Some New Versions of Fractional Inequalities for Exponential Trigonometric Convex Mappings via Ordered Relation on Interval-Valued Settings

Abstract

This paper’s main goal is to introduce left and right exponential trigonometric convex interval-valued mappings and to go over some of their important characteristics. Additionally, we demonstrate the Hermite–Hadamard inequality for interval-valued functions by utilizing fractional integrals with exponential kernels. Moreover, we use the idea of left and right exponential trigonometric convex interval-valued mappings to show various findings for midpoint- and Pachpatte-type inequalities. Additionally, we show that the results provided in this paper are expansions of several of the results already demonstrated in prior publications The suggested research generates variants that are applicable for conducting in-depth analyses of fractal theory, optimization, and research challenges in several practical domains, such as computer science, quantum mechanics, and quantum physics.

1. Introduction

It is common knowledge that mathematical subjects such as mathematical economy, probability theory, optimal control theory, and others depend heavily on convex function and convexity. Classical convexity has been expanded and generalized over time to include harmonic convexity, h-convexity, and p-convexity, among others. In reality, inequality is the basis for the ideas of convexity and convex function, and its significance cannot be overstated. One of the most significant classical inequalities, the Hermite–Hadamard (HH) inequality below, has recently received a lot of attention.

For a convex mapping $\mathsf{Ϣ}:K\to ℝ$ on an interval $K=\left[ȥ, \mathsf{ѵ}\right]$, the HH inequality is written as:

$$\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\le \frac{1}{\mathsf{ѵ}-ȥ}{{\displaystyle \int }}_{ȥ}^{\mathsf{ѵ}}\mathsf{Ϣ}\left(\varkappa \right)d\varkappa \le \frac{\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)}{2}.$$

(1)

For all $ȥ, \mathsf{ѵ}\in K,$ with $K$ being a convex set. If $\mathsf{Ϣ}$ is concave, then (1) is reversed.

The following inequality as the weighted generalization of (1) was established by Fejér in [1]. This important generalization of the HH inequality is known as the HH–Fejér inequality.

Let us consider $\mathsf{Ϣ}:K=\left[ȥ, \mathsf{ѵ}\right]\to ℝ$ a convex mapping on a convex set $K$, and $ȥ, \mathsf{ѵ}\in K$. Then, we have

$$\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\le \frac{1}{{{\displaystyle \int }}_{ȥ}^{\mathsf{ѵ}}\mathcal{C}\left(\varkappa \right)d\varkappa }{{\displaystyle \int }}_{ȥ}^{\mathsf{ѵ}}\mathsf{Ϣ}\left(\varkappa \right)\mathcal{C}\left(\varkappa \right)d\varkappa \le \frac{\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)}{2}.$$

(2)

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

Integral inequality (1) and (2) in various variants have also been extensively examined in [2,3,4,5,6,7,8,9,10] due to the differences between the ideas of convexity. In order to further their study and take advantage of the growing significance of fractional integrals, numerous writers have combined fractional integrals and Hermite–Hadamard-type inequalities. Recent advances in this field in different areas of mathematics can easily be seen and we refer readers to references [11,12,13,14,15,16,17,18,19,20,21,22].

Some fractional Hermite–Hadamard-type inequalities have been discovered in this way; for more information, see references [23,24,25,26,27,28,29,30,31,32]. This field of inequalities has many applications. Similarly, various other types of inequalities have found the bounds of mean inequalities. For more information, see also [33,34,35,36,37,38,39,40,41,42,43].

On the other hand, Moore initially presented interval analysis as a key method to manage interval uncertainty [44]. This has a wide range of applications [45,46,47,48,49,50,51,52,53,54]. Recently, Khan et al. also contributed to this field and defined different types of inequalities using crip theory and fuzzy theory, see [55,56,57,58].

In particular, researchers such as Chalco-Cano et al. [59,60], Costa and Román-Flores [61], Zhao et al. [62,63], An et al. [64], and others have studied a number of classical inequalities with interval-valued functions. Budak et al. [65] demonstrated the fractional Hermite–Hadamard inequality for the interval convex function as an additional extension. Since then, the authors of [66,67,68,69,70,71,72,73,74,75,76] have extensively investigated various additional improvements to and expansions of Hermite–Hadamard inequalities for different convex fuzzy-valued functions. Additionally, in [77], some Hermite–Hadamard- and Jensen-type inequalities for up and down convex fuzzy-number-valued functions were discovered. In this study, several Hermite–Hadamard-type inequalities for interval-valued left and right exponential trigonometric functions are established. The earlier inequalities described in [78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93] are generalized by our findings. For more information, see [94,95,96,97,98,99].

We establish some additional modifications for interval fractional Hermite–Hadamard-type inequalities as a result of [77,78,85,86]. Our findings clarify some previous questions. Furthermore, it is possible that the findings will be acknowledged as important approaches to investigating the study of interval-valued differential equations, interval optimization, and interval vector spaces, among other things. In Section 2, we provide an introduction. The idea of left and right exponential trigonometric I-V∙M is introduced in Section 3 along with several intervals fractional Hermite–Hadamard-type inequalities that are proven. Finally, several examples are provided in Section 4.

2. Preliminaries

Let $X_C$ be the space of all closed and bounded intervals of $ℝ$ and $\mathsf{Л}\in X_C$ defined by

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

(3)

If ${\mathsf{Л}}_{*}={\mathsf{Л}}^{*}$, then $\mathsf{Л}$ is said to be degenerate. In this article, all intervals will be non-degenerate intervals. If ${\mathsf{Л}}_{*}\ge 0$, then $\left[{\mathsf{Л}}_{*}, {\mathsf{Л}}^{*}\right]$ is called a positive interval. The set of all positive intervals is denoted by $X_C^{+}$ and defined as

$$X_C^{+}=\left\{\left[{\mathsf{Л}}_{*}, {\mathsf{Л}}^{*}\right]:\left[{\mathsf{Л}}_{*}, {\mathsf{Л}}^{*}\right]\in X_C \mathrm{and} {\mathsf{Л}}_{*}\ge 0\right\}.$$

Let $\lambda \in ℝ$ and $\lambda \cdot \mathsf{Л}$ be defined by

$$\lambda \cdot \mathsf{Л}=\{\begin{array}{cc}\left[\lambda {\mathsf{Л}}_{*}, \lambda {\mathsf{Л}}^{*}\right]& \mathrm{if} \lambda >0,\\ \left\{0\right\}& \mathrm{if} \lambda =0,\\ \left[\lambda {\mathsf{Л}}^{*}{\mathsf{Л}}_{*}\right]& \mathrm{if} \lambda <0.\end{array}$$

(4)

Then, the Minkowski difference $\mathsf{Ʋ}-\mathsf{Л}$, addition $\mathsf{Л}+\mathsf{Ʋ}$ and $\mathsf{Л}\times \mathsf{Ʋ}$ for $\mathsf{Л},\mathsf{Ʋ}\in X_C$ are defined by

$$\left[{\mathsf{Ʋ}}_{*}, {\mathsf{Ʋ}}^{*}\right]+\left[{\mathsf{Л}}_{*}, {\mathsf{Л}}^{*}\right]=\left[{\mathsf{Ʋ}}_{*}+{\mathsf{Л}}_{*}, {\mathsf{Ʋ}}^{*}+{\mathsf{Л}}^{*}\right],$$

(5)

$$\left[{\mathsf{Ʋ}}_{*}, {\mathsf{Ʋ}}^{*}\right]\times \left[{\mathsf{Л}}_{*}, {\mathsf{Л}}^{*}\right]=\left[\min \left\{{\mathsf{Ʋ}}_{*}{\mathsf{Л}}_{*}, {\mathsf{Ʋ}}^{*}{\mathsf{Л}}_{*}, {\mathsf{Ʋ}}_{*}{\mathsf{Л}}^{*}, {\mathsf{Ʋ}}^{*}{\mathsf{Л}}^{*}\right\}, \max \left\{{\mathsf{Ʋ}}_{*}{\mathsf{Л}}_{*}, {\mathsf{Ʋ}}^{*}{\mathsf{Л}}_{*}, {\mathsf{Ʋ}}_{*}{\mathsf{Л}}^{*}, {\mathsf{Ʋ}}^{*}{\mathsf{Л}}^{*}\right\}\right]$$

(6)

$$\left[{\mathsf{Ʋ}}_{*}, {\mathsf{Ʋ}}^{*}\right]-\left[{\mathsf{Л}}_{*}, {\mathsf{Л}}^{*}\right]=\left[{\mathsf{Ʋ}}_{*}-{\mathsf{Л}}^{*}, {\mathsf{Ʋ}}^{*}-{\mathsf{Л}}_{*}\right].$$

(7)

Remark 1.

For given $\left[{\mathsf{Ʋ}}_{*}, {\mathsf{Ʋ}}^{*}\right], \left[{\mathsf{Л}}_{*},{ Л}^{*}\right]\in X_C,$ we say that $\left[{\mathsf{Ʋ}}_{*}, {\mathsf{Ʋ}}^{*}\right]{\le }_p\left[{\mathsf{Л}}_{*},{ \mathsf{Л}}^{*}\right]$ if and only if ${\mathsf{Ʋ}}_{*}\le {\mathsf{Л}}_{*}, {\mathsf{Ʋ}}^{*}\le {\mathsf{Л}}^{*}$ is a partial interval order relation [84].

For $\left[{\mathsf{Ʋ}}_{*}, {\mathsf{Ʋ}}^{*}\right], \left[{\mathsf{Л}}_{*}, {\mathsf{Л}}^{*}\right]\in X_C,$ the Hausdorff–Pompeiu distance between intervals $\left[{\mathsf{Ʋ}}_{*}, {\mathsf{Ʋ}}^{*}\right]$ and $\left[{\mathsf{Л}}_{*}, {\mathsf{Л}}^{*}\right]$ is defined by

$$d_H\left(\left[{\mathsf{Ʋ}}_{*}, {\mathsf{Ʋ}}^{*}\right], \left[{\mathsf{Л}}_{*}, {\mathsf{Л}}^{*}\right]\right)=\mathrm{ma}\varkappa \left\{\left|{\mathsf{Ʋ}}_{*}-{\mathsf{Л}}_{*}\right|, \left|{\mathsf{Ʋ}}^{*}-{\mathsf{Л}}^{*}\right|\right\}.$$

(8)

It is a familiar fact that $\left(X_C,d_H\right)$ is a complete metric space, see [79,82,83].

3. Fractional Integral Operators of Real- and Interval-Valued Mappings

Now, we define and discuss some properties of fractional integral operators of real- and interval-valued mappings.

Theorem 1.

If $\mathsf{Ϣ}:\left[ȥ,\mathsf{ѵ}\right]\subset ℝ\to X_C$ is an interval-valued mapping (I∙V∙M) satisfying that $\mathsf{Ϣ}\left(\varkappa \right)=\left[{\mathsf{Ϣ}}_{*}\left(\varkappa \right),{ \mathsf{Ϣ}}^{*}\left(\varkappa \right)\right]$, then $\mathsf{Ϣ}$ is Aumann integrable (IA-integrable) over $\left[ȥ,\mathsf{ѵ}\right]$ when and only when ${\mathsf{Ϣ}}_{*}\left(\varkappa \right)$ and ${\mathsf{Ϣ}}^{*}\left(\varkappa \right)$ are both integrable over  $\left[ȥ,\mathsf{ѵ}\right]$ such that [79,81]

$$\left(IA\right){{\displaystyle \int }}_{ȥ}^{\mathsf{ѵ}}\mathsf{Ϣ}\left(\varkappa \right)d\varkappa =\left[{{\displaystyle \int }}_{ȥ}^{\begin{array}{c} \\ \mathsf{ѵ}\end{array}}{\mathsf{Ϣ}}_{*}\left(\varkappa \right)d\varkappa , {{\displaystyle \int }}_{ȥ}^{\mathsf{ѵ}}{\mathsf{Ϣ}}^{*}\left(\varkappa \right)d\varkappa \right].$$

(9)

Definition 1.

Let  $\alpha >0$ and $L\left(\left[ȥ, \mathsf{ѵ}\right], ℝ\right)$ be the collection of all Lebesgue-measurable mapping on $\left[ȥ,\mathsf{ѵ}\right]$. Then, the left and right Riemann–Liouville fractional integral with exponential kernels in connection of $\mathsf{Ϣ}\in$ $L\left(\left[ȥ, \mathsf{ѵ}\right],ℝ\right)$ with order $\alpha >0$ are, respectively, defined by [85]: 

$${ℐ}_{{ȥ}^{+}}^{\alpha }\mathsf{Ϣ}\left(\varkappa \right)=\frac{1}{\alpha }{{\displaystyle \int }}_{ȥ}^{\varkappa }{e}^{\left(-\frac{1-\alpha }{\alpha }\left(\varkappa -\mathfrak{v}\right)\right)}\mathsf{Ϣ}\left(\mathfrak{v}\right)\mathrm{d}\mathfrak{v},\left(\varkappa >ȥ\right),$$

(10)

 and

$${ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha }\mathsf{Ϣ}\left(\varkappa \right)=\frac{1}{\alpha }{{\displaystyle \int }}_{\varkappa }^{\mathsf{ѵ}}{e}^{\left(-\frac{1-\alpha }{\alpha }\left(\mathfrak{v}-\varkappa \right)\right)}\mathsf{Ϣ}\left(\mathfrak{v}\right)d\mathfrak{v},(\varkappa <\mathsf{ѵ})$$

(11)

Definition 2.

Let $\alpha >0$ and  $L\left(\left[ȥ, \mathsf{ѵ}\right],X_C^{ }\right)$ be the collection of all Lebesgue-measurable interval-valued mapping on $\left[ȥ,\mathsf{ѵ}\right]$. Then, the left and right Riemann–Liouville fractional integral with exponential kernels in connection of $\mathsf{Ϣ}\in$ $L\left(\left[ȥ, \mathsf{ѵ}\right],X_C^{ }\right)$ with order $\alpha >0$ are, respectively, defined by [86]

$${ℐ}_{{ȥ}^{+}}^{\alpha }\mathsf{Ϣ}\left(\varkappa \right)=\left[{ℐ}_{{ȥ}^{+}}^{\alpha }{\mathsf{Ϣ}}_{*}\left(\varkappa \right),{ℐ}_{{ȥ}^{+}}^{\alpha }{\mathsf{Ϣ}}^{*}\left(\varkappa \right)\right]=\frac{1}{\alpha }{{\displaystyle \int }}_{ȥ}^{\varkappa }{e}^{\left(-\frac{1-\alpha }{\alpha }\left(\varkappa -\mathfrak{v}\right)\right)}\left[{\mathsf{Ϣ}}_{*}\left(\mathfrak{v}\right),{\mathsf{Ϣ}}^{*}\left(\mathfrak{v}\right)\right]d\mathfrak{v},\left(\varkappa >ȥ\right),$$

(12)

and

$${ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha }\mathsf{Ϣ}\left(\varkappa \right)=\left[{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha }{\mathsf{Ϣ}}_{*}\left(\varkappa \right),{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha }{\mathsf{Ϣ}}^{*}\left(\varkappa \right)\right]=\frac{1}{\alpha }{{\displaystyle \int }}_{\varkappa }^{\mathsf{ѵ}}{e}^{\left(-\frac{1-\alpha }{\alpha }\left(\mathfrak{v}-\varkappa \right)\right)}\left[{\mathsf{Ϣ}}_{*}\left(\mathfrak{v}\right),{\mathsf{Ϣ}}^{*}\left(\mathfrak{v}\right)\right]d\mathfrak{v},(\varkappa <\mathsf{ѵ}),$$

(13)

Definition 3.

The mapping $\mathsf{Ϣ}:\left[ȥ, \mathsf{ѵ}\right]\to ℝ$ is called exponential trigonometric convex mapping on $\left[ȥ, \mathsf{ѵ}\right]$ if [78]

$$\mathsf{Ϣ}\left(\mathfrak{v}\varkappa +\left(1-\mathfrak{v}\right)s\right)\le \frac{\sin \frac{\pi \mathfrak{v}}{2}}{e^{1-\mathfrak{v}}}\mathsf{Ϣ}\left(\varkappa \right)+\frac{\cos \frac{\pi \mathfrak{v}}{2}}{e^{\mathfrak{v}}}\mathsf{Ϣ}\left(s\right).$$

(14)

For all $\varkappa , s\in \left[ȥ, \mathsf{ѵ}\right], \mathfrak{v}\in \left[0, 1\right],$ and $\varkappa \in \left[ȥ, \mathsf{ѵ}\right].$ If (14) is reversed, then $\mathsf{Ϣ}$ is called exponential trigonometric concave mapping on $\left[ȥ, \mathsf{ѵ}\right]$.

4. Left and Right Exponential Trigonometric Convex Interval-Valued Functions

In following results, we will use left and right Riemann–Liouville fractional integrals with left and right exponential kernels, and some nontrivial examples are also given to prove the validity of these integrals and results.

Definition 4.

The I-V∙M $\mathsf{Ϣ}:\left[ȥ, \mathsf{ѵ}\right]\to X_C^{ }$ is called a left and right exponential trigonometric convex I-V∙M on $\left[ȥ, \mathsf{ѵ}\right]$ if

$$\mathsf{Ϣ}\left(\mathfrak{v}\varkappa +\left(1-\mathfrak{v}\right)s\right){\le }_p\frac{\mathit{sin}\frac{\mathsf{\pi }\mathfrak{v}}{2}}{e^{1-\mathfrak{v}}}\mathsf{Ϣ}\left(\varkappa \right)+\frac{\mathit{cos}\frac{\mathsf{\pi }\mathfrak{v}}{2}}{e^{\mathfrak{v}}}\mathsf{Ϣ}\left(s\right).$$

(15)

For all$\varkappa , s\in \left[ȥ, \mathsf{ѵ}\right], \mathfrak{v}\in \left[0, 1\right],$ where $\mathsf{Ϣ}\left(\varkappa \right){\ge }_{p}0$ for all $\varkappa \in \left[ȥ, \mathsf{ѵ}\right].$ If (15) is reversed, then $\mathsf{Ϣ}$ is called a left and right exponential trigonometric concave I-V∙M on $\left[ȥ, \mathsf{ѵ}\right]$.

Theorem 2.

Let $K$ be an invex set and $\mathsf{Ϣ}:K\to X_C^{ }$ be a F-N-V∙M given by

$$\mathsf{Ϣ}\left(\varkappa \right)=\left[{\mathsf{Ϣ}}_{*}\left(\varkappa \right),{ \mathsf{Ϣ}}^{*}\left(\varkappa \right)\right], \forall \varkappa \in K.$$

(16)

For all $\varkappa \in K$. Then $\mathsf{Ϣ}$ is a left and right exponential trigonometric convex F-N-V∙M on $K$ if and only if ${\mathsf{Ϣ}}_{*}\left(\varkappa \right)$ and ${\mathsf{Ϣ}}^{*}\left(\varkappa \right)$ are both exponential trigonometric convex mappings.

Proof. 

Consider that ${\mathsf{Ϣ}}_{*}\left(\varkappa \right)$ and ${\mathsf{Ϣ}}^{*}\left(\varkappa \right)$ are both exponential trigonometric convex and concave mappings on $K$, respectively. Then, from (14), we have

$${\mathsf{Ϣ}}_{*}\left(\mathfrak{v}\varkappa +\left(1-\mathfrak{v}\right)s\right)\le \frac{\sin \frac{\pi \mathfrak{v}}{2}}{e^{1-\mathfrak{v}}}{\mathsf{Ϣ}}_{*}\left(\varkappa \right)+\frac{\cos \frac{\pi \mathfrak{v}}{2}}{e^{\mathfrak{v}}}{\mathsf{Ϣ}}_{*}\left(s\right),\forall \varkappa ,s\in K, \mathfrak{v}\in \left[0, 1\right],$$

and

$${\mathsf{Ϣ}}^{*}\left(\mathfrak{v}\varkappa +\left(1-\mathfrak{v}\right)s\right)\le \frac{\sin \frac{\pi \mathfrak{v}}{2}}{e^{1-\mathfrak{v}}}{\mathsf{Ϣ}}^{*}\left(\varkappa \right)+\frac{\cos \frac{\pi \mathfrak{v}}{2}}{e^{\mathfrak{v}}}{\mathsf{Ϣ}}^{*}\left(s\right),\forall \varkappa ,s\in K, \mathfrak{v}\in \left[0, 1\right].$$

Then, by (16), (8), and (10), we obtain

$$\begin{array}{l}\mathsf{Ϣ}\left(\mathfrak{v}\varkappa +\left(1-\mathfrak{v}\right)s\right)\\ =\left[{\mathsf{Ϣ}}_{*}\left(\mathfrak{v}\varkappa +\left(1-\mathfrak{v}\right)s\right), {\mathsf{Ϣ}}^{*}\left(\mathfrak{v}\varkappa +\left(1-\mathfrak{v}\right)s\right)\right],\\ {\le }_p\frac{\sin \frac{\pi \mathfrak{v}}{2}}{e^{1-\mathfrak{v}}}\left[{\mathsf{Ϣ}}_{*}\left(\varkappa \right), {\mathsf{Ϣ}}^{*}\left(\varkappa \right)\right]+\frac{\cos \frac{\pi \mathfrak{v}}{2}}{e^{\mathfrak{v}}}\left[{\mathsf{Ϣ}}_{*}\left(s\right), {\mathsf{Ϣ}}^{*}\left(s\right)\right],\end{array}$$

that is

$$\mathsf{Ϣ}\left(\mathfrak{v}\varkappa +\left(1-\mathfrak{v}\right)s\right){\le }_p\frac{\sin \frac{\pi \mathfrak{v}}{2}}{e^{1-\mathfrak{v}}}\mathsf{Ϣ}\left(\varkappa \right)+\frac{\cos \frac{\pi \mathfrak{v}}{2}}{e^{\mathfrak{v}}}\mathsf{Ϣ}\left(s\right),\forall \varkappa ,s\in K, \mathfrak{v}\in \left[0, 1\right].$$

Hence, $\mathsf{Ϣ}$ is a left and right exponential trigonometric convex F-N-V∙M on $K$.

