Fuzzy Milne, Ostrowski, and Hermite–Hadamard-Type Inequalities for ħ-Godunova–Levin Convexity and Their Applications

Abstract

In this paper, we establish several Milne-type inequalities for fuzzy number mappings and investigate their relationships with other inequalities. Specifically, we utilize Aumann’s integral and the fuzzy Kulisch–Miranker order, as well as the newly defined class, $ħ$-Godunova–Levin convex fuzzy number mappings, to derive Ostrowski’s and Hermite–Hadamard-type inequalities for fuzzy number mappings. Using the fuzzy Kulisch–Miranker order, we also establish connections with Hermite–Hadamard-type inequalities. Furthermore, we explore novel ideas and results based on Hermite–Hadamard–Fejér and provide examples and applications to illustrate our findings. Some very interesting examples are also provided to discuss the validation of the main results. Additionally, some new exceptional and classical outcomes have been obtained, which can be considered as applications of our main results.

1. Introduction

The inequalities derived by Hermite and Hadamard for convex functions hold significant importance in the literature (refer to, for instance, [1] p. 137). These inequalities assert that if $\mathcal{P}:K\to \mathfrak{N}$ is a convex function defined on the interval $K$ of real numbers, and $\theta , \lambda \in K$ with $\theta <\lambda$, then

$$\mathcal{P}\left(\frac{\theta +\lambda }{2}\right)\le \frac{1}{\lambda -\theta }{\int }_{\theta }^{\lambda }\mathcal{P}\left(\mathsf{б}\right)d\mathsf{б}\le \frac{\mathcal{P}\left(\theta \right)+\mathcal{P}\left(\lambda \right)}{2}.$$

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Both inequalities are valid in the opposite direction if f exhibits concavity. It is worth mentioning that Hadamard’s inequality can be considered a further specification of convexity and can be readily derived from Jensen’s inequality; see [2,3]. Hadamard’s inequality for convex functions has garnered renewed interest in recent years, leading to a notable array of enhancements and extensions. These can be explored in various sources, such as [4,5,6,7,8], and the references provided therein.

The classical Hermite–Hadamard inequality provides estimates for the mean value of a continuous convex function $\mathcal{P}:\left[\theta , \lambda \right]\subseteq \mathfrak{N}\to \mathfrak{N}$.

$$\mathcal{P}\left(\kappa \mathsf{б}+\left(1-\kappa \right)s\right)\le \kappa \mathcal{P}\left(\mathsf{б}\right)+\left(1-\kappa \right)\mathcal{P}\left(s\right),$$

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For every $\mathsf{б}$ and $s$ within the interval $\left[\theta , \lambda \right]$ and for every $\kappa$ belonging to $\left[0, 1\right]$, we define $\mathcal{P}$ as concave if its $\left(-\mathcal{P}\right)$ is convex.

On the other hand, an issue in astrophysics, particularly regarding stellar absorption, led to Edward Arthur Milne [9] establishing the following intriguing integral inequality in 1925, inspired by a paper by Rosseland, a Norwegian astrophysicist, on this subject from 1924.

This Milne inequality applies to all positive and integrable functions $\mathcal{P}$ and $\mathcal{J}$ defined on the interval $\left[\theta , \lambda \right]$ such that

$${\int }_{\theta }^{\lambda }\frac{\mathcal{P}\left(\mathsf{б}\right)\mathcal{J}\left(\mathsf{б}\right)}{\mathcal{P}\left(\mathsf{б}\right)+\mathcal{J}\left(\mathsf{б}\right)}d\mathsf{б}{\int }_{\theta }^{\lambda }\left(\mathcal{P}\left(\mathsf{б}\right)+\mathcal{J}\left(\mathsf{б}\right)\right)d\mathsf{б}{\le }_{\mathbb{F}}{\int }_{\theta }^{\lambda }\mathcal{P}\left(\mathsf{б}\right)d\mathsf{б}{\int }_{\theta }^{\lambda }\mathcal{J}\left(\mathsf{б}\right)d\mathsf{б}.$$

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In 1938, Ostrowski [10] explored the following compelling integral inequalities:

Let ${\mathcal{P}}_{\mathrm{*}}:I=\left[\theta ,\lambda \right]\to \mathfrak{N}$ be a differentiable function on $I_0$ with $\left(\theta ,\lambda \right)$. If $\mathcal{P}\in \mathbb{E}\left[\theta ,\lambda \right]$ and $\left|{\mathcal{P}}^{\prime }\left(\mathrm{б}\right)\right|\le M$, for all $\mathrm{б}\in \left[\theta ,\lambda \right]$, then

$$\left|\mathcal{P}\left(\mathsf{б}\right)-\frac{1}{\lambda -\theta }{\int }_{\theta }^{\lambda }\mathcal{P}\left(u\right)du\right|\le M\left(\lambda -\theta \right)\left[\frac{1}{4}-\frac{{\left(\mathsf{б}-\frac{\theta +\lambda }{2}\right)}^2}{{\left(\lambda -\theta \right)}^2}\right].$$

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The renowned integral inequalities of the Ostrowski, Čebyšev, and Grüss varieties permeate numerous branches of mathematics (for historical context and generalizations, refer to the seminal monograph [11], as well as works [12,13]). The Čebyšev and Ostrowski-type inequalities, closely intertwined (refer to [14] for elaboration), hold significance in various mathematical applications and have garnered considerable attention from scholars. Ujević [15] derived the subsequent Ostrowski-type inequality:

$$\left|\mathcal{P}\left(\mathsf{б}\right)-\frac{1}{\lambda -\theta }{\int }_{\theta }^{\lambda }\mathcal{P}\left(u\right)du-\frac{\mathcal{P}\left(\lambda \right)-\mathcal{P}\left(\theta \right)}{\lambda -\theta }\left(\mathsf{б}-\frac{\theta +\lambda }{2}\right)\right|\le \frac{\sqrt{\left(\lambda -\theta \right){‖{\mathcal{P}}^{\prime }‖}_2^2-{\left(\mathcal{P}\left(\lambda \right)-\mathcal{P}\left(\theta \right)\right)}^2}}{2\sqrt{3}}.$$

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where $\mathcal{P}:\left[\theta , \lambda \right]\to \mathfrak{N}$ is a differentiable function with ${\mathcal{P}}^{\prime }\in L_2^{ }\left[\theta , \lambda \right]$ and $\frac{1}{2\sqrt{3}}$ is the best possible. For further information on additional Ostrowski-type inequalities, we direct interested readers to [16,17,18,19,20,21].

Developing a variety of integral inequalities has become a contemporary focus. In recent years, significant advancements have been achieved through the utilization of diverse integrals, including the Sugeno integral [22,23], the pseudo integral [24], and the Choquet integral [25], among others. Interval-valued functions [26], which extend beyond traditional functions, have emerged as a crucial mathematical area, particularly in addressing practical issues, notably in mathematical economics [27]. Recent studies have extended certain classical integral inequalities to encompass interval-valued functions.

Costa et al. [28] introduced novel interval adaptations of Minkowski and Beckenbach’s integral inequalities. They generalized Hermite–Hadamard, Jensen, and Ostrowski-type inequalities within this framework [29]. Additionally, they addressed Hermite–Hadamard and Hermite–Hadamard-type inequalities using interval-valued Riemann–Liouville fractional integrals [30]. Zhao et al. [31,32,33] investigated Chebyshev-type inequalities, Opial-type integral inequalities, and Jensen and Hermite–Hadamard-type inequalities for interval-valued functions, utilizing the concepts of gH-differentiability or h-convexity. Budak et al. [34] derived innovative fractional inequalities of the Ostrowski type for interval-valued functions, drawing on the definitions of gH-derivatives. Basic concepts related to fuzzy numbers and fuzzy Aumman’s integral are in the following literature: [35,36,37] and the references therein. Nanda [38] was the first to introduce the concept of convexity in fuzzy environments. Breckner [39] was the first to introduce the concept of continuity as well as to propose the class of convex mappings which is known as interval convex mapping. Khan et al. [40,41,42,43] introduced log-h-convex and ($h_1$,$h_2$)–convex fuzzy-interval-valued functions by using fuzzy Kulisch–Miranker, up- and down-, and left- and right-order relations as a distinct class of convex fuzzy-interval-valued functions. This class facilitated the establishment of Jensen and Hermite–Hadamard inequalities; see [44,45,46] and the references therein.

Notice that the left Hermite–Hadamard inequality can be approximated using Ostrowski’s inequality, well known for its estimation of the deviation of smooth function values from their mean. Drawing from current research trends, we once more examine the category of $ħ$-Godunova–Levin convex functions. This article’s primary aim is to establish connections between $ħ$-Godunova–Levin convex functions and integral inequalities resembling Ostrowski’s. Additionally, we derive Ostrowski-like inequalities from other sets of Godunova–Levin convex functions, which are essentially special instances of our principal findings. Furthermore, we discuss some applications of our primary results to special means. We anticipate that the concepts and methodologies presented herein will inspire further exploration in this domain; see [47,48,49,50,51,52] and the references therein.

The rest of the paper is structured as follows: In Section 2, we present a literature review. In Section 3, we introduce a novel definition of an $ħ$-Godunova–Levin convexity over a fuzzy codomain and derive several classical special cases of this class of convex fuzzy mappings. The Milne-type inequality has been introduced in Section 4. In Section 5, we established the Ostrowski-type inequality for $ħ$-Godunova–Levin convexity. In Section 6, other main findings related to Hermite–Hadamard-type inequalities and associated corollaries are presented, and some illustrative examples are also examined in this section. Lastly, we summarize the results and outline potential avenues for future research in Section 7.

