New Versions of Fuzzy-Valued Integral Inclusion over p-Convex Fuzzy Number-Valued Mappings and Related Fuzzy Aumman’s Integral Inequalities

Abstract

It is well known that both concepts of symmetry and convexity are directly connected. Similarly, in fuzzy theory, both ideas behave alike. It is important to note that real and interval-valued mappings are exceptional cases of fuzzy number-valued mappings ($\mathrm{F}\mathrm{N}\mathrm{V}\mathrm{M}$s) because fuzzy theory depends upon the unit interval that make a significant contribution to overcoming the issues that arise in the theory of interval analysis and fuzzy number theory. In this paper, the new class of $p$-convexity over up and down ($\mathrm{U}\mathrm{D}$) fuzzy relation has been introduced which is known as $\mathrm{U}\mathrm{D}$-$p$-convex fuzzy number-valued mappings ($\mathrm{U}\mathrm{D}$-$p$-convex $\mathrm{F}\mathrm{N}\mathrm{V}\mathrm{M}$s). We offer a thorough analysis of Hermite–Hadamard-type inequalities for $\mathrm{F}\mathrm{N}\mathrm{V}\mathrm{M}$s that are $\mathrm{U}\mathrm{D}$-$p$-convex using the fuzzy Aumann integral. Some previous results from the literature are expanded upon and broadly applied in our study. Additionally, we offer precise justifications for the key theorems that Kunt and İşcan first deduced in their article titled “Hermite–Hadamard–Fejer type inequalities for p-convex functions”. Some new and classical exceptional cases are also discussed. Finally, we illustrate our findings with well-defined examples.

1. Introduction

Since the discovery of the first convex inequality, also referred to as the Jensen inequality, convex inequalities have been a hotly debated subject in mathematics. There are many inequalities that are derived using convexity; for example, see the books [1,2]. See the works [3,4,5,6,7,8,9,10,11,12,13,14] for further information on the applications of inequality to diverse areas of mathematics, such as numerical analysis, probability density functions, and optimization. It should be noted that L’Hospital and Leibniz first proposed the concept of fractional calculus in 1695. Numerous mathematicians, including Riemann, Grunwald, Letnikov, Hadamard, and Weyl, expanded on this idea. These mathematicians contributed significantly to fractional calculus and its many applications. For further information on fractional calculus, see [15,16,17,18,19,20,21,22,23,24,25,26,27]. In the modern era, fractional calculus is frequently used to describe a variety of phenomena, such as the fractional conservation of mass, and the fractional Schrodinger equation in quantum theory. One of the inequalities that has garnered the most interest in the mathematical community is the Hermite–Hadamard inequality [28], which was independently proven by Charles Hermite and Jacques Hadamard. Numerous mathematicians have generalized this inequality in numerous ways. For more related results, see [29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44]. The inequality is written as if $𝒯:I\to \mathbb{R}$ is a convex function on $I$ and $𝚤, 𝚥\in I$ with $𝚤\le 𝚥$ such that:

$$𝒯\left(\frac{𝚤+𝚥}{2}\right)\le \frac{1}{𝚥-𝚤}{\int }_{𝚤}^{𝚥}𝒯\left(ⱷ\right)dⱷ\le \frac{𝒯\left(𝚤\right)+𝒯\left(𝚥\right)}{2}.$$

(1)

New variations of these disparities have been obtained in recent years using various creative ways. For instance, the Hermite–Hadamard inequality’s first fractional analog was discovered by Sarikaya et al. [45]. Ramik [46] used fuzzy numbers to derive inequalities in 1985 and applied the inequality in fuzzy optimization. The concept of $\mathrm{F}\mathrm{N}\mathrm{V}\mathrm{M}$s was created by the authors in [47] using Jensen-type inequalities. Costa et al.’s [48] computation of fresh integral inequality uses the idea of $\mathrm{F}\mathrm{N}\mathrm{V}\mathrm{M}$s. For more information, see [49,50,51,52,53,54,55,56,57,58,59] and the references therein. In order to look at inequalities of the Jensen and Hermite–Hadamard types, Zhao et al. [60] introduced the concept of generalized interval-valued convexity. Liu et al. used J-inclusions to study the modular inequalities of interval-valued soft sets in [61]. Yang et al.’s [62] formulation of novel Hermite–Hadamard-type inequalities in conjunction with exponential $\mathrm{F}\mathrm{N}\mathrm{V}\mathrm{M}$ was published in 2021. The authors of [63] computed Ostrowski-type inequalities and applied them to numerical integration using the concept of interval-valued mappings. Santos–Gomez used fuzzy number-valued pre-invex functions in [64] to study coordinated inequalities. In [65], Khan et al. developed new harmonically $\mathrm{F}\mathrm{N}\mathrm{V}\mathrm{M}$-based Hermite–Hadamard inclusions. According to the authors of [66], some Hermite–Hadamard inequalities and their weighted variants, referred to as Fejer-type inclusions, involve generalized fractional operators with an exponential kernel. See [67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82] for contemporary research and applications involving the Hermite–Hadamard inequality.

This inequality was discovered by many scholars as a result of the generalization employing various types of convexity with fractional operators [83,84,85,86,87,88,89]. The works go into further detail concerning the Hermite–Hadamard inequality and $\mathrm{U}\mathrm{D}$-$p$-convex inequality. In the current study, several $\mathrm{U}\mathrm{D}$-$p$-convex inequalities are derived together with fuzzy Aumann integral operators using the fuzzy number-valued settings and newly defined fuzzy $\mathrm{U}\mathrm{D}$-convexity. In this study, the recent findings of Kunt and İşcan [90] are generalized and several exceptional cases are discussed.

2. Preliminaries

We will go through the fundamental terminologies and findings in this section, which aid in comprehending the ideas behind our fresh findings.

Let ${\mathcal{L}}_C$ be the space of all closed and bounded intervals of $\mathbb{R}$, and $₼\in {\mathcal{L}}_C$ be defined by

$$\mathsf{₼}=\left[{\mathsf{₼}}_{*},{\mathsf{₼}}^{*}\right]=\left\{\mathbb{ⱷ}\in \mathbb{R}|{\mathsf{₼}}_{*}\le \mathrm{ⱷ}\le {\mathsf{₼}}^{*}\right\},\left({\mathsf{₼}}_{*},{\mathsf{₼}}^{*}\in \mathbb{R}\right).$$

(2)

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

$${\mathcal{L}}_{\mathrm{C}}^{+}=\left\{\left[{\mathsf{₼}}_{*},{\mathsf{₼}}^{*}\right]:\left[{\mathsf{₼}}_{*},{\mathsf{₼}}^{*}\right]\in {\mathcal{L}}_{\mathrm{C}} a\mathcal{n}d {\mathsf{₼}}_{*}\ge 0\right\}.$$

(3)

Let $\tau \in \mathbb{R}$ and $\tau \cdot ₼$ be defined by

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

(4)

Then the Minkowski difference $\mathsf{Ɖ}-₼$, addition $₼+\mathsf{Ɖ},$ and $₼\times \mathsf{Ɖ}$ for $₼,\mathsf{Ɖ}\in {\mathcal{L}}_C$ are defined by

$$\left[{\mathsf{Ɖ}}_{*},{\mathsf{Ɖ}}^{*}\right]+\left[{\mathsf{₼}}_{*},{\mathsf{₼}}^{*}\right]=\left[{\mathsf{Ɖ}}_{*}+{\mathsf{₼}}_{*},{\mathsf{Ɖ}}^{*}+{\mathsf{₼}}^{*}\right],$$

(5)

$$\left[{\mathsf{Ɖ}}_{*},{\mathsf{Ɖ}}^{*}\right]\times \left[{\mathsf{₼}}_{*},{\mathsf{₼}}^{*}\right]=\left[mi\mathcal{n}\left\{{\mathsf{Ɖ}}_{*}{\mathsf{₼}}_{*},{\mathsf{Ɖ}}^{*}{\mathsf{₼}}_{*},{\mathsf{Ɖ}}_{*}{\mathsf{₼}}^{*},{\mathsf{Ɖ}}^{*}{\mathsf{₼}}^{*}\right\},max\left\{{\mathsf{Ɖ}}_{*}{\mathsf{₼}}_{*},{\mathsf{Ɖ}}^{*}{\mathsf{₼}}_{*},{\mathsf{Ɖ}}_{*}{\mathsf{₼}}^{*},{\mathsf{Ɖ}}^{*}{\mathsf{₼}}^{*}\right\}\right],$$

(6)

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

(7)

Remark 1. 

(i) For given $\left[{\mathsf{Ɖ}}_{*}, {\mathsf{Ɖ}}^{*}\right], \left[{₼}_{*}, {₼}^{*}\right]\in {\mathcal{L}}_C,$ the relation $“{\supseteq }_I”$ defined on ${\mathcal{L}}_C$ by $\left[{₼}_{*}, {₼}^{*}\right]{\supseteq }_I\left[{\mathsf{Ɖ}}_{*}, {\mathsf{Ɖ}}^{*}\right]$ if and only if ${₼}_{*}\le {\mathsf{Ɖ}}_{*}, {\mathsf{Ɖ}}^{*}{\le ₼}^{*}$ for all $\left[{\mathsf{Ɖ}}_{*}, {\mathsf{Ɖ}}^{*}\right], \left[{₼}_{*}, {₼}^{*}\right]\in {\mathcal{L}}_C$ is a partial interval inclusion relation. The relation $\left[{₼}_{*}, {₼}^{*}\right]{\supseteq }_I\left[{\mathsf{Ɖ}}_{*}, {\mathsf{Ɖ}}^{*}\right]$ is coincident to $\left[{₼}_{*}, {₼}^{*}\right]\supseteq \left[{\mathsf{Ɖ}}_{*}, {\mathsf{Ɖ}}^{*}\right]$ on ${\mathcal{L}}_C.$ It can be easily seen that “${\supseteq }_I$” looks like “up and down” on the real line $\mathbb{R},$ so we call $“{\supseteq }_I”$ “up and down” (or “$UD$” order, in short) [80]. For $\left[{\mathsf{Ɖ}}_{*}, {\mathsf{Ɖ}}^{*}\right], \left[{₼}_{*}, {₼}^{*}\right]\in {\mathcal{L}}_C,$ the Hausdorff–Pompeiu distance between intervals $\left[{\mathsf{Ɖ}}_{*}, {\mathsf{Ɖ}}^{*}\right]$ and $\left[{₼}_{*}, {₼}^{*}\right]$ is defined by

$$d_H\left(\left[{\mathsf{Ɖ}}_{*},{\mathsf{Ɖ}}^{*}\right],\left[{\mathsf{₼}}_{*},{\mathsf{₼}}^{*}\right]\right)=max\left\{\left|{\mathsf{Ɖ}}_{*}-{\mathsf{₼}}_{*}\right|,\left|{\mathsf{Ɖ}}^{*}-{\mathsf{₼}}^{*}\right|\right\}.$$

(8)

It is a familiar fact that $\left({\mathcal{L}}_C,d_H\right)$ is a complete metric space [74,75,76].

Noting that we will be using the traditional definitions of fuzzy set and fuzzy numbers, we will only review some fundamental ideas about fuzzy set and fuzzy numbers. Be mindful that we refer to the set of all fuzzy subsets and fuzzy numbers of $\mathbb{R}$ as $\mathcal{L}$, and ${\mathcal{L}}_C$.

Definition 1 

([78,79]). Given $\tilde{f}\in {\mathcal{L}}_C$, the level sets or cut sets are given by ${\left[\tilde{f}\right]}^{\tau }=\left\{ⱷ\in \mathbb{R}|\tilde{f}\left(ⱷ\right)\ge \tau \right\}$ for all $\tau \in \left(\left.0, 1\right]\right.$ and by ${\left[\tilde{f}\right]}^0=\mathit{c}\mathit{l}\left\{\mathbb{ⱷ}\in \mathbb{R}|\tilde{f}\left(ⱷ\right)>0\right\}$, where ${\left[\tilde{f}\right]}^0$ is known as support of $\tilde{f}$. These sets are known as $\tau$-level sets or $\tau$-cut sets of $\tilde{f}$.

Definition 2 

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

Definition 3 

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

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

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

(9)

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

(10)

$${\left[𝒯\odot \tilde{f}\right]}^{\tau }=𝒯{\left[\tilde{f}\right]}^{\tau },$$

(11)

Equations (4)–(6) directly relate to these processes.

Definition 4 

([69]). Let $H$ be a Hausdorff metric. Then a supremum metric is handled by the space ${\mathcal{L}}_C$; that is, for each $\tilde{f}, \tilde{g}\in {\mathcal{L}}_C$, the whole metric space is represented by the formula

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

(12)

Theorem 1 

([75,78]). $𝒯$ is an Aumann integrable (IA integrable) over $[𝚤, 𝚥]$ when and only when ${𝒯}_{*}\left(ⱷ\right)$ and ${𝒯}^{*}\left(ⱷ\right)$ both are integrable over $\left[𝚤, 𝚥\right],$ such that

$$\left(IA\right){\int }_{𝚤}^{𝚥}𝒯\left(ⱷ\right)dⱷ=\left[{\int }_{𝚤}^{𝚥}{𝒯}_{*}\left(ⱷ\right)dⱷ, {\int }_{𝚤}^{𝚥}{𝒯}^{*}\left(ⱷ\right)dⱷ\right],$$

(13)

where $𝒯:[𝚤, 𝚥]\subset \mathbb{R}\to {\mathcal{L}}_C$ is an interval-valued mapping ($IVM$) fulfilling that $𝒯\left(ⱷ\right)=\left[{𝒯}_{*}\left(ⱷ\right), {𝒯}^{*}\left(ⱷ\right)\right]$.

The literature supports the following inferences [47,48,70,72,73]:

Definition 5. 

([48]). A fuzzy interval-valued map$\widetilde{𝒯}:K\subset \mathbb{R}\to {\mathcal{L}}_C$ is called $FNVM$. For each $\tau \in (0, 1],$ its $IVM$s are classified according to their $\tau$-cuts ${𝒯}_{\tau }:K\subset \mathbb{R}\to {\mathcal{L}}_C$ are given by ${𝒯}_{\tau }\left(ⱷ\right)=\left[{𝒯}_{*}\left(ⱷ,\tau \right), {𝒯}^{*}\left(ⱷ,\tau \right)\right]$ for all $ⱷ\in K.$ Here, for each $\tau \in (0, 1],$ the end point real functions ${𝒯}_{*}\left(.,\tau \right), {𝒯}^{*}\left(.,\tau \right):K\to \mathbb{R}$ are called lower and upper functions of $\widetilde{𝒯}(ⱷ)$.

Definition 6. 

Let  $\widetilde{𝒯}:[𝚤, 𝚥]\subset \mathbb{R}\to {\mathcal{L}}_C$   be a   $FNVM$  . Then, fuzzy integral of   $\widetilde{𝒯}$   over   $\left[𝚤, 𝚥\right],$  denoted by   $\left(FA\right){\int }_{𝚤}^{𝚥}\widetilde{𝒯}\left(ⱷ\right)dⱷ$  , is given level-wise by

$${\left[\left(FA\right){\int }_{𝚤}^{𝚥}\widetilde{𝒯}\left(ⱷ\right)dⱷ\right]}^{\begin{array}{c} \\ \tau \end{array}}=\left(IA\right){\int }_{𝚤}^{𝚥}{𝒯}_{\tau }\left(ⱷ\right)dⱷ=\left\{{\int }_{𝚤}^{𝚥}𝒯\left(ⱷ,\tau \right)dⱷ:𝒯\left(ⱷ,\tau \right)\in {\mathcal{R}}_{\left(\left[𝚤, 𝚥\right], \tau \right)}\right\},$$

(14)

 for all $\tau \in (0, 1],$ where ${\mathcal{R}}_{\left(\left[𝚤, 𝚥\right], \tau \right)}$ denotes the collection of Riemannian integrable functions of $IVM$s. The $FNVM$ $\widetilde{𝒯}$ is $FA$-integrable over $[𝚤, 𝚥]$ if $\left(FA\right){\int }_{𝚤}^{𝚥}\widetilde{𝒯}\left(ⱷ\right)dⱷ\in {\mathcal{L}}_C.$ Note that, if ${𝒯}_{*}\left(ⱷ,\tau \right), {𝒯}^{*}\left(ⱷ,\tau \right)$ are Lebesgue-integrable, then $𝒯$ is fuzzy Aumann-integrable function over $[𝚤, 𝚥]$, see [47].

Theorem 2. 

Let   $\widetilde{𝒯}:[𝚤, 𝚥]\subset \mathbb{R}\to {\mathcal{L}}_C$  be a   $FNVM$  , its   $IVM$  s are classified according to their   $\tau$  -cuts   ${𝒯}_{\tau }:[𝚤, 𝚥]\subset \mathbb{R}\to {\mathcal{L}}_C$  are given by   ${𝒯}_{\tau }\left(ⱷ\right)=\left[{𝒯}_{*}\left(ⱷ,\tau \right), {𝒯}^{*}\left(ⱷ,\tau \right)\right]$  for all   $ⱷ\in [𝚤, 𝚥]$  and for all   $\tau \in (0, 1].$  Then,   $\widetilde{𝒯}$  is   $FA$  -integrable over   $[𝚤, 𝚥]$  if and only if,   ${𝒯}_{*}\left(ⱷ,\tau \right)$  and   ${𝒯}^{*}\left(ⱷ,\tau \right)$  are both   $A$  -integrable over   $[𝚤, 𝚥]$  . Moreover, if   $\widetilde{𝒯}$  is   $FA$  -integrable over   $\left[𝚤, 𝚥\right],$  then

$$\begin{gathered} {\left[\left(FA\right){\int }_{𝚤}^{𝚥}\widetilde{𝒯}\left(ⱷ\right)dⱷ\right]}^{\tau }=\left[\left(A\right){\int }_{𝚤}^{𝚥}{𝒯}_{*}\left(ⱷ,\tau \right)dⱷ, \left(A\right){\int }_{𝚤}^{𝚥}{𝒯}^{*}\left(ⱷ,\tau \right)dⱷ\right] \\ =\left(IA\right){\int }_{𝚤}^{𝚥}{𝒯}_{\tau }\left(ⱷ\right)dⱷ, \end{gathered}$$

(15)

 for all $\tau \in (0, 1].$ For all $\tau \in \left(0, 1\right],$ ${\mathcal{F}\mathcal{A}}_{\left(\left[𝚤, 𝚥\right], \tau \right)}$ denotes the collection of all $FA$-integrable $FNVM$s over $[𝚤, 𝚥]$.