Conversely, let $\mathsf{Ϣ}$ be a left and right exponential trigonometric convex F-N-V∙M on $K.$ Then, for all $\varkappa ,s\in K$ and $\mathfrak{v}\in \left[0, 1\right],$ we have

$$\mathsf{Ϣ}\left(\mathfrak{v}\varkappa +\left(1-\mathfrak{v}\right)s\right){\le }_p\frac{\sin \frac{\pi \mathfrak{v}}{2}}{e^{1-\mathfrak{v}}}\mathsf{Ϣ}\left(\varkappa \right)+\frac{\cos \frac{\pi \mathfrak{v}}{2}}{e^{\mathfrak{v}}}\mathsf{Ϣ}\left(s\right).$$

Therefore, from (16), we have

$$\mathsf{Ϣ}\left(\mathfrak{v}\varkappa +\left(1-\mathfrak{v}\right)s\right)=\left[{\mathsf{Ϣ}}_{*}\left(\mathfrak{v}\varkappa +\left(1-\mathfrak{v}\right)s\right), {\mathsf{Ϣ}}^{*}\left(\mathfrak{v}\varkappa +\left(1-\mathfrak{v}\right)s\right)\right].$$

Again, from (16), (6), and (8), we obtain

$$\frac{\sin \frac{\pi \mathfrak{v}}{2}}{e^{1-\mathfrak{v}}}\mathsf{Ϣ}\left(\varkappa \right)+\frac{\cos \frac{\pi \mathfrak{v}}{2}}{e^{\mathfrak{v}}}\mathsf{Ϣ}\left(\varkappa \right)=\left[\frac{\sin \frac{\pi \mathfrak{v}}{2}}{e^{1-\mathfrak{v}}}{\mathsf{Ϣ}}_{*}\left(\varkappa \right), \frac{\sin \frac{\pi \mathfrak{v}}{2}}{e^{1-\mathfrak{v}}}{\mathsf{Ϣ}}^{*}\left(\varkappa \right)\right]+\left[\frac{\cos \frac{\pi \mathfrak{v}}{2}}{e^{\mathfrak{v}}}{\mathsf{Ϣ}}_{*}\left(s\right), \frac{\cos \frac{\pi \mathfrak{v}}{2}}{e^{\mathfrak{v}}}{\mathsf{Ϣ}}^{*}\left(s\right)\right].$$

For all $\varkappa ,s\in K$ and $\mathfrak{v}\in \left[0, 1\right].$ Then, by left and right exponential trigonometric convexity of $\mathsf{Ϣ}$, we have for all $\varkappa ,s\in K$ and $\mathfrak{v}\in \left[0, 1\right]$ such that

$${\mathsf{Ϣ}}_{*}\left(\mathfrak{v}\varkappa +\left(1-\mathfrak{v}\right)s\right)\le \frac{\sin \frac{\pi \mathfrak{v}}{2}}{e^{1-\mathfrak{v}}}{\mathsf{Ϣ}}_{*}\left(\varkappa \right)+\frac{\cos \frac{\pi \mathfrak{v}}{2}}{e^{\mathfrak{v}}}{\mathsf{Ϣ}}_{*}\left(s\right),$$

and

$${\mathsf{Ϣ}}^{*}\left(\mathfrak{v}\varkappa +\left(1-\mathfrak{v}\right)s\right)\le \frac{\sin \frac{\pi \mathfrak{v}}{2}}{e^{1-\mathfrak{v}}}{\mathsf{Ϣ}}^{*}\left(\varkappa \right)+\frac{\cos \frac{\pi \mathfrak{v}}{2}}{e^{\mathfrak{v}}}{\mathsf{Ϣ}}^{*}\left(s\right).$$

Hence, the result follows.□

Remark 2.

If ${\mathsf{Ϣ}}_{*}\left(\varkappa \right)={\mathsf{Ϣ}}^{*}\left(\varkappa \right)$, then we obtain the classical definition of exponential trigonometric convex mappings, see [78].

We obtained some new definitions from the literature which will be helpful in investigating some classical and new results as special cases of the main results.

Definition 5.

Let $\mathsf{Ϣ}:\left[ȥ,\mathsf{ѵ}\right]\to X_C^{ }$ be an I-V∙M. Then, $\mathsf{Ϣ}\left(\varkappa \right)$ is given by

$$\mathsf{Ϣ}\left(\varkappa \right)=\left[{\mathsf{Ϣ}}_{*}\left(\varkappa \right),{ \mathsf{Ϣ}}^{*}\left(\varkappa \right)\right].$$

 For all $\varkappa \in \left[ȥ,\mathsf{ѵ}\right]$. Then, $\mathsf{Ϣ}$ is a lower left and right exponential trigonometric convex (concave) I-V∙M on $\left[ȥ,\mathsf{ѵ}\right]$ if and only if

$${\mathsf{Ϣ}}_{*}\left(\mathfrak{v}\varkappa +\left(1-\mathfrak{v}\right)s\right)\le (\ge )\frac{\sin \frac{\pi \mathfrak{v}}{2}}{e^{1-\mathfrak{v}}}{\mathsf{Ϣ}}_{*}\left(\varkappa \right)+\frac{\cos \frac{\pi \mathfrak{v}}{2}}{e^{\mathfrak{v}}}{\mathsf{Ϣ}}_{*}\left(s\right),$$

and

$${\mathsf{Ϣ}}^{*}\left(\mathfrak{v}\varkappa +\left(1-\mathfrak{v}\right)s\right)=\frac{\sin \frac{\pi \mathfrak{v}}{2}}{e^{1-\mathfrak{v}}}{\mathsf{Ϣ}}^{*}\left(\varkappa \right)+\frac{\cos \frac{\pi \mathfrak{v}}{2}}{e^{\mathfrak{v}}}{\mathsf{Ϣ}}^{*}\left(s\right)$$

Definition 6.

Suppose that $\mathsf{Ϣ}:\left[ȥ,\mathsf{ѵ}\right]\to X_C^{ }$ is an I-V∙M that is defined by

$$\mathsf{Ϣ}\left(\varkappa \right)=\left[{\mathsf{Ϣ}}_{*}\left(\varkappa \right), {\mathsf{Ϣ}}^{*}\left(\varkappa \right)\right]$$

For all $\varkappa \in \left[ȥ,\mathsf{ѵ}\right].$ Then, $\mathsf{Ϣ}$ is an upper left and right exponential trigonometric convex (concave) I-V∙M on $\left[ȥ,\mathsf{ѵ}\right]$ if and only if

$${\mathsf{Ϣ}}_{*}\left(\mathfrak{v}\varkappa +\left(1-\mathfrak{v}\right)s\right)=\frac{\sin \frac{\pi \mathfrak{v}}{2}}{e^{1-\mathfrak{v}}}{\mathsf{Ϣ}}_{*}\left(\varkappa \right)+\frac{\cos \frac{\pi \mathfrak{v}}{2}}{e^{\mathfrak{v}}}{\mathsf{Ϣ}}_{*}\left(s\right),$$

 and

$${\mathsf{Ϣ}}^{*}\left(\mathfrak{v}\varkappa +\left(1-\mathfrak{v}\right)s\right)\le (\ge )\frac{\sin \frac{\pi \mathfrak{v}}{2}}{e^{1-\mathfrak{v}}}{\mathsf{Ϣ}}^{*}\left(\varkappa \right)+\frac{\cos \frac{\pi \mathfrak{v}}{2}}{e^{\mathfrak{v}}}{\mathsf{Ϣ}}^{*}\left(s\right)$$

5. Riemann–Liouville Fractional Integrals Hermite–Hadamard-Type Inequalities

In the following section, we use the new concept of left and right exponential trigonometric convex interval-valued mapping to illustrate a few Riemann–Liouville fractional integrals Hermite–Hadamard-type inequalities having exponential kernels.

Theorem 3.

Let $\mathsf{Ϣ}:\left[ȥ, \mathsf{ѵ}\right]\to X_C^{+}$ be an I-V∙M on $\left[ȥ, \mathsf{ѵ}\right]$ given by $\mathsf{Ϣ}\left(\varkappa \right)=\left[{\mathsf{Ϣ}}_{*}\left(\varkappa \right),{ \mathsf{Ϣ}}^{*}\left(\varkappa \right)\right]$ for all $\varkappa \in \left[ȥ, \mathsf{ѵ}\right]$. If $\mathsf{Ϣ}$ is a left and right exponential trigonometric convex I-V∙M on $\left[ȥ, \mathsf{ѵ}\right]$ and $\mathsf{Ϣ}\in L\left(\left[ȥ, \mathsf{ѵ}\right],X_C^{+}\right)$, then

$$\sqrt{\frac{e}{2}}\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right){\le }_p\frac{1-\alpha }{2\left(1-e^{-\rho }\right)}\left[{ℐ}_{{ȥ}^{+}}^{\alpha }\mathsf{Ϣ}\left(\mathsf{ѵ}\right)+{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha }\mathsf{Ϣ}(ȥ)\right]{\le }_p\frac{\rho }{1-e^{-\rho }}C\left(\rho \right)\frac{\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)}{2}.$$

(17)

If  $\mathsf{Ϣ}\left(\varkappa \right)$ is a left and right exponential trigonometric concave I-V∙M, then 

$$\begin{array}{cc}\hfill \sqrt{\frac{e}{2}}\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)& {\ge }_p\frac{1-\alpha }{2\left(1-e^{-\rho }\right)}[{ℐ}_{{ȥ}^{+}}^{\alpha }\mathsf{Ϣ}\left(\mathsf{ѵ}\right)\hfill \\ & +{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha }\mathsf{Ϣ}(ȥ)]{\ge }_p\frac{\rho }{1-e^{-\rho }}C\left(\rho \right)\frac{\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)}{2}\end{array}$$

(18)

 where 

$$C\left(\rho \right)=\frac{4\rho +2\pi e^{-\rho -1}+4}{4{\rho }^2+8\rho +{\pi }^2+4}+\frac{2\pi e^{-1}+4e^{-\rho -1}\left(e+\rho e\right)}{4{\rho }^2-8\rho +{\pi }^2+4}, \rho =\frac{1-\alpha }{\alpha }\left(\mathsf{ѵ}-ȥ\right) \mathrm{and} 1>\alpha >0.$$

Proof. 

Let $\mathsf{Ϣ}:\left[ȥ, \mathsf{ѵ}\right]\to X_C^{+}$ be a left and right exponential trigonometric convex I-V∙M. Then, by hypothesis, we have

$$\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right){\le }_p\frac{\sin \frac{\pi }{4}}{\sqrt{e}}\mathsf{Ϣ}\left(\mathfrak{v}ȥ+\left(1-\mathfrak{v}\right)\mathsf{ѵ}\right)+\frac{\cos \frac{\pi }{4}}{\sqrt{e}}\mathsf{Ϣ}\left(\left(1-\mathfrak{v}\right)ȥ+\mathfrak{v}\mathsf{ѵ}\right).$$

After simplification, we find that

$$2\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right){\le }_p\sqrt{\frac{2}{e}}\left[\mathsf{Ϣ}\left(\mathfrak{v}ȥ+\left(1-\mathfrak{v}\right)\mathsf{ѵ}\right)+\mathsf{Ϣ}\left(\left(1-\mathfrak{v}\right)ȥ+\mathfrak{v}\mathsf{ѵ}\right)\right].$$

Therefore, we have

$$\begin{array}{c}2{\mathsf{Ϣ}}_{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\le \sqrt{\frac{2}{e}}\left[{\mathsf{Ϣ}}_{*}\left(\mathfrak{v}ȥ+\left(1-\mathfrak{v}\right)\mathsf{ѵ}\right)+{\mathsf{Ϣ}}_{*}\left(\left(1-\mathfrak{v}\right)ȥ+\mathfrak{v}\mathsf{ѵ}\right)\right]\end{array},$$

$$2{\mathsf{Ϣ}}^{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\le \sqrt{\frac{2}{e}}\left[{\mathsf{Ϣ}}^{*}\left(\mathfrak{v}ȥ+\left(1-\mathfrak{v}\right)\mathsf{ѵ}\right)+{\mathsf{Ϣ}}^{*}\left(\left(1-\mathfrak{v}\right)ȥ+\mathfrak{v}\mathsf{ѵ}\right)\right].$$

Taking ${\mathsf{Ϣ}}_{*}(.)$ and multiplying both sides by $e^{-\rho \mathfrak{v}}$ and integrating the obtained result with respect to $\mathfrak{v}$ from $0$ to $1$, we have

$$2{{\displaystyle \int }}_0^1{e}^{-\rho \mathfrak{v}}{\mathsf{Ϣ}}_{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)d\mathfrak{v} \le \sqrt{\frac{2}{e}}\left[{{\displaystyle \int }}_0^1{e}^{-\rho \mathfrak{v}}{\mathsf{Ϣ}}_{*}\left(\mathfrak{v}ȥ+\left(1-\mathfrak{v}\right)\mathsf{ѵ}\right)d\mathfrak{v}+{{\displaystyle \int }}_0^1{e}^{-\rho \mathfrak{v}}{\mathsf{Ϣ}}_{*}\left(\left(1-\mathfrak{v}\right)ȥ+\mathfrak{v}\mathsf{ѵ}\right)d\mathfrak{v}\right].$$

Let $u=\mathfrak{v}ȥ+\left(1-\mathfrak{v}\right)\mathsf{ѵ}$ and $\varkappa =\left(1-\mathfrak{v}\right)ȥ+\mathfrak{v}\mathsf{ѵ}.$ Then, we have

$$\begin{gathered} 2{{\displaystyle \int }}_0^1{e}^{-\rho \mathfrak{v}}{\mathsf{Ϣ}}_{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)d\mathfrak{v} \le \sqrt{\frac{2}{e}}\frac{1}{\mathsf{ѵ}-ȥ} {{\displaystyle \int }}_{ȥ}^{\mathsf{ѵ}}{e}^{\left(-\frac{1-\alpha }{\alpha }\left(\mathsf{ѵ}-u\right)\right)}{\mathsf{Ϣ}}_{*}\left(u\right)du+\frac{1}{\mathsf{ѵ}-ȥ}{{\displaystyle \int }}_{ȥ}^{\mathsf{ѵ}}{e}^{\left(-\frac{1-\alpha }{\alpha }\left(\varkappa -ȥ\right)\right)}{\mathsf{Ϣ}}_{*}\left(\varkappa \right)d\varkappa \\ =\sqrt{\frac{2}{e}}\frac{\alpha }{\mathsf{ѵ}-ȥ}\left[{ℐ}_{{ȥ}^{+}}^{\alpha } {\mathsf{Ϣ}}_{*}\left(\mathsf{ѵ}\right)+{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha } {\mathsf{Ϣ}}_{*}(ȥ)\right]. \end{gathered}$$

(19)

Now, taking the right side of Equation (19), we have

$${{\displaystyle \int }}_0^1{e}^{-\rho \mathfrak{v}}{\mathsf{Ϣ}}_{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)d\mathfrak{v}=\frac{1-e^{-\rho }}{\rho }{\mathsf{Ϣ}}_{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right).$$

(20)

From (19) and (20), we have

$$\frac{2}{\alpha }·\frac{1-e^{-\rho }}{\rho }{\mathsf{Ϣ}}_{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\le \sqrt{\frac{2}{e}}·\frac{1}{\mathsf{ѵ}-ȥ}\left[{ℐ}_{{ȥ}^{+}}^{\alpha } {\mathsf{Ϣ}}_{*}\left(\mathsf{ѵ}\right)+{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha } {\mathsf{Ϣ}}_{*}(ȥ)\right].$$

(21)

Similarly, for ${\mathsf{Ϣ}}^{*}\left(\varkappa \right)$, we have

$$\frac{2}{\alpha }·\frac{1-e^{-\rho }}{\rho }{\mathsf{Ϣ}}^{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\le \sqrt{\frac{2}{e}}·\frac{1}{\mathsf{ѵ}-ȥ}\left[{ℐ}_{{ȥ}^{+}}^{\alpha } {\mathsf{Ϣ}}^{*}\left(\mathsf{ѵ}\right)+{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha } {\mathsf{Ϣ}}^{*}(ȥ)\right].$$

(22)

From (21) and (22), we have

$$\frac{2}{\alpha }·\frac{1-e^{-\rho }}{\rho }\left[{\mathsf{Ϣ}}_{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right), {\mathsf{Ϣ}}^{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\right] {\le }_p\sqrt{\frac{2}{e}}·\frac{1}{\mathsf{ѵ}-ȥ}\left[\left[{ℐ}_{{ȥ}^{+}}^{\alpha } {\mathsf{Ϣ}}_{*}\left(\mathsf{ѵ}\right)+{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha } {\mathsf{Ϣ}}_{*}(ȥ)\right], \left[{ℐ}_{{ȥ}^{+}}^{\alpha } {\mathsf{Ϣ}}^{*}\left(\mathsf{ѵ}\right)+{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha } {\mathsf{Ϣ}}^{*}(ȥ)\right]\right].$$

That is

$$\frac{2}{\alpha }·\frac{1-e^{-\rho }}{\rho } \mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right){\le }_p\sqrt{\frac{2}{e}}·\frac{1}{\mathsf{ѵ}-ȥ}\left[{ℐ}_{{ȥ}^{+}}^{\alpha }\mathsf{Ϣ}\left(\mathsf{ѵ}\right)+{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha }\mathsf{Ϣ}(ȥ)\right].$$

(23)

For the right side of Equation (17), since $\mathsf{Ϣ}$ is a left and right exponential trigonometric convex I-V∙M, we can deduce that

$$\mathsf{Ϣ}\left(\mathfrak{v}ȥ+\left(1-\mathfrak{v}\right)\mathsf{ѵ}\right){\le }_p\frac{\sin \frac{\pi \mathfrak{v}}{2}}{e^{1-\mathfrak{v}}}\mathsf{Ϣ}(ȥ)+\frac{\cos \frac{\pi \mathfrak{v}}{2}}{e^{\mathfrak{v}}}\mathsf{Ϣ}\left(\mathsf{ѵ}\right),$$

(24)

and

$$\mathsf{Ϣ}\left(\left(1-\mathfrak{v}\right)ȥ+\mathfrak{v}\mathsf{ѵ}\right){\le }_p\frac{\cos \frac{\pi \mathfrak{v}}{2}}{e^{\mathfrak{v}}}\mathsf{Ϣ}(ȥ)+\frac{\sin \frac{\pi \mathfrak{v}}{2}}{e^{1-\mathfrak{v}}}\mathsf{Ϣ}\left(\mathsf{ѵ}\right).$$

(25)

Adding (24) and (25), we have

$$\mathsf{Ϣ}\left(\mathfrak{v}ȥ+\left(1-\mathfrak{v}\right)\mathsf{ѵ}\right)+\mathsf{Ϣ}\left(\left(1-\mathfrak{v}\right)ȥ+\mathfrak{v}\mathsf{ѵ}\right){\le }_p\left[\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)\right]\left[\frac{\sin \frac{\pi \mathfrak{v}}{2}}{e^{1-\mathfrak{v}}}+\frac{\cos \frac{\pi \mathfrak{v}}{2}}{e^{\mathfrak{v}}}\right].$$

(26)

Since $\mathsf{Ϣ}$ is I-V∙M, then we have

$$\begin{array}{c}{\mathsf{Ϣ}}_{*}\left(\mathfrak{v}ȥ+\left(1-\mathfrak{v}\right)\mathsf{ѵ}\right)+{\mathsf{Ϣ}}_{*}\left(\left(1-\mathfrak{v}\right)ȥ+\mathfrak{v}\mathsf{ѵ}\right)\le \left[{\mathsf{Ϣ}}_{*}(ȥ)+{\mathsf{Ϣ}}_{*}\left(\mathsf{ѵ}\right)\right]\left[\frac{\sin \frac{\pi \mathfrak{v}}{2}}{e^{1-\mathfrak{v}}}+\frac{\cos \frac{\pi \mathfrak{v}}{2}}{e^{\mathfrak{v}}}\right], \\ {\mathsf{Ϣ}}^{*}\left(\mathfrak{v}ȥ+\left(1-\mathfrak{v}\right)\mathsf{ѵ}\right)+{\mathsf{Ϣ}}^{*}\left(\left(1-\mathfrak{v}\right)ȥ+\mathfrak{v}\mathsf{ѵ}\right)\le \left[{\mathsf{Ϣ}}^{*}(ȥ)+{\mathsf{Ϣ}}^{*}\left(\mathsf{ѵ}\right)\right]\left[\frac{\sin \frac{\pi \mathfrak{v}}{2}}{e^{1-\mathfrak{v}}}+\frac{\cos \frac{\pi \mathfrak{v}}{2}}{e^{\mathfrak{v}}}\right]. \end{array}$$

(27)

Taking ${\mathsf{Ϣ}}_{*}(.)$ from (27) and multiplying the inequality with $e^{-\rho \mathfrak{v}}$, and integrating the resultant with $\mathfrak{v}$ from $0$ to $1$, we have

$$\begin{array}{cc}\hfill {{\displaystyle \int }}_0^1{e}^{-\rho \mathfrak{v}}{\mathsf{Ϣ}}_{*}\left(\mathfrak{v}ȥ+\left(1-\mathfrak{v}\right)\mathsf{ѵ}\right)& d\mathfrak{v}+{{\displaystyle \int }}_0^1{e}^{-\rho \mathfrak{v}}{\mathsf{Ϣ}}_{*}\left(\left(1-\mathfrak{v}\right)ȥ+\mathfrak{v}\mathsf{ѵ}\right)d\mathfrak{v}\hfill \\ & \le \left[{\mathsf{Ϣ}}_{*}(ȥ)+{\mathsf{Ϣ}}_{*}\left(\mathsf{ѵ}\right)\right]{{\displaystyle \int }}_0^1{e}^{-\rho \mathfrak{v}}\left[\frac{\sin \frac{\pi \mathfrak{v}}{2}}{e^{1-\mathfrak{v}}}+\frac{\cos \frac{\pi \mathfrak{v}}{2}}{e^{\mathfrak{v}}}\right]d\mathfrak{v},\\ & =\frac{4\rho +2\pi e^{-\rho -1}+4}{4{\rho }^2+8\rho +{\pi }^2+4}+\frac{2\pi e^{-1}+4e^{-\rho -1}\left(e+\rho e\right)}{4{\rho }^2-8\rho +{\pi }^2+4}\left[{\mathsf{Ϣ}}_{*}(ȥ)+{\mathsf{Ϣ}}_{*}\left(\mathsf{ѵ}\right)\right].\end{array}$$

(28)

In a similar way to the above, for ${\mathsf{Ϣ}}_{*}(.)$ we have

$$\begin{array}{cc}\hfill {{\displaystyle \int }}_0^1{e}^{-\rho \mathfrak{v}}{\mathsf{Ϣ}}^{*}\left(\mathfrak{v}ȥ+\left(1-\mathfrak{v}\right)\mathsf{ѵ}\right)& d\mathfrak{v}+{{\displaystyle \int }}_0^1{e}^{-\rho \mathfrak{v}}{\mathsf{Ϣ}}^{*}\left(\left(1-\mathfrak{v}\right)ȥ+\mathfrak{v}\mathsf{ѵ}\right)d\mathfrak{v}\hfill \\ & \le \left[{\mathsf{Ϣ}}^{*}(ȥ)+{\mathsf{Ϣ}}^{*}\left(\mathsf{ѵ}\right)\right]{{\displaystyle \int }}_0^1{e}^{-\rho \mathfrak{v}}\left[\frac{\sin \frac{\pi \mathfrak{v}}{2}}{e^{1-\mathfrak{v}}}+\frac{\cos \frac{\pi \mathfrak{v}}{2}}{e^{\mathfrak{v}}}\right]d\mathfrak{v},\\ & =\frac{4\rho +2\pi e^{-\rho -1}+4}{4{\rho }^2+8\rho +{\pi }^2+4}+\frac{2\pi e^{-1}+4e^{-\rho -1}\left(e+\rho e\right)}{4{\rho }^2-8\rho +{\pi }^2+4}\left[{\mathsf{Ϣ}}^{*}(ȥ)+{\mathsf{Ϣ}}^{*}\left(\mathsf{ѵ}\right)\right].\end{array}$$

(29)

From (28) and (29), we have

$$\begin{array}{cc}\hfill {{\displaystyle \int }}_0^1{e}^{-\rho \mathfrak{v}}\mathsf{Ϣ}\left(\mathfrak{v}ȥ+\left(1-\mathfrak{v}\right)\mathsf{ѵ} \right)& d\mathfrak{v}+{{\displaystyle \int }}_0^1{e}^{-\rho \mathfrak{v}}\mathsf{Ϣ}\left(\left(1-\mathfrak{v}\right)ȥ+\mathfrak{v}\mathsf{ѵ} \right)d\mathfrak{v}\hfill \\ & {\le }_p\frac{4\rho +2\pi e^{-\rho -1}+4}{4{\rho }^2+8\rho +{\pi }^2+4}+\frac{2\pi e^{-1}+4e^{-\rho -1}\left(e+\rho e\right)}{4{\rho }^2-8\rho +{\pi }^2+4}\left[\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)\right].\end{array}$$

(30)

From (37) and (30), we have

$$\sqrt{\frac{e}{2}}\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right){\le }_p\frac{1-\alpha }{2\left(1-e^{-\rho }\right)}\left[{ℐ}_{{ȥ}^{+}}^{\alpha }\mathsf{Ϣ}\left(\mathsf{ѵ}\right)+{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha }\mathsf{Ϣ}(ȥ)\right]{\le }_p\frac{\rho }{1-e^{-\rho }}C\left(\rho \right)\frac{\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)}{2}.$$

Hence, the required result. □

If we consider some mild restrictions on Theorem 3, then the following new and classical outcomes can be obtained.

