon ,\varsigma \right)\right)\mathfrak{X}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon ,\varsigma \right)\right)d\rho \\ ={{\displaystyle \int }}_0^1{\mathfrak{S}}_{*}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)\mathfrak{X}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)d\rho ,\\ =\frac{1}{\gamma \left(\upsilon , \varsigma \right)} {{\displaystyle \int }}_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}{\mathfrak{S}}_{*}\left(𝓸\right)\mathfrak{X}\left(𝓸\right)d𝓸,\\ {{\displaystyle \int }}_0^1{\mathfrak{S}}^{*}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)\mathfrak{X}\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)d\rho \\ ={{\displaystyle \int }}_0^1{\mathfrak{S}}^{*}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon ,\varsigma \right)\right)\mathfrak{X}\left(\varsigma +\left(1-\rho \right)\gamma \left(\upsilon ,\varsigma \right)\right)d\rho ,\\ =\frac{1}{\gamma \left(\upsilon , \varsigma \right)} {{\displaystyle \int }}_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}{\mathfrak{S}}^{*}\left(𝓸\right)\mathfrak{X}\left(𝓸\right)d𝓸,\end{array}$$

(30)

From (29) and (30), we have

$$\begin{array}{c}{\mathfrak{S}}_{*}\left(\varsigma +\frac{1}{2}\gamma \left(\upsilon , \varsigma \right)\right) \le \frac{2\mathfrak{w}\left(\frac{1}{2}\right)}{{\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}\mathfrak{X}\left(𝓸\right)d𝓸} {\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}{\mathfrak{S}}_{*}\left(𝓸\right)\mathfrak{X}\left(𝓸\right)d𝓸,\\ {\mathfrak{S}}^{*}\left(\varsigma +\frac{1}{2}\gamma \left(\upsilon , \varsigma \right)\right) \ge \frac{2\mathfrak{w}\left(\frac{1}{2}\right)}{{\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}\mathfrak{X}\left(𝓸\right)d𝓸} {\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}{\mathfrak{S}}^{*}\left(𝓸\right)\mathfrak{X}\left(𝓸\right)d𝓸.\end{array}$$

From which, we have

$$\begin{array}{l}\left[{\mathfrak{S}}_{*}\left(\varsigma +\frac{1}{2}\gamma \left(\upsilon , \varsigma \right)\right), {\mathfrak{S}}^{*}\left(\varsigma +\frac{1}{2}\gamma \left(\upsilon , \varsigma \right)\right)\right]\\ \supseteq \frac{2\mathfrak{w}\left(\frac{1}{2}\right)}{{\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}\mathfrak{X}\left(𝓸\right)d𝓸}\left[ {\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}{\mathfrak{S}}_{*}\left(𝓸\right)\mathfrak{X}\left(𝓸\right)d𝓸, {\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}{\mathfrak{S}}^{*}\left(𝓸\right)\mathfrak{X}\left(𝓸\right)d𝓸\right],\end{array}$$

that is

$$\mathfrak{S}\left(\varsigma +\frac{1}{2}\gamma \left(\upsilon , \varsigma \right)\right)\supseteq \frac{2\mathfrak{w}\left(\frac{1}{2}\right)}{{{\displaystyle \int }}_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}\mathfrak{X}\left(𝓸\right)d𝓸} \left(IR\right){\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}\mathfrak{S}\left(𝓸\right)\mathfrak{X}\left(𝓸\right)d𝓸,$$

Then the proof is complete.  □

Remark 4.

If one takes $\mathfrak{w}\left(\rho \right)=\rho$, then (22) and (27) reduces to inequalities for preinvex IVFs.

If one takes ${\mathfrak{S}}_{*}\left(𝓸\right)={\mathfrak{S}}^{*}\left(𝓸\right)$, then Inequalities (22) and (27) reduces to the classical first and second H–H Fejér inequality for 𝔴-preinvex function, see [59].