Definition 7 

([80]). Let $\left[𝚤, 𝚥\right]$ be a convex interval. Then, $FNVM$ $\widetilde{𝒯}:\left[𝚤, 𝚥\right]\to {\mathcal{L}}_C$ is said to be $UD$-convex on$\left[𝚤, 𝚥\right]$ if

$$\widetilde{𝒯}\left(\hslash ⱷ+\left(1-\hslash \right)ⱴ\right){\supseteq }_{\mathbb{F}}\hslash \odot \widetilde{𝒯}\left(ⱷ\right)\oplus (1-\hslash )\odot \widetilde{𝒯}\left(ⱴ\right),$$

(16)

 for all  $ⱷ, ⱴ\in \left[𝚤, 𝚥\right], \hslash \in \left[0, 1\right],$  where   $\widetilde{𝒯}\left(ⱷ\right){\ge }_{\mathbb{F}}\tilde{0}$  , for all   $ⱷ\in \left[𝚤, 𝚥\right]$  . If inequality (16) is reversed, then   $\widetilde{𝒯}$  is said to be   $UD$  -concave   $FNVM$  on   $\left[𝚤, 𝚥\right]$  . The set of all   $UD$  -convex (  $UD$  -concave)   $FNVM$  s is denoted by

$$UDFSX\left(\left[𝚤, 𝚥\right], {\mathcal{L}}_C\right), \left(UDFSV\left(\left[𝚤, 𝚥\right], {\mathcal{L}}_C\right)\right).$$

Definition 8. 

Let $\left[𝚤, 𝚥\right]$ be a $p$-convex interval. Then, $FNVM$ $\widetilde{𝒯}:\left[𝚤, 𝚥\right]\to {\mathcal{L}}_C$ is said to be $UD$-$p$-convex on$\left[𝚤, 𝚥\right]$ if

$$\widetilde{𝒯}\left({\left[\hslash {ⱷ}^p+\left(1-\hslash \right){ⱴ}^p\right]}^{\frac{1}{p}} \right){\supseteq }_{\mathbb{F}}\hslash \odot \widetilde{𝒯}\left(ⱷ\right)\oplus (1-\hslash )\odot \widetilde{𝒯}\left(ⱴ\right),$$

(17)

 for all  $ⱷ, ⱴ\in \left[𝚤, 𝚥\right], \hslash \in \left[0, 1\right],$  where   $\widetilde{𝒯}\left(ⱷ\right){\ge }_{\mathbb{F}}\tilde{0}$  , for all   $ⱷ\in \left[𝚤, 𝚥\right]$  . If inequality (17) is reversed, then   $\widetilde{𝒯}$  is said to be   $UD$-$p$-concave   $FNVM$  on   $\left[𝚤, 𝚥\right]$  . The set of all   $UD$-$p$-convex ($UD$-$p$-concave)   $FNVM$  s is denoted by

$$UDFSX\left(\left[𝚤,\mathit{ȷ}\right], {\mathcal{L}}_C, p\right), \left(UDFSV\left(\left[𝚤, 𝚥\right], {\mathcal{L}}_C, p\right)\right).$$

Remark 2. 

If $p\equiv 1$, then $UD$-$p$-convex $FNVM$ becomes $UD$-convex $FNVM$, see Definition 7.

When $p\equiv -1$, then inequality (17) is converted into inequality obtained from the definition of harmonically $\mathrm{U}\mathrm{D}$-convex $\mathrm{F}\mathrm{N}\mathrm{V}\mathrm{M}$s.

The following results discuss the characterization of definition of $\mathrm{U}\mathrm{D}$-convex $\mathrm{F}\mathrm{N}\mathrm{V}\mathrm{M}$

Theorem 3. 

Let $\left[𝚤, 𝚥\right]$ be a convex set, and $\widetilde{𝒯}:\left[𝚤, 𝚥\right]\to {\mathcal{L}}_C$ be a $FNVM$. The family of $IVMs$ is defined by $\tau$-cuts ${𝒯}_{\tau }:\left[𝚤, 𝚥\right]\subset \mathbb{R}\to {{\mathcal{K}}_C}^{+}\subset {\mathcal{K}}_C$ are given by

$${𝒯}_{\tau }\left(ⱷ\right)=\left[{𝒯}_{*}\left(ⱷ,\tau \right), {𝒯}^{*}\left(ⱷ,\tau \right)\right], \forall ⱷ\in \left[𝚤, 𝚥\right]$$

(18)

 for all $ⱷ\in \left[𝚤, 𝚥\right]$ and for all $\tau \in \left[0, 1\right]$. Then, $\widetilde{𝒯}$ is $UD$-$p$-convex on $\left[𝚤, 𝚥\right],$ if and only if, for all $\tau \in \left[0, 1\right],$ ${𝒯}_{*}\left(ⱷ, \tau \right)$ is $p$-convex and ${𝒯}^{*}\left(ⱷ, \tau \right)$ is $p$-concave functions$.$

Proof. 

Assume that for each $\tau \in \left[0, 1\right],$ ${𝒯}_{*}\left(ⱷ, \tau \right)$ is $p$-convex and ${𝒯}^{*}\left(ⱷ, \tau \right)$ is $p$-concave functions on $\left[𝚤, 𝚥\right].$ Then, from (17) we have

$${𝒯}_{*}\left({\left[\hslash {ⱷ}^p+\left(1-\hslash \right){ⱴ}^p\right]}^{\frac{1}{p}} , \tau \right)\le \hslash {𝒯}_{*}\left(ⱷ, \tau \right)+\left(1-\hslash \right){𝒯}_{*}\left(ⱴ, \tau \right), \forall ⱷ,ⱴ\in \left[𝚤, 𝚥\right], \hslash \in \left[0, 1\right],$$

and

$${𝒯}^{*}\left({\left[\hslash {ⱷ}^p+\left(1-\hslash \right){ⱴ}^p\right]}^{\frac{1}{p}} , \tau \right)\ge \hslash {𝒯}^{*}\left(ⱷ, \tau \right)+\left(1-\hslash \right){𝒯}^{*}\left(ⱴ, \tau \right),\forall ⱷ,ⱴ\in \left[𝚤, 𝚥\right], \hslash \in \left[0, 1\right].$$

Then, by (9), (11), and (18), we obtain

$$\begin{gathered} {𝒯}_{\tau }\left({\left[\hslash {ⱷ}^p+\left(1-\hslash \right){ⱴ}^p\right]}^{\frac{1}{p}} \right)=\left[{𝒯}_{*}\left({\left[\hslash {ⱷ}^p+\left(1-\hslash \right){ⱴ}^p\right]}^{\frac{1}{p}} , \tau \right), {𝒯}^{*}\left({\left[\hslash {ⱷ}^p+\left(1-\hslash \right){ⱴ}^p\right]}^{\frac{1}{p}}, \tau \right)\right], \\ {\supseteq }_I\left[\hslash {𝒯}_{*}\left(ⱷ, \tau \right), {\mathcal{\hslash }𝒯}^{*}\left(ⱷ, \tau \right)\right]+\left[\left(1-\hslash \right){𝒯}_{*}\left(ⱴ, \tau \right), {\left(1-\hslash \right)𝒯}^{*}\left(ⱴ, \tau \right)\right], \end{gathered}$$

that is

$$\widetilde{𝒯}\left({\left[\hslash {ⱷ}^p+\left(1-\hslash \right){ⱴ}^p\right]}^{\frac{1}{p}} \right){\supseteq }_{\mathbb{F}}\hslash \odot \widetilde{𝒯}\left(ⱷ\right)\oplus \left(1-\hslash \right)\odot \widetilde{𝒯}\left(ⱴ\right),\forall ⱷ,ⱴ\in \left[𝚤, 𝚥\right], \hslash \in \left[0, 1\right].$$

Hence, $\widetilde{𝒯}$ is $\mathrm{U}\mathrm{D}$-$p$-convex $\mathrm{F}\mathrm{N}\mathrm{V}\mathrm{M}$ on $\left[𝚤,𝚥\right].$

Conversely, let $\widetilde{𝒯}$ be $\mathrm{U}\mathrm{D}$-$p$-convex $\mathrm{F}\mathrm{N}\mathrm{V}\mathrm{M}$ on $\left[𝚤, 𝚥\right].$ Then, for all $ⱷ,ⱴ\in \left[𝚤, 𝚥\right]$, and $\hslash \in \left[0, 1\right],$ we have

$$\widetilde{𝒯}\left({\left[\hslash {ⱷ}^p+\left(1-\hslash \right){ⱴ}^p\right]}^{\frac{1}{p}} \right){\supseteq }_{\mathbb{F}}\hslash \odot \widetilde{𝒯}\left(ⱷ\right)\oplus \left(1-\hslash \right)\odot \widetilde{𝒯}\left(ⱴ\right).$$

Therefore, from (18), we have

$${𝒯}_{\tau }\left({\left[\hslash {ⱷ}^p+\left(1-\hslash \right){ⱴ}^p\right]}^{\frac{1}{p}} \right)=\left[{𝒯}_{*}\left({\left[\hslash {ⱷ}^p+\left(1-\hslash \right){ⱴ}^p\right]}^{\frac{1}{p}} , \tau \right), {𝒯}^{*}\left({\left[\hslash {ⱷ}^p+\left(1-\hslash \right){ⱴ}^p\right]}^{\frac{1}{p}} , \tau \right)\right].$$

Again, from (9), (11), and (18), we obtain

$$\hslash {𝒯}_{\tau }\left(ⱷ\right)+\left(1-\hslash \right){𝒯}_{\tau }\left(ⱴ\right)=\left[\hslash {𝒯}_{*}\left(ⱷ, \tau \right), {\mathcal{\hslash }𝒯}^{*}\left(ⱷ, \tau \right)\right]+\left[\left(1-\hslash \right){𝒯}_{*}\left(ⱴ, \tau \right), {\left(1-\hslash \right)𝒯}^{*}\left(ⱴ, \tau \right)\right],$$

for all $ⱷ,ⱴ\in \left[𝚤, 𝚥\right]$ and $\hslash \in \left[0, 1\right].$ Then, by $\mathrm{U}\mathrm{D}$-$p$-convexity of $\widetilde{𝒯}$, we have for all $ⱷ,ⱴ\in \left[𝚤, 𝚥\right]$ and $\hslash \in \left[0, 1\right]$such that

$${𝒯}_{*}\left({\left[\hslash {ⱷ}^p+\left(1-\hslash \right){ⱴ}^p\right]}^{\frac{1}{p}} , \tau \right)\le \hslash {𝒯}_{*}\left(ⱷ, \tau \right)+\left(1-\hslash \right){𝒯}_{*}\left(ⱴ, \tau \right),$$

and

$${𝒯}^{*}\left({\left[\hslash {ⱷ}^p+\left(1-\hslash \right){ⱴ}^p\right]}^{\frac{1}{p}} , \tau \right)\ge \hslash {𝒯}^{*}\left(ⱷ, \tau \right)+\left(1-\hslash \right){𝒯}^{*}\left(ⱴ, \tau \right),$$

for each $\tau \in \left[0, 1\right].$ Hence, the result follows. □

Example 1. 

We consider the $FNVM$ $𝒯:[0, 1]\to {\mathcal{L}}_C$ defined by,

$$𝒯\left(ⱷ\right)\left(\lambda \right)=\left\{\begin{array}{c}\frac{\lambda }{{2ⱷ}^2}, \lambda \in \left[0, {2ⱷ}^2\right]\\ \frac{{4ⱷ}^2-\lambda }{{2ⱷ}^2}, \lambda \in ({2ⱷ}^2, {4ⱷ}^2]\\ 0, otherwise,\end{array}\right.$$

then, for each $\tau \in \left[0, 1\right],$ we have ${𝒯}_{\tau }\left(ⱷ\right)=\left[2\tau {ⱷ}^2,(4-2\tau ){ⱷ}^2 \right]$. Since end point functions, ${𝒯}_{*}\left(ⱷ, \tau \right)$ is $p$-convex and ${𝒯}^{*}\left(ⱷ, \tau \right)$ is $p$-concave functions$.$for each $\tau \in [0, 1]$. Hence $\widetilde{𝒯}\left(ⱷ\right)$ is $UD$-$p$-convex $FNVM$.

Remark 3. 

If ${𝒯}_{*}\left(ⱷ , \tau \right)={𝒯}^{*}\left(ⱷ , \tau \right)$, then Definition 6 cuts down to the definition of classical $p$-convex function, [90].

If  ${𝒯}_{*}\left(ⱷ, \tau \right)={𝒯}^{*}\left(ⱷ, \tau \right)$  and  $p\equiv 1$  , then definition 6 cuts down to the definition of classical convex function.

3. Jensen’s and Schur’s Type Inequalities

We first provide a new ideal inequality known as discrete Jensen’s type inequality for $\mathrm{U}\mathrm{D}$-$p$-convex $\mathrm{F}\mathrm{N}\mathrm{V}\mathrm{M}$. This is how it is explained.

Theorem 4. 

Let ${\hslash }_{\mathcal{l}}\in {\mathbb{R}}^{+}$, ${𝚤}_{\mathcal{l}}\in \left[𝚤, 𝚥\right],$ $\left(\mathcal{l}=1, 2, 3, \dots ,k, k\ge 2\right)$ and $\widetilde{𝒯}\in UDFSX\left(\left[𝚤, 𝚥\right], {\mathcal{L}}_C, p\right)$ and for all $\tau \in \left[0, 1\right]$, the family of $IVMs$ is defined by $\tau$-cuts ${𝒯}_{\tau }:\left[𝚤, 𝚥\right]\subset \mathbb{R}\to {{\mathcal{K}}_C}^{+}$ are given by ${𝒯}_{\tau }\left(ⱷ\right)=\left[{𝒯}_{*}\left(ⱷ,\tau \right), {𝒯}^{*}\left(ⱷ,\tau \right)\right]$ for all $ⱷ\in \left[𝚤, 𝚥\right]$. Then,

$$\widetilde{𝒯}\left({\left[\frac{1}{W_k}{\sum }_{\mathcal{l}=1}^k{\hslash }_{\mathcal{l}}{{𝚤}_{\mathcal{l}}}^p\right]}^{\frac{1}{p}}\right){\supseteq }_{\mathbb{F}}{\sum }_{\mathcal{l}}^k\frac{{\hslash }_{\mathcal{l}}}{W_k}\odot \widetilde{𝒯}\left({𝚤}_{\mathcal{l}}\right),$$

(19)

where  $W_k=\sum _{\mathcal{l}=1}^k{\hslash }_{\mathcal{l}}.$If  $\widetilde{𝒯}$  is  $UD$-$p$-concave, then inequality (19) is reversed.

Proof. 

When $k=2$, then inequality (19) is true. Consider inequality (19) is true for $k=\mathcal{n}-1,$ then

$$\widetilde{𝒯}\left({\left[\frac{1}{W_{\mathcal{n}-1}}{\sum }_{\mathcal{l}=1}^{\mathcal{n}-1}{\hslash }_{\mathcal{l}}{{𝚤}_{\mathcal{l}}}^p\right]}^{\frac{1}{p}}\right){\supseteq }_{\mathbb{F}}{\sum }_{\mathcal{l}=1}^{\mathcal{n}-1}\frac{{\hslash }_{\mathcal{l}}}{W_{\mathcal{n}-1}}\widetilde{𝒯}\left({𝚤}_{\mathcal{l}}\right).$$

Now, let us prove that inequality (19) holds for $k=\mathcal{n}.$

$$\widetilde{𝒯}\left({\left[\frac{1}{W_{\mathcal{n}}}{\sum }_{\mathcal{l}=1}^{\mathcal{n}}{\hslash }_{\mathcal{l}}{{𝚤}_{\mathcal{l}}}^p\right]}^{\frac{1}{p}}\right)=\widetilde{𝒯}\left({\left[\frac{1}{W_{\mathcal{n}}}{\sum }_{\mathcal{l}=1}^{\mathcal{n}-2}{\hslash }_{\mathcal{l}}{{𝚤}_{\mathcal{l}}}^p+\frac{{\hslash }_{\mathcal{n}-1}+{\hslash }_{\mathcal{n}}}{W_{\mathcal{n}}}(\frac{{\hslash }_{\mathcal{n}-1}}{{\hslash }_{\mathcal{n}-1}+{\hslash }_{\mathcal{n}}}{{𝚤}_{\mathcal{n}-1}}^p+\frac{{\hslash }_{\mathcal{n}}}{{\hslash }_{\mathcal{n}-1}+{\hslash }_{\mathcal{n}}}{{𝚤}_{\mathcal{n}}}^p\right]}^{\frac{1}{p}}\right).$$