2. Preliminaries

Firstly, we recall some basic notations that will be helpful in this section such as the following:

Gamma and Beta functions are, respectively, characterized as

$$\Gamma \left(\mathcal{y}\right)={\int }_0^{\infty }{\kappa }^{\mathcal{y}-1}{e}^{-\kappa }d\kappa ,$$

for $\mathcal{R}\left(\mathcal{y}\right)>0$

$$ß\left(\mathcal{y},\mathcal{z}\right)={\int }_0^1{\kappa }^{\mathcal{y}-1}{\left(1-\kappa \right)}^{\mathcal{z}-1}d\kappa =\frac{\Gamma \left(\mathcal{y}\right)\Gamma \left(\mathcal{z}\right)}{\Gamma \left(\mathcal{y}+\mathcal{z}\right)},$$

for $\mathcal{R}\left(\mathcal{y}\right)>0$, $\mathcal{R}\left(\mathrm{ʑ}\right)>0$.

The integral representation of the hypergeometric function is

$${}_2{}^{ }{F}_1\left(\mathcal{y},\mathcal{z};c;\mathsf{б}\right)=\frac{1}{ß\left(\mathcal{z},c-\mathcal{z}\right)}{\int }_0^1{\kappa }^{\mathcal{z}-1}{\left(1-\kappa \right)}^{c-\mathcal{z}-1}{\left(1-x\kappa \right)}^{-\mathcal{y}}d\kappa ,$$

for $\left|x\right|<1$, $\mathcal{R}\left(c\right)>0$, $\mathcal{R}\left(\mathrm{ʑ}\right)>0$.

Consider ${\mathbb{E}}_C$ as the set comprising all closed and bounded intervals of $\mathfrak{N}$, and let $\mathsf{Ѵ}$ belong to ${\mathbb{E}}_C$, defined as

$$\mathrm{Ѵ}=\left[{\mathrm{Ѵ}}_{\mathrm{*}},{\mathrm{Ѵ}}^{\mathrm{*}}\right]=\left\{\mathfrak{б}\in \mathfrak{N}|{\mathrm{Ѵ}}_{\mathrm{*}}\le \mathrm{б}\le {\mathrm{Ѵ}}^{\mathrm{*}}\right\},\left({\mathrm{Ѵ}}_{\mathrm{*}},{\mathrm{Ѵ}}^{\mathrm{*}}\in \mathfrak{N}\right)$$

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It is named a positive interval $\left[{\mathsf{Ѵ}}_{*}, {\mathsf{Ѵ}}^{*}\right]$ if ${\mathsf{Ѵ}}_{*}\ge 0$. The definition of ${\mathbb{E}}_C^{+}$, which represents the set of all positive intervals, is

$${\mathbb{E}}_{\mathrm{C}}^{+}=\left\{\left[{\mathrm{Ѵ}}_{\mathrm{*}},{\mathrm{Ѵ}}^{\mathrm{*}}\right]:\left[{\mathrm{Ѵ}}_{\mathrm{*}},{\mathrm{Ѵ}}^{\mathrm{*}}\right]\in {\mathbb{E}}_{\mathrm{C}} and {\mathrm{Ѵ}}_{\mathrm{*}}\ge 0\right\}.$$

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Let $\mathcal{ı}\in \mathfrak{N}$ and $\mathcal{ı}\cdot \mathsf{Ѵ}$ be defined by

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

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Subsequently, the Minkowski difference $\mathsf{Ӄ}-\mathsf{Ѵ}$, addition $\mathsf{Ѵ}+\mathsf{Ӄ}$, and multiplication $\mathsf{Ѵ}\times \mathsf{Ӄ}$ for $\mathsf{Ѵ},\mathsf{Ӄ}$ belong to ${\mathbb{E}}_C$ and are delineated as follows:

$$\left[{\mathsf{Ӄ}}_{*}, {\mathsf{Ӄ}}^{*}\right]+\left[{\mathsf{Ѵ}}_{*}, {\mathsf{Ѵ}}^{*}\right]=\left[{\mathsf{Ӄ}}_{*}+{\mathsf{Ѵ}}_{*}, {\mathsf{Ӄ}}^{*}+{\mathsf{Ѵ}}^{*}\right],$$

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

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

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Remark 1.

For given $\left[{\mathsf{Ӄ}}_{*}, {\mathsf{Ӄ}}^{*}\right], \left[{\mathsf{Ѵ}}_{*}, {\mathsf{Ѵ}}^{*}\right]\in {\mathbb{E}}_C,$ the relation $“{\le }_I”$ is defined on ${\mathbb{E}}_C$ by $\left[{\mathsf{Ѵ}}_{*}, {\mathsf{Ѵ}}^{*}\right]{\le }_I\left[{\mathsf{Ӄ}}_{*}, {\mathsf{Ӄ}}^{*}\right]$ if and only if ${\mathsf{Ѵ}}_{*}\le {\mathsf{Ӄ}}_{*}, {\mathsf{Ӄ}}^{*}{\le \mathsf{Ѵ}}^{*}$ for all $\left[{\mathsf{Ӄ}}_{*}, {\mathsf{Ӄ}}^{*}\right], \left[{\mathsf{Ѵ}}_{*}, {\mathsf{Ѵ}}^{*}\right]\in {\mathbb{E}}_C$ is a fuzzy-order relation. The relation $\left[{\mathsf{Ѵ}}_{*}, {\mathsf{Ѵ}}^{*}\right]{\le }_I\left[{\mathsf{Ӄ}}_{*}, {\mathsf{Ӄ}}^{*}\right]$ is coincident to $\left[{\mathsf{Ѵ}}_{*}, {\mathsf{Ѵ}}^{*}\right]\le \left[{\mathsf{Ӄ}}_{*}, {\mathsf{Ӄ}}^{*}\right]$ on ${\mathbb{E}}_C$; see [46]. This fuzzy-order relation is known as fuzzy Kulisch–Miranker order.

For $\left[{\mathrm{Ӄ}}_{\mathrm{*}},{\mathrm{Ӄ}}^{\mathrm{*}}\right],\left[{\mathrm{Ѵ}}_{\mathrm{*}},{\mathrm{Ѵ}}^{\mathrm{*}}\right]\in {\mathbb{E}}_{\mathrm{C}},$ the Hausdorff–Pompeiu distance between intervals $\left[{\mathrm{Ӄ}}_{\mathrm{*}},{\mathrm{Ӄ}}^{\mathrm{*}}\right]$ and $\left[{\mathrm{Ѵ}}_{\mathrm{*}},{\mathrm{Ѵ}}^{\mathrm{*}}\right]$ is defined by

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

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It is a familiar fact that $\left({\mathbb{E}}_C,d_H\right)$ is a complete metric space [35,36,37].

We will briefly review some essential concepts regarding fuzzy sets and fuzzy numbers since we will rely on the standard definitions of these sets.

Please note that we refer to $\mathbb{E}$ and ${\mathbb{E}}_C$ as the set of all fuzzy subsets and fuzzy numbers of $\mathfrak{N}$.

Definition 1 [35].

Given $\widetilde{\mathsf{Ӄ}}\in {\mathbb{E}}_C$, the level sets or cut sets are given by ${\left[\widetilde{\mathsf{Ӄ}}\right]}^{\mathcal{ı}}=\left\{\mathsf{б}\in \mathfrak{N}|\widetilde{\mathsf{Ӄ}}\left(\mathsf{б}\right)>\mathcal{ı}\right\}$ for all $\mathcal{ı}\in [0, 1]$ and by ${\left[\widetilde{\mathsf{Ӄ}}\right]}^0=\left\{\mathfrak{б}\in \mathfrak{N}|\widetilde{\mathsf{Ӄ}}\left(\mathsf{б}\right)>0\right\}$. These sets are known as $\mathcal{ı}$-level sets or $\mathcal{ı}$-cut sets of $\widetilde{\mathsf{Ӄ}}$.

Proposition 1 [28].

Let $\widetilde{\mathsf{Ӄ}},\widetilde{\mathsf{Ѵ}}\in {\mathbb{E}}_C$. Then, relation $“{\le }_{\mathbb{F}}”$ is given on ${\mathbb{E}}_C$ by $\widetilde{\mathsf{Ӄ}}{\le }_{\mathbb{F}}\widetilde{\mathsf{Ѵ}}$ when and only when ${{\left[\widetilde{\mathsf{Ӄ}}\right]}^{\mathcal{ı}}\le }_I{\left[\widetilde{\mathsf{Ѵ}}\right]}^{\mathcal{ı}}$, for every $\mathcal{ı}\in [0, 1],$ which are left- and right-order relations or fuzzy Kulisch–Miranker-order relations.

Proposition 2 [49].

Let $\widetilde{\mathsf{Ӄ}},\widetilde{\mathsf{Ѵ}}\in {\mathbb{E}}_C$. Then, relation $“{\supseteq }_{\mathbb{F}}”$ is given on ${\mathbb{E}}_C$ by $\widetilde{\mathsf{Ӄ}}{\supseteq }_{\mathbb{F}}\widetilde{\mathsf{Ѵ}}$ when and only when ${\left[\widetilde{\mathsf{Ӄ}}\right]}^{\mathcal{ı}}{\supseteq }_I{\left[\widetilde{\mathsf{Ѵ}}\right]}^{\mathcal{ı}}$ for every $\mathcal{ı}\in [0, 1],$ which is the $UD-$ order relation on ${\mathbb{E}}_C$.