Remark 3.

From Theorem 3, we can clearly see the following.

If one lays $\mathsf{Ϣ}$ which is an upper left and right exponential trigonometric concave I-V∙M on $\left[ȥ,\mathsf{ѵ}\right],$ then one acquires the following inequality [98]:

$$\sqrt{\frac{e}{2}}\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right){\supseteq }_p\frac{1-\alpha }{2\left(1-e^{-\rho }\right)}\left[{ℐ}_{{ȥ}^{+}}^{\alpha }\mathsf{Ϣ}\left(\mathsf{ѵ}\right)+{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha }\mathsf{Ϣ}(ȥ)\right]{\supseteq }_p\frac{\rho }{1-e^{-\rho }}C\left(\rho \right)\frac{\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)}{2}.$$

(31)

If $\alpha \to 1$, then

$$\underset{\alpha \to 1}{\lim }\rho =\underset{\alpha \to 1}{\lim }\frac{1-\alpha }{\alpha }\left(\mathsf{ѵ}-ȥ\right)=0,\mathrm{then}$$

$$\underset{\alpha \to 1}{\lim }\left(\frac{4\rho +2\pi e^{-\rho -1}+4}{4{\rho }^2+8\rho +{\pi }^2+4}+\frac{2\pi e^{-1}+4e^{-\rho -1}\left(e+\rho e\right)}{4{\rho }^2-8\rho +{\pi }^2+4}\right)=\frac{2\pi e^{-1}+4}{{\pi }^2+4}, \underset{\alpha \to 1}{\lim }\frac{1-\alpha }{2\left(1-e^{-\rho }\right)}=\frac{1}{2\left(\mathsf{ѵ}-ȥ\right).}$$

Now from Theorem 3, we acquire the following result, which is also a new one:

$$\sqrt{\frac{e}{2}}\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right){\le }_p\frac{1}{\mathsf{ѵ}-ȥ}{{\displaystyle \int }}_{ȥ}^{\mathsf{ѵ}}\mathsf{Ϣ}\left(\varkappa \right)d\varkappa {\le }_p\frac{2\pi e^{-1}+4}{{\pi }^2+4}\left[\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)\right].$$

(32)

If one lays $\alpha \to 1$ and $\mathsf{Ϣ}$ which is an upper left and right exponential trigonometric concave I-V∙M on $\left[ȥ,\mathsf{ѵ}\right],$ then one can acquire the following inequality [98]:

$$\sqrt{\frac{e}{2}}\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right){\supseteq }_p\frac{1}{\mathsf{ѵ}-ȥ}{{\displaystyle \int }}_{ȥ}^{\mathsf{ѵ}}\mathsf{Ϣ}\left(\varkappa \right)d\varkappa {\supseteq }_p\frac{2\pi e^{-1}+4}{{\pi }^2+4}\left[\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)\right].$$

(33)

Let $\alpha \to 1$ and ${\mathsf{Ϣ}}_{*}\left(\varkappa \right)\ne {\mathsf{Ϣ}}^{*}\left(\varkappa \right)$. Then, from Theorem 3, we achieve the Hermite–Hadamard inequality for the interval-valued left and right exponential trigonometric convex mapping, which is also a new one:

$$\sqrt{\frac{e}{2}}\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right){\le }_p\frac{1}{\mathsf{ѵ}-ȥ}{{\displaystyle \int }}_{ȥ}^{\mathsf{ѵ}}\mathsf{Ϣ}\left(\varkappa \right)d\varkappa {\le }_p\frac{2\pi e^{-1}+4}{{\pi }^2+4}\left[\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)\right].$$

(34)

If ${\mathsf{Ϣ}}_{*}\left(\varkappa \right)={\mathsf{Ϣ}}^{*}\left(\varkappa \right),$ then, from Theorem 3, we arrive at classical fractional Hermite–Hadamard inequality for the exponential trigonometric convex mapping.

$$\sqrt{\frac{e}{2}}\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\le \frac{1-\alpha }{2\left(1-e^{-\rho }\right)}\left[{ℐ}_{{ȥ}^{+}}^{\alpha } \mathsf{Ϣ}\left(\mathsf{ѵ}\right)+{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha } \mathsf{Ϣ}(ȥ)\right]\le \frac{\rho }{1-e^{-\rho }}C\left(\rho \right)\frac{\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)}{2}.$$

(35)

Let $\alpha \to 1$ and ${\mathsf{Ϣ}}_{*}\left(\varkappa \right)={\mathsf{Ϣ}}^{*}\left(\varkappa \right)$. Then, from Theorem 3, we achieve the classical Hermite–Hadamard inequality for the exponential trigonometric convex mapping, see [78].

$$\sqrt{\frac{e}{2}}\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\le \frac{1}{\mathsf{ѵ}-ȥ}{{\displaystyle \int }}_{ȥ}^{\mathsf{ѵ}}\mathsf{Ϣ}\left(\varkappa \right)d\varkappa \le \frac{2\pi e^{-1}+4}{{\pi }^2+4}\left[\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)\right].$$

(36)

Example 1.

Let $\alpha =\frac{1}{2}, \varkappa \in \left[0,1\right]$, and the I-V∙M $\mathsf{Ϣ}:\left[ȥ, \mathsf{ѵ}\right]=\left[0, 1\right]\to X_C^{+},$ defined by $\mathsf{Ϣ}\left(\varkappa \right)=\left[2{\varkappa }^2,4{\varkappa }^2\right]$. Since left and right end point mappings ${\mathsf{Ϣ}}_{*}\left(\varkappa \right)=2{\varkappa }^2,$ ${\mathsf{Ϣ}}^{*}\left(\varkappa \right)=4{\varkappa }^2$ are exponential trigonometric convex mappings, then $\mathsf{Ϣ}\left(\varkappa \right)$ is a left and right exponential trigonometric convex I-V∙M. We can clearly see that $\mathsf{Ϣ}\in L\left(\left[ȥ, \mathsf{ѵ}\right],X_C^{+}\right)$ and

$$\sqrt{\frac{e}{2}}{\mathsf{Ϣ}}_{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)={\mathsf{Ϣ}}_{*}\left(\frac{5}{2}\right)=\frac{\sqrt{e}}{2\sqrt{2}}$$

$$\sqrt{\frac{e}{2}}{\mathsf{Ϣ}}^{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)={\mathsf{Ϣ}}^{*}\left(\frac{5}{2}\right)=\frac{\sqrt{e}}{\sqrt{2}}$$

$$\frac{\rho }{1-e^{-\rho }}C\left(\rho \right)\frac{{\mathsf{Ϣ}}_{*}(ȥ)+{\mathsf{Ϣ}}_{*}\left(\mathsf{ѵ}\right)}{2}=\frac{8+2{\mathsf{\pi }e}^{-2}}{2\left(16+{\mathsf{\pi }}^2\right)\left(1-e^{-1}\right)}$$

$$\frac{\rho }{1-e^{-\rho }}C\left(\rho \right)\frac{{\mathsf{Ϣ}}^{*}(ȥ)+{\mathsf{Ϣ}}^{*}\left(\mathsf{ѵ}\right)}{2}=\frac{8+2{\mathsf{\pi }e}^{-2}}{\left(16+{\mathsf{\pi }}^2\right)\left(1-e^{-1}\right)}.$$

Note that

$$\begin{array}{l}\frac{1-\alpha }{2\left(1-e^{-\rho }\right)}\left[{ℐ}_{{ȥ}^{+}}^{\alpha } {\mathsf{Ϣ}}_{*}\left(\mathsf{ѵ}\right)+{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha } {\mathsf{Ϣ}}_{*}(ȥ)\right]\\ =\frac{1}{2\left(1-e^{-1}\right)}{{\displaystyle \int }}_0^1{e}^{-\left(1-\varkappa \right)} . 2{\varkappa }^{2}d\varkappa \\ +\frac{1}{2\left(1-e^{-1}\right)}{{\displaystyle \int }}_0^1{e}^{-\varkappa } .2{\varkappa }^{2}d\varkappa \\ =\frac{1}{1-e^{-1}}\left[1-2e^{-1}+2-5e^{-1}\right]\\ =\frac{3-7e^{-1}}{1-e^{-1}}.\end{array}$$

$$\begin{array}{l}\frac{1-\alpha }{2\left(1-e^{-\rho }\right)}\left[{ℐ}_{{ȥ}^{+}}^{\alpha } {\mathsf{Ϣ}}^{*}\left(\mathsf{ѵ}\right)+{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha } {\mathsf{Ϣ}}^{*}(ȥ)\right]\\ =\frac{1}{2\left(1-e^{-1}\right)}{{\displaystyle \int }}_0^1{e}^{-\left(1-\varkappa \right)} . 4{\varkappa }^{2}d\varkappa \\ +\frac{1}{2\left(1-e^{-1}\right)}{{\displaystyle \int }}_0^1{e}^{-\varkappa } . 4{\varkappa }^{2}d\varkappa \\ =\frac{2}{1-e^{-1}}\left[1-2e^{-1}+2-5e^{-1}\right]\\ =\frac{2\left(3-7e^{-1}\right)}{1-e^{-1}}.\end{array}$$

Therefore,

$$\left[\frac{\sqrt{e}}{\sqrt{2}},\frac{\sqrt{e}}{\sqrt{2}}\right]{\le }_p\left[\frac{8+2\pi e^{-2}}{2\left(16+{\pi }^2\right)\left(1-e^{-1}\right)},\frac{8+2\pi e^{-2}}{\left(16+{\pi }^2\right)\left(1-e^{-1}\right)}\right]{\le }_p\left[\frac{3-7e^{-1}}{1-e^{-1}},\frac{2\left(3-7e^{-1}\right)}{1-e^{-1}}\right]$$

 and Theorem 3 is verified.

The fractional integrals with exponential kernels can be used to describe Hermite–Hadamard-type inclusions involving midpoint as follows:

Theorem 4.

Let $\mathsf{Ϣ}:\left[ȥ, \mathsf{ѵ}\right]\subset ℝ\to X_C^{+}$ be an I-V∙M on $\left[ȥ, \mathsf{ѵ}\right]$ given by $\mathsf{Ϣ}\left(\varkappa \right)=\left[{\mathsf{Ϣ}}_{*}\left(\varkappa \right),{ \mathsf{Ϣ}}^{*}\left(\varkappa \right)\right]$ for all $\varkappa \in \left[ȥ, \mathsf{ѵ}\right]$. If $\mathsf{Ϣ}$ is a left and right exponential trigonometric convex I-V∙M on $\left[ȥ, \mathsf{ѵ}\right]$ and $\mathsf{Ϣ}\in L\left(\left[ȥ, \mathsf{ѵ}\right],X_C^{+}\right)$, then

$$\sqrt{\frac{e}{2}}\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right){\le }_p\frac{1-\mathsf{\alpha }}{2\left(1-e^{-\frac{\rho }{2}}\right)}\left[{ℐ}_{{\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)}^{+}}^{\mathsf{\alpha }}\mathsf{Ϣ}\left(\mathsf{ѵ}\right)+{ℐ}_{{\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)}^{-}}^{\mathsf{\alpha }}\mathsf{Ϣ}(ȥ)\right]{\le }_p\frac{\rho }{2\left(1-e^{-\frac{\rho }{2}}\right)}B\left(\rho \right)\frac{\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)}{2}.$$

(37)

If $\mathsf{Ϣ}\left(\varkappa \right)$ is a left and right exponential trigonometric concave I-V∙M, then

$$\sqrt{\frac{e}{2}}\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right){\ge }_p\frac{1-\alpha }{2\left(1-e^{-\frac{\rho }{2}}\right)}\left[{ℐ}_{{\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)}^{+}}^{\alpha }\mathsf{Ϣ}\left(\mathsf{ѵ}\right)+{ℐ}_{{\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)}^{-}}^{\alpha }\mathsf{Ϣ}(ȥ)\right]{\ge }_p\frac{\rho }{1-e^{-\frac{\rho }{2}}}B\left(\rho \right)\frac{\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)}{2},$$

(38)

where $B\left(\rho \right)=\frac{\pi \left(4e^{-1}-2^{\frac{3}{2}}{e}^{-\frac{\rho +1}{2}}\right)-2^{\frac{5}{2}}\rho e^{-\frac{\rho +1}{2}}+2^{\frac{5}{2}}{e}^{-\frac{\rho +1}{2}}}{4{\rho }^2-8\rho +{\pi }^2+4}+\frac{8\rho +2^{\frac{3}{2}}\pi e^{-\frac{\rho +1}{2}}-2^{\frac{5}{2}}\left(\rho +1\right)e^{-\frac{\rho +1}{2}}+8}{4{\rho }^2+8\rho +{\pi }^2+4}$  , $\rho =\frac{1-\alpha }{\alpha }\left(\mathsf{ѵ}-ȥ\right)$, and $1>\alpha >0$.

Proof. 

Let $\mathsf{Ϣ}:\left[ȥ, \mathsf{ѵ}\right]\to X_C^{+}$ be a left and right exponential trigonometric convex I-V∙M. Then, by hypothesis, we have

$$\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right){\le }_p\frac{\sin \frac{\pi }{4}}{\sqrt{e}}\mathsf{Ϣ}\left(\frac{\mathfrak{v}}{2}ȥ+\frac{2-\mathfrak{v}}{2}\mathsf{ѵ}\right)+\frac{\cos \frac{\pi }{4}}{\sqrt{e}}\mathsf{Ϣ}\left(\frac{2-\mathfrak{v}}{2}ȥ+\frac{\mathfrak{v}}{2}\mathsf{ѵ}\right).$$

After simplification, we find that

$$2\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right){\le }_p\sqrt{\frac{2}{e}}\left[\mathsf{Ϣ}\left(\frac{\mathfrak{v}}{2}ȥ+\frac{2-\mathfrak{v}}{2}\mathsf{ѵ}\right)+\mathsf{Ϣ}\left(\frac{2-\mathfrak{v}}{2}ȥ+\frac{\mathfrak{v}}{2}\mathsf{ѵ}\right)\right]$$

Therefore, we have

$$\begin{array}{c}2{\mathsf{Ϣ}}_{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\le \sqrt{\frac{2}{e}}\left[{\mathsf{Ϣ}}_{*}\left(\frac{\mathfrak{v}}{2}ȥ+\frac{2-\mathfrak{v}}{2}\mathsf{ѵ}\right)+{\mathsf{Ϣ}}_{*}\left(\frac{2-\mathfrak{v}}{2}ȥ+\frac{\mathfrak{v}}{2}\mathsf{ѵ}\right)\right]\end{array},$$

$$2{\mathsf{Ϣ}}^{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\le \sqrt{\frac{2}{e}}\left[{\mathsf{Ϣ}}^{*}\left(\frac{\mathfrak{v}}{2}ȥ+\frac{2-\mathfrak{v}}{2}\mathsf{ѵ}\right)+{\mathsf{Ϣ}}^{*}\left(\frac{2-\mathfrak{v}}{2}ȥ+\frac{\mathfrak{v}}{2}\mathsf{ѵ}\right)\right].$$

Taking ${\mathsf{Ϣ}}_{*}(.)$ and multiplying both sides by $e^{-\frac{\rho \mathfrak{v}}{2}}$ and integrating the obtained result with respect to $\mathfrak{v}$ from $0$ to $1$, we have

$$2{{\displaystyle \int }}_0^1{e}^{-\frac{\rho \mathfrak{v}}{2}}{\mathsf{Ϣ}}_{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)d\mathfrak{v} \le \sqrt{\frac{2}{e}}\left[{{\displaystyle \int }}_0^1{e}^{-\frac{\rho \mathfrak{v}}{2}}{\mathsf{Ϣ}}_{*}\left(\frac{\mathfrak{v}}{2}ȥ+\frac{2-\mathfrak{v}}{2}\mathsf{ѵ}\right)d\mathfrak{v}+{{\displaystyle \int }}_0^1{e}^{-\frac{\rho \mathfrak{v}}{2}}{\mathsf{Ϣ}}_{*}\left(\frac{2-\mathfrak{v}}{2}ȥ+\frac{\mathfrak{v}}{2}\mathsf{ѵ}\right)d\mathfrak{v}\right].$$

Let $u=\frac{\mathfrak{v}}{2}ȥ+\frac{2-\mathfrak{v}}{2}\mathsf{ѵ}$ and $\varkappa =\frac{2-\mathfrak{v}}{2}ȥ+\frac{\mathfrak{v}}{2}\mathsf{ѵ}.$ Then, we have

$$\begin{gathered} 2{{\displaystyle \int }}_0^1{e}^{-\frac{\rho \mathfrak{v}}{2}}{\mathsf{Ϣ}}_{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)d\mathfrak{v} \le \sqrt{\frac{2}{e}}\frac{1}{\mathsf{ѵ}-ȥ} {{\displaystyle \int }}_{\frac{ȥ+\mathsf{ѵ}}{2}}^{\mathsf{ѵ}}{e}^{\left(-\frac{1-\alpha }{\alpha }\left(\mathsf{ѵ}-u\right)\right)}{\mathsf{Ϣ}}_{*}\left(u\right)du+\frac{1}{\mathsf{ѵ}-ȥ}{{\displaystyle \int }}_{\frac{ȥ+\mathsf{ѵ}}{2}}^{\mathsf{ѵ}}{e}^{\left(-\frac{1-\alpha }{\alpha }\left(\varkappa -ȥ\right)\right)}{\mathsf{Ϣ}}_{*}\left(\varkappa \right)d\varkappa \\ \begin{array}{c}=\sqrt{\frac{2}{e}}\frac{\alpha }{\mathsf{ѵ}-ȥ}\left[{ℐ}_{{\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)}^{+}}^{\alpha } {\mathsf{Ϣ}}_{*}\left(\mathsf{ѵ}\right)+{ℐ}_{{\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)}^{-}}^{\alpha } {\mathsf{Ϣ}}_{*}(ȥ)\right].\end{array} \end{gathered}$$

(39)

Now, taking the right side of Equation (39), we have

$${{\displaystyle \int }}_0^1{e}^{-\frac{\rho \mathfrak{v}}{2}}{\mathsf{Ϣ}}_{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)d\mathfrak{v}=\frac{2\left(1-e^{-\rho }\right)}{\rho }{\mathsf{Ϣ}}_{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right).$$

(40)

From (39) and (40), we have

$$4·\frac{1-e^{-\rho }}{\rho }{\mathsf{Ϣ}}_{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\le \sqrt{\frac{2}{e}}·\frac{2\left(1-ȥ\right)}{\mathsf{ѵ}-ȥ}\left[{ℐ}_{{\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)}^{+}}^{\alpha } {\mathsf{Ϣ}}_{*}\left(\mathsf{ѵ}\right)+{ℐ}_{{\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)}^{-}}^{\alpha } {\mathsf{Ϣ}}_{*}(ȥ)\right].$$

(41)

Similarly, for ${\mathsf{Ϣ}}^{*}\left(\varkappa \right)$, we have

$$4·\frac{1-e^{-\rho }}{\rho }{\mathsf{Ϣ}}^{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\le \sqrt{\frac{2}{e}}·\frac{2\left(1-ȥ\right)}{\mathsf{ѵ}-ȥ}\left[{ℐ}_{{\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)}^{+}}^{\alpha } {\mathsf{Ϣ}}^{*}\left(\mathsf{ѵ}\right)+{ℐ}_{{\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)}^{-}}^{\alpha } {\mathsf{Ϣ}}^{*}(ȥ)\right].$$

(42)

From (41) and (42), we have

$$\begin{array}{c}2·\frac{1-e^{-\rho }}{\rho }\left[{\mathsf{Ϣ}}_{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right), {\mathsf{Ϣ}}^{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\right]\hfill \\ {\le }_p\sqrt{\frac{2}{e}}·\frac{1-ȥ}{\mathsf{ѵ}-ȥ}\left[\left[{ℐ}_{{\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)}^{+}}^{\alpha } {\mathsf{Ϣ}}_{*}\left(\mathsf{ѵ}\right)+{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha } {\mathsf{Ϣ}}_{*}(ȥ)\right], \left[{ℐ}_{{\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)}^{+}}^{\alpha } {\mathsf{Ϣ}}^{*}\left(\mathsf{ѵ}\right)+{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha } {\mathsf{Ϣ}}^{*}(ȥ)\right]\right].\end{array}$$

That is

$$2·\frac{1-e^{-\rho }}{\rho } \mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right){\le }_p\sqrt{\frac{2}{e}}·\frac{1-ȥ}{\mathsf{ѵ}-ȥ}\left[{ℐ}_{{\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)}^{+}}^{\alpha }\mathsf{Ϣ}\left(\mathsf{ѵ}\right)+{ℐ}_{{\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)}^{-}}^{\alpha }\mathsf{Ϣ}(ȥ)\right].$$

(43)

For the right side of Equation (37), since $\mathsf{Ϣ}$ is a left and right exponential trigonometric convex I-V∙M, we can deduce that

$$\mathsf{Ϣ}\left(\frac{\mathfrak{v}}{2}ȥ+\frac{2-\mathfrak{v}}{2}\mathsf{ѵ}\right){\le }_p\frac{\sin \frac{\pi \mathfrak{v}}{4}}{e^{\frac{2-\mathfrak{v}}{2}}}\mathsf{Ϣ}(ȥ)+\frac{\cos \frac{\pi \mathfrak{v}}{4}}{e^{\frac{\mathfrak{v}}{2}}}\mathsf{Ϣ}\left(\mathsf{ѵ}\right),$$