Therefore, for each $\tau \in \left[0, 1\right],$ we have

$$\begin{gathered} \begin{array}{c}{𝒯}_{*}\left({\left[\frac{1}{W_{\mathcal{n}}}{\displaystyle \sum _{\mathcal{l}=1}^{\mathcal{n}}}{\hslash }_{\mathcal{l}}{{𝚤}_{\mathcal{l}}}^p\right]}^{\frac{1}{p}},\tau \right)\\ {𝒯}^{*}\left({\left[\frac{1}{W_{\mathcal{n}}}{\displaystyle \sum _{\mathcal{l}=1}^{\mathcal{n}}}{\hslash }_{\mathcal{l}}{{𝚤}_{\mathcal{l}}}^p\right]}^{\frac{1}{p}},\tau \right)\end{array} \\ \begin{array}{c}={𝒯}_{*}\left({\left[\frac{1}{W_{\mathcal{n}}}{\displaystyle \sum _{\mathcal{l}=1}^{\mathcal{n}-2}}{\hslash }_{\mathcal{l}}{{𝚤}_{\mathcal{l}}}^p+\frac{{\hslash }_{\mathcal{n}-1}+{\hslash }_{\mathcal{n}}}{W_{\mathcal{n}}}(\frac{{\hslash }_{\mathcal{n}-1}}{{\hslash }_{\mathcal{n}-1}+{\hslash }_{\mathcal{n}}}{{𝚤}_{\mathcal{n}-1}}^p+\frac{{\hslash }_{\mathcal{n}}}{{\hslash }_{\mathcal{n}-1}+{\hslash }_{\mathcal{n}}}{{𝚤}_{\mathcal{n}}}^p\right]}^{\frac{1}{p}},\tau \right),\\ ={𝒯}^{*}\left({\left[\frac{1}{W_{\mathcal{n}}}{\displaystyle \sum _{\mathcal{l}=1}^{\mathcal{n}-2}}{\hslash }_{\mathcal{l}}{{𝚤}_{\mathcal{l}}}^p+\frac{{\hslash }_{\mathcal{n}-1}+{\hslash }_{\mathcal{n}}}{W_{\mathcal{n}}}(\frac{{\hslash }_{\mathcal{n}-1}}{{\hslash }_{\mathcal{n}-1}+{\hslash }_{\mathcal{n}}}{{𝚤}_{\mathcal{n}-1}}^p+\frac{{\hslash }_{\mathcal{n}}}{{\hslash }_{\mathcal{n}-1}+{\hslash }_{\mathcal{n}}}{{𝚤}_{\mathcal{n}}}^p\right]}^{\frac{1}{p}},\tau \right),\end{array} \\ \begin{array}{c}\le {\sum }_{\mathcal{l}=1}^{\mathcal{n}-2}\frac{{\hslash }_{\mathcal{l}}}{W_{\mathcal{n}}}{𝒯}_{*}\left({𝚤}_{\mathcal{l}},\tau \right)+\frac{{\hslash }_{\mathcal{n}-1}+{\hslash }_{\mathcal{n}}}{W_{\mathcal{n}}}{𝒯}_{*}\left({\left[\frac{{\hslash }_{\mathcal{n}-1}}{{\hslash }_{\mathcal{n}-1}+{\hslash }_{\mathcal{n}}}{{𝚤}_{\mathcal{n}-1}}^p+\frac{{\hslash }_{\mathcal{n}}}{{\hslash }_{\mathcal{n}-1}+{\hslash }_{\mathcal{n}}}{{𝚤}_{\mathcal{n}}}^p\right]}^{\frac{1}{p}},\tau \right), \\ \ge {\sum }_{\mathcal{l}=1}^{\mathcal{n}-2}\frac{{\hslash }_{\mathcal{l}}}{W_{\mathcal{n}}}{𝒯}^{*}\left({𝚤}_{\mathcal{l}},\tau \right)+\frac{{\hslash }_{\mathcal{n}-1}+{\hslash }_{\mathcal{n}}}{W_{\mathcal{n}}}{𝒯}^{*}\left({\left[\frac{{\hslash }_{\mathcal{n}-1}}{{\hslash }_{\mathcal{n}-1}+{\hslash }_{\mathcal{n}}}{{𝚤}_{\mathcal{n}-1}}^p+\frac{{\hslash }_{\mathcal{n}}}{{\hslash }_{\mathcal{n}-1}+{\hslash }_{\mathcal{n}}}{{𝚤}_{\mathcal{n}}}^p\right]}^{\frac{1}{p}},\tau \right),\end{array} \\ \begin{array}{c}\le {\sum }_{\mathcal{l}=1}^{\mathcal{n}-2}\frac{{\hslash }_{\mathcal{l}}}{W_{\mathcal{n}}}{𝒯}_{*}\left({𝚤}_{\mathcal{l}},\tau \right)+\frac{{\hslash }_{\mathcal{n}-1}+{\hslash }_{\mathcal{n}}}{W_{\mathcal{n}}}\left[ \frac{{\hslash }_{\mathcal{n}-1}}{{\hslash }_{\mathcal{n}-1}+{\hslash }_{\mathcal{n}}}{𝒯}_{*}\left({𝚤}_{\mathcal{n}-1},\tau \right)+\frac{{\hslash }_{\mathcal{n}}}{{\hslash }_{\mathcal{n}-1}+{\hslash }_{\mathcal{n}}}{𝒯}_{*}\left({𝚤}_{\mathcal{n}},\tau \right)\right], \\ \ge {\sum }_{\mathcal{l}=1}^{\mathcal{n}-2}\frac{{\hslash }_{\mathcal{l}}}{W_{\mathcal{n}}}{𝒯}^{*}\left({𝚤}_{\mathcal{l}},\tau \right)+\frac{{\hslash }_{\mathcal{n}-1}+{\hslash }_{\mathcal{n}}}{W_{\mathcal{n}}}\left[\frac{{\hslash }_{\mathcal{n}-1}}{{\hslash }_{\mathcal{n}-1}+{\hslash }_{\mathcal{n}}}{𝒯}^{*}\left({𝚤}_{\mathcal{n}-1},\tau \right)+\frac{{\hslash }_{\mathcal{n}}}{{\hslash }_{\mathcal{n}-1}+{\hslash }_{\mathcal{n}}}{𝒯}^{*}\left({𝚤}_{\mathcal{n}},\tau \right)\right],\end{array} \\ \begin{array}{c}\le {\sum }_{\mathcal{l}=1}^{\mathcal{n}-2}\frac{{\hslash }_{\mathcal{l}}}{W_{\mathcal{n}}}{𝒯}_{*}\left({𝚤}_{\mathcal{l}},\tau \right)+\left[\frac{{\hslash }_{\mathcal{n}-1}}{W_{\mathcal{n}}}{𝒯}_{*}\left({𝚤}_{\mathcal{n}-1},\tau \right)+\frac{{\hslash }_{\mathcal{n}}}{W_{\mathcal{n}}}{𝒯}_{*}\left({𝚤}_{\mathcal{n}},\tau \right)\right], \\ \ge {\sum }_{\mathcal{l}=1}^{\mathcal{n}-2}\frac{{\hslash }_{\mathcal{l}}}{W_{\mathcal{n}}}{𝒯}^{*}\left({𝚤}_{\mathcal{l}},\tau \right)+\left[\frac{{\hslash }_{\mathcal{n}-1}}{W_{\mathcal{n}}}{𝒯}^{*}\left({𝚤}_{\mathcal{n}-1},\tau \right)+\frac{{\hslash }_{\mathcal{n}}}{W_{\mathcal{n}}}{𝒯}^{*}\left({𝚤}_{\mathcal{n}},\tau \right)\right],\end{array} \\ \begin{array}{c}={\sum }_{\mathcal{l}=1}^{\mathcal{n}}\frac{{\hslash }_{\mathcal{l}}}{W_{\mathcal{n}}}{𝒯}_{*}\left({𝚤}_{\mathcal{l}},\tau \right), \\ ={\sum }_{\mathcal{l}=1}^{\mathcal{n}}\frac{{\hslash }_{\mathcal{l}}}{W_{\mathcal{n}}}{𝒯}^{*}\left({𝚤}_{\mathcal{l}},\tau \right).\end{array} \end{gathered}$$

From which, we have

$$\left[{𝒯}_{*}\left({\left[\frac{1}{W_{\mathcal{n}}}{\sum }_{\mathcal{l}=1}^{\mathcal{n}}{\hslash }_{\mathcal{l}}{{𝚤}_{\mathcal{l}}}^p\right]}^{\frac{1}{p}},\tau \right), {𝒯}^{*}\left({\left[\frac{1}{W_{\mathcal{n}}}{\sum }_{\mathcal{l}=1}^{\mathcal{n}}{\hslash }_{\mathcal{l}}{{𝚤}_{\mathcal{l}}}^p\right]}^{\frac{1}{p}},\tau \right)\right]{\supseteq }_I\left[{\sum }_{\mathcal{l}=1}^{\mathcal{n}}\frac{{\hslash }_{\mathcal{l}}}{W_{\mathcal{n}}}{𝒯}_{*}\left({𝚤}_{\mathcal{l}},\tau \right), {\sum }_{\mathcal{l}=1}^{\mathcal{n}}\frac{{\hslash }_{\mathcal{l}}}{W_{\mathcal{n}}}{𝒯}^{*}\left({𝚤}_{\mathcal{l}},\tau \right)\right],$$

that is

$$\widetilde{𝒯}\left({\left[\frac{1}{W_{\mathcal{n}}}\sum _{\mathcal{l}=1}^{\mathcal{n}}{\hslash }_{\mathcal{l}}{{𝚤}_{\mathcal{l}}}^p\right]}^{\frac{1}{p}}\right){\supseteq }_{\mathbb{F}}{\sum }_{\mathcal{l}=1}^{\mathcal{n}}\frac{{\hslash }_{\mathcal{l}}}{W_{\mathcal{n}}}\odot \widetilde{𝒯}\left({𝚤}_{\mathcal{l}}\right),$$

and the result follows. □

If ${\hslash }_1={\hslash }_2={\hslash }_3=\dots ={\hslash }_k=1,$ then from (19) we obtain following outcome:

Corollary 1. 

Let ${𝚤}_{\mathcal{l}}\in \left[𝚤, 𝚥\right],$ $\left(\mathcal{l}=1, 2, 3, \dots ,k, k\ge 2\right)$, and $\widetilde{𝒯}\in UDFSX\left(\left[𝚤, 𝚥\right], {\mathcal{L}}_C, p\right)$. The family of $IVMs$ is defined by $\tau$-cuts ${𝒯}_{\tau }:\left[𝚤, 𝚥\right]\subset \mathbb{R}\to {{\mathcal{K}}_C}^{+}$ are given by ${𝒯}_{\tau }\left(ⱷ\right)=\left[{𝒯}_{*}\left(ⱷ,\tau \right), {𝒯}^{*}\left(ⱷ,\tau \right)\right]$ for all $ⱷ\in \left[𝚤, 𝚥\right]$, and for all $\tau \in \left[0, 1\right]$. Then,

$$\widetilde{𝒯}\left({\left[\frac{1}{W_k}{\sum }_{\mathcal{l}=1}^k{\hslash }_{\mathcal{l}}{{𝚤}_{\mathcal{l}}}^p\right]}^{\frac{1}{p}}\right){\supseteq }_{\mathbb{F}}{\sum }_{𝚥=1}^k\frac{1}{k}\odot \widetilde{𝒯}\left({𝚤}_{\mathcal{l}}\right).$$

(20)

If  $\widetilde{𝒯}$  is an  $p$-concave, then inequality (20) is reversed.

Here is the generalized form of discrete Schur’s type inequality for $\mathrm{U}\mathrm{D}$-$p$-convex $\mathrm{F}\mathrm{N}\mathrm{V}\mathrm{M}$.

Theorem 5. 

Let $\widetilde{𝒯}\in UDFSX\left(\left[𝚤, 𝚥\right], {\mathcal{L}}_C, p\right)$. The family of $IVMs$ is defined by $\tau$-cuts ${𝒯}_{\tau }:\left[𝚤, 𝚥\right]\subset \mathbb{R}\to {{\mathcal{K}}_C}^{+}$ are given by ${𝒯}_{\tau }\left(ⱷ\right)=\left[{𝒯}_{*}\left(ⱷ,\tau \right), {𝒯}^{*}\left(ⱷ,\tau \right)\right]$ for all $ⱷ\in \left[𝚤, 𝚥\right]$, and for all $\tau \in \left[0, 1\right]$. If ${𝚤}_1, {𝚤}_2, {𝚤}_3\in \left[𝚤, 𝚥\right]$, such that ${𝚤}_1<{𝚤}_2<{𝚤}_3$ and ${{𝚤}_3}^p-{{𝚤}_1}^p$, ${{𝚤}_3}^p-{{𝚤}_2}^p,$ ${{𝚤}_2}^p-{{𝚤}_1}^p\in \left[0, 1\right]$, then we have

$$\left({{𝚤}_3}^p-{{𝚤}_1}^p\right)\odot \widetilde{𝒯}\left({𝚤}_2\right){\supseteq }_{\mathbb{F}}\left({{𝚤}_3}^p-{{𝚤}_2}^p\right)\odot \widetilde{𝒯}\left({𝚤}_1\right)\oplus \left({{𝚤}_2}^p-{{𝚤}_1}^p\right)\odot \widetilde{𝒯}\left({𝚤}_3\right).$$

(21)

If  $\widetilde{𝒯}$  is a  $UD$-$p$-concave, then inequality (21) is reversed.

Proof. 

Let ${𝚤}_1, {𝚤}_2, {𝚤}_3\in \left[𝚤, 𝚥\right]$ and ${{𝚤}_3}^p-{{𝚤}_1}^p>0.$ Consider$\hslash =\frac{{{𝚤}_3}^p-{{𝚤}_2}^p}{{{𝚤}_3}^p-{{𝚤}_1}^p}$ , then ${{𝚤}_2}^p=\hslash {{𝚤}_1}^p+\left(1-\hslash \right){{𝚤}_3}^p.$ Since $\widetilde{𝒯}$ is a $\mathrm{U}\mathrm{D}$-$p$-convex $\mathrm{F}\mathrm{N}\mathrm{V}\mathrm{M}$, then by hypothesis, we have

$$\widetilde{𝒯}\left({𝚤}_2\right){\supseteq }_{\mathbb{F}}\left(\frac{{{𝚤}_3}^p-{{𝚤}_2}^p}{{{𝚤}_3}^p-{{𝚤}_1}^p}\right)\odot \widetilde{𝒯}\left({𝚤}_1\right)\oplus \left(\frac{{{𝚤}_2}^p-{{𝚤}_1}^p}{{{𝚤}_3}^p-{{𝚤}_1}^p}\right)\odot \widetilde{𝒯}\left({𝚤}_3\right).$$

Therefore, for each $\tau \in \left[0, 1\right],$ we have

$$\begin{array}{c}{𝒯}_{*}\left({𝚤}_2,\tau \right)\le \left(\frac{{{𝚤}_3}^p-{{𝚤}_2}^p}{{{𝚤}_3}^p-{{𝚤}_1}^p}\right){𝒯}_{*}\left({𝚤}_1,\tau \right)+\left(\frac{{{𝚤}_2}^p-{{𝚤}_1}^p}{{{𝚤}_3}^p-{{𝚤}_1}^p}\right){𝒯}_{*}\left({𝚤}_3,\tau \right),\\ {𝒯}^{*}\left({𝚤}_2,\tau \right)\ge \left(\frac{{{𝚤}_3}^p-{{𝚤}_2}^p}{{{𝚤}_3}^p-{{𝚤}_1}^p}\right){𝒯}^{*}\left({𝚤}_1,\tau \right)+\left(\frac{{{𝚤}_2}^p-{{𝚤}_1}^p}{{{𝚤}_3}^p-{{𝚤}_1}^p}\right){𝒯}^{*}\left({𝚤}_3,\tau \right),\end{array}$$

(22)

$$\begin{array}{c}=\frac{\left({{𝚤}_3}^p-{{𝚤}_2}^p\right)}{\left({{𝚤}_3}^p-{{𝚤}_1}^p\right)}{𝒯}_{*}\left({𝚤}_1,\tau \right)+\frac{\left({{𝚤}_2}^p-{{𝚤}_1}^p\right)}{\left({{𝚤}_3}^p-{{𝚤}_1}^p\right)}{𝒯}_{*}\left({𝚤}_3,\tau \right), \\ =\frac{\left({{𝚤}_3}^p-{{𝚤}_2}^p\right)}{\left({{𝚤}_3}^p-{{𝚤}_1}^p\right)}{𝒯}^{*}\left({𝚤}_1,\tau \right)+\frac{\left({{𝚤}_2}^p-{{𝚤}_1}^p\right)}{\left({{𝚤}_3}^p-{{𝚤}_1}^p\right)}{𝒯}^{*}\left({𝚤}_3,\tau \right).\end{array}$$

(23)

From (23), we have

$$\begin{array}{c}\left({{𝚤}_3}^p-{{𝚤}_1}^p\right){𝒯}_{*}\left({𝚤}_2,\tau \right)\le \left({{𝚤}_3}^p-{{𝚤}_2}^p\right){𝒯}_{*}\left({𝚤}_1,\tau \right)+\left({{𝚤}_2}^p-{{𝚤}_1}^p\right){𝒯}_{*}\left({𝚤}_3,\tau \right),\\ \left({{𝚤}_3}^p-{{𝚤}_1}^p\right){𝒯}^{*}\left({𝚤}_2,\tau \right)\ge \left({{𝚤}_3}^p-{{𝚤}_2}^p\right){𝒯}^{*}\left({𝚤}_1,\tau \right)+\left({{𝚤}_2}^p-{{𝚤}_1}^p\right){𝒯}^{*}\left({𝚤}_3,\tau \right),\end{array}$$

that is

$$\begin{gathered} \left[\left({{𝚤}_3}^p-{{𝚤}_1}^p\right){𝒯}_{*}\left({𝚤}_2,\tau \right), \left({{𝚤}_3}^p-{{𝚤}_1}^p\right){𝒯}^{*}\left({𝚤}_2,\tau \right)\right]{\supseteq }_I\left[\left({{𝚤}_3}^p-{{𝚤}_2}^p\right){𝒯}_{*}\left({𝚤}_1,\tau \right)+ \\ \left({{𝚤}_3}^p-{{𝚤}_2}^p\right){𝒯}_{*}\left({𝚤}_3,\tau \right), \left({{𝚤}_2}^p-{{𝚤}_1}^p\right){𝒯}^{*}\left({𝚤}_1,\tau \right)+\left({{𝚤}_2}^p-{{𝚤}_1}^p\right){𝒯}^{*}\left({𝚤}_3,\tau \right)\right], \end{gathered}$$

hence

$$\left({{𝚤}_3}^p-{{𝚤}_1}^p\right)\odot \widetilde{𝒯}\left({𝚤}_2\right){\supseteq }_{\mathbb{F}}{{𝚤}_3}^p-{{𝚤}_2}^p\odot \widetilde{𝒯}\left({𝚤}_1\right)\oplus \left({{𝚤}_2}^p-{{𝚤}_1}^p\right)\odot \widetilde{𝒯}\left({𝚤}_3\right).$$

□

The following theorem provides a clarification of Jensen’s type inequality for $\mathrm{U}\mathrm{D}$-$p$-convex $\mathrm{F}\mathrm{N}\mathrm{V}\mathrm{M}$s.

Theorem 6. 

Let ${\hslash }_{\mathcal{l}}\in {\mathbb{R}}^{+}$, ${𝚤}_{\mathcal{l}}\in \left[𝚤, 𝚥\right],$ $\left(\mathcal{l}=1, 2, 3, \dots ,k, k\ge 2\right)$ and $\widetilde{𝒯}\in UDFSX\left(\left[𝚤, 𝚥\right], {\mathcal{L}}_C, p\right)$. The family of $IVMs$ is defined by $\tau$-cuts ${𝒯}_{\tau }:\left[𝚤, 𝚥\right]\subset \mathbb{R}\to {{\mathcal{K}}_C}^{+}$ are given by ${𝒯}_{\tau }\left(ⱷ\right)=\left[{𝒯}_{*}\left(ⱷ,\tau \right), {𝒯}^{*}\left(ⱷ,\tau \right)\right]$ for all $ⱷ\in \left[𝚤, 𝚥\right]$ and for all $\tau \in \left[0, 1\right]$. If $\left(𝚤, U\right)\subseteq [𝚤,𝚥]$, then

$${\sum }_{\mathcal{l}=1}^k\left(\frac{{\hslash }_{\mathcal{l}}}{W_k}\right)\odot \widetilde{𝒯}\left({𝚤}_{\mathcal{l}}\right){\supseteq }_{\mathbb{F}}{\sum }_{\mathcal{l}=1}^k\left(\left(\frac{U^p-{{𝚤}_{\mathcal{l}}}^p}{U^p-{𝚤}^p}\right)\left(\frac{{\hslash }_{\mathcal{l}}}{W_k}\right)\odot \widetilde{𝒯}\left(𝚤,\tau \right)\oplus \left(\frac{{{𝚤}_{\mathcal{l}}}^p-{𝚤}^p}{U^p-{𝚤}^p}\right)\left(\frac{{\hslash }_{\mathcal{l}}}{W_k}\right)\odot \widetilde{𝒯}\left(U,\tau \right)\right),$$

(24)

where  $W_k=\sum _{\mathcal{l}=1}^k{\hslash }_{\mathcal{l}}.$  If  $\widetilde{𝒯}$  is  $UD$-$p$-concave, then inequality (24) is reversed.

