Some New Properties of Exponential Trigonometric Convex Functions Using up and down Relations over Fuzzy Numbers and Related Inequalities through Fuzzy Fractional Integral Operators Having Exponential Kernels

Abstract

The concept of convexity is fundamental in order to produce various types of inequalities. Thus, convexity and integral inequality are closely related. The objectives of this paper are to present a new class of up and down convex fuzzy number valued functions known as up and down exponential trigonometric convex fuzzy number valued mappings ($UDET\text{-}$convex FNVMs) and, with the help of this newly defined class, Hermite–Hadamard-type inequalities (H–H-type inequalities) via fuzzy inclusion relation and fuzzy fractional integral operators having exponential kernels. This fuzzy inclusion relation is level-wise defined by the interval-based inclusion relation. Furthermore, we have shown that our findings apply to a significant class of both novel and well-known inequalities for $UDET\text{-}\mathrm{convex} FNVM$s. The application of the theory developed in this study is illustrated with useful instances. Some very interesting examples are provided to discuss the validation of our main results. These results and other approaches may open up new avenues for modeling, interval-valued functions, and fuzzy optimization problems.

1. Introduction

The convexity of functions is a strong technique used to solve a variety of pure and applied scientific problems. Many scientists have recently devoted themselves to investigating the properties and inequalities of convexity in various directions (see [1,2,3,4,5,6,7,8,9,10] and the references therein). $H–H$ inequality is one of the most important mathematical inequalities relevant to convex maps, and it is also frequently used in many other parts of practical mathematics, particularly in optimization and probability. Let us elicit it as follows.

The H–H inequality for convex mapping ${\rm Y}:K\to ℝ$ on an interval $K=\left[\mu , b\right]$ is

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

(1)

for all $\mu , b\in K,$where $K$ is a convex set.

This classical inequality gives error boundaries for the mean value of a continuous convex mapping ${\rm Y}:K\to ℝ$, which has piqued the interest of numerous scholars. Many investigations have been conducted on $H–H\text{-}$type inequalities for additional forms of convex mappings. For example, Kórus [11] can be used to find s-convex mappings, Abramovich and Persson [12] for N-quasi-convex mappings, Delavar and De La Sen [13] for h-convex mappings, and so on. Ahmad et al. [14] created a fractional variation of the $H–H\text{-}$type inequality by combining fractional integrals with exponential kernels. For more recent developments on this topic, see Khan et al. [15], Marinescu and Monea [16], şcan [17], Kadakal et al. [18], and Kadakal and Bekar [19], as well as [20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35] and the references therein.

As a very helpful tool, fractional calculus has proven to be a fundamental cornerstone in mathematics and applied sciences. Many researchers have been drawn to this field to investigate this issue. The H–H inequality for Reimann–Liouville fractional integrals [36], the H–H Fejér-type inequality for Katugampola fractional integrals [37], and extensions of trapezium inequalities for k-fractional integrals [38] are just a few examples of the prominent integral inequalities that have been studied by numerous authors through the successful interaction of different fractional calculus approaches. For further significant results relating to fractional integral operators, we direct interested readers to [39,40,41,42,43,44,45,46,47] and the references therein.

Set-valued analysis includes interval analysis. As shown by [48], interval analysis is unquestionably significant in both theoretical and applied research. Computing the error bounds of numerical finite-state machine solutions was one of the early uses of interval analysis. Over the past 50 years, however, interval analysis—one method for addressing interval uncertainty—has grown to be a crucial component of mathematical and computer models. Additionally, certain applications in computer graphics [49], automatic error analysis [50], and neural network output optimization [51] have been detailed here. Furthermore, [52,53,54,55,56,57,58,59,60,61,62,63] discussed several applications in optimization theory involving interval-valued functions. The interested reader is directed to Zhao et al. [64] and Román-Flores et al. [65] and the references therein for recent developments in association with interval-valued functions.

Khan and his colleagues recently extended the concept of convex interval-valued mappings (convex I∙V∙Ms) and fuzzy number-valued mappings by using fuzzy–order relations such as the convex $FNVMs$ (apparently new) concept to include (h1, h2)-convex $FNVMs$ (see [66]) and harmonic-convex $FNVMs$ (see [67]). To demonstrate the inequalities of the $H–H\text{-}$, $H–H\text{-}$Fej'er, and Pachpatte types, his team used h-preinvex $FNVMs$ [68], (h1, h2)-preinvex $FNVMs$ [69], and higher-order preinvex $FNVMs$ [70]. Khan et al. [71] recently developed novel $H–H\text{-}$ and $H–H–$Fej’er-type inequalities based on the recently introduced concept of fuzzy Reimann–Liouville fractional integrals via $UD\text{-}FNVM$s. We refer interested readers to investigate certain basic ideas connected to fuzzy calculus (see [72,73,74,75,76,77,78,79,80,81,82,83,84,85,86]) and the references therein for different recent breakthroughs linked to the notion of the fuzzy interval-valued analysis of several well-known integral inequalities.

In this study, we employed a novel fractional integral operator to provide more broadly applicable results. This is a result of the exponential kernel of this fuzzy fractional operator. Our findings differ from previous generalizations in that the aforementioned fractional analogous functional inequalities do not follow our conclusions. There is no exponential kernel in the results of experts’ expansions of the $H–H$inequality using different fuzzy fractional integral operators.

This work stimulated interest in the development of more generalized fuzzy fractional inequalities with an exponential kernel. A new direction in the study of inequalities is also introduced by the use of fuzzy number analysis in the key findings. To propose new inequalities, we have combined the ideas of a fuzzy analysis with a fuzzy $UD$relation. There are still a lot of unanswered questions regarding fuzzy fractional integral inequalities involving different kinds of fuzzy convex number valued mappings, despite the fact that a lot of research has been done on the growth of fuzzy fractional order integral inequalities using convex fuzzy number valued functions. We develop new fuzzy $H–H\text{-}$, Pachpatte, and Fej’er-type inequalities for $UDET\text{-}\mathrm{convex} FNVM$s via fuzzy fractional integral operators having exponential kernels, which is this study’s main objective.

2. Preliminaries

Let ${\mathcal{X}}_C$ be the space of all closed and bounded intervals of $ℝ$ and $\mathrm{Y}\in {\mathcal{X}}_C$ be defined by

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

(2)

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

$${\mathcal{X}}_C^{+}=\left\{\left[{\mathrm{Y}}_{*}, {\mathrm{Y}}^{*}\right]:\left[{\mathrm{Y}}_{*}, {\mathrm{Y}}^{*}\right]\in {\mathcal{X}}_C \mathrm{and} {\mathrm{Y}}_{*}\ge 0\right\}.$$

(3)

Let $\theta \in ℝ$ and θ· Y be defined by

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

(4)

Then, the Minkowski difference $\mathrm{Z}-\mathrm{Y}$, addition $\mathrm{Y}+\mathrm{Z},$ and $\mathrm{Y}\times \mathrm{Z}$ for $\mathrm{Y},\mathrm{Z}\in {\mathcal{X}}_C$ are defined by

$$\left[{\mathrm{Z}}_{*}, {\mathrm{Z}}^{*}\right]+\left[{\mathrm{Y}}_{*}, {\mathrm{Y}}^{*}\right]=\left[{\mathrm{Z}}_{*}+{\mathrm{Y}}_{*}, {\mathrm{Z}}^{*}+{\mathrm{Y}}^{*}\right],$$

(5)

$$\left[{\mathrm{Z}}_{*}, {\mathrm{Z}}^{*}\right]\times \left[{\mathrm{Y}}_{*}, {\mathrm{Y}}^{*}\right]=\left[\min \left\{{\mathrm{Z}}_{*}{\mathrm{Y}}_{*}, {\mathrm{Z}}^{*}{\mathrm{Y}}_{*}, {\mathrm{Z}}_{*}{\mathrm{Y}}^{*}, {\mathrm{Z}}^{*}{\mathrm{Y}}^{*}\right\}, \max \left\{{\mathrm{Z}}_{*}{\mathrm{Y}}_{*}, {\mathrm{Z}}^{*}{\mathrm{Y}}_{*}, {\mathrm{Z}}_{*}{\mathrm{Y}}^{*}, {\mathrm{Z}}^{*}{\mathrm{Y}}^{*}\right\}\right],$$

(6)

$$\left[{\mathrm{Z}}_{*}, {\mathrm{Z}}^{*}\right]-\left[{\mathrm{Y}}_{*}, {\mathrm{Y}}^{*}\right]=\left[{\mathrm{Z}}_{*}-{\mathrm{Y}}^{*}, {\mathrm{Z}}^{*}-{\mathrm{Y}}_{*}\right].$$

(7)

Remark 1.

 (i) For given  $\left[{\mathrm{Z}}_{*}, {\mathrm{Z}}^{*}\right], \left[{\mathrm{Y}}_{*}, {\mathrm{Y}}^{*}\right]\in {\mathcal{X}}_C,$ the relation  $“{\supseteq }_I”$ defined on ${\mathcal{X}}_C$ by $\left[{\mathrm{Y}}_{*}, {\mathrm{Y}}^{*}\right]{\supseteq }_I\left[{\mathrm{Z}}_{*}, {\mathrm{Z}}^{*}\right]$ if and only if ${\mathrm{Y}}_{*}\le {\mathrm{Z}}_{*}, {\mathrm{Z}}^{*}\le {\mathrm{Y}}^{*}$ for all $\left[{\mathrm{Z}}_{*}, {\mathrm{Z}}^{*}\right], \left[{\mathrm{Y}}_{*}, {\mathrm{Y}}^{*}\right]\in {\mathcal{X}}_C$ is a partial interval inclusion relation. The relation $\left[{\mathrm{Y}}_{*}, {\mathrm{Y}}^{*}\right]{\supseteq }_I\left[{\mathrm{Z}}_{*}, {\mathrm{Z}}^{*}\right]$ is coincident to $\left[{\mathrm{Y}}_{*}, {\mathrm{Y}}^{*}\right]\supseteq \left[{\mathrm{Z}}_{*}, {\mathrm{Z}}^{*}\right]$ on ${\mathcal{X}}_C.$ It can be easily seen that “${\supseteq }_I$” looks like “up and down” on the real line $ℝ,$ so we call $“{\supseteq }_I”$ “up and down ” (or “$UD$” order, in short) [84]. (ii) For given $\left[{\mathrm{Z}}_{*}, {\mathrm{Z}}^{*}\right], \left[{\mathrm{Y}}_{*}, {\mathrm{Y}}^{*}\right]\in {\mathcal{X}}_C,$ we say that $\left[{\mathrm{Z}}_{*}, {\mathrm{Z}}^{*}\right]{\le }_I\left[{\mathrm{Y}}_{*}, {\mathrm{Y}}^{*}\right]$ if and only if ${\mathrm{Z}}_{*}\le {\mathrm{Y}}_{*}, {\mathrm{Z}}^{*}\le {\mathrm{Y}}^{*}$ or ${\mathrm{Z}}_{*}\le {\mathrm{Y}}_{*}, {\mathrm{Z}}^{*}<{\mathrm{Y}}^{*}$, which is a partial interval order relation. The relation $\left[{\mathrm{Z}}_{*}, {\mathrm{Z}}^{*}\right]{\le }_I\left[{\mathrm{Y}}_{*}, {\mathrm{Y}}^{*}\right]$ is coincident to $\left[{\mathrm{Z}}_{*}, {\mathrm{Z}}^{*}\right]\le \left[{\mathrm{Y}}_{*}, {\mathrm{Y}}^{*}\right]$ on  ${\mathcal{X}}_C.$ It can be easily seen that $“{\le }_I”$ looks like “left and right” on the real line $ℝ,$ so we call $“{\le }_I”$ “left and right” (or “LR” order, in short) [82,84]. For $\left[{\mathrm{Z}}_{*}, {\mathrm{Z}}^{*}\right], \left[{\mathrm{Y}}_{*}, {\mathrm{Y}}^{*}\right]\in {\mathcal{X}}_C,$ the Hausdorff–Pompeiu distance between intervals $\left[{\mathrm{Z}}_{*}, {\mathrm{Z}}^{*}\right]$ and $\left[{\mathrm{Y}}_{*}, {\mathrm{Y}}^{*}\right]$ is defined by

$$d_H\left(\left[{\mathrm{Z}}_{*}, {\mathrm{Z}}^{*}\right], \left[{\mathrm{Y}}_{*}, {\mathrm{Y}}^{*}\right]\right)=max\left\{\left|{\mathrm{Z}}_{*}-{\mathrm{Y}}_{*}\right|, \left|{\mathrm{Z}}^{*}-{\mathrm{Y}}^{*}\right|\right\}.$$

(8)

It is a familiar fact that $\left({\mathcal{X}}_C,d_H\right)$ is a complete metric space [37,40,41].

Note that, here, we are going to use the classical concepts of fuzzy set and fuzzy number so just we will recall some basic concepts related to fuzzy set and fuzzy numbers. Be aware that we designate the terms $R$ and $R_C$ to signify, respectively, the set of all fuzzy subsets and fuzzy numbers of $ℝ$.

Definition 1 

 ([75,76]). Given $\tilde{f}\in R_C$ , the level sets or cut sets are given by ${\left[\tilde{f}\right]}^{\theta }=\left\{\varkappa \in ℝ| \tilde{f}\left(\varkappa \right)>\theta \right\}$ for all $\theta \in \left[0, 1\right]$ and by ${\left[\tilde{f}\right]}^0=\left\{\varkappa \in ℝ| \tilde{f}\left(\varkappa \right)>0\right\}$. These sets are known as $\theta$-level sets or  $\theta$-cut sets of $\tilde{f}$.

Proposition 1 

 ([82]). Let $\tilde{f},\tilde{g}\in R_C$. Then, relation $“{\le }_{\mathbb{F}}”$ is given on R_c by $\tilde{f}{\le }_{\mathbb{F}}\tilde{g}$ when and only when ${\left[\tilde{f}\right]}^{\theta }{\le }_I{\left[\tilde{g}\right]}^{\theta }$, for every $\theta \in \left[0, 1\right],$ which are left- and right-order relations.

Proposition 2 

([72]). Let $\tilde{f},\tilde{g}\in R_C$. Then, relation $“{\supseteq }_{\mathbb{F}}”$ is given on $R_C$ by $\tilde{f}{\supseteq }_{\mathbb{F}}\tilde{g}$ when and only when ${\left[\tilde{f}\right]}^{\theta }{\supseteq }_I{\left[\tilde{g}\right]}^{\theta }$ for every $\theta \in \left[0, 1\right],$ which is $the UD\text{-}$order relation on $R_C$.

Remember the approaching notions, which are offered in the literature. If $\tilde{f},\tilde{g}\in R_C$ and $\theta \in ℝ$, then, for every $\theta \in \left[0, 1\right],$ the arithmetic operations addition “$\oplus ”$, multiplication “$\otimes ”$, and scaler multiplication “$\odot ”$ are defined by

$${\left[\tilde{f}\oplus \tilde{g}\right]}^{\theta }={\left[\tilde{f}\right]}^{\theta }+{\left[\tilde{g}\right]}^{\theta },$$

(9)

$${\left[\tilde{f}\otimes \tilde{g}\right]}^{\theta }={\left[\tilde{f}\right]}^{\theta }\times {\left[ \tilde{g}\right]}^{\theta },$$

(10)

$${\left[𝓉\odot \tilde{f}\right]}^{\theta }=𝓉,$$

(11)

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

Theorem 1 

 ([75]). The space $R_C$ dealing with a supremum metric, i.e., for $\tilde{f}, \tilde{g}\in R_C$

$$d_{\infty }\left(\tilde{f}, \tilde{g}\right)=\underset{0\le \theta \le 1}{\sup }{d}_H\left({\left[\tilde{f}\right]}^{\theta }, {\left[\tilde{g}\right]}^{\theta }\right),$$

(12)

is a complete metric space, where $H$  denotes the well-known Hausdorff metric on the space of intervals.

Fractional Integral Operators of Interval- and Fuzzy Number-Valued Mappings

Now, we define and discuss some properties of fractional integral operators for single-valued, interval-valued, and fuzzy number-valued mappings.

Theorem 2 

 ([75,79]). If ${\rm Y}:\left[\mu ,b\right]\subset ℝ\to {\mathcal{X}}_C$ is an interval-valued mapping (i-v-m) satisfying that ${\rm Y}\left(\varkappa \right)=\left[{{\rm Y}}_{*}\left(\varkappa \right), {{\rm Y}}^{*}\left(\varkappa \right)\right]$, then ${\rm Y}$ is an Aumann integrable (IA integrable) over $\left[\mu ,b\right]$ when and only when ${{\rm Y}}_{*}\left(\varkappa \right)$ and ${{\rm Y}}^{*}\left(\varkappa \right)$ both are integrable over $\left[\mu ,b\right],$ such that

$$\left(IA\right){\int }_{\mu }^b{\rm Y}\left(\varkappa \right)d\varkappa =\left[{\int }_{\mu }^b{{\rm Y}}_{*}\left(\varkappa \right)d\varkappa , {\int }_{\mu }^b{{\rm Y}}^{*}\left(\varkappa \right)d\varkappa \right].$$

(13)

Definition 2

 ([83]). Let $\widetilde{{\rm Y}}:\mathbb{I}\subset ℝ\to R_C$ be called $FNVM$. Then, for every $\theta \in \left[0, 1\right],$ as well as $\theta$ levels, define the family of i-v-ms ${{\rm Y}}_{\theta }:\mathbb{I}\subset ℝ\to {\mathcal{X}}_C,$ satisfying that ${{\rm Y}}_{\theta }\left(\varkappa \right)=\left[{{\rm Y}}_{*}\left(\varkappa ,\theta \right), {{\rm Y}}^{*}\left(\varkappa ,\theta \right)\right]$ for every $\varkappa \in \mathbb{I}.$ Here, for every $\theta \in \left[0, 1\right],$ the end point real valued mappings ${{\rm Y}}_{*}\left(·,\theta \right), {{\rm Y}}^{*}\left(·,\theta \right):\mathbb{I}\to ℝ$ are called lower and upper mappings of ${\rm Y}$.

Definition 3

 ([83]). Let $\widetilde{{\rm Y}}:\mathbb{I}\subset ℝ\to R_C$ be a $FNVM$. Then, $\widetilde{{\rm Y}}\left(\varkappa \right)$ is said to be continuous at $\varkappa \in \mathbb{I}$ if, for every $\theta \in \left[0, 1\right],$ ${{\rm Y}}_{\theta }\left(\varkappa \right), it$ is continuous when and only when both end point mappings ${{\rm Y}}_{*}\left(\varkappa ,\theta \right)$ and ${{\rm Y}}^{*}\left(\varkappa ,\theta \right)$ are continuous at $\varkappa \in \mathbb{I}.$

Definition 4

 ([79]). Let $\widetilde{{\rm Y}}:\left[\mu , b\right]\subset ℝ\to R_C$ be $FNVM$. The fuzzy Aumann integral ($\left(FA\right)$ integral) of $\widetilde{{\rm Y}}$ over $\left[\mu , b\right],$ denoted by $\left(FA\right){\int }_{\mu }^b\widetilde{{\rm Y}}\left(\varkappa \right)d\varkappa$, is defined level-wise by

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

(14)

where $S\left({{\rm Y}}_{\theta }\right)=\left\{{\rm Y}\left(.,\theta \right)\to ℝ:{\rm Y}\left(.,\theta \right) is integrable and {\rm Y}\left(\varkappa ,\theta \right)\in {{\rm Y}}_{\theta }\left(\varkappa \right)\right\}$for every $\theta \in \left[0, 1\right]$. $\widetilde{{\rm Y}}$ is the $\left(FA\right)$ integrable over $\left[\mu , b\right]$ if $\left(FA\right){\int }_{\mu }^b\widetilde{{\rm Y}}\left(\varkappa \right)d\varkappa \in R_C.$

Theorem 3

 ([80]). Let ${\rm Y}:\left[\mu , b\right]\subset ℝ\to R_C$ be a $FNVM,$ as well as $\theta$ levels, define the family of i-v-ms ${{\rm Y}}_{\theta }:\left[\mu , b\right]\subset ℝ\to {\mathcal{X}}_C$, satisfying that ${{\rm Y}}_{\theta }\left(\varkappa \right)=\left[{{\rm Y}}_{*}\left(\varkappa ,\theta \right), {{\rm Y}}^{*}\left(\varkappa ,\theta \right)\right]$ for every $\varkappa \in \left[\mu , b\right]$ and for every $\theta \in \left[0, 1\right].$ Then, Υ is $\left(FA\right)$ integrable over $\left[\mu , b\right]$ when and only when, ${{\rm Y}}_{*}\left(\varkappa ,\theta \right)$ and ${{\rm Y}}^{*}\left(\varkappa ,\theta \right)$ both are integrable over $\left[\mu , b\right]$. Moreover, if ${\rm Y}$ is $\left(FA\right)$ integrable over $\left[\mu , b\right],$ then

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

(15)

for every θ∈[0,1].

Allahviranloo et al. [74] introduced the following fuzzy Reimann–Liouville fractional integral operators:

Definition 5.

Let $\alpha >0$ and $L\left(\left[\mu , b\right],R_C\right)$ be the collection of all Lebesgue measurable $FNVMs$ on $\left[\mu ,b\right]$. Then, the fuzzy left and right Reimann–Liouville fractional integrals of $\widetilde{{\rm Y}}\in$ $L\left(\left[\mu , b\right],R_C\right)$ with order $\alpha >0$ are defined by

$${ℐ}_{{\mu }^{+}}^{\alpha }\widetilde{{\rm Y}}\left(\varkappa \right)=\frac{1}{\Gamma \left(\alpha \right)}{\int }_{\mu }^{\varkappa }{\left(\varkappa -𝓉\right)}^{\alpha -1}\widetilde{{\rm Y}}\left(𝓉\right)d𝓉, \left(\varkappa >\mu \right),$$

(16)

and

$${ℐ}_{b^{-}}^{\alpha }\widetilde{{\rm Y}}\left(\varkappa \right)=\frac{1}{\Gamma \left(\alpha \right)}{\int }_{\varkappa }^b{\left(𝓉-\varkappa \right)}^{\alpha -1}\widetilde{{\rm Y}}\left(𝓉\right)d𝓉, (\varkappa <b),$$

(17)

respectively, where $\Gamma \left(\varkappa \right)={\int }_0^{\infty }{𝓉}^{\varkappa -1}{e}^{-𝓉}d𝓉$  is the Euler gamma mapping. The fuzzy left and right Reimann–Liouville fractional integral $\varkappa$  based on left and right end point mappings can be defined as

$$\begin{array}{c}{\left[{ℐ}_{{\mu }^{+}}^{\alpha } \widetilde{{\rm Y}}\left(\varkappa \right)\right]}^{\theta }=\frac{1}{\Gamma \left(\alpha \right)}{\int }_{\mu }^{\varkappa }{\left(\varkappa -𝓉\right)}^{\alpha -1}{{\rm Y}}_{\theta }\left(𝓉\right)d𝓉\\ =\frac{1}{\Gamma \left(\alpha \right)}{\int }_{\mu }^{\varkappa }{\left(\varkappa -𝓉\right)}^{\alpha -1}\left[{{\rm Y}}_{*}\left(𝓉, \theta \right),{{\rm Y}}^{*}\left(𝓉, \theta \right)\right]d𝓉, \left(\varkappa >\mu \right), \end{array}$$

(18)

where

$${ℐ}_{{\mu }^{+}}^{\alpha } {{\rm Y}}_{*}\left(\varkappa , \theta \right)=\frac{1}{\Gamma \left(\alpha \right)}{\int }_{\mu }^{\varkappa }{\left(\varkappa -𝓉\right)}^{\alpha -1}{{\rm Y}}_{*}\left(𝓉, \theta \right)d𝓉, \left(\varkappa >\mu \right),$$

(19)

and

$${ℐ}_{{\mu }^{+}}^{\alpha } {{\rm Y}}^{*}\left(\varkappa , \theta \right)=\frac{1}{\Gamma \left(\alpha \right)}{\int }_{\mu }^{\varkappa }{\left(\varkappa -𝓉\right)}^{\alpha -1}{{\rm Y}}^{*}\left(𝓉, \theta \right)d𝓉, \left(\varkappa >\mu \right),$$

(20)

Similarly, we can define the right Reimann–Liouville fractional integral ${\rm Y}$ of $\varkappa$ based on the left and right end point mappings.

Definition 6

 ([73]). The mapping of ${\rm Y}:\left[\mu , b\right]\to ℝ$ is called exponential trigonometric convex mapping on $\left[\mu , b\right]$ if

$${\rm Y}\left(𝓉\varkappa +\left(1-𝓉\right)s\right)\le \frac{sin\frac{\pi 𝓉}{2}}{e^{1-𝓉}}\odot {\rm Y}\left(\varkappa \right)\oplus \frac{cos\frac{\pi 𝓉}{2}}{e^{𝓉}}\odot {\rm Y}\left(s\right),$$

(21)

for all$\varkappa , s\in \left[\mu , b\right], 𝓉\in \left[0, 1\right],$  and $\varkappa \in \left[\mu , b\right].$ If (21) is reversed, then ${\rm Y}$ is called exponential trigonometric concave mapping on [μ,b].

Definition 7

 ([85]). $FNVM \widetilde{{\rm Y}}:\left[\mu , b\right]\to R_C$ is called UD–convex FNVM on$\left[\mu , b\right]$ if

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

(22)

for all$\varkappa , s\in \left[\mu , b\right], 𝓉\in \left[0, 1\right],$ where $\widetilde{{\rm Y}}\left(\varkappa \right){\ge }_{\mathbb{F}}\tilde{0}$ for all $\varkappa \in \left[\mu , b\right].$ If (22) is reversed, then $\widetilde{{\rm Y}}$ is called concave $FNVM$ on $\left[\mu , b\right]$. $\widetilde{{\rm Y}}$ is affinal if and only if it is both convex and concave $FNVM$.

Definition 8

 ([86]). The $FNVM$ $\widetilde{{\rm Y}}:\left[\mu , b\right]\to R_C$ is called convex $FNVM$ on$\left[\mu , b\right]$ if

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

(23)

for all$\varkappa , s\in \left[\mu , b\right], 𝓉\in \left[0, 1\right],$ where $\widetilde{{\rm Y}}\left(\varkappa \right){\ge }_{\mathbb{F}}\tilde{0}$ for all $\varkappa \in \left[\mu , b\right].$ If (23) is reversed, then $\widetilde{{\rm Y}}$ is called $UD-$concave $FNVM$ on $\left[\mu , b\right]$. $\widetilde{{\rm Y}}$ is $UD\text{-}\mathrm{affine} FNVM$ if and only if it is both $UD\text{-}$convex and UD-concave FNVM.