If one takes ${\mathfrak{S}}_{*}\left(𝓸\right)={\mathfrak{S}}^{*}\left(𝓸\right)$ $\gamma \left(\upsilon , \varsigma \right)=\upsilon -\varsigma$ then Inequalities (22) and (27) reduces to classical first and second H–H–Fejér inequality for 𝔴-convex function, see [58].

Example 5.

We consider $\mathfrak{w}\left(\rho \right)=\rho ,$ for $\rho \in \left[0, 1\right]$ and the IVF $\mathfrak{S}:\left[1, 1+\partial \left(4, 1\right)\right]\to {ℝ}_{\mathbb{I}}^{+}$ defined by $\mathfrak{S}\left(𝓸\right)=\left[\frac{1}{𝓸},𝓸\right]$. Since ${\mathfrak{S}}_{*}\left(𝓸\right)$ and ${\mathfrak{S}}^{*}\left(𝓸\right)$ are 𝔴-preinvex functions $\gamma \left(y,𝓸\right)=y-𝓸$, then $\mathfrak{S}\left(𝓸\right)$ is 𝔴-preinvex IVF. If

$$\mathfrak{X}\left(𝓸\right)=\{\begin{array}{c}𝓸-1,\hspace{1em}\hspace{1em}\varrho \in \left[1,\frac{5}{2}\right]\\ 4-𝓸,\hspace{1em}\varrho \in \left(\frac{5}{2}, 4\right],\end{array}$$

then, we have

$$\begin{array}{l}\frac{1}{\gamma \left(4, 1\right)}{\int }_1^{1+ \gamma \left(4, 1\right)}{\mathfrak{S}}_{*}\left(𝓸\right)\mathfrak{X}\left(𝓸\right)d𝓸=\frac{1}{3}{\int }_1^4{\mathfrak{S}}_{*}\left(𝓸\right)\mathfrak{X}\left(𝓸\right)d𝓸=\frac{1}{3}{\int }_1^{\frac{5}{2}}{\mathfrak{S}}_{*}\left(𝓸\right)\mathfrak{X}\left(𝓸\right)d𝓸\\ +\frac{1}{3}{\int }_{\frac{5}{2}}^4{\mathfrak{S}}_{*}\left(𝓸\right)\mathfrak{X}\left(𝓸\right)d𝓸, \\ \frac{1}{\gamma \left(4, 1\right)}{\int }_1^{1+ \gamma \left(4, 1\right)}{\mathfrak{S}}^{*}\left(𝓸\right)\mathfrak{X}\left(𝓸\right)d𝓸=\frac{1}{3}{\int }_1^4{\mathfrak{S}}^{*}\left(𝓸\right)\mathfrak{X}\left(𝓸\right)d𝓸=\frac{1}{3}{\int }_1^{\frac{5}{2}}{\mathfrak{S}}^{*}\left(𝓸\right)\mathfrak{X}\left(𝓸\right)d𝓸\\ +\frac{1}{3}{\int }_{\frac{5}{2}}^4{\mathfrak{S}}^{*}\left(𝓸\right)\mathfrak{X}\left(𝓸\right)d𝓸,\end{array}$$

$$\begin{array}{l}=\frac{1}{3}{\int }_1^{\frac{5}{2}}\frac{1}{𝓸}\left(𝓸-1\right)d𝓸+\frac{1}{3}{\int }_{\frac{5}{2}}^4\frac{1}{𝓸}\left(4-𝓸\right)d𝓸=\frac{1}{3}\left(4log\left(\frac{8}{5}\right)+log\left(\frac{5}{2}\right)\right),\\ =\frac{1}{3}{\int }_1^{\frac{5}{2}}𝓸\left(𝓸-1\right)d𝓸+\frac{1}{3}{\int }_{\frac{5}{2}}^4𝓸\left(4-𝓸\right)d𝓸=\frac{15}{8},\end{array}$$

(31)