Proof. 

Consider $𝚤={𝚤}_1, {𝚤}_{\mathcal{l}}={𝚤}_2,$ $\left(\mathcal{l}=1, 2, 3, \dots ,k\right)$, $U={𝚤}_3$. Then, by hypothesis and inequality (24), we have

$$\widetilde{𝒯}\left({𝚤}_{\mathcal{l}}\right){\supseteq }_{\mathbb{F}}\left(\frac{U^p-{{𝚤}_{\mathcal{l}}}^p}{U^p-{𝚤}^p}\right)\odot \widetilde{𝒯}\left(𝚤,\tau \right)\oplus \left(\frac{{{𝚤}_{\mathcal{l}}}^p-{𝚤}^p}{U^p-{𝚤}^p}\right)\odot \widetilde{𝒯}\left(U,\tau \right).$$

Therefore, for each $\tau \in \left[0, 1\right]$, we have

$$\begin{array}{c}{𝒯}_{*}\left({𝚤}_{\mathcal{l}},\tau \right)\le \left(\frac{U^p-{{𝚤}_{\mathcal{l}}}^p}{U^p-{𝚤}^p}\right){𝒯}_{*}\left(𝚤,\tau \right)+\left(\frac{{{𝚤}_{\mathcal{l}}}^p-{𝚤}^p}{U^p-{𝚤}^p}\right){𝒯}_{*}\left(U,\tau \right),\\ {𝒯}^{*}\left({𝚤}_{\mathcal{l}},\tau \right)\ge \left(\frac{U-{{𝚤}_{\mathcal{l}}}^p}{U^p-{𝚤}^p}\right){𝒯}^{*}\left(𝚤,\tau \right)+\left(\frac{{{𝚤}_{\mathcal{l}}}^p-{𝚤}^p}{U^p-{𝚤}^p}\right){𝒯}^{*}\left(U,\tau \right).\end{array}$$

The above inequality can be written as

$$\begin{array}{c}\left(\frac{{\hslash }_{\mathcal{l}}}{W_k}\right){𝒯}_{*}\left({𝚤}_{\mathcal{l}},\tau \right)\le \left(\frac{U^p-{{𝚤}_{\mathcal{l}}}^p}{U^p-{𝚤}^p}\right)\left(\frac{{\hslash }_{\mathcal{l}}}{W_k}\right){𝒯}_{*}\left(𝚤,\tau \right)+\left(\frac{{{𝚤}_{\mathcal{l}}}^p-{𝚤}^p}{U^p-{𝚤}^p}\right)\left(\frac{{\hslash }_{\mathcal{l}}}{W_k}\right){𝒯}_{*}\left(U,\tau \right),\\ \left(\frac{{\hslash }_{\mathcal{l}}}{W_k}\right){𝒯}^{*}\left({𝚤}_{\mathcal{l}},\tau \right)\ge \left(\frac{U^p-{{𝚤}_{\mathcal{l}}}^p}{U^p-{𝚤}^p}\right)\left(\frac{{\hslash }_{\mathcal{l}}}{W_k}\right){𝒯}^{*}\left(𝚤,\tau \right)+\left(\frac{{{𝚤}_{\mathcal{l}}}^p-{𝚤}^p}{U^p-{𝚤}^p}\right)\left(\frac{{\hslash }_{\mathcal{l}}}{W_k}\right){𝒯}^{*}\left(U,\tau \right).\end{array}$$

(25)

Taking the sum of all inequalities (25) for $\mathcal{l}=1, 2, 3, \dots ,k,$ we have

$$\begin{array}{c}{\sum }_{\mathcal{l}=1}^k\left(\frac{{\hslash }_{\mathcal{l}}}{W_k}\right){𝒯}_{*}\left({𝚤}_{\mathcal{l}},\tau \right)\le {\sum }_{\mathcal{l}=1}^k\left(\left(\frac{U^p-{{𝚤}_{\mathcal{l}}}^p}{U^p-{𝚤}^p}\right)\left(\frac{{\hslash }_{\mathcal{l}}}{W_k}\right){𝒯}_{*}\left(𝚤,\tau \right)+\left(\frac{{{𝚤}_{\mathcal{l}}}^p-{𝚤}^p}{U^p-{𝚤}^p}\right)\left(\frac{{\hslash }_{\mathcal{l}}}{W_k}\right){𝒯}_{*}\left(U,\tau \right)\right),\\ {\sum }_{\mathcal{l}=1}^k\left(\frac{{\hslash }_{\mathcal{l}}}{W_k}\right){𝒯}^{*}\left({𝚤}_{\mathcal{l}},\tau \right)\ge {\sum }_{\mathcal{l}=1}^k\left(\left(\frac{U^p-{{𝚤}_{\mathcal{l}}}^p}{U^p-{𝚤}^p}\right)\left(\frac{{\hslash }_{\mathcal{l}}}{W_k}\right){𝒯}^{*}\left(𝚤,\tau \right)+\left(\frac{{{𝚤}_{\mathcal{l}}}^p-{𝚤}^p}{U^p-{𝚤}^p}\right)\left(\frac{{\hslash }_{\mathcal{l}}}{W_k}\right){𝒯}^{*}\left(U,\tau \right)\right).\end{array}$$

That is

$$\begin{array}{c}{\sum }_{\mathcal{l}=1}^k\left(\frac{{\hslash }_{\mathcal{l}}}{W_k}\right)𝒯\left({𝚤}_{\mathcal{l}}\right)=\left[{\sum }_{\mathcal{l}=1}^k\left(\frac{{\hslash }_{\mathcal{l}}}{W_k}\right){𝒯}_{*}\left({𝚤}_{\mathcal{l}},\tau \right), {\sum }_{\mathcal{l}=1}^k\left(\frac{{\hslash }_{\mathcal{l}}}{W_k}\right){𝒯}^{*}\left({𝚤}_{\mathcal{l}},\tau \right)\right]\hfill \\ {\supseteq }_I\left[{\sum }_{\mathcal{l}=1}^k\left(\begin{array}{c}\left(\frac{U^p-{{𝚤}_{\mathcal{l}}}^p}{U^p-{𝚤}^p}\right)\left(\frac{{\hslash }_{\mathcal{l}}}{W_k}\right){𝒯}_{*}\left(𝚤,\tau \right)\\ +\left(\frac{{{𝚤}_{\mathcal{l}}}^p-{𝚤}^p}{U^p-{𝚤}^p}\right)\left(\frac{{\hslash }_{\mathcal{l}}}{W_k}\right){𝒯}_{*}\left(U,\tau \right)\end{array}\right), {\sum }_{\mathcal{l}=1}^k\left(\begin{array}{c}\left(\frac{U^p-{{𝚤}_{\mathcal{l}}}^p}{U^p-{𝚤}^p}\right)\left(\frac{{\hslash }_{\mathcal{l}}}{W_k}\right){𝒯}^{*}\left(𝚤,\tau \right)\\ +\left(\frac{{{𝚤}_{\mathcal{l}}}^p-{𝚤}^p}{U^p-{𝚤}^p}\right)\left(\frac{{\hslash }_{\mathcal{l}}}{W_k}\right){𝒯}^{*}\left(U,\tau \right)\end{array}\right)\right],\hfill \\ {\supseteq }_I{\sum }_{\mathcal{l}=1}^k\left(\frac{U^p-{{𝚤}_{\mathcal{l}}}^p}{U^p-{𝚤}^p}\right)\left(\frac{{\hslash }_{\mathcal{l}}}{W_k}\right)\left[{𝒯}_{*}\left(𝚤,\tau \right), {𝒯}^{*}\left(𝚤,\tau \right)\right]+{\sum }_{\mathcal{l}=1}^k\left(\frac{U^p-{{𝚤}_{\mathcal{l}}}^p}{U^p-{𝚤}^p}\right)\left(\frac{{\hslash }_{\mathcal{l}}}{W_k}\right)\left[{𝒯}_{*}\left(U,\tau \right), {𝒯}^{*}\left(U,\tau \right)\right].\hfill \\ ={\sum }_{\mathcal{l}=1}^k\left(\frac{U^p-{{𝚤}_{\mathcal{l}}}^p}{U^p-{𝚤}^p}\right)\left(\frac{{\hslash }_{\mathcal{l}}}{W_k}\right)𝒯\left(𝚤,\tau \right)+{\sum }_{\mathcal{l}=1}^k\left(\frac{U^p-{{𝚤}_{\mathcal{l}}}^p}{U^p-{𝚤}^p}\right)\left(\frac{{\hslash }_{\mathcal{l}}}{W_k}\right)𝒯\left(U,\tau \right).\hfill \end{array}$$

Thus,

$${\sum }_{\mathcal{l}=1}^k\left(\frac{{\hslash }_{\mathcal{l}}}{W_k}\right)\odot \widetilde{𝒯}\left({𝚤}_{\mathcal{l}}\right){\supseteq }_{\mathbb{F}}{\sum }_{\mathcal{l}=1}^k\left(\left(\frac{U^p-{{𝚤}_{\mathcal{l}}}^p}{U^p-{𝚤}^p}\right)\left(\frac{{\hslash }_{\mathcal{l}}}{W_k}\right)\odot \widetilde{𝒯}\left(𝚤\right)\oplus \left(\frac{{{𝚤}_{\mathcal{l}}}^p-{𝚤}^p}{U^p-{𝚤}^p}\right)\left(\frac{{\hslash }_{\mathcal{l}}}{W_k}\right)\odot \widetilde{𝒯}\left(U\right)\right),$$

completes the proof. □

In the next outcomes, we shall discuss the exceptional cases that are acquired from the Theorems 6 and 7.

If ${𝒯}_{*}\left(ⱷ,\tau \right)={𝒯}_{*}\left(ⱷ,\tau \right)$ with$\tau =1$, then Theorems 5 and 6 cuts down to the following results:

Corollary 2 

([81]). Let${\hslash }_{\mathcal{l}}\in {\mathbb{R}}^{+}$, ${𝚤}_{\mathcal{l}}\in \left[𝚤, 𝚥\right],$ $\left(\mathcal{l}=1, 2, 3, \dots ,k, k\ge 2\right)$ and let $𝒯:\left[𝚤, 𝚥\right]\to {\mathbb{R}}^{+}$ be a non-negative real-valued function. If $𝒯$ is a $p$-convex function, then

$$𝒯\left({\left[\frac{1}{W_k}{\sum }_{\mathcal{l}=1}^k{\hslash }_{\mathcal{l}}{{𝚤}_{\mathcal{l}}}^p\right]}^{\frac{1}{p}}\right)\le {\sum }_{\mathcal{l}=1}^k\left(\frac{{\hslash }_{\mathcal{l}}}{W_k}\right)𝒯\left({𝚤}_{\mathcal{l}}\right),$$

(26)

where  $W_k=\sum _{\mathcal{l}=1}^k{\hslash }_{\mathcal{l}}.$  If   $𝒯$  is  $p$-concave function, then inequality (26) is reversed.

Corollary 3 

([81]). Let ${\hslash }_{\mathcal{l}}\in {\mathbb{R}}^{+}$, ${𝚤}_{\mathcal{l}}\in \left[𝚤, 𝚥\right],$ $\left(\mathcal{l}=1, 2, 3, \dots , k, k\ge 2\right)$ and $𝒯:\left[𝚤, 𝚥\right]\to {\mathbb{R}}^{+}$ be a non-negative real-valued function. If $𝒯$ is a $p$-convex function and ${𝚤}_1, {𝚤}_2,\dots ,{𝚤}_{\mathcal{l}}\in \left(𝚤, U\right)\subseteq [𝚤,𝚥]$then,

$${\sum }_{\mathcal{l}=1}^k\left(\frac{{\hslash }_{\mathcal{l}}}{W_k}\right)𝒯\left({𝚤}_{\mathcal{l}}\right)\le {\sum }_{\mathcal{l}=1}^k\left(\left(\frac{U^p-{{𝚤}_{\mathcal{l}}}^p}{U^p-{𝚤}^p}\right)\left(\frac{{\hslash }_{\mathcal{l}}}{W_k}\right)𝒯\left(𝚤\right)+\left(\frac{{{𝚤}_{\mathcal{l}}}^p-{𝚤}^p}{U^p-{𝚤}^p}\right)\left(\frac{{\hslash }_{\mathcal{l}}}{W_k}\right)𝒯\left(U\right)\right),$$

(27)

where  $W_k=\sum _{\mathcal{l}=1}^k{\hslash }_{\mathcal{l}}.$  If  $𝒯$  is a  $p$-concave function, then inequality (27) is reversed.

4. Fuzzy Aumann’s Integral Hermite–Hadamard Type Inequalities

Primary goal and focus of this section is to establish a novel version of the H-H-type inequalities in the mode of $\mathrm{U}\mathrm{D}$-$p$-convex $\mathrm{F}\mathrm{N}\mathrm{V}\mathrm{M}$s via fuzzy Aumann’s integrals.

Theorem 7. 

Let  $\widetilde{𝒯}\in UDFSX\left(\left[𝚤, 𝚥\right], {\mathcal{L}}_C, p\right)$  . The family of  $IVMs$  is defined by  $\tau$-cuts  ${𝒯}_{\tau }:\left[𝚤, 𝚥\right]\subset \mathbb{R}\to {{\mathcal{K}}_C}^{+}$  are given by  ${𝒯}_{\tau }\left(ⱷ\right)=\left[{𝒯}_{*}\left(ⱷ,\tau \right), {𝒯}^{*}\left(ⱷ,\tau \right)\right]$  for all  $ⱷ\in \left[𝚤, 𝚥\right]$  and for all  $\tau \in \left[0, 1\right]$  . If  $\widetilde{𝒯}\in {\mathcal{F}\mathcal{A}}_{\left(\left[𝚤, 𝚥\right], \tau \right)}$, then

$$\widetilde{𝒯}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}\right){\supseteq }_{\mathbb{F}}\frac{p}{{𝚥}^p-{𝚤}^p}\odot \left(FA\right){\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}\odot \widetilde{𝒯}\left(ⱷ\right)dⱷ{\supseteq }_{\mathbb{F}}\frac{\widetilde{𝒯}\left(𝚤\right)\oplus \widetilde{𝒯}\left(𝚥\right)}{2}.$$

(28)

If $\widetilde{𝒯}(ⱷ)$ is $\mathrm{U}\mathrm{D}$-$p$-concave $\mathrm{F}\mathrm{N}\mathrm{V}\mathrm{M}$, then

$$\widetilde{𝒯}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}\right){\subseteq }_{\mathbb{F}}\frac{p}{{𝚥}^p-{𝚤}^p}\odot \left(FA\right){\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}\odot \widetilde{𝒯}\left(ⱷ\right)dⱷ{\subseteq }_{\mathbb{F}}\frac{\widetilde{𝒯}\left(𝚤\right)\oplus \widetilde{𝒯}\left(𝚥\right)}{2}$$

(29)

Proof. 

Let $\widetilde{𝒯}:\left[𝚤, 𝚥\right]\to {\mathcal{L}}_C$ be an $\mathrm{U}\mathrm{D}$-$p$-convex $\mathrm{F}\mathrm{N}\mathrm{V}\mathrm{M}$. Then, by hypothesis, we have

$$2\odot \widetilde{𝒯}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}\right){\supseteq }_{\mathbb{F}}\widetilde{𝒯}\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right)v^p\right]}^{\frac{1}{p}}\right)\oplus \widetilde{𝒯}\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}\right).$$

Therefore, for every $\tau \in [0, 1]$, we have

$$\begin{array}{c}2{𝒯}_{*}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}, \tau \right)\le {𝒯}_{*}\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)+{𝒯}_{*}\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}, \tau \right), \\ 2{𝒯}^{*}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}, \tau \right)\ge {𝒯}^{*}\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)+{𝒯}^{*}\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}},\tau \right).\end{array}$$

Then

$$\begin{array}{c}2{\int }_0^1{𝒯}_{*}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}, \tau \right)d\hslash \le {\int }_0^1{𝒯}_{*}\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}},\tau \right)d\hslash +{\int }_0^1{𝒯}_{*}\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)d\hslash , \\ 2{\int }_0^1{𝒯}^{*}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}},\tau \right)d\hslash \ge {\int }_0^1{𝒯}^{*}\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)d\hslash +{\int }_0^1{𝒯}^{*}\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}},\tau \right)d\hslash .\end{array}$$

It follows that

$$\begin{array}{c}{𝒯}_{*}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}, \tau \right)\le \frac{p}{{𝚥}^p-{𝚤}^p}{\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}{𝒯}_{*}\left(ⱷ, \tau \right)dⱷ,\\ {𝒯}^{*}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}, \tau \right)\ge \frac{p}{{𝚥}^p-{𝚤}^p}{\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}{𝒯}^{*}\left(ⱷ, \tau \right)dⱷ.\end{array}$$

That is

$$\left[{𝒯}_{*}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}, \tau \right), {𝒯}^{*}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}, \tau \right)\right]{\supseteq }_I\frac{p}{{𝚥}^p-{𝚤}^p}\left[{\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}{𝒯}_{*}\left(ⱷ, \tau \right)dⱷ, {\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}{𝒯}^{*}\left(ⱷ, \tau \right)dⱷ\right].$$

Thus,

$$\widetilde{𝒯}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}\right){\supseteq }_{\mathbb{F}}\frac{p}{{𝚥}^p-{𝚤}^p}\odot \left(FA\right){\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}\odot \widetilde{𝒯}\left(ⱷ\right)dⱷ.$$

(30)

In a similar way as above, we have

$$\frac{p}{{𝚥}^p-{𝚤}^p}\odot \left(FA\right){\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}\odot \widetilde{𝒯}\left(ⱷ\right)dⱷ{\supseteq }_{\mathbb{F}}\frac{\widetilde{𝒯}\left(𝚤\right)\oplus \widetilde{𝒯}\left(𝚥\right)}{2}.$$

(31)

Combining (21) and (22), we have

$$\widetilde{𝒯}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}\right){\supseteq }_{\mathbb{F}}\frac{p}{{𝚥}^p-{𝚤}^p}\odot \left(FA\right){\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}\odot \widetilde{𝒯}\left(ⱷ\right)dⱷ{\supseteq }_{\mathbb{F}}\frac{\widetilde{𝒯}\left(𝚤\right)\oplus \widetilde{𝒯}\left(𝚥\right)}{2}.$$

Hence, the required result. □

Remark 4. 