Remember the approaching notions, which are offered in the literature. If $\widetilde{\mathsf{Ӄ}},\widetilde{\mathsf{Ѵ}}\in {\mathbb{E}}_C$ and $\mathcal{ı}\in \mathfrak{N}$, then, for every $\mathcal{ı}\in \left[0, 1\right],$ the arithmetic operations addition “$\oplus ”$, multiplication “$\otimes ”$, and scaler multiplication “$\odot ”$ are defined by

$${\left[\widetilde{\mathsf{Ӄ}}\oplus \widetilde{\mathsf{Ѵ}}\right]}^{\mathcal{ı}}={\left[\widetilde{\mathsf{Ӄ}}\right]}^{\mathcal{ı}}+{\left[\widetilde{\mathsf{Ѵ}}\right]}^{\mathcal{ı}},$$

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$${\left[\widetilde{\mathsf{Ӄ}}\otimes \widetilde{\mathsf{Ѵ}}\right]}^{\mathcal{ı}}={\left[\widetilde{\mathsf{Ӄ}}\right]}^{\mathcal{ı}}\times {\left[ \widetilde{\mathsf{Ѵ}}\right]}^{\mathcal{ı}},$$

(14)

$${\left[\mathcal{t}\odot \widetilde{\mathsf{Ӄ}}\right]}^{\mathcal{ı}}=\mathcal{t}{\left[\widetilde{\mathsf{Ӄ}}\right]}^{\mathcal{ı}},$$

(15)

over $[\theta ,\lambda ]$.

Aumann Integral Operators for Interval and $F$·$N$·Ms

Now, we define and discuss some properties of Aumann integral operators for interval and $F$·$N$·$M$s.

Theorem 1 [35,37].

If $\mathcal{P}:[\theta , \lambda ]\subset \mathfrak{N}\to {\mathbb{E}}_C$ is an interval-valued mapping ($\mathit{\text{Ι-V-M}}$) which satisfies $\mathcal{P}\left(\mathsf{б}\right)=\left[{\mathcal{P}}_{*}\left(\mathsf{б}\right), {\mathcal{P}}^{*}\left(\mathsf{б}\right)\right]$, then $\mathcal{P}$ is an Aumann integrable (IA integrable) over $[\theta , \lambda ]$ when and only when ${\mathcal{P}}_{*}\left(\mathsf{б}\right)$ and ${\mathcal{P}}^{*}\left(\mathsf{б}\right)$ both are integrable over $\left[\theta , \lambda \right],$ such that

$$\left(IA\right){\int }_{\theta }^{\lambda }\mathcal{P}\left(\mathrm{б}\right)d\mathrm{б}=\left[{\int }_{\theta }^{\begin{array}{c} \\ \lambda \end{array}}{\mathcal{P}}_{\mathrm{*}}\left(\mathrm{б}\right)d\mathrm{б},{\int }_{\theta }^{\lambda }{\mathcal{P}}^{\mathrm{*}}\left(\mathrm{б}\right)d\mathrm{б}\right].$$

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The literature suggests the following conclusions; see [36,38,46]:

Definition 2 [46].

A fuzzy-number-valued map $\tilde{\mathcal{P}}:\Lambda \subset \mathfrak{N}\to {\mathbb{E}}_C$ is named fuzzy number mapping ($F$·$N$·$M$). For each $\mathcal{ı}\in (0, 1],$ its $\mathit{\text{Ι-V-M}}$s are classified according to their $\mathcal{ı}$ -levels. ${\mathcal{P}}_{\mathcal{ı}}:\Lambda \subset \mathfrak{N}\to {\mathbb{E}}_C$ are given by ${\mathcal{P}}_{\mathcal{ı}}\left(\mathsf{б}\right)=\left[{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right), {\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\right]$ for all $\mathsf{б}\in \Lambda .$ Here, for each $\mathcal{ı}\in (0, 1],$ the end-point real mappings ${\mathcal{P}}_{*}\left(.,\mathcal{ı}\right), {\mathcal{P}}^{*}\left(.,\mathcal{ı}\right):\Lambda \to \mathfrak{N}$ are called lower and upper mappings of $\tilde{\mathcal{P}}\left(\mathsf{б}\right)$.

Definition 3.

Let $\tilde{\mathcal{P}}:[\theta , \lambda ]\subset \mathfrak{N}\to {\mathbb{E}}_C$ be an $F$ ·$N$ ·$M$. Then, the fuzzy integral of $\tilde{\mathcal{P}}$ over $\left[\theta , \lambda \right],$ denoted by $\left(FA\right){\int }_{\theta }^{\lambda }\tilde{\mathcal{P}}\left(\mathsf{б}\right)d\mathsf{б}$, is given level-wise by

$${\left[\left(FA\right){\int }_{\theta }^{\lambda }\tilde{\mathcal{P}}\left(\mathsf{б}\right)d\mathsf{б}\right]}^{\begin{array}{c} \\ \mathcal{ı}\end{array}}=\left(IA\right){\int }_{\theta }^{\lambda }{\mathcal{P}}_{\mathcal{ı}}\left(\mathsf{б}\right)d\mathsf{б}=\left\{{\int }_{\theta }^{\lambda }\mathcal{P}\left(\mathsf{б},\mathcal{ı}\right)d\mathsf{б}:\mathcal{P}\left(\mathsf{б},\mathcal{ı}\right)\in {\mathcal{R}}_{\left(\left[\theta , \lambda \right], \mathcal{ı}\right)}\right\},$$

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for all $\mathcal{ı}\in (0, 1],$ where ${\mathcal{R}}_{\left(\left[\theta , \lambda \right], \mathcal{ı}\right)}$ denotes the collection of Riemannian integrable mappings of $\mathit{\text{Ι-V-M}}$s. The $F$·$N$·$M$ $\tilde{\mathcal{P}}$ is $FA$-integrable over $[\theta , \lambda ]$ if $\left(FA\right){\int }_{\theta }^{\lambda }\tilde{\mathcal{P}}\left(\mathsf{б}\right)d\mathsf{б}\in {\mathbb{E}}_C.$ Note that, if ${\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right), {\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)$ are Lebesgue-integrable, then $\mathcal{P}$ is fuzzy Aumann-integrable mapping over $[\theta , \lambda ]$; see [46].

Theorem 2 [37].

Let $\tilde{\mathcal{P}}:[\theta , \lambda ]\subset \mathfrak{N}\to {\mathbb{E}}_C$ be an $F$·$N$·$M$; $\mathit{\text{Ι-V-M}}$s are classified according to their $\mathcal{ı}$-levels. ${\mathcal{P}}_{\mathcal{ı}}:[\theta , \lambda ]\subset \mathfrak{N}\to {\mathbb{E}}_C$ are given by ${\mathcal{P}}_{\mathcal{ı}}\left(\mathsf{б}\right)=\left[{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right), {\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\right]$ for all $\mathsf{б}\in [\theta , \lambda ]$ and for all $\mathcal{ı}\in (0, 1].$ Then, $\tilde{\mathcal{P}}$ is $FA$-integrable over $[\theta , \lambda ]$ if and only if ${\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)$ and ${\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)$ are both $A$-integrable over $[\theta , \lambda ]$. Moreover, if $\tilde{\mathcal{P}}$ is $FA$-integrable over $\left[\theta , \lambda \right],$ then

$$\begin{gathered} {\left[\left(FA\right){\int }_{\theta }^{\lambda }\tilde{\mathcal{P}}\left(\mathsf{б}\right)d\mathsf{б}\right]}^{\mathcal{ı}}=\left[\left(A\right){\int }_{\theta }^{\lambda }{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)d\mathsf{б}, \left(A\right){\int }_{\theta }^{\lambda }{\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)d\mathsf{б}\right] \\ =\left(IA\right){\int }_{\theta }^{\lambda }{\mathcal{P}}_{\mathcal{ı}}\left(\mathsf{б}\right)d\mathsf{б} \end{gathered}$$

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for all $\mathcal{ı}\in (0, 1].$ For all $\mathcal{ı}\in \left(0, 1\right],$ ${\mathcal{F}\mathcal{R}}_{\left(\left[\theta , \lambda \right], \mathcal{ı}\right)}$ denotes the collection of all $FA$-integrable $F$·$N$·$M$s.

The family of all $FA$-integrable $F$·$N$·$M$s over $[\theta , \lambda ]$ is denoted by ${\mathcal{F}\mathcal{A}}_{\left(\left[\theta , \lambda \right], \mathcal{ı}\right)}.$

Breckner discussed the emerging idea of interval-valued convexity in [39].

A $\mathit{\text{Ι-V-M}}$ $\mathcal{P}:\mathbb{I}=\left[\theta ,\lambda \right]\to {\mathcal{X}}_I$ is called convex $\mathit{\text{Ι-V-M}}$ if

$$\mathcal{P}\left(\kappa \mathsf{б}+\left(1-\kappa \right)s\right)\supseteq \kappa \mathcal{P}\left(\mathsf{б}\right)+(1-\kappa )\mathcal{P}\left(s\right),$$

(19)

for all $\mathsf{б}, y\in \left[\theta , \lambda \right], \kappa \in \left[0, 1\right]$, where ${\mathcal{X}}_I$ is the collection of all real-valued intervals. If (19) is reversed, then $\mathcal{P}$ is called concave.

Definition 4 [38].

The $F$·$N$·$M$ $\mathcal{P}:\left[\theta , \lambda \right]\to {\mathbb{E}}_C$ is called convex $F$·$N$·$M$ on$\left[\theta , \lambda \right]$ if

$$\tilde{\mathcal{P}}\left(\kappa \mathsf{б}+\left(1-\kappa \right)s\right){\le }_{\mathbb{F}}\kappa \odot \tilde{\mathcal{P}}\left(\mathsf{б}\right)\oplus (1-\kappa )\odot \tilde{\mathcal{P}}\left(s\right),$$

(20)

for all $\mathsf{б},s\in \left[\theta , \lambda \right], \kappa \in \left[0, 1\right],$ where $\tilde{\mathcal{P}}\left(\mathsf{б}\right){\ge }_{\mathbb{F}}\tilde{0}$ for all $\mathsf{б}\in \left[\theta , \lambda \right].$ If (20) is reversed, then $\tilde{\mathcal{P}}$ is called concave $F$·$N$·$M$ on $\left[\theta , \lambda \right]$. $\tilde{\mathcal{P}}$ is affine if and only if it is both convex and concave $F$·$N$·$M$.