And

$$\begin{array}{c}\left[{\mathfrak{S}}_{*}\left(\varsigma \right)+{\mathfrak{S}}_{*}\left(\upsilon \right)\right]{\int }_0^1\begin{array}{c}\mathfrak{w}\left(\rho \right)X\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)\end{array}d\rho \\ \left[{\mathfrak{S}}^{*}\left(\varsigma \right)+{\mathfrak{S}}^{*}\left(\upsilon \right)\right]{\int }_0^1\begin{array}{c}\mathfrak{w}\left(\rho \right)X\left(\varsigma +\rho \gamma \left(\upsilon , \varsigma \right)\right)\end{array}d\rho \end{array}$$

$$\begin{array}{l}=\frac{5}{4} \left[{\int }_1^{\frac{5}{2}}3{\rho }^{2}d𝓸+{\int }_{\frac{5}{2}}^4\rho \left(3-3\rho \right)d\rho \right]=\frac{15}{32}.\\ =5\left[{\int }_1^{\frac{5}{2}}3{\rho }^{2}d𝓸+{\int }_{\frac{5}{2}}^4\rho \left(3-3\rho \right)d\rho \right]=\frac{15}{8}.\end{array}$$

(32)

From (31) and (32), we have

$$\left[\frac{1}{3}\left(4log\left(\frac{8}{5}\right)+log\left(\frac{5}{2}\right)\right), \frac{15}{8}\right]\supseteq \left[\frac{15}{32}, \frac{15}{8}\right]$$

Hence, Theorem 6 is verified.

For Theorem 7, we have

$$\begin{array}{c}{\mathfrak{S}}_{*}\left(\varsigma +\frac{1}{2}\gamma \left(\upsilon , \varsigma \right)\right)=\frac{2}{5},\\ {\mathfrak{S}}^{*}\left(\varsigma +\frac{1}{2}\gamma \left(\upsilon , \varsigma \right)\right)=\frac{5}{2},\end{array}$$

(33)

$${\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}\mathfrak{X}\left(𝓸\right)d𝓸={\int }_1^{\frac{5}{2}}\left(𝓸-1\right)d𝓸+{\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}\left(4-𝓸\right)d𝓸=\frac{9}{4},$$

$$\begin{array}{c}\frac{2\mathfrak{w}\left(\frac{1}{2}\right)}{{\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}\mathfrak{X}\left(𝓸\right)d𝓸} {\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}{\mathfrak{S}}_{*}\left(𝓸\right)\mathfrak{X}\left(𝓸\right)d𝓸=\frac{4}{9}\left(4log\left(\frac{8}{5}\right)+log\left(\frac{5}{2}\right)\right)\\ \frac{2\mathfrak{w}\left(\frac{1}{2}\right)}{{\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}\mathfrak{X}\left(𝓸\right)d𝓸} {\int }_{\varsigma }^{\varsigma +\gamma \left(\upsilon , \varsigma \right)}{\mathfrak{S}}^{*}\left(𝓸\right)\mathfrak{X}\left(𝓸\right)d𝓸=\frac{5}{2}\end{array}$$

(34)

From (33) and (34), we have

$$\left[\frac{2}{5}, \frac{5}{2}\right]\supseteq \left[\frac{4}{9}\left(4log\left(\frac{8}{5}\right)+log\left(\frac{5}{2}\right)\right), \frac{5}{2}\right].$$

Hence, Theorem 7 is verified.

4. Conclusions

For 𝔴-preinvex interval-valued functions, we have found the Hermite–Hadamard type inclusions in this paper. In addition, we have demonstrated Hermite–Hadamard–Fejer’ type inclusions for symmetric functions and Pachpatte type inclusions for the product of two 𝔴-preinvex interval-valued functions. In the future, we will investigate the quantum (or q-) calculus and 𝔴-preinvex interval-valued functions on coordinates. This new study aims to motivate researchers in interval analysis, fractional calculus, and other important areas.