If $p=1$, then Theorem 7, cuts down to the outcome for $UD$-convex $FNVM$, as shown in [77]:

$$\widetilde{𝒯}\left(\frac{𝚤+𝚥}{2}\right){\supseteq }_{\mathbb{F}}\frac{1}{𝚥-𝚤}\odot \left(FA\right){\int }_{𝚤}^{𝚥}\widetilde{𝒯}\left(ⱷ\right)dⱷ{\supseteq }_{\mathbb{F}}\frac{\widetilde{𝒯}\left(𝚤\right)\oplus \widetilde{𝒯}\left(𝚥\right)}{2}$$

(32)

If ${𝒯}_{*}\left(ⱷ, \tau \right)={𝒯}^{*}\left(ⱷ, \tau \right)$ with$\tau =1$, then Theorem 7, cuts down to the finding for $p$-convex function, as shown in [90]:

$$𝒯\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}\right)\le \frac{p}{{𝚥}^p-{𝚤}^p}\left(A\right){\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}𝒯\left(ⱷ\right)dⱷ\le \frac{𝒯\left(𝚤\right)+𝒯\left(𝚥\right)}{2}.$$

(33)

If  ${𝒯}_{*}\left(ⱷ, \tau \right)={𝒯}^{*}\left(ⱷ, \tau \right)$  with  $\tau =1$  and  $p=1$  , then Theorem 7 cuts down to the outcome for classical convex function:

$$𝒯\left(\frac{𝚤+𝚥}{2}\right)\le \frac{1}{𝚥-𝚤}\left(A\right){\int }_{𝚤}^{𝚥}𝒯\left(ⱷ\right)dⱷ\le \frac{𝒯\left(𝚤\right)+𝒯\left(𝚥\right)}{2}.$$

(34)

Example 2. 

Let $p$ be an odd number and the $FNVM$ $\widetilde{𝒯}:\left[𝚤, 𝚥\right]=[2, 3]\to {\mathcal{L}}_C$ defined by,

$$\widetilde{𝒯}\left(ⱷ\right)\left(\lambda \right)=\left\{\begin{array}{c}\frac{\lambda -2+{ⱷ}^{\frac{p}{2}}}{1-{ⱷ}^{\frac{p}{2}}} \lambda \in \left[2-{ⱷ}^{\frac{p}{2}}, 3\right],\\ \frac{2+{ⱷ}^{\frac{p}{2}}-\lambda }{{ⱷ}^{\frac{p}{2}}-1} \lambda \in \left(3, 2+{ⱷ}^{\frac{p}{2}}\right], \\ 0 \mathrm{otherwise}.\end{array}\right.$$

(35)

Then, for each $\tau \in \left[0, 1\right],$ we have ${𝒯}_{\tau }\left(ⱷ\right)=\left[\left(1-\tau \right)\left(2-{ⱷ}^{\frac{p}{2}}\right)+3\tau , \left(1-\tau \right)\left(2+{ⱷ}^{\frac{p}{2}}\right)+3\tau \right]$. Since endpoint functions ${𝒯}_{*}\left(ⱷ,\tau \right)=\left(1-\tau \right)\left(2-{ⱷ}^{\frac{p}{2}}\right)+3\tau ,$ ${𝒯}^{*}\left(ⱷ, \tau \right)=\left(1-\tau \right)\left(2+{ⱷ}^{\frac{p}{2}}\right)+3\tau$ are $p$-convex functions for each $\tau \in [0, 1]$. Then, $\widetilde{𝒯}\left(ⱷ\right)$ is $UD$-$p$-convex $FNVM$.

We now computing the following

$$\begin{gathered} \begin{array}{c}{𝒯}_{*}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}},\tau \right)=\left(1-\tau \right)\frac{4-\sqrt{10}}{2}+3\tau ,\\ {𝒯}^{*}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}},\tau \right)=\left(1-\tau \right)\frac{4+\sqrt{10}}{2}+3\tau ,\end{array} \\ \begin{array}{c}\frac{p}{{𝚥}^p-{𝚤}^p} {\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}{𝒯}_{*}\left(ⱷ, \tau \right)dⱷ={\int }_2^3\left(\left(1-\tau \right)\left(2-{ⱷ}^{\frac{p}{2}}\right)+3\tau \right)dⱷ\approx \frac{843}{2000}\left(1-\tau \right)+3\tau ,\\ \frac{p}{{𝚥}^p-{𝚤}^p} {\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}{𝒯}^{*}\left(ⱷ, \tau \right)dⱷ={\int }_2^3\left(\left(1-\tau \right)\left(2+{ⱷ}^{\frac{p}{2}}\right)+3\tau \right)dⱷ\approx \frac{179}{50}\left(1-\tau \right)+3\tau ,\end{array} \\ \begin{array}{c}\frac{{𝒯}_{*}\left(𝚤, \tau \right)+{𝒯}_{*}\left(𝚥, \tau \right)}{2}=\left(1-\tau \right)\left(\frac{4-\sqrt{2}-\sqrt{3}}{2}\right)+3\tau ,\\ \frac{{𝒯}^{*}\left(𝚤, \tau \right)+{𝒯}^{*}\left(𝚥, \tau \right)}{2}=\left(1-\tau \right)\left(\frac{4+\sqrt{2}+\sqrt{3}}{2}\right)+3\tau ,\end{array} \end{gathered}$$

for all  $\tau \in \left[0, 1\right].$  That means

$$\begin{gathered} \left[\left(1-\tau \right)\frac{4-\sqrt{10}}{2}+3\tau ,\left(1-\tau \right)\frac{4+\sqrt{10}}{2}+3\tau \right]{\supseteq }_I\left[\frac{843}{2000}\left(1-\tau \right)+3\tau ,\frac{179}{50}\left(1-\tau \right)+3\tau \right] \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\supseteq }_I\left[\left(1-\tau \right)\left(\frac{4-\sqrt{2}-\sqrt{3}}{2}\right)+3\tau ,\left(1-\tau \right)\left(\frac{4+\sqrt{2}+\sqrt{3}}{2}\right)+3\tau \right], \end{gathered}$$

 for all  $\tau \in \left[0, 1\right],$  and Theorem 7 has been verified.

Theorem 8 presents the extended version of Aumann’s integral Hermite–Hadamard type inequalities.

Theorem 8. 

Let  $\widetilde{𝒯}\in UDFSX\left(\left[𝚤, 𝚥\right], {\mathcal{L}}_C, p\right)$. The family of  $IVMs$  is defined by$\tau$-cuts  ${𝒯}_{\tau }:\left[𝚤, 𝚥\right]\subset \mathbb{R}\to {{\mathcal{K}}_C}^{+}$  are given by  ${𝒯}_{\tau }\left(ⱷ\right)=\left[{𝒯}_{*}\left(ⱷ,\tau \right), {𝒯}^{*}\left(ⱷ,\tau \right)\right]$  for all  $ⱷ\in \left[𝚤, 𝚥\right]$  and for all  $\tau \in \left[0, 1\right]$  . If  $\widetilde{𝒯}\in {\mathcal{F}\mathcal{A}}_{\left(\left[𝚤, 𝚥\right], \tau \right)}$, then

$$\widetilde{𝒯}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}\right){\supseteq }_{\mathbb{F}} {⪧}_2{\supseteq }_{\mathbb{F}}\frac{p}{{𝚥}^p-{𝚤}^p}\odot \left(\mathrm{F}\mathsf{A}\right){\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}\odot \widetilde{𝒯}\left(ⱷ\right)dⱷ{\supseteq }_{\mathbb{F}}{⪧}_1{\supseteq }_{\mathbb{F}}\frac{\widetilde{𝒯}\left(𝚤\right)\oplus \widetilde{𝒯}\left(𝚥\right)}{2},$$

(36)

where

$${⪧}_1=\frac{\frac{\widetilde{𝒯}\left(𝚤\right)\oplus \widetilde{𝒯}\left(𝚥\right)}{2}\oplus \widetilde{𝒯}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}\right)}{2}, {⪧}_2=\frac{\widetilde{𝒯}\left({\left[\frac{3{𝚤}^p+{𝚥}^p}{4}\right]}^{\frac{1}{p}}\right)\oplus \widetilde{𝒯}\left({\left[\frac{{𝚤}^p+3{𝚥}^p}{4}\right]}^{\frac{1}{p}}\right)}{2}\mathbf{,} \mathrm{and} {⪧}_1=\left[{{⪧}_1}_{*}, {{⪧}_1}^{*}\right], {⪧}_2=\left[{{⪧}_2}_{*}, {{⪧}_2}^{*}\right].$$

Proof. 

Take $\left[{𝚤}^p, \frac{{𝚤}^p+{𝚥}^p}{2}\right],$ we have

$$2\odot \widetilde{𝒯}\left(\frac{{\left[\hslash {𝚤}^p+\left(1-\hslash \right)\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}}{2}+\frac{{\left[\left(1-\hslash \right){𝚤}^p+\hslash \frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}}{2}\right){\supseteq }_{\mathbb{F}}\widetilde{𝒯}\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right)\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}} \right)\oplus \widetilde{𝒯}\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash \frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}\right).$$

Therefore, for every $\tau \in [0, 1]$, we have

$$2{𝒯}_{*}\left(\frac{{\left[\hslash {𝚤}^p+\left(1-\hslash \right)\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}}{2}+\frac{{\left[\left(1-\hslash \right){𝚤}^p+\hslash \frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}}{2}, \tau \right)\le {𝒯}_{*}\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right)\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}, \tau \right)+{𝒯}_{*}\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash \frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}, \tau \right),$$

$$2{𝒯}^{*}\left(\frac{{\left[\hslash {𝚤}^p+\left(1-\hslash \right)\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}}{2}+\frac{{\left[\left(1-\hslash \right){𝚤}^p+\hslash \frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}}{2}, \tau \right)\ge {𝒯}^{*}\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right)\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}, \tau \right)\begin{array}{c} \\ +{𝒯}^{*}\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash \frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}},\tau \right).\end{array}$$

In consequence, we obtain

$$\begin{array}{c}\frac{{𝒯}_{*}\left({\left[\frac{3{𝚤}^p+{𝚥}^p}{4}\right]}^{\frac{1}{p}}, \tau \right)}{2}\le \frac{p}{{𝚥}^p-{𝚤}^p} {\int }_{𝚤}^{\frac{{𝚤}^p+{𝚥}^p}{2}}{𝒯}_{*}\left(ⱷ, \tau \right)dⱷ,\\ \frac{{𝒯}^{*}\left({\left[\frac{3{𝚤}^p+{𝚥}^p}{4}\right]}^{\frac{1}{p}}, \tau \right)}{2}\ge \frac{p}{{𝚥}^p-{𝚤}^p}{\int }_{𝚤}^{\frac{{𝚤}^p+{𝚥}^p}{2}}{𝒯}^{*}\left(ⱷ, \tau \right)dⱷ.\end{array}$$

That is

$$\frac{\left[{𝒯}_{*}\left({\left[\frac{3{𝚤}^p+{𝚥}^p}{4}\right]}^{\frac{1}{p}}, \tau \right), {𝒯}^{*}\left({\left[\frac{3{𝚤}^p+{𝚥}^p}{4}\right]}^{\frac{1}{p}}, \tau \right)\right]}{2}{\supseteq }_I\frac{p}{{𝚥}^p-{𝚤}^p}\left[{\int }_{𝚤}^{\frac{{𝚤}^p+{𝚥}^p}{2}}{𝒯}_{*}\left(ⱷ, \tau \right)dⱷ, {\int }_{𝚤}^{\frac{{𝚤}^p+{𝚥}^p}{2}}{𝒯}^{*}\left(ⱷ, \tau \right)dⱷ\right].$$

It follows that

$$\frac{\widetilde{𝒯}\left({\left[\frac{3{𝚤}^p+{𝚥}^p}{4}\right]}^{\frac{1}{p}}\right)}{2}{\supseteq }_{\mathbb{F}}\frac{p}{{𝚥}^p-{𝚤}^p}\odot (FA){\int }_{𝚤}^{\frac{{𝚤}^p+{𝚥}^p}{2}}\widetilde{𝒯}\left(ⱷ\right)dⱷ.$$

(37)

In a similar way as above, we have

$$\frac{\widetilde{𝒯}\left({\left[\frac{{𝚤}^p+3{𝚥}^p}{4}\right]}^{\frac{1}{p}}\right)}{2}{\supseteq }_{\mathbb{F}}\frac{p}{{𝚥}^p-{𝚤}^p}\odot (FA){\int }_{\frac{{𝚤}^p+{𝚥}^p}{2}}^{𝚥}\widetilde{𝒯}\left(ⱷ\right)dⱷ.$$

(38)

Combining (37) and (38), we have

$$\frac{\left[\widetilde{𝒯}\left({\left[\frac{3{𝚤}^p+{𝚥}^p}{4}\right]}^{\frac{1}{p}}\right)\oplus \widetilde{𝒯}\left({\left[\frac{{𝚤}^p+3{𝚥}^p}{4}\right]}^{\frac{1}{p}}\right)\right]}{2}{\supseteq }_{\mathbb{F}}\frac{p}{{𝚥}^p-{𝚤}^p}\odot (FA){\int }_{𝚤}^{𝚥}\widetilde{𝒯}\left(ⱷ\right)dⱷ.$$

By using Theorem 7, we have

$$\widetilde{𝒯}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}\right)=\widetilde{𝒯}\left({\left[\frac{1}{2}.\frac{3{𝚤}^p+{𝚥}^p}{4}+\frac{1}{2}.\frac{{𝚤}^p+3{𝚥}^p}{4}\right]}^{\frac{1}{p}}\right).$$

Therefore, for every $\tau \in [0, 1]$, we have

$$\begin{array}{c}{𝒯}_{*}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}, \tau \right)={𝒯}_{*}\left({\left[\frac{1}{2}.\frac{3{𝚤}^p+{𝚥}^p}{4}+\frac{1}{2}.\frac{{𝚤}^p+3{𝚥}^p}{4}\right]}^{\frac{1}{p}}, \tau \right),\hfill \\ {𝒯}^{*}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}, \tau \right)={𝒯}^{*}\left({\left[\frac{1}{2}.\frac{3{𝚤}^p+{𝚥}^p}{4}+\frac{1}{2}.\frac{{𝚤}^p+3{𝚥}^p}{4}\right]}^{\frac{1}{p}}, \tau \right),\hfill \\ \le \left[\frac{1}{2}{𝒯}_{*}\left({\left[\frac{3{𝚤}^p+{𝚥}^p}{4}\right]}^{\frac{1}{p}}, \tau \right)+\frac{1}{2}{𝒯}_{*}\left({\left[\frac{{𝚤}^p+3{𝚥}^p}{4}\right]}^{\frac{1}{p}}, \tau \right)\right],\hfill \\ \ge \left[\frac{1}{2}{𝒯}^{*}\left({\left[\frac{3{𝚤}^p+{𝚥}^p}{4}\right]}^{\frac{1}{p}}, \tau \right)+\frac{1}{2}{𝒯}^{*}\left({\left[\frac{{𝚤}^p+3{𝚥}^p}{4}\right]}^{\frac{1}{p}}, \tau \right)\right],\hfill \\ ={{⪧}_2}_{*},\hfill \\ ={{⪧}_2}^{*},\hfill \\ \le \frac{p}{{𝚥}^p-{𝚤}^p} {\int }_{𝚤}^{𝚥}{𝒯}_{*}\left(ⱷ, \tau \right)dⱷ, \hfill \\ \ge \frac{p}{{𝚥}^p-{𝚤}^p} {\int }_{𝚤}^{𝚥}{𝒯}^{*}\left(ⱷ, \tau \right)dⱷ,\hfill \\ \le \frac{1}{2}\left[\frac{{𝒯}_{*}\left(𝚤, \tau \right)+{𝒯}_{*}\left(𝚥, \tau \right)}{2}+{𝒯}_{*}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}, \tau \right)\right], \hfill \\ \ge \frac{1}{2}\left[\frac{{𝒯}^{*}\left(𝚤, \tau \right)+{𝒯}^{*}\left(𝚥, \tau \right)}{2}+{𝒯}^{*}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}, \tau \right)\right],\hfill \\ ={{⪧}_1}_{*},\le \frac{1}{2}\left[\frac{{𝒯}_{*}\left(𝚤, \tau \right)+{𝒯}_{*}\left(𝚥, \tau \right)}{2}+\frac{{𝒯}_{*}\left(𝚤, \tau \right)+{𝒯}_{*}\left(𝚥, \tau \right)}{2}\right], \hfill \\ ={{⪧}_1}^{*},\ge \frac{1}{2}\left[\frac{{𝒯}^{*}\left(𝚤, \tau \right)+{𝒯}^{*}\left(𝚥, \tau \right)}{2}+\frac{{𝒯}^{*}\left(𝚤,\tau \right)+{𝒯}^{*}\left(𝚥,\tau \right)}{2}\right],\hfill \\ =\frac{{𝒯}_{*}\left(𝚤, \tau \right)+{𝒯}_{*}\left(𝚥, \tau \right)}{2}, \hfill \\ =\frac{{𝒯}^{*}\left(𝚤, \tau \right)+{𝒯}^{*}\left(𝚥, \tau \right)}{2},\hfill \end{array}$$

that is

$$\widetilde{𝒯}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}\right){\supseteq }_{\mathbb{F}} {⪧}_2{\supseteq }_{\mathbb{F}}\frac{p}{{𝚥}^p-{𝚤}^p}\odot (FA){\int }_{𝚤}^{𝚥}\widetilde{𝒯}\left(ⱷ\right)dⱷ{\supseteq }_{\mathbb{F}}{⪧}_1{\supseteq }_{\mathbb{F}}\frac{\widetilde{𝒯}\left(𝚤\right)\oplus \widetilde{𝒯}\left(𝚥\right)}{2},$$

hence, the result follows. □

Example 3. 