3. $\mathbf{ħ}$-Godunova–Levin Convex $\mathsf{F}$·$\mathsf{N}$·$\mathsf{M}$

In this section, we start with the main definition of $ħ$-Godunova–Levin convexity over the fuzzy domain that will be helpful for the upcoming results.

Definition 5.

Let $K$ be a convex set and $ħ:[0, 1]\subseteq K\to {\mathfrak{N}}^{+}$ such that $ħ\not\equiv 0$. Then, the mapping $\tilde{\mathcal{P}}:K\to {\mathbb{E}}_C$ is said to be $ħ$-Godunova–Levin convex $F$·$N$·$M$  on $K$ if

$$\tilde{\mathcal{P}}\left(\kappa \mathsf{б}+\left(1-\kappa \right)s\right){\le }_{\mathbb{F}}\frac{\tilde{\mathcal{P}}\left(\mathsf{б}\right)}{ħ\left(\kappa \right)}\oplus \frac{\tilde{\mathcal{P}}\left(s\right)}{ħ\left(1-\kappa \right)},$$

(21)

for all $\mathsf{б},s\in K, \kappa \in \left[0, 1\right],$ where $\tilde{\mathcal{P}}\left(\mathsf{б}\right){\ge }_{\mathbb{F}}\tilde{0}.$ The mapping $\tilde{\mathcal{P}}:K\to {\mathbb{E}}_C$ is said to be $ħ$-Godunova–Levin concave $F$·$N$·$M$ on $K$ if inequality (21) is reversed. Moreover, $\tilde{\mathcal{P}}$ is known as $ħ$-Godunova–Levin affine $F$·$N$·$M$ on$K$ if

$$\tilde{\mathcal{P}}\left(\kappa \mathsf{б}+\left(1-\kappa \right)s\right)=\frac{\tilde{\mathcal{P}}\left(\mathsf{б}\right)}{ħ\left(\kappa \right)}\oplus \frac{\tilde{\mathcal{P}}\left(s\right)}{ħ\left(1-\kappa \right)},$$

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for all $\mathsf{б},s\in K, \kappa \in \left[0, 1\right],$ where $\tilde{\mathcal{P}}\left(\mathsf{б}\right){\ge }_{\mathbb{F}}\tilde{0}$.

Remark 2.

The $ħ$-Godunova–Levin convex $F$·$N$·$M$s have some very nice properties, similar to convex $F$·$N$·$M$.

(1)

If $\tilde{\mathcal{P}}$ is an $ħ$-Godunova–Levin convex $F$·$N$·$M$, then $\alpha \tilde{\mathcal{P}}$ is also $ħ$-Godunova–Levin convex for $\alpha \ge 0$.

(2)

If $\tilde{\mathcal{P}}$ and $\tilde{\mathcal{T}}$ both are $ħ$-Godunova–Levin convex $F$·$N$·$M$s, then $max\left(\tilde{\mathcal{P}}\left(\mathsf{б}\right),\tilde{\mathcal{T}}\left(\mathsf{б}\right)\right)$ is also $ħ$-Godunova–Levin convex $F$·$N$·$M$.

Here, we will go through a few unique exceptional cases of $ħ$-Godunova–Levin convex $F$·$N$·$M$s.

(i)

If $ħ\left(\kappa \right)={\kappa }^s,$ then $ħ$-Godunova–Levin convex $F$·$N$·$M$ becomes $s$-Godunova–Levin convex $F$·$N$·$M$, that is,

$$\tilde{\mathcal{P}}\left(\kappa \mathsf{б}+\left(1-\kappa \right)s\right){\le }_{\mathbb{F}}\frac{\tilde{\mathcal{P}}\left(\mathsf{б}\right)}{{\kappa }^s}\oplus \frac{\tilde{\mathcal{P}}\left(s\right)}{{\left(1-\kappa \right)}^s},\forall \mathsf{б},s\in K, \kappa \in \left[0, 1\right].$$

(ii)

If $ħ\left(\kappa \right)=\kappa ,$ then $ħ$-Godunova–Levin convex $F$·$N$·$M$ becomes Godunova–Levin convex $F$·$N$·$M$—see [41]—that is,

$$\tilde{\mathcal{P}}\left(\kappa \mathsf{б}+\left(1-\kappa \right)s\right){\le }_{\mathbb{F}}\frac{\tilde{\mathcal{P}}\left(\mathsf{б}\right)}{\kappa }\oplus \frac{\tilde{\mathcal{P}}\left(s\right)}{1-\kappa },\forall \mathsf{б},s\in K, \kappa \in \left[0, 1\right].$$

(iii)

If $ħ\left(\kappa \right)\equiv 1,$ then $ħ$-Godunova–Levin convex $F$·$N$·$M$ becomes Godunova–Levin $P$-$F$·$N$·$M$, that is,

$$\tilde{\mathcal{P}}\left(\kappa \mathsf{б}+\left(1-\kappa \right)s\right){\le }_{\mathbb{F}}\tilde{\mathcal{P}}\left(\mathsf{б}\right)\oplus \tilde{\mathcal{P}}\left(s\right),\forall \mathsf{б},s\in K, \kappa \in \left[0, 1\right].$$

Note that there are also new special cases (i) and (iii) as well.

Theorem 3.

Let $K$ be convex set, non-negative real-valued function $ħ:[0, 1]\subseteq K\to \mathfrak{N}$ such that $ħ\not\equiv 0$ and let $\tilde{\mathcal{P}}:K\to {\mathbb{E}}_C$ be an $F$·$N$·$M$, and $\mathit{\text{Ι-V-M}}$s are classified according to their $\mathcal{ı}$-levels, such that ${\mathcal{P}}_{\mathcal{ı}}:K\subset \mathfrak{N}\to {\mathcal{X}}_I^{+}\subset {\mathcal{X}}_I$ are given by

$${\mathcal{P}}_{\mathcal{ı}}\left(\mathsf{б}\right)=\left[{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right), {\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\right],$$

(23)

for all $\mathsf{б}\in K$ and for all $\mathcal{ı}\in \left[0, 1\right]$. Then, $\tilde{\mathcal{P}}$ is an $ħ$-Godunova–Levin convex on $K,$ if and only if, for all $\mathcal{ı}\in \left[0, 1\right],$ ${\mathcal{P}}_{*}\left(\mathsf{б}, \mathcal{ı}\right)$ and ${\mathcal{P}}^{*}\left(\mathsf{б}, \mathcal{ı}\right)$ are $ħ$-Godunova–Levin convex mappings.

Proof .

Assume that for each $\mathcal{ı}\in \left[0, 1\right],$ ${\mathcal{P}}_{*}\left(\mathsf{б}, \mathcal{ı}\right)$ and ${\mathcal{P}}^{*}\left(\mathsf{б}, \mathcal{ı}\right)$ are both $ħ$-Godunova–Levin convex on $K$, respectively. Then, we have

$${\mathcal{P}}_{*}\left(\kappa \mathsf{б}+\left(1-\kappa \right)s,\mathcal{ı}\right)\le \frac{{\mathcal{P}}_{*}\left(\mathsf{б}, \mathcal{ı}\right)}{ħ\left(\kappa \right)}+\frac{{\mathcal{P}}_{*}\left(s,\mathcal{ı}\right)}{ħ\left(1-\kappa \right)}, \forall \mathsf{б},s\in K, \kappa \in \left[0, 1\right],$$

and

$${\mathcal{P}}^{*}\left(\kappa \mathsf{б}+\left(1-\kappa \right)s,\mathcal{ı}\right)\le \frac{{\mathcal{P}}^{*}\left(\mathsf{б}, \mathcal{ı}\right)}{ħ\left(\kappa \right)}+\frac{{\mathcal{P}}^{*}\left(s,\mathcal{ı}\right)}{ħ\left(1-\kappa \right)},\forall \mathsf{б},s\in K, \kappa \in \left[0, 1\right].$$

Then, by (23), (8), and (9), we obtain

$${\mathcal{P}}_{\mathcal{ı}}\left(\kappa \mathsf{б}+\left(1-\kappa \right)s\right)=\left[{\mathcal{P}}_{*}\left(\kappa \mathsf{б}+\left(1-\kappa \right)s,\mathcal{ı}\right), {\mathcal{P}}^{*}\left(\kappa \mathsf{б}+\left(1-\kappa \right)s,\mathcal{ı}\right)\right],$$

$${\le }_I\left[\frac{{\mathcal{P}}_{*}\left(s,\mathcal{ı}\right)}{ħ\left(\kappa \right)}, \frac{{\mathcal{P}}^{*}\left(s,\mathcal{ı}\right)}{ħ\left(\kappa \right)}\right]+\left[\frac{{\mathcal{P}}_{*}\left(s,\mathcal{ı}\right)}{ħ\left(1-\kappa \right)}, \frac{{\mathcal{P}}^{*}\left(s,\mathcal{ı}\right)}{ħ\left(1-\kappa \right)}\right],$$

that is,

$$\tilde{\mathcal{P}}\left(\kappa \mathsf{б}+\left(1-\kappa \right)s\right){\le }_{\mathbb{F}}\frac{\tilde{\mathcal{P}}\left(\mathsf{б}\right)}{ħ\left(\kappa \right)}\oplus \frac{\tilde{\mathcal{P}}\left(s\right)}{ħ\left(1-\kappa \right)},\forall \mathsf{б},s\in K, \kappa \in \left[0, 1\right].$$