(44)

and

$$\mathsf{Ϣ}\left(\frac{2-\mathfrak{v}}{2}ȥ+\frac{\mathfrak{v}}{2}\mathsf{ѵ}\right){\le }_p\frac{\cos \frac{\pi \mathfrak{v}}{4}}{e^{\frac{\mathfrak{v}}{2}}}\mathsf{Ϣ}(ȥ)+\frac{\sin \frac{\pi \mathfrak{v}}{4}}{e^{\frac{2-\mathfrak{v}}{2}}}\mathsf{Ϣ}\left(\mathsf{ѵ}\right).$$

(45)

Adding (44) and (45), we have

$$\mathsf{Ϣ}\left(\frac{\mathfrak{v}}{2}ȥ+\frac{2-\mathfrak{v}}{2}\mathsf{ѵ}\right)+\mathsf{Ϣ}\left(\frac{2-\mathfrak{v}}{2}ȥ+\frac{\mathfrak{v}}{2}\mathsf{ѵ}\right){\le }_p\left[\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)\right]\left[\frac{\sin \frac{\pi \mathfrak{v}}{4}}{e^{\frac{2-\mathfrak{v}}{2}}}+\frac{\cos \frac{\pi \mathfrak{v}}{4}}{e^{\frac{\mathfrak{v}}{2}}}\right].$$

(46)

Since $\mathsf{Ϣ}$ is I-V∙M, then we have

$$\begin{array}{c}{\mathsf{Ϣ}}_{*}\left(\frac{\mathfrak{v}}{2}ȥ+\frac{2-\mathfrak{v}}{2}\mathsf{ѵ}\right)+{\mathsf{Ϣ}}_{*}\left(\frac{2-\mathfrak{v}}{2}ȥ+\frac{\mathfrak{v}}{2}\mathsf{ѵ}\right)\le \left[{\mathsf{Ϣ}}_{*}(ȥ)+{\mathsf{Ϣ}}_{*}\left(\mathsf{ѵ}\right)\right]\left[\frac{\sin \frac{\pi \mathsf{ѵ}}{4}}{e^{\frac{2-\mathsf{ѵ}}{2}}}+\frac{\cos \frac{\pi \mathsf{ѵ}}{4}}{e^{\frac{\mathsf{ѵ}}{2}}}\right],\\ {\mathsf{Ϣ}}^{*}\left(\frac{\mathfrak{v}}{2}ȥ+\frac{2-\mathfrak{v}}{2}\mathsf{ѵ}\right)+{\mathsf{Ϣ}}^{*}\left(\frac{2-\mathfrak{v}}{2}ȥ+\frac{\mathfrak{v}}{2}\mathsf{ѵ}\right)\le \left[{\mathsf{Ϣ}}^{*}(ȥ)+{\mathsf{Ϣ}}^{*}\left(\mathsf{ѵ}\right)\right]\left[\frac{\sin \frac{\pi \mathsf{ѵ}}{4}}{e^{\frac{2-\mathsf{ѵ}}{2}}}+\frac{\cos \frac{\pi \mathsf{ѵ}}{4}}{e^{\frac{\mathsf{ѵ}}{2}}}\right].\end{array}$$

(47)

Taking ${\mathsf{Ϣ}}_{*}(.)$ from (47) and multiplying the inequality by $e^{e^{-\frac{\rho \mathfrak{v}}{2}}}$, and integrating the resultant with $\mathfrak{v}$ from $0$ to $1$, we have

$$\begin{array}{c}{{\displaystyle \int }}_0^1{e}^{e^{-\frac{\rho \mathfrak{v}}{2}}}{\mathsf{Ϣ}}_{*}\left(\frac{\mathfrak{v}}{2}ȥ+\frac{2-\mathfrak{v}}{2}\mathsf{ѵ}\right)d\mathfrak{v}+{{\displaystyle \int }}_0^1{e}^{e^{-\frac{\rho \mathfrak{v}}{2}}}{\mathsf{Ϣ}}_{*}\left(\frac{2-\mathfrak{v}}{2}ȥ+\frac{\mathfrak{v}}{2}\mathsf{ѵ} \right)d\mathfrak{v}\hfill \\ \le \left[{\mathsf{Ϣ}}_{*}(ȥ)+{\mathsf{Ϣ}}_{*}\left(\mathsf{ѵ}\right)\right]{{\displaystyle \int }}_0^1{e}^{e^{-\frac{\rho \mathfrak{v}}{2}}}\left[\frac{\sin \frac{\pi \mathfrak{v}}{4}}{\frac{2-\mathfrak{v}}{2}}+\frac{\cos \frac{\pi \mathfrak{v}}{4}}{e^{\frac{\mathfrak{v}}{2}}}\right]d\mathfrak{v},\hfill \\ =\left(\frac{\pi \left(4e^{-1}-2^{\frac{3}{2}}{e}^{-\frac{\rho +1}{2}}\right)-2^{\frac{5}{2}}\rho e^{-\frac{\rho +1}{2}}+2^{\frac{5}{2}}{e}^{-\frac{\rho +1}{2}}}{4{\rho }^2-8\rho +{\pi }^2+4}+\frac{8\rho +2^{\frac{3}{2}}\pi e^{-\frac{\rho +1}{2}}-2^{\frac{5}{2}}\left(\rho +1\right)e^{-\frac{\rho +1}{2}}+8}{4{\rho }^2+8\rho +{\pi }^2+4}\right)\left[{\mathsf{Ϣ}}_{*}(ȥ)+{\mathsf{Ϣ}}_{*}\left(\mathsf{ѵ}\right)\right].\end{array}$$

(48)

In a similar way as above, for ${\mathsf{Ϣ}}_{*}(.)$ we have

$$\begin{array}{l}{{\displaystyle \int }}_0^1{e}^{e^{-\frac{\rho \mathfrak{v}}{2}}}{\mathsf{Ϣ}}^{*}\left(\frac{\mathfrak{v}}{2}ȥ+\frac{2-\mathfrak{v}}{2}\mathsf{ѵ}\right)d\mathfrak{v}+{{\displaystyle \int }}_0^1{e}^{e^{-\frac{\rho \mathfrak{v}}{2}}}{\mathsf{Ϣ}}^{*}\left(\frac{2-\mathfrak{v}}{2}ȥ+\frac{\mathfrak{v}}{2}\mathsf{ѵ}\right)d\mathfrak{v}\\ \le \left[{\mathsf{Ϣ}}^{*}(ȥ)+{\mathsf{Ϣ}}^{*}\left(\mathsf{ѵ}\right)\right]{{\displaystyle \int }}_0^1{e}^{-\rho \mathfrak{v}}\left[\frac{\sin \frac{\pi \mathfrak{v}}{4}}{e^{\frac{2-\mathfrak{v}}{2}}}+\frac{\cos \frac{\pi \mathfrak{v}}{4}}{e^{\frac{\mathfrak{v}}{2}}}\right]d\mathfrak{v}\\ =\left(\frac{\pi \left(4e^{-1}-2^{\frac{3}{2}}{e}^{-\frac{\rho +1}{2}}\right)-2^{\frac{5}{2}}\rho e^{-\frac{\rho +1}{2}}+2^{\frac{5}{2}}{e}^{-\frac{\rho +1}{2}}}{4{\rho }^2-8\rho +{\pi }^2+4}+\frac{8\rho +2^{\frac{3}{2}}\pi e^{-\frac{\rho +1}{2}}-2^{\frac{5}{2}}\left(\rho +1\right)e^{-\frac{\rho +1}{2}}+8}{4{\rho }^2+8\rho +{\pi }^2+4}\right)\left[{\mathsf{Ϣ}}^{*}(ȥ)+{\mathsf{Ϣ}}^{*}\left(\mathsf{ѵ}\right)\right].\end{array}$$

(49)

From (48) and (49), we have

$$\begin{array}{l}{{\displaystyle \int }}_0^1{e}^{-\rho \mathfrak{v}}\mathsf{Ϣ}\left(\frac{\mathfrak{v}}{2}ȥ+\frac{2-\mathfrak{v}}{2}\mathsf{ѵ} \right)d\mathfrak{v}+{{\displaystyle \int }}_0^1{e}^{-\rho \mathfrak{v}}\mathsf{Ϣ}\left(\frac{2-\mathfrak{v}}{2}ȥ+\frac{\mathfrak{v}}{2}\mathsf{ѵ} \right)d\mathfrak{v}\\ {\le }_p\left(\frac{\pi \left(4e^{-1}-2^{\frac{3}{2}}{e}^{-\frac{\rho +1}{2}}\right)-2^{\frac{5}{2}}\rho e^{-\frac{\rho +1}{2}}+2^{\frac{5}{2}}{e}^{-\frac{\rho +1}{2}}}{4{\rho }^2-8\rho +{\pi }^2+4}+\frac{8\rho +2^{\frac{3}{2}}\pi e^{-\frac{\rho +1}{2}}-2^{\frac{5}{2}}\left(\rho +1\right)e^{-\frac{\rho +1}{2}}+8}{4{\rho }^2+8\rho +{\pi }^2+4}\right)\left[\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)\right].\end{array}$$

(50)

From (43) and (50), we have

$$\sqrt{\frac{e}{2}}\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right){\le }_p\frac{1-\alpha }{2\left(1-e^{-\frac{\rho }{2}}\right)}\left[{ℐ}_{{\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)}^{+}}^{\alpha }\mathsf{Ϣ}\left(\mathsf{ѵ}\right)+{ℐ}_{{\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)}^{-}}^{\alpha }\mathsf{Ϣ}(ȥ)\right]{\le }_p\frac{\rho }{2\left(1-e^{-\frac{\rho }{2}}\right)}B\left(\rho \right)\frac{\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)}{2}.$$

Hence, the required result. □

Remark 4.

From Theorem 4, we can clearly see the following.

If one lays $\mathsf{Ϣ}$ which is an upper left and right exponential trigonometric concave I-V∙M on $\left[ȥ,\mathsf{ѵ}\right],$ then one acquires the following inequality [98]:

$$\sqrt{\frac{e}{2}}\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right){\supseteq }_p\frac{1-\alpha }{2\left(1-e^{-\frac{\rho }{2}}\right)}\left[{ℐ}_{{\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)}^{+}}^{\alpha }\mathsf{Ϣ}\left(\mathsf{ѵ}\right)+{ℐ}_{{\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)}^{-}}^{\alpha }\mathsf{Ϣ}(ȥ)\right]{\supseteq }_p\frac{\rho }{2\left(1-e^{-\frac{\rho }{2}}\right)}B\left(\rho \right)\frac{\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)}{2}.$$

(51)

If $\alpha \to 1$  , that is

$$\underset{\alpha \to 1}{\lim }\rho =\underset{\alpha \to 1}{\lim }\frac{1-\alpha }{\alpha }\left(\mathsf{ѵ}-ȥ\right)=0,\mathrm{then}$$

$$\underset{\alpha \to 1}{\lim }\frac{\rho }{1-e^{-\frac{\rho }{2}}}=\left(\frac{\pi \left(4e^{-1}-2^{\frac{3}{2}}{e}^{-\frac{\rho +1}{2}}\right)-2^{\frac{5}{2}}\rho e^{-\frac{\rho +1}{2}}+2^{\frac{5}{2}}{e}^{-\frac{\rho +1}{2}}}{4{\rho }^2-8\rho +{\pi }^2+4}+\frac{8\rho +2^{\frac{3}{2}}\pi e^{-\frac{\rho +1}{2}}-2^{\frac{5}{2}}\left(\rho +1\right)e^{-\frac{\rho +1}{2}}+8}{4{\rho }^2+8\rho +{\pi }^2+4}\right)=\frac{4\left(2\pi +4e\right)}{e\left({\pi }^2+4\right)}, \underset{\alpha \to 1}{\lim }\frac{1-\alpha }{2\left(1-e^{e^{-\frac{\rho }{2}}}\right)}=\frac{1}{\mathsf{ѵ}-ȥ}$$

Then, we acquire the following result, which is also a new one:

$$\sqrt{\frac{e}{2}}\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right){\le }_p\frac{1}{\mathsf{ѵ}-ȥ}{{\displaystyle \int }}_{ȥ}^{\mathsf{ѵ}}\mathsf{Ϣ}\left(\varkappa \right)d\varkappa {\le }_p\frac{2\pi +4e}{e\left({\pi }^2+4\right)}\left[\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)\right]$$

(52)

If one lays $\alpha \to 1$ and $\mathsf{Ϣ}$ which is an upper left and right exponential trigonometric concave I-V∙M on $\left[ȥ,\mathsf{ѵ}\right],$ then one acquires the following inequality [98]:

$$\sqrt{\frac{e}{2}}\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right){\supseteq }_p\frac{1}{\mathsf{ѵ}-ȥ}{{\displaystyle \int }}_{ȥ}^{\mathsf{ѵ}}\mathsf{Ϣ}\left(\varkappa \right)d\varkappa {\supseteq }_p\frac{2\pi +4e}{e\left({\pi }^2+4\right)}\left[\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)\right]$$

(53)

Let $\alpha \to 1$ and ${\mathsf{Ϣ}}_{*}\left(\varkappa \right)\ne {\mathsf{Ϣ}}^{*}\left(\varkappa \right)$  . Then, from Theorem 4, we achieve the Hermite–Hadamard inequality for interval-valued   left and right exponential trigonometric convex mapping, which is also a new one:

$$\sqrt{\frac{e}{2}}\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right){\le }_p\frac{1}{\mathsf{ѵ}-ȥ}{{\displaystyle \int }}_{ȥ}^{\mathsf{ѵ}}\mathsf{Ϣ}\left(\varkappa \right)d\varkappa {\le }_p\frac{2\pi +4e}{e\left({\pi }^2+4\right)}\left[\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)\right]$$

(54)

If  ${\mathsf{Ϣ}}_{*}\left(\varkappa \right)={\mathsf{Ϣ}}^{*}\left(\varkappa \right),$  then, from Theorem 4, we arrive at classical fractional Hermite–Hadamard inequality for   exponential trigonometric convex mapping.

$$\sqrt{\frac{e}{2}}\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\le \frac{1-\alpha }{2\left(1-e^{-\frac{\rho }{2}}\right)}\left[{ℐ}_{{\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)}^{+}}^{\alpha }\mathsf{Ϣ}\left(\mathsf{ѵ}\right)+{ℐ}_{{\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)}^{-}}^{\alpha }\mathsf{Ϣ}(ȥ)\right]\le \frac{\rho }{2\left(1-e^{-\frac{\rho }{2}}\right)}B\left(\rho \right)\frac{\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)}{2}$$

(55)

Let $\alpha \to 1$ and ${\mathsf{Ϣ}}_{*}\left(\varkappa \right)={\mathsf{Ϣ}}^{*}\left(\varkappa \right)$. Then, from Theorem 4, we achieve the classical Hermite–Hadamard inequality for exponential trigonometric convex mapping, see [78].

$$\sqrt{\frac{e}{2}}\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\le \frac{1}{\mathsf{ѵ}-ȥ}{{\displaystyle \int }}_{ȥ}^{\mathsf{ѵ}}\mathsf{Ϣ}\left(\varkappa \right)d\varkappa \le \frac{2\pi +4e}{e\left({\pi }^2+4\right)}\left[\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)\right].$$

(56)

Finally, we present the Pachpatte-type fractional integral inclusions. Moreover, in Theorem 5 we will establish a fractional integral inclusion, and discuss the several inclusions via a left and right exponential trigonometric convex I-V∙M.

Theorem 5.

Let $\mathsf{Ϣ},\mathsf{{\rm T}} :\left[ȥ, \mathsf{ѵ}\right]\to X_C^{+}$ be two I-V∙Ms on $\left[ȥ, \mathsf{ѵ}\right]$ defined by $\mathsf{Ϣ}\left(\varkappa \right)=\left[{\mathsf{Ϣ}}_{*}\left(\varkappa \right),{ \mathsf{Ϣ}}^{*}\left(\varkappa \right)\right]$ and $T\left(\varkappa \right)=\left[{\mathsf{{\rm T}}}_{*}\left(\varkappa \right),{ \mathsf{{\rm T}}}^{*}\left(\varkappa \right)\right]$ for all $\varkappa \in \left[ȥ, \mathsf{ѵ}\right]$. If $\mathsf{Ϣ}$ and $\mathsf{{\rm T}}$ are two left and right exponential trigonometric convex I-V∙Ms on $\left[ȥ, \mathsf{ѵ}\right]$ and $\mathsf{Ϣ}\times \mathsf{{\rm T}}\in L\left(\left[ȥ, \mathsf{ѵ}\right],X_C^{+}\right)$, then

$$\frac{\alpha }{\mathsf{ѵ}-ȥ}\left[{ℐ}_{{ȥ}^{+}}^{\alpha }\mathsf{Ϣ}\left(\mathsf{ѵ}\right)\times \mathsf{{\rm T}}\left(\mathsf{ѵ}\right)+{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha }\mathsf{Ϣ}(ȥ)\times \mathsf{{\rm T}}(ȥ)\right]{\le }_p\mathrm{D}\left(\rho \right)\mathsf{\Delta }\left(ȥ,\mathsf{ѵ}\right)+\frac{{\mathsf{\pi }e}^{-\rho -1}\left(e^{\rho }+1\right)}{{\rho }^2+{\mathsf{\pi }}^2}\nabla \left(ȥ,\mathsf{ѵ}\right).$$

(57)

If $\mathsf{Ϣ}\left(\varkappa \right)$ and $T\left(\varkappa \right)$ are left and right exponential trigonometric concave I-V∙Ms, then

$$\frac{\alpha }{\mathsf{ѵ}-ȥ}\left[{ℐ}_{{ȥ}^{+}}^{\alpha }\mathsf{Ϣ}\left(\mathsf{ѵ}\right)\times {\rm T}\left(\mathsf{ѵ}\right)+{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha }\mathsf{Ϣ}(ȥ)\times {\rm T}(ȥ)\right]{\ge }_{p}D\left(\rho \right)\Delta \left(ȥ,\mathsf{ѵ}\right)+\frac{\pi e^{-\rho -1}\left(e^{\rho }+1\right)}{{\rho }^2+{\pi }^2}\nabla \left(ȥ,\mathsf{ѵ}\right),$$

(58)

where $D\left(\rho \right)=-\frac{e^{-\rho -2}\left(8e^2-8\rho e^2+{\pi }^2{e}^2+2{\rho }^2{e}^2-{\pi }^2{e}^{\rho }\right)}{2\left(\rho -2\right)\left({\rho }^2-4\rho +{\pi }^2+4\right)}+\frac{8\rho -{\pi }^2{e}^{-\rho -2}+{\pi }^2+2{\rho }^2+8}{2\left(\rho +2\right)\left({\rho }^2+4\rho +{\pi }^2+4\right)}$, $\rho =\frac{1-\alpha }{\alpha }\left(\mathsf{ѵ}-ȥ\right)$, $1>\alpha >0$, $\Delta \left(ȥ,\mathsf{ѵ}\right)=\left[{\Delta }_{*}\left(ȥ,\mathsf{ѵ}\right), {\Delta }^{*}\left(ȥ,\mathsf{ѵ}\right)\right]$ and $\nabla \left(ȥ,\mathsf{ѵ}\right)=\left[{\nabla }_{*}\left(ȥ,\mathsf{ѵ}\right), {\nabla }^{*}\left(ȥ,\mathsf{ѵ}\right)\right].$

Proof. 

Since $\mathsf{Ϣ},{\rm T}$ are both left and right exponential trigonometric convex I-V∙Ms, taking left end points mappings, we have

$$\begin{array}{c}{\mathsf{Ϣ}}_{*}\left(\mathfrak{v}ȥ+\left(1-\mathfrak{v}\right)\mathsf{ѵ}\right)\le \frac{\sin \frac{\pi \mathfrak{v}}{2}}{e^{1-\mathfrak{v}}}{\mathsf{Ϣ}}_{*}(ȥ)+\frac{\cos \frac{\pi \mathfrak{v}}{2}}{e^{\mathfrak{v}}}{\mathsf{Ϣ}}_{*}\left(\mathsf{ѵ}\right),\end{array}$$

and

$${{\rm T}}_{*}\left(\mathfrak{v}ȥ+\left(1-\mathfrak{v}\right)\mathsf{ѵ}\right)\le \mathfrak{v}\frac{\sin \frac{\pi \mathfrak{v}}{2}}{e^{1-\mathfrak{v}}}{{\rm T}}_{*}(ȥ)+\frac{\cos \frac{\pi \mathfrak{v}}{2}}{e^{\mathfrak{v}}}{{\rm T}}_{*}\left(\mathsf{ѵ}\right).$$

From the definition of left and right exponential trigonometric convex I-V∙Ms, it follows that $0{\le }_p\mathsf{Ϣ}\left(\varkappa \right)$ and $0{\le }_{\mathrm{I}}{\rm T}\left(\varkappa \right)$, so

$$\begin{array}{l}{\mathsf{Ϣ}}_{*}\left(\mathfrak{v}ȥ+\left(1-\mathfrak{v}\right)\mathsf{ѵ}\right)\times {{\rm T}}_{*}\left(\mathfrak{v}ȥ+\left(1-\mathfrak{v}\right)\mathsf{ѵ}\right)\\ \le (\frac{\sin \frac{\pi \mathfrak{v}}{2}}{e^{1-\mathfrak{v}}}{\mathsf{Ϣ}}_{*}(ȥ)+\frac{\cos \frac{\pi \mathfrak{v}}{2}}{e^{\mathfrak{v}}}{\mathsf{Ϣ}}_{*}\left(\mathsf{ѵ}\right))(\frac{\sin \frac{\pi \mathfrak{v}}{2}}{e^{1-\mathfrak{v}}}{{\rm T}}_{*}(ȥ)+\frac{\cos \frac{\pi \mathfrak{v}}{2}}{e^{\mathfrak{v}}}{{\rm T}}_{*}\left(\mathsf{ѵ}\right))\\ ={\left(\frac{\sin \frac{\pi \mathfrak{v}}{2}}{e^{1-\mathfrak{v}}}\right)}^2{\mathsf{Ϣ}}_{*}(ȥ)\times {{\rm T}}_{*}(ȥ)+{\left(\frac{\cos \frac{\pi \mathfrak{v}}{2}}{e^{\mathfrak{v}}}\right)}^2{\mathsf{Ϣ}}_{*}(ѵ)\times {{\rm T}}_{*}(ѵ)\\ +\left(\frac{\cos \frac{\pi \mathfrak{v}}{2}\sin \frac{\pi \mathfrak{v}}{2}}{e}\right){\mathsf{Ϣ}}_{*}(ȥ)\times {{\rm T}}_{*}(ѵ)+\left(\frac{\cos \frac{\pi \mathfrak{v}}{2}\sin \frac{\pi \mathfrak{v}}{2}}{e}\right){\mathsf{Ϣ}}_{*}(ѵ)\times {{\rm T}}_{*}(ȥ)\end{array}$$

(59)