Let $p$ be an odd number and the $FNVM$ $\widetilde{𝒯}:\left[𝚤, 𝚥\right]=[2, 3]\to {\mathcal{L}}_C$ defined by, ${𝒯}_{\tau }\left(ⱷ\right)=\left[\left(1-\tau \right)\left(2-{ⱷ}^{\frac{p}{2}}\right)+3\tau , \left(1-\tau \right)\left(2+{ⱷ}^{\frac{p}{2}}\right)+3\tau \right],$ as in Example 2, then $\widetilde{𝒯}(ⱷ)$ is $UD$-$p$-convex $FNVM$ and satisfying (38). We have ${𝒯}_{*}\left(ⱷ, \tau \right)=\left(1-\tau \right)\left(2-{ⱷ}^{\frac{p}{2}}\right)+3\tau$and ${𝒯}^{*}\left(ⱷ, \tau \right)=\left(1-\tau \right)\left(2+{ⱷ}^{\frac{p}{2}}\right)+3\tau$. We now computing the following

$$\begin{gathered} \begin{array}{c}\left[\frac{{𝒯}_{*}\left(𝚤, \tau \right)+{𝒯}_{*}\left(𝚥, \tau \right)}{2}\right]=\frac{4+2ʚ-\left(1-ʚ\right)\left(\sqrt{2}+\sqrt{3}\right)}{2}, \\ \left[\frac{{𝒯}^{*}\left(𝚤, \tau \right)+{𝒯}^{*}\left(𝚥, \tau \right)}{2}\right]=\frac{4+10ʚ+\left(1+ʚ\right)\left(\sqrt{2}+\sqrt{3}\right)}{2},\end{array} \\ \begin{array}{c}{{⪧}_1}_{*}=\frac{\frac{{𝒯}_{*}\left(𝚤, \tau \right)+{𝒯}_{*}\left(𝚥, \tau \right)}{2}+{𝒯}_{*}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}, \tau \right)}{2}=\frac{8+4ʚ-\left(1-ʚ\right)\left(\sqrt{2}+\sqrt{3}+\sqrt{2}\times \sqrt{5}\right)}{4}, \\ {{⪧}_1}^{*}=\frac{\frac{{𝒯}^{*}\left(𝚤, \tau \right)+{𝒯}^{*}\left(𝚥, \tau \right)}{2}+{𝒯}^{*}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}, \tau \right)}{2}=\frac{8+20ʚ+\left(1+ʚ\right)\left(\sqrt{2}+\sqrt{3}+\sqrt{2}\times \sqrt{5}\right)}{4},\end{array} \\ \begin{array}{c}{{⪧}_2}_{*}=\frac{1}{2} \left[{𝒯}_{*}\left({\left[\frac{3{𝚤}^p+{𝚥}^p}{4}\right]}^{\frac{1}{p}}, \tau \right)+{𝒯}_{*}\left({\left[\frac{{𝚤}^p+3{𝚥}^p}{4}\right]}^{\frac{1}{p}}, \tau \right)\right]=5+7ʚ-\sqrt{11}\left(1-ʚ\right), \\ {{⪧}_2}^{*}=\frac{1}{2}\left[{𝒯}^{*}\left({\left[\frac{3{𝚤}^p+{𝚥}^p}{4}\right]}^{\frac{1}{p}}, \tau \right)+{𝒯}^{*}\left({\left[\frac{{𝚤}^p+3{𝚥}^p}{4}\right]}^{\frac{1}{p}}, \tau \right)\right]=\frac{11+23ʚ+\sqrt{11}\left(1+ʚ\right)}{4},\end{array} \\ \begin{array}{c}{𝒯}_{*}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}},\tau \right)=\left(1-\tau \right)\frac{4-\sqrt{10}}{2}+3\tau ,\\ {𝒯}^{*}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}},\tau \right)=\left(1-\tau \right)\frac{4+\sqrt{10}}{2}+3\tau .\end{array} \end{gathered}$$

Then, we obtain that

$$\begin{array}{c}\left(1-ʚ\right)\frac{4-\sqrt{10}}{2}+3ʚ\le \frac{5+7ʚ-\sqrt{11}\left(1-ʚ\right)}{4}\le \frac{843}{2000}\left(1-ʚ\right)+3ʚ \\ \le \frac{8+4ʚ-\left(1-ʚ\right)\left(\sqrt{2}+\sqrt{3}+\sqrt{2}\times \sqrt{5}\right)}{4}\le \left(1-ʚ\right)\left(\frac{4-\sqrt{2}-\sqrt{3}}{2}\right)+3ʚ\\ \left(1+ʚ\right)\frac{4+\sqrt{10}}{2}+3ʚ\ge \frac{11+23ʚ+\sqrt{11}\left(1+ʚ\right)}{4}\ge \frac{179}{50}\left(1+ʚ\right)+3ʚ\\ \ge \frac{8+20ʚ+\left(1+ʚ\right)\left(\sqrt{2}+\sqrt{3}+\sqrt{2}\times \sqrt{5}\right)}{4}\ge \left(1+ʚ\right)\left(\frac{4+\sqrt{2}+\sqrt{3}}{2}\right)+3ʚ.\end{array}$$

Hence, Theorem 8 is verified.

The next outcomes will give us the Aumann’s integral Hermite–Hadamard type inequalities for the product of two $\mathrm{U}\mathrm{D}$-$p$-convex $\mathrm{F}\mathrm{N}\mathrm{V}\mathrm{M}$s

Theorem 9. 

Let  $\widetilde{𝒯},\widetilde{{\rm Y}}\in UDFSX\left(\left[𝚤, 𝚥\right], {\mathcal{L}}_C, p\right)$  . Then,  $\tau$-cuts  ${𝒯}_{\tau }, {{\rm Y}}_{\tau }:\left[𝚤, 𝚥\right]\subset \mathbb{R}\to {{\mathcal{K}}_C}^{+}$  are defined by  ${𝒯}_{\tau }\left(ⱷ\right)=\left[{𝒯}_{*}\left(ⱷ,\tau \right), {𝒯}^{*}\left(ⱷ,\tau \right)\right]$  and  ${{\rm Y}}_{\tau }\left(ⱷ\right)=\left[{{\rm Y}}_{*}\left(ⱷ,\tau \right), {{\rm Y}}^{*}\left(ⱷ,\tau \right)\right]$  for all  $ⱷ\in \left[𝚤, 𝚥\right]$  and for all  $\tau \in \left[0, 1\right]$  . If  $\widetilde{𝒯}\otimes \widetilde{{\rm Y}}\in {\mathcal{F}\mathcal{A}}_{\left(\left[𝚤, 𝚥\right], \tau \right)}$, then

$$\frac{p}{{𝚥}^p-{𝚤}^p}\odot \left(FA\right){\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}\odot \widetilde{𝒯}\left(ⱷ\right)\otimes \widetilde{{\rm Y}}\left(ⱷ\right)dⱷ{\supseteq }_{\mathbb{F}}\frac{\tilde{\mathcal{M}}\left(𝚤,𝚥\right)}{3}\oplus \frac{\tilde{\mathcal{N}}\left(𝚤,𝚥\right)}{6}.$$

(39)

where $\tilde{\mathcal{M}}\left(𝚤,𝚥\right)=\widetilde{𝒯}\left(𝚤\right)\otimes \widetilde{{\rm Y}}\left(𝚤\right)\oplus \widetilde{𝒯}\left(𝚥\right)\otimes \widetilde{{\rm Y}}\left(𝚥\right),$ $\tilde{\mathcal{N}}\left(𝚤,𝚥\right)=\widetilde{𝒯}\left(𝚤\right)\otimes \widetilde{{\rm Y}}\left(𝚥\right)\oplus \widetilde{𝒯}\left(𝚥\right)\otimes \widetilde{{\rm Y}}\left(𝚤\right),$ and ${\mathcal{M}}_{\tau }\left(𝚤,𝚥\right)=\left[{\mathcal{M}}_{*}\left(\left(𝚤,𝚥\right), \tau \right), {\mathcal{M}}^{*}\left(\left(𝚤,𝚥\right), \tau \right)\right]$ and ${\mathcal{N}}_{\tau }\left(𝚤,𝚥\right)=\left[{\mathcal{N}}_{*}\left(\left(𝚤,𝚥\right), \tau \right), {\mathcal{N}}^{*}\left(\left(𝚤,𝚥\right), \tau \right)\right].$

Example 4. 

Let $p$ be an odd number, and $UD$-$p$-convex $FNVM$s $\widetilde{𝒯},\widetilde{{\rm Y}}:\left[𝚤, 𝚥\right]=[2, 3]\to {\mathcal{L}}_C$ are, respectively, defined by ${𝒯}_{\tau }\left(ⱷ\right)=\left[\left(1-\tau \right)\left(2-{ⱷ}^{\frac{p}{2}}\right)+3\tau , \left(1-\tau \right)\left(2+{ⱷ}^{\frac{p}{2}}\right)+3\tau \right],$ as in example 3, and taking ${{\rm Y}}_{\tau }\left(ⱷ\right)=\left[\tau {ⱷ}^p,(2-\tau ){ⱷ}^p \right]$. Since $\widetilde{𝒯}(ⱷ)$ and $\widetilde{{\rm Y}}(ⱷ)$ both are $UD$-$p$-convex $FNVM$s and ${𝒯}_{*}\left(ⱷ, \tau \right)=\left(1-\tau \right)\left(2-{ⱷ}^{\frac{p}{2}}\right)+3\tau$, ${𝒯}^{*}\left(ⱷ, \tau \right)=\left(1-\tau \right)\left(2+{ⱷ}^{\frac{p}{2}}\right)+3\tau$, and ${{\rm Y}}_{*}\left(ⱷ, \tau \right)=\tau {ⱷ}^p$, ${{\rm Y}}^{*}\left(ⱷ, \tau \right)=(2-\tau ){ⱷ}^p$, then we computing the following, where $p=1$

$$\begin{gathered} \begin{array}{c}\frac{p}{{𝚥}^p-{𝚤}^p}{\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}{𝒯}_{*}\left(ⱷ,\tau \right)\times {{\rm Y}}_{*}\left(ⱷ,\tau \right)dⱷ=\frac{\tau }{10}\left[\left(25-16\sqrt{2}+36\sqrt{3}\right)\tau -36\sqrt{3}+16\sqrt{2}+50\right],\\ \frac{p}{{𝚥}^p-{𝚤}^p}{\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}{𝒯}^{*}\left(ⱷ,\tau \right)\times {{\rm Y}}^{*}\left(ⱷ,\tau \right)dⱷ=\frac{\left(2-\tau \right)}{10}\left[\left(25+16\sqrt{2}-36\sqrt{3}\right)\tau +36\sqrt{3}-16\sqrt{2}+50\right],\end{array} \\ \begin{array}{c}\frac{{\mathcal{M}}_{*}\left(\left(𝚤,𝚥\right), \tau \right)}{3}=\left(5+2\sqrt{2}+3\sqrt{3}\right)\frac{{\tau }^{\begin{array}{c} \\ 2\end{array}}}{3}+\frac{\tau }{3}\left(10-2\sqrt{2}-3\sqrt{3}\right), \\ \frac{{\mathcal{M}}^{*}\left(\left(𝚤,𝚥\right), \tau \right)}{3}=\sqrt{3}{\left(\tau -\frac{1}{2}\right)}^2+\frac{2\sqrt{2}}{3}{\left(\tau -\frac{1}{2}\right)}^2-\frac{10}{3}{\left(\tau -\frac{3}{4}\right)}^2-\frac{\sqrt{3}}{2}-\frac{1}{3\sqrt{2}}+\frac{125}{24},\end{array} \\ \begin{array}{c}\frac{{\mathcal{N}}_{*}\left(\left(𝚤,𝚥\right), \tau \right)}{6}=\frac{\tau }{6}\left[15\tau +\left(1-\tau \right)\left\{10+3\sqrt{2}+2\sqrt{3}\right\}\right]\\ \frac{{\mathcal{N}}^{*}\left(\left(𝚤,𝚥\right), \tau \right)}{6}=\frac{\left(2-\tau \right)}{6}\left(10+3\sqrt{2}+2\sqrt{3}-\left(3\sqrt{2}+2\sqrt{3}\right)\tau +15\tau \right),\end{array} \end{gathered}$$

for each  $\tau \in \left[0, 1\right],$  that means

$$\begin{array}{c}\frac{\tau }{10}\left[\left(25-16\sqrt{2}+36\sqrt{3}\right)\tau -36\sqrt{3}+16\sqrt{2}+50\right] \\ \le \left(5+2\sqrt{2}+3\sqrt{3}\right)\frac{{\tau }^{\begin{array}{c} \\ 2\end{array}}}{3}+\frac{\tau }{3}\left(10-2\sqrt{2}-3\sqrt{3}\right)+\frac{\tau }{6}\left[15\tau +\left(1-\tau \right)\left\{10+3\sqrt{2}+2\sqrt{3}\right\}\right], \\ \frac{\left(2-\tau \right)}{10}\left[\left(25+16\sqrt{2}-36\sqrt{3}\right)\tau +36\sqrt{3}-16\sqrt{2}+50\right] \\ \ge \sqrt{3}{\left(\tau -\frac{1}{2}\right)}^2+\frac{2\sqrt{2}}{3}{\left(\tau -\frac{1}{2}\right)}^2-\frac{10}{3}{\left(\tau -\frac{3}{4}\right)}^2-\frac{\sqrt{3}}{2}-\frac{1}{3\sqrt{2}}+\frac{125}{24}\\ +\frac{\left(2-\tau \right)}{6}\left(10+3\sqrt{2}+2\sqrt{3}-\left(3\sqrt{2}+2\sqrt{3}\right)\tau +15\tau \right).\end{array}$$

Hence, Theorem 9 has been verified.

Theorem 10. 

Let  $\widetilde{𝒯},\widetilde{{\rm Y}}\in UDFSX\left(\left[𝚤, 𝚥\right], {\mathcal{L}}_C, p\right)$. The family of  $IVMs$  is defined by  $\tau$-cuts  ${𝒯}_{\tau }, {{\rm Y}}_{\tau }:\left[𝚤, 𝚥\right]\subset \mathbb{R}\to {{\mathcal{K}}_C}^{+}$  are given by  ${𝒯}_{\tau }\left(ⱷ\right)=\left[{𝒯}_{*}\left(ⱷ,\tau \right), {𝒯}^{*}\left(ⱷ,\tau \right)\right]$  and  ${{\rm Y}}_{\tau }\left(ⱷ\right)=\left[{{\rm Y}}_{*}\left(ⱷ,\tau \right), {{\rm Y}}^{*}\left(ⱷ,\tau \right)\right]$  for all  $ⱷ\in \left[𝚤, 𝚥\right]$  and for all  $\tau \in \left[0, 1\right]$. If  $\widetilde{𝒯}\otimes \widetilde{{\rm Y}}\in {\mathcal{F}\mathcal{A}}_{\left(\left[𝚤, 𝚥\right], \tau \right)}$, then

$$\begin{gathered} 2\odot \widetilde{𝒯}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}\right)\otimes \widetilde{{\rm Y}}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}\right) \\ {\supseteq }_{\mathbb{F}}\frac{p}{{𝚥}^p-{𝚤}^p}\odot \left(FA\right){\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}\odot \widetilde{𝒯}\left(ⱷ\right)\otimes \widetilde{{\rm Y}}\left(ⱷ\right)dⱷ\oplus \frac{\tilde{\mathcal{M}}\left(𝚤,𝚥\right)}{6}\oplus \frac{\tilde{\mathcal{N}}\left(𝚤,𝚥\right)}{3}. \end{gathered}$$

(40)

where$\tilde{\mathcal{M}}\left(𝚤,𝚥\right)=\widetilde{𝒯}\left(𝚤\right)\otimes \widetilde{{\rm Y}}\left(𝚤\right)\oplus \widetilde{𝒯}\left(𝚥\right)\otimes \widetilde{{\rm Y}}\left(𝚥\right),$ $\tilde{\mathcal{N}}\left(𝚤,𝚥\right)=\widetilde{𝒯}\left(𝚤\right)\otimes \widetilde{{\rm Y}}\left(𝚥\right)\oplus \widetilde{𝒯}\left(𝚥\right)\otimes \widetilde{{\rm Y}}\left(𝚤\right),$ and ${\mathcal{M}}_{\tau }\left(𝚤,𝚥\right)=\left[{\mathcal{M}}_{*}\left(\left(𝚤,𝚥\right), \tau \right), {\mathcal{M}}^{*}\left(\left(𝚤,𝚥\right), \tau \right)\right]$ and ${\mathcal{N}}_{\tau }\left(𝚤,𝚥\right)=\left[{\mathcal{N}}_{*}\left(\left(𝚤,𝚥\right), \tau \right), {\mathcal{N}}^{*}\left(\left(𝚤,𝚥\right), \tau \right)\right].$

Proof. 