Hence, $\tilde{\mathcal{P}}$ is an $ħ$-Godunova–Levin convex $F$·$N$·$M$ on $K.$

Conversely, let $\tilde{\mathcal{P}}$ be $ħ$-Godunova–Levin convex $F$·$N$·$M$ on $K.$ Then, for all $\mathsf{б},s\in K$ and $\kappa \in \left[0, 1\right],$ we have $\tilde{\mathcal{P}}\left(\kappa \mathsf{б}+\left(1-\kappa \right)s\right){\le }_{\mathbb{F}}\frac{\tilde{\mathcal{P}}\left(\mathsf{б}\right)}{ħ\left(\kappa \right)}\oplus \frac{\tilde{\mathcal{P}}\left(s\right)}{ħ\left(1-\kappa \right)}.$ Therefore, from (21), we have

$${\mathcal{P}}_{\mathcal{ı}}\left(\kappa \mathsf{б}+\left(1-\kappa \right)s\right)=\left[{\mathcal{P}}_{*}\left(\kappa \mathsf{б}+\left(1-\kappa \right)s,\mathcal{ı}\right), {\mathcal{P}}^{*}\left(\kappa \mathsf{б}+\left(1-\kappa \right)s,\mathcal{ı}\right)\right].$$

Again, from (23), (8) and (9), we obtain

$$\frac{{\mathcal{P}}_{\mathcal{ı}}\left(\mathsf{б}\right)}{ħ\left(\kappa \right)}+\frac{{\mathcal{P}}_{\mathcal{ı}}\left(\mathsf{б}\right)}{ħ\left(1-\kappa \right)}=\left[\frac{{\mathcal{P}}_{*}\left(s,\mathcal{ı}\right)}{ħ\left(\kappa \right)}, \frac{{\mathcal{P}}^{*}\left(s,\mathcal{ı}\right)}{ħ\left(\kappa \right)}\right]+\left[\frac{{\mathcal{P}}_{*}\left(s,\mathcal{ı}\right)}{ħ\left(1-\kappa \right)}, \frac{{\mathcal{P}}^{*}\left(s,\mathcal{ı}\right)}{ħ\left(1-\kappa \right)}\right],$$

for all $\mathsf{б},s\in K$ and $\kappa \in \left[0, 1\right].$ Then, by $ħ$-Godunova–Levin convexity of $\tilde{\mathcal{P}}$, we have for all $\mathsf{б},s\in K$ and $\kappa \in \left[0, 1\right]$ such that

$${\mathcal{P}}_{*}\left(\kappa \mathsf{б}+\left(1-\kappa \right)s,\mathcal{ı}\right)\le \frac{{\mathcal{P}}_{*}\left(\mathsf{б}, \mathcal{ı}\right)}{ħ\left(\kappa \right)}+\frac{{\mathcal{P}}_{*}\left(s,\mathcal{ı}\right)}{ħ\left(1-\kappa \right)},$$

and

$${\mathcal{P}}^{*}\left(\kappa \mathsf{б}+\left(1-\kappa \right)s,\mathcal{ı}\right)\le \frac{{\mathcal{P}}^{*}\left(\mathsf{б}, \mathcal{ı}\right)}{ħ\left(\kappa \right)}+\frac{{\mathcal{P}}^{*}\left(s,\mathcal{ı}\right)}{ħ\left(1-\kappa \right)},$$

for each $\mathcal{ı}\in \left[0, 1\right].$ Hence, the result follows. □

Remark 3.

If ${\mathcal{P}}_{*}\left(\mathsf{б}, \mathcal{ı}\right)={\mathcal{P}}^{*}\left(\mathsf{б}, \mathcal{ı}\right)$ with $\mathcal{ı}=1,$ then $ħ$-Godunova–Levin convex $F$·$N$·$M$ reduces to the $ħ$-Godunova–Levin convex function.

If ${\mathcal{P}}_{*}\left(\mathsf{б}, \mathcal{ı}\right)={\mathcal{P}}^{*}\left(\mathsf{б}, \mathcal{ı}\right)$ with $\mathcal{ı}=1$ and $ħ\left(\kappa \right)={\kappa }^s$ with $s\in (0, 1)$, then $ħ$-Godunova–Levin convex $F$·$N$·$M$ reduces to the $s$-Godunova–Levin convex function.

If ${\mathcal{P}}_{*}\left(\mathsf{б}, \mathcal{ı}\right)={\mathcal{P}}^{*}\left(\mathsf{б}, \mathcal{ı}\right)$ with $\mathcal{ı}=1$ and $ħ\left(\kappa \right)=\kappa$, then $ħ$-Godunova–Levin convex $F$·$N$·$M$ reduces to the Godunova–Levin convex function.

If ${\mathcal{P}}_{*}\left(\mathsf{б}, \mathcal{ı}\right)={\mathcal{P}}^{*}\left(\mathsf{б}, \mathcal{ı}\right)$ with $\mathcal{ı}=1$ and $ħ\left(\kappa \right)=1$, then $ħ$-Godunova–Levin convex $F$·$N$·$M$ reduces to the $P$-convex function.

Example 1.

We consider $ħ\left(\kappa \right)=\kappa$ for $\kappa \in \left[0, 1\right]$ and the $F$·$N$·$M$ $\tilde{\mathcal{P}}:[0, 1]\to {\mathbb{E}}_C$ defined by

$$\tilde{\mathcal{P}}\left(\mathsf{б}\right)\left(\sigma \right)=\left\{\begin{array}{c}\frac{\sigma }{{2\mathsf{б}}^2}\hspace{1em}\sigma \in \left[0, {2\mathsf{б}}^2\right]\\ \frac{{4\mathsf{б}}^2-\sigma }{{2\mathsf{б}}^2} \sigma \in ({2\mathsf{б}}^2, {4\mathsf{б}}^2] \\ 0\hspace{1em}\mathrm{otherwise},\end{array}\right.$$

Then, for each $\mathcal{ı}\in \left[0, 1\right],$ we have ${\mathcal{P}}_{\mathcal{ı}}\left(\mathsf{б}\right)=\left[2\mathcal{ı}{\mathsf{б}}^2,(4-2\mathcal{ı}){\mathsf{б}}^2 \right]$, since end-point functions ${\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right),$ ${\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)$ are $ħ$-Godunova–Levin convex and $ħ$-Godunova–Levin concave functions for each $\mathcal{ı}\in [0, 1]$, respectively. Hence, $\tilde{\mathcal{P}}\left(\mathsf{б}\right)$ is an $ħ$-Godunova–Levin convex $F$·$N$·$M$.

Definition 6.

Let $\mathcal{P}:[\theta , \lambda ]\to {\mathbb{E}}_C$ be an $F$·$N$·$M$, and $\mathit{\text{Ι-V-M}}$s are classified according to their $\mathcal{ı}$-levels, such that ${\mathcal{P}}_{\mathcal{ı}}:[\theta , \lambda ]\to {\mathcal{X}}_C^{+}\subset {\mathcal{X}}_C$ are given by

$${\mathcal{P}}_{\mathcal{ı}}\left(\mathsf{б}\right)=\left[{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right), {\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\right],$$

for all $\varkappa \in [\theta , \lambda ]$ and for all $\mathcal{ı}\in \left[0, 1\right]$. Then, $\mathcal{P}$ is a lower $ħ$-Godunova–Levin convex ($ħ$-Godunova–Levin concave) $F$·$N$·$M$ on $[\theta , \lambda ],$ if and only if, for all $\mathcal{ı}\in \left[0, 1\right],$ ${\mathcal{P}}_{*}\left(\varkappa , \mathcal{ı}\right)$ is a $ħ$-Godunova–Levin convex ($ħ$-Godunova–Levin concave) mapping and ${\mathcal{P}}^{*}\left(\varkappa , \mathcal{ı}\right)$ is a $ħ$-Godunova–Levin affine mapping.

Definition 7.

Let $\mathcal{P}:[\theta , \lambda ]\to {\mathbb{E}}_C$ be an $F$·$N$·$M$, and $\mathit{\text{Ι-V-M}}$s are classified according to their $\mathcal{ı}$-levels such that ${\mathcal{P}}_{\mathcal{ı}}:[\theta , \lambda ]\to {\mathcal{X}}_C^{+}\subset {\mathcal{X}}_C$ are given by

$${\mathcal{P}}_{\mathcal{ı}}\left(\mathsf{б}\right)=\left[{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right), {\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\right],$$

for all $\varkappa \in [\theta , \lambda ]$ and for all $\mathcal{ı}\in \left[0, 1\right]$. Then, $\mathcal{P}$ is an upper $ħ$-Godunova–Levin convex ($ħ$-Godunova–Levin concave) $F$·$N$·$M$ on $[\theta , \lambda ],$ if and only if, for all $\mathcal{ı}\in \left[0, 1\right],$ ${\mathcal{P}}_{*}\left(\varkappa , \mathcal{ı}\right)$ is an $ħ$-Godunova–Levin affine mapping and ${\mathcal{P}}^{*}\left(\varkappa , \mathcal{ı}\right)$ is an $ħ$-Godunova–Levin convex ($ħ$-Godunova–Levin concave) mapping.

Remark 4.

If $ħ\left(\kappa \right)=\kappa$, then both concepts, “$ħ$-Godunova–Levin convex $F$·$N$·$M$” and classical “$ħ$-Godunova–Levin convex $F$·$N$·$M^{″}$, behave alike when $\mathcal{P}$ is a lower $ħ$-Godunova–Levin convex $F$·$N$·$M$.

Both concepts, “$ħ$-Godunova–Levin convex interval-valued mapping” and “left and right $ħ$-Godunova–Levinconvex interval-valued mapping” are coincident when $\mathcal{P}$ is a lower $ħ$-Godunova–Levin convex $F$·$N$·$M$ with $\mathcal{ı}=1$.

4. Fuzzy Version of Milne-Type Inequality

This section just proposes the following new estimation of Milne-type inequality in a fuzzy environment.

Theorem 4.