Analogously, we have

$$\begin{array}{l}{\mathsf{Ϣ}}_{*}\left(\left(1-\mathfrak{v}\right)ȥ+\mathfrak{v}\mathsf{ѵ}\right)\times {{\rm T}}_{*}\left(\left(1-\mathfrak{v}\right)ȥ+\mathfrak{v}\mathsf{ѵ}\right)\\ \le {\left(\frac{\cos \frac{\pi \mathfrak{v}}{2}}{e^{\mathfrak{v}}}\right)}^2{\mathsf{Ϣ}}_{*}(ȥ)\times {{\rm T}}_{*}(ȥ)+{\left(\frac{\sin \frac{\pi \mathfrak{v}}{2}}{e^{1-\mathfrak{v}}}\right)}^2{\mathsf{Ϣ}}_{*}(ѵ)\times {{\rm T}}_{*}(ѵ)\\ +\left(\frac{\cos \frac{\pi \mathfrak{v}}{2}\sin \frac{\pi \mathfrak{v}}{2}}{e}\right){\mathsf{Ϣ}}_{*}(ȥ)\times {{\rm T}}_{*}(ѵ)+\left(\frac{\cos \frac{\pi \mathfrak{v}}{2}\sin \frac{\pi \mathfrak{v}}{2}}{e}\right){\mathsf{Ϣ}}_{*}(ѵ)\times {{\rm T}}_{*}(ȥ)\end{array}$$

(60)

Adding (59) and (60), we have

$$\begin{array}{l}{\mathsf{Ϣ}}_{*}\left(\mathfrak{v}ȥ+\left(1-\mathfrak{v}\right)\mathsf{ѵ}\right)\times {{\rm T}}_{*}\left(\mathfrak{v}ȥ+\left(1-\mathfrak{v}\right)\mathsf{ѵ}\right)\\ +{\mathsf{Ϣ}}_{*}\left(\left(1-\mathfrak{v}\right)ȥ+\mathfrak{v}\mathsf{ѵ}\right)\times {{\rm T}}_{*}\left(\left(1-\mathfrak{v}\right)ȥ+\mathfrak{v}\mathsf{ѵ}\right)\\ \le \left[{\left(\frac{\sin \frac{\pi \mathfrak{v}}{2}}{e^{1-\mathfrak{v}}}\right)}^2+{\left(\frac{\sin \frac{\pi \mathfrak{v}}{2}}{e^{1-\mathfrak{v}}}\right)}^2\right][{\mathsf{Ϣ}}_{*}(ȥ)\times {{\rm T}}_{*}(ȥ)+{\mathsf{Ϣ}}_{*}\left(\mathsf{ѵ}\right)\times {{\rm T}}_{*}\left(\mathsf{ѵ}\right)]\\ +\frac{2\cos \frac{\pi \mathfrak{v}}{2} \sin \frac{\pi \mathfrak{v}}{2}}{e}[{\mathsf{Ϣ}}_{*}\left(\mathsf{ѵ}\right)\times {{\rm T}}_{*}(ȥ)+{\mathsf{Ϣ}}_{*}(ȥ)\times {{\rm T}}_{*}\left(\mathsf{ѵ}\right)]\end{array}$$

(61)

Multiplying (61) by $e^{-\rho \mathfrak{v}}$ and integrating the obtained result with respect to $\mathfrak{v}$ over (0,1), we have

$$\begin{array}{l}{{\displaystyle \int }}_0^1{e}^{-\rho \mathfrak{v}}{\mathsf{Ϣ}}_{*}\left(\mathfrak{v}ȥ+\left(1-\mathfrak{v}\right)\mathsf{ѵ}\right)\times {{\rm T}}_{*}\left(\mathfrak{v}ȥ+\left(1-\mathfrak{v}\right)\mathsf{ѵ}\right)\\ +e^{-\rho \mathfrak{v}}{\mathsf{Ϣ}}_{*}\left(\left(1-\mathfrak{v}\right)ȥ+\mathfrak{v}\mathsf{ѵ}\right)\times {{\rm T}}_{*}\left(\left(1-\mathfrak{v}\right)ȥ+\mathfrak{v}\mathsf{ѵ}\right)d\mathfrak{v}\\ \le {\Delta }_{*}\left(\left(ȥ,\mathsf{ѵ}\right)\right){{\displaystyle \int }}_0^1{e}^{-\rho \mathfrak{v}}\left[{\left(\frac{\sin \frac{\pi \mathfrak{v}}{2}}{e^{1-\mathfrak{v}}}\right)}^2+{\left(\frac{\sin \frac{\pi \mathfrak{v}}{2}}{e^{1-\mathfrak{v}}}\right)}^2\right]d\mathfrak{v}+2{\nabla }_{*}\left(\left(ȥ,\mathsf{ѵ}\right)\right){{\displaystyle \int }}_0^1{e}^{-\rho \mathfrak{v}}\frac{\cos \frac{\pi \mathfrak{v}}{2} \sin \frac{\pi \mathfrak{v}}{2}}{e}d\mathfrak{v}\end{array}$$

It follows that

$$\begin{array}{l}\frac{\alpha }{\mathsf{ѵ}-ȥ}\left[{ℐ}_{{ȥ}^{+}}^{\alpha } {\mathsf{Ϣ}}_{*}\left(\mathsf{ѵ}\right)\times {{\rm T}}_{*}\left(\mathsf{ѵ}\right)+{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha } {\mathsf{Ϣ}}_{*}(ȥ)\times {{\rm T}}_{*}(ȥ)\right]\\ \le D\left(\rho \right){\Delta }_{*}\left(\left(ȥ,\mathsf{ѵ}\right)\right)+\frac{\pi e^{-\rho -1}\left(e^{\rho }+1\right)}{{\rho }^2+{\pi }^2}{\nabla }_{*}\left(\left(ȥ,\mathsf{ѵ}\right)\right)\end{array}$$

(62)

Similarly, for ${\mathsf{Ϣ}}^{*}\left(\varkappa \right)$, we have

$$\begin{array}{l}\frac{\alpha }{\mathsf{ѵ}-ȥ}\left[{ℐ}_{{ȥ}^{+}}^{\alpha } {\mathsf{Ϣ}}^{*}\left(\mathsf{ѵ}\right)\times {{\rm T}}^{*}\left(\mathsf{ѵ}\right)+{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha } {\mathsf{Ϣ}}^{*}(ȥ)\times {{\rm T}}^{*}(ȥ)\right]\\ \le D\left(\rho \right){\Delta }^{*}\left(\left(ȥ,\mathsf{ѵ}\right)\right)+\frac{\pi e^{-\rho -1}\left(e^{\rho }+1\right)}{{\rho }^2+{\pi }^2}{\nabla }^{*}\left(\left(ȥ,\mathsf{ѵ}\right)\right)\end{array}$$

(63)

where

$$\begin{array}{cc}\hfill D\left(\rho \right)& ={{\displaystyle \int }}_0^1{e}^{-\rho \mathfrak{v}}\left[{\left(\frac{\sin \frac{\pi \mathfrak{v}}{2}}{e^{1-\mathfrak{v}}}\right)}^2+{\left(\frac{\sin \frac{\pi \mathfrak{v}}{2}}{e^{1-\mathfrak{v}}}\right)}^2\right]d\mathfrak{v}\hfill \\ & =-\frac{e^{-\rho -2}\left(8e^2-8\rho e^2+{\pi }^2{e}^2+2{\rho }^2{e}^2-{\pi }^2{e}^{\rho }\right)}{2\left(\rho -2\right)\left({\rho }^2-4\rho +{\pi }^2+4\right)}+\frac{8\rho -{\pi }^2{e}^{-\rho -2}+{\pi }^2+2{\rho }^2+8}{2\left(\rho +2\right)\left({\rho }^2+4\rho +{\pi }^2+4\right)}\end{array}$$

and

$${{\displaystyle \int }}_0^1{e}^{-\rho \mathfrak{v}}\frac{\cos \frac{\pi \mathfrak{v}}{2} \sin \frac{\pi \mathfrak{v}}{2}}{e}d\mathfrak{v}=\frac{\pi e^{-\rho -1}\left(e^{\rho }+1\right)}{{\rho }^2+{\pi }^2}$$

From (62) and (63), we have

$$\begin{array}{cc}\hfill \frac{\alpha }{\mathsf{ѵ}-ȥ}[{ℐ}_{{ȥ}^{+}}^{\alpha } {\mathsf{Ϣ}}_{*}\left(\mathsf{ѵ}\right)\times {{\rm T}}_{*}\left(\mathsf{ѵ}\right)& +{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha } {\mathsf{Ϣ}}_{*}(ȥ)\times {{\rm T}}_{*}(ȥ), {ℐ}_{{ȥ}^{+}}^{\alpha } {\mathsf{Ϣ}}^{*}\left(\mathsf{ѵ}\right)\times {{\rm T}}^{*}\left(\mathsf{ѵ}\right)+{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha } {\mathsf{Ϣ}}^{*}(ȥ)\times {{\rm T}}^{*}(ȥ)]\hfill \\ & {\le }_{p}D\left(\rho \right)\left[{\Delta }_{*}\left(\left(ȥ,\mathsf{ѵ}\right)\right), {\Delta }^{*}\left(\left(ȥ,\mathsf{ѵ}\right)\right)\right]+\frac{\pi e^{-\rho -1}\left(e^{\rho }+1\right)}{{\rho }^2+{\pi }^2}\left[{\nabla }_{*}\left(\left(ȥ,\mathsf{ѵ}\right)\right), {\nabla }^{*}\left(\left(ȥ,\mathsf{ѵ}\right)\right)\right].\end{array}$$

That is

$$\frac{\alpha }{\mathsf{ѵ}-ȥ}\left[{ℐ}_{{ȥ}^{+}}^{\alpha }\mathsf{Ϣ}\left(\mathsf{ѵ}\right)\times {\rm T}\left(\mathsf{ѵ}\right)+{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha }\mathsf{Ϣ}(ȥ)\times {\rm T}(ȥ)\right]{\le }_{p}D\left(\rho \right)\Delta \left(ȥ,\mathsf{ѵ}\right)+\frac{\pi e^{-\rho -1}\left(e^{\rho }+1\right)}{{\rho }^2+{\pi }^2}\nabla \left(ȥ,\mathsf{ѵ}\right).$$

and the theorem has been established. □

Remark 5.

From Theorem 5 we can clearly see the following.

If one lays $\mathsf{Ϣ}$ which is an upper left and right exponential trigonometric concave I-V∙M on $\left[ȥ,\mathsf{ѵ}\right],$ then one acquires the following inequality [98]:

$$\frac{\alpha }{\mathsf{ѵ}-ȥ}\left[{ℐ}_{{ȥ}^{+}}^{\alpha }\mathsf{Ϣ}\left(\mathsf{ѵ}\right)\times {\rm T}\left(\mathsf{ѵ}\right)+{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha }\mathsf{Ϣ}(ȥ)\times {\rm T}(ȥ)\right]{\supseteq }_{p}D\left(\rho \right)\Delta \left(ȥ,\mathsf{ѵ}\right)+\frac{\pi e^{-\rho -1}\left(e^{\rho }+1\right)}{{\rho }^2+{\pi }^2}\nabla \left(ȥ,\mathsf{ѵ}\right).$$

(64)

If $\alpha \to 1$, that is

$$\begin{array}{l}\underset{\alpha \to 1}{\lim }\rho =\underset{\alpha \to 1}{\lim }\frac{1-\alpha }{\alpha }\left(\mathsf{ѵ}-ȥ\right)=0,\mathrm{then}\\ \underset{\alpha \to 1}{\lim }\left(-\frac{e^{-\rho -2}\left(8e^2-8\rho e^2+{\pi }^2{e}^2+2{\rho }^2{e}^2-{\pi }^2{e}^{\rho }\right)}{2\left(\rho -2\right)\left({\rho }^2-4\rho +{\pi }^2+4\right)}+\frac{8\rho -{\pi }^2{e}^{-\rho -2}+{\pi }^2+2{\rho }^2+8}{2\left(\rho +2\right)\left({\pi }^2+4\right)}\right)=\frac{{\pi }^2-{\pi }^2{e}^2+8}{2\left({\pi }^2+4\right),}\\ \underset{\alpha \to 1}{\lim }\frac{\pi e^{-\rho -1}\left(e^{\rho }+1\right)}{{\rho }^2+{\pi }^2}=\frac{2}{\pi e}\end{array}$$

Then, we acquire the following result, which is also a new one:

$$\frac{1}{\mathsf{ѵ}-ȥ}{{\displaystyle \int }}_{ȥ}^{\mathsf{ѵ}}\mathsf{Ϣ}\left(\varkappa \right)\times {\rm T}\left(\varkappa \right)d\varkappa {\le }_p\frac{{\pi }^2-{\pi }^2{e}^2+8}{4\left({\pi }^2+4\right)}\Delta \left(ȥ,\mathsf{ѵ}\right)+\frac{2}{\pi e}\nabla \left(ȥ,\mathsf{ѵ}\right)$$

(65)

If one lays $\alpha \to 1$ and $\mathsf{Ϣ}$ which is an upper left and right exponential trigonometric concave I-V∙M on $\left[ȥ,\mathsf{ѵ}\right],$ then one acquires the following inequality [98]:

$$\frac{1}{\mathsf{ѵ}-ȥ}{{\displaystyle \int }}_{ȥ}^{\mathsf{ѵ}}\mathsf{Ϣ}\left(\varkappa \right)\times {\rm T}\left(\varkappa \right)d\varkappa {\supseteq }_p\frac{{\pi }^2-{\pi }^2{e}^2+8}{4\left({\pi }^2+4\right)}\Delta \left(ȥ,\mathsf{ѵ}\right)+\frac{2}{\pi e}\nabla \left(ȥ,\mathsf{ѵ}\right)$$

(66)

Let $\alpha \to 1$ and ${\mathsf{Ϣ}}_{*}\left(\varkappa \right)\ne {\mathsf{Ϣ}}^{*}\left(\varkappa \right)$  . Then, from Theorem 5, we achieve the Hermite–Hadamard inequality for interval-valued   left and right exponential trigonometric convex mapping, which is also a new one:

$$\frac{1}{\mathsf{ѵ}-ȥ}{{\displaystyle \int }}_{ȥ}^{\mathsf{ѵ}}\mathsf{Ϣ}\left(\varkappa \right)\times {\rm T}\left(\varkappa \right)d\varkappa {\le }_p\frac{{\pi }^2-{\pi }^2{e}^2+8}{4\left({\pi }^2+4\right)}\Delta \left(ȥ,\mathsf{ѵ}\right)+\frac{2}{\pi e}\nabla \left(ȥ,\mathsf{ѵ}\right).$$

(67)

If ${\mathsf{Ϣ}}_{*}\left(\varkappa \right)={\mathsf{Ϣ}}^{*}\left(\varkappa \right)$  , then, from Theorem 5, we arrive at the classical fractional Hermite–Hadamard inequality for exponential trigonometric convex mapping:

$$\frac{\alpha }{\mathsf{ѵ}-ȥ}\left[{ℐ}_{{ȥ}^{+}}^{\alpha } \mathsf{Ϣ}\left(\mathsf{ѵ}\right)\times {\rm T}\left(\mathsf{ѵ}\right)+{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha }\mathsf{Ϣ}(ȥ)\times {\rm T}(ȥ)\right]\le D\left(\rho \right)\Delta \left(ȥ,\mathsf{ѵ}\right)+\frac{\pi e^{-\rho -1}\left(e^{\rho }+1\right)}{{\rho }^2+{\pi }^2}\nabla \left(ȥ,\mathsf{ѵ}\right).$$

(68)

Let $\alpha \to 1$ and ${\mathsf{Ϣ}}_{*}\left(\varkappa \right)={\mathsf{Ϣ}}^{*}\left(\varkappa \right)$. Then, from Theorem 5, we achieve the classical Hermite–Hadamard inequality for exponential trigonometric convex mapping, see [78].

$$\frac{1}{\mathsf{ѵ}-ȥ}{{\displaystyle \int }}_{ȥ}^{\mathsf{ѵ}}\mathsf{Ϣ}\left(\varkappa \right)\times {\rm T}\left(\varkappa \right)d\varkappa \le \frac{{\pi }^2-{\pi }^2{e}^2+8}{4\left({\pi }^2+4\right)}\Delta \left(ȥ,\mathsf{ѵ}\right)+\frac{2}{\pi e}\nabla \left(ȥ,\mathsf{ѵ}\right).$$

(69)

Example 2.

Let $\left[ȥ,\mathsf{ѵ}\right]=\left[0,1\right]$, $\alpha =\frac{1}{4}$, $\mathsf{Ϣ}\left(\varkappa \right)=\left[{\varkappa }^2,2{\varkappa }^2\right]$ and $T\left(\varkappa \right)=\left[2{\varkappa }^3,4{\varkappa }^3\right].$ Since left and right end point mappings ${\mathsf{Ϣ}}_{*}\left(\varkappa \right)={\varkappa }^2,$ ${\mathsf{Ϣ}}^{*}\left(\varkappa \right)=2{\varkappa }^2$, $T_{*}\left(\varkappa \right)=2{\varkappa }^3$ and $T^{*}\left(\varkappa \right)=4{\varkappa }^3$ are exponential trigonometric convex mappings, then $\mathsf{Ϣ}\left(\varkappa \right)$ and $T\left(\varkappa \right)$ are both exponential trigonometric convex I-V∙Ms. We can clearly see that $\mathsf{Ϣ}\left(\varkappa \right)\times T\left(\varkappa \right)\in L\left(\left[ȥ, \mathsf{ѵ}\right],X_C^{+}\right)$ and

$$\begin{array}{cc}\hfill \frac{\alpha }{\mathsf{ѵ}-ȥ}[{ℐ}_{{ȥ}^{+}}^{\alpha }{ \mathsf{Ϣ}}_{*}\left(\mathsf{ѵ}\right)\times & T_{*}\left(\mathsf{ѵ}\right)+{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha }{ \mathsf{Ϣ}}_{*}(ȥ)\times T_{*}(ȥ)]\hfill \\ & ={{\displaystyle \int }}_0^1{e}^{-3\left(1-\varkappa \right)}\left(2{\varkappa }^5\right)\mathrm{d}\varkappa +{{\displaystyle \int }}_0^1{e}^{-3\varkappa }\left(2{\varkappa }^5\right)\mathrm{d}\varkappa \hfill \\ & =\left(\frac{80e^{-3}}{243}+\frac{52}{243}\right)+\left(\frac{80}{243}-\frac{1472e^{-3}}{243}\right)\hfill \\ & =\left(\frac{44}{81}-\frac{464e^{-3}}{81}\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill \frac{\alpha }{\mathsf{ѵ}-ȥ}[{ℐ}_{{ȥ}^{+}}^{\alpha }{ \mathsf{Ϣ}}^{*}\left(\mathsf{ѵ}\right)& \times T^{*}\left(\mathsf{ѵ}\right)+{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha }{ \mathsf{Ϣ}}^{*}(ȥ)\times T^{*}(ȥ)]\hfill \\ & ={{\displaystyle \int }}_0^1{e}^{-3\left(1-\varkappa \right)}\left(8{\varkappa }^5\right)\mathrm{d}\varkappa +{{\displaystyle \int }}_0^1{e}^{-3\varkappa }\left(8{\varkappa }^5\right)\mathrm{d}\varkappa \hfill \\ & =4\left[\frac{44}{81}-\frac{464e^{-3}}{81}\right].\hfill \end{array}$$

Note that

$$\begin{array}{cc}\hfill D\left(\rho \right){\Delta }_{*}\left(\left(ȥ,\mathsf{ѵ}\right)\right)=(-\frac{e^{-5}\left(2e^2+{\pi }^2{e}^2-{\pi }^2{e}^3\right)}{2\left(1+{\pi }^2\right)}& +\frac{50-{\pi }^2{e}^{-5}+{\pi }^2}{10\left(25+{\pi }^2\right)})\left[{\mathsf{Ϣ}}_{*}\left(0\right)\times T_{*}\left(1\right)+{\mathsf{Ϣ}}_{*}\left(1\right)\times T_{*}\left(1\right)\right]\hfill \\ & =2\left(\frac{50-{\pi }^2{e}^{-5}+{\pi }^2}{10\left(25+{\pi }^2\right)}-\frac{e^{-5}\left(2e^2+{\pi }^2{e}^2-{\pi }^2{e}^3\right)}{2\left(1+{\pi }^2\right)}\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill D\left(\rho \right){\Delta }^{*}\left(\left(ȥ,\mathsf{ѵ}\right)\right)=(-\frac{e^{-5}\left(2e^2+{\pi }^2{e}^2-{\pi }^2{e}^3\right)}{2\left(1+{\pi }^2\right)}& +\frac{50-{\pi }^2{e}^{-5}+{\pi }^2}{10\left(25+{\pi }^2\right)})\left[{\mathsf{Ϣ}}^{*}(ȥ)\times T^{*}(ȥ)+{\mathsf{Ϣ}}^{*}\left(\mathsf{ѵ}\right)\times T^{*}\left(\mathsf{ѵ}\right)\right]\hfill \\ & =8\left(\frac{50-{\pi }^2{e}^{-5}+{\pi }^2}{10\left(25+{\pi }^2\right)}-\frac{e^{-5}\left(2e^2+{\pi }^2{e}^2-{\pi }^2{e}^3\right)}{2\left(1+{\pi }^2\right)}\right),\hfill \end{array}$$

$$\frac{\pi e^{-\rho -1}\left(e^{\rho }+1\right)}{{\rho }^2+{\pi }^2}{\nabla }_{*}\left(\left(ȥ,\mathsf{ѵ}\right)\right)=\frac{\pi e^{-\rho -1}\left(e^{\rho }+1\right)}{{\rho }^2+{\pi }^2}\left[{\mathsf{Ϣ}}_{*}\left(0\right)\times T_{*}\left(1\right)+{Ʊ}_{*}\left(1\right)\times T_{*}\left(0\right)\right]=0,$$

$$\frac{\pi e^{-\rho -1}\left(e^{\rho }+1\right)}{{\rho }^2+{\pi }^2}{\nabla }_{*}\left(\left(ȥ,\mathsf{ѵ}\right)\right)=\frac{\pi e^{-\rho -1}\left(e^{\rho }+1\right)}{{\rho }^2+{\pi }^2}\left[{Ʊ}^{*}\left(1\right)\times T^{*}\left(1\right)+{Ʊ}^{*}\left(1\right)\times T^{*}\left(0\right)\right]=0.$$

Therefore, we have

$$\begin{array}{l}D\left(\rho \right)\Delta \left(ȥ,\mathsf{ѵ}\right)+\frac{\pi e^{-\rho -1}\left(e^{\rho }+1\right)}{{\rho }^2+{\pi }^2}\nabla \left(ȥ,\mathsf{ѵ}\right)\\ =\left(\frac{50-{\pi }^2{e}^{-5}+{\pi }^2}{10\left(25+{\pi }^2\right)}-\frac{e^{-5}\left(2e^2+{\pi }^2{e}^2-{\pi }^2{e}^3\right)}{2\left(1+{\pi }^2\right)}\right)\left[2,8\right]+\frac{\pi e^{-\rho -1}\left(e^{\rho }+1\right)}{{\rho }^2+{\pi }^2}\left[0,0\right]\\ =\left(\frac{50-{\pi }^2{e}^{-5}+{\pi }^2}{10\left(25+{\pi }^2\right)}-\frac{e^{-5}\left(2e^2+{\pi }^2{e}^2-{\pi }^2{e}^3\right)}{2\left(1+{\pi }^2\right)}\right)\left[2,8\right]\end{array}$$

It follows that

$$\left(\frac{44}{81}-\frac{464e^{-3}}{81}\right)\left[2,4\right]{\le }_p\left(\frac{50-{\pi }^2{e}^{-5}+{\pi }^2}{10\left(25+{\pi }^2\right)}-\frac{e^{-5}\left(2e^2+{\pi }^2{e}^2-{\pi }^2{e}^3\right)}{2\left(1+{\pi }^2\right)}\right)\left[2,8\right].$$

and Theorem 5 has been demonstrated.