By hypothesis, for each $\tau \in \left[0, 1\right],$ we have

$$\begin{gathered} \begin{array}{c}{𝒯}_{*}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}},\tau \right)\times {{\rm Y}}_{*}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}},\tau \right)\\ {𝒯}^{*}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}},\tau \right)\times {{\rm Y}}^{*}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}},\tau \right)\end{array} \\ \le \frac{1}{4}\left[\begin{array}{c}{𝒯}_{*}\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)\times {{\rm Y}}_{*}\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)\\ +{𝒯}_{*}\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)\times {{\rm Y}}_{*}\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)\end{array}\right] \\ +\frac{1}{4}\left[\begin{array}{c}{𝒯}_{*}\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}, \tau \right){\times {\rm Y}}_{*}\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)\\ +{𝒯}_{*}\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}, \tau \right){\times {\rm Y}}_{*}\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)\end{array}\right] \\ \begin{array}{c} \\ , \\ \ge \frac{1}{4}\left[\begin{array}{c}{𝒯}^{*}\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}, \tau \right){\times {\rm Y}}^{*}\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)\\ +{𝒯}^{*}\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}, \tau \right){\times {\rm Y}}^{*}\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)\end{array}\right] \\ +\frac{1}{4}\left[\begin{array}{c}{𝒯}^{*}\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}, \tau \right){\times {\rm Y}}^{*}\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)\\ +{𝒯}^{*}\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}, \tau \right){\times {\rm Y}}^{*}\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)\end{array}\right], \end{array} \\ \begin{array}{c}\le \frac{1}{4}\left[\begin{array}{c}{𝒯}_{*}\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}, \tau \right){\times {\rm Y}}_{*}\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)\\ +{𝒯}_{*}\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}, \tau \right){\times {\rm Y}}_{*}\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)\end{array}\right] \\ +\frac{1}{4}\left[\begin{array}{c}\left(\hslash {𝒯}_{*}\left(𝚤, \tau \right)+\left(1-\hslash \right){𝒯}_{*}\left(𝚥, \tau \right)\right)\\ \times \left(\left(1-\hslash \right){{\rm Y}}_{*}\left(𝚤, \tau \right)+\hslash {{\rm Y}}_{*}\left(𝚥, \tau \right)\right)\\ +\left({\left(1-\hslash \right)𝒯}_{*}\left(𝚤, \tau \right)+\hslash {𝒯}_{*}\left(𝚥, \tau \right)\right)\\ \times \left(\hslash {{\rm Y}}_{*}\left(𝚤, \tau \right)+\left(1-\hslash \right){{\rm Y}}_{*}\left(𝚥, \tau \right)\right)\end{array}\right], \\ \ge \frac{1}{4}\left[\begin{array}{c}{𝒯}^{*}\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}, \tau \right){\times {\rm Y}}^{*}\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)\\ +{𝒯}^{*}\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}, \tau \right){\times {\rm Y}}^{*}\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)\end{array}\right]\\ +\frac{1}{4}\left[\begin{array}{c}\left(\hslash {𝒯}^{*}\left(𝚤, \tau \right)+\left(1-\hslash \right){𝒯}^{*}\left(𝚥, \tau \right)\right)\\ \times \left(\left(1-\hslash \right){{\rm Y}}^{*}\left(𝚤, \tau \right)+\hslash {{\rm Y}}^{*}\left(𝚥, \tau \right)\right)\\ +\left(\left(1-\hslash \right){𝒯}^{*}\left(𝚤, \tau \right)+\hslash {𝒯}^{*}\left(𝚥, \tau \right)\right)\\ \times \left(\hslash {{\rm Y}}^{*}\left(𝚤, \tau \right)+\left(1-\hslash \right){{\rm Y}}^{*}\left(𝚥, \tau \right)\right)\end{array}\right], \end{array} \\ \begin{array}{c}=\frac{1}{4}\left[\begin{array}{c}{𝒯}_{*}\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}, \tau \right){\times {\rm Y}}_{*}\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)\\ +{𝒯}_{*}\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}, \tau \right){\times {\rm Y}}_{*}\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)\end{array}\right] \\ +\frac{1}{2}\left[\begin{array}{c}\left\{{\hslash }^2+{\left(1-\hslash \right)}^2\right\}{\mathcal{N}}_{*}\left(\left(𝚤,𝚥\right), \tau \right)\\ +\left\{\hslash \left(1-\hslash \right)+\left(1-\hslash \right)\hslash \right\}{\mathcal{M}}_{*}\left(\left(𝚤,𝚥\right), \tau \right)\end{array}\right], \\ =\frac{1}{4}\left[\begin{array}{c}{𝒯}^{*}\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}, \tau \right){\times {\rm Y}}^{*}\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)\\ +{𝒯}^{*}\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}, \tau \right){\times {\rm Y}}^{*}\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)\end{array}\right] \\ +\frac{1}{2}\left[\begin{array}{c}\left\{{\hslash }^2+{\left(1-\hslash \right)}^2\right\}{\mathcal{N}}^{*}\left(\left(𝚤,𝚥\right), \tau \right)\\ +\left\{\hslash \left(1-\hslash \right)+\left(1-\hslash \right)\hslash \right\}{\mathcal{M}}^{*}\left(\left(𝚤,𝚥\right), \tau \right)\end{array}\right], \end{array} \end{gathered}$$

$A$-integrating over $\left[0, 1\right],$ we have

$$\begin{array}{c}2 {𝒯}_{*}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}},\tau \right){\times {\rm Y}}_{*}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}},\tau \right)\le \frac{p}{{𝚥}^p-{𝚤}^p}{\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}{𝒯}_{*}\left(ⱷ,\tau \right){\times {\rm Y}}_{*}\left(ⱷ,\tau \right)dⱷ+\frac{{\mathcal{M}}_{*}\left(\left(𝚤,𝚥\right), \tau \right)}{6}\\ +\frac{{\mathcal{N}}_{*}\left(\left(𝚤,𝚥\right), \tau \right)}{3}, \\ 2 {𝒯}^{*}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}},\tau \right){\times {\rm Y}}^{*}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}},\tau \right)\ge \frac{p}{{𝚥}^p-{𝚤}^p}{\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}{𝒯}^{*}\left(ⱷ,\tau \right){\times {\rm Y}}^{*}\left(ⱷ,\tau \right)dⱷ+\frac{{\mathcal{M}}^{*}\left(\left(𝚤,𝚥\right), \tau \right)}{6}\\ +\frac{{\mathcal{N}}^{*}\left(\left(𝚤,𝚥\right), \tau \right)}{3},\end{array}$$

that is

$$2\odot \widetilde{𝒯}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}\right)\otimes \widetilde{{\rm Y}}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}\right){\supseteq }_{\mathbb{F}}\frac{p}{{𝚥}^p-{𝚤}^p}\odot \left(FA\right){\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}\odot \widetilde{𝒯}\left(ⱷ\right)\otimes \widetilde{{\rm Y}}\left(ⱷ\right)dⱷ\oplus \frac{\tilde{\mathcal{M}}\left(𝚤,𝚥\right)}{6}\oplus \frac{\tilde{\mathcal{N}}\left(𝚤,𝚥\right)}{3}.$$

Hence, the required result. □

Example 5. 

Let $p$ be an odd number, and $UD$-$p$-convex $FNVM$s $\widetilde{𝒯},\widetilde{{\rm Y}}:\left[𝚤, 𝚥\right]=[2, 3]\to {\mathcal{L}}_C$ are, respectively, defined by ${𝒯}_{\tau }\left(ⱷ\right)=\left[\left(1-\tau \right)\left(2-{ⱷ}^{\frac{p}{2}}\right)+3\tau , \left(1-\tau \right)\left(2+{ⱷ}^{\frac{p}{2}}\right)+3\tau \right],$ as in Example 3, and ${{\rm Y}}_{\tau }\left(ⱷ\right)=\left[\tau {ⱷ}^p,(2-\tau ){ⱷ}^p \right]$. Since $\widetilde{𝒯}(ⱷ)$ and $\widetilde{{\rm Y}}(ⱷ)$ both are $UD$-$p$-convex $FNVM$s and ${𝒯}_{*}\left(ⱷ, \tau \right)=\left(1-\tau \right)\left(2-{ⱷ}^{\frac{p}{2}}\right)+3\tau$, ${𝒯}^{*}\left(ⱷ, \tau \right)=\left(1-\tau \right)\left(2+{ⱷ}^{\frac{p}{2}}\right)+3\tau$, and ${{\rm Y}}_{*}\left(ⱷ, \tau \right)=\tau {ⱷ}^p$, ${{\rm Y}}^{*}\left(ⱷ, \tau \right)=(2-\tau ){ⱷ}^p$, then we computing the following, where $p=1$

$$\begin{gathered} \begin{array}{c}2{𝒯}_{*}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}},\tau \right)\times {{\rm Y}}_{*}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}},\tau \right)=\frac{5}{2}\left[\left(2+\sqrt{10}\right){\tau }^2+\left(4-\sqrt{10}\right)\tau \right], \\ 2{𝒯}^{*}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}},\tau \right)\times {{\rm Y}}^{*}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}},\tau \right)=\frac{5}{2}\left(2-\tau \right)\left[4+\sqrt{10}+\left(2-\sqrt{10}\right)\tau \right],\end{array} \\ \begin{array}{c}\frac{{\mathcal{M}}_{*}\left(\left(𝚤,𝚥\right), \tau \right)}{6}=\left(5+2\sqrt{2}+3\sqrt{3}\right)\frac{{\tau }^{\begin{array}{c} \\ 2\end{array}}}{6}+\frac{\tau }{6}\left(10-2\sqrt{2}-3\sqrt{3}\right), \\ \frac{{\mathcal{M}}^{*}\left(\left(𝚤,𝚥\right), \tau \right)}{6}=\frac{\sqrt{3}}{2}{\left(\tau -\frac{1}{2}\right)}^2+\frac{2\sqrt{2}}{6}{\left(\tau -\frac{1}{2}\right)}^2-\frac{5}{3}{\left(\tau -\frac{3}{4}\right)}^2-\frac{\sqrt{3}}{4}-\frac{1}{6\sqrt{2}}+\frac{125}{48},\end{array} \\ \begin{array}{c}\frac{{\mathcal{N}}_{*}\left(\left(𝚤,𝚥\right), \tau \right)}{3}=\frac{\tau }{3}\left[15\tau +\left(1-\tau \right)\left\{10+3\sqrt{2}+2\sqrt{3}\right\}\right]\\ \frac{{\mathcal{N}}^{*}\left(\left(𝚤,𝚥\right), \tau \right)}{3}=\frac{\left(2-\tau \right)}{3}\left(10+3\sqrt{2}+2\sqrt{3}-\left(3\sqrt{2}+2\sqrt{3}\right)\tau +15\tau \right),\end{array} \end{gathered}$$

for each  $\tau \in \left[0, 1\right],$  that means

$$\begin{array}{c}\frac{5}{2}\left[\left(2+\sqrt{10}\right){\tau }^2+\left(4-\sqrt{10}\right)\tau \right] \\ \le \frac{\tau }{10}\left[\left(25-16\sqrt{2}+36\sqrt{3}\right)\tau -36\sqrt{3}+16\sqrt{2}+50\right]\\ +\left(5+2\sqrt{2}+3\sqrt{3}\right)\frac{{\tau }^{\begin{array}{c} \\ 2\end{array}}}{6}+\frac{\tau }{6}\left(10-2\sqrt{2}-3\sqrt{3}\right)\\ +\frac{\tau }{3}\left[15\tau +\left(1-\tau \right)\left\{10+3\sqrt{2}+2\sqrt{3}\right\}\right],\\ \frac{5}{2}\left(2-\tau \right)\left[4+\sqrt{10}+\left(2-\sqrt{10}\right)\tau \right] \\ \ge \frac{\left(2-\tau \right)}{10}\left[\left(25+16\sqrt{2}-36\sqrt{3}\right)\tau +36\sqrt{3}-16\sqrt{2}+50\right]\\ +\frac{\sqrt{3}}{2}{\left(\tau -\frac{1}{2}\right)}^2+\frac{2\sqrt{2}}{6}{\left(\tau -\frac{1}{2}\right)}^2-\frac{5}{3}{\left(\tau -\frac{3}{4}\right)}^2-\frac{\sqrt{3}}{4}-\frac{1}{6\sqrt{2}}+\frac{125}{48}\\ +\frac{\left(2-\tau \right)}{3}\left(10+3\sqrt{2}+2\sqrt{3}-\left(3\sqrt{2}+2\sqrt{3}\right)\tau +15\tau \right),\end{array}$$

 hence, Theorem 10 is verified.

𝐻-𝐻 Fejér inequality for UD-p-convex FNVM

Theorem 11. 

(𝐻-𝐻  Fejér inequality for  $UD$-$p$-convex  $FNVM$) Let  $\widetilde{𝒯}\in UDFSX\left(\left[𝚤, 𝚥\right], {\mathcal{L}}_C, p\right)$. The family of  $IVMs$  is defined by  $\tau$-cuts  ${𝒯}_{\tau }:\left[𝚤, 𝚥\right]\subset \mathbb{R}\to {{\mathcal{K}}_C}^{+}$  are given by  ${𝒯}_{\tau }\left(ⱷ\right)=\left[{𝒯}_{*}\left(ⱷ,\tau \right), {𝒯}^{*}\left(ⱷ,\tau \right)\right]$  for all  $ⱷ\in \left[𝚤, 𝚥\right]$  and for all  $\tau \in \left[0, 1\right]$  . If  $\widetilde{𝒯}\in {\mathcal{F}\mathcal{A}}_{\left(\left[𝚤, 𝚥\right], \tau \right)}$  and  $Ɲ:\left[𝚤, 𝚥\right]\to \mathbb{R},Ɲ(ⱷ)\ge 0,$  symmetric with respect to  ${\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}},$  then

$$\frac{p}{{𝚥}^p-{𝚤}^p}\odot \left(FA\right){\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}\odot \widetilde{𝒯}\left(ⱷ\right)Ɲ(ⱷ)dⱷ{\supseteq }_{\mathbb{F}}\left[\widetilde{𝒯}\left(𝚤\right)\oplus \widetilde{𝒯}\left(𝚥\right)\right]\odot {\int }_0^1\hslash Ɲ\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}\right)d\hslash .$$

(41)

If  $\widetilde{𝒯}$  is  $UD$-$p$-concave  $FNVM$, then inequality (41) is reversed.

Proof. 

Let $\widetilde{𝒯}$ be an $\mathrm{U}\mathrm{D}$-$p$-convex $\mathrm{F}\mathrm{N}\mathrm{V}\mathrm{M}$. Then, for each $\tau \in \left[0, 1\right],$ we have

$${𝒯}_{*}\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)Ɲ\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}\right)\le \left(\hslash {𝒯}_{*}\left(𝚤, \tau \right)+\left(1-\hslash \right){𝒯}_{*}\left(𝚥, \tau \right)\right)Ɲ\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}\right),$$

(42)

$${𝒯}^{*}\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)Ɲ\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}\right)\ge \left(\hslash {𝒯}^{*}\left(𝚤, \tau \right)+\left(1-\hslash \right){𝒯}^{*}\left(𝚥, \tau \right)\right)Ɲ\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}\right).$$

And

$$\begin{array}{c} \\ {𝒯}_{*}\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)Ɲ\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}\right)\\ \le \left(\left(1-\hslash \right){𝒯}_{*}\left(𝚤, \tau \right)+\hslash {𝒯}_{*}\left(𝚥, \tau \right)\right)Ɲ\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}\right),\\ {𝒯}^{*}\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)Ɲ\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}\right)\\ \ge \left(\left(1-\hslash \right){𝒯}^{*}\left(𝚤, \tau \right)+\hslash {𝒯}^{*}\left(𝚥, \tau \right)\right)Ɲ\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}\right).\end{array}$$

(43)

After adding (42) and (43), and integrating over $\left[0, 1\right],$ we get

$$\begin{gathered} \begin{array}{c} \\ {\int }_0^1{𝒯}_{*}\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)Ɲ\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}\right)d\hslash \\ +{\int }_0^1{𝒯}_{*}\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)Ɲ\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}\right)d\hslash \\ \le {\int }_0^1\left[\begin{array}{c}{𝒯}_{*}\left(𝚤, \tau \right)\left\{\hslash Ɲ\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}\right)+\left(1-\hslash \right)Ɲ\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}\right)\right\}\\ +{𝒯}_{*}\left(𝚥, \tau \right)\left\{\left(1-\hslash \right)Ɲ\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}\right)+\hslash Ɲ\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}\right)\right\}\end{array}\right]d\hslash ,\\ {\int }_0^1{𝒯}^{*}\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)Ɲ\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}\right)d\hslash \\ +{\int }_0^1{𝒯}^{*}\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)Ɲ\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}\right)d\hslash \\ \ge {\int }_0^1\left[\begin{array}{c}{𝒯}^{*}\left(𝚤, \tau \right)\left\{\hslash Ɲ\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}\right)+\left(1-\hslash \right)Ɲ\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}\right)\right\}\\ +{𝒯}^{*}\left(𝚥, \tau \right)\left\{\left(1-\hslash \right)Ɲ\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}\right)+\hslash Ɲ\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}\right)\right\}\end{array}\right]d\hslash ,\end{array} \\ \begin{array}{c} \\ =2{𝒯}_{*}\left(𝚤, \tau \right){\int }_0^1\begin{array}{c}\hslash Ɲ\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}\right)\end{array}d\hslash +2{𝒯}_{*}\left(𝚥, \tau \right){\int }_0^1\begin{array}{c}\hslash Ɲ\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}\right)\end{array}d\hslash ,\\ =2{𝒯}^{*}\left(𝚤, \tau \right){\int }_0^1\begin{array}{c}\hslash Ɲ\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}\right)\end{array}d\hslash +2{𝒯}^{*}\left(𝚥, \tau \right){\int }_0^1\begin{array}{c}\hslash Ɲ\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}\right)\end{array}d\hslash .\end{array} \end{gathered}$$

Since $Ɲ$ is symmetric, then

$$\begin{array}{c} \\ =2\left[{𝒯}_{*}\left(𝚤, \tau \right)+{𝒯}_{*}\left(𝚥, \tau \right)\right]{\int }_0^1\begin{array}{c}\hslash Ɲ\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}\right)\end{array}d\hslash ,\\ =2\left[{𝒯}^{*}\left(𝚤, \tau \right)+{𝒯}^{*}\left(𝚥, \tau \right)\right]{\int }_0^1\begin{array}{c}\hslash Ɲ\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}\right)\end{array}d\hslash .\end{array}$$

(44)

Since

$$\begin{array}{c} \\ {\int }_0^1{𝒯}_{*}\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)Ɲ\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}\right)d\hslash \\ ={\int }_0^1{𝒯}_{*}\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)Ɲ\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}\right)d\hslash \\ =\frac{p}{{𝚥}^p-{𝚤}^p} {\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}{𝒯}_{*}\left(ⱷ,\tau \right)Ɲ(ⱷ)dⱷ \\ {\int }_0^1{𝒯}^{*}\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)Ɲ\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}\right)d\hslash \\ ={\int }_0^1{𝒯}^{*}\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)Ɲ\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}\right)d\hslash \\ =\frac{p}{{𝚥}^p-{𝚤}^p} {\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}{𝒯}^{*}\left(ⱷ, \tau \right)Ɲ(ⱷ)dⱷ. \end{array}$$

(45)

Then, from (44), we have

$$\begin{array}{c}\frac{p}{{𝚥}^p-{𝚤}^p} {\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}{𝒯}_{*}\left(ⱷ,\tau \right)Ɲ(ⱷ)dⱷ\le \left[{𝒯}_{*}\left(𝚤, \tau \right)+{𝒯}_{*}\left(𝚥, \tau \right)\right]{\int }_0^1\begin{array}{c}\hslash Ɲ\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}\right)\end{array}d\hslash , \\ \\ \frac{p}{{𝚥}^p-{𝚤}^p} {\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}{𝒯}^{*}\left(ⱷ,\tau \right)Ɲ(ⱷ)dⱷs\ge \left[{𝒯}^{*}\left(𝚤, \tau \right)+{𝒯}^{*}\left(𝚥, \tau \right)\right]{\int }_0^1\begin{array}{c}\hslash Ɲ\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}\right)\end{array}d\hslash ,\end{array}$$

that is

$$\left[\frac{p}{{𝚥}^p-{𝚤}^p} {\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}{𝒯}_{*}\left(ⱷ,\tau \right)Ɲ\left(ⱷ\right)dⱷ, \frac{p}{{𝚥}^p-{𝚤}^p} {\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}{𝒯}^{*}\left(ⱷ,\tau \right)Ɲ\left(ⱷ\right)dⱷ\right]$$

$${\supseteq }_I\left[{𝒯}_{*}\left(𝚤, \tau \right)+{𝒯}_{*}\left(𝚥, \tau \right), {𝒯}^{*}\left(𝚤, \tau \right)+{𝒯}^{*}\left(𝚥, \tau \right)\right]{\int }_0^1\begin{array}{c}\hslash Ɲ\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}\right)\end{array}d\hslash ,$$

hence

$$\frac{p}{{𝚥}^p-{𝚤}^p}\odot \left(FA\right){\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}\odot \widetilde{𝒯}\left(ⱷ\right)Ɲ(ⱷ)dⱷ{\supseteq }_{\mathbb{F}}\left[\widetilde{𝒯}\left(𝚤\right)\oplus \widetilde{𝒯}\left(𝚥\right)\right]\odot {\int }_0^1\hslash Ɲ\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}\right)d\hslash .$$

□

Theorem 12. 