Theorem 4

 ([86]). Let $\widetilde{{\rm Y}}:\left[\mu ,b\right]\to R_C$ be a  $FNVM$, with $\theta$ levels defined by the family of interval-valued mappings (i-v-ms) ${{\rm Y}}_{\theta }:\left[\mu ,b\right]\to {\mathcal{X}}_C^{+}\subset {\mathcal{X}}_C$ given by

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

(24)

for all $\varkappa \in \left[\mu ,b\right]$ and for all $\theta \in \left[0, 1\right]$. Then, $\widetilde{{\rm Y}}$ is $UD\text{-}convex FNVM$ on $\left[\mu ,b\right]$ if and only if, for all $\theta \in \left[0, 1\right],$ ${{\rm Y}}_{*}\left(\varkappa , \theta \right)$ is a convex mapping and ${{\rm Y}}^{*}\left(\varkappa , \theta \right)$ is a concave mapping.

Definition 9

 ([74]). Let $\alpha >0$ and $L\left(\left[\mu , b\right], ℝ\right)$ be the collection of all Lebesgue measurable mappings on $\left[\mu ,b\right]$. Then, the left and right Reimann–Liouville fractional integrals with exponential kernels in connection of Υ∈L ([μ,b], ℝ) with order $\alpha >0$ are defined by

$${ℐ}_{{\mu }^{+}}^{\alpha } {\rm Y}\left(\varkappa \right)=\frac{1}{\alpha }{\int }_{\mu }^{\varkappa }{e}^{\left(-\frac{1-\alpha }{\alpha }\left(\varkappa -𝓉\right)\right)}{\rm Y}\left(𝓉\right)d𝓉, \left(\varkappa >\mu \right),$$

and

$${ℐ}_{b^{-}}^{\alpha } {\rm Y}\left(\varkappa \right)=\frac{1}{\alpha }{\int }_{\varkappa }^b{e}^{\left(-\frac{1-\alpha }{\alpha }\left(𝓉-\varkappa \right)\right)}{\rm Y}\left(𝓉\right)d𝓉, \left(\varkappa <b\right),$$

respectively.

Now, we present newly defined the fuzzy left and right Reimann–Liouville fractional integral with exponential kernels with respect to $FNVM$ $\widetilde{{\rm Y}}\in$ $L\left(\left[\mu , b\right],R_C\right)$.

Definition 10

 ([78]). Let $\widetilde{{\rm Y}}:\left[\mu , b\right]\subset ℝ\to R_C$ be $FNVM,$ and $\widetilde{{\rm Y}}$ is the fuzzy Aumann integrable over $\left[\mu , b\right].$ Then, the fuzzy left and right Reimann–Liouville fractional integrals with exponential kernels in connection of $\widetilde{{\rm Y}}\in$ $L\left(\left[\mu , b\right],R_C\right)$ with order $\alpha >0$ are defined by

$${\left[{ℐ}_{{\mu }^{+}}^{\alpha } \widetilde{{\rm Y}}\left(\varkappa \right)\right]}^{\begin{array}{c} \\ \theta \end{array}}={ℐ}_{{\mu }^{+}}^{\alpha }{{\rm Y}}_{\theta }\left(\varkappa \right)=\left\{{ℐ}_{{\mu }^{+}}^{\alpha }{\rm Y}\left(\varkappa ,\theta \right):{\rm Y}\left(\varkappa ,\theta \right)\in S\left({{\rm Y}}_{\theta }\right)\right\},$$

(25)

and

$${\left[{ℐ}_{b^{-}}^{\alpha } \widetilde{{\rm Y}}\left(\varkappa \right)\right]}^{\begin{array}{c} \\ \theta \end{array}}={ℐ}_{b^{-}}^{\alpha }{{\rm Y}}_{\theta }\left(\varkappa \right)=\left\{{ℐ}_{b^{-}}^{\alpha }{\rm Y}\left(\varkappa ,\theta \right):{\rm Y}\left(\varkappa ,\theta \right)\in S\left({{\rm Y}}_{\theta }\right)\right\},$$

(26)

where $S\left({{\rm Y}}_{\theta }\right)=$ {${\rm Y}\left(.,\theta \right)\to ℝ:{\rm Y}\left(.,\theta \right)$ is integrable and Υ(ϰ,θ)∈Υ_θ(ϰ)} for every θ∈ [0,1].  $S\left({{\rm Y}}_{\theta }\right)=$ {${\rm Y}\left(.,\theta \right)\to ℝ:{\rm Y}\left(.,\theta \right)$ are the left and right Reimann–Liouville fractional integrable with exponential kernels and order $\alpha >0$ for every $\theta \in \left[0, 1\right]$}. $\widetilde{{\rm Y}}$ is fuzzy left and right Reimann–Liouville fractional integrals with exponential Kernels and order $\alpha >0$ over $\left[\mu , b\right]$ if ${ℐ}_{{\mu }^{+}}^{\alpha } \widetilde{{\rm Y}}\left(\varkappa \right), {ℐ}_{b^{-}}^{\alpha }\widetilde{{\rm Y}}\left(\varkappa \right)\in R_C$.

Definition 11

 ([78]). Let $\alpha >0$ and $L\left(\left[\mu , b\right],R_C\right)$ be the collection of all Lebesgue measurable $FNVM$ on $\left[\mu ,b\right]$. Then, the fuzzy left and right Reimann–Liouville fractional integrals with exponential kernels in connection with $\widetilde{{\rm Y}}\in L\left(\right[\mu ,b],R_C)$ with order $\alpha >0$ are defined by

$${ℐ}_{{\mu }^{+}}^{\alpha }\widetilde{{\rm Y}}\left(\varkappa \right)=\frac{1}{\alpha }{\int }_{\mu }^{\varkappa }{e}^{\left(-\frac{1-\alpha }{\alpha }\left(\varkappa -𝓉\right)\right)}\widetilde{{\rm Y}}\left(𝓉\right)d𝓉, \left(\varkappa >\mu \right),$$

(27)

and

$${ℐ}_{b^{-}}^{\alpha }\widetilde{{\rm Y}}\left(\varkappa \right)=\frac{1}{\alpha }{\int }_{\varkappa }^b{e}^{\left(-\frac{1-\alpha }{\alpha }\left(𝓉-\varkappa \right)\right)}\widetilde{{\rm Y}}\left(𝓉\right)d𝓉, (\varkappa <b),$$

(28)

respectively. The fuzzy left Reimann–Liouville fractional integrals with exponential kernels in connection with left and right end point mappings can be defined as

$$\begin{array}{ll}{\left[{ℐ}_{{\mu }^{+}}^{\alpha } \widetilde{{\rm Y}}\left(\varkappa \right)\right]}^{\theta }& =\frac{1}{\alpha }{\int }_{\mu }^{\varkappa }{e}^{\left(-\frac{1-\alpha }{\alpha }\left(\varkappa -𝓉\right)\right)}{{\rm Y}}_{\theta }\left(𝓉\right)d𝓉\\ & =\frac{1}{\alpha }{\int }_{\mu }^{\varkappa }{e}^{\left(-\frac{1-\alpha }{\alpha }\left(\varkappa -𝓉\right)\right)}\left[{{\rm Y}}_{\ast }\right(𝓉,\theta ),{{\rm Y}}^{\ast }(𝓉,\theta \left)\right]d𝓉,\left(\varkappa >\mu \right),\end{array}$$

(29)

where

$${ℐ}_{{\mu }^{+}}^{\alpha } {{\rm Y}}_{*}\left(\varkappa , \theta \right)=\frac{1}{\alpha }{\int }_{\mu }^{\varkappa }{e}^{\left(-\frac{1-\alpha }{\alpha }\left(\varkappa -𝓉\right)\right)}{{\rm Y}}_{*}\left(𝓉, \theta \right)d𝓉, \left(\varkappa >\mu \right),$$

(30)

and

$${ℐ}_{{\mu }^{+}}^{\alpha } {{\rm Y}}^{*}\left(\varkappa , \theta \right)=\frac{1}{\alpha }{\int }_{\mu }^{\varkappa }{e}^{\left(-\frac{1-\alpha }{\alpha }\left(\varkappa -𝓉\right)\right)}{{\rm Y}}^{*}\left(𝓉, \theta \right)d𝓉, \left(\varkappa >\mu \right).$$

(31)

Similarly, the fuzzy right Reimann–Liouville fractional integral with exponential kernels in connection of left and right end point mappings can be defined as

$$\begin{array}{ll}{\left[{ℐ}_{b^{-}}^{\alpha } \widetilde{{\rm Y}}\left(\varkappa \right)\right]}^{\theta }& =\frac{1}{\alpha }{\int }_{\varkappa }^b{e}^{\left(-\frac{1-\alpha }{\alpha }\left(𝓉-\varkappa \right)\right)}{{\rm Y}}_{\theta }\left(𝓉\right)d𝓉, \\ & =\frac{1}{\alpha }{\int }_{\varkappa }^b{e}^{\left(-\frac{1-\alpha }{\alpha }\left(𝓉-\varkappa \right)\right)}\left[{{\rm Y}}_{*}\left(𝓉, \theta \right),{{\rm Y}}^{*}\left(𝓉, \theta \right)\right]d𝓉, (\varkappa <b)\end{array}$$

(32)

where

$${ℐ}_{b^{-}}^{\alpha } {{\rm Y}}_{*}\left(\varkappa , \theta \right)=\frac{1}{\alpha }{\int }_{\varkappa }^b{e}^{\left(-\frac{1-\alpha }{\alpha }\left(𝓉-\varkappa \right)\right)}{{\rm Y}}_{*}\left(𝓉, \theta \right)d𝓉, (\varkappa <b)$$

(33)

and

$${ℐ}_{b^{-}}^{\alpha } {{\rm Y}}^{*}\left(\varkappa , \theta \right)=\frac{1}{\alpha }{\int }_{\varkappa }^b{e}^{\left(-\frac{1-\alpha }{\alpha }\left(𝓉-\varkappa \right)\right)}{{\rm Y}}^{*}\left(𝓉, \theta \right)d𝓉, \left(\varkappa <b\right).$$

(34)

Now, we introduce the new class of convex $FNVM,$ which is known as $UDET\text{-}\mathrm{convex} FNVM$.

Definition 12.

The $FNVM$ $\widetilde{{\rm Y}}:\left[\mu , b\right]\to R_C$is called$UDET\text{-}$convex$FNVM$ on$\left[\mu , b\right]$ if

$$\widetilde{{\rm Y}}\left(𝓉\varkappa +\left(1-𝓉\right)s\right){\supseteq }_{\mathbb{F}}\frac{sin\frac{\pi 𝓉}{2}}{e^{1-𝓉}}\odot \widetilde{{\rm Y}}\left(\varkappa \right)\oplus \frac{cos\frac{\pi 𝓉}{2}}{e^{𝓉}}\odot \widetilde{{\rm Y}}\left(s\right),$$

(35)

for all$\varkappa , s\in \left[\mu , b\right], 𝓉\in \left[0, 1\right],$ where $\widetilde{{\rm Y}}\left(\varkappa \right){\ge }_{\mathbb{F}}\tilde{0}$ for all $\varkappa \in \left[\mu , b\right].$ If (35) is reversed, then $\widetilde{{\rm Y}}$ is called $UDET\text{-}concave FNVM$ on $\left[\mu , b\right]$.

Theorem 5.

Let $K$ be an invex set and $\widetilde{{\rm Y}}:K\to R_C$ be a $FNVM$, with $\theta$ levels defined by the family of IVFs ${{\rm Y}}_{\theta }:K\subset ℝ\to {\mathcal{K}}_C{}^{+}\subset {\mathcal{K}}_C$ given by

$${{\rm Y}}_{\theta }\left(x\right)=\left[{{\rm Y}}_{*}\left(x,\theta \right), {{\rm Y}}^{*}\left(x,\theta \right)\right], \forall x\in K.$$

(36)

for all $x\in K$ and for all $\theta \in \left[0, 1\right]$. Then, $\widetilde{{\rm Y}}$ is the $UDET\text{-}$convex $FNVM$ on $K$ if and only if, for all $\theta \in \left[0, 1\right],$ ${{\rm Y}}_{*}\left(x, \theta \right)$ and ${{\rm Y}}^{*}\left(x, \theta \right)$ are exponential trigonometric convex and concave functions, respectively.

Proof. 

Assume that, for each $\theta \in \left[0, 1\right],$  ${{\rm Y}}_{*}\left(x, \theta \right)$ and ${{\rm Y}}^{*}\left(x, \theta \right)$ are exponential trigonometric convex and concave functions on $K$, respectively. Then, from (21), we have

$${{\rm Y}}_{*}\left(𝓉\varkappa +\left(1-𝓉\right)s, \theta \right)\le \frac{\sin \frac{\pi 𝓉}{2}}{e^{1-𝓉}}{{\rm Y}}_{*}\left(x, \theta \right)+\frac{\cos \frac{\pi 𝓉}{2}}{e^{𝓉}}{{\rm Y}}_{*}\left(s, \theta \right),\forall x,s\in K, 𝓉\in \left[0, 1\right],$$

and

$${{\rm Y}}^{*}\left(𝓉\varkappa +\left(1-𝓉\right)s, \theta \right)\ge \frac{\sin \frac{\pi 𝓉}{2}}{e^{1-𝓉}}{{\rm Y}}^{*}\left(x, \theta \right)+\frac{\cos \frac{\pi 𝓉}{2}}{e^{𝓉}}{{\rm Y}}^{*}\left(s, \theta \right),\forall x,s\in K, 𝓉\in \left[0, 1\right].$$

Then, using (36), (5), and (7), we obtain

$$\begin{array}{l}{{\rm Y}}_{\theta }\left(𝓉\varkappa +\left(1-𝓉\right)s\right) \\ =\left[{{\rm Y}}_{*}\left(𝓉\varkappa +\left(1-𝓉\right)s, \theta \right), {{\rm Y}}^{*}\left(𝓉\varkappa +\left(1-𝓉\right)s, \theta \right)\right],\\ {\supseteq }_I\frac{\sin \frac{\pi 𝓉}{2}}{e^{1-𝓉}}\left[{{\rm Y}}_{*}\left(x, \theta \right), {{\rm Y}}^{*}\left(x, \theta \right)\right]+\frac{\cos \frac{\pi 𝓉}{2}}{e^{𝓉}}\left[{{\rm Y}}_{*}\left(s, \theta \right), {{\rm Y}}^{*}\left(s, \theta \right)\right], \end{array}$$

which is

$$\widetilde{{\rm Y}}\left(𝓉\varkappa +\left(1-𝓉\right)s\right){\supseteq }_{\mathbb{F}}\frac{\sin \frac{\pi 𝓉}{2}}{e^{1-𝓉}}\odot \widetilde{{\rm Y}}\left(x\right)\oplus \frac{\cos \frac{\pi 𝓉}{2}}{e^{𝓉}}\odot \widetilde{{\rm Y}}\left(s\right),\forall x,s\in K, 𝓉\in \left[0, 1\right].$$

Hence, $\widetilde{{\rm Y}}$ is the $UDET\text{-}$convex $FNVM$ on $K.$

Conversely, let $\widetilde{{\rm Y}}$ be a $UDET-$convex $FNVM$ on $K.$ Then, for all $x,s\in K$ and $𝓉\in \left[0, 1\right],$ we have

$$\widetilde{{\rm Y}}\left(𝓉\varkappa +\left(1-𝓉\right)s\right){\supseteq }_{\mathbb{F}}\frac{\sin \frac{\pi 𝓉}{2}}{e^{1-𝓉}}\odot \widetilde{{\rm Y}}\left(x\right)\oplus \frac{\cos \frac{\pi 𝓉}{2}}{e^{𝓉}}\odot \widetilde{{\rm Y}}\left(s\right).$$

Therefore, from (36), we have

$${{\rm Y}}_{\theta }\left(𝓉\varkappa +\left(1-𝓉\right)s\right)=\left[{{\rm Y}}_{*}\left(𝓉\varkappa +\left(1-𝓉\right)s, \theta \right), {{\rm Y}}^{*}\left(𝓉\varkappa +\left(1-𝓉\right)s, \theta \right)\right].$$

Again, from (36), (5), and (7), we obtain

$$\frac{\sin \frac{\pi 𝓉}{2}}{e^{1-𝓉}}{{\rm Y}}_{\theta }\left(x\right)+\frac{\cos \frac{\pi 𝓉}{2}}{e^{𝓉}}{{\rm Y}}_{\theta }\left(x\right)=\left[\frac{\sin \frac{\pi 𝓉}{2}}{e^{1-𝓉}}{{\rm Y}}_{*}\left(x, \theta \right), \frac{\sin \frac{\pi 𝓉}{2}}{e^{1-𝓉}}{{\rm Y}}^{*}\left(x, \theta \right)\right]+\left[\frac{\cos \frac{\pi 𝓉}{2}}{e^{𝓉}}{{\rm Y}}_{*}\left(s, \theta \right), \frac{\cos \frac{\pi 𝓉}{2}}{e^{𝓉}}{{\rm Y}}^{*}\left(s, \theta \right)\right],$$

for all $x,s\in K$ and $𝓉\in \left[0, 1\right].$ Then, by $UDET\text{-}$convexity of $\widetilde{{\rm Y}}$, we have, for all, x,s ∈ K and $𝓉\in \left[0, 1\right],$such that

$${{\rm Y}}_{*}\left(𝓉\varkappa +\left(1-𝓉\right)s, \theta \right)\le \frac{\sin \frac{\pi 𝓉}{2}}{e^{1-𝓉}}{{\rm Y}}_{*}\left(x, \theta \right)+\frac{\cos \frac{\pi 𝓉}{2}}{e^{𝓉}}{{\rm Y}}_{*}\left(s, \theta \right),$$

and

$${{\rm Y}}^{*}\left(𝓉\varkappa +\left(1-𝓉\right)s, \theta \right)\ge \frac{\sin \frac{\pi 𝓉}{2}}{e^{1-𝓉}}{{\rm Y}}^{*}\left(x, \theta \right)+\frac{\cos \frac{\pi 𝓉}{2}}{e^{𝓉}}{{\rm Y}}^{*}\left(s, \theta \right),$$

for each $\theta \in \left[0, 1\right].$ Hence, the results follow. □

Remark 2.

If ${{\rm Y}}_{*}\left(\varkappa ,\theta \right)\ne {{\rm Y}}^{*}\left(\varkappa ,\theta \right)$ and $\theta =1$, then we obtain the definition of an exponential trigonometric convex interval-valued function, which is also a new one:

$${\rm Y}\left(𝓉\varkappa +\left(1-𝓉\right)s\right)\supseteq \frac{sin\frac{\pi 𝓉}{2}}{e^{1-𝓉}}{\rm Y}\left(\varkappa \right)+\frac{cos\frac{\pi 𝓉}{2}}{e^{𝓉}}{\rm Y}\left(s\right).$$

(37)

If ${{\rm Y}}_{*}\left(\varkappa ,\theta \right)={{\rm Y}}^{*}\left(\varkappa ,\theta \right)$ and $\theta =1$, then we obtain the classical definition of exponential trigonometric convex functions.

Here are some new definitions that will support us in acquiring some classical results.

Definition 13.

Let $\widetilde{{\rm Y}}:\left[\mu ,b\right]\to R_C$ be a $FNVM$, with $\theta$ levels defined by the family of I-V∙Ms ${{\rm Y}}_{\theta }:\left[\mu ,b\right]\to {\mathcal{X}}_C^{+}\subset {\mathcal{X}}_C$ given by

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

(38)

for all  $\varkappa \in \left[\mu ,b\right]$ and for all $\theta \in \left[0, 1\right]$. Then, $\widetilde{{\rm Y}}$ is a lower $UD$ exponential trigonometric convex (concave)  $FNVM$ on $\left[\mu ,b\right]$ if and only if, for all,  $\theta \in \left[0, 1\right],$

$${{\rm Y}}_{*}\left(𝓉\varkappa +\left(1-𝓉\right)s, \theta \right)\le (\ge )\frac{\sin \frac{\pi 𝓉}{2}}{e^{1-𝓉}}{{\rm Y}}_{*}\left(\varkappa ,\theta \right)+\frac{\cos \frac{\pi 𝓉}{2}}{e^{𝓉}}{{\rm Y}}_{*}\left(s,\theta \right),$$

and

$${{\rm Y}}^{*}\left(𝓉\varkappa +\left(1-𝓉\right)s, \theta \right)=\frac{\sin \frac{\pi 𝓉}{2}}{e^{1-𝓉}}{{\rm Y}}^{*}\left(\varkappa ,\theta \right)+\frac{\cos \frac{\pi 𝓉}{2}}{e^{𝓉}}{{\rm Y}}^{*}\left(s,\theta \right).$$

Definition 14.

Let $\widetilde{{\rm Y}}:\left[\mu ,b\right]\to R_C$ be a $FNVM$, with  $\theta$ levels defined by the family of I-V∙Ms ${{\rm Y}}_{\theta }:\left[\mu ,b\right]\to {\mathcal{X}}_C^{+}\subset {\mathcal{X}}_C$ given by

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

(39)

for all  $\varkappa \in \left[\mu ,b\right]$ and for all $\theta \in \left[0, 1\right]$. Then,  $\widetilde{{\rm Y}}$ is the $UD$ exponential trigonometric convex (concave) $FNVM$ on if and only if, for all, $\theta \in \left[0, 1\right],$

$${{\rm Y}}_{*}\left(𝓉\varkappa +\left(1-𝓉\right)s, \theta \right)=\frac{\sin \frac{\pi 𝓉}{2}}{e^{1-𝓉}}{{\rm Y}}_{*}\left(\varkappa ,\theta \right)+\frac{\cos \frac{\pi 𝓉}{2}}{e^{𝓉}}{{\rm Y}}_{*}\left(s,\theta \right),$$

and

$${{\rm Y}}^{*}\left(𝓉\varkappa +\left(1-𝓉\right)s, \theta \right)\le (\ge )\frac{\sin \frac{\pi 𝓉}{2}}{e^{1-𝓉}}{{\rm Y}}^{*}\left(\varkappa ,\theta \right)+\frac{\cos \frac{\pi 𝓉}{2}}{e^{𝓉}}{{\rm Y}}^{*}\left(s,\theta \right).$$

Remark 3.

Both concepts “$UDET\text{-}$convex $FNVM$” and “exponential trigonometric convex $FNVM$” behave alike when $\widetilde{{\rm Y}}$ is $UDET\text{-}$convex $FNVM$.

3. Main Results

In this part, we will suggest a novel version of fuzzy H–H inequalities, and we will use nontrivial examples to demonstrate how these fuzzy up and down relations work.

Theorem 6.

Let $\widetilde{{\rm Y}}:\left[\mu , b\right]\to R_C$ be an $UDET\text{-}$convex $FNVM$ on $\left[\mu , b\right],$ with $\theta$ levels defined by the family of i-v-ms ${{\rm Y}}_{\theta }:\left[\mu , b\right]\subset ℝ\to {\mathcal{X}}_C^{+}$ given by ${{\rm Y}}_{\theta }\left(\varkappa \right)=\left[{{\rm Y}}_{*}\left(\varkappa ,\theta \right), {{\rm Y}}^{*}\left(\varkappa ,\theta \right)\right]$ for all $\varkappa \in \left[\mu , b\right]$ and for all $\theta \in \left[0, 1\right]$. If $\widetilde{{\rm Y}}\in L\left(\left[\mu , b\right],R_C\right)$, then

$$\sqrt{\frac{e}{2}}\odot \widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right){\supseteq }_{\mathbb{F}}\frac{1-\alpha }{2\left(1-e^{-\nu }\right)}\odot \left[{ℐ}_{{\mu }^{+}}^{\alpha } \widetilde{{\rm Y}}\left(b\right)\oplus {ℐ}_{b^{-}}^{\alpha } \widetilde{{\rm Y}}\left(\mu \right)\right]{\supseteq }_{\mathbb{F}}\frac{\nu }{1-e^{-\nu }}S\left(\nu \right)\odot \frac{\widetilde{{\rm Y}}\left(\mu \right)\oplus \widetilde{{\rm Y}}\left(b\right)}{2}.$$

(40)

If$\widetilde{{\rm Y}}\left(\varkappa \right)$ is  $UDET-$concave  $FNVM$, then

$$\sqrt{\frac{e}{2}}\odot \widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right){\subseteq }_{\mathbb{F}}\frac{1-\alpha }{2\left(1-e^{-\nu }\right)}\odot \left[{ℐ}_{{\mu }^{+}}^{\alpha }\widetilde{{\rm Y}}\left(b\right)\oplus {ℐ}_{b^{-}}^{\alpha }\widetilde{{\rm Y}}\left(\mu \right)\right]{\subseteq }_{\mathbb{F}}\frac{\nu }{1-e^{-\nu }}S\left(\nu \right)\odot \frac{\widetilde{{\rm Y}}\left(\mu \right)\oplus \widetilde{{\rm Y}}\left(b\right)}{2}.$$

(41)

where

$$S\left(\nu \right)=\frac{4\nu +2\pi e^{-\nu -1}+4}{4{\nu }^2+8\nu +{\pi }^2+4}+\frac{2\pi e^{-1}+4e^{-\nu -1}\left(e+\nu e\right)}{4{\nu }^2-8\nu +{\pi }^2+4},\nu =\frac{1-\alpha }{\alpha }\left(b-\mu \right) \mathit{and} 1>\alpha >0$$

Proof. 