Theorem 6.

Let $\mathsf{Ϣ},\mathsf{{\rm T}} :\left[ȥ, \mathsf{ѵ}\right]\to X_C^{+}$ be two I-V∙Ms on $\left[ȥ, \mathsf{ѵ}\right]$ defined by $\mathsf{Ϣ}\left(\varkappa \right)=\left[{\mathsf{Ϣ}}_{*}\left(\varkappa \right),{ \mathsf{Ϣ}}^{*}\left(\varkappa \right)\right]$ and $T\left(\varkappa \right)=\left[{\mathsf{{\rm T}}}_{*}\left(\varkappa \right),{ \mathsf{{\rm T}}}^{*}\left(\varkappa \right)\right]$ for all $\varkappa \in \left[ȥ, \mathsf{ѵ}\right]$. If $\mathsf{Ϣ}$ and $\mathsf{{\rm T}}$ are two left and right exponential trigonometric convex I-V∙Ms on $\left[ȥ, \mathsf{ѵ}\right]$ and $\mathsf{Ϣ}\times \mathsf{{\rm T}}\in L\left(\left[ȥ, \mathsf{ѵ}\right],X_C^{+}\right)$, then

$$\begin{array}{cc}\hfill 2\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\times {\rm T}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)& {\le }_p\frac{1-\alpha }{e\left(1-e^{-\rho }\right)}\left[{ℐ}_{{ȥ}^{+}}^{\alpha }\mathsf{Ϣ}\left(\mathsf{ѵ}\right)\times {\rm T}\left(\mathsf{ѵ}\right)+{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha }\mathsf{Ϣ}(ȥ)\times {\rm T}(ȥ)\right]\hfill \\ & +\frac{\rho \pi e^{-\rho -1}\left(e^{\rho }+1\right)}{e\left(1-e^{-\rho }\right)\left({\rho }^2+{\pi }^2\right)}\nabla \left(ȥ,\mathsf{ѵ}\right)+\frac{\rho }{e\left(1-e^{-\rho }\right)}D\left(\rho \right)\Delta \left(ȥ,\mathsf{ѵ}\right).\hfill \end{array}$$

(70)

If $\mathsf{Ϣ}\left(\varkappa \right)$ and ${\rm T}\left(\varkappa \right)$  are left and right exponential trigonometric concave I-V∙Ms, then

$$\begin{array}{cc}\hfill 2\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\times {\rm T}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)& {\ge }_p\frac{1-\alpha }{e\left(1-e^{-\rho }\right)}\left[{ℐ}_{{ȥ}^{+}}^{\alpha }\mathsf{Ϣ}\left(\mathsf{ѵ}\right)\times {\rm T}\left(\mathsf{ѵ}\right)+{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha }\mathsf{Ϣ}(ȥ)\times {\rm T}(ȥ)\right]\hfill \\ & +\frac{\rho \pi e^{-\rho -1}\left(e^{\rho }+1\right)}{e\left(1-e^{-\rho }\right)\left({\rho }^2+{\pi }^2\right)}\nabla \left(ȥ,\mathsf{ѵ}\right)+\frac{\rho }{e\left(1-e^{-\rho }\right)}D\left(\rho \right)\Delta \left(ȥ,\mathsf{ѵ}\right).\hfill \end{array}$$

(71)

where $D\left(\rho \right)=-\frac{e^{-\rho -2}\left(8e^2-8\rho e^2+{\pi }^2{e}^2+2{\rho }^2{e}^2-{\pi }^2{e}^{\rho }\right)}{2\left(\rho -2\right)\left({\rho }^2-4\rho +{\pi }^2+4\right)}+\frac{8\rho -{\pi }^2{e}^{-\rho -2}+{\pi }^2+2{\rho }^2+8}{2\left(\rho +2\right)\left({\rho }^2+4\rho +{\pi }^2+4\right)}$, $\rho =\frac{1-\alpha }{\alpha }\left(\mathsf{ѵ}-ȥ\right)$, $1>\alpha >0$, $\Delta \left(ȥ,\mathsf{ѵ}\right)=\left[{\Delta }_{*}\left(ȥ,\mathsf{ѵ}\right), {\Delta }^{*}\left(ȥ,\mathsf{ѵ}\right)\right]$ and $\nabla \left(ȥ,\mathsf{ѵ}\right)=\left[{\nabla }_{*}\left(ȥ,\mathsf{ѵ}\right), {\nabla }^{*}\left(ȥ,\mathsf{ѵ}\right)\right].$

Proof. 

Consider $\mathsf{Ϣ},{\rm T}:\left[ȥ, \mathsf{ѵ}\right]\to X_C^{+}$ are left and right exponential trigonometric convex I-V∙Ms. Then, by hypothesis, we have

$$\begin{array}{l}{\mathsf{Ϣ}}_{*}\left(\frac{ȥ+\mathsf{ѵ}}{2} \right)\times {{\rm T}}_{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right) \\ \le \frac{1}{2e}\left[\begin{array}{c}{\mathsf{Ϣ}}_{*}\left(\mathfrak{v}ȥ+\left(1-\mathfrak{v}\right)\mathsf{ѵ}\right)\times {{\rm T}}_{*}\left(\mathfrak{v}ȥ+\left(1-\mathfrak{v}\right)\mathsf{ѵ}\right)\\ +{\mathsf{Ϣ}}_{*}\left(\left(1-\mathfrak{v}\right)ȥ+\left(1-\mathfrak{v}\right)\mathsf{ѵ}\right)\times {{\rm T}}_{*}\left(\left(1-\mathfrak{v}\right)ȥ+\mathfrak{v}\mathsf{ѵ}\right)\end{array}\right] \\ +\frac{1}{2e}\left[\begin{array}{c}{\mathsf{Ϣ}}_{*}\left(\left(1-\mathfrak{v}\right)ȥ+\mathfrak{v}\mathsf{ѵ}\right)\times {{\rm T}}_{*}\left(\mathfrak{v}ȥ+\left(1-\mathfrak{v}\right)\mathsf{ѵ}\right)\\ +{\mathsf{Ϣ}}_{*}\left(\mathfrak{v}ȥ+\left(1-\mathfrak{v}\right)\mathsf{ѵ}\right)\times {{\rm T}}_{*}\left(\left(1-\mathfrak{v}\right)ȥ+\mathfrak{v}\mathsf{ѵ}\right)\end{array}\right] \\ \le \frac{1}{2e}\left[\begin{array}{c}{\mathsf{Ϣ}}_{*}\left(\mathfrak{v}ȥ+\left(1-\mathfrak{v}\right)\mathsf{ѵ}\right)\times {{\rm T}}_{*}\left(\mathfrak{v}ȥ+\left(1-\mathfrak{v}\right)\mathsf{ѵ}\right)\\ +{\mathsf{Ϣ}}_{*}\left(\left(1-\mathfrak{v}\right)ȥ+\mathfrak{v}\mathsf{ѵ}\right)\times {{\rm T}}_{*}\left(\left(1-\mathfrak{v}\right)ȥ+\mathfrak{v}\mathsf{ѵ}\right)\end{array}\right] \\ +\frac{1}{2e}\left[\begin{array}{c}(\frac{\cos \frac{\pi \mathfrak{v}}{2}}{e^{\mathfrak{v}}}{\mathsf{Ϣ}}_{*}(ȥ)+\frac{\sin \frac{\pi \mathfrak{v}}{2}}{e^{1-\mathfrak{v}}}{\mathsf{Ϣ}}_{*}\left(\mathsf{ѵ}\right))\\ \times (\frac{\sin \frac{\pi \mathfrak{v}}{2}}{e^{1-\mathfrak{v}}}{{\rm T}}_{*}(ȥ)+\frac{\cos \frac{\pi \mathfrak{v}}{2}}{e^{\mathfrak{v}}}{{\rm T}}_{*}\left(\mathsf{ѵ}\right))\\ +(\frac{\sin \frac{\pi \mathfrak{v}}{2}}{e^{1-\mathfrak{v}}}{\mathsf{Ϣ}}_{*}(ȥ)+\frac{\cos \frac{\pi \mathfrak{v}}{2}}{e^{\mathfrak{v}}}{\mathsf{Ϣ}}_{*}\left(\mathsf{ѵ}\right))\\ \times (\frac{\cos \frac{\pi \mathfrak{v}}{2}}{e^{\mathfrak{v}}}{{\rm T}}_{*}(ȥ)+\frac{\sin \frac{\pi \mathfrak{v}}{2}}{e^{1-\mathfrak{v}}}{{\rm T}}_{*}\left(\mathsf{ѵ}\right))\end{array}\right]\\ \le \frac{1}{2e}\left[\begin{array}{c}{\mathsf{Ϣ}}_{*}\left(\mathfrak{v}ȥ+\left(1-\mathfrak{v}\right)\mathsf{ѵ}\right)\times {{\rm T}}_{*}\left(\mathfrak{v}ȥ+\left(1-\mathfrak{v}\right)\mathsf{ѵ}\right)\\ +{\mathsf{Ϣ}}_{*}\left(\left(1-\mathfrak{v}\right)ȥ+\mathfrak{v}\mathsf{ѵ}\right)\times {{\rm T}}_{*}\left(\left(1-\mathfrak{v}\right)ȥ+\mathfrak{v}\mathsf{ѵ}\right)\end{array}\right] \\ +\frac{1}{2e}\left[\begin{array}{c}\frac{2\cos \frac{\pi \mathfrak{v}}{2}\sin \frac{\pi \mathfrak{v}}{2}}{e}{\nabla }_{*}\left(\left(ȥ,\mathsf{ѵ}\right)\right)\\ +[{\left(\frac{\sin \frac{\pi \mathfrak{v}}{2}}{e^{1-\mathfrak{v}}}\right)}^2+{\left(\frac{\sin \frac{\pi \mathfrak{v}}{2}}{e^{1-\mathfrak{v}}}\right)}^2]{\Delta }_{*}\left(\left(ȥ,\mathsf{ѵ}\right)\right)\end{array}\right]\end{array}$$

(72)

Multiplying (72) by $e^{-\rho \mathfrak{v}}$ and integrating over $\left(0, 1\right),$ we find

$$\begin{array}{l}{{\displaystyle \int }}_0^1{e}^{-\rho \mathfrak{v}} {\mathsf{Ϣ}}_{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\times {{\rm T}}_{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)d\mathfrak{v}\\ \le \frac{1}{2e}[{{\displaystyle \int }}_{ȥ}^{\mathsf{ѵ}}{e}^{-\rho \mathfrak{v}}{\mathsf{Ϣ}}_{*}\left(\varkappa \right)\times {{\rm T}}_{*}\left(\varkappa \right)d\mathfrak{v}+{{\displaystyle \int }}_{ȥ}^{\mathsf{ѵ}}{e}^{-\rho \mathfrak{v}}{\mathsf{Ϣ}}_{*}\left(s\right)\times {{\rm T}}_{*}\left(s\right)d\mathfrak{v}]\\ +\frac{{\nabla }^{*}\left(\left(ȥ,\mathsf{ѵ}\right)\right)}{2e}{{\displaystyle \int }}_0^1{e}^{-\rho \mathfrak{v}}\frac{2\cos \frac{\pi \mathfrak{v}}{2} \sin \frac{\pi \mathfrak{v}}{2}}{e}d\mathfrak{v}+\frac{{\Delta }_{*}\left(\left(ȥ,\mathsf{ѵ}\right)\right)}{2e}{{\displaystyle \int }}_0^1{e}^{-\rho \mathfrak{v}}\left[{\left(\frac{\sin \frac{\pi \mathfrak{v}}{2}}{e^{1-\mathfrak{v}}}\right)}^2+{\left(\frac{\sin \frac{\pi \mathfrak{v}}{2}}{e^{1-\mathfrak{v}}}\right)}^2\right]d\mathfrak{v}\end{array}$$

$$\begin{array}{l}\frac{1-e^{-\rho }}{\rho }{{\displaystyle \int }}_0^1{e}^{-\rho \mathfrak{v}} {\mathsf{Ϣ}}_{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\times {{\rm T}}_{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\\ \le \frac{\alpha }{2e\left(\mathsf{ѵ}-ȥ\right)}\left[{ℐ}_{{ȥ}^{+}}^{\alpha } {\mathsf{Ϣ}}_{*}\left(\mathsf{ѵ}\right)\times {{\rm T}}_{*}\left(\mathsf{ѵ}\right)+{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha } {\mathsf{Ϣ}}_{*}(ȥ)\times {{\rm T}}_{*}(ȥ)\right]\\ +\frac{\pi e^{-\rho -1}\left(e^{\rho }+1\right)}{2e\left({\rho }^2+{\pi }^2\right)}\frac{{\nabla }_{*}\left(\left(ȥ,\mathsf{ѵ}\right)\right)}{2e}+\frac{1}{2e}D\left(\rho \right)\frac{{\Delta }_{*}\left(\left(ȥ,\mathsf{ѵ}\right)\right)}{2e}.\end{array}$$

(73)

Similarly, for ${\mathsf{Ϣ}}^{*}\left(\varkappa \right)$, we have

$$\begin{array}{l}\frac{1-e^{-\rho }}{\rho } {\mathsf{Ϣ}}^{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\times {{\rm T}}^{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\\ \le \frac{\alpha }{2e\left(\mathsf{ѵ}-ȥ\right)}\left[{ℐ}_{{ȥ}^{+}}^{\alpha } {\mathsf{Ϣ}}^{*}\left(\mathsf{ѵ}\right)\times {{\rm T}}^{*}\left(\mathsf{ѵ}\right)+{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha } {\mathsf{Ϣ}}^{*}(ȥ)\times {{\rm T}}^{*}(ȥ)\right]\\ +\frac{\pi e^{-\rho -1}\left(e^{\rho }+1\right)}{2e\left({\rho }^2+{\pi }^2\right)}\frac{{\nabla }^{*}\left(\left(ȥ,\mathsf{ѵ}\right)\right)}{2e}+\frac{1}{2e}D\left(\rho \right)\frac{{\Delta }^{*}\left(\left(ȥ,\mathsf{ѵ}\right)\right)}{2e}.\end{array}$$

(74)

From (73) and (74), we have

$$\begin{array}{cc}\hfill 2[{\mathsf{Ϣ}}_{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)& \times {{\rm T}}_{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right), {\mathsf{Ϣ}}^{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\times {{\rm T}}^{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)]\hfill \\ & {\le }_p\frac{1-\alpha }{e\left(\mathsf{ѵ}-ȥ\right)}\left[{ℐ}_{{ȥ}^{+}}^{\alpha } {\mathsf{Ϣ}}_{*}\left(\mathsf{ѵ}\right)\times {{\rm T}}_{*}\left(\mathsf{ѵ}\right)+{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha } {\mathsf{Ϣ}}_{*}(ȥ)\times {{\rm T}}_{*}(ȥ), {ℐ}_{{ȥ}^{+}}^{\alpha } {\mathsf{Ϣ}}^{*}\left(\mathsf{ѵ}\right)\times {{\rm T}}^{*}\left(\mathsf{ѵ}\right)+{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha } {\mathsf{Ϣ}}^{*}(ȥ)\times {{\rm T}}^{*}(ȥ)\right]\hfill \\ & +\frac{\rho \pi e^{-\rho -1}\left(e^{\rho }+1\right)}{e\left(1-e^{-\rho }\right)\left({\rho }^2+{\pi }^2\right)}\left[{\nabla }_{*}\left(\left(ȥ,\mathsf{ѵ}\right)\right), {\nabla }^{*}\left(\left(ȥ,\mathsf{ѵ}\right)\right)\right]+\frac{\rho }{e\left(1-e^{-\rho }\right)}D\left(\rho \right)\left[{\nabla }_{*}\left(\left(ȥ,\mathsf{ѵ}\right)\right), {\nabla }^{*}\left(\left(ȥ,\mathsf{ѵ}\right)\right)\right],\hfill \end{array}$$

where

$$D\left(\rho \right)=-\frac{e^{-\rho -2}\left(8e^2-8\rho e^2+{\pi }^2{e}^2+2{\rho }^2{e}^2-{\pi }^2{e}^{\rho }\right)}{2\left(\rho -2\right)\left({\rho }^2-4\rho +{\pi }^2+4\right)}+\frac{8\rho -{\pi }^2{e}^{-\rho -2}+{\pi }^2+2{\rho }^2+8}{2\left(\rho +2\right)\left({\rho }^2+4\rho +{\pi }^2+4\right).}$$

Hence, the required result. □

Remark 6.

From Theorem 6 we can clearly see the following.

If one lays $\mathsf{Ϣ}$ and ${\rm T}$ which are upper left and right exponential trigonometric concave I-V∙Ms on $\left[ȥ,\mathsf{ѵ}\right],$ then one acquires the following inequality [98]:

$$\begin{array}{cc}\hfill 2\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\times {\rm T}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)& {\supseteq }_p\frac{\alpha }{e\left(\mathsf{ѵ}-ȥ\right)}\left[{ℐ}_{{ȥ}^{+}}^{\alpha }\mathsf{Ϣ}\left(\mathsf{ѵ}\right)\times {\rm T}\left(\mathsf{ѵ}\right)+{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha }\mathsf{Ϣ}(ȥ)\times {\rm T}(ȥ)\right]\hfill \\ & +\frac{\rho \pi e^{-\rho -1}\left(e^{\rho }+1\right)}{e\left(1-e^{-\rho }\right)\left({\rho }^2+{\pi }^2\right)}\nabla \left(ȥ,\mathsf{ѵ}\right)+\frac{\rho }{e\left(1-e^{-\rho }\right)}D\left(\rho \right)\Delta \left(ȥ,\mathsf{ѵ}\right).\hfill \end{array}$$

(75)

If $\alpha \to 1$  , that is

$$\underset{\alpha \to 1}{\lim }\rho =\underset{\alpha \to 1}{\lim }\frac{1-\alpha }{\alpha }\left(\mathsf{ѵ}-ȥ\right)=0,then \underset{\alpha \to 1}{\lim }\frac{1-\alpha }{e\left(1-e^{-\rho }\right)}=\frac{1}{e\left(\mathsf{ѵ}-ȥ\right),}$$

$$\underset{\alpha \to 1}{\lim }\frac{\rho }{e\left(1-e^{-\rho }\right)}\left(-\frac{e^{-\rho -2}\left(8e^2-8\rho e^2+{\pi }^2{e}^2+2{\rho }^2{e}^2-{\pi }^2{e}^{\rho }\right)}{2\left(\rho -2\right)\left({\rho }^2-4\rho +{\pi }^2+4\right)}+\frac{8\rho -{\pi }^2{e}^{-\rho -2}+{\pi }^2+2{\rho }^2+8}{2\left(\rho +2\right)\left({\pi }^2+4\right)}\right)=\frac{{\pi }^2-{\pi }^2{e}^{-2}+8}{2e\left({\pi }^2+4\right)},$$

$$\underset{\alpha \to 1}{\lim }\frac{\rho \pi e^{-\rho -1}\left(e^{\rho }+1\right)}{e\left(1-e^{-\rho }\right)\left({\rho }^2+{\pi }^2\right)}=\frac{2}{\pi e^2}.$$

Then, we acquire the following result, which is also a new one:

$$2\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\times {\rm T}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right){\le }_p\frac{2}{e\left(\mathsf{ѵ}-ȥ\right)}{{\displaystyle \int }}_{ȥ}^{\mathsf{ѵ}}\mathsf{Ϣ}\left(\varkappa \right)\times {\rm T}\left(\varkappa \right)d\varkappa +\frac{2}{\pi e^2}\Delta \left(ȥ,\mathsf{ѵ}\right)+\frac{{\pi }^2-{\pi }^2{e}^2+8}{2e\left({\pi }^2+4\right)}\nabla \left(ȥ,\mathsf{ѵ}\right).$$

(76)

If one lays $\alpha \to 1$ and $\mathsf{Ϣ}$ and $T$ which are upper left and right exponential trigonometric concave I-V∙Ms on $\left[ȥ,\mathsf{ѵ}\right],$ then one acquires the following inequality [98]:

$$2\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\times {\rm T}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right){\supseteq }_p\frac{2}{e\left(\mathsf{ѵ}-ȥ\right)}{{\displaystyle \int }}_{ȥ}^{\mathsf{ѵ}}\mathsf{Ϣ}\left(\varkappa \right)\times {\rm T}\left(\varkappa \right)d\varkappa +\frac{2}{\pi e^2}\Delta \left(ȥ,\mathsf{ѵ}\right)+\frac{{\pi }^2-{\pi }^2{e}^2+8}{2e\left({\pi }^2+4\right)}\nabla \left(ȥ,\mathsf{ѵ}\right).$$

(77)

Let $\alpha \to 1$, ${\mathsf{Ϣ}}_{*}\left(\varkappa \right)\ne {\mathsf{Ϣ}}^{*}\left(\varkappa \right)$ and $T_{*}\left(\varkappa \right)\ne T^{*}\left(\varkappa \right)$  . Then, from Theorem 6 we achieve the Hermite–Hadamard inequality for interval-valued   left and right exponential trigonometric convex mapping, which is also a new one:

$$2\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\times {\rm T}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right){\le }_p\frac{2}{e\left(\mathsf{ѵ}-ȥ\right)}{{\displaystyle \int }}_{ȥ}^{\mathsf{ѵ}}\mathsf{Ϣ}\left(\varkappa \right)\times {\rm T}\left(\varkappa \right)d\varkappa +\frac{2}{\pi e^2}\Delta \left(ȥ,\mathsf{ѵ}\right)+\frac{{\pi }^2-{\pi }^2{e}^2+8}{2e\left({\pi }^2+4\right)}\nabla \left(ȥ,\mathsf{ѵ}\right).$$

(78)

If ${\mathsf{Ϣ}}_{*}\left(\varkappa \right)={\mathsf{Ϣ}}^{*}\left(\varkappa \right)$ and $T_{*}\left(\varkappa \right)\ne T^{*}\left(\varkappa \right)$  , then, from Theorem 6, we achieve the classical fractional Hermite–Hadamard inequality for   exponential trigonometric convex mapping

$$\begin{array}{cc}\hfill 2 \mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\times {\rm T}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)& \le \frac{\alpha }{e\left(\mathsf{ѵ}-ȥ\right)}\left[{ℐ}_{{ȥ}^{+}}^{\alpha } \mathsf{Ϣ}\left(\mathsf{ѵ}\right)\times {\rm T}\left(\mathsf{ѵ}\right)+{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha } \mathsf{Ϣ}(ȥ)\times {\rm T}(ȥ)\right]\hfill \\ & +\frac{\rho \pi e^{-\rho -1}\left(e^{\rho }+1\right)}{e\left(1-e^{-\rho }\right)\left({\rho }^2+{\pi }^2\right)}\nabla \left(ȥ,\mathsf{ѵ}\right)+\frac{\rho }{e\left(1-e^{-\rho }\right)}D\left(\rho \right)\Delta \left(ȥ,\mathsf{ѵ}\right).\hfill \end{array}$$

(79)

Let $\alpha \to 1$, ${\mathsf{Ϣ}}_{*}\left(\varkappa \right)={\mathsf{Ϣ}}^{*}\left(\varkappa \right)$ and $T_{*}\left(\varkappa \right)\ne T^{*}\left(\varkappa \right)$. Then, from Theorem 6, we achieve the classical Hermite–Hadamard inequality for exponential trigonometric convex mapping, see [78]

$$2\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\times {\rm T}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\le \frac{2}{e\left(\mathsf{ѵ}-ȥ\right)}{{\displaystyle \int }}_{ȥ}^{\mathsf{ѵ}}\mathsf{Ϣ}\left(\varkappa \right)\times {\rm T}\left(\varkappa \right)d\varkappa +\frac{2}{\pi e^2}\Delta \left(ȥ,\mathsf{ѵ}\right)+\frac{{\pi }^2-{\pi }^2{e}^2+8}{2e\left({\pi }^2+4\right)}\nabla \left(ȥ,\mathsf{ѵ}\right).$$

(80)

Theorem 7.