(𝐻-𝐻  Fejér inequality for  $UD$-$p$-convex  $FNVM$)  Let  $\widetilde{𝒯}\in UDFSX\left(\left[𝚤, 𝚥\right], {\mathcal{L}}_C, p\right)$. The family of  $IVMs$  is defined by  $\tau$-cuts  ${𝒯}_{\tau }:\left[𝚤, 𝚥\right]\subset \mathbb{R}\to {{\mathcal{K}}_C}^{+}$  are given by  ${𝒯}_{\tau }\left(ⱷ\right)=\left[{𝒯}_{*}\left(ⱷ,\tau \right), {𝒯}^{*}\left(ⱷ,\tau \right)\right]$  for all  $ⱷ\in \left[𝚤, 𝚥\right]$  and for all  $\tau \in \left[0, 1\right]$. If  $\widetilde{𝒯}\in {\mathcal{F}\mathcal{A}}_{\left(\left[𝚤, 𝚥\right], \tau \right)}$  and  $Ɲ:\left[𝚤, 𝚥\right]\to \mathbb{R},Ɲ(ⱷ)\ge 0,$  symmetric with respect to  ${\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}},$  and  ${\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}Ɲ(ⱷ)dⱷ>0$, then

$$\widetilde{𝒯}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}\right){\supseteq }_{\mathbb{F}}\frac{1}{{\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}Ɲ(ⱷ)dⱷ}\odot \left(FA\right){\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}\odot \widetilde{𝒯}\left(ⱷ\right)Ɲ(ⱷ)dⱷ.$$

(46)

If  $\widetilde{𝒯}$  is  $UD$-$p$-concave  $FNVM$, then inequality (46) is reversed.

Proof. 

Since $\widetilde{𝒯}$ is an $\mathrm{U}\mathrm{D}$-convex, then for $\tau \in \left[0, 1\right],$ we have

$$\begin{array}{c}{𝒯}_{*}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}, \tau \right)\le \frac{1}{2}\left({𝒯}_{*}\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)+{𝒯}_{*}\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)\right),\\ {𝒯}^{*}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}, \tau \right)\ge \frac{1}{2}\left({𝒯}^{*}\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)+{𝒯}^{*}\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)\right).\end{array}$$

(47)

Since $\mathrm{Ɲ}\left({\left[\mathrm{\hslash }{𝚤}^p+\left(1-\mathrm{\hslash }\right){𝚥}^p\right]}^{\frac{1}{p}}\right)=\mathrm{Ɲ}\left({\left[\left(1-\mathrm{\hslash }\right){𝚤}^p+\mathrm{\hslash }{𝚥}^p\right]}^{\frac{1}{p}}\right)$, then by multiplying (47) by $\mathrm{Ɲ}\left({\left[\left(1-\mathrm{\hslash }\right){𝚤}^p+\mathrm{\hslash }{𝚥}^p\right]}^{\frac{1}{p}}\right)$ and integrate it with respect to $\mathrm{\hslash }$ over $\left[0,1\right],$ we obtain

$$\begin{array}{c} \\ {𝒯}_{*}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}, \tau \right){\int }_0^1Ɲ\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}\right)d\hslash \\ \le \frac{1}{2}\left(\begin{array}{c}{\int }_0^1{𝒯}_{*}\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)Ɲ\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}\right)d\hslash \\ +{\int }_0^1{𝒯}_{*}\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)d\hslash Ɲ\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}\right)\end{array}\right),\\ {𝒯}^{*}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}, \tau \right){\int }_0^1Ɲ\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}\right)d\hslash \\ \ge \frac{1}{2}\left(\begin{array}{c}{\int }_0^1{𝒯}^{*}\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)Ɲ\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}\right)d\hslash \\ +{\int }_0^1{𝒯}^{*}\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)Ɲ\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}\right)d\hslash \end{array}\right).\end{array}$$

(48)

Since

$$\begin{array}{c} \\ {\int }_0^1{𝒯}_{*}\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)Ɲ\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}\right)d\hslash \\ ={\int }_0^1{𝒯}_{*}\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)Ɲ\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}\right)d\hslash \\ =\frac{p}{{𝚥}^p-{𝚤}^p} {\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}{𝒯}_{*}\left(ⱷ,\tau \right)Ɲ(ⱷ)dⱷ, \\ \\ {\int }_0^1{𝒯}^{*}\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)Ɲ\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}\right)d\hslash \\ ={\int }_0^1{𝒯}^{*}\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}, \tau \right)Ɲ\left({\left[\hslash {𝚤}^p+\left(1-\hslash \right){𝚥}^p\right]}^{\frac{1}{p}}\right)d\hslash \\ =\frac{p}{{𝚥}^p-{𝚤}^p} {\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}{𝒯}^{*}\left(ⱷ,\tau \right)Ɲ(ⱷ)dⱷ. \\ \end{array}$$

(49)

Then, from (49) we have

$$\begin{array}{c} \\ {𝒯}_{*}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}, \tau \right)\le \frac{1}{{\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}Ɲ\left(ⱷ\right)dⱷ} {\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}{𝒯}_{*}\left(ⱷ,\tau \right)Ɲ\left(ⱷ\right)dⱷ, \\ {𝒯}^{*}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}, \tau \right)\ge \frac{1}{{\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}Ɲ\left(ⱷ\right)dⱷ} {\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}{𝒯}^{*}\left(ⱷ,\tau \right)Ɲ\left(ⱷ\right)dⱷ,\end{array}$$

from which, we have

$$\begin{array}{c} \\ \left[{𝒯}_{*}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}, \tau \right), {𝒯}^{*}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}, \tau \right)\right] \\ {\supseteq }_I\frac{1}{{\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}Ɲ\left(ⱷ\right)dⱷ}\left[ {\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}{𝒯}_{*}\left(ⱷ,\tau \right)Ɲ\left(ⱷ\right)dⱷ, {\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}{𝒯}^{*}\left(ⱷ,\tau \right)Ɲ\left(ⱷ\right)dⱷ\right], \\ \end{array}$$

that is

$$\widetilde{𝒯}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}\right){\supseteq }_{\mathbb{F}}\frac{1}{{\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}Ɲ(ⱷ)dⱷ}\odot \left(FA\right){\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}\odot \widetilde{𝒯}\left(ⱷ\right)Ɲ(ⱷ)dⱷ,$$

this completes the proof. □

Remark 5. 

If in Theorems 11 and 12, $p=1$, then we obtain the appropriate theorems for $UD$-convex fuzzy-$IVMs$ [77].

If in the Theorems 11 and 12, ${𝒯}_{*}\left(ⱷ,\gamma \right)={𝒯}^{*}\left(ⱷ,\gamma \right)$ with$\gamma =1$, then we obtain the appropriate theorems for $p$-convex function [84].

If in the Theorems 11 and 12, ${𝒯}_{*}\left(ⱷ,\gamma \right)={𝒯}^{*}\left(ⱷ,\gamma \right)$ with$\gamma =1$ and $p=1$, then we obtain the appropriate theorems for convex function [90].

If  $Ɲ\left(ⱷ\right)=1,$  then combining Theorems 11 and 12, we get Theorem 7.

Example 6. 

We consider the $FNVM$ $\widetilde{𝒯}:\left[0, 2\right]\to {\mathcal{L}}_C$ defined by,

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

Then, for each $\tau \in \left[0, 1\right],$ we have ${𝒯}_{\tau }\left(ⱷ\right)=\left[\left(1-\tau \right)\left(2-{ⱷ}^{\frac{p}{2}}\right)+\frac{3}{2}\tau ,\left(1+\tau \right)\left(2+{ⱷ}^{\frac{p}{2}}\right)+\frac{3}{2}\tau \right]$. Since end point mappings ${𝒯}_{*}\left(ⱷ,\tau \right),$ and ${𝒯}^{*}\left(ⱷ,\tau \right)$ are convex and concave mappings, respectively, for each $\tau \in [0, 1]$, then $\widetilde{𝒯}\left(ⱷ\right)$ is $\mathrm{U}\mathcal{D}$-convex $\mathrm{F}\mathrm{N}\mathrm{V}\mathrm{M}$. If

$$Ɲ\left(ⱷ\right)=\left\{\begin{array}{c}{ⱷ}^{\frac{p}{2}}, \sigma \in \left[0,1\right],\\ {\left(2-ⱷ\right)}^{\frac{p}{2}}, \sigma \in \left(1, 2\right], \end{array}\right.$$

then $Ɲ\left({\left(2-ⱷ\right)}^p\right)=Ɲ\left(ⱷ\right)\ge 0$, for all $ⱷ\in \left[0, 2\right]$.

Since ${𝒯}_{*}\left(\mathrm{ⱷ},\tau \right)=\left(1-\tau \right)\left(2-{\mathrm{ⱷ}}^{\frac{p}{2}}\right)+\frac{3}{2}\tau$ and ${𝒯}^{*}\left(\mathrm{ⱷ},\tau \right)=\left(1+\tau \right)\left(2+{\mathrm{ⱷ}}^{\frac{p}{2}}\right)+\frac{3}{2}\tau$.

Now we compute the following:

$$\begin{gathered} \begin{array}{c}\frac{p}{{𝚥}^p-{𝚤}^p}{\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}{𝒯}_{*}\left(ⱷ,\tau \right)Ɲ\left(ⱷ\right)dⱷ=\frac{1}{2}{\int }_0^2{ⱷ}^{p-1}{𝒯}_{*}\left(ⱷ,\tau \right)Ɲ\left(ⱷ\right)dⱷ \hfill \\ =\frac{1}{2}{\int }_0^1{ⱷ}^{p-1}{𝒯}_{*}\left(ⱷ,\tau \right)Ɲ\left(ⱷ\right)dⱷ+\frac{1}{2}{\int }_1^2{ⱷ}^{p-1}{𝒯}_{*}\left(ⱷ,\tau \right)Ɲ\left(ⱷ\right)dⱷ,\hfill \\ \frac{p}{{𝚥}^p-{𝚤}^p}{\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}{𝒯}^{*}\left(ⱷ,\tau \right)Ɲ\left(ⱷ\right)dⱷ=\frac{1}{2}{\int }_0^2{ⱷ}^{p-1}{𝒯}^{*}\left(ⱷ,\tau \right)Ɲ\left(ⱷ\right)dⱷ \hfill \\ =\frac{1}{2}{\int }_0^1{ⱷ}^{p-1}{𝒯}^{*}\left(ⱷ,\tau \right)Ɲ\left(ⱷ\right)dⱷ+\frac{1}{2}{\int }_1^2{ⱷ}^{p-1}{𝒯}^{*}\left(ⱷ,\tau \right)Ɲ\left(ⱷ\right)dⱷ,\hfill \end{array} \\ \begin{array}{c}=\frac{1}{2}{\int }_0^1\left[\left(1-\tau \right)\left(2-{ⱷ}^{\frac{1}{2}}\right)+\frac{3}{2}\tau \right]\left(\sqrt{ⱷ}\right)dⱷ+\frac{1}{2}{\int }_1^2\left[\left(1-\tau \right)\left(2-{ⱷ}^{\frac{1}{2}}\right)+\frac{3}{2}\tau \right]\left(\sqrt{2-ⱷ}\right)dⱷ\hfill \\ =\frac{1}{4}\left[\frac{13}{3}-\frac{\pi }{2}\right]+\tau \left[\frac{\pi }{8}-\frac{1}{12}\right], \\ =\frac{1}{2}{\int }_0^1\left[\left(1+\tau \right)\left(2+{ⱷ}^{\frac{1}{2}}\right)+\frac{3}{2}\tau \right]\left(\sqrt{ⱷ}\right)dⱷ+\frac{1}{2}{\int }_1^2\left[\left(1+\tau \right)\left(2+{ⱷ}^{\frac{1}{2}}\right)+\frac{3}{2}\tau \right]\left(\sqrt{2-ⱷ}\right)dⱷ\hfill \\ =\frac{1}{4}\left[\frac{19}{3}+\frac{\pi }{2}\right]+\tau \left[\frac{\pi }{8}+\frac{31}{12}\right]. \end{array} \end{gathered}$$

(50)

And

$$\begin{array}{c}\left[{𝒯}_{*}\left(𝚤, \tau \right)+{𝒯}_{*}\left(𝚥, \tau \right)\right]{\int }_0^1\begin{array}{c}\hslash Ɲ\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}\right)\end{array}d\hslash \\ =\left[4\left(1-\tau \right)-\sqrt{2}\left(1-\tau \right)+3\tau \right]\left[{\int }_0^{\frac{1}{2}}\hslash \sqrt{2\hslash }d\hslash +{\int }_{\frac{1}{2}}^1\hslash \sqrt{2\left(1-\hslash \right)}d\hslash \right]\\ =\frac{1}{3}\left(4\left(1-\tau \right)-\sqrt{2}\left(1-\tau \right)+3\tau \right), \\ \left[{𝒯}^{*}\left(𝚤, \tau \right)+{𝒯}^{*}\left(𝚥, \tau \right)\right]{\int }_0^1\hslash Ɲ\left({\left[\left(1-\hslash \right){𝚤}^p+\hslash {𝚥}^p\right]}^{\frac{1}{p}}\right)d\hslash \\ =\left[4\left(1+\tau \right)+\sqrt{2}\left(1+\tau \right)+3\tau \right]\left[{\int }_0^{\frac{1}{2}}\hslash \sqrt{2\hslash }d\hslash +{\int }_{\frac{1}{2}}^1\hslash \sqrt{2\left(1-\hslash \right)}d\hslash \right]\\ =\frac{1}{3}\left(4\left(1+\tau \right)+\sqrt{2}\left(1+\tau \right)+3\tau \right). \end{array}$$

(51)

From (50) and (51), we have

$$\begin{gathered} \begin{array}{l}\left[\frac{1}{4}\left[\frac{13}{3}-\frac{\pi }{2}\right]+\tau \left[\frac{\pi }{4}-\frac{7}{6}\right], \frac{1}{4}\left[\frac{19}{3}+\frac{\pi }{2}\right]+\tau \left[\frac{\pi }{4}+\frac{25}{6}\right]\right]\hfill \end{array} \\ \begin{array}{c}{\supseteq }_I\left[\frac{1}{3}\left(4\left(1-\tau \right)-\sqrt{2}\left(1-\tau \right)+3\tau \right), \frac{1}{3}\left(4\left(1+\tau \right)+\sqrt{2}\left(1+\tau \right)+3\tau \right)\right], \mathrm{for} \mathrm{all} \tau \in \left[0, 1\right].\hfill \end{array} \end{gathered}$$

Hence, Theorem 11 is verified.

For Theorem 12, we have

$$\begin{array}{c}{𝒯}_{*}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}, \tau \right)={𝒯}_{*}\left(1, \tau \right)=\frac{2+\tau }{2}, \\ {𝒯}^{*}\left({\left[\frac{{𝚤}^p+{𝚥}^p}{2}\right]}^{\frac{1}{p}}, \tau \right)={𝒯}^{*}\left(1, \tau \right)=\frac{3\left(2+3\tau \right)}{2},\\ \end{array}$$

(52)

$${\int }_{𝚤}^{𝚥}Ɲ\left(ⱷ\right)dⱷ={\int }_0^1\sqrt{ⱷ}dⱷ+{\int }_1^2\sqrt{2-ⱷ}dⱷ=\frac{4}{3},$$

$$\begin{array}{c} \\ \frac{p}{{\int }_{𝚤}^{𝚥}Ɲ\left(ⱷ\right)dⱷ} {\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}{𝒯}_{*}\left(ⱷ,\tau \right)Ɲ\left(ⱷ\right)dⱷ=\frac{3}{8}\left[\frac{13}{3}-\frac{\pi }{2}\right]+\frac{3\tau }{2}\left[\frac{\pi }{8}-\frac{1}{12}\right], \\ \frac{p}{{\int }_{𝚤}^{𝚥}Ɲ\left(ⱷ\right)dⱷ} {\int }_{𝚤}^{𝚥}{ⱷ}^{p-1}{𝒯}^{*}\left(ⱷ,\tau \right)Ɲ\left(ⱷ\right)dⱷ=\frac{3}{8}\left[\frac{19}{3}+\frac{\pi }{2}\right]+\frac{3\tau }{2}\left[\frac{\pi }{8}+\frac{31}{12}\right].\\ \end{array}$$

(53)

From (52) and (53), we have

$$\left[\frac{2+\tau }{2}, \frac{3\left(2+3\tau \right)}{2}\right]{\supseteq }_I\left[\frac{3}{8}\left[\frac{13}{3}-\frac{\pi }{2}\right]+\frac{3\tau }{2}\left[\frac{\pi }{8}-\frac{1}{12}\right], \frac{3}{8}\left[\frac{19}{3}+\frac{\pi }{2}\right]+\frac{3\tau }{2}\left[\frac{\pi }{8}+\frac{31}{12}\right]\right].$$

Hence, Theorem 12 has been verified.

5. Conclusions

This work examines the new class of $p$-convexity over up and down fuzzy relation which is known as $\mathrm{U}\mathrm{D}$-$p$-convex $\mathrm{F}\mathrm{N}\mathrm{V}\mathrm{M}$s. The usage of $\mathrm{F}\mathrm{N}\mathrm{V}\mathrm{M}$s in probability density functions and numerical integration makes the subject intriguing. Kunt and İşcan’s (see, [90]) indings are generalized in the context of convex $\mathrm{F}\mathrm{N}\mathrm{V}\mathrm{M}$s. We obtain both a novel Hermite–Hadamard-type inequality and a Hermite–Hadamard–Fejer-type inequality. There are no doubts regarding the feasibility of generalizing the fuzzy Aumann integral type inequalities found in this article because we have given some exceptional cases that can be viewed as applications of main outcomes. Some new examples have been provided to discuss the validity of main results.