Let $\widetilde{{\rm Y}}:\left[\mu , b\right]\to R_C$ be an $UDET\text{-}$convex $FNVM$. Then, by hypothesis, we have

$$\widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right){\supseteq }_{\mathbb{F}}\frac{\sin \frac{\pi }{4}}{\sqrt{e}}\odot \widetilde{{\rm Y}}\left(𝓉\mu +\left(1-𝓉\right)b\right)\oplus \frac{\cos \frac{\pi }{4}}{\sqrt{e}}\odot \widetilde{{\rm Y}}\left(\left(1-𝓉\right)\mu +𝓉b\right).$$

After simplification, we get that

$$2\widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right){\supseteq }_{\mathbb{F}}\sqrt{\frac{2}{e}}\odot \left[\widetilde{{\rm Y}}\left(𝓉\mu +\left(1-𝓉\right)b\right)\oplus \widetilde{{\rm Y}}\left(\left(1-𝓉\right)\mu +𝓉b\right)\right].$$

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

$$2{{\rm Y}}_{*}\left(\frac{\mu +b}{2}, \theta \right)\le \sqrt{\frac{2}{e}}\left[{{\rm Y}}_{*}\left(𝓉\mu +\left(1-𝓉\right)b, \theta \right)+{{\rm Y}}_{*}\left(\left(1-𝓉\right)\mu +𝓉b, \theta \right)\right],$$

$$2{{\rm Y}}^{*}\left(\frac{\mu +b}{2}, \theta \right)\ge \sqrt{\frac{2}{e}}\left[{{\rm Y}}^{*}\left(𝓉\mu +\left(1-𝓉\right)b, \theta \right)+{{\rm Y}}^{*}\left(\left(1-𝓉\right)\mu +𝓉b, \theta \right)\right].$$

Taking ${{\rm Y}}_{*}\left(., \theta \right)$ and multiplying both sides by $e^{-\nu 𝓉}$ and integrating the obtained results with respect to $𝓉$ from $0$ to $1$, we have

$$\begin{array}{c}2{\int }_0^1{e}^{-\nu 𝓉}{{\rm Y}}_{*}\left(\frac{\mu +b}{2}, \theta \right)d𝓉\\ \le \sqrt{\frac{2}{e}}\left[{\int }_0^1{e}^{-\nu 𝓉}{{\rm Y}}_{*}\left(𝓉\mu +\left(1-𝓉\right)b,\theta \right)d𝓉+{\int }_0^1{e}^{-\nu 𝓉}{{\rm Y}}_{*}\left(\left(1-𝓉\right)\mu +𝓉b, \theta \right)d𝓉\right]. \end{array}$$

Let u = 𝓉μ+ (1−𝓉)b and $x=\left(1-𝓉\right)\mu +𝓉b.$ Then, we have

$$\begin{array}{ll}2{\int }_0^1{e}^{-\nu 𝓉}{{\rm Y}}_{*}\left(\frac{\mu +b}{2}, \theta \right)d𝓉& \le \sqrt{\frac{2}{e}}\frac{1}{b-\mu } {\int }_{\mu }^b{e}^{\left(-\frac{1-\alpha }{\alpha }\left(b-u\right)\right)}{{\rm Y}}_{*}\left(u,\theta \right)du+\frac{1}{b-\mu }{\int }_{\mu }^b{e}^{\left(-\frac{1-\alpha }{\alpha }\left(x-\mu \right)\right)}{{\rm Y}}_{*}\left(\varkappa ,\theta \right)d\varkappa \\ & =\sqrt{\frac{2}{e}}\frac{\alpha }{b-\mu }\left[{ℐ}_{{\mu }^{+}}^{\alpha } {{\rm Y}}_{*}\left(b, \theta \right)+{ℐ}_{b^{-}}^{\alpha } {{\rm Y}}_{*}\left(\mu , \theta \right)\right].\end{array}$$

(42)

Now, taking the right side of Equation (42), we have

$${\int }_0^1{e}^{-\nu 𝓉}{{\rm Y}}_{*}\left(\frac{\mu +b}{2}, \theta \right)d𝓉=\frac{1-e^{-\nu }}{\nu }{{\rm Y}}_{*}\left(\frac{\mu +b}{2}, \theta \right)$$

(43)

From (42) and (43), we have

$$\frac{2}{\alpha }·\frac{1-e^{-\nu }}{\nu }{{\rm Y}}_{*}\left(\frac{\mu +b}{2}, \theta \right)\le \sqrt{\frac{2}{e}}·\frac{1}{b-\mu }\left[{ℐ}_{{\mu }^{+}}^{\alpha } {{\rm Y}}_{*}\left(b, \theta \right)+{ℐ}_{b^{-}}^{\alpha }{{\rm Y}}^{*}\left(\mu , \theta \right)\right].$$

(44)

Similarly, for ${{\rm Y}}^{*}\left(\varkappa ,\theta \right)$, we have

$$\frac{2}{\alpha }·\frac{1-e^{-\nu }}{\nu }{{\rm Y}}^{*}\left(\frac{\mu +b}{2}, \theta \right)\ge \sqrt{\frac{2}{e}}·\frac{1}{b-\mu }\left[{ℐ}_{{\mu }^{+}}^{\alpha } {{\rm Y}}^{*}\left(b, \theta \right)+{ℐ}_{b^{-}}^{\alpha }{{\rm Y}}^{*}\left(\mu , \theta \right)\right].$$

(45)

From (44) and (45), we have

$$\begin{array}{c}\frac{2}{\alpha }·\frac{1-e^{-\nu }}{\nu }\left[{{\rm Y}}_{*}\right(\frac{\mu +b}{2}, \theta ),{{\rm Y}}^{*}(\frac{\mu +b}{2}, \theta \left)\right]\\ {\supseteq }_I\sqrt{\frac{2}{e}}·\frac{1}{b-\mu }\left[\left[{ℐ}_{{\mu }^{+}}^{\alpha } {{\rm Y}}_{*}\left(b, \theta \right)+{ℐ}_{b^{-}}^{\alpha } {{\rm Y}}_{*}\left(\mu , \theta \right)\right], \left[{ℐ}_{{\mu }^{+}}^{\alpha } {{\rm Y}}^{*}\left(b, \theta \right)+{ℐ}_{b^{-}}^{\alpha } {{\rm Y}}^{*}\left(\mu , \theta \right)\right]\right].\end{array}$$

That is

$$\frac{2}{\alpha }·\frac{1-e^{-\nu }}{\nu } {{\rm Y}}_{\theta }\left(\frac{\mu +b}{2}\right){\supseteq }_I\sqrt{\frac{2}{e}}·\frac{1}{b-\mu }\left[{ℐ}_{{\mu }^{+}}^{\alpha } {{\rm Y}}_{\theta }\left(b\right)+{ℐ}_{b^{-}}^{\alpha } {{\rm Y}}_{\theta }\left(\mu \right)\right].$$

(46)

For the right side of Equation (40), since $\widetilde{{\rm Y}}$ is an $UDET\text{-}$convex $FNVM$, then we deduce that

$$\widetilde{{\rm Y}}\left(𝓉\mu +\left(1-𝓉\right)b\right){\supseteq }_{\mathbb{F}}\frac{\sin \frac{\pi 𝓉}{2}}{e^{1-𝓉}}\odot \widetilde{{\rm Y}}\left(\mu \right)\oplus \frac{\cos \frac{\pi 𝓉}{2}}{e^{𝓉}}\odot \widetilde{{\rm Y}}\left(b\right),$$

(47)

and

$$\widetilde{{\rm Y}}\left(\left(1-𝓉\right)\mu +𝓉b \right){\supseteq }_{\mathbb{F}}\frac{\cos \frac{\pi 𝓉}{2}}{e^{𝓉}}\odot \widetilde{{\rm Y}}\left(\mu \right)\oplus \frac{\sin \frac{\pi 𝓉}{2}}{e^{1-𝓉}}\odot \widetilde{{\rm Y}}\left(b\right).$$

(48)

Adding (47) and (48), we have

$$\widetilde{{\rm Y}}\left(𝓉\mu +\left(1-𝓉\right)b \right)\oplus \widetilde{{\rm Y}}\left(\left(1-𝓉\right)\mu +𝓉b \right){\supseteq }_{\mathbb{F}}\left[\widetilde{{\rm Y}}\left(\mu \right)\oplus \widetilde{{\rm Y}}\left(b\right)\right]\odot \left[\frac{\sin \frac{\pi 𝓉}{2}}{e^{1-𝓉}}+\frac{\cos \frac{\pi 𝓉}{2}}{e^{𝓉}}\right].$$

(49)

Since $\widetilde{{\rm Y}}$ is $FNVM$, then, for each$\theta \in \left[0, 1\right]$, we have

$$\begin{array}{c}{{\rm Y}}_{*}\left(𝓉\mu +\left(1-𝓉\right)b, \theta \right)+{{\rm Y}}_{*}\left(\left(1-𝓉\right)\mu +𝓉b, \theta \right)\le \left[{{\rm Y}}_{*}\left(\mu , \theta \right)+{{\rm Y}}_{*}\left(b, \theta \right)\right]\left[\frac{\sin \frac{\pi 𝓉}{2}}{e^{1-𝓉}}+\frac{\cos \frac{\pi 𝓉}{2}}{e^{𝓉}}\right], \\ {{\rm Y}}^{*}\left(𝓉\mu +\left(1-𝓉\right)b, \theta \right)+{{\rm Y}}^{*}\left(\left(1-𝓉\right)\mu +𝓉b, \theta \right)\ge \left[{{\rm Y}}^{*}\left(\mu , \theta \right)+{{\rm Y}}^{*}\left(b, \theta \right)\right]\left[\frac{\sin \frac{\pi 𝓉}{2}}{e^{1-𝓉}}+\frac{\cos \frac{\pi 𝓉}{2}}{e^{𝓉}}\right]. \end{array}$$

(50)

Taking ${{\rm Y}}_{*}\left(., \theta \right)$ from (50) and multiplying the inequality with $e^{-\nu 𝓉}$ and integrating the results with $𝓉$ from $0$ to $1$, we have

$$\begin{array}{c}{\int }_0^1{e}^{-\nu 𝓉}{{\rm Y}}_{*}\left(𝓉\mu +\left(1-𝓉\right)b, \theta \right)d𝓉+{\int }_0^1{e}^{-\nu 𝓉}{{\rm Y}}_{*}\left(\left(1-𝓉\right)\mu +𝓉b, \theta \right)d𝓉\\ \le \left[{{\rm Y}}_{*}\left(\mu , \theta \right)+{{\rm Y}}_{*}\left(b, \theta \right)\right]{\int }_0^1{e}^{-\nu 𝓉}\left[\frac{\sin \frac{\pi 𝓉}{2}}{e^{1-𝓉}}+\frac{\cos \frac{\pi 𝓉}{2}}{e^{𝓉}}\right]d𝓉,\\ =\frac{4\nu +2\pi e^{-\nu -1}+4}{4{\nu }^2+8\nu +{\pi }^2+4}+\frac{2\pi e^{-1}+4e^{-\nu -1}\left(e+\nu e\right)}{4{\nu }^2-8\nu +{\pi }^2+4}\left[{{\rm Y}}_{*}\left(\mu , \theta \right)+{{\rm Y}}_{*}\left(b, \theta \right)\right].\end{array}$$

(51)

In a manner identical to that described previously, we have ${{\rm Y}}^{*}\left(., \theta \right)$

$$\begin{array}{c}{\int }_0^1{e}^{-\nu 𝓉}{{\rm Y}}^{*}\left(𝓉\mu +\left(1-𝓉\right)b, \theta \right)d𝓉+{\int }_0^1{e}^{-\nu 𝓉}{{\rm Y}}^{*}\left(\left(1-𝓉\right)\mu +𝓉b, \theta \right)d𝓉\\ \ge \left[{{\rm Y}}^{*}\left(\mu , \theta \right)+{{\rm Y}}^{*}\left(b, \theta \right)\right]{\int }_0^1{e}^{-\nu 𝓉}\left[\frac{\sin \frac{\pi 𝓉}{2}}{e^{1-𝓉}}+\frac{\cos \frac{\pi 𝓉}{2}}{e^{𝓉}}\right]d𝓉,\\ =\frac{4\nu +2\pi e^{-\nu -1}+4}{4{\nu }^2+8\nu +{\pi }^2+4}+\frac{2\pi e^{-1}+4e^{-\nu -1}\left(e+\nu e\right)}{4{\nu }^2-8\nu +{\pi }^2+4}\left[{{\rm Y}}^{*}\left(\mu , \theta \right)+{{\rm Y}}^{*}\left(b, \theta \right)\right].\end{array}$$

(52)

From (51) and (52), we have

$$\begin{array}{c}{\int }_0^1{e}^{-\nu 𝓉}{{\rm Y}}_{\theta }\left(𝓉\mu +\left(1-𝓉\right)b \right)d𝓉+{\int }_0^1{e}^{-\nu 𝓉}{{\rm Y}}_{\theta }\left(\left(1-𝓉\right)\mu +𝓉b \right)d𝓉\\ {\supseteq }_I\frac{4\nu +2\pi e^{-\nu -1}+4}{4{\nu }^2+8\nu +{\pi }^2+4}+\frac{2\pi e^{-1}+4e^{-\nu -1}\left(e+\nu e\right)}{4{\nu }^2-8\nu +{\pi }^2+4}\left[{{\rm Y}}_{\theta }\left(\mu \right)+{{\rm Y}}_{\theta }\left(b\right)\right].\end{array}$$

(53)

From (46) and (53), we have

$$\sqrt{\frac{e}{2}}{{\rm Y}}_{\theta }\left(\frac{\mu +b}{2}\right){\supseteq }_I\frac{1-\alpha }{2\left(1-e^{-\nu }\right)}\left[{ℐ}_{{\mu }^{+}}^{\alpha } {{\rm Y}}_{\theta }\left(b\right)+{ℐ}_{b^{-}}^{\alpha } {{\rm Y}}_{\theta }\left(\mu \right)\right]{\supseteq }_I\frac{\nu }{1-e^{-\nu }}S\left(\nu \right)\frac{{{\rm Y}}_{\theta }\left(\mu \right)+{{\rm Y}}_{\theta }\left(b\right)}{2},$$

that is

$$\sqrt{\frac{e}{2}}\odot \widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right){\supseteq }_{\mathbb{F}}\frac{1-\alpha }{2\left(1-e^{-\nu }\right)}\odot \left[{ℐ}_{{\mu }^{+}}^{\alpha } \widetilde{{\rm Y}}\left(b\right)\oplus {ℐ}_{b^{-}}^{\alpha } \widetilde{{\rm Y}}\left(\mu \right)\right]{\supseteq }_{\mathbb{F}}\frac{\nu }{1-e^{-\nu }}S\left(\nu \right)\odot \frac{\widetilde{{\rm Y}}\left(\mu \right)\oplus \widetilde{{\rm Y}}\left(b\right)}{2}.$$

Hence, the required results. □

If we use a few minor limits on Theorem 6, then the novel and conventional results can be as follows:

Remark 4.

We readily see, from Theorem 6, that

If $\widetilde{{\rm Y}}$ is a lower $UDET\text{-}$convex $FNVM$ on $\left[\mu ,b\right],$ then we have (see [78])

$$\sqrt{\frac{e}{2}}\odot \widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right){\le }_{\mathbb{F}}\frac{1-\alpha }{2\left(1-e^{-\nu }\right)}\odot \left[{ℐ}_{{\mu }^{+}}^{\alpha }\widetilde{{\rm Y}}\left(b\right)\oplus {ℐ}_{b^{-}}^{\alpha }\widetilde{{\rm Y}}\left(\mu \right)\right]{\le }_{\mathbb{F}}\frac{\nu }{1-e^{-\nu }}S\left(\nu \right)\odot \frac{\widetilde{{\rm Y}}\left(\mu \right)\oplus \widetilde{{\rm Y}}\left(b\right)}{2}.$$

(54)

Let ${{\rm Y}}_{*}\left(\varkappa ,\theta \right)\ne {{\rm Y}}^{*}\left(\varkappa ,\theta \right)$ and $\theta =1$,$.$ Then, from Theorem 6, we have (see [78])

$$\sqrt{\frac{e}{2}}{\rm Y}\left(\frac{\mu +b}{2}\right)\supseteq \frac{1-\alpha }{2\left(1-e^{-\nu }\right)}\left[{ℐ}_{{\mu }^{+}}^{\alpha } {\rm Y}\left(b\right)+{ℐ}_{b^{-}}^{\alpha } {\rm Y}\left(\mu \right)\right]\supseteq \frac{\nu }{1-e^{-\nu }}S\left(\nu \right)\frac{{\rm Y}\left(\mu \right)+{\rm Y}\left(b\right)}{2}.$$

(55)

If $\alpha \to 1$, then

$$\underset{\alpha \to 1}{\lim }\nu =\underset{\alpha \to 1}{\lim }\frac{1-\alpha }{\alpha }\left(b-\mu \right)=0, \mathrm{then}$$

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

Now, from Theorem 6, we acquire the following result, which is also new one:

$$\sqrt{\frac{e}{2}}\odot \widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right){\supseteq }_{\mathbb{F}}\frac{1}{b-\mu }\odot {\int }_{\mu }^b\widetilde{{\rm Y}}\left(\varkappa \right)d\varkappa {\supseteq }_{\mathbb{F}}\frac{2\pi e^{-1}+4}{{\pi }^2+4}\odot \left[\widetilde{{\rm Y}}\left(\mu \right)\oplus \widetilde{{\rm Y}}\left(b\right)\right].$$

(56)

If one lays $\alpha \to 1,$ and $\widetilde{{\rm Y}}$ is a lower $UDET\text{-}$convex $FNVM$ on $\left[\mu ,b\right],$ then we get (see [78])

$$\sqrt{\frac{e}{2}}\odot \widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right){\le }_{\mathbb{F}}\frac{1}{b-\mu }\odot {\int }_{\mu }^b\widetilde{{\rm Y}}\left(\varkappa \right)d\varkappa {\le }_{\mathbb{F}}\frac{2\pi e^{-1}+4}{{\pi }^2+4}\odot \left[\widetilde{{\rm Y}}\left(\mu \right)\oplus \widetilde{{\rm Y}}\left(b\right)\right].$$

(57)

Let $\alpha \to 1$ and ${{\rm Y}}_{*}\left(\varkappa , \theta \right)\ne {{\rm Y}}^{*}\left(\varkappa , \theta \right)$ with $\theta =1$. Then, from Theorem 6, we acquire (see [78])

$$\sqrt{\frac{e}{2}}{\rm Y}\left(\frac{\mu +b}{2}\right)\supseteq \frac{1}{b-\mu }{\int }_{\mu }^b{\rm Y}\left(\varkappa \right)d\varkappa \supseteq \frac{2\pi e^{-1}+4}{{\pi }^2+4}\left[{\rm Y}\left(\mu \right)+{\rm Y}\left(b\right)\right].$$

(58)

If ${{\rm Y}}_{*}\left(\varkappa , \theta \right)={{\rm Y}}^{*}\left(\varkappa , \theta \right)$ and $\theta =1,$ then, from Theorem 6, we have

$$\sqrt{\frac{e}{2}}{\rm Y}\left(\frac{\mu +b}{2}\right)\le \frac{1-\alpha }{2\left(1-e^{-\nu }\right)}\left[{ℐ}_{{\mu }^{+}}^{\alpha } {\rm Y}\left(b\right)+{ℐ}_{b^{-}}^{\alpha } {\rm Y}\left(\mu \right)\right]\le \frac{\nu }{1-e^{-\nu }}S\left(\nu \right)\frac{{\rm Y}\left(\mu \right)+{\rm Y}\left(b\right)}{2}.$$

(59)

Let $\alpha \to 1$ and ${{\rm Y}}_{*}\left(\varkappa , \theta \right)={{\rm Y}}^{*}\left(\varkappa , \theta \right)$ with $\theta =1$. Then, from Theorem 6, we achieve (see [73])

$$\sqrt{\frac{e}{2}}{\rm Y}\left(\frac{\mu +b}{2}\right)\le \frac{1}{b-\mu }{\int }_{\mu }^b{\rm Y}\left(\varkappa \right)d\varkappa \le \frac{2\pi e^{-1}+4}{{\pi }^2+4}\left[{\rm Y}\left(\mu \right)+{\rm Y}\left(b\right)\right].$$

(60)

Example 1.

Let $\alpha =\frac{1}{2}, \varkappa \in \left[0,1\right]$, and the $FNVM$ $\widetilde{{\rm Y}}:\left[\mu , b\right]=\left[0, 1\right]\to R_C,$ defined by

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

(61)

then, for each $\theta \in \left[0, 1\right],$ we have ${{\rm Y}}_{\theta }\left(\varkappa \right)=\left[2\left(1-\theta \right){\varkappa }^2+3\theta ,2\left(1-\theta \right)\left(1+e^x\right)+3\theta \right]$. Since, for each $\theta \in \left[0, 1\right]$, the left and right end point functions ${{\rm Y}}_{*}\left(\varkappa ,\theta \right)=2\left(1-\theta \right){\varkappa }^2+3\theta ,$ ${{\rm Y}}^{*}\left(\varkappa , \theta \right)=2\left(1-\theta \right)\left(1+e^x\right)+3\theta$ are exponential trigonometric convex and concave functions, respectively, then $\widetilde{{\rm Y}}\left(\varkappa \right)$ is the $UDET\text{-}$convex $FNVM$. We readily see that $\widetilde{{\rm Y}}\in L\left(\left[\mu , b\right],R_C\right)$ and $\nu =\frac{1-\alpha }{\alpha }\left(b-\mu \right)=1$. Now, we calculate the following:

$$\sqrt{\frac{e}{2}}{{\rm Y}}_{*}\left(\frac{\mu +b}{2}, \theta \right)=\sqrt{\frac{e}{2}}{{\rm Y}}_{*}(\frac{1}{2}, \theta )=\frac{\sqrt{e}}{2\sqrt{2}}(1-\theta )+\frac{3\sqrt{e}}{\sqrt{2}}\theta$$

$$\sqrt{\frac{e}{2}}{{\rm Y}}^{*}\left(\frac{\mu +b}{2}, \theta \right)={{\rm Y}}^{*}\left(\frac{5}{2}, \theta \right)=2\left(1-\theta \right)\frac{1+\sqrt{e}}{\sqrt{2}}+\frac{3\sqrt{e}}{\sqrt{2}}\theta$$

$$\frac{\nu }{1-e^{-\nu }}S\left(\nu \right)\frac{{{\rm Y}}_{*}\left(\mu ,\theta \right)+{{\rm Y}}_{*}\left(b, \theta \right)}{2}=\frac{8+2\pi e^{-2}}{\left(16+{\pi }^2\right)\left(1-e^{-1}\right)}\left(1+2\theta \right)$$

$$\frac{\nu }{1-e^{-\nu }}S\left(\nu \right)\frac{{{\rm Y}}^{*}\left(\mu ,\theta \right)+{{\rm Y}}^{*}\left(b, \theta \right)}{2}=\frac{8+2\pi e^{-2}}{\left(16+{\pi }^2\right)\left(1-e^{-1}\right)}\left(\left(1-\theta \right)\left(3+e\right)+3\theta \right).$$

Note that

$$\begin{array}{ll}\frac{1-\alpha }{2\left(1-e^{-\nu }\right)}& \left[{ℐ}_{{\mu }^{+}}^{\alpha } {{\rm Y}}_{*}\left(b, \theta \right)+{ℐ}_{b^{-}}^{\alpha } {{\rm Y}}_{*}\left(\mu ,\theta \right)\right]\\ & =\frac{1}{2\left(1-e^{-1}\right)}{\int }_0^1{e}^{-x} .\left(2\left(1-\theta \right){\varkappa }^2+3\theta \right)d\varkappa \\ & +\frac{1}{2\left(1-e^{-1}\right)}{\int }_0^1{e}^{-x} .\left(2\left(1-\theta \right){\varkappa }^2+3\theta \right)d\varkappa \\ & =\left(1-\theta \right)\frac{1}{1-e^{-1}}\left[1-2e^{-1}+2-5e^{-1}\right]+\frac{3}{2}\theta \\ & =\left(1-\theta \right)\frac{3-7e^{-1}}{1-e^{-1}}\frac{3}{2}\theta .\end{array}$$

$$\begin{array}{ll}\frac{1-\alpha }{2\left(1-e^{-\nu }\right)}& \left[{ℐ}_{{\mu }^{+}}^{\alpha } {{\rm Y}}^{*}\left(b, \theta \right)+{ℐ}_{b^{-}}^{\alpha } {{\rm Y}}^{*}\left(\mu ,\theta \right)\right]\\ & =\frac{1}{2\left(1-e^{-1}\right)}{\int }_0^1{e}^{-\left(1-\varkappa \right)} . \left(2\left(1-\theta \right)\left(1+e^x\right)+3\theta \right)d\varkappa \\ & +\frac{1}{2\left(1-e^{-1}\right)}{\int }_0^1{e}^{-x} . \left(2\left(1-\theta \right)\left(1+e^x\right)+3\theta \right)d\varkappa \\ & =\left(1-\theta \right)\frac{1}{1-e^{-1}}[\frac{e}{2}-\frac{3e^{-1}}{2}+3-e^{-1}]+\frac{3}{2}\theta \\ & =\left(1-\theta \right)\frac{e-5e^{-1}+6}{2\left(1-e^{-1}\right)}+\frac{3}{2}\theta .\end{array}$$

Therefore,

$$\begin{gathered} \left[\frac{\sqrt{e}}{2\sqrt{2}}\left(1-\theta \right)+\frac{3\sqrt{e}}{\sqrt{2}}\theta ,2\left(1-\theta \right)\frac{1+\sqrt{e}}{\sqrt{2}}+\frac{3\sqrt{e}}{\sqrt{2}}\right]{\supseteq }_I\left[\left(1-\theta \right)\frac{3-7e^{-1}}{1-e^{-1}}+\frac{3}{2}\theta ,\left(1-\theta \right)\frac{e-5e^{-1}+6}{2\left(1-e^{-1}\right)}+\frac{3}{2}\theta \right] \\ {\supseteq }_I\left[\frac{8+2\pi e^{-2}}{\left(16+{\pi }^2\right)\left(1-e^{-1}\right)}\left(1+2\theta \right),\frac{8+2\pi e^{-2}}{\left(16+{\pi }^2\right)\left(1-e^{-1}\right)}\left(\left(1-\theta \right)\left(3+e\right)+3\theta \right)\right], \end{gathered}$$

and Theorem 6 is verified.

Here are midpoint $H–H\text{-}$type inequalities for $UDET\text{-}$convex $FNVM$.

Theorem 7.

Let $\widetilde{{\rm Y}}:\left[\mu , b\right]\to R_C$ be an $UDET\text{-}$convex $FNVM$ on $\left[\mu , b\right],$ with  $\theta$ levels defined by the family of i-v-ms ${{\rm Y}}_{\theta }:\left[\mu , b\right]\subset ℝ\to {\mathcal{X}}_C^{+}$ given by ${{\rm Y}}_{\theta }\left(\varkappa \right)=\left[{{\rm Y}}_{*}\left(\varkappa ,\theta \right), {{\rm Y}}^{*}\left(\varkappa ,\theta \right)\right]$ for all $\varkappa \in \left[\mu , b\right]$ and for all $\theta \in \left[0, 1\right]$. If $\widetilde{{\rm Y}}\in L\left(\left[\mu , b\right],R_C\right)$; then,

$$\sqrt{\frac{e}{2}}\odot \widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right){\supseteq }_{\mathbb{F}}\frac{1-\alpha }{2\left(1-e^{-\frac{\nu }{2}}\right)}\odot \left[{ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{+}}^{\alpha } \widetilde{{\rm Y}}\left(b\right)\oplus {ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{-}}^{\alpha } \widetilde{{\rm Y}}\left(\mu \right)\right]{\supseteq }_{\mathbb{F}}\frac{\nu }{2\left(1-e^{-\frac{\nu }{2}}\right)}P\left(\nu \right)\odot \frac{\widetilde{{\rm Y}}\left(\mu \right)\oplus \widetilde{{\rm Y}}\left(b\right)}{2}.$$

(62)

If $\widetilde{{\rm Y}}\left(\varkappa \right)$ is $UDET-$concave $FNVM$, then

$$\sqrt{\frac{e}{2}}\odot \widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right){\subseteq }_{\mathbb{F}}\frac{1-\alpha }{2\left(1-e^{-\frac{\nu }{2}}\right)}\odot \left[{ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{+}}^{\alpha } \widetilde{{\rm Y}}\left(b\right)\oplus {ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{-}}^{\alpha } \widetilde{{\rm Y}}\left(\mu \right)\right]{\subseteq }_{\mathbb{F}}\frac{\nu }{1-e^{-\frac{\nu }{2}}}P\left(\nu \right)\odot \frac{\widetilde{{\rm Y}}\left(\mu \right)\oplus \widetilde{{\rm Y}}\left(b\right)}{2}.$$

(63)

where

$$P\left(\nu \right)=\frac{\pi \left(4e^{-1}-2^{\frac{3}{2}}{e}^{-\frac{\nu +1}{2}}\right)-2^{\frac{5}{2}}\nu e^{-\frac{\nu +1}{2}}+2^{\frac{5}{2}}{e}^{-\frac{\nu +1}{2}}}{4{\nu }^2-8\nu +{\pi }^2+4}+\frac{8\nu +2^{\frac{3}{2}}\pi e^{-\frac{\nu +1}{2}}-2^{\frac{5}{2}}\left(\nu +1\right)e^{-\frac{\nu +1}{2}}+8}{4{\nu }^2+8\nu +{\pi }^2+4}, \nu =\frac{1-\alpha }{\alpha }\left(b-\mu \right), \mathrm{and} 1>\alpha >0.$$

Proof. 