Let $\tilde{\mathcal{P}},\tilde{\mathcal{J}} :\left[\theta , \lambda \right]\to {\mathbb{E}}_C^{+}$ be two $F$·$N$·$M$s, and $\mathit{\text{Ι-V-M}}$s are classified according to their $\mathcal{ı}$-levels such that, ${\mathcal{P}}_{\mathcal{ı}}, {\mathcal{J}}_{\mathcal{ı}}:\left[\theta , \lambda \right]\subset \mathfrak{N}\to {\mathcal{X}}_I^{+}$ are given, respectively, by ${\mathcal{P}}_{\mathcal{ı}}\left(\mathsf{б}\right)=\left[{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right), {\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\right]$ and ${\mathcal{J}}_{\mathcal{ı}}\left(\mathsf{б}\right)=\left[{\mathcal{J}}_{*}\left(\mathsf{б},\mathcal{ı}\right), {\mathcal{J}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\right]$ for all $\mathsf{б}\in \left[\theta , \lambda \right]$ and for all $\mathcal{ı}\in \left[0, 1\right]$. Then,

$$\begin{gathered} \left[{\int }_{\theta }^{\lambda }\frac{{\mathcal{P}}_{*}\left(\mathsf{б}\right)\times {\mathcal{J}}_{*}\left(\mathsf{б}\right)}{{\mathcal{P}}_{*}\left(\mathsf{б}\right)+{\mathcal{J}}_{*}\left(\mathsf{б}\right)}d\mathsf{б}\times {\int }_{\theta }^{\lambda }\left({\mathcal{P}}_{*}\left(\mathsf{б}\right)+{\mathcal{J}}_{*}\left(\mathsf{б}\right)\right)d\mathsf{б}, {\int }_{\theta }^{\lambda }\frac{{\mathcal{P}}^{*}\left(\mathsf{б}\right)\times {\mathcal{J}}^{*}\left(\mathsf{б}\right)}{{\mathcal{P}}^{*}\left(\mathsf{б}\right)+{\mathcal{J}}^{*}\left(\mathsf{б}\right)}d\mathsf{б}\times {\int }_{\theta }^{\lambda }\left({\mathcal{P}}^{*}\left(\mathsf{б}\right)+{\mathcal{J}}^{*}\left(\mathsf{б}\right)\right)d\mathsf{б}\right] \\ {\le }_I\left[{\int }_{\theta }^{\lambda }{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)d\mathsf{б}{\int }_{\theta }^{\lambda }{\mathcal{J}}_{*}\left(\mathsf{б},\mathcal{ı}\right)d\mathsf{б}, {\int }_{\theta }^{\lambda }\frac{{\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\times {\mathcal{J}}^{*}\left(\mathsf{б},\mathcal{ı}\right)}{{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)+{\mathcal{J}}_{*}\left(\mathsf{б},\mathcal{ı}\right)}d\mathsf{б}{\int }_{\theta }^{\lambda }\left({\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)+{\mathcal{J}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\right)d\mathsf{б}\right] \end{gathered}$$

Proof. 

Since, for each $\mathcal{ı}\in \left[0, 1\right]$ we have

$$\begin{gathered} {\int }_{\theta }^{\lambda }\frac{\tilde{\mathcal{P}}\left(\mathsf{б}\right)\otimes \tilde{\mathcal{J}}\left(\mathsf{б}\right)}{\tilde{\mathcal{P}}\left(\mathsf{б}\right)\oplus \tilde{\mathcal{J}}\left(\mathsf{б}\right)}d\mathsf{б}\otimes {\int }_{\theta }^{\lambda }\left(\tilde{\mathcal{P}}\left(\mathsf{б}\right)\oplus \tilde{\mathcal{J}}\left(\mathsf{б}\right)\right)d\mathsf{б} \\ ={\int }_{\theta }^{\lambda }\frac{{\mathcal{P}}_{\mathcal{ı}}\left(\mathsf{б}\right)\times {\mathcal{J}}_{\mathcal{ı}}\left(\mathsf{б}\right)}{{\mathcal{P}}_{\mathcal{ı}}\left(\mathsf{б}\right)+{\mathcal{J}}_{\mathcal{ı}}\left(\mathsf{б}\right)}d\mathsf{б}\times {\int }_{\theta }^{\lambda }\left({\mathcal{P}}_{\mathcal{ı}}\left(\mathsf{б}\right)+{\mathcal{J}}_{\mathcal{ı}}\left(\mathsf{б}\right)\right)d\mathsf{б} \\ ={\int }_{\theta }^{\lambda }\frac{\left[{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)\times {\mathcal{J}}_{*}\left(\mathsf{б},\mathcal{ı}\right), {\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\times {\mathcal{J}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\right]}{\left[{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)+{\mathcal{J}}_{*}\left(\mathsf{б},\mathcal{ı}\right), {\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)+{\mathcal{J}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\right]}d\mathsf{б}\times {\int }_{\theta }^{\lambda }\left[{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)+{\mathcal{J}}_{*}\left(\mathsf{б},\mathcal{ı}\right), {\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)+{\mathcal{J}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\right]d\mathsf{б} \\ ={\int }_{\theta }^{\lambda }\frac{\left[{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)\times {\mathcal{J}}_{*}\left(\mathsf{б},\mathcal{ı}\right), {\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\times {\mathcal{J}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\right]}{\left[{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)+{\mathcal{J}}_{*}\left(\mathsf{б},\mathcal{ı}\right), {\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)+{\mathcal{J}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\right]}d\mathsf{б}\times {\int }_{\theta }^{\lambda }\left[{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)+{\mathcal{J}}_{*}\left(\mathsf{б},\mathcal{ı}\right), {\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)+{\mathcal{J}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\right]d\mathsf{б} \\ =\left[{\int }_{\theta }^{\lambda }\frac{{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)\times {\mathcal{J}}_{*}\left(\mathsf{б},\mathcal{ı}\right)}{{\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)+{\mathcal{J}}^{*}\left(\mathsf{б},\mathcal{ı}\right)}d\mathsf{б},{\int }_{\theta }^{\lambda }\frac{{\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\times {\mathcal{J}}^{*}\left(\mathsf{б},\mathcal{ı}\right)}{{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)+{\mathcal{J}}_{*}\left(\mathsf{б},\mathcal{ı}\right)}d\mathsf{б}\right]\times \left[{\int }_{\theta }^{\lambda }\left({\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)+{\mathcal{J}}_{*}\left(\mathsf{б},\mathcal{ı}\right)\right)d\mathsf{б}, {\int }_{\theta }^{\lambda }\left({\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)+{\mathcal{J}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\right)d\mathsf{б}\right] \\ =\left[{\int }_{\theta }^{\lambda }\frac{{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)\times {\mathcal{J}}_{*}\left(\mathsf{б},\mathcal{ı}\right)}{{\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)+{\mathcal{J}}^{*}\left(\mathsf{б},\mathcal{ı}\right)}d\mathsf{б}\times {\int }_{\theta }^{\lambda }\left({\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)+{\mathcal{J}}_{*}\left(\mathsf{б},\mathcal{ı}\right)\right)d\mathsf{б},{\int }_{\theta }^{\lambda }\frac{{\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\times {\mathcal{J}}^{*}\left(\mathsf{б},\mathcal{ı}\right)}{{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)+{\mathcal{J}}_{*}\left(\mathsf{б},\mathcal{ı}\right)}d\mathsf{б}\times {\int }_{\theta }^{\lambda }\left({\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)+{\mathcal{J}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\right)d\mathsf{б}\right] \\ \le \left[{\int }_{\theta }^{\lambda }\frac{{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)\times {\mathcal{J}}_{*}\left(\mathsf{б},\mathcal{ı}\right)}{{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)+{\mathcal{J}}_{*}\left(\mathsf{б},\mathcal{ı}\right)}d\mathsf{б}\times {\int }_{\theta }^{\lambda }\left({\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)+{\mathcal{J}}_{*}\left(\mathsf{б},\mathcal{ı}\right)\right)d\mathsf{б},{\int }_{\theta }^{\lambda }\frac{{\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\times {\mathcal{J}}^{*}\left(\mathsf{б},\mathcal{ı}\right)}{{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)+{\mathcal{J}}_{*}\left(\mathsf{б},\mathcal{ı}\right)}d\mathsf{б}\times {\int }_{\theta }^{\lambda }\left({\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)+{\mathcal{J}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\right)d\mathsf{б}\right] \\ \le \left[{\int }_{\theta }^{\lambda }{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)d\mathsf{б}{\int }_{\theta }^{\lambda }{\mathcal{J}}_{*}\left(\mathsf{б},\mathcal{ı}\right)d\mathsf{б},{\int }_{\theta }^{\lambda }\frac{{\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\times {\mathcal{J}}^{*}\left(\mathsf{б},\mathcal{ı}\right)}{{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)+{\mathcal{J}}_{*}\left(\mathsf{б},\mathcal{ı}\right)}d\mathsf{б}\times {\int }_{\theta }^{\lambda }\left({\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)+{\mathcal{J}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\right)d\mathsf{б},\right] \end{gathered}$$

and hence the required result. □

The subsequent outcome indicates that the interval integral inequality stated in Theorem 4 entails the classical Milne’s inequality (3).

Corollary 1.