Let $\mathsf{Ϣ}:\left[ȥ, \mathsf{ѵ}\right]\subset ℝ\to X_C^{+}$ be an I-V∙M on $\left[ȥ, \mathsf{ѵ}\right]$ given by $\mathsf{Ϣ}\left(\varkappa \right)=\left[{\mathsf{Ϣ}}_{*}\left(\varkappa \right),{ \mathsf{Ϣ}}^{*}\left(\varkappa \right)\right]$ for all $\varkappa \in \left[ȥ, \mathsf{ѵ}\right]$. If $\mathsf{Ϣ}$ is a left and right exponential trigonometric convex I-V∙M on $\left[ȥ, \mathsf{ѵ}\right]$ and $\mathsf{Ϣ}\in L\left(\left[ȥ, \mathsf{ѵ}\right],X_C^{+}\right)$, then

$$\begin{array}{cc}\hfill e\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)& {\le }_p\sqrt{\frac{e}{2}}\left[\mathsf{Ϣ}\left(\frac{3ȥ+\mathsf{ѵ}}{2}\right)+\mathsf{Ϣ}\left(\frac{ȥ+3\mathsf{ѵ}}{2}\right)\right]\hfill \\ & {\le }_p\frac{1-\alpha }{2\left(1-e^{-\frac{\rho }{2}}\right)}\left[{ℐ}_{{ȥ}^{+}}^{\alpha } \mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)+{ℐ}_{{\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)}^{+}}^{\alpha }\mathsf{Ϣ}\left(\mathsf{ѵ}\right)+{ℐ}_{{\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)}^{-}}^{\alpha }\mathsf{Ϣ}(ȥ)+{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha }\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\right]\hfill \\ & {\le }_p\frac{\rho }{2\left(1-e^{-\frac{\rho }{2}}\right)}K\left(\rho \right)\left(\frac{\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)}{2}+\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\right)\hfill \\ & {\le }_p\frac{\rho }{2\left(1-e^{-\frac{\rho }{2}}\right)}\left(1+\sqrt{\frac{e}{2}}\right)K\left(\rho \right)\frac{\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)}{2}.\hfill \end{array}$$

(81)

If $\mathsf{Ϣ}\left(\varkappa \right)$  is a left and right exponential trigonometric concave I-V∙M, then

$$\begin{array}{cc}\hfill e\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)& {\ge }_p\sqrt{\frac{e}{2}}\left[\mathsf{Ϣ}\left(\frac{3ȥ+\mathsf{ѵ}}{4}\right)+\mathsf{Ϣ}\left(\frac{ȥ+3\mathsf{ѵ}}{4}\right)\right]\hfill \\ & {\ge }_p\frac{1-\alpha }{2\left(1-e^{-\frac{\rho }{2}}\right)}\left[{ℐ}_{{ȥ}^{+}}^{\alpha }\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)+{ℐ}_{{\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)}^{+}}^{\alpha }\mathsf{Ϣ}\left(\mathsf{ѵ}\right)+{ℐ}_{{\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)}^{-}}^{\alpha }\mathsf{Ϣ}(ȥ)+{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha }\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\right]\hfill \\ & {\ge }_p\frac{\rho }{2\left(1-e^{-\frac{\rho }{2}}\right)}K\left(\rho \right)\left(\frac{\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)}{2}+\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\right)\hfill \\ & {\ge }_p\frac{\rho }{2\left(1-e^{-\frac{\rho }{2}}\right)}\left(1+\sqrt{\frac{e}{2}}\right)K\left(\rho \right)\frac{\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)}{2},\hfill \end{array}$$

(82)

where

$$K\left(\rho \right)=\frac{2\rho +2\pi e^{-\frac{\rho +2}{2}}+4}{{\rho }^2+4\rho +{\pi }^2+4}+\frac{2\pi e^{-1}+2e^{-\frac{\rho }{2}}\left(2+\rho \right)}{{\rho }^2-4\rho +{\pi }^2+4},\rho =\frac{1-\alpha }{\alpha }\left(\mathsf{ѵ}-ȥ\right),\mathrm{and}1>\alpha >0.$$

Proof. 

Taking $\left[ȥ, \frac{ȥ+\mathsf{ѵ}}{2}\right],$ we deduce that

$$\mathsf{Ϣ}\left(\frac{3ȥ+\mathsf{ѵ}}{4}\right){\le }_p\frac{\sin \frac{\pi }{4}}{\sqrt{e}}\mathsf{Ϣ}\left(\mathfrak{v}ȥ+\left(1-\mathfrak{v}\right)\frac{ȥ+\mathsf{ѵ}}{2}\right)+\frac{\cos \frac{\pi }{4}}{\sqrt{e}}\mathsf{Ϣ}\left(\left(1-\mathfrak{v}\right)ȥ+\mathfrak{v}\frac{ȥ+\mathsf{ѵ}}{2}\right).$$

After simplification, we find that

$$2\mathsf{Ϣ}\left(\frac{3ȥ+\mathsf{ѵ}}{4}\right){\le }_p\sqrt{\frac{2}{e}}\left[\mathsf{Ϣ}\left(\mathfrak{v}ȥ+\left(1-\mathfrak{v}\right)\frac{ȥ+\mathsf{ѵ}}{2}\right)+\mathsf{Ϣ}\left(\left(1-\mathfrak{v}\right)ȥ+\mathfrak{v}\frac{ȥ+\mathsf{ѵ}}{2}\right)\right].$$

Therefore, we have

$$\begin{array}{c}2{\mathsf{Ϣ}}_{*}\left(\frac{3ȥ+\mathsf{ѵ}}{4}\right)\le \sqrt{\frac{2}{e}}\left[{\mathsf{Ϣ}}_{*}\left(\mathfrak{v}ȥ+\left(1-\mathfrak{v}\right)\frac{ȥ+\mathsf{ѵ}}{2}\right)+{\mathsf{Ϣ}}_{*}\left(\left(1-\mathfrak{v}\right)ȥ+\mathfrak{v}\frac{ȥ+\mathsf{ѵ}}{2}\right)\right]\end{array},$$

$$2{\mathsf{Ϣ}}^{*}\left(\frac{3ȥ+\mathsf{ѵ}}{4}\right)\le \sqrt{\frac{2}{e}}\left[{\mathsf{Ϣ}}^{*}\left(\mathfrak{v}ȥ+\left(1-\mathfrak{v}\right)\frac{ȥ+\mathsf{ѵ}}{2}\right)+{\mathsf{Ϣ}}^{*}\left(\left(1-\mathfrak{v}\right)ȥ+\mathfrak{v}\frac{ȥ+\mathsf{ѵ}}{2}\right)\right].$$

Taking ${\mathsf{Ϣ}}_{*}(.)$ and multiplying both sides by $e^{-\frac{\rho \mathfrak{v}}{2}}$ and integrating the obtained result with respect to $\mathfrak{v}$ from $0$ to $1$, we have

$${{\displaystyle \int }}_0^1{e}^{-\frac{\rho \mathfrak{v}}{2}}{\mathsf{Ϣ}}_{*}\left(\frac{3ȥ+\mathsf{ѵ}}{4}\right)d\mathfrak{v} \le \sqrt{\frac{2}{e}}\left[{{\displaystyle \int }}_0^1{e}^{-\rho \mathfrak{v}}{\mathsf{Ϣ}}_{*}\left(\mathfrak{v}ȥ+\left(1-\mathfrak{v}\right)\frac{ȥ+\mathsf{ѵ}}{2}\right)d\mathfrak{v}+{{\displaystyle \int }}_0^1{e}^{-\rho \mathfrak{v}}{\mathsf{Ϣ}}_{*}\left(\left(1-\mathfrak{v}\right)ȥ+\mathfrak{v}\frac{ȥ+\mathsf{ѵ}}{2}\right)d\mathfrak{v}\right].$$

Let $u=\mathfrak{v}ȥ+\left(1-\mathfrak{v}\right)\frac{ȥ+\mathsf{ѵ}}{2}$ and $\varkappa =\left(1-\mathfrak{v}\right)ȥ+\mathfrak{v}\frac{ȥ+\mathsf{ѵ}}{2}.$ Then, we have

$$\begin{array}{cc}\hfill {{\displaystyle \int }}_0^1{e}^{-\frac{\rho \mathfrak{v}}{2}}{\mathsf{Ϣ}}_{*}\left(\frac{3ȥ+\mathsf{ѵ}}{4}\right)d\mathfrak{v} & \le \sqrt{\frac{2}{e}}\frac{1}{\mathsf{ѵ}-ȥ} {{\displaystyle \int }}_{\frac{ȥ+\mathsf{ѵ}}{2}}^{\mathsf{ѵ}}{e}^{\left(-\frac{1-\alpha }{\alpha }\left(\frac{ȥ+\mathsf{ѵ}}{2}-u\right)\right)}{\mathsf{Ϣ}}_{*}\left(u\right)du+\frac{1}{\mathsf{ѵ}-ȥ}{{\displaystyle \int }}_{\frac{ȥ+\mathsf{ѵ}}{2}}^{\mathsf{ѵ}}{e}^{\left(-\frac{1-\alpha }{\alpha }\left(\varkappa -ȥ\right)\right)}{\mathsf{Ϣ}}_{*}\left(\varkappa \right)d\varkappa \hfill \\ & =\sqrt{\frac{2}{e}}\frac{\alpha }{\mathsf{ѵ}-ȥ}\left[{ℐ}_{{ȥ}^{+}}^{\alpha } {\mathsf{Ϣ}}_{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)+{ℐ}_{{\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)}^{-}}^{\alpha } {\mathsf{Ϣ}}_{*}(ȥ)\right].\hfill \end{array}$$

(83)

Now, taking the right side of Equation (83), we have

$${{\displaystyle \int }}_0^1{e}^{-\frac{\rho \mathfrak{v}}{2}}{\mathsf{Ϣ}}_{*}\left(\frac{3ȥ+\mathsf{ѵ}}{2}\right)d\mathfrak{v}=\frac{2\left(1-e^{-\rho }\right)}{\rho }{\mathsf{Ϣ}}_{*}\left(\frac{3ȥ+\mathsf{ѵ}}{4}\right).$$

(84)

From (83) and (84), we deduce that

$$\frac{2\left(1-e^{-\rho }\right)}{\rho }{\mathsf{Ϣ}}_{*}\left(\frac{3ȥ+\mathsf{ѵ}}{4}\right)\begin{array}{c}\le \sqrt{\frac{2}{e}}\frac{1-\alpha }{\mathsf{ѵ}-ȥ}\left[{ℐ}_{{ȥ}^{+}}^{\alpha } {\mathsf{Ϣ}}_{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)+{ℐ}_{{\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)}^{-}}^{\alpha } {\mathsf{Ϣ}}_{*}(ȥ)\right].\end{array}$$

(85)

Similarly, for ${\mathsf{Ϣ}}^{*}(.)$, from (85), we have

$$\frac{2\left(1-e^{-\rho }\right)}{\rho }{\mathsf{Ϣ}}^{*}\left(\frac{3ȥ+\mathsf{ѵ}}{4}\right)\begin{array}{c}\le \sqrt{\frac{2}{e}}\frac{1-\alpha }{\mathsf{ѵ}-ȥ}\left[{ℐ}_{{ȥ}^{+}}^{\alpha }{\mathsf{Ϣ}}^{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)+{ℐ}_{{\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)}^{-}}^{\alpha }{\mathsf{Ϣ}}^{*}(ȥ)\right].\end{array}$$

(86)

From (85) and (86), we deduce that

$$\frac{2\left(1-e^{-\rho }\right)}{\rho }\mathsf{Ϣ}\left(\frac{3ȥ+\mathsf{ѵ}}{4}\right)\begin{array}{c}{\le }_p\sqrt{\frac{2}{e}}\frac{1-\alpha }{\mathsf{ѵ}-ȥ}\left[{ℐ}_{{ȥ}^{+}}^{\alpha }\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)+{ℐ}_{{\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)}^{-}}^{\alpha }\mathsf{Ϣ}(ȥ)\right].\end{array}$$

(87)

For the right side of Equation (81), since $\mathsf{Ϣ}$ is a left and right exponential trigonometric convex I-V∙M, then we can deduce that

$$\mathsf{Ϣ}\left(\mathfrak{v}ȥ+\left(1-\mathfrak{v}\right)\frac{ȥ+\mathsf{ѵ}}{2}\right){\le }_p\frac{\sin \frac{\pi \mathfrak{v}}{4}}{e^{1-\mathfrak{v}}}\mathsf{Ϣ}(ȥ)+\frac{\cos \frac{\pi \mathfrak{v}}{4}}{e^{\mathfrak{v}}}\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right),$$

(88)

and

$$\mathsf{Ϣ}\left(\left(1-\mathfrak{v}\right)ȥ+\mathfrak{v}\frac{ȥ+\mathsf{ѵ}}{2}\right){\le }_p\frac{\cos \frac{\pi \mathfrak{v}}{4}}{e^{\mathfrak{v}}}\mathsf{Ϣ}(ȥ)+\frac{\sin \frac{\pi \mathfrak{v}}{4}}{e^{1-\mathfrak{v}}}\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)$$

(89)

Adding (88) and (89), we have

$$\mathsf{Ϣ}\left(\mathfrak{v}ȥ+\left(1-\mathfrak{v}\right)\frac{ȥ+\mathsf{ѵ}}{2}\right)+\mathsf{Ϣ}\left(\left(1-\mathfrak{v}\right)ȥ+\mathfrak{v}\frac{ȥ+\mathsf{ѵ}}{2}\right){\le }_p\left[\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\right]\left[\frac{\sin \frac{\pi \mathfrak{v}}{4}}{e^{1-\mathfrak{v}}}+\frac{\cos \frac{\pi \mathfrak{v}}{4}}{e^{\mathfrak{v}}}\right].$$

(90)

Since $\mathsf{Ϣ}$ is I-V∙M, then we have

$$\begin{array}{c}{\mathsf{Ϣ}}_{*}\left(\mathfrak{v}ȥ+\left(1-\mathfrak{v}\right)\frac{ȥ+\mathsf{ѵ}}{2}\right)+{\mathsf{Ϣ}}_{*}\left(\left(1-\mathfrak{v}\right)ȥ+\mathfrak{v}\frac{ȥ+\mathsf{ѵ}}{2}\right)\le \left[{\mathsf{Ϣ}}_{*}(ȥ)+{\mathsf{Ϣ}}_{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\right]\left[\frac{\sin \frac{\pi \mathfrak{v}}{4}}{e^{1-\mathfrak{v}}}+\frac{\cos \frac{\pi \mathfrak{v}}{4}}{e^{\mathfrak{v}}}\right], \\ {\mathsf{Ϣ}}^{*}\left(\mathfrak{v}ȥ+\left(1-\mathfrak{v}\right)\frac{ȥ+\mathsf{ѵ}}{2}\right)+{\mathsf{Ϣ}}^{*}\left(\left(1-\mathfrak{v}\right)ȥ+\mathfrak{v}\frac{ȥ+\mathsf{ѵ}}{2}\right)\le \left[{\mathsf{Ϣ}}^{*}(ȥ)+{\mathsf{Ϣ}}^{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\right]\left[\frac{\sin \frac{\pi \mathfrak{v}}{4}}{e^{1-\mathfrak{v}}}+\frac{\cos \frac{\pi \mathfrak{v}}{4}}{e^{\mathfrak{v}}}\right]. \end{array}$$

(91)

Taking ${\mathsf{Ϣ}}_{*}(.)$ from (91) and multiplying the inequality by $e^{-\frac{\rho \mathfrak{v}}{2}}$, and integrating the resultant with $\mathfrak{v}$ from $0$ to $1$, we have

$$\begin{array}{cc}\hfill {{\displaystyle \int }}_0^1{e}^{e^{-\frac{\rho \mathfrak{v}}{2}}}{\mathsf{Ϣ}}_{*}(\mathfrak{v}ȥ& +\left(1-\mathfrak{v}\right)\frac{ȥ+\mathsf{ѵ}}{2} )d\mathfrak{v}+{{\displaystyle \int }}_0^1{e}^{e^{-\frac{\rho \mathfrak{v}}{2}}}{\mathsf{Ϣ}}_{*}\left(\left(1-\mathfrak{v}\right)ȥ+\mathfrak{v}\frac{ȥ+\mathsf{ѵ}}{2}\right)d\mathfrak{v}\hfill \\ & \le \left[{\mathsf{Ϣ}}_{*}(ȥ)+{\mathsf{Ϣ}}_{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\right]{{\displaystyle \int }}_0^1{e}^{e^{-\frac{\rho \mathfrak{v}}{2}}}\left[\frac{\sin \frac{\pi \mathfrak{v}}{4}}{e^{1-\mathfrak{v}}}+\frac{\cos \frac{\pi \mathfrak{v}}{4}}{e^{\mathfrak{v}}}\right]d\mathfrak{v},\hfill \\ & =\left(\frac{2\rho +2\pi e^{-\frac{\rho +2}{2}}+4}{{\rho }^2+4\rho +{\pi }^2+4}+\frac{2\pi e^{-1}+2e^{-\frac{\rho }{2}}\left(2+\rho \right)}{{\rho }^2-4\rho +{\pi }^2+4}\right)\left[{\mathsf{Ϣ}}_{*}(ȥ)+{\mathsf{Ϣ}}_{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\right].\hfill \end{array}$$

(92)

In a similar way to the above, for ${\mathsf{Ϣ}}_{*}(.)$ we have

$$\begin{array}{cc}\hfill {{\displaystyle \int }}_0^1{e}^{e^{-\frac{\rho \mathfrak{v}}{2}}}{\mathsf{Ϣ}}^{*}(\mathfrak{v}ȥ& +\left(1-\mathfrak{v}\right)\frac{ȥ+\mathsf{ѵ}}{2} )d\mathfrak{v}+{{\displaystyle \int }}_0^1{e}^{e^{-\frac{\rho \mathfrak{v}}{2}}}{\mathsf{Ϣ}}^{*}\left(\left(1-\mathfrak{v}\right)ȥ+\mathfrak{v}\frac{ȥ+\mathsf{ѵ}}{2}\right)d\mathfrak{v}\hfill \\ & \le \left[{\mathsf{Ϣ}}^{*}(ȥ)+{\mathsf{Ϣ}}^{*}\left(\mathsf{ѵ}\right)\right]{{\displaystyle \int }}_0^1{e}^{-\rho \mathfrak{v}}\left[\frac{\sin \frac{\pi \mathfrak{v}}{4}}{e^{\frac{2-\mathfrak{v}}{2}}}+\frac{\cos \frac{\pi \mathfrak{v}}{4}}{e^{\frac{\mathfrak{v}}{2}}}\right]d\mathfrak{v},\hfill \\ & =\left(\frac{2\rho +2\pi e^{-\frac{\rho +2}{2}}+4}{{\rho }^2+4\rho +{\pi }^2+4}+\frac{2\pi e^{-1}+2e^{-\frac{\rho }{2}}\left(2+\rho \right)}{{\rho }^2-4\rho +{\pi }^2+4}\right)\left[{\mathsf{Ϣ}}^{*}(ȥ)+{\mathsf{Ϣ}}^{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\right].\hfill \end{array}$$

(93)

From (92) and (93), we have

$$\begin{array}{cc}\hfill {{\displaystyle \int }}_0^1{e}^{-\rho \mathfrak{v}}\mathsf{Ϣ}(\mathfrak{v}ȥ+& \left(1-\mathfrak{v}\right)\frac{ȥ+\mathsf{ѵ}}{2})d\mathfrak{v}+{{\displaystyle \int }}_0^1{e}^{-\rho \mathfrak{v}}\mathsf{Ϣ}\left(\left(1-\mathfrak{v}\right)ȥ+\mathfrak{v}\frac{ȥ+\mathsf{ѵ}}{2}\right)d\mathfrak{v}\hfill \\ & {\le }_p\left(\frac{2\rho +2\pi e^{-\frac{\rho +2}{2}}+4}{{\rho }^2+4\rho +{\pi }^2+4}+\frac{2\pi e^{-1}+2e^{-\frac{\rho }{2}}\left(2+\rho \right)}{{\rho }^2-4\rho +{\pi }^2+4}\right)\left[\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\right].\hfill \end{array}$$

(94)

Combining (87) and (94), we have

$$\begin{array}{cc}\hfill \sqrt{\frac{e}{2}}\mathsf{Ϣ}\left(\frac{3ȥ+\mathsf{ѵ}}{2}\right)& {\le }_p\frac{1-\alpha }{2\left(1-e^{-\frac{\rho }{2}}\right)}\left[{ℐ}_{{ȥ}^{+}}^{\alpha }\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)+{ℐ}_{{\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)}^{-}}^{\alpha }\mathsf{Ϣ}(ȥ)\right]\hfill \\ & {\le }_p\frac{\rho }{2\left(1-e^{-\frac{\rho }{2}}\right)}K\left(\rho \right)\left(\frac{\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)}{2}\right),\hfill \end{array}$$

(95)

where

$$K\left(\rho \right)=\frac{2\rho +2\pi e^{-\frac{\rho +2}{2}}+4}{{\rho }^2+4\rho +{\pi }^2+4}+\frac{2\pi e^{-1}+2e^{-\frac{\rho }{2}}\left(2+\rho \right)}{{\rho }^2-4\rho +{\pi }^2+4}.$$

Similarly, if we take the interval $\left[\frac{ȥ+\mathsf{ѵ}}{2}, \mathsf{ѵ}\right],$ then, from (38), we find that

$$\begin{array}{cc}\hfill \sqrt{\frac{e}{2}}\left[\mathsf{Ϣ}\left(\frac{ȥ+3\mathsf{ѵ}}{2}\right)\right]& {\le }_p\frac{1-\alpha }{2\left(1-e^{-\frac{\rho }{2}}\right)}\left[{ℐ}_{{\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)}^{+}}^{\alpha }\mathsf{Ϣ}\left(\mathsf{ѵ}\right)+{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha }\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\right]\hfill \\ & {\le }_p\frac{\rho }{2\left(1-e^{-\frac{\rho }{2}}\right)}K\left(\rho \right)\left(\frac{\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)}{2}\right).\hfill \end{array}$$