Let $\widetilde{{\rm Y}}:\left[\mu , b\right]\to R_C$ be an $UDET\text{-}$convex $FNVM$. Then, by hypothesis, we have

$$\widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right){\supseteq }_{\mathbb{F}}\frac{\sin \frac{\pi }{4}}{\sqrt{e}}\odot \widetilde{{\rm Y}}\left(\frac{𝓉}{2}\mu +\frac{2-𝓉}{2}b\right)\oplus \frac{\cos \frac{\pi }{4}}{\sqrt{e}}\odot \widetilde{{\rm Y}}\left(\frac{2-𝓉}{2}\mu +\frac{𝓉}{2}b\right).$$

After simplification, we get that

$$2\widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right){\supseteq }_{\mathbb{F}}\sqrt{\frac{2}{e}}\odot \left[\widetilde{{\rm Y}}\left(\frac{𝓉}{2}\mu +\frac{2-𝓉}{2}b\right)\oplus \widetilde{{\rm Y}}\left(\frac{2-𝓉}{2}\mu +\frac{𝓉}{2}b\right)\right].$$

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

$$\begin{array}{c}\begin{array}{c}2{{\rm Y}}_{*}\left(\frac{\mu +b}{2}, \theta \right)\le \sqrt{\frac{2}{e}}\left[{{\rm Y}}_{*}\left(\frac{𝓉}{2}\mu +\frac{2-𝓉}{2}b, \theta \right)+{{\rm Y}}_{*}\left(\frac{2-𝓉}{2}\mu +\frac{𝓉}{2}b, \theta \right)\right]\end{array},\\ 2{{\rm Y}}^{*}\left(\frac{\mu +b}{2}, \theta \right)\ge \sqrt{\frac{2}{e}}\left[{{\rm Y}}^{*}\left(\frac{𝓉}{2}\mu +\frac{2-𝓉}{2}b, \theta \right)+{{\rm Y}}^{*}\left(\frac{2-𝓉}{2}\mu +\frac{𝓉}{2}b, \theta \right)\right].\end{array}$$

Taking ${{\rm Y}}_{*}\left(., \theta \right)$ and multiplying both sides by $e^{-\frac{\nu 𝓉}{2}}$ and integrating the obtained results with respect to $𝓉$ from $0$ to $1$, we have

$$2{\int }_0^1{e}^{-\frac{\nu 𝓉}{2}}{{\rm Y}}_{*}\left(\frac{\mu +b}{2}, \theta \right)d𝓉\le \sqrt{\frac{2}{e}}\left[{\int }_0^1{e}^{-\frac{\nu 𝓉}{2}}{{\rm Y}}_{*}\left(\frac{𝓉}{2}\mu +\frac{2-𝓉}{2}b,\theta \right)d𝓉+{\int }_0^1{e}^{-\frac{\nu 𝓉}{2}}{{\rm Y}}_{*}\left(\frac{2-𝓉}{2}\mu +\frac{𝓉}{2}b, \theta \right)d𝓉\right].$$

Let $u=\frac{𝓉}{2}\mu +\frac{2-𝓉}{2}b$ and $x=\frac{2-𝓉}{2}\mu +\frac{𝓉}{2}b.$ Then, we have

$$\begin{array}{ll}2{\int }_0^1{e}^{-\frac{\nu 𝓉}{2}}{{\rm Y}}_{*}\left(\frac{\mu +b}{2}, \theta \right)d𝓉 & \le \sqrt{\frac{2}{e}}\frac{1}{b-\mu } {\int }_{\frac{\mu +b}{2}}^b{e}^{\left(-\frac{1-\alpha }{\alpha }\left(b-u\right)\right)}{{\rm Y}}_{*}\left(u,\theta \right)du+\frac{1}{b-\mu }{\int }_{\frac{\mu +b}{2}}^b{e}^{\left(-\frac{1-\alpha }{\alpha }\left(x-\mu \right)\right)}{{\rm Y}}_{*}\left(\varkappa ,\theta \right)d\varkappa \\ & =\sqrt{\frac{2}{e}}\frac{\alpha }{b-\mu }\left[{\mathcal{I}}_{{\left(\frac{\mu +b}{2}\right)}^{+}}^{\alpha } {{\rm Y}}_{\ast }\left(b, \theta \right)+{\mathcal{I}}_{{\left(\frac{\mu +b}{2}\right)}^{-}}^{\alpha } {{\rm Y}}_{\ast }\left(\mu , \theta \right)\right].\end{array}$$

(64)

Now, taking the right side of Equation (64), we have

$${\int }_0^1{e}^{-\frac{\nu 𝓉}{2}}{{\rm Y}}_{*}\left(\frac{\mu +b}{2}, \theta \right)d𝓉=\frac{2\left(1-e^{-\nu }\right)}{\nu }{{\rm Y}}_{*}\left(\frac{\mu +b}{2}, \theta \right).$$

(65)

From (64) and (65), we have

$$4·\frac{1-e^{-\nu }}{\nu }{{\rm Y}}_{*}\left(\frac{\mu +b}{2}, \theta \right)\le \sqrt{\frac{2}{e}}·\frac{2\left(1-\mu \right)}{b-\mu }\left[{ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{+}}^{\alpha } {{\rm Y}}_{*}\left(b, \theta \right)+{ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{-}}^{\alpha } {{\rm Y}}_{*}\left(\mu , \theta \right)\right].$$

(66)

Similarly, for ${{\rm Y}}^{*}\left(\varkappa ,\theta \right)$, we have

$$4·\frac{1-e^{-\nu }}{\nu }{{\rm Y}}^{*}\left(\frac{\mu +b}{2}, \theta \right)\ge \sqrt{\frac{2}{e}}·\frac{2\left(1-\mu \right)}{b-\mu }\left[{ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{+}}^{\alpha } {{\rm Y}}^{*}\left(b, \theta \right)+{ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{-}}^{\alpha } {{\rm Y}}^{*}\left(\mu , \theta \right)\right].$$

(67)

From (66) and (67), we have

$$\begin{array}{c}2·\frac{1-e^{-\nu }}{\nu }\left[{{\rm Y}}_{*}\left(\frac{\mu +b}{2}, \theta \right), {{\rm Y}}^{*}\left(\frac{\mu +b}{2}, \theta \right)\right]\\ {\supseteq }_I\sqrt{\frac{2}{e}}·\frac{1-\mu }{b-\mu }\left[{ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{+}}^{\alpha } {{\rm Y}}_{*}\left(b, \theta \right)+{ℐ}_{b^{-}}^{\alpha } {{\rm Y}}_{*}\left(\mu , \theta \right), {ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{+}}^{\alpha } {{\rm Y}}^{*}\left(b, \theta \right)+{ℐ}_{b^{-}}^{\alpha } {{\rm Y}}^{*}\left(\mu , \theta \right)\right].\end{array}$$

That is

$$2·\frac{1-e^{-\nu }}{\nu } {{\rm Y}}_{\theta }\left(\frac{\mu +b}{2}\right){\supseteq }_I\sqrt{\frac{2}{e}}·\frac{1-\mu }{b-\mu }\left[{ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{+}}^{\alpha } {{\rm Y}}_{\theta }\left(b\right)+{ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{-}}^{\alpha } {{\rm Y}}_{\theta }\left(\mu \right)\right].$$

(68)

For the right side of Equation (62), since $\widetilde{{\rm Y}}$ is an $UDET\text{-}$convex $FNVM$, then we deduce that

$$\widetilde{{\rm Y}}\left(\frac{𝓉}{2}\mu +\frac{2-𝓉}{2}b\right){\supseteq }_{\mathbb{F}}\frac{\sin \frac{\pi 𝓉}{4}}{e^{\frac{2-𝓉}{2}}}\odot \widetilde{{\rm Y}}\left(\mu \right)\oplus \frac{\cos \frac{\pi 𝓉}{4}}{e^{\frac{𝓉}{2}}}\odot \widetilde{{\rm Y}}\left(b\right),$$

(69)

and

$$\widetilde{{\rm Y}}\left(\frac{2-𝓉}{2}\mu +\frac{𝓉}{2}b\right){\supseteq }_{\mathbb{F}}\frac{\cos \frac{\pi 𝓉}{4}}{e^{\frac{𝓉}{2}}}\odot \widetilde{{\rm Y}}\left(\mu \right)\oplus \frac{\sin \frac{\pi 𝓉}{4}}{e^{\frac{2-𝓉}{2}}}\odot \widetilde{{\rm Y}}\left(b\right).$$

(70)

Adding (69) and (70), we have

$$\widetilde{{\rm Y}}\left(\frac{𝓉}{2}\mu +\frac{2-𝓉}{2}b\right)\oplus \widetilde{{\rm Y}}\left(\frac{2-𝓉}{2}\mu +\frac{𝓉}{2}b\right){\supseteq }_{\mathbb{F}}\left[\widetilde{{\rm Y}}\left(\mu \right)\oplus \widetilde{{\rm Y}}\left(b\right)\right]\odot \left[\frac{\sin \frac{\pi 𝓉}{4}}{e^{\frac{2-𝓉}{2}}}+\frac{\cos \frac{\pi 𝓉}{4}}{e^{\frac{𝓉}{2}}}\right].$$

(71)

Since $\widetilde{{\rm Y}}$ is $FNVM$, then, for each$\theta \in \left[0, 1\right]$, we have

$$\begin{array}{c}{{\rm Y}}_{\ast }\left(\frac{𝓉}{2}\mu +\frac{2-𝓉}{2}b, \theta \right)+{{\rm Y}}_{\ast }\left(\frac{2-𝓉}{2}\mu +\frac{𝓉}{2}b, \theta \right)\le \left[{{\rm Y}}_{\ast }\left(\mu , \theta \right)+{{\rm Y}}_{\ast }\left(b, \theta \right)\right]\left[\frac{\mathrm{s}\mathrm{i}\mathrm{n}\frac{\pi 𝓉}{4}}{e^{\frac{2-𝓉}{2}}}+\frac{\mathrm{c}\mathrm{o}\mathrm{s}\frac{\pi 𝓉}{4}}{e^{\frac{𝓉}{2}}}\right], \\ {{\rm Y}}^{\ast }\left(\frac{𝓉}{2}\mu +\frac{2-𝓉}{2}b, \theta \right)+{{\rm Y}}^{\ast }\left(\frac{2-𝓉}{2}\mu +\frac{𝓉}{2}b, \theta \right)\ge \left[{{\rm Y}}^{\ast }\left(\mu , \theta \right)+{{\rm Y}}^{\ast }\left(b, \theta \right)\right]\left[\frac{\mathrm{s}\mathrm{i}\mathrm{n}\frac{\pi 𝓉}{4}}{e^{\frac{2-𝓉}{2}}}+\frac{\mathrm{c}\mathrm{o}\mathrm{s}\frac{\pi 𝓉}{4}}{e^{\frac{𝓉}{2}}}\right]. \end{array}$$

(72)

Taking ${{\rm Y}}_{*}\left(., \theta \right)$ from (72) and multiplying the inequality with $e^{-\frac{\nu 𝓉}{2}}$ and integrating the results with $𝓉$ from $0$ to $1$, we have

$$\begin{array}{l}{\int }_0^1{e}^{e^{-\frac{\nu 𝓉}{2}}}{{\rm Y}}_{*}\left(\frac{𝓉}{2}\mu +\frac{2-𝓉}{2}b, \theta \right)d𝓉+{\int }_0^1{e}^{e^{-\frac{\nu 𝓉}{2}}}{{\rm Y}}_{*}\left(\frac{2-𝓉}{2}\mu +\frac{𝓉}{2}b , \theta \right)d𝓉\\ \le \left[{{\rm Y}}_{*}\left(\mu , \theta \right)+{{\rm Y}}_{*}\left(b, \theta \right)\right]{\int }_0^1{e}^{e^{-\frac{\nu 𝓉}{2}}}\left[\frac{\sin \frac{\pi 𝓉}{4}}{\frac{2-𝓉}{2}}+\frac{\cos \frac{\pi 𝓉}{4}}{e^{\frac{𝓉}{2}}}\right]d𝓉,\\ =\left(\frac{\pi \left(4e^{-1}-2^{\frac{3}{2}}{e}^{-\frac{\nu +1}{2}}\right)-2^{\frac{5}{2}}\nu e^{-\frac{\nu +1}{2}}+2^{\frac{5}{2}}{e}^{-\frac{\nu +1}{2}}}{4{\nu }^2-8\nu +{\pi }^2+4}+\frac{8\nu +2^{\frac{3}{2}}\pi e^{-\frac{\nu +1}{2}}-2^{\frac{5}{2}}\left(\nu +1\right)e^{-\frac{\nu +1}{2}}+8}{4{\nu }^2+8\nu +{\pi }^2+4}\right)\left[{{\rm Y}}_{*}\left(\mu , \theta \right)+{{\rm Y}}_{*}\left(b, \theta \right)\right].\end{array}$$

(73)

In a manner identical to that described previously, we have ${{\rm Y}}^{*}\left(., \theta \right)$

$$\begin{array}{l}{\int }_0^1{e}^{e^{-\frac{\nu 𝓉}{2}}}{{\rm Y}}^{*}\left(\frac{𝓉}{2}\mu +\frac{2-𝓉}{2}b, \theta \right)d𝓉+{\int }_0^1{e}^{e^{-\frac{\nu 𝓉}{2}}}{{\rm Y}}^{*}\left(\frac{2-𝓉}{2}\mu +\frac{𝓉}{2}b, \theta \right)d𝓉\\ \ge \left[{{\rm Y}}^{*}\left(\mu , \theta \right)+{{\rm Y}}^{*}\left(b, \theta \right)\right]{\int }_0^1{e}^{-\nu 𝓉}\left[\frac{\sin \frac{\pi 𝓉}{4}}{e^{\frac{2-𝓉}{2}}}+\frac{\cos \frac{\pi 𝓉}{4}}{e^{\frac{𝓉}{2}}}\right]d𝓉,\\ =\left(\frac{\pi \left(4e^{-1}-2^{\frac{3}{2}}{e}^{-\frac{\nu +1}{2}}\right)-2^{\frac{5}{2}}\nu e^{-\frac{\nu +1}{2}}+2^{\frac{5}{2}}{e}^{-\frac{\nu +1}{2}}}{4{\nu }^2-8\nu +{\pi }^2+4}+\frac{8\nu +2^{\frac{3}{2}}\pi e^{-\frac{\nu +1}{2}}-2^{\frac{5}{2}}\left(\nu +1\right)e^{-\frac{\nu +1}{2}}+8}{4{\nu }^2+8\nu +{\pi }^2+4}\right)\left[{{\rm Y}}^{*}\left(\mu , \theta \right)+{{\rm Y}}^{*}\left(b, \theta \right)\right].\end{array}$$

(74)

From (73) and (74), we have

$$\begin{array}{c}{\int }_0^1{e}^{-\nu 𝓉}{{\rm Y}}_{\theta }\left(\frac{𝓉}{2}\mu +\frac{2-𝓉}{2}b \right)d𝓉+{\int }_0^1{e}^{-\nu 𝓉}{{\rm Y}}_{\theta }\left(\frac{2-𝓉}{2}\mu +\frac{𝓉}{2}b \right)d𝓉\\ {\supseteq }_I\left(\frac{\pi \left(4e^{-1}-2^{\frac{3}{2}}{e}^{-\frac{\nu +1}{2}}\right)-2^{\frac{5}{2}}\nu e^{-\frac{\nu +1}{2}}+2^{\frac{5}{2}}{e}^{-\frac{\nu +1}{2}}}{4{\nu }^2-8\nu +{\pi }^2+4}+\frac{8\nu +2^{\frac{3}{2}}\pi e^{-\frac{\nu +1}{2}}-2^{\frac{5}{2}}\left(\nu +1\right)e^{-\frac{\nu +1}{2}}+8}{4{\nu }^2+8\nu +{\pi }^2+4}\right)\left[{{\rm Y}}_{\theta }\left(\mu \right)+{{\rm Y}}_{\theta }\left(b\right)\right].\end{array}$$

(75)

From (68) and (75), we have

$$\sqrt{\frac{e}{2}}{{\rm Y}}_{\theta }\left(\frac{\mu +b}{2}\right){\supseteq }_I\frac{1-\alpha }{2\left(1-e^{-\frac{\nu }{2}}\right)}\left[{ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{+}}^{\alpha }{{\rm Y}}_{\theta }\left(b\right)+{ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{-}}^{\alpha }{{\rm Y}}_{\theta }\left(\mu \right)\right]{\supseteq }_I\frac{\nu }{2\left(1-e^{-\frac{\nu }{2}}\right)}P\left(\nu \right)\frac{{{\rm Y}}_{\theta }\left(\mu \right)+{{\rm Y}}_{\theta }\left(b\right)}{2}.$$

that is

$$\sqrt{\frac{e}{2}}\odot \widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right){\supseteq }_{\mathbb{F}}\frac{1-\alpha }{2\left(1-e^{-\frac{\nu }{2}}\right)}\odot \left[{ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{+}}^{\alpha } \widetilde{{\rm Y}}\left(b\right)\oplus {ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{-}}^{\alpha } \widetilde{{\rm Y}}\left(\mu \right)\right]{\supseteq }_{\mathbb{F}}\frac{\nu }{2\left(1-e^{-\frac{\nu }{2}}\right)}P\left(\nu \right)\odot \frac{\widetilde{{\rm Y}}\left(\mu \right)\oplus \widetilde{{\rm Y}}\left(b\right)}{2}.$$

Hence, the required results. □

Remark 5.

From Theorem 7, we see that

If one lays $\widetilde{{\rm Y}}$ is a lower $UDET\text{-}$convex $FNVM$ on $\left[\mu ,b\right],$ then one acquires the following outcome (see [78]):

$$\sqrt{\frac{e}{2}}\odot \widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right){\le }_{\mathbb{F}}\frac{1-\alpha }{2\left(1-e^{-\frac{\nu }{2}}\right)}\odot \left[{ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{+}}^{\alpha } \widetilde{{\rm Y}}\left(b\right)\oplus {ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{-}}^{\alpha } \widetilde{{\rm Y}}\left(\mu \right)\right]{\le }_{\mathbb{F}}\frac{\nu }{2\left(1-e^{-\frac{\nu }{2}}\right)}P\left(\nu \right)\odot \frac{\widetilde{{\rm Y}}\left(\mu \right)\oplus \widetilde{{\rm Y}}\left(b\right)}{2}.$$

(76)

Let ${{\rm Y}}_{\ast }\left(\varkappa ,\theta \right)\ne {{\rm Y}}^{\ast }\left(\varkappa ,\theta \right)$ and $\theta =1$,$.$ Then, from Theorem 7, we have (see [78])

$$\sqrt{\frac{e}{2}}{\rm Y}\left(\frac{\mu +b}{2}\right)\supseteq \frac{1-\alpha }{2\left(1-e^{-\frac{\nu }{2}}\right)}\left[{ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{+}}^{\alpha } {\rm Y}\left(b\right)+{ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{-}}^{\alpha } {\rm Y}\left(\mu \right)\right]\supseteq \frac{\nu }{2\left(1-e^{-\frac{\nu }{2}}\right)}P\left(\nu \right)\frac{{\rm Y}\left(\mu \right)+{\rm Y}\left(b\right)}{2}.$$

(77)

If $\alpha \to 1$, that is

$$\underset{\alpha \to 1}{\lim }\nu =\underset{\alpha \to 1}{\lim }\frac{1-\alpha }{\alpha }\left(b-\mu \right)=0,$$

then

$$\begin{array}{c}\underset{\alpha \to 1}{\lim }\frac{\nu }{1-e^{-\frac{\nu }{2}}}\left(\frac{\pi \left(4e^{-1}-2^{\frac{3}{2}}{e}^{-\frac{\nu +1}{2}}\right)-2^{\frac{5}{2}}\nu e^{-\frac{\nu +1}{2}}+2^{\frac{5}{2}}{e}^{-\frac{\nu +1}{2}}}{4{\nu }^2-8\nu +{\pi }^2+4}+\frac{8\nu +2^{\frac{3}{2}}\pi e^{-\frac{\nu +1}{2}}-2^{\frac{5}{2}}\left(\nu +1\right)e^{-\frac{\nu +1}{2}}+8}{4{\nu }^2+8\nu +{\pi }^2+4}\right)=\frac{4\left(2\pi +4e\right)}{e\left({\pi }^2+4\right)}\\ \underset{\alpha \to 1}{\lim }\frac{1-\alpha }{2\left(1-e^{e^{-\frac{\nu }{2}}}\right)}=\frac{1}{b-\mu }.\end{array}$$

The next finding, which is likewise novel, is as follows:

$$\sqrt{\frac{e}{2}}\odot \widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right){\supseteq }_{\mathbb{F}}\frac{1}{b-\mu }\odot {\int }_{\mu }^b\widetilde{{\rm Y}}\left(\varkappa \right)d\varkappa {\supseteq }_{\mathbb{F}}\frac{2\pi +4e}{e\left({\pi }^2+4\right)}\odot \left[\widetilde{{\rm Y}}\left(\mu \right)\oplus \widetilde{{\rm Y}}\left(b\right)\right].$$

(78)

If one lays $\alpha \to 1,$ and $\widetilde{{\rm Y}}$ is a lower $UDET\text{-}$convex $FNVM$ on $\left[\mu ,b\right],$ then we have (see [78])

$$\sqrt{\frac{e}{2}}\odot \widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right){\le }_{\mathbb{F}}\frac{1}{b-\mu }\odot {\int }_{\mu }^b\widetilde{{\rm Y}}\left(\varkappa \right)d\varkappa {\le }_{\mathbb{F}}\frac{2\pi +4e}{e\left({\pi }^2+4\right)}\odot \left[\widetilde{{\rm Y}}\left(\mu \right)\oplus \widetilde{{\rm Y}}\left(b\right)\right].$$

(79)

Let $\alpha \to 1$ and ${{\rm Y}}_{*}\left(\varkappa , \theta \right)\ne {{\rm Y}}^{*}\left(\varkappa , \theta \right)$ with $\theta =1$. Then, from Theorem 7, we have (see [78])

$$\sqrt{\frac{e}{2}}{\rm Y}\left(\frac{\mu +b}{2}\right)\supseteq \frac{1}{b-\mu }{\int }_{\mu }^b{\rm Y}\left(\varkappa \right)d\varkappa \supseteq \frac{2\pi +4e}{e\left({\pi }^2+4\right)}\left[{\rm Y}\left(\mu \right)+{\rm Y}\left(b\right)\right].$$

(80)

If ${{\rm Y}}_{*}\left(\varkappa , \theta \right)={{\rm Y}}^{*}\left(\varkappa , \theta \right)$ and $\theta =1,$ then, from Theorem 7, we have

$$\sqrt{\frac{e}{2}}{\rm Y}\left(\frac{\mu +b}{2}\right)\le \frac{1-\alpha }{2\left(1-e^{-\frac{\nu }{2}}\right)}\left[{ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{+}}^{\alpha } {\rm Y}\left(b\right)+{ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{-}}^{\alpha } {\rm Y}\left(\mu \right)\right]\le \frac{\nu }{2\left(1-e^{-\frac{\nu }{2}}\right)}P\left(\nu \right)\frac{{\rm Y}\left(\mu \right)+{\rm Y}\left(b\right)}{2}.$$

(81)

Let $\alpha \to 1$ and ${{\rm Y}}_{*}\left(\varkappa , \theta \right)={{\rm Y}}^{*}\left(\varkappa , \theta \right)$ with $\theta =1$. Then, from Theorem 7, we have (see [73])

$$\sqrt{\frac{e}{2}}{\rm Y}\left(\frac{\mu +b}{2}\right)\le \frac{1}{b-\mu }{\int }_{\mu }^b{\rm Y}\left(\varkappa \right)d\varkappa \le \frac{2\pi +4e}{e\left({\pi }^2+4\right)}\left[{\rm Y}\left(\mu \right)+{\rm Y}\left(b\right)\right].$$

(82)

Here are some results on $H–H\text{-}$type inequalities for the products of $UDET\text{-}$convex FNVM, which are known as Pachpatte-type inequalities.

Theorem 8.