Let $\tilde{\mathcal{P}}$ and $\tilde{\mathcal{J}}$ be two integrable $F$·$N$·$M$s from $\left[\theta , \lambda \right]$ to ${\mathbb{E}}_C^{+}$, such that ${\mathcal{P}}_{\mathcal{ı}}\left(\mathsf{б}\right)=\left[\mathcal{P}\left(\mathsf{б},\mathcal{ı}\right), \mathcal{P}\left(\mathsf{б},\mathcal{ı}\right)\right]$ and ${\mathcal{J}}_{\mathcal{ı}}\left(\mathsf{б}\right)=\left[\mathcal{J}\left(\mathsf{б},\mathcal{ı}\right),\mathcal{J}\left(\mathsf{б},\mathcal{ı}\right)\right]$ for all $\mathsf{б}\in \left[\theta , \lambda \right]$ for all $\mathcal{ı}\in \left[0, 1\right]$. Then,

$$\left[{\int }_{\theta }^{\lambda }\frac{{\mathcal{P}}_{*}\left(\mathsf{б}\right)\times {\mathcal{J}}_{*}\left(\mathsf{б}\right)}{{\mathcal{P}}_{*}\left(\mathsf{б}\right)+{\mathcal{J}}_{*}\left(\mathsf{б}\right)}d\mathsf{б}\times {\int }_{\theta }^{\lambda }\left({\mathcal{P}}_{*}\left(\mathsf{б}\right)+{\mathcal{J}}_{*}\left(\mathsf{б}\right)\right)d\mathsf{б}, {\int }_{\theta }^{\lambda }\frac{{\mathcal{P}}^{*}\left(\mathsf{б}\right)\times {\mathcal{J}}^{*}\left(\mathsf{б}\right)}{{\mathcal{P}}^{*}\left(\mathsf{б}\right)+{\mathcal{J}}^{*}\left(\mathsf{б}\right)}d\mathsf{б}\times {\int }_{\theta }^{\lambda }\left({\mathcal{P}}^{*}\left(\mathsf{б}\right)+{\mathcal{J}}^{*}\left(\mathsf{б}\right)\right)d\mathsf{б}\right]$$

$${\le }_I\left[{\int }_{\theta }^{\lambda }\mathcal{P}\left(\mathsf{б},\mathcal{ı}\right)d\mathsf{б}{\int }_{\theta }^{\lambda }\mathcal{J}\left(\mathsf{б},\mathcal{ı}\right)d\mathsf{б}, {\int }_{\theta }^{\lambda }\mathcal{P}\left(\mathsf{б},\mathcal{ı}\right)d\mathsf{б}{\int }_{\theta }^{\lambda }\mathcal{J}\left(\mathsf{б},\mathcal{ı}\right)d\mathsf{б}.\right]$$

5. Fuzzy Version of Ostrowski-Type Inequality via $\mathbf{ħ}$-Godunova–Levin $\mathsf{F}$·$\mathsf{N}$·$\mathsf{M}$s

Here, an Ostrowski-type inequality was formulated in conjunction with several illustrations for Godunova–Levin functions within a broader category.

The subsequent lemma aids in achieving our goal.

Lemma 1.

Let ${\mathcal{P}}_{*}\left(.,\mathcal{i}\right),{\mathcal{P}}^{*}\left(.,\mathcal{i}\right):\left[\theta , \lambda \right]\to {\mathfrak{N}}^{+}$ be a differentiable function on $I_0$ with $\left(\theta ,\lambda \right)$, where $\mathcal{i}\in [0, 1]$. If $\mathcal{P}$ is integrable over $\left[\theta , \lambda \right]$, then

$$\begin{gathered} \left[{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)-\frac{1}{\lambda -\theta }{\int }_{\theta }^{\lambda }{\mathcal{P}}_{*}\left(\phi ,\mathcal{ı}\right)d\phi ,{\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)-\frac{1}{\lambda -\theta }{\int }_{\theta }^{\lambda }{\mathcal{P}}^{*}\left(\phi ,\mathcal{ı}\right)d\phi \right] \\ =\frac{{\left(\mathsf{б}-\theta \right)}^2}{\lambda -\theta }{\int }_0^1\kappa \left[{{\mathcal{P}}_{*}}^{\prime }\left(\kappa \mathsf{б}+\left(1-\kappa \right)\theta ,\mathcal{ı}\right),{{\mathcal{P}}^{*}}^{\prime }\left(\kappa \mathsf{б}+\left(1-\kappa \right)\theta ,\mathcal{ı}\right) \right]d\kappa \\ +\frac{{\left(\lambda -\mathsf{б}\right)}^2}{\lambda -\theta }{\int }_0^1\kappa \left[{{\mathcal{P}}_{*}}^{\prime }\left(\kappa \lambda +\left(1-\kappa \right)\mathsf{б},\mathcal{ı}\right),{{\mathcal{P}}^{*}}^{\prime }\left(\kappa \lambda +\left(1-\kappa \right)\mathsf{б},\mathcal{ı}\right) \right]d\kappa . \end{gathered}$$

Proof. 

Integration by parts finalizes the proof. □

Now, employing Lemma 1, we derive the principal outcomes.

Theorem 5.

Let $\tilde{\mathcal{P}}:\left[\theta , \lambda \right]\to {\mathbb{F}}_0^{+}$ be a differentiable function on $I_0$ with $\mathsf{б}\in \left(\theta ,\lambda \right)$, where $\mathcal{i}\in [0, 1]$, and let $\tilde{\mathcal{P}}$ be a integrable over $\left[\theta , \lambda \right]$. If $M>0$ and $\left|{\tilde{\mathcal{P}}}^{\prime }\right|$ is an $ħ$-Godunova–Levin $F$·$N$·$M$, with $\left|{\tilde{\mathcal{P}}}^{\prime }\right|{\ge }_{\mathbb{F}}\tilde{0}$, then for

$$\begin{gathered} \left[\left|{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)-\frac{1}{\lambda -\theta }{\int }_{\theta }^{\lambda }{\mathcal{P}}_{*}\left(\phi ,\mathcal{ı}\right)d\phi \right|,\left|{\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)-\frac{1}{\lambda -\theta }{\int }_{\theta }^{\lambda }{\mathcal{P}}^{*}\left(\phi ,\mathcal{ı}\right)d\phi \right|\right] \\ {\le }_I\frac{M\left[{\left(\mathsf{б}-\theta \right)}^2+{\left(\lambda -\mathsf{б}\right)}^2\right]}{\lambda -\theta }{\int }_0^1\left[\frac{1}{ƛ\left(\kappa \right)ħ\left(\kappa \right)}+\frac{1}{ƛ\left(\kappa \right)ħ\left(1-\kappa \right)}\right]d\kappa , \end{gathered}$$

for all $\mathsf{б}\in \left[\theta , \lambda \right].$

Proof. 

In accordance with Lemma 1 and when $\left|{\mathcal{P}}^{\prime }\right|$ is an $ħ$-Godunova–Levin $F$·$N$·$M$, for $\mathcal{ı}\in \left[0, 1\right]$, we have

$$\begin{gathered} {\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)-\frac{1}{\lambda -\theta }{\int }_{\theta }^{\lambda }{\mathcal{P}}_{*}\left(\phi ,\mathcal{ı}\right)d\phi \\ =\frac{{\left(\mathsf{б}-\theta \right)}^2}{\lambda -\theta }{\int }_0^1\kappa {{\mathcal{P}}_{*}}^{\prime }\left(\kappa \mathsf{б}+\left(1-\kappa \right)\theta ,\mathcal{ı}\right)d\kappa +\frac{{\left(\lambda -\mathsf{б}\right)}^2}{\lambda -\theta }{\int }_0^1\kappa {{\mathcal{P}}_{*}}^{\prime }\left(\kappa \lambda +\left(1-\kappa \right)\mathsf{б},\mathcal{ı}\right)d\kappa \end{gathered}$$

and

$$\begin{gathered} {\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)-\frac{1}{\lambda -\theta }{\int }_{\theta }^{\lambda }{\mathcal{P}}^{*}\left(\phi ,\mathcal{ı}\right)d\phi \\ =\frac{{\left(\mathsf{б}-\theta \right)}^2}{\lambda -\theta }{\int }_0^1\kappa {{\mathcal{P}}^{*}}^{\prime }\left(\kappa \mathsf{б}+\left(1-\kappa \right)\theta ,\mathcal{ı}\right)d\kappa +\frac{{\left(\lambda -\mathsf{б}\right)}^2}{\lambda -\theta }{\int }_0^1\kappa {{\mathcal{P}}^{*}}^{\prime }\left(\kappa \lambda +\left(1-\kappa \right)\mathsf{б},\mathcal{ı}\right)d\kappa \end{gathered}$$

From the above equations, we have

$$\begin{gathered} \left|{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)-\frac{1}{\lambda -\theta }{\int }_{\theta }^{\lambda }{\mathcal{P}}_{*}\left(\phi ,\mathcal{ı}\right)d\phi \right| \\ \le \frac{{\left(\mathsf{б}-\theta \right)}^2}{\lambda -\theta }{\int }_0^1\kappa \left[\frac{\left|{{\mathcal{P}}_{*}}^{\prime }\left(\mathsf{б},\mathcal{ı}\right)\right|}{ħ\left(\kappa \right)}+\frac{\left|{{\mathcal{P}}_{*}}^{\prime }\left(\theta ,\mathcal{ı}\right)\right|}{ħ\left(1-\kappa \right)}\right]d\kappa +\frac{{\left(\lambda -\mathsf{б}\right)}^2}{\lambda -\theta }{\int }_0^1\kappa \left[\frac{\left|{{\mathcal{P}}_{*}}^{\prime }\left(\lambda ,\mathcal{ı}\right)\right|}{ħ\left(\kappa \right)}+\frac{\left|{{\mathcal{P}}_{*}}^{\prime }\left(\mathsf{б},\mathcal{ı}\right)\right|}{ħ\left(1-\kappa \right)}\right]d\kappa . \end{gathered}$$

and

$$\begin{gathered} \left|{\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)-\frac{1}{\lambda -\theta }{\int }_{\theta }^{\lambda }{\mathcal{P}}^{*}\left(\phi ,\mathcal{ı}\right)d\phi \right| \\ \le \frac{{\left(\mathsf{б}-\theta \right)}^2}{\lambda -\theta }{\int }_0^1\kappa \left[\frac{\left|{{\mathcal{P}}^{*}}^{\prime }\left(\mathsf{б},\mathcal{ı}\right)\right|}{ħ\left(\kappa \right)}+\frac{\left|{{\mathcal{P}}^{*}}^{\prime }\left(\theta ,\mathcal{ı}\right)\right|}{ħ\left(1-\kappa \right)}\right]d\kappa +\frac{{\left(\lambda -\mathsf{б}\right)}^2}{\lambda -\theta }{\int }_0^1\kappa \left[\frac{\left|{{\mathcal{P}}^{*}}^{\prime }\left(\lambda ,\mathcal{ı}\right)\right|}{ħ\left(\kappa \right)}+\frac{\left|{{\mathcal{P}}^{*}}^{\prime }\left(\mathsf{б},\mathcal{ı}\right)\right|}{ħ\left(1-\kappa \right)}\right]d\kappa . \end{gathered}$$