(96)

Adding (95) and (96), we have

$$\begin{array}{c}\sqrt{\frac{e}{2}}\left[\mathsf{Ϣ}\left(\frac{3ȥ+\mathsf{ѵ}}{2}\right)+\mathsf{Ϣ}\left(\frac{ȥ+3\mathsf{ѵ}}{2}\right)\right]\hfill \\ {\le }_p\frac{1-\alpha }{2\left(1-e^{-\frac{\rho }{2}}\right)}\left[{ℐ}_{{ȥ}^{+}}^{\alpha } \mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)+{ℐ}_{{\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)}^{+}}^{\alpha } \mathsf{Ϣ}\left(\mathsf{ѵ}\right)+{ℐ}_{{\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)}^{-}}^{\alpha }\mathsf{Ϣ}(ȥ)+{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha }\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\right]\hfill \\ {\le }_p\frac{\rho }{2\left(1-e^{-\frac{\rho }{2}}\right)}K\left(\rho \right)\left(\frac{\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)}{2}+\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\right).\hfill \end{array}$$

(97)

To achieve the first and fourth order relations in (81), again by taking

$$\begin{array}{cc}\hfill \mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)& =\mathsf{Ϣ}\left(\frac{\frac{3ȥ+\mathsf{ѵ}}{4}+\frac{ȥ+3\mathsf{ѵ}}{4}}{2}\right)\hfill \\ & {\le }_p\frac{\sin \frac{\pi }{4}}{\sqrt{e}}\mathsf{Ϣ}(ȥ)+\frac{\cos \frac{\pi }{4}}{\sqrt{e}}\mathsf{Ϣ}\left(\mathsf{ѵ}\right)\hfill \\ & =\frac{1}{2}\sqrt{\frac{2}{e}}\mathsf{Ϣ}(ȥ)+\frac{1}{2}\sqrt{\frac{2}{e}}\mathsf{Ϣ}\left(\mathsf{ѵ}\right)\hfill \end{array}$$

(98)

and

$$\begin{array}{cc}\hfill \mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)& =\mathsf{Ϣ}\left(\frac{\frac{3ȥ+\mathsf{ѵ}}{4}+\frac{ȥ+3\mathsf{ѵ}}{4}}{2}\right)\hfill \\ & {\le }_p\frac{\sin \frac{\pi }{4}}{\sqrt{e}}\mathsf{Ϣ}\left(\frac{3ȥ+\mathsf{ѵ}}{4}\right)+\frac{\cos \frac{\pi }{4}}{\sqrt{e}}\mathsf{Ϣ}\left(\frac{ȥ+3\mathsf{ѵ}}{4}\right)\hfill \\ & =\frac{1}{2}\sqrt{\frac{2}{e}}\mathsf{Ϣ}\left(\frac{3ȥ+\mathsf{ѵ}}{4}\right)+\frac{1}{2}\sqrt{\frac{2}{e}}\mathsf{Ϣ}\left(\frac{ȥ+3\mathsf{ѵ}}{4}\right).\hfill \end{array}$$

(99)

By using the inclusion relation (98) and (99), we obtain the first and fourth inclusions of (81). By combining the resultant inclusion and (97), we obtain the following relation:

$$\begin{array}{cc}\hfill e\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)& {\le }_p\sqrt{\frac{e}{2}}\left[\mathsf{Ϣ}\left(\frac{3ȥ+\mathsf{ѵ}}{2}\right)+\mathsf{Ϣ}\left(\frac{ȥ+3\mathsf{ѵ}}{2}\right)\right]\hfill \\ & {\le }_p\frac{1-\alpha }{2\left(1-e^{-\frac{\rho }{2}}\right)}\left[{ℐ}_{{ȥ}^{+}}^{\alpha }\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)+{ℐ}_{{\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)}^{+}}^{\alpha }\mathsf{Ϣ}\left(\mathsf{ѵ}\right)+{ℐ}_{{\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)}^{-}}^{\alpha }\mathsf{Ϣ}(ȥ)+{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha }\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\right]\hfill \\ & {\le }_p\frac{\rho }{2\left(1-e^{-\frac{\rho }{2}}\right)}K\left(\rho \right)\left(\frac{\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)}{2}+\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\right)\hfill \\ & {\le }_p\frac{\rho }{2\left(1-e^{-\frac{\rho }{2}}\right)}\left(1+\sqrt{\frac{e}{2}}\right)K\left(\rho \right)\frac{\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)}{2}.\hfill \end{array}$$

Hence, the required result. □

Remark 7.

From Theorem 7 we can clearly see the following.

If one lays $\mathsf{Ϣ}$ which is an upper left and right exponential trigonometric concave I-V∙M on $\left[ȥ,\mathsf{ѵ}\right],$ then one acquires the following inequality [98]:

$$\begin{array}{cc}\hfill e\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)& {\supseteq }_p\sqrt{\frac{e}{2}}\left[\mathsf{Ϣ}\left(\frac{3ȥ+\mathsf{ѵ}}{2}\right)\mathsf{Ϣ}\left(\frac{ȥ+3\mathsf{ѵ}}{2}\right)\right]\hfill \\ & {\supseteq }_p\frac{1-\alpha }{2\left(1-e^{-\frac{\rho }{2}}\right)}\left[{ℐ}_{{ȥ}^{+}}^{\alpha }\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)+{ℐ}_{{\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)}^{+}}^{\alpha }\mathsf{Ϣ}\left(\mathsf{ѵ}\right)+{ℐ}_{{\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)}^{-}}^{\alpha }\mathsf{Ϣ}(ȥ)+{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha }\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\right]\hfill \\ & {\supseteq }_p\frac{\rho }{2\left(1-e^{-\frac{\rho }{2}}\right)}K\left(\rho \right)\left(\frac{\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)}{2}+\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\right)\hfill \\ & {\supseteq }_p\frac{\rho }{2\left(1-e^{-\frac{\rho }{2}}\right)}\left(1+\sqrt{\frac{e}{2}}\right)K\left(\rho \right)\frac{\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)}{2}.\hfill \end{array}$$

(100)

If $\alpha \to 1$, that is

$$\underset{\alpha \to 1}{\lim }\rho =\underset{\alpha \to 1}{\lim }\frac{1-\alpha }{\alpha }\left(\mathsf{ѵ}-ȥ\right)=0,\mathrm{then}$$

$$\underset{\alpha \to 1}{\lim }\frac{\rho }{1-e^{-\frac{\rho }{2}}}\left(\frac{2\rho +2\pi e^{-\frac{\rho +2}{2}}+4}{{\rho }^2+4\rho +{\pi }^2+4}+\frac{2\pi e^{-1}+2e^{-\frac{\rho }{2}}\left(2+\rho \right)}{{\rho }^2-4\rho +{\pi }^2+4}\right)=\frac{4\left(2\pi e^{-1}+4\right)}{{\pi }^2+4}, \underset{\alpha \to 1}{\lim }\frac{1-\alpha }{2\left(1-e^{e^{-\frac{\rho }{2}}}\right)}=\frac{1}{\mathsf{ѵ}-ȥ}$$

Then, we acquire the following result, which is also a new one:

$$\begin{array}{cc}\hfill \frac{e}{2}\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)& {\le }_p\frac{1}{2}\sqrt{\frac{e}{2}}\left[\mathsf{Ϣ}\left(\frac{3ȥ+\mathsf{ѵ}}{2}\right)+\mathsf{Ϣ}\left(\frac{ȥ+3\mathsf{ѵ}}{2}\right)\right]\hfill \\ & {\le }_p\frac{1}{\mathsf{ѵ}-ȥ}{{\displaystyle \int }}_{ȥ}^{\mathsf{ѵ}}\mathsf{Ϣ}\left(\varkappa \right)d\varkappa \hfill \\ & {\le }_p\frac{2\pi e^{-1}+4}{{\pi }^2+4}\left(\frac{\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)}{2}+\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\right)\hfill \\ & {\le }_p\frac{2\pi e^{-1}+4}{{\pi }^2+4}\left(1+\sqrt{\frac{e}{2}}\right)\frac{\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)}{2}.\hfill \end{array}$$

(101)

If one lays $\alpha \to 1$ and $\mathsf{Ϣ}$ which is an upper left and right exponential trigonometric concave I-V∙M on $\left[ȥ,\mathsf{ѵ}\right],$ then one acquires the following inequality [98]:

$$\begin{array}{cc}\hfill \frac{e}{2}\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)& {\supseteq }_p\frac{1}{2}\sqrt{\frac{e}{2}}\left[\mathsf{Ϣ}\left(\frac{3ȥ+\mathsf{ѵ}}{2}\right)+\mathsf{Ϣ}\left(\frac{ȥ+3\mathsf{ѵ}}{2}\right)\right]\hfill \\ & {\supseteq }_p\frac{1}{\mathsf{ѵ}-ȥ}{{\displaystyle \int }}_{ȥ}^{\mathsf{ѵ}}\mathsf{Ϣ}\left(\varkappa \right)d\varkappa \hfill \\ & {\supseteq }_p\frac{2\pi e^{-1}+4}{{\pi }^2+4}\left(\frac{\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)}{2}+\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\right)\hfill \\ & {\supseteq }_p\frac{2\pi e^{-1}+4}{{\pi }^2+4}\left(1+\sqrt{\frac{e}{2}}\right)\frac{\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)}{2}.\hfill \end{array}$$

(102)

Let $\alpha \to 1$ and ${\mathsf{Ϣ}}_{*}\left(\varkappa \right)\ne {\mathsf{Ϣ}}^{*}\left(\varkappa \right)$  . Then, from Theorem 7, we achieve the Hermite–Hadamard inequality for interval-valued   left and right exponential trigonometric convex mapping, which is also a new one:

$$\begin{array}{cc}\hfill \frac{e}{2}\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)& {\le }_p\frac{1}{2}\sqrt{\frac{e}{2}}\left[\mathsf{Ϣ}\left(\frac{3ȥ+\mathsf{ѵ}}{2}\right)+\mathsf{Ϣ}\left(\frac{ȥ+3\mathsf{ѵ}}{2}\right)\right]\hfill \\ & {\le }_p\frac{1}{\mathsf{ѵ}-ȥ}{{\displaystyle \int }}_{ȥ}^{\mathsf{ѵ}}\mathsf{Ϣ}\left(\varkappa \right)d\varkappa \hfill \\ & {\le }_p\frac{2\pi e^{-1}+4}{{\pi }^2+4}\left(\frac{\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)}{2}+\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\right)\hfill \\ & {\le }_p\frac{2\pi e^{-1}+4}{{\pi }^2+4}\left(1+\sqrt{\frac{e}{2}}\right)\frac{\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)}{2}.\hfill \end{array}$$

(103)

If ${\mathsf{Ϣ}}_{*}\left(\varkappa \right)={\mathsf{Ϣ}}^{*}\left(\varkappa \right)$, then, from Theorem 7, we arrive at the classical fractional Hermite–Hadamard inequality for   exponential trigonometric convex mapping:

$$\begin{array}{cc}\hfill e\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)& \le \sqrt{\frac{e}{2}}\left[\mathsf{Ϣ}\left(\frac{3ȥ+\mathsf{ѵ}}{2}\right)+\mathsf{Ϣ}\left(\frac{ȥ+3\mathsf{ѵ}}{2}\right)\right]\hfill \\ & \le \frac{1-\alpha }{2\left(1-e^{-\frac{\rho }{2}}\right)}\left[{ℐ}_{{ȥ}^{+}}^{\alpha } \mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)+{ℐ}_{{\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)}^{+}}^{\alpha } \mathsf{Ϣ}\left(\mathsf{ѵ}\right)+{ℐ}_{{\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)}^{-}}^{\alpha } \mathsf{Ϣ}(ȥ)+{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha } \mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\right]\hfill \\ & \le \frac{\rho }{2\left(1-e^{-\frac{\rho }{2}}\right)}K\left(\rho \right)\left(\frac{\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)}{2}+\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\right)\hfill \\ & \le \frac{\rho }{2\left(1-e^{-\frac{\rho }{2}}\right)}\left(1+\sqrt{\frac{e}{2}}\right)K\left(\rho \right)\frac{\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)}{2}.\hfill \end{array}$$

(104)

Let $\alpha \to 1$ and ${\mathsf{Ϣ}}_{*}\left(\varkappa \right)={\mathsf{Ϣ}}^{*}\left(\varkappa \right)$. Then, from Theorem 7, we arrive at the classical Hermite–Hadamard inequality for exponential trigonometric convex mapping, see [78].

$$\begin{array}{cc}\hfill \frac{e}{2}\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)& \le \frac{1}{2}\sqrt{\frac{e}{2}}\left[\mathsf{Ϣ}\left(\frac{3ȥ+\mathsf{ѵ}}{2}\right)+\mathsf{Ϣ}\left(\frac{ȥ+3\mathsf{ѵ}}{2}\right)\right]\hfill \\ & \le \frac{1}{\mathsf{ѵ}-ȥ}{{\displaystyle \int }}_{ȥ}^{\mathsf{ѵ}}\mathsf{Ϣ}\left(\varkappa \right)d\varkappa \hfill \\ & \le \frac{2\pi e^{-1}+4}{{\pi }^2+4}\left(\frac{\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)}{2}+\mathsf{Ϣ}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\right)\hfill \\ & \le \frac{2\pi e^{-1}+4}{{\pi }^2+4}\left(1+\sqrt{\frac{e}{2}}\right)\frac{\mathsf{Ϣ}(ȥ)+\mathsf{Ϣ}\left(\mathsf{ѵ}\right)}{2}.\hfill \end{array}$$

(105)

To validate Theorem 7, we provide the following nontrivial example:

Example 3.

Let $\alpha =\frac{1}{3}, \varkappa \in \left[0,1\right]$, and the I-V∙M $\mathsf{Ϣ}:\left[ȥ, \mathsf{ѵ}\right]=\left[0, 1\right]\to X_C^{+},$ defined by

$$\mathsf{Ϣ}\left(\varkappa \right)=\left[2{\varkappa }^4,4{\varkappa }^4\right].$$

Since left and right end point mappings ${\mathsf{Ϣ}}_{*}\left(\varkappa \right)=2{\varkappa }^4,$ ${\mathsf{Ϣ}}^{*}\left(\varkappa \right)=4{\varkappa }^4$, are exponential trigonometric convex mappings then $\mathsf{Ϣ}\left(\varkappa \right)$ is an exponential trigonometric convex I-V∙M. We can clearly see that $\mathsf{Ϣ}\in L\left(\left[ȥ, \mathsf{ѵ}\right],X_C^{+}\right)$ and

$$e{\mathsf{Ϣ}}_{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)=\frac{e}{8}$$

$$e{\mathsf{Ϣ}}^{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)=\frac{e}{4}$$

$$\sqrt{\frac{e}{2}}\left[{\mathsf{Ϣ}}_{*}\left(\frac{3ȥ+\mathsf{ѵ}}{2}\right)+{\mathsf{Ϣ}}_{*}\left(\frac{ȥ+3\mathsf{ѵ}}{2}\right)\right]=\sqrt{\frac{e}{2}}\frac{41}{64}$$

$$\sqrt{\frac{e}{2}}\left[{\mathsf{Ϣ}}^{*}\left(\frac{3ȥ+\mathsf{ѵ}}{2}\right)+{\mathsf{Ϣ}}^{*}\left(\frac{ȥ+3\mathsf{ѵ}}{2}\right)\right]=\sqrt{\frac{e}{2}}\frac{41}{32}$$

$$\begin{array}{cc}\hfill \frac{1-\alpha }{2\left(1-e^{-\frac{\rho }{2}}\right)}[{ℐ}_{{ȥ}^{+}}^{\alpha }{\mathsf{Ϣ}}_{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)& +{ℐ}_{{\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)}^{+}}^{\alpha }{\mathsf{Ϣ}}_{*}\left(\mathsf{ѵ}\right)+{ℐ}_{{\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)}^{-}}^{\alpha }{\mathsf{Ϣ}}_{*}(ȥ)+{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha } {\mathsf{Ϣ}}_{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)]\hfill \\ & =\frac{1}{1-e^{-1}}\left\{{{\displaystyle \int }}_0^{\frac{1}{2}}{e}^{-2\left(\frac{1}{2}-\varkappa \right)} \left( 2{\varkappa }^4\right)d\varkappa +{{\displaystyle \int }}_0^{\frac{1}{2}}{e}^{-2\varkappa }\left(2{\varkappa }^4\right)d\varkappa \right\}\hfill \\ & +\frac{1}{1-e^{-1}}\left\{{{\displaystyle \int }}_{\frac{1}{2}}^1{e}^{-2\left(1-\varkappa \right)} \left( 2{\varkappa }^4\right)d\varkappa +{{\displaystyle \int }}_{\frac{1}{2}}^1{e}^{-2\left(\varkappa -\frac{1}{2}\right)}\left(2{\varkappa }^4\right)d\varkappa \right\}\hfill \\ & =\frac{1}{1-e^{-1}}\left\{\frac{3e}{2}-\frac{121e^{-1}}{8}+2\right\}\hfill \end{array}$$

$$\begin{array}{cc}\hfill \frac{1-\alpha }{2\left(1-e^{-\frac{\rho }{2}}\right)}[{ℐ}_{{ȥ}^{+}}^{\alpha }{\mathsf{Ϣ}}^{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)& +{ℐ}_{{\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)}^{+}}^{\alpha }{\mathsf{Ϣ}}^{*}\left(\mathsf{ѵ}\right)+{ℐ}_{{\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)}^{-}}^{\alpha }{\mathsf{Ϣ}}^{*}(ȥ)+{ℐ}_{{\mathsf{ѵ}}^{-}}^{\alpha } {\mathsf{Ϣ}}^{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)]\hfill \\ & =\frac{1}{1-e^{-1}}\left\{{{\displaystyle \int }}_0^{\frac{1}{2}}{e}^{-2\left(\frac{1}{2}-\varkappa \right)} \left( 4{\varkappa }^4\right)d\varkappa +{{\displaystyle \int }}_0^{\frac{1}{2}}{e}^{-2\varkappa }\left(4{\varkappa }^4\right)d\varkappa \right\}\hfill \\ & +\frac{1}{1-e^{-1}}\left\{{{\displaystyle \int }}_{\frac{1}{2}}^1{e}^{-2\left(1-\varkappa \right)} \left( 4{\varkappa }^4\right)d\varkappa +{{\displaystyle \int }}_{\frac{1}{2}}^1{e}^{-2\left(\varkappa -\frac{1}{2}\right)}\left(4{\varkappa }^4\right)d\varkappa \right\}\hfill \\ & =\frac{2}{1-e^{-1}}\left\{\frac{3e}{2}-\frac{121e^{-1}}{8}+2\right\}.\hfill \end{array}$$

$$\frac{\rho }{2\left(1-e^{-\frac{\rho }{2}}\right)}K\left(\rho \right)\left(\frac{{\mathsf{Ϣ}}_{*}(ȥ)+{\mathsf{Ϣ}}_{*}\left(\mathsf{ѵ}\right)}{2}+{\mathsf{Ϣ}}_{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\right)=\frac{9}{8\left(1-e^{-1}\right)}\left(\frac{6+2\pi e^{-2}}{16+{\pi }^2}+\frac{2e^{-1}}{\pi }\right)$$

$$\frac{\rho }{2\left(1-e^{-\frac{\rho }{2}}\right)}K\left(\rho \right)\left(\frac{{\mathsf{Ϣ}}^{*}(ȥ)+{\mathsf{Ϣ}}^{*}\left(\mathsf{ѵ}\right)}{2}+{\mathsf{Ϣ}}^{*}\left(\frac{ȥ+\mathsf{ѵ}}{2}\right)\right)=\frac{9}{4\left(1-e^{-1}\right)}\left(\frac{6+2\pi e^{-2}}{16+{\pi }^2}+\frac{2e^{-1}}{\pi }\right)$$

$$\frac{\rho }{2\left(1-e^{-\frac{\rho }{2}}\right)}\left(1+\sqrt{\frac{e}{2}}\right)K\left(\rho \right)\frac{{\mathsf{Ϣ}}_{*}(ȥ)+{\mathsf{Ϣ}}_{*}\left(\mathsf{ѵ}\right)}{2}=\frac{1}{1-e^{-1}}\left(1+\sqrt{\frac{e}{2}}\right)\left(\frac{6+2\pi e^{-2}}{16+{\pi }^2}+\frac{2e^{-1}}{\pi }\right)$$

$$\frac{\rho }{2\left(1-e^{-\frac{\rho }{2}}\right)}\left(1+\sqrt{\frac{e}{2}}\right)K\left(\rho \right)\frac{{\mathsf{Ϣ}}^{*}(ȥ)+{\mathsf{Ϣ}}^{*}\left(\mathsf{ѵ}\right)}{2}=\frac{2}{1-e^{-1}}\left(1+\sqrt{\frac{e}{2}}\right)\left(\frac{6+2\pi e^{-2}}{16+{\pi }^2}+\frac{2e^{-1}}{\pi }\right).$$

That is

$$\begin{array}{cc}\hfill \left[\frac{e}{8},\frac{e}{4}\right]& {\le }_p\left[\sqrt{\frac{e}{2}}\frac{41}{64},\sqrt{\frac{e}{2}}\frac{41}{32}\right]\hfill \\ & {\le }_p[\frac{1}{\begin{array}{c}1-e^{-1}\\ \\ \end{array}}\left\{\frac{3e}{2}-\frac{121e^{-1}}{8}+2\right\},\frac{2}{1-e^{-1}}\left\{\frac{3e}{2}-\frac{121e^{-1}}{8}+2\right\}]\hfill \\ & {\le }_p\left[\frac{9}{8\left(1-e^{-1}\right)}\left(\frac{6+2\pi e^{-2}}{16+{\pi }^2}+\frac{2e^{-1}}{\pi }\right),\frac{9}{4\left(1-e^{-1}\right)}\left(\frac{6+2\pi e^{-2}}{16+{\pi }^2}+\frac{2e^{-1}}{\pi }\right)\right]\hfill \\ & {\le }_p\left[\frac{1}{1-e^{-1}}\left(1+\sqrt{\frac{e}{2}}\right)\left(\frac{6+2\pi e^{-2}}{16+{\pi }^2}+\frac{2e^{-1}}{\pi }\right),\frac{2}{1-e^{-1}}\left(1+\sqrt{\frac{e}{2}}\right)\left(\frac{6+2\pi e^{-2}}{16+{\pi }^2}+\frac{2e^{-1}}{\pi }\right)\right].\hfill \end{array}$$

Hence, Theorem 7 is verified.

6. Conclusions

This study discusses some fundamental properties and introduces the concepts of left and right exponential trigonometric interval-valued convex mappings. Furthermore, by utilizing the idea of fractional integrals having exponential kernels, we established some novel Hermite–Hadamard-type inequalities and proved certain conclusions for midpoint- and Pachpatte-type inequalities. Further research is necessary in this important area of interval-valued analysis that includes fractional integral operators. By utilizing the -integral, we plan to investigate the integral inequalities of fuzzy-interval-valued functions and some applications in interval optimizations.