Let $\widetilde{{\rm Y}},\widetilde{{\rm T}} :\left[\mu , b\right]\to R_C$ be two $UDET-$convex $FNVM$s on $\left[\mu , b\right],$ with $\theta$ levels ${{\rm Y}}_{\theta }, {{\rm T}}_{\theta }:\left[\mu , b\right]\subset ℝ\to {\mathcal{X}}_C^{+}$ defined by ${{\rm Y}}_{\theta }\left(\varkappa \right)=\left[{{\rm Y}}_{*}\left(\varkappa ,\theta \right), {{\rm Y}}^{*}\left(\varkappa ,\theta \right)\right]$ and ${{\rm T}}_{\theta }\left(\varkappa \right)=\left[{{\rm T}}_{*}\left(\varkappa ,\theta \right), {{\rm T}}^{*}\left(\varkappa ,\theta \right)\right]$ for all $\varkappa \in \left[\mu , b\right]$ and for all $\theta \in \left[0, 1\right]$. If $\widetilde{{\rm Y}}\otimes \widetilde{{\rm T}}\in L\left(\left[\mu , b\right],R_C\right)$, then

$$\frac{\alpha }{b-\mu }\odot \left[{ℐ}_{{\mu }^{+}}^{\alpha } \widetilde{{\rm Y}}\left(b\right)\otimes \widetilde{{\rm T}}\left(b\right)\oplus {ℐ}_{b^{-}}^{\alpha }\widetilde{{\rm Y}}\left(\mu \right)\otimes \widetilde{{\rm T}}\left(\mu \right)\right]{\supseteq }_{\mathbb{F}}A\left(\nu \right)\odot \mathrm{\Delta }\left(\mu ,b\right)\oplus \frac{\pi e^{-\nu -1}\left(e^{\nu }+1\right)}{{\nu }^2+{\pi }^2}\odot \nabla \left(\mu ,b\right).$$

(83)

If $\widetilde{{\rm Y}}\left(\varkappa \right)$ is $UDET\text{-}$concave FNVM, then

$$\frac{\alpha }{b-\mu }\odot \left[{ℐ}_{{\mu }^{+}}^{\alpha } \widetilde{{\rm Y}}\left(b\right)\otimes \widetilde{{\rm T}}\left(b\right)\oplus {ℐ}_{b^{-}}^{\alpha }\widetilde{{\rm Y}}\left(\mu \right)\otimes \widetilde{{\rm T}}\left(\mu \right)\right]{\subseteq }_{\mathbb{F}}A\left(\nu \right)\odot \mathrm{\Delta }\left(\mu ,b\right)\oplus \frac{\pi e^{-\nu -1}\left(e^{\nu }+1\right)}{{\nu }^2+{\pi }^2}\odot \nabla \left(\mu ,b\right).$$

(84)

where

$$\begin{array}{c}\mathsf{A}\left(\nu \right)=-\frac{e^{-\nu -2}\left(8e^2-8{\nu e}^2+{\pi }^2{e}^2+2{\nu }^2{e}^2-{\pi }^2{e}^{\nu }\right)}{2\left(\nu -2\right)\left({\nu }^2-4\nu +{\pi }^2+4\right)}+\frac{8\nu -{\pi }^2{e}^{-\nu -2}+{\pi }^2+2{\nu }^2+8}{2\left(\nu +2\right)\left({\nu }^2+4\nu +{\pi }^2+4\right)}, \nu =\frac{1-\alpha }{\alpha }\left(b-\mu \right), 1>\alpha >0,\\ \mathrm{\Delta }\left(\mu ,b\right)=\widetilde{{\rm Y}}\left(\mu \right)\otimes \widetilde{{\rm T}}\left(\mu \right) \oplus \widetilde{{\rm Y}}\left(b\right)\otimes \widetilde{{\rm T}}\left(b\right),\\ \nabla \left(\mu ,b\right)=\widetilde{{\rm Y}}\left(\mu \right)\otimes \widetilde{{\rm T}}\left(b\right)\oplus \widetilde{{\rm Y}}\left(b\right)\otimes \widetilde{{\rm T}}\left(\mu \right),\end{array}$$

and

$${\mathrm{\Delta }}_{\theta }\left(\mu ,b\right)=\left[{\mathrm{\Delta }}_{*}\left(\left(\mu ,b\right), \theta \right), {\mathrm{\Delta }}^{*}\left(\left(\mu ,b\right), \theta \right)\right]$$

and

$${\nabla }_{\theta }\left(\mu ,b\right)=\left[{\nabla }_{*}\left(\left(\mu ,b\right), \theta \right), {\nabla }^{*}\left(\left(\mu ,b\right), \theta \right)\right].$$

Proof. 

Since $\widetilde{{\rm Y}}, \widetilde{{\rm T}}$ both are $UDET\text{-}$convex FNVMs, for each $\theta \in \left[0, 1\right]$,taking the left end points functions, we have

$$\begin{array}{c} \\ {{\rm Y}}_{\ast }\left(𝓉\mu +\left(1-𝓉\right)b, \theta \right)\le \frac{\sin \frac{\pi 𝓉}{2}}{e^{1-𝓉}}{{\rm Y}}_{\ast }\left(\mu , \theta \right)+\frac{\cos \frac{\pi 𝓉}{2}}{e^{𝓉}}{{\rm Y}}_{\ast }\left(b, \theta \right).\end{array}$$

and

$$\begin{array}{c} \\ {{\rm T}}_{\ast }\left(𝓉\mu +\left(1-𝓉\right)b, \theta \right)\le t\frac{\sin \frac{\pi 𝓉}{2}}{e^{1-𝓉}}{{\rm T}}_{\ast }\left(\mu , \theta \right)+\frac{\cos \frac{\pi 𝓉}{2}}{e^{𝓉}}{{\rm T}}_{\ast }\left(b, \theta \right). \end{array}$$

From the definition of $UDET\text{-}$convex $FNVM$s, it follows that $\tilde{0}{\le }_{\mathbb{F}}\widetilde{{\rm Y}}\left(\varkappa \right)$ and $\tilde{0}{\le }_{\mathbb{F}}\widetilde{{\rm T}}\left(\varkappa \right)$, so

$$\begin{array}{c}{{\rm Y}}_{\ast }\left(𝓉\mu +\left(1-𝓉\right)b, \theta \right)\times {{\rm T}}_{\ast }\left(𝓉\mu +\left(1-𝓉\right)b, \theta \right) \\ \le \left(\frac{\mathrm{s}\mathrm{i}\mathrm{n}\frac{\pi 𝓉}{2}}{e^{1-𝓉}}{{\rm Y}}_{\ast }\left(\mu ,\theta \right)+\frac{\mathrm{c}\mathrm{o}\mathrm{s}\frac{\pi 𝓉}{2}}{e^{𝓉}}{{\rm Y}}_{\ast }\left(b, \theta \right)\right)\left(\frac{\mathrm{s}\mathrm{i}\mathrm{n}\frac{\pi 𝓉}{2}}{e^{1-𝓉}}{{\rm T}}_{\ast }\left(\mu ,\theta \right)+\frac{\mathrm{c}\mathrm{o}\mathrm{s}\frac{\pi 𝓉}{2}}{e^{𝓉}}{{\rm T}}_{\ast }\left(b, \theta \right)\right)\\ ={\left(\frac{\mathrm{s}\mathrm{i}\mathrm{n}\frac{\pi 𝓉}{2}}{e^{1-𝓉}}\right)}^2{{\rm Y}}_{\ast }\left(\mu ,\theta \right)\times {{\rm T}}_{\ast }\left(\mu ,\theta \right)+{\left(\frac{\mathrm{c}\mathrm{o}\mathrm{s}\frac{\pi 𝓉}{2}}{e^{𝓉}}\right)}^2{{\rm Y}}_{\ast }\left(b, \theta \right)\times {{\rm T}}_{\ast }\left(b, \theta \right) \\ +\left(\frac{\mathrm{c}\mathrm{o}\mathrm{s}\frac{\pi 𝓉}{2}\mathrm{s}\mathrm{i}\mathrm{n}\frac{\pi 𝓉}{2}}{e}\right){{\rm Y}}_{\ast }\left(\mu ,\theta \right)\times {{\rm T}}_{\ast }\left(b, \theta \right)+\left(\frac{\mathrm{c}\mathrm{o}\mathrm{s}\frac{\pi 𝓉}{2}\mathrm{s}\mathrm{i}\mathrm{n}\frac{\pi 𝓉}{2}}{e}\right){{\rm Y}}_{\ast }\left(b, \theta \right)\times {{\rm T}}_{\ast }\left(\mu ,\theta \right).\end{array}$$

(85)

Analogously, we have

$$\begin{array}{c}{{\rm Y}}_{\ast }\left(\left(1-𝓉\right)\mu +𝓉b, \theta \right){{\rm T}}_{\ast }\left(\left(1-𝓉\right)\mu +𝓉b, \theta \right)\\ \le {\left(\frac{\mathrm{c}\mathrm{o}\mathrm{s}\frac{\pi 𝓉}{2}}{e^{𝓉}}\right)}^2{{\rm Y}}_{\ast }\left(\mu ,\theta \right)\times {{\rm T}}_{\ast }\left(\mu ,\theta \right)+{\left(\frac{\mathrm{s}\mathrm{i}\mathrm{n}\frac{\pi 𝓉}{2}}{e^{1-𝓉}}\right)}^2{{\rm Y}}_{\ast }\left(b, \theta \right)\times {{\rm T}}_{\ast }\left(b, \theta \right)\\ +\left(\frac{\mathrm{c}\mathrm{o}\mathrm{s}\frac{\pi 𝓉}{2}\mathrm{s}\mathrm{i}\mathrm{n}\frac{\pi 𝓉}{2}}{e}\right){{\rm Y}}_{\ast }\left(\mu ,\theta \right)\times {{\rm T}}_{\ast }\left(b, \theta \right)+\left(\frac{\mathrm{c}\mathrm{o}\mathrm{s}\frac{\pi 𝓉}{2}\mathrm{s}\mathrm{i}\mathrm{n}\frac{\pi 𝓉}{2}}{e}\right){{\rm Y}}_{\ast }\left(b, \theta \right)\times {{\rm T}}_{\ast }\left(\mu ,\theta \right).\end{array}$$

(86)

Adding (85) and (86), we have

$$\begin{array}{c}{{\rm Y}}_{\ast }\left(𝓉\mu +\left(1-𝓉\right)b, \theta \right)\times {{\rm T}}_{\ast }\left(𝓉\mu +\left(1-𝓉\right)b, \theta \right)\\ \hspace{1em} +{{\rm Y}}_{\ast }\left(\left(1-𝓉\right)\mu +𝓉b, \theta \right)\times {{\rm T}}_{\ast }\left(\left(1-𝓉\right)\mu +𝓉b, \theta \right) \\ \le \left[{\left(\frac{\mathrm{s}\mathrm{i}\mathrm{n}\frac{\pi 𝓉}{2}}{e^{1-𝓉}}\right)}^2+{\left(\frac{\mathrm{s}\mathrm{i}\mathrm{n}\frac{\pi 𝓉}{2}}{e^{1-𝓉}}\right)}^2\right]\left[{{\rm Y}}_{\ast }\left(\mu ,\theta \right)\times {{\rm T}}_{\ast }\left(\mu ,\theta \right)+{{\rm Y}}_{\ast }\left(b, \theta \right)\times {{\rm T}}_{\ast }\left(b, \theta \right)\right]\\ +\frac{2\mathrm{c}\mathrm{o}\mathrm{s}\frac{\pi 𝓉}{2}\mathrm{s}\mathrm{i}\mathrm{n}\frac{\pi 𝓉}{2}}{e}\left[{{\rm Y}}_{\ast }\left(b, \theta \right)\times {{\rm T}}_{\ast }\left(\mu ,\theta \right)+{{\rm Y}}_{\ast }\left(\mu ,\theta \right)\times {{\rm T}}_{\ast }\left(b, \theta \right)\right].\end{array}$$

(87)

Taking the multiplication of (87) by $e^{-\nu 𝓉}$ and integrating the obtained result with respect to $𝓉$ over (0,1), we have

$$\begin{array}{c}{\int }_0^1{e}^{-\nu 𝓉}{{\rm Y}}_{\ast }\left(𝓉\mu +\left(1-𝓉\right)b, \theta \right)\times {{\rm T}}_{\ast }\left(𝓉\mu +\left(1-𝓉\right)b, \theta \right)\\ +e^{-\nu 𝓉}{{\rm Y}}_{\ast }\left(\left(1-𝓉\right)\mu +𝓉b, \theta \right)\times {{\rm T}}_{\ast }\left(\left(1-𝓉\right)\mu +𝓉b, \theta \right)dt\\ \le {\mathrm{\Delta }}_{\ast }\left(\left(\mu ,b\right), \theta \right){\int }_0^1{e}^{-\nu 𝓉}\left[{\left(\frac{\mathrm{s}\mathrm{i}\mathrm{n}\frac{\pi 𝓉}{2}}{e^{1-𝓉}}\right)}^2+{\left(\frac{\mathrm{s}\mathrm{i}\mathrm{n}\frac{\pi 𝓉}{2}}{e^{1-𝓉}}\right)}^2\right]dt+2{\nabla }_{\ast }\left(\left(\mu ,b\right), \theta \right){\int }_0^1{e}^{-\nu 𝓉}\frac{\mathrm{c}\mathrm{o}\mathrm{s}\frac{\pi 𝓉}{2}\mathrm{s}\mathrm{i}\mathrm{n}\frac{\pi 𝓉}{2}}{e}d𝓉.\end{array}$$

It follows that

$$\begin{array}{c}\frac{\alpha }{b-\mu }\left[{\mathcal{I}}_{{\mu }^{+}}^{\alpha } {{\rm Y}}_{\ast }\left(b, \theta \right)\times {{\rm T}}_{\ast }\left(b, \theta \right)+{\mathcal{I}}_{b^{-}}^{\alpha } {{\rm Y}}_{\ast }\left(\mu ,\theta \right)\times {{\rm T}}_{\ast }\left(\mu ,\theta \right)\right] \\ \le A\left(\nu \right){\mathrm{\Delta }}_{\ast }\left(\left(\mu ,b\right), \theta \right)+\frac{\pi e^{-\nu -1}\left(e^{\nu }+1\right)}{{\nu }^2+{\pi }^2}{\nabla }_{\ast }\left(\left(\mu ,b\right), \theta \right).\\ \end{array}$$

(88)

Similarly, for ${{\rm Y}}^{*}\left(\varkappa ,\theta \right)$, we have

$$\frac{\alpha }{b-\mu }\left[{ℐ}_{{\mu }^{+}}^{\alpha } {{\rm Y}}^{*}\left(b, \theta \right)\times {{\rm T}}^{*}\left(b, \theta \right)+{ℐ}_{b^{-}}^{\alpha } {{\rm Y}}^{*}\left(\mu ,\theta \right)\times {{\rm T}}^{*}\left(\mu ,\theta \right)\right]\ge A\left(\nu \right){\mathrm{\Delta }}^{*}\left(\left(\mu ,b\right), \theta \right)+\frac{\pi e^{-\nu -1}\left(e^{\nu }+1\right)}{{\nu }^2+{\pi }^2}{\nabla }^{*}\left(\left(\mu ,b\right), \theta \right),$$

(89)

where

$$A\left(\nu \right)={\int }_0^1{e}^{-\nu 𝓉}\left[{\left(\frac{\sin \frac{\pi 𝓉}{2}}{e^{1-𝓉}}\right)}^2+{\left(\frac{\sin \frac{\pi 𝓉}{2}}{e^{1-𝓉}}\right)}^2\right]d𝓉=-\frac{e^{-\nu -2}\left(8e^2-8\nu e^2+{\pi }^2{e}^2+2{\nu }^2{e}^2-{\pi }^2{e}^{\nu }\right)}{2\left(\nu -2\right)\left({\nu }^2-4\nu +{\pi }^2+4\right)}+\frac{8\nu -{\pi }^2{e}^{-\nu -2}+{\pi }^2+2{\nu }^2+8}{2\left(\nu +2\right)\left({\nu }^2+4\nu +{\pi }^2+4\right)},$$

and

$${\int }_0^1{e}^{-\nu 𝓉}\frac{\cos \frac{\pi 𝓉}{2} \sin \frac{\pi 𝓉}{2}}{e}d𝓉=\frac{\pi e^{-\nu -1}\left(e^{\nu }+1\right)}{{\nu }^2+{\pi }^2}.$$

From (88) and (89), we have

$$\begin{array}{c}\frac{\alpha }{b-\mu }\left[{ℐ}_{{\mu }^{+}}^{\alpha } {{\rm Y}}_{*}\left(b, \theta \right)\times {{\rm T}}_{*}\left(b, \theta \right)+{ℐ}_{b^{-}}^{\alpha } {{\rm Y}}_{*}\left(\mu ,\theta \right)\times {{\rm T}}_{*}\left(\mu ,\theta \right), {ℐ}_{{\mu }^{+}}^{\alpha } {{\rm Y}}^{*}\left(b, \theta \right)\times {{\rm T}}^{*}\left(b, \theta \right)+{ℐ}_{b^{-}}^{\alpha } {{\rm Y}}^{*}\left(\mu ,\theta \right)\times {{\rm T}}^{*}\left(\mu , \theta \right)\right]\\ {\supseteq }_{I}A\left(\nu \right)\left[{\mathrm{\Delta }}_{*}\left(\left(\mu ,b\right), \theta \right), {\mathrm{\Delta }}^{*}\left(\left(\mu ,b\right), \theta \right)\right]+\frac{\pi e^{-\nu -1}\left(e^{\nu }+1\right)}{{\nu }^2+{\pi }^2}\left[{\nabla }_{*}\left(\left(\mu ,b\right), \theta \right), {\nabla }^{*}\left(\left(\mu ,b\right), \theta \right)\right].\end{array}$$

That is

$$\frac{\alpha }{b-\mu }\odot \left[{ℐ}_{{\mu }^{+}}^{\alpha } \widetilde{{\rm Y}}\left(b\right)\otimes \widetilde{{\rm T}}\left(b\right)\oplus {ℐ}_{b^{-}}^{\alpha }\widetilde{{\rm Y}}\left(\mu \right)\otimes \widetilde{{\rm T}}\left(\mu \right)\right]{\supseteq }_{\mathbb{F}}A\left(\nu \right)\odot \mathrm{\Delta }\left(\mu ,b\right)\oplus \frac{\pi e^{-\nu -1}\left(e^{\nu }+1\right)}{{\nu }^2+{\pi }^2}\odot \nabla \left(\mu ,b\right).$$

and the theorem has been established. □

Remark 6.

We readily see from Theorem 8 that

If one lays $\widetilde{{\rm Y}}$ is a lower $UDET\text{-}$convex $FNVM$ on $\left[\mu ,b\right],$ then one obtains (see [78])

$$\frac{\alpha }{b-\mu }\odot \left[{ℐ}_{{\mu }^{+}}^{\alpha } \widetilde{{\rm Y}}\left(b\right)\otimes \widetilde{{\rm T}}\left(b\right)\oplus {ℐ}_{b^{-}}^{\alpha }\widetilde{{\rm Y}}\left(\mu \right)\otimes \widetilde{{\rm T}}\left(\mu \right)\right]{\le }_{\mathbb{F}}A\left(\nu \right)\odot \mathrm{\Delta }\left(\mu ,b\right)\oplus \frac{\pi e^{-\nu -1}\left(e^{\nu }+1\right)}{{\nu }^2+{\pi }^2}\odot \nabla \left(\mu ,b\right).$$

(90)

Let ${{\rm Y}}_{\ast }\left(\varkappa ,\theta \right)\ne {{\rm Y}}^{\ast }\left(\varkappa ,\theta \right)$ and $\theta =1$,$.$ Then, from Theorem 8, we have (see [86])

$$\frac{\alpha }{b-\mu }\left[{ℐ}_{{\mu }^{+}}^{\alpha } {\rm Y}\left(b\right)\times {\rm T}\left(b\right)+{ℐ}_{b^{-}}^{\alpha }{\rm Y}\left(\mu \right)\times {\rm T}\left(\mu \right)\right]\supseteq A\left(\nu \right)\mathrm{\Delta }\left(\mu ,b\right)+\frac{\pi e^{-\nu -1}\left(e^{\nu }+1\right)}{{\nu }^2+{\pi }^2}\nabla \left(\mu ,b\right).$$

(91)

If $\alpha \to 1$, that is

$$\underset{\alpha \to 1}{\lim }\nu =\underset{\alpha \to 1}{\lim }\frac{1-\alpha }{\alpha }\left(b-\mu \right)=0,$$

then

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

The next finding, which is likewise novel, is as follows:

$$\frac{1}{b-\mu }\odot {\int }_{\mu }^b\widetilde{{\rm Y}}\left(x\right)\otimes \widetilde{{\rm T}}\left(x\right)d\varkappa {\supseteq }_{\mathbb{F}}\frac{{\pi }^2-{\pi }^2{e}^2+8}{4\left({\pi }^2+4\right)}\odot \mathrm{\Delta }\left(\mu ,b\right)\oplus \frac{2}{\pi e}\odot \nabla \left(\mu ,b\right).$$

(92)

If one lays $\alpha \to 1$ and $\widetilde{{\rm Y}}$ is a lower $UDET\text{-}$convex $FNVM$ on $\left[\mu ,b\right],$ then one can get (see [78])

$$\frac{1}{b-\mu }\odot {\int }_{\mu }^b\widetilde{{\rm Y}}\left(x\right)\otimes \widetilde{{\rm T}}\left(x\right)d\varkappa {\le }_{\mathbb{F}}\frac{{\pi }^2-{\pi }^2{e}^2+8}{4\left({\pi }^2+4\right)}\odot \mathrm{\Delta }\left(\mu ,b\right)\oplus \frac{2}{\pi e}\odot \nabla \left(\mu ,b\right).$$

(93)

Let $\alpha \to 1$ and ${{\rm Y}}_{*}\left(\varkappa , \theta \right)\ne {{\rm Y}}^{*}\left(\varkappa , \theta \right)$ with $\theta =1$. Then, from Theorem 8, we have (see [86])

$$\frac{1}{b-\mu }{\int }_{\mu }^b{\rm Y}\left(x\right)\times {\rm T}\left(x\right)d\varkappa \supseteq \frac{{\pi }^2-{\pi }^2{e}^2+8}{4\left({\pi }^2+4\right)}\mathrm{\Delta }\left(\mu ,b\right)+\frac{2}{\pi e}\nabla \left(\mu ,b\right).$$

(94)

If ${{\rm Y}}_{*}\left(\varkappa , \theta \right)={{\rm Y}}^{*}\left(\varkappa , \theta \right)$ and $\theta =1,$ then, from Theorem 8, we achieve

$$\frac{\alpha }{b-\mu }\left[{ℐ}_{{\mu }^{+}}^{\alpha } {\rm Y}\left(b\right)\times {\rm T}\left(b\right)+{ℐ}_{b^{-}}^{\alpha }{\rm Y}\left(\mu \right)\times {\rm T}\left(\mu \right)\right]\le A\left(\nu \right)\mathrm{\Delta }\left(\mu ,b\right)+\frac{\pi e^{-\nu -1}\left(e^{\nu }+1\right)}{{\nu }^2+{\pi }^2}\nabla \left(\mu ,b\right).$$

(95)

Let $\alpha \to 1$ and ${{\rm Y}}_{*}\left(\varkappa , \theta \right)={{\rm Y}}^{*}\left(\varkappa , \theta \right)$ with $\theta =1$. Then, from Theorem 8, we have (see [73])

$$\frac{1}{b-\mu }{\int }_{\mu }^b{\rm Y}\left(x\right)\times {\rm T}\left(x\right)d\varkappa \le \frac{{\pi }^2-{\pi }^2{e}^2+8}{4\left({\pi }^2+4\right)}\mathrm{\Delta }\left(\mu ,b\right)+\frac{2}{\pi e}\nabla \left(\mu ,b\right).$$

(96)

Example 2.

Let $\left[\mu ,b\right]=\left[0,1\right]$, $\alpha =\frac{1}{4}$ and

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

(97)

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

(98)

Then, for each $\theta \in \left[0, 1\right],$ we have ${{\rm Y}}_{\theta }\left(x\right)=\left[\left(1-\theta \right)x^2+\frac{3}{2}\theta ,\left(1-\theta \right)\left(x+1\right)+\frac{3}{2}\theta \right]$ and $T_{\theta }\left(x\right)=\left[2\left(1-\theta \right)x^3+3\theta ,\left(1-\theta \right)\left(1+e^x\right)+3\theta \right].$ It can be easily seen that, for each $\theta \in \left[0, 1\right]$, the left and right end point functions ${{\rm Y}}_{*}\left(x,\theta \right)=\left(1-\theta \right)x^2+\frac{3}{2}\theta ,$ ${{\rm Y}}^{*}\left(x, \theta \right)=\left(1-\theta \right)\left(x+1\right)+\frac{3}{2}\theta$, $T_{*}\left(x,\theta \right)=2\left(1-\theta \right)x^3+3\theta ,$ and $T^{*}\left(x, \theta \right)=\left(1-\theta \right)\left(1+e^x\right)+3\theta$ are exponential trigonometric convex and concave functions; then, ${\rm Y}\left(x\right)$ and $T\left(x\right)$ both are $UDET-$convex $FNVM$s. We readily see that $\widetilde{{\rm Y}}\left(x\right)\otimes \widetilde{{\rm T}}\left(x\right)\in L\left(\left[\mu , b\right],R_C\right)$ and $\nu =\frac{1-\alpha }{\alpha }\left(b-\mu \right)=3$.

$$\begin{array}{l}\frac{\alpha }{b-\mu }\left[{\mathcal{I}}_{{\mu }^{+}}^{\alpha } {{\rm Y}}_{\ast }\left(b, \theta \right)\times T_{\ast }\left(b, \theta \right)+{\mathcal{I}}_{b^{-}}^{\alpha } {{\rm Y}}_{\ast }\left(\mu , \theta \right)\times T_{\ast }\left(\mu , \theta \right)\right]\\ ={\int }_0^1{e}^{-3\left(1-x\right)}\left(\left(1-\theta \right)x^2+\frac{3}{2}\theta \right)\left(2\left(1-\theta \right)x^3+3\theta \right)dx\\ +{\int }_0^1{e}^{-3x}\left(\left(1-\theta \right)x^2+\frac{3}{2}\theta \right)\left(2\left(1-\theta \right)x^3+3\theta \right)dx\\ =\frac{\left(1020{\theta }^2+174\theta +264\right)e^3-1920{\theta }^2+3246\theta -2784}{486e^3}\end{array}$$

$$\begin{array}{l}\frac{\alpha }{b-\mu }\left[{\mathcal{I}}_{{\mu }^{+}}^{\alpha } {{\rm Y}}^{\ast }\left(b, \theta \right)\times T^{\ast }\left(b, \theta \right)+{\mathcal{I}}_{b^{-}}^{\alpha } {{\rm Y}}^{\ast }\left(\mu , \theta \right)\times T^{\ast }\left(\mu , \theta \right)\right]\\ ={\int }_0^1{e}^{-3\left(1-x\right)}\left(\left(1-\theta \right)\left(x+1\right)+\frac{3}{2}\theta \right)\left(\left(1-\theta \right)\left(1+e^x\right)+3\theta \right)dx\\ +{\int }_0^1{e}^{-3x}\left(\left(1-\theta \right)\left(x+1\right)+\frac{3}{2}\theta \right)\left(\left(1-\theta \right)\left(1+e^x\right)+3\theta \right)dx\\ =\frac{-9e^4\left(\theta -1\right)\left(\theta -7\right)+4e^3\left(8{\theta }^2-31\theta +23\right)-36e\left(2{\theta }^2-7\theta +5\right)+133{\theta }^2-80\theta -37}{144e^3}\end{array}$$

Note that

$$\begin{array}{c}A\left(\nu \right){\mathrm{\Delta }}_{*}\left(\left(\mu ,b\right), \theta \right)=\left(-\frac{e^{-5}\left(2e^2+{\pi }^2{e}^2-{\pi }^2{e}^3\right)}{2\left(1+{\pi }^2\right)}+\frac{50-{\pi }^2{e}^{-5}+{\pi }^2}{10\left(25+{\pi }^2\right)}\right)\left[{{\rm Y}}_{*}\left(0, \theta \right)\times T_{*}\left(0, \theta \right)+{{\rm Y}}_{*}\left(1, \theta \right)\times T_{*}\left(1\right)\right]\\ =\left(\frac{50-{\pi }^2{e}^{-5}+{\pi }^2}{10\left(25+{\pi }^2\right)}-\frac{e^{-5}\left(2e^2+{\pi }^2{e}^2-{\pi }^2{e}^3\right)}{2\left(1+{\pi }^2\right)}\right)\left(5{\theta }^2+2\theta +2\right),\end{array}$$

$$\begin{array}{c}A\left(\nu \right){\mathrm{\Delta }}^{*}\left(\left(\mu ,b\right), \theta \right)=\left(-\frac{e^{-5}\left(2e^2+{\pi }^2{e}^2-{\pi }^2{e}^3\right)}{2\left(1+{\pi }^2\right)}+\frac{50-{\pi }^2{e}^{-5}+{\pi }^2}{10\left(25+{\pi }^2\right)}\right)\left[{{\rm Y}}^{*}\left(0, \theta \right)\times T^{*}\left(0, \theta \right)+{{\rm Y}}^{*}\left(1, \theta \right)\times T^{*}\left(1, \theta \right)\right]\\ =\left(\frac{50-{\pi }^2{e}^{-5}+{\pi }^2}{10\left(25+{\pi }^2\right)}-\frac{e^{-5}\left(2e^2+{\pi }^2{e}^2-{\pi }^2{e}^3\right)}{2\left(1+{\pi }^2\right)}\right)\left(e{\theta }^2-{\theta }^2-5e\theta +11\theta +4e+8\right),\end{array}$$

$$\frac{\pi e^{-\nu -1}\left(e^{\nu }+1\right)}{{\nu }^2+{\pi }^2}{\nabla }_{*}\left(\left(\mu ,b\right), \theta \right)=\frac{\pi e^{-\nu -1}\left(e^{\nu }+1\right)}{{\nu }^2+{\pi }^2}\left[{{\rm Y}}_{*}\left(0, \theta \right)\times T_{*}\left(1, \theta \right)+{{\rm Y}}_{*}\left(1, \theta \right)\times T_{*}\left(0, \theta \right)\right]=3\theta \left(\theta +2\right),$$

$$\begin{array}{c}\frac{\pi e^{-\nu -1}\left(e^{\nu }+1\right)}{{\nu }^2+{\pi }^2}{\nabla }_{*}\left(\left(\mu ,b\right), \theta \right)=\frac{\pi e^{-\nu -1}\left(e^{\nu }+1\right)}{{\nu }^2+{\pi }^2}\left[{{\rm Y}}^{*}\left(1, \theta \right)\times T^{*}\left(1, \theta \right)+{{\rm Y}}^{*}\left(1, \theta \right)\times T^{*}\left(0, \theta \right)\right]\\ =\frac{1}{2}\frac{\pi e^{-\nu -1}\left(e^{\nu }+1\right)}{{\nu }^2+{\pi }^2}\left(\theta +2\right)\left(5+e+\theta -\theta e\right).\end{array}$$

Therefore, we have

$$\begin{array}{c}A\left(\nu \right){\mathrm{\Delta }}_{\theta }\left(\mu ,b\right)+\frac{\pi e^{-\nu -1}\left(e^{\nu }+1\right)}{{\nu }^2+{\pi }^2}{\nabla }_{\theta }\left(\mu ,b\right)\\ =\left(\frac{50-{\pi }^2{e}^{-5}+{\pi }^2}{10\left(25+{\pi }^2\right)}-\frac{e^{-5}\left(2e^2+{\pi }^2{e}^2-{\pi }^2{e}^3\right)}{2\left(1+{\pi }^2\right)}\right)\left[5{\theta }^2+2\theta +2,e{\theta }^2-{\theta }^2-5e\theta +11\theta +4e+8\right]\\ +\frac{\pi e^{-\nu -1}\left(e^{\nu }+1\right)}{{\nu }^2+{\pi }^2}\left[3\theta \left(\theta +2\right),\frac{1}{2}\left(\theta +2\right)\left(5+e+\theta -\theta e\right)\right]\end{array}$$

It follows that

$$\begin{array}{c}\left[\begin{array}{c}\frac{\left(1020{\theta }^2+174\theta +264\right)e^3-1920{\theta }^2+3246\theta -2784}{486e^3},\\ \frac{-9e^4\left(\theta -1\right)\left(\theta -7\right)+4e^3\left(8{\theta }^2-31\theta +23\right)-36e\left(2{\theta }^2-7\theta +5\right)+133{\theta }^2-80\theta -37}{144e^3}\end{array}\right]\\ {\supseteq }_I\left(\frac{50-{\pi }^2{e}^{-5}+{\pi }^2}{10\left(25+{\pi }^2\right)}-\frac{e^{-5}\left(2e^2+{\pi }^2{e}^2-{\pi }^2{e}^3\right)}{2\left(1+{\pi }^2\right)}\right)\left[5{\theta }^2+2\theta +2,e{\theta }^2-{\theta }^2-5e\theta +11\theta +4e+8\right]\\ +\frac{\pi e^{-\nu -1}\left(e^{\nu }+1\right)}{{\nu }^2+{\pi }^2}\left[3\theta \left(\theta +2\right),\frac{1}{2}\left(\theta +2\right)\left(5+e+\theta -\theta e\right)\right]\end{array}$$

and Theorem 8 has been validated.

Theorem 9.

Let $\widetilde{{\rm Y}},\widetilde{{\rm T}} :\left[\mu , b\right]\to R_C$ be two $UDET\text{-}$convex $FNVM$s, with $\theta$ levels defined by the family of i-v-ms given by ${{\rm Y}}_{\theta }\left(\varkappa \right)=\left[{{\rm Y}}_{*}\left(\varkappa ,\theta \right), {{\rm Y}}^{*}\left(\varkappa ,\theta \right)\right]$ and ${{\rm T}}_{\theta }\left(\varkappa \right)=\left[{{\rm T}}_{*}\left(\varkappa ,\theta \right), {{\rm T}}^{*}\left(\varkappa ,\theta \right)\right]$ for all $\varkappa \in \left[\mu , b\right]$ and for all $\theta \in \left[0, 1\right]$. If $\widetilde{{\rm Y}}\otimes {\rm T}\in L\left(\left[\mu , b\right],R_C\right)$, then

$$\begin{array}{c}2\odot \widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right)\otimes \widetilde{{\rm T}}\left(\frac{\mu +b}{2}\right){\supseteq }_{\mathbb{F}}\frac{1-\alpha }{e\left(1-e^{-\nu }\right)}\odot \left[{ℐ}_{{\mu }^{+}}^{\alpha } \widetilde{{\rm Y}}\left(b\right)\otimes \widetilde{{\rm T}}\left(b\right)\oplus {ℐ}_{b^{-}}^{\alpha } \widetilde{{\rm Y}}\left(\mu \right)\otimes \widetilde{{\rm T}}\left(\mu \right)\right]\\ +\frac{\nu \pi e^{-\nu -1}\left(e^{\nu }+1\right)}{e\left(1-e^{-\nu }\right)\left({\nu }^2+{\pi }^2\right)}\odot \nabla \left(\mu ,b\right)\oplus \frac{\nu }{e\left(1-e^{-\nu }\right)}A\left(\nu \right)\odot \mathrm{\Delta }\left(\mu ,b\right). \end{array}$$

(99)

If $\widetilde{{\rm Y}}\left(\varkappa \right)$ is $UDET\text{-}$concave $FNVM$, then

$$\begin{array}{c}2\odot \widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right)\otimes \widetilde{{\rm T}}\left(\frac{\mu +b}{2}\right){\subseteq }_{\mathbb{F}}\frac{1-\alpha }{e\left(1-e^{-\nu }\right)}\odot \left[{ℐ}_{{\mu }^{+}}^{\alpha } \widetilde{{\rm Y}}\left(b\right)\otimes \widetilde{{\rm T}}\left(b\right)\oplus {ℐ}_{b^{-}}^{\alpha } \widetilde{{\rm Y}}\left(\mu \right)\otimes \widetilde{{\rm T}}\left(\mu \right)\right]\\ +\frac{\nu \pi e^{-\nu -1}\left(e^{\nu }+1\right)}{e\left(1-e^{-\nu }\right)\left({\nu }^2+{\pi }^2\right)}\odot \nabla \left(\mu ,b\right)\oplus \frac{\nu }{e\left(1-e^{-\nu }\right)}A\left(\nu \right)\odot \mathrm{\Delta }\left(\mu ,b\right).\end{array}$$

(100)

where

$$A\left(\nu \right)=-\frac{e^{-\nu -2}\left(8e^2-8\nu e^2+{\pi }^2{e}^2+2{\nu }^2{e}^2-{\pi }^2{e}^{\nu }\right)}{2\left(\nu -2\right)\left({\nu }^2-4\nu +{\pi }^2+4\right)}+\frac{8\nu -{\pi }^2{e}^{-\nu -2}+{\pi }^2+2{\nu }^2+8}{2\left(\nu +2\right)\left({\nu }^2+4\nu +{\pi }^2+4\right)}, \nu =\frac{1-\alpha }{\alpha }\left(b-\mu \right), 1>\alpha >0,$$

and

$$\begin{gathered} \mathrm{\Delta }\left(\mu ,b\right)=\widetilde{{\rm Y}}\left(\mu \right)\otimes \widetilde{{\rm T}}\left(\mu \right)\oplus \widetilde{{\rm Y}}\left(b\right)\otimes \widetilde{{\rm T}}\left(b\right), \\ \nabla \left(\mu ,b\right)=\widetilde{{\rm Y}}\left(\mu \right)\otimes \widetilde{{\rm T}}\left(b\right)\oplus \widetilde{{\rm Y}}\left(b\right)\otimes \widetilde{{\rm T}}\left(\mu \right), \end{gathered}$$

and

$${\mathrm{\Delta }}_{\theta }\left(\mu ,b\right)=\left[{\mathrm{\Delta }}_{*}\left(\left(\mu ,b\right), \theta \right), {\mathrm{\Delta }}^{*}\left(\left(\mu ,b\right), \theta \right)\right]$$

and

$${\nabla }_{\theta }\left(\mu ,b\right)=\left[{\nabla }_{*}\left(\left(\mu ,b\right), \theta \right), {\nabla }^{*}\left(\left(\mu ,b\right), \theta \right)\right].$$

Proof. 

Consider $\widetilde{{\rm Y}},\widetilde{{\rm T}} :\left[\mu , b\right]\to R_C$ are $UDET\text{-}$convex $FNVM$s. Then, by hypothesis, for each $\theta \in \left[0, 1\right],$ we have

$$\begin{array}{l}{{\rm Y}}_{\ast }\left(\frac{\mu +b}{2} ,\theta \right)\times {{\rm T}}_{\ast }\left(\frac{\mu +b}{2},\theta \right)\\ \le \frac{1}{2e}\left[\begin{array}{c}{{\rm Y}}_{\ast }\left(𝓉\mu +\left(1-𝓉\right)b, \theta \right)\times {{\rm T}}_{\ast }\left(𝓉\mu +\left(1-𝓉\right)b, \theta \right)\\ +{{\rm Y}}_{\ast }\left(\left(1-𝓉\right)\mu +\left(1-𝓉\right)b, \theta \right)\times {{\rm T}}_{\ast }\left(\left(1-𝓉\right)\mu +𝓉b, \theta \right)\end{array}\right] \\ +\frac{1}{2e}\left[\begin{array}{c}{{\rm Y}}_{\ast }\left(\left(1-𝓉\right)\mu +𝓉b, \theta \right){\times {\rm T}}_{\ast }\left(𝓉\mu +\left(1-𝓉\right)b, \theta \right)\\ +{{\rm Y}}_{\ast }\left(𝓉\mu +\left(1-𝓉\right)b, \theta \right){\times {\rm T}}_{\ast }\left(\left(1-𝓉\right)\mu +𝓉b, \theta \right)\end{array}\right] \\ \le \frac{1}{2e}\left[\begin{array}{c}{{\rm Y}}_{\ast }\left(𝓉\mu +\left(1-𝓉\right)b, \theta \right){\times {\rm T}}_{\ast }\left(𝓉\mu +\left(1-𝓉\right)b, \theta \right)\\ +{{\rm Y}}_{\ast }\left(\left(1-𝓉\right)\mu +𝓉b, \theta \right){\times {\rm T}}_{\ast }\left(\left(1-𝓉\right)\mu +𝓉b, \theta \right)\end{array}\right] \\ +\frac{1}{2e}\left[\begin{array}{c}\left(\frac{\mathrm{c}\mathrm{o}\mathrm{s}\frac{\pi 𝓉}{2}}{e^{𝓉}}{{\rm Y}}_{\ast }\left(\mu , \theta \right)+\frac{\mathrm{s}\mathrm{i}\mathrm{n}\frac{\pi 𝓉}{2}}{e^{1-𝓉}}{{\rm Y}}_{\ast }\left(b, \theta \right)\right)\\ \times \left(\frac{\mathrm{s}\mathrm{i}\mathrm{n}\frac{\pi 𝓉}{2}}{e^{1-𝓉}}{{\rm T}}_{\ast }\left(\mu , \theta \right)+\frac{\mathrm{c}\mathrm{o}\mathrm{s}\frac{\pi 𝓉}{2}}{e^{𝓉}}{{\rm T}}_{\ast }\left(b, \theta \right)\right)\\ +\left({\frac{\mathrm{s}\mathrm{i}\mathrm{n}\frac{\pi 𝓉}{2}}{e^{1-𝓉}}{\rm Y}}_{\ast }\left(\mu , \theta \right)+\frac{\mathrm{c}\mathrm{o}\mathrm{s}\frac{\pi 𝓉}{2}}{e^{𝓉}}{{\rm Y}}_{\ast }\left(b, \theta \right)\right)\\ \times \left(\frac{\mathrm{c}\mathrm{o}\mathrm{s}\frac{\pi 𝓉}{2}}{e^{𝓉}}{{\rm T}}_{\ast }\left(\mu , \theta \right)+\frac{\mathrm{s}\mathrm{i}\mathrm{n}\frac{\pi 𝓉}{2}}{e^{1-𝓉}}{{\rm T}}_{\ast }\left(b, \theta \right)\right)\end{array}\right] \\ =\frac{1}{2e}\left[\begin{array}{c}{{\rm Y}}_{\ast }\left(𝓉\mu +\left(1-𝓉\right)b, \theta \right){\times {\rm T}}_{\ast }\left(𝓉\mu +\left(1-𝓉\right)b, \theta \right)\\ +{{\rm Y}}_{\ast }\left(\left(1-𝓉\right)\mu +𝓉b, \theta \right){\times {\rm T}}_{\ast }\left(\left(1-𝓉\right)\mu +𝓉b, \theta \right)\end{array}\right]\\ +\frac{1}{2e}\left[\begin{array}{c}\frac{2\mathrm{c}\mathrm{o}\mathrm{s}\frac{\pi 𝓉}{2}\mathrm{s}\mathrm{i}\mathrm{n}\frac{\pi 𝓉}{2}}{e}{\nabla }_{\ast }\left(\left(\mu ,b\right), \theta \right)\\ +\left[{\left(\frac{\mathrm{s}\mathrm{i}\mathrm{n}\frac{\pi 𝓉}{2}}{e^{1-𝓉}}\right)}^2+{\left(\frac{\mathrm{s}\mathrm{i}\mathrm{n}\frac{\pi 𝓉}{2}}{e^{1-𝓉}}\right)}^2\right]{\mathrm{\Delta }}_{\ast }\left(\left(\mu ,b\right), \theta \right)\end{array}\right]. \end{array}$$

(101)

Taking the multiplication of (101) with $e^{-\nu 𝓉}$ and integrating over $\left(0, 1\right),$ we get

$$\begin{array}{c} {\int }_0^1{e}^{-\nu 𝓉}{{\rm Y}}_{\ast }\left(\frac{\mu +b}{2},\theta \right){\times {\rm T}}_{\ast }\left(\frac{\mu +b}{2},\theta \right)d𝓉 \\ \le \frac{1}{2e} \left[{\int }_{\mu }^b{e}^{-\nu 𝓉}{{\rm Y}}_{\ast }\left(\varkappa ,\theta \right){\times {\rm T}}_{\ast }\left(\varkappa ,\theta \right)d𝓉+{\int }_{\mu }^b{e}^{-\nu 𝓉}{{\rm Y}}_{\ast }\left(\mathsf{s},\theta \right){\times {\rm T}}_{\ast }\left(\mathsf{s},\theta \right)d𝓉 \right]\\ +\frac{{\nabla }_{\ast }\left(\left(\mu ,b\right), \theta \right)}{2e}{\int }_0^1{e}^{-\nu 𝓉}\frac{2\mathrm{c}\mathrm{o}\mathrm{s}\frac{\pi 𝓉}{2}\mathrm{s}\mathrm{i}\mathrm{n}\frac{\pi 𝓉}{2}}{e}dt+\frac{{\mathrm{\Delta }}_{\ast }\left(\left(\mu ,b\right), \theta \right)}{2e}{\int }_0^1{e}^{-\nu 𝓉}\left[{\left(\frac{\mathrm{s}\mathrm{i}\mathrm{n}\frac{\pi 𝓉}{2}}{e^{1-𝓉}}\right)}^2+{\left(\frac{\mathrm{s}\mathrm{i}\mathrm{n}\frac{\pi 𝓉}{2}}{e^{1-𝓉}}\right)}^2\right]dt.\\ \frac{ 1-e^{-\nu }}{\nu } {\int }_0^1{e}^{-\nu 𝓉}{{\rm Y}}_{\ast }\left(\frac{\mu +b}{2},\theta \right){\times {\rm T}}_{\ast }\left(\frac{\mu +b}{2},\theta \right) \\ \le \frac{\alpha }{2e\left(b-\mu \right)}\left[{\mathcal{I}}_{{\mu }^{+}}^{\alpha } {{\rm Y}}_{\ast }\left(b, \theta \right){\times {\rm T}}_{\ast }\left(b, \theta \right)+{\mathcal{I}}_{b^{-}}^{\alpha } {{\rm Y}}_{\ast }\left(\mu , \theta \right){\times {\rm T}}_{\ast }\left(\mu , \theta \right)\right]\\ +\frac{\pi e^{-\nu -1}\left(e^{\nu }+1\right)}{2e\left({\nu }^2+{\pi }^2\right)}\frac{{\nabla }_{\ast }\left(\left(\mu ,b\right), \theta \right)}{2e}+\frac{1}{2e}A\left(\nu \right)\frac{{\mathrm{\Delta }}_{\ast }\left(\left(\mu ,b\right), \theta \right)}{2e}.\end{array}$$

(102)

Similarly, for , we have

$$\begin{array}{c}\frac{1-e^{-\nu }}{\nu } {{\rm Y}}^{*}\left(\frac{\mu +b}{2},\theta \right)\times {{\rm T}}^{*}\left(\frac{\mu +b}{2},\theta \right)\\ \ge \frac{\alpha }{2e\left(b-\mu \right)}\left[{ℐ}_{{\mu }^{+}}^{\alpha } {{\rm Y}}^{*}\left(b, \theta \right)\times {{\rm T}}^{*}\left(b, \theta \right)+{ℐ}_{b^{-}}^{\alpha } {{\rm Y}}^{*}\left(\mu , \theta \right)\times {{\rm T}}^{*}\left(\mu , \theta \right)\right] \\ +\frac{\pi e^{-\nu -1}\left(e^{\nu }+1\right)}{2e\left({\nu }^2+{\pi }^2\right)}\frac{{\nabla }^{*}\left(\left(\mu ,b\right), \theta \right)}{2e}+\frac{1}{2e}A\left(\nu \right)\frac{{\mathrm{\Delta }}^{*}\left(\left(\mu ,b\right), \theta \right)}{2e}.\end{array}$$

(103)

From (102) and (103), we have

$$\begin{array}{c}2\left[{{\rm Y}}_{*}\left(\frac{\mu +b}{2},\theta \right)\times {{\rm T}}_{*}\left(\frac{\mu +b}{2},\theta \right), {{\rm Y}}^{*}\left(\frac{\mu +b}{2},\theta \right)\times {{\rm T}}^{*}\left(\frac{\mu +b}{2},\theta \right)\right]\\ {\supseteq }_I\frac{1-\alpha }{e\left(b-\mu \right)}\left[{ℐ}_{{\mu }^{+}}^{\alpha } {{\rm Y}}_{*}\left(b, \theta \right)\times {{\rm T}}_{*}\left(b, \theta \right)+{ℐ}_{b^{-}}^{\alpha } {{\rm Y}}_{*}\left(\mu , \theta \right)\times {{\rm T}}_{*}\left(\mu , \theta \right), {ℐ}_{{\mu }^{+}}^{\alpha } {{\rm Y}}^{*}\left(b, \theta \right)\times {{\rm T}}^{*}\left(b, \theta \right)+{ℐ}_{b^{-}}^{\alpha } {{\rm Y}}^{*}\left(\mu , \theta \right)\times {{\rm T}}^{*}\left(\mu , \theta \right)\right].\\ +\frac{\nu \pi e^{-\nu -1}\left(e^{\nu }+1\right)}{e\left(1-e^{-\nu }\right)\left({\nu }^2+{\pi }^2\right)}\left[{\nabla }_{*}\left(\left(\mu ,b\right), \theta \right), {\nabla }^{*}\left(\left(\mu ,b\right), \theta \right)\right]+\frac{\nu }{e\left(1-e^{-\nu }\right)}A\left(\nu \right)\left[{\nabla }_{*}\left(\left(\mu ,b\right), \theta \right), {\nabla }^{*}\left(\left(\mu ,b\right), \theta \right)\right],\end{array}$$

where

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

That is

$$\begin{array}{c}2\odot \widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right)\otimes \widetilde{{\rm T}}\left(\frac{\mu +b}{2}\right){\supseteq }_{\mathbb{F}}\frac{\alpha }{e\left(b-\mu \right)}\odot \left[{ℐ}_{{\mu }^{+}}^{\alpha } \widetilde{{\rm Y}}\left(b\right)\otimes \widetilde{{\rm T}}\left(b\right)\oplus {ℐ}_{b^{-}}^{\alpha } \widetilde{{\rm Y}}\left(\mu \right)\otimes \widetilde{{\rm T}}\left(\mu \right)\right]\\ \oplus \frac{\nu \pi e^{-\nu -1}\left(e^{\nu }+1\right)}{e\left(1-e^{-\nu }\right)\left({\nu }^2+{\pi }^2\right)}\odot \nabla \left(\mu ,b\right)\oplus \frac{\nu }{e\left(1-e^{-\nu }\right)}A\left(\nu \right)\odot \mathrm{\Delta }\left(\mu ,b\right).\end{array}$$

Hence, the required results. □

Remark 7.

We readily see from Theorem 9 that

If one lays $\widetilde{{\rm Y}}$ is a lower $UDET\text{-}$convex $FNVM$ on $\left[\mu ,b\right],$ then we have (see [78])

$$\begin{array}{c}2\odot \widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right)\otimes \widetilde{{\rm T}}\left(\frac{\mu +b}{2}\right){\le }_{\mathbb{F}}\frac{\alpha }{e\left(b-\mu \right)}\odot \left[{ℐ}_{{\mu }^{+}}^{\alpha } \widetilde{{\rm Y}}\left(b\right)\otimes \widetilde{{\rm T}}\left(b\right)\oplus {ℐ}_{b^{-}}^{\alpha }\widetilde{{\rm Y}}\left(\mu \right)\otimes \widetilde{{\rm T}}\left(\mu \right)\right]\\ \oplus \frac{\nu \pi e^{-\nu -1}\left(e^{\nu }+1\right)}{e\left(1-e^{-\nu }\right)\left({\nu }^2+{\pi }^2\right)}\odot \nabla \left(\mu ,b\right)\oplus \frac{\nu }{e\left(1-e^{-\nu }\right)}A\left(\nu \right)\odot \mathrm{\Delta }\left(\mu ,b\right).\end{array}$$

(104)

Let ${{\rm Y}}_{*}\left(\varkappa ,\theta \right)\ne {{\rm Y}}^{*}\left(\varkappa ,\theta \right)$ and $\theta =1$,$.$ Then, we have (see [86])

$$\begin{array}{c}2{\rm Y}\left(\frac{\mu +b}{2}\right)\times {\rm T}\left(\frac{\mu +b}{2}\right)\supseteq \frac{\alpha }{e\left(b-\mu \right)}\left[{ℐ}_{{\mu }^{+}}^{\alpha } {\rm Y}\left(b\right)\times {\rm T}\left(b\right)+{ℐ}_{b^{-}}^{\alpha } {\rm Y}\left(\mu \right)\times {\rm T}\left(\mu \right)\right]\\ +\frac{\nu \pi e^{-\nu -1}\left(e^{\nu }+1\right)}{e\left(1-e^{-\nu }\right)\left({\nu }^2+{\pi }^2\right)}\nabla \left(\mu ,b\right)+\frac{\nu }{e\left(1-e^{-\nu }\right)}A\left(\nu \right)\mathrm{\Delta }\left(\mu ,b\right).\end{array}$$

(105)

If $\alpha \to 1$, that is

$$\underset{\alpha \to 1}{\lim }\nu =\underset{\alpha \to 1}{\lim }\frac{1-\alpha }{\alpha }\left(b-\mu \right)=0,$$

then

$$\underset{\alpha \to 1}{\lim }\frac{1-\alpha }{e\left(1-e^{-\nu }\right)}=\frac{1}{e\left(b-\mu \right)},$$

$$\begin{array}{c}\underset{\alpha \to 1}{\lim }\frac{\nu }{e\left(1-e^{-\nu }\right)}\left(-\frac{e^{-\nu -2}\left(8e^2-8\nu e^2+{\pi }^2{e}^2+2{\nu }^2{e}^2-{\pi }^2{e}^{\nu }\right)}{2\left(\nu -2\right)\left({\nu }^2-4\nu +{\pi }^2+4\right)}+\frac{8\nu -{\pi }^2{e}^{-\nu -2}+{\pi }^2+2{\nu }^2+8}{2\left(\nu +2\right)\left({\pi }^2+4\right)}\right)=\frac{{\pi }^2-{\pi }^2{e}^{-2}+8}{2e\left({\pi }^2+4\right)},\\ \underset{\alpha \to 1}{\lim }\frac{\nu \pi e^{-\nu -1}\left(e^{\nu }+1\right)}{e\left(1-e^{-\nu }\right)\left({\nu }^2+{\pi }^2\right)}=\frac{2}{\pi e^2}.\end{array}$$

The next finding, which is likewise novel, is as follows:

$$2\odot \widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right)\otimes \widetilde{{\rm T}}\left(\frac{\mu +b}{2}\right){\supseteq }_{\mathbb{F}}\frac{2}{e\left(b-\mu \right)}\odot {\int }_{\mu }^b\widetilde{{\rm Y}}\left(x\right)\otimes \widetilde{{\rm T}}\left(x\right)d\varkappa \oplus \frac{2}{\pi e^2}\odot \mathrm{\Delta }\left(\mu ,b\right)\oplus \frac{{\pi }^2-{\pi }^2{e}^2+8}{2e\left({\pi }^2+4\right)}\odot \nabla \left(\mu ,b\right).$$

(106)

If one lays $\alpha \to 1$ and $\widetilde{{\rm Y}}$ is a lower $UDET\text{-}$convex $FNVM$ on then we achieve (see [78])

$$2\odot \widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right)\otimes \widetilde{{\rm T}}\left(\frac{\mu +b}{2}\right){\le }_{\mathbb{F}}\frac{2}{e\left(b-\mu \right)}\odot {\int }_{\mu }^b\widetilde{{\rm Y}}\left(x\right)\otimes \widetilde{{\rm T}}\left(x\right)d\varkappa \oplus \frac{2}{\pi e^2}\odot \mathrm{\Delta }\left(\mu ,b\right)\oplus \frac{{\pi }^2-{\pi }^2{e}^2+8}{2e\left({\pi }^2+4\right)}\odot \nabla \left(\mu ,b\right).$$

(107)

Let $\alpha \to 1$ and ${{\rm Y}}_{*}\left(\varkappa , \theta \right)\ne {{\rm Y}}^{*}\left(\varkappa , \theta \right)$ with . Then, from Theorem 9, we have (see [86])

$$2{\rm Y}\left(\frac{\mu +b}{2}\right)\times {\rm T}\left(\frac{\mu +b}{2}\right)\supseteq \frac{2}{e\left(b-\mu \right)}{\int }_{\mu }^b{\rm Y}\left(x\right)\times {\rm T}\left(x\right)d\varkappa +\frac{2}{\pi e^2}\mathrm{\Delta }\left(\mu ,b\right)+\frac{{\pi }^2-{\pi }^2{e}^2+8}{2e\left({\pi }^2+4\right)}\nabla \left(\mu ,b\right).$$

(108)

If ${{\rm Y}}_{*}\left(\varkappa , \theta \right)={{\rm Y}}^{*}\left(\varkappa , \theta \right)$ and then, from Theorem 9, we obtain

$$\begin{array}{c}2{\rm Y}\left(\frac{\mu +b}{2}\right)\times {\rm T}\left(\frac{\mu +b}{2}\right)\le \frac{\alpha }{e\left(b-\mu \right)}\left[{ℐ}_{{\mu }^{+}}^{\alpha } {\rm Y}\left(b\right)\times {\rm T}\left(b\right)+{ℐ}_{b^{-}}^{\alpha } {\rm Y}\left(\mu \right)\times {\rm T}\left(\mu \right)\right]\\ +\frac{\nu \pi e^{-\nu -1}\left(e^{\nu }+1\right)}{e\left(1-e^{-\nu }\right)\left({\nu }^2+{\pi }^2\right)}\nabla \left(\mu ,b\right)+\frac{\nu }{e\left(1-e^{-\nu }\right)}A\left(\nu \right)\mathrm{\Delta }\left(\mu ,b\right).\end{array}$$

(109)

Let $\alpha \to 1$ and ${{\rm Y}}_{*}\left(\varkappa , \theta \right)={{\rm Y}}^{*}\left(\varkappa , \theta \right)$ with $\theta =1$. Then, from Theorem 9, we have (see [73])

$$2{\rm Y}\left(\frac{\mu +b}{2}\right)\times {\rm T}\left(\frac{\mu +b}{2}\right)\le \frac{2}{e\left(b-\mu \right)}{\int }_{\mu }^b{\rm Y}\left(x\right)\times {\rm T}\left(x\right)d\varkappa +\frac{2}{\pi e^2}\mathrm{\Delta }\left(\mu ,b\right)+\frac{{\pi }^2-{\pi }^2{e}^2+8}{2e\left({\pi }^2+4\right)}\nabla \left(\mu ,b\right).$$

(110)

Here, we have presented some multiple inequalities in one result with the help of the midpoint and end points of the interval $\left[\mu , b\right],$ and this outcome is very interesting because this is a generalization of some new and classical inequalities.

Theorem 10.

Let $\widetilde{{\rm Y}}:\left[\mu , b\right]\to R_C$ be an UDET-convex $FNVM$ on $\left[\mu , b\right],$ with $\theta$ levels defined by the family of i-v-ms ${{\rm Y}}_{\theta }:\left[\mu , b\right]\subset ℝ\to {\mathcal{X}}_C^{+}$ given by ${{\rm Y}}_{\theta }\left(\varkappa \right)=\left[{{\rm Y}}_{*}\left(\varkappa ,\theta \right), {{\rm Y}}^{*}\left(\varkappa ,\theta \right)\right]$ for all $\varkappa \in \left[\mu , b\right]$ and for all $\theta \in \left[0, 1\right]$. If $\widetilde{{\rm Y}}\in L\left(\left[\mu , b\right],R_C\right)$, then

$$\begin{array}{c}e\odot \widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right){\supseteq }_{\mathbb{F}}\sqrt{\frac{e}{2}}\odot \left[\widetilde{{\rm Y}}\left(\frac{3\mu +b}{2}\right)\oplus \widetilde{{\rm Y}}\left(\frac{\mu +3b}{2}\right)\right]\\ {\supseteq }_{\mathbb{F}}\frac{1-\alpha }{2\left(1-e^{-\frac{\nu }{2}}\right)}\odot \left[{ℐ}_{{\mu }^{+}}^{\alpha } \widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right)\oplus {ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{+}}^{\alpha } \widetilde{{\rm Y}}\left(b\right)\oplus {ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{-}}^{\alpha } \widetilde{{\rm Y}}\left(\mu \right)\oplus {ℐ}_{b^{-}}^{\alpha } \widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right)\right]\\ {\supseteq }_{\mathbb{F}}\frac{\nu }{2\left(1-e^{-\frac{\nu }{2}}\right)}K\left(\nu \right)\odot \left(\frac{\widetilde{{\rm Y}}\left(\mu \right)\oplus \widetilde{{\rm Y}}\left(b\right)}{2}\oplus \widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right)\right)\\ {\supseteq }_{\mathbb{F}}\frac{\nu }{2\left(1-e^{-\frac{\nu }{2}}\right)}\left(1+\sqrt{\frac{e}{2}}\right)K\left(\nu \right)\odot \frac{\widetilde{{\rm Y}}\left(\mu \right)\oplus \widetilde{{\rm Y}}\left(b\right)}{2}. \end{array}$$

(111)

If $\widetilde{{\rm Y}}\left(\varkappa \right)$ is $UDET\text{-}$concave $FNVM$, then

$$\begin{array}{c}e\odot \widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right){\subseteq }_{\mathbb{F}}\sqrt{\frac{e}{2}}\odot \left[\widetilde{{\rm Y}}\left(\frac{3\mu +b}{4}\right)\oplus \widetilde{{\rm Y}}\left(\frac{\mu +3b}{4}\right)\right]\\ {\subseteq }_{\mathbb{F}}\frac{1-\alpha }{2\left(1-e^{-\frac{\nu }{2}}\right)}\odot \left[{ℐ}_{{\mu }^{+}}^{\alpha } \widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right)\oplus {ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{+}}^{\alpha } \widetilde{{\rm Y}}\left(b\right)\oplus {ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{-}}^{\alpha } \widetilde{{\rm Y}}\left(\mu \right)\oplus {ℐ}_{b^{-}}^{\alpha } \widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right)\right]\\ {\subseteq }_{\mathbb{F}}\frac{\nu }{2\left(1-e^{-\frac{\nu }{2}}\right)}K\left(\nu \right)\odot \left(\frac{\widetilde{{\rm Y}}\left(\mu \right)\oplus \widetilde{{\rm Y}}\left(b\right)}{2}\oplus \widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right)\right)\\ {\supseteq }_{\mathbb{F}}\frac{\nu }{2\left(1-e^{-\frac{\nu }{2}}\right)}\left(1+\sqrt{\frac{e}{2}}\right)K\left(\nu \right)\odot \frac{\widetilde{{\rm Y}}\left(\mu \right)\oplus \widetilde{{\rm Y}}\left(b\right)}{2}.\end{array}$$

(112)

where

$$K\left(\nu \right)=\frac{2\nu +2\pi e^{-\frac{\nu +2}{2}}+4}{{\nu }^2+4\nu +{\pi }^2+4}+\frac{2\pi e^{-1}+2e^{-\frac{\nu }{2}}\left(2+\nu \right)}{{\nu }^2-4\nu +{\pi }^2+4},\nu =\frac{1-\alpha }{\alpha }\left(b-\mu \right), \mathrm{and} 1>\alpha >0.$$

Proof. 

Take $\left[\mu , \frac{\mu +b}{2}\right],$ we deduce that

$$\widetilde{{\rm Y}}\left(\frac{3\mu +b}{4}\right){\supseteq }_{\mathbb{F}}\frac{\mathrm{s}\mathrm{i}\mathrm{n}\frac{\pi }{4}}{\sqrt{e}}\odot \widetilde{{\rm Y}}\left(𝓉\mu +\left(1-𝓉\right)\frac{\mu +b}{2}\right)\oplus \frac{\mathrm{c}\mathrm{o}\mathrm{s}\frac{\pi }{4}}{\sqrt{e}}\odot \widetilde{{\rm Y}}\left(\left(1-𝓉\right)\mu +𝓉\frac{\mu +b}{2}\right).$$

After simplification, we get that

$$2\widetilde{{\rm Y}}\left(\frac{3\mu +b}{4}\right){\supseteq }_{\mathbb{F}}\sqrt{\frac{2}{e}}\odot \left[\widetilde{{\rm Y}}\left(𝓉\mu +\left(1-𝓉\right)\frac{\mu +b}{2}\right)\oplus \widetilde{{\rm Y}}\left(\left(1-𝓉\right)\mu +𝓉\frac{\mu +b}{2}\right)\right].$$

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

$$2{{\rm Y}}_{*}\left(\frac{3\mu +b}{4}, \theta \right)\le \sqrt{\frac{2}{e}}\left[{{\rm Y}}_{*}\left(𝓉\mu +\left(1-𝓉\right)\frac{\mu +b}{2}, \theta \right)+{{\rm Y}}_{*}\left(\left(1-𝓉\right)\mu +𝓉\frac{\mu +b}{2}, \theta \right)\right],$$

$$2{{\rm Y}}^{*}\left(\frac{3\mu +b}{4}, \theta \right)\ge \sqrt{\frac{2}{e}}\left[{{\rm Y}}^{*}\left(𝓉\mu +\left(1-𝓉\right)\frac{\mu +b}{2}, \theta \right)+{{\rm Y}}^{*}\left(\left(1-𝓉\right)\mu +𝓉\frac{\mu +b}{2}, \theta \right)\right].$$

Taking ${{\rm Y}}_{*}\left(., \theta \right)$ and multiplying both sides by $e^{-\frac{\nu 𝓉}{2}}$ and integrating the obtained results with respect to $𝓉$ from $0$ to $1$, we have

$$\begin{array}{c}{\int }_0^1{e}^{-\frac{\nu 𝓉}{2}}{{\rm Y}}_{*}\left(\frac{3\mu +b}{4}, \theta \right)d𝓉\\ \le \sqrt{\frac{2}{e}}\left[{\int }_0^1{e}^{-\nu 𝓉}{{\rm Y}}_{*}\left(𝓉\mu +\left(1-𝓉\right)\frac{\mu +b}{2},\theta \right)d𝓉+{\int }_0^1{e}^{-\nu 𝓉}{{\rm Y}}_{*}\left(\left(1-𝓉\right)\mu +𝓉\frac{\mu +b}{2}, \theta \right)d𝓉\right]. \end{array}$$

Let $u=𝓉\mu +\left(1-𝓉\right)\frac{\mu +b}{2}$ and $x=\left(1-𝓉\right)\mu +𝓉\frac{\mu +b}{2}.$ Then, we have

$$\begin{array}{c}{\int }_0^1{e}^{-\frac{\nu 𝓉}{2}}{{\rm Y}}_{*}\left(\frac{3\mu +b}{4}, \theta \right)d𝓉 \le \sqrt{\frac{2}{e}}\frac{1}{b-\mu } {\int }_{\frac{\mu +b}{2}}^b{e}^{\left(-\frac{1-\alpha }{\alpha }\left(\frac{\mu +b}{2}-u\right)\right)}{{\rm Y}}_{*}\left(u,\theta \right)du+\frac{1}{b-\mu }{\int }_{\frac{\mu +b}{2}}^b{e}^{\left(-\frac{1-\alpha }{\alpha }\left(x-\mu \right)\right)}{{\rm Y}}_{*}\left(\varkappa ,\theta \right)d\varkappa \\ =\sqrt{\frac{2}{e}}\frac{\alpha }{b-\mu }\left[{ℐ}_{{\mu }^{+}}^{\alpha } {{\rm Y}}_{*}\left(\frac{\mu +b}{2}, \theta \right)+{ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{-}}^{\alpha } {{\rm Y}}_{*}\left(\mu , \theta \right)\right].\end{array}$$

(113)

Now, taking the right side of Equation (113), we have

$${\int }_0^1{e}^{-\frac{\nu 𝓉}{2}}{{\rm Y}}_{*}\left(\frac{3\mu +b}{2}, \theta \right)d𝓉=\frac{2\left(1-e^{-\nu }\right)}{\nu }{{\rm Y}}_{*}\left(\frac{3\mu +b}{4}, \theta \right).$$

(114)

From (113) and (114), we deduce that

$$\frac{2\left(1-e^{-\nu }\right)}{\nu }{{\rm Y}}_{*}\left(\frac{3\mu +b}{4}, \theta \right)\begin{array}{c}\le \sqrt{\frac{2}{e}}\frac{1-\alpha }{b-\mu }\left[{ℐ}_{{\mu }^{+}}^{\alpha } {{\rm Y}}_{*}\left(\frac{\mu +b}{2}, \theta \right)+{ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{-}}^{\alpha } {{\rm Y}}_{*}\left(\mu , \theta \right)\right].\end{array}$$

(115)

Similarly, for ${{\rm Y}}^{*}\left(., \theta \right)$ from (115), we have

$$\frac{2\left(1-e^{-\nu }\right)}{\nu }{{\rm Y}}^{*}\left(\frac{3\mu +b}{4}, \theta \right)\begin{array}{c}\ge \sqrt{\frac{2}{e}}\frac{1-\alpha }{b-\mu }\left[{ℐ}_{{\mu }^{+}}^{\alpha }{{\rm Y}}^{*}\left(\frac{\mu +b}{2}, \theta \right)+{ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{-}}^{\alpha }{{\rm Y}}^{*}\left(\mu , \theta \right)\right].\end{array}$$

(116)

From (115) and (116), we deduce that

$$\frac{2\left(1-e^{-\nu }\right)}{\nu }{{\rm Y}}_{\theta }\left(\frac{3\mu +b}{4}, \theta \right)\begin{array}{c}{\supseteq }_I\sqrt{\frac{2}{e}}\frac{1-\alpha }{b-\mu }\left[{ℐ}_{{\mu }^{+}}^{\alpha } {{\rm Y}}_{\theta }\left(\frac{\mu +b}{2}, \theta \right)+{ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{-}}^{\alpha } {{\rm Y}}_{\theta }\left(\mu , \theta \right)\right].\end{array}$$

(117)

For the right side of Equation (111), since $\widetilde{{\rm Y}}$ be an $UDET\text{-}$convex $FNVM$, we deduce that

$$\widetilde{{\rm Y}}\left(𝓉\mu +\left(1-𝓉\right)\frac{\mu +b}{2}\right){\supseteq }_{\mathbb{F}}\frac{\sin \frac{\pi 𝓉}{4}}{e^{1-𝓉}}\odot \widetilde{{\rm Y}}\left(\mu \right)\oplus \frac{\cos \frac{\pi 𝓉}{4}}{e^{𝓉}}\odot \widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right),$$

(118)

and

$$\widetilde{{\rm Y}}\left(\left(1-𝓉\right)\mu +𝓉\frac{\mu +b}{2}\right){\supseteq }_{\mathbb{F}}\frac{\cos \frac{\pi 𝓉}{4}}{e^{𝓉}}\odot \widetilde{{\rm Y}}\left(\mu \right)\oplus \frac{\sin \frac{\pi 𝓉}{4}}{e^{1-𝓉}}\odot \widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right).$$

(119)

Adding (118) and (119), we have

$$\widetilde{{\rm Y}}\left(𝓉\mu +\left(1-𝓉\right)\frac{\mu +b}{2}\right)\oplus \widetilde{{\rm Y}}\left(\left(1-𝓉\right)\mu +𝓉\frac{\mu +b}{2}\right){\supseteq }_{\mathbb{F}}\left[\widetilde{{\rm Y}}\left(\mu \right)\oplus \widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right)\right]\odot \left[\frac{\mathrm{s}\mathrm{i}\mathrm{n}\frac{\pi 𝓉}{4}}{e^{1-𝓉}}+\frac{\mathrm{c}\mathrm{o}\mathrm{s}\frac{\pi 𝓉}{4}}{e^{𝓉}}\right].$$

(120)

Since $\widetilde{{\rm Y}}$ is $FNVM$, then, for each$\theta \in \left[0, 1\right]$, we have

$$\begin{array}{c}{{\rm Y}}_{\ast }\left(𝓉\mu +\left(1-𝓉\right)\frac{\mu +b}{2}, \theta \right)+{{\rm Y}}_{\ast }\left(\left(1-𝓉\right)\mu +𝓉\frac{\mu +b}{2}, \theta \right)\le \left[{{\rm Y}}_{\ast }\left(\mu , \theta \right)+{{\rm Y}}_{\ast }\left(\frac{\mu +b}{2}, \theta \right)\right]\left[\frac{\mathrm{s}\mathrm{i}\mathrm{n}\frac{\pi 𝓉}{4}}{e^{1-𝓉}}+\frac{\mathrm{c}\mathrm{o}\mathrm{s}\frac{\pi 𝓉}{4}}{e^{𝓉}}\right], \\ {{\rm Y}}^{\ast }\left(𝓉\mu +\left(1-𝓉\right)\frac{\mu +b}{2}, \theta \right)+{{\rm Y}}^{\ast }\left(\left(1-𝓉\right)\mu +𝓉\frac{\mu +b}{2}, \theta \right)\ge \left[{{\rm Y}}^{\ast }\left(\mu , \theta \right)+{{\rm Y}}^{\ast }\left(\frac{\mu +b}{2}, \theta \right)\right]\left[\frac{\mathrm{s}\mathrm{i}\mathrm{n}\frac{\pi 𝓉}{4}}{e^{1-𝓉}}+\frac{\mathrm{c}\mathrm{o}\mathrm{s}\frac{\pi 𝓉}{4}}{e^{𝓉}}\right]. \end{array}$$

(121)

Taking ${{\rm Y}}_{*}\left(., \theta \right)$ from (121) and multiplying the inequality with $e^{-\frac{\nu 𝓉}{2}}$ and integrating the results with $𝓉$ from $0$ to $1$, we have

$$\begin{array}{c}{\int }_0^1{e}^{e^{-\frac{\nu 𝓉}{2}}}{{\rm Y}}_{*}\left(𝓉\mu +\left(1-𝓉\right)\frac{\mu +b}{2}, \theta \right)d𝓉+{\int }_0^1{e}^{e^{-\frac{\nu 𝓉}{2}}}{{\rm Y}}_{*}\left(\left(1-𝓉\right)\mu +𝓉\frac{\mu +b}{2}, \theta \right)d𝓉\\ \le \left[{{\rm Y}}_{*}\left(\mu , \theta \right)+{{\rm Y}}_{*}\left(\frac{\mu +b}{2}, \theta \right)\right]{\int }_0^1{e}^{e^{-\frac{\nu 𝓉}{2}}}\left[\frac{\sin \frac{\pi 𝓉}{4}}{e^{1-𝓉}}+\frac{\cos \frac{\pi 𝓉}{4}}{e^{𝓉}}\right]d𝓉,\\ =\left(\frac{2\nu +2\pi e^{-\frac{\nu +2}{2}}+4}{{\nu }^2+4\nu +{\pi }^2+4}+\frac{2\pi e^{-1}+2e^{-\frac{\nu }{2}}\left(2+\nu \right)}{{\nu }^2-4\nu +{\pi }^2+4}\right)\left[{{\rm Y}}_{*}\left(\mu , \theta \right)+{{\rm Y}}_{*}\left(\frac{\mu +b}{2}, \theta \right)\right].\end{array}$$

(122)

In a manner identical to that described previously, we have ${{\rm Y}}^{*}\left(., \theta \right)$

$$\begin{array}{c}{\int }_0^1{e}^{e^{-\frac{\nu 𝓉}{2}}}{{\rm Y}}^{*}\left(𝓉\mu +\left(1-𝓉\right)\frac{\mu +b}{2} , \theta \right)d𝓉+{\int }_0^1{e}^{e^{-\frac{\nu 𝓉}{2}}}{{\rm Y}}^{*}\left(\left(1-𝓉\right)\mu +𝓉\frac{\mu +b}{2}, \theta \right)d𝓉\\ \ge \left[{{\rm Y}}^{*}\left(\mu , \theta \right)+{{\rm Y}}^{*}\left(b, \theta \right)\right]{\int }_0^1{e}^{-\nu 𝓉}\left[\frac{\sin \frac{\pi 𝓉}{4}}{e^{\frac{2-𝓉}{2}}}+\frac{\cos \frac{\pi 𝓉}{4}}{e^{\frac{𝓉}{2}}}\right]d𝓉,\\ =\left(\frac{2\nu +2\pi e^{-\frac{\nu +2}{2}}+4}{{\nu }^2+4\nu +{\pi }^2+4}+\frac{2\pi e^{-1}+2e^{-\frac{\nu }{2}}\left(2+\nu \right)}{{\nu }^2-4\nu +{\pi }^2+4}\right)\left[{{\rm Y}}^{*}\left(\mu , \theta \right)+{{\rm Y}}^{*}\left(\frac{\mu +b}{2}, \theta \right)\right].\end{array}$$

(123)

From (122) and (123), we have

$$\begin{array}{c}{\int }_0^1{e}^{-\nu 𝓉}{{\rm Y}}_{\theta }\left(𝓉\mu +\left(1-𝓉\right)\frac{\mu +b}{2}\right)d𝓉+{\int }_0^1{e}^{-\nu 𝓉}{{\rm Y}}_{\theta }\left(\left(1-𝓉\right)\mu +𝓉\frac{\mu +b}{2}\right)d𝓉\\ {\supseteq }_I\left(\frac{2\nu +2\pi e^{-\frac{\nu +2}{2}}+4}{{\nu }^2+4\nu +{\pi }^2+4}+\frac{2\pi e^{-1}+2e^{-\frac{\nu }{2}}\left(2+\nu \right)}{{\nu }^2-4\nu +{\pi }^2+4}\right)\left[{{\rm Y}}_{\theta }\left(\mu \right)+{{\rm Y}}_{\theta }\left(\frac{\mu +b}{2}\right)\right].\end{array}$$

(124)

Combining (117) and (124), we have

$$\sqrt{\frac{e}{2}}{{\rm Y}}_{\theta }\left(\frac{3\mu +b}{2}\right){\supseteq }_I\frac{1-\alpha }{2\left(1-e^{-\frac{\nu }{2}}\right)}\left[{ℐ}_{{\mu }^{+}}^{\alpha } {{\rm Y}}_{\theta }\left(\frac{\mu +b}{2}\right)+{ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{-}}^{\alpha }{{\rm Y}}_{\theta }\left(\mu \right)\right]{\supseteq }_I\frac{\nu }{2\left(1-e^{-\frac{\nu }{2}}\right)}K\left(\nu \right)\left(\frac{{{\rm Y}}_{\theta }\left(\mu \right)+{{\rm Y}}_{\theta }\left(\frac{\mu +b}{2}\right)}{2}\right),$$

(125)

where

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

Similarly, if we take the interval $\left[\frac{\mu +b}{2}, b\right],$ then, from (38), we acquire that

$$\sqrt{\frac{e}{2}}\left[{{\rm Y}}_{\theta }\left(\frac{\mu +3b}{2}\right)\right]{\supseteq }_I\frac{1-\alpha }{2\left(1-e^{-\frac{\nu }{2}}\right)}\left[{ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{+}}^{\alpha } {{\rm Y}}_{\theta }\left(b\right)+{ℐ}_{b^{-}}^{\alpha } {{\rm Y}}_{\theta }\left(\frac{\mu +b}{2}\right)\right]{\supseteq }_I\frac{\nu }{2\left(1-e^{-\frac{\nu }{2}}\right)}K\left(\nu \right)\left(\frac{{{\rm Y}}_{\theta }\left(\frac{\mu +b}{2}\right)+{{\rm Y}}_{\theta }\left(b\right)}{2}\right).$$

(126)

Adding (125) and (126), we have

$$\begin{array}{c}\sqrt{\frac{e}{2}}\left[{{\rm Y}}_{\theta }\left(\frac{3\mu +b}{2}\right)+{{\rm Y}}_{\theta }\left(\frac{\mu +3b}{2}\right)\right]\\ {\supseteq }_I\frac{1-\alpha }{2\left(1-e^{-\frac{\nu }{2}}\right)}\left[{ℐ}_{{\mu }^{+}}^{\alpha } {{\rm Y}}_{\theta }\left(\frac{\mu +b}{2}\right)+{ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{+}}^{\alpha } {{\rm Y}}_{\theta }\left(b\right)+{ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{-}}^{\alpha }{{\rm Y}}_{\theta }\left(\mu \right)+{ℐ}_{b^{-}}^{\alpha } {{\rm Y}}_{\theta }\left(\frac{\mu +b}{2}\right)\right]\\ {\supseteq }_I\frac{\nu }{2\left(1-e^{-\frac{\nu }{2}}\right)}K\left(\nu \right)\left(\frac{{{\rm Y}}_{\theta }\left(\mu \right)+{{\rm Y}}_{\theta }\left(b\right)}{2}+{{\rm Y}}_{\theta }\left(\frac{\mu +b}{2}\right)\right).\end{array}$$

(127)

To achieve the first and fourth fuzzy inclusion relations in (111), we take

$${{\rm Y}}_{\theta }\left(\frac{\mu +b}{2}\right)={{\rm Y}}_{\theta }\left(\frac{\frac{3\mu +b}{4}+\frac{\mu +3b}{4}}{2}\right){\supseteq }_I\frac{\sin \frac{\pi }{4}}{\sqrt{e}}{{\rm Y}}_{\theta }\left(\mu \right)+\frac{\cos \frac{\pi }{4}}{\sqrt{e}}{{\rm Y}}_{\theta }\left(b\right)=\frac{1}{2}\sqrt{\frac{2}{e}}{{\rm Y}}_{\theta }\left(\mu \right)+\frac{1}{2}\sqrt{\frac{2}{e}}{{\rm Y}}_{\theta }\left(b\right),$$

(128)

and

$${{\rm Y}}_{\theta }\left(\frac{\mu +b}{2}\right)={{\rm Y}}_{\theta }\left(\frac{\frac{3\mu +b}{4}+\frac{\mu +3b}{4}}{2}\right){\supseteq }_I\frac{\sin \frac{\pi }{4}}{\sqrt{e}}{{\rm Y}}_{\theta }\left(\frac{3\mu +b}{4}\right)+\frac{\cos \frac{\pi }{4}}{\sqrt{e}}{{\rm Y}}_{\theta }\left(\frac{\mu +3b}{4}\right)=\frac{1}{2}\sqrt{\frac{2}{e}}{{\rm Y}}_{\theta }\left(\frac{3\mu +b}{4}\right)+\frac{1}{2}\sqrt{\frac{2}{e}}{{\rm Y}}_{\theta }\left(\frac{\mu +3b}{4}\right).$$

(129)

By using the inclusion relations in (128) and (129), we obtain the first and fourth fuzzy inclusions of (111). By combining the resultant inclusion and (127), we obtain the following relations

$$\begin{array}{c}e{{\rm Y}}_{\theta }\left(\frac{\mu +b}{2}\right){\supseteq }_I\sqrt{\frac{e}{2}}\left[{{\rm Y}}_{\theta }\left(\frac{3\mu +b}{2}\right)+{{\rm Y}}_{\theta }\left(\frac{\mu +3b}{2}\right)\right]\\ {\supseteq }_I\frac{1-\alpha }{2\left(1-e^{-\frac{\nu }{2}}\right)}\left[{ℐ}_{{\mu }^{+}}^{\alpha } {{\rm Y}}_{\theta }\left(\frac{\mu +b}{2}\right)+{ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{+}}^{\alpha } {{\rm Y}}_{\theta }\left(b\right)+{ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{-}}^{\alpha } {{\rm Y}}_{\theta }\left(\mu \right)+{ℐ}_{b^{-}}^{\alpha } {{\rm Y}}_{\theta }\left(\frac{\mu +b}{2}\right)\right]\\ {\supseteq }_I\frac{\nu }{2\left(1-e^{-\frac{\nu }{2}}\right)}K\left(\nu \right)\left(\frac{{{\rm Y}}_{\theta }\left(\mu \right)+{{\rm Y}}_{\theta }\left(b\right)}{2}+{{\rm Y}}_{\theta }\left(\frac{\mu +b}{2}\right)\right)\\ {\supseteq }_I\frac{\nu }{2\left(1-e^{-\frac{\nu }{2}}\right)}\left(1+\sqrt{\frac{e}{2}}\right)K\left(\nu \right)\frac{{{\rm Y}}_{\theta }\left(\mu \right)+{{\rm Y}}_{\theta }\left(b\right)}{2}.\end{array}$$

That is

$$\begin{array}{c}e\odot \widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right){\supseteq }_{\mathbb{F}}\sqrt{\frac{e}{2}}\odot \left[\widetilde{{\rm Y}}\left(\frac{3\mu +b}{2}\right)\oplus \widetilde{{\rm Y}}\left(\frac{\mu +3b}{2}\right)\right]\\ {\supseteq }_{\mathbb{F}}\frac{1-\alpha }{2\left(1-e^{-\frac{\nu }{2}}\right)}\odot \left[{ℐ}_{{\mu }^{+}}^{\alpha } \widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right)\oplus {ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{+}}^{\alpha } \widetilde{{\rm Y}}\left(b\right)\oplus {ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{-}}^{\alpha } \widetilde{{\rm Y}}\left(\mu \right)\oplus {ℐ}_{b^{-}}^{\alpha } \widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right)\right]\\ {\supseteq }_{\mathbb{F}}\frac{\nu }{2\left(1-e^{-\frac{\nu }{2}}\right)}K\left(\nu \right)\odot \left(\frac{\widetilde{{\rm Y}}\left(\mu \right)\oplus \widetilde{{\rm Y}}\left(b\right)}{2}\oplus \widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right)\right)\\ {\supseteq }_{\mathbb{F}}\frac{\nu }{2\left(1-e^{-\frac{\nu }{2}}\right)}\left(1+\sqrt{\frac{e}{2}}\right)K\left(\nu \right)\odot \frac{\widetilde{{\rm Y}}\left(\mu \right)\oplus \widetilde{{\rm Y}}\left(b\right)}{2},\end{array}$$

hence the required results. □

Remark 8. 

We readily see from Theorem 10 that

If one lays $\widetilde{{\rm Y}}$ is a lower $UDET\text{-}$convex $FNVM$ on $\left[\mu ,b\right],$ then one can acquire (see [78])

$$\begin{array}{c}e\odot \widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right){\le }_{\mathbb{F}}\sqrt{\frac{e}{2}}\odot \left[\widetilde{{\rm Y}}\left(\frac{3\mu +b}{2}\right)\oplus \widetilde{{\rm Y}}\left(\frac{\mu +3b}{2}\right)\right]\\ {\le }_{\mathbb{F}}\frac{1-\alpha }{2\left(1-e^{-\frac{\nu }{2}}\right)}\odot \left[{ℐ}_{{\mu }^{+}}^{\alpha } \widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right)\oplus {ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{+}}^{\alpha } \widetilde{{\rm Y}}\left(b\right)\oplus {ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{-}}^{\alpha } \widetilde{{\rm Y}}\left(\mu \right)\oplus {ℐ}_{b^{-}}^{\alpha } \widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right)\right]\\ {\le }_{\mathbb{F}}\frac{\nu }{2\left(1-e^{-\frac{\nu }{2}}\right)}K\left(\nu \right)\odot \left(\frac{\widetilde{{\rm Y}}\left(\mu \right)\oplus \widetilde{{\rm Y}}\left(b\right)}{2}\oplus \widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right)\right)\\ {\le }_{\mathbb{F}}\frac{\nu }{2\left(1-e^{-\frac{\nu }{2}}\right)}\left(1+\sqrt{\frac{e}{2}}\right)K\left(\nu \right)\odot \frac{\widetilde{{\rm Y}}\left(\mu \right)\oplus \widetilde{{\rm Y}}\left(b\right)}{2}.\end{array}$$

(130)

Let ${{\rm Y}}_{*}\left(\varkappa ,\theta \right)\ne {{\rm Y}}^{*}\left(\varkappa ,\theta \right)$ and $\theta =1$. Then, from Theorem 10, we have (see [86])

$$\begin{array}{c}e{\rm Y}\left(\frac{\mu +b}{2}\right)\supseteq \sqrt{\frac{e}{2}}\left[{\rm Y}\left(\frac{3\mu +b}{2}\right)+{\rm Y}\left(\frac{\mu +3b}{2}\right)\right]\\ \supseteq \frac{1-\alpha }{2\left(1-e^{-\frac{\nu }{2}}\right)}\left[{ℐ}_{{\mu }^{+}}^{\alpha } {\rm Y}\left(\frac{\mu +b}{2}\right)+{ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{+}}^{\alpha } {\rm Y}\left(b\right)+{ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{-}}^{\alpha } {\rm Y}\left(\mu \right)+{ℐ}_{b^{-}}^{\alpha } {\rm Y}\left(\frac{\mu +b}{2}\right)\right]\\ \supseteq \frac{\nu }{2\left(1-e^{-\frac{\nu }{2}}\right)}K\left(\nu \right)\left(\frac{{\rm Y}\left(\mu \right)+{\rm Y}\left(b\right)}{2}+{\rm Y}\left(\frac{\mu +b}{2}\right)\right)\\ \supseteq \frac{\nu }{2\left(1-e^{-\frac{\nu }{2}}\right)}\left(1+\sqrt{\frac{e}{2}}\right)K\left(\nu \right)\frac{{\rm Y}\left(\mu \right)+{\rm Y}\left(b\right)}{2}.\end{array}$$

(131)

If $\alpha \to 1$, that is

$$\underset{\alpha \to 1}{\lim }\nu =\underset{\alpha \to 1}{\lim }\frac{1-\alpha }{\alpha }\left(b-\mu \right)=0,$$

then

$$\underset{\alpha \to 1}{\lim }\frac{\nu }{1-e^{-\frac{\nu }{2}}}\left(\frac{2\nu +2\pi e^{-\frac{\nu +2}{2}}+4}{{\nu }^2+4\nu +{\pi }^2+4}+\frac{2\pi e^{-1}+2e^{-\frac{\nu }{2}}\left(2+\nu \right)}{{\nu }^2-4\nu +{\pi }^2+4}\right)=\frac{4\left(2\pi e^{-1}+4\right)}{{\pi }^2+4}, \underset{\alpha \to 1}{\lim }\frac{1-\alpha }{2\left(1-e^{e^{-\frac{\nu }{2}}}\right)}=\frac{1}{b-\mu }.$$

The next finding, which is likewise novel, is as follows:

$$\begin{array}{c}\frac{e}{2}\odot \widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right){\supseteq }_{\mathbb{F}}\frac{1}{2}\sqrt{\frac{e}{2}}\odot \left[\widetilde{{\rm Y}}\left(\frac{3\mu +b}{2}\right)\oplus \widetilde{{\rm Y}}\left(\frac{\mu +3b}{2}\right)\right]\\ {\supseteq }_{\mathbb{F}}\frac{1}{b-\mu }\odot {\int }_{\mu }^b\widetilde{{\rm Y}}\left(\varkappa \right)d\varkappa \\ {\supseteq }_{\mathbb{F}}\frac{2\pi e^{-1}+4}{{\pi }^2+4}\odot \left(\frac{\widetilde{{\rm Y}}\left(\mu \right)\oplus \widetilde{{\rm Y}}\left(b\right)}{2}\oplus \widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right)\right)\\ {\supseteq }_{\mathbb{F}}\frac{2\pi e^{-1}+4}{{\pi }^2+4}\left(1+\sqrt{\frac{e}{2}}\right)\odot \frac{\widetilde{{\rm Y}}\left(\mu \right)\oplus \widetilde{{\rm Y}}\left(b\right)}{2}.\end{array}$$

(132)

If one lays $\alpha \to 1$ and $\widetilde{{\rm Y}}$ is a lower $UDET\text{-}$convex $FNVM$ on $\left[\mu ,b\right],$ then one can achieve (see [78])

$$\begin{array}{c}\frac{e}{2}\odot \widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right){\le }_{\mathbb{F}}\frac{1}{2}\sqrt{\frac{e}{2}}\odot \left[\widetilde{{\rm Y}}\left(\frac{3\mu +b}{2}\right)\oplus \widetilde{{\rm Y}}\left(\frac{\mu +3b}{2}\right)\right]\\ {\le }_{\mathbb{F}}\frac{1}{b-\mu }\odot {\int }_{\mu }^b\widetilde{{\rm Y}}\left(\varkappa \right)d\varkappa \\ {\le }_{\mathbb{F}}\frac{2\pi e^{-1}+4}{{\pi }^2+4}\odot \left(\frac{\widetilde{{\rm Y}}\left(\mu \right)\oplus \widetilde{{\rm Y}}\left(b\right)}{2}\oplus \widetilde{{\rm Y}}\left(\frac{\mu +b}{2}\right)\right)\\ {\le }_{\mathbb{F}}\frac{2\pi e^{-1}+4}{{\pi }^2+4}\left(1+\sqrt{\frac{e}{2}}\right)\odot \frac{\widetilde{{\rm Y}}\left(\mu \right)\oplus \widetilde{{\rm Y}}\left(b\right)}{2}.\end{array}$$

(133)

Let $\alpha \to 1$ and ${{\rm Y}}_{*}\left(\varkappa , \theta \right)\ne {{\rm Y}}^{*}\left(\varkappa , \theta \right)$ with . Then, from Theorem 10, we have (see [86])

$$\begin{array}{c}\frac{e}{2}{\rm Y}\left(\frac{\mu +b}{2}\right)\supseteq \frac{1}{2}\sqrt{\frac{e}{2}}\left[{\rm Y}\left(\frac{3\mu +b}{2}\right)+{\rm Y}\left(\frac{\mu +3b}{2}\right)\right]\\ \supseteq \frac{1}{b-\mu }{\int }_{\mu }^b{\rm Y}\left(\varkappa \right)d\varkappa \\ \supseteq \frac{2\pi e^{-1}+4}{{\pi }^2+4}\left(\frac{{\rm Y}\left(\mu \right)+{\rm Y}\left(b\right)}{2}+{\rm Y}\left(\frac{\mu +b}{2}\right)\right)\\ \supseteq \frac{2\pi e^{-1}+4}{{\pi }^2+4}\left(1+\sqrt{\frac{e}{2}}\right)\frac{{\rm Y}\left(\mu \right)+{\rm Y}\left(b\right)}{2}.\end{array}$$

(134)

If ${{\rm Y}}_{*}\left(\varkappa , \theta \right)={{\rm Y}}^{*}\left(\varkappa , \theta \right)$ with $\theta =1,$ then, from Theorem 10, we acquire

$$\begin{array}{c}e{\rm Y}\left(\frac{\mu +b}{2}\right)\le \sqrt{\frac{e}{2}}\left[{\rm Y}\left(\frac{3\mu +b}{2}\right)+{\rm Y}\left(\frac{\mu +3b}{2}\right)\right]\\ \le \frac{1-\alpha }{2\left(1-e^{-\frac{\nu }{2}}\right)}\left[{ℐ}_{{\mu }^{+}}^{\alpha } {\rm Y}\left(\frac{\mu +b}{2}\right)+{ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{+}}^{\alpha } {\rm Y}\left(b\right)+{ℐ}_{{\left(\frac{\mu +b}{2}\right)}^{-}}^{\alpha } {\rm Y}\left(\mu \right)+{ℐ}_{b^{-}}^{\alpha } {\rm Y}\left(\frac{\mu +b}{2}\right)\right]\\ \le \frac{\nu }{2\left(1-e^{-\frac{\nu }{2}}\right)}K\left(\nu \right)\left(\frac{{\rm Y}\left(\mu \right)+{\rm Y}\left(b\right)}{2}+{\rm Y}\left(\frac{\mu +b}{2}\right)\right)\\ \le \frac{\nu }{2\left(1-e^{-\frac{\nu }{2}}\right)}\left(1+\sqrt{\frac{e}{2}}\right)K\left(\nu \right)\frac{{\rm Y}\left(\mu \right)+{\rm Y}\left(b\right)}{2}.\end{array}$$

(135)

Let $\alpha \to 1$ and ${{\rm Y}}_{*}\left(\varkappa , \theta \right)={{\rm Y}}^{*}\left(\varkappa , \theta \right)$ with $\theta =1$. Then, from Theorem 10, we achieve (see [73])

$$\begin{array}{c}\frac{e}{2}{\rm Y}\left(\frac{\mu +b}{2}\right)\le \frac{1}{2}\sqrt{\frac{e}{2}}\left[{\rm Y}\left(\frac{3\mu +b}{2}\right)+{\rm Y}\left(\frac{\mu +3b}{2}\right)\right]\\ \le \frac{1}{b-\mu }{\int }_{\mu }^b{\rm Y}\left(\varkappa \right)d\varkappa \\ \le \frac{2\pi e^{-1}+4}{{\pi }^2+4}\left(\frac{{\rm Y}\left(\mu \right)+{\rm Y}\left(b\right)}{2}+{\rm Y}\left(\frac{\mu +b}{2}\right)\right)\\ \le \frac{2\pi e^{-1}+4}{{\pi }^2+4}\left(1+\sqrt{\frac{e}{2}}\right)\frac{{\rm Y}\left(\mu \right)+{\rm Y}\left(b\right)}{2}.\end{array}$$

(136)

Example 3. 

Let $\alpha =\frac{1}{3}, \varkappa \in \left[0,1\right]$ and the $FNVM$ $\widetilde{{\rm Y}}:\left[\mu , b\right]=\left[0, 1\right]\to R_C,$ defined by

$$\widetilde{{\rm Y}}\left(\varkappa \right)\left(\theta \right)=\{\begin{array}{c}\frac{\theta -2{\varkappa }^4}{\frac{3}{2}-2{\varkappa }^4} \theta \in \left[2{\varkappa }^4,\frac{3}{2}\right],\\ \frac{ 1+x-\theta }{x-\frac{1}{2}} \theta \in \left(\frac{3}{2}, 1+x\right], \\ 0 \end{array}$$

(137)

then, for each $\theta \in \left[0, 1\right],$ we have ${{\rm Y}}_{\theta }\left(\varkappa \right)=\left[2\left(1-\theta \right){\varkappa }^4+\frac{3}{2}\theta ,\left(1-\theta \right)\left(1+x\right)+\frac{3}{2}\theta \right]$. It can be easily seen that, for each $\theta \in \left[0, 1\right]$, the left and right end point functions ${{\rm Y}}_{*}\left(\varkappa ,\theta \right)=2\left(1-\theta \right){\varkappa }^4+\frac{3}{2}\theta ,$ ${{\rm Y}}^{*}\left(\varkappa , \theta \right)=\left(1-\theta \right)\left(1+x\right)+\frac{3}{2}\theta$ are exponential trigonometric convex and concave functions, respectively. Then $\widetilde{{\rm Y}}\left(\varkappa \right)$ is $a UDET-$convex $FNVM$. We readily see that $\widetilde{{\rm Y}}\in L\left(\left[\mu , b\right],R_C\right)$ and $\nu =\frac{1-\alpha }{\alpha }\left(b-\mu \right)=2$.

$$\begin{array}{c}e{{\rm Y}}_{\ast }\left(\frac{\mu +b}{2},\theta \right)=e\left[\frac{1}{8}\left(1-\theta \right)+\frac{3}{2}\theta \right]\\ e{{\rm Y}}^{\ast }\left(\frac{\mu +b}{2},\theta \right)=e\left[\frac{3}{2}\left(1-\theta \right)+\frac{3}{2}\theta \right]\\ \sqrt{\frac{e}{2}}\left[{{\rm Y}}_{\ast }\left(\frac{3\mu +b}{2},\theta \right)+{{\rm Y}}_{\ast }\left(\frac{\mu +3b}{2},\theta \right)\right]=\sqrt{\frac{e}{2}}\left[\frac{41}{64}\left(1-\theta \right)+\frac{3}{2}\theta \right]\\ \sqrt{\frac{e}{2}}\left[{{\rm Y}}^{\ast }\left(\frac{3\mu +b}{2},\theta \right)+{{\rm Y}}^{\ast }\left(\frac{\mu +3b}{2},\theta \right)\right]=\sqrt{\frac{e}{2}}\left[3\left(1-\theta \right)+\frac{3}{2}\theta \right]\\ \frac{1-\alpha }{2\left(1-e^{-\frac{\nu }{2}}\right)}\left[{\mathcal{I}}_{{\mu }^{+}}^{\alpha }{{\rm Y}}_{\ast }\left(\frac{\mu +b}{2},\theta \right)+{\mathcal{I}}_{{\left(\frac{\mu +b}{2}\right)}^{+}}^{\alpha }{{\rm Y}}_{\ast }\left(b,\theta \right)+{\mathcal{I}}_{{\left(\frac{\mu +b}{2}\right)}^{-}}^{\alpha }{{\rm Y}}_{\ast }\left(\mu ,\theta \right)+{\mathcal{I}}_{b^{-}}^{\alpha } {{\rm Y}}_{\ast }\left(\frac{\mu +b}{2},\theta \right)\right]\\ =\frac{1}{1-e^{-1}}\left\{{\int }_0^{\frac{1}{2}}{e}^{-2\left(\frac{1}{2}-x\right)} \left(2\left(1-\theta \right){\varkappa }^4+\frac{3}{2}\theta \right)dx+{\int }_0^{\frac{1}{2}}{e}^{-2x}\left(2\left(1-\theta \right){\varkappa }^4+\frac{3}{2}\theta \right)dx\right\}\\ +\frac{1}{1-e^{-1}}\left\{{\int }_{\frac{1}{2}}^1{e}^{-2\left(1-x\right)} \left( 2\left(1-\theta \right){\varkappa }^4+\frac{3}{2}\theta \right)dx+{\int }_{\frac{1}{2}}^1{e}^{-2\left(x-\frac{1}{2}\right)}\left(2\left(1-\theta \right){\varkappa }^4+\frac{3}{2}\theta \right)dx\right\}\\ =\frac{1}{1-e^{-1}}\left(1-\theta \right)\left\{\frac{3e}{2}-\frac{121e^{-1}}{8}+2\right\}+12\theta .\\ \frac{1-\alpha }{2\left(1-e^{-\frac{\nu }{2}}\right)}\left[{\mathcal{I}}_{{\mu }^{+}}^{\alpha }{{\rm Y}}^{\ast }\left(\frac{\mu +b}{2},\theta \right)+{\mathcal{I}}_{{\left(\frac{\mu +b}{2}\right)}^{+}}^{\alpha }{{\rm Y}}^{\ast }\left(b,\theta \right)+{\mathcal{I}}_{{\left(\frac{\mu +b}{2}\right)}^{-}}^{\alpha }{{\rm Y}}^{\ast }\left(\mu ,\theta \right)+{\mathcal{I}}_{b^{-}}^{\alpha } {{\rm Y}}^{\ast }\left(\frac{\mu +b}{2},\theta \right)\right]\\ =\frac{1}{1-e^{-1}}\left\{{\int }_0^{\frac{1}{2}}{e}^{-2\left(\frac{1}{2}-x\right)} \left( \left(1-\theta \right)\left(1+x\right)+\frac{3}{2}\theta \right)dx+{\int }_0^{\frac{1}{2}}{e}^{-2x}\left(\left(1-\theta \right)\left(1+x\right)+\frac{3}{2}\theta \right)dx\right\}\\ +\frac{1}{1-e^{-1}}\left\{{\int }_{\frac{1}{2}}^1{e}^{-2\left(1-x\right)} \left(\left(1-\theta \right)\left(1+x\right)+\frac{3}{2}\theta \right)dx+{\int }_{\frac{1}{2}}^1{e}^{-2\left(x-\frac{1}{2}\right)}\left(\left(1-\theta \right)\left(1+x\right)+\frac{3}{2}\theta \right)dx\right\}\\ =\frac{1}{1-e^{-1}}\left(1-\theta \right)\left\{3-3e^{-1}\right\}+12\theta \\ \frac{\nu }{2\left(1-e^{-\frac{\nu }{2}}\right)}\mathsf{K}\left(\nu \right)\left(\frac{{{\rm Y}}_{\ast }\left(\mu ,\theta \right)+{{\rm Y}}_{\ast }\left(b,\theta \right)}{2}+{{\rm Y}}_{\ast }\left(\frac{\mu +b}{2},\theta \right)\right)=\frac{1}{\left(1-e^{-1}\right)}\left(\frac{6+2\pi e^{-2}}{16+{\pi }^2}+\frac{2e^{-1}}{\pi }\right)\left[\frac{9}{8}\left(1-\theta \right)+3\theta \right]\\ \frac{\nu }{2\left(1-e^{-\frac{\nu }{2}}\right)}\mathsf{K}\left(\nu \right)\left(\frac{{{\rm Y}}^{\ast }\left(\mu ,\theta \right)+{{\rm Y}}^{\ast }\left(b,\theta \right)}{2}+{{\rm Y}}^{\ast }\left(\frac{\mu +b}{2},\theta \right)\right)=\frac{1}{\left(1-e^{-1}\right)}\left(\frac{6+2\pi e^{-2}}{16+{\pi }^2}+\frac{2e^{-1}}{\pi }\right)\left[3\left(1-\theta \right)+3\theta \right]\\ \frac{\nu }{2\left(1-e^{-\frac{\nu }{2}}\right)}\left(1+\sqrt{\frac{e}{2}}\right)\mathsf{K}\left(\nu \right)\frac{{{\rm Y}}_{\ast }\left(\mu ,\theta \right)+{{\rm Y}}_{\ast }\left(b,\theta \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)\left[\left(1-\theta \right)+\frac{3}{2}\theta \right]\\ \frac{\nu }{2\left(1-e^{-\frac{\nu }{2}}\right)}\left(1+\sqrt{\frac{e}{2}}\right)\mathsf{K}\left(\nu \right)\frac{{{\rm Y}}^{\ast }\left(\mu ,\theta \right)+{{\rm Y}}^{\ast }\left(b,\theta \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)\left[\frac{3}{2}\left(1-\theta \right)+\frac{3}{2}\theta \right]\\ e\left[\frac{1}{8}\left(1-\theta \right)+\frac{3}{2}\theta ,\frac{3}{2}\left(1-\theta \right)+\frac{3}{2}\theta \right]\supseteq \sqrt{\frac{e}{2}}\left[\frac{41}{64}\left(1-\theta \right)+\frac{3}{2}\theta ,3\left(1-\theta \right)+\frac{3}{2}\theta \right]\\ {\supseteq }_I\frac{1}{1-e^{-1}}\left[\left(1-\theta \right)\left\{\frac{3e}{2}-\frac{121e^{-1}}{8}+2\right\}+12\theta ,\left(1-\theta \right)\left\{3-3e^{-1}\right\}+12\theta \right]\\ {\supseteq }_I\frac{1}{\left(1-e^{-1}\right)}\left(\frac{6+2\pi e^{-2}}{16+{\pi }^2}+\frac{2e^{-1}}{\pi }\right)\left[\frac{9}{8}\left(1-\theta \right)+3\theta ,3\left(1-\theta \right)+3\theta \right]\\ {\supseteq }_I\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)\left[\left(1-\theta \right)+\frac{3}{2}\theta ,\frac{3}{2}\left(1-\theta \right)+\frac{3}{2}\theta \right],\end{array}$$

Hence, Theorem 10 has been verified.

4. Conclusions

The $UDET-$convex fuzzy number valued mappings are a novel class of fuzzy convex functions that we introduced in this paper. We also looked at some of their algebraic characteristics. For the newly constructed convex function and fuzzy fractional integral operators, we created a brand new inequality of $H–H$types. Additionally, for the newly specified convex function, we proved a new midpoint $H–H$type and a few associated integral inequalities. In Remarks 2–8, we explored a number of particular situations that were discovered as a result of the major findings and made note of their presence in the literature. Finally, we applied our findings to particular values and functions to demonstrate some applications of Bessel functions and special means. In any event, we anticipate that these findings will help us better comprehend the fundamentals of fractional calculus and its various applications.