As a result, we obtain

$$\begin{gathered} \left|{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)-\frac{1}{\lambda -\theta }{\int }_{\theta }^{\lambda }{\mathcal{P}}_{*}\left(\phi ,\mathcal{ı}\right)d\phi \right| \\ \le \frac{{\left(\mathsf{б}-\theta \right)}^2}{\lambda -\theta }{\int }_0^1\left[\frac{\left|{{\mathcal{P}}_{*}}^{\prime }\left(\mathsf{б},\mathcal{ı}\right)\right|}{ƛ\left(\kappa \right)ħ\left(\kappa \right)}+\frac{\left|{{\mathcal{P}}_{*}}^{\prime }\left(\theta ,\mathcal{ı}\right)\right|}{ƛ\left(\kappa \right)ħ\left(1-\kappa \right)}\right]d\kappa +\frac{{\left(\lambda -\mathsf{б}\right)}^2}{\lambda -\theta }{\int }_0^1\left[\frac{\left|{{\mathcal{P}}_{*}}^{\prime }\left(\lambda ,\mathcal{ı}\right)\right|}{ƛ\left(\kappa \right)ħ\left(\kappa \right)}+\frac{\left|{{\mathcal{P}}_{*}}^{\prime }\left(\mathsf{б},\mathcal{ı}\right)\right|}{ƛ\left(\kappa \right)ħ\left(1-\kappa \right)}\right]d\kappa \\ \le \frac{M{\left(\mathsf{б}-\theta \right)}^2}{\lambda -\theta }{\int }_0^1\left[\frac{1}{ƛ\left(\kappa \right)ħ\left(\kappa \right)}+\frac{1}{ƛ\left(\kappa \right)ħ\left(1-\kappa \right)}\right]d\kappa +\frac{M{\left(\lambda -\mathsf{б}\right)}^2}{\lambda -\theta }{\int }_0^1\left[\frac{1}{ƛ\left(\kappa \right)ħ\left(\kappa \right)}+\frac{1}{ƛ\left(\kappa \right)ħ\left(1-\kappa \right)}\right]d\kappa , \end{gathered}$$

and

$$\begin{gathered} \left|{\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)-\frac{1}{\lambda -\theta }{\int }_{\theta }^{\lambda }{\mathcal{P}}^{*}\left(\phi ,\mathcal{ı}\right)d\phi \right| \\ \le \frac{{\left(\mathsf{б}-\theta \right)}^2}{\lambda -\theta }{\int }_0^1\left[\frac{\left|{{\mathcal{P}}^{*}}^{\prime }\left(\mathsf{б},\mathcal{ı}\right)\right|}{ƛ\left(\kappa \right)ħ\left(\kappa \right)}+\frac{\left|{{\mathcal{P}}^{*}}^{\prime }\left(\theta ,\mathcal{ı}\right)\right|}{ƛ\left(\kappa \right)ħ\left(1-\kappa \right)}\right]d\kappa +\frac{{\left(\lambda -\mathsf{б}\right)}^2}{\lambda -\theta }{\int }_0^1\left[\frac{\left|{{\mathcal{P}}^{*}}^{\prime }\left(\lambda ,\mathcal{ı}\right)\right|}{ƛ\left(\kappa \right)ħ\left(\kappa \right)}+\frac{\left|{{\mathcal{P}}^{*}}^{\prime }\left(\mathsf{б},\mathcal{ı}\right)\right|}{ƛ\left(\kappa \right)ħ\left(1-\kappa \right)}\right]d\kappa \\ \le \frac{M{\left(\mathsf{б}-\theta \right)}^2}{\lambda -\theta }{\int }_0^1\left[\frac{1}{ƛ\left(\kappa \right)ħ\left(\kappa \right)}+\frac{1}{ƛ\left(\kappa \right)ħ\left(1-\kappa \right)}\right]d\kappa +\frac{M{\left(\lambda -\mathsf{б}\right)}^2}{\lambda -\theta }{\int }_0^1\left[\frac{1}{ƛ\left(\kappa \right)ħ\left(\kappa \right)}+\frac{1}{ƛ\left(\kappa \right)ħ\left(1-\kappa \right)}\right]d\kappa . \end{gathered}$$

That is,

$$\begin{gathered} \left[\left|{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)-\frac{1}{\lambda -\theta }{\int }_{\theta }^{\lambda }{\mathcal{P}}_{*}\left(\phi ,\mathcal{ı}\right)d\phi \right|,\left|{\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)-\frac{1}{\lambda -\theta }{\int }_{\theta }^{\lambda }{\mathcal{P}}^{*}\left(\phi ,\mathcal{ı}\right)d\phi \right|\right] \\ {\le }_I\frac{M\left[{\left(\mathsf{б}-\theta \right)}^2+{\left(\lambda -\mathsf{б}\right)}^2\right]}{\lambda -\theta }{\int }_0^1\left[\frac{1}{ƛ\left(\kappa \right)ħ\left(\kappa \right)}+\frac{1}{ƛ\left(\kappa \right)ħ\left(1-\kappa \right)}\right]d\kappa \square \end{gathered}$$

Example 2.

We consider $ħ\left(\kappa \right)=\frac{1}{\kappa },$ for $\kappa \in \left[0, 1\right]$, and the $F$·$N$·$M$ $\mathcal{P}:\left[0, 2\right]\to {\mathbb{E}}_C$ defined by

$$\mathcal{P}\left(\mathsf{б}\right)\left(\theta \right)=\left\{\begin{array}{c}\frac{\theta }{{\mathsf{б}}^2}, \theta \in \left[0, {\mathsf{б}}^2\right]\\ \frac{ 2{\mathsf{б}}^2-\theta }{{\mathsf{б}}^2}, \theta \in \left.\left({\mathsf{б}}^2,2{\mathsf{б}}^2\right.\right] \\ 0, \mathrm{otherwise},\end{array}\right.$$

Then, for each $\mathcal{ı}\in \left[0, 1\right],$ we have ${\mathcal{P}}_{\mathcal{ı}}\left(\mathsf{б}\right)=\left[\mathcal{ı}{\mathsf{б}}^2,\left(2-\mathcal{ı}\right){\mathsf{б}}^2\right]$. Since left and right end-point mappings, ${\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)=\mathcal{ı}{\mathsf{б}}^2,$ ${\mathcal{P}}^{*}\left(\mathsf{б}, \mathcal{ı}\right)=\left(2-\mathcal{ı}\right){\mathsf{б}}^2$, are $ħ$-Godunova–Levin convex mappings for each $\mathcal{ı}\in [0, 1]$, $\tilde{\mathcal{P}}\left(\mathsf{б}\right)$ is an $ħ$-Godunova–Levin convex $F$·$N$·$M$. We clearly see that $\mathcal{P}\in L\left(\left[\theta , \lambda \right],{\mathbb{E}}_C\right)$.

Applying the Ostrowski-type inequality, we have the following:

Taking $\mathsf{б}=1$,

$$\left|{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)-\frac{1}{\lambda -\theta }{\int }_{\theta }^{\lambda }{\mathcal{P}}_{*}\left(\phi ,\mathcal{ı}\right)d\phi \right|=\frac{2\mathcal{ı}}{3}.$$

Since $\left|{{\mathcal{P}}_{\mathrm{*}}}^{\prime }\left(\mathrm{б},\mathcal{ı}\right)\right|\le M=2\mathcal{ı}$, we have

$$\frac{M\left[{\left(\mathsf{б}-\theta \right)}^2+{\left(\lambda -\mathsf{б}\right)}^2\right]}{\lambda -\theta }{\int }_0^1\left[\frac{1}{ƛ\left(\kappa \right)ħ\left(\kappa \right)}+\frac{1}{ƛ\left(\kappa \right)ħ\left(1-\kappa \right)}\right]=\mathcal{ı}$$

On the other hand, for ${\mathcal{P}}^{*}$, we have

$$\left|{\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)-\frac{1}{\lambda -\theta }{\int }_{\theta }^{\lambda }{\mathcal{P}}^{*}\left(\phi ,\mathcal{ı}\right)d\phi \right|=\frac{2\left(2-\mathcal{ı}\right)}{3}.$$

Since $\left|{{\mathcal{P}}_{\mathrm{*}}}^{\prime }\left(\mathrm{б},\mathcal{ı}\right)\right|\le M=2\left(2-\mathcal{ı}\right)$, we have

$$\frac{M\left[{\left(\mathsf{б}-\theta \right)}^2+{\left(\lambda -\mathsf{б}\right)}^2\right]}{\lambda -\theta }{\int }_0^1\left[\frac{1}{ƛ\left(\kappa \right)ħ\left(\kappa \right)}+\frac{1}{ƛ\left(\kappa \right)ħ\left(1-\kappa \right)}\right]=\left(2-\mathcal{ı}\right).$$

That is,

$$\left[\frac{2\mathcal{ı}}{3},\frac{2\left(2-\mathcal{ı}\right)}{3}\right]{\le }_I\left[\mathcal{ı},\left(2-\mathcal{ı}\right).\right]$$

6. Fuzzy-Valued Hermite–Hadamard Inequalities

The fuzzy-valued Hermite–Hadamard inequalities for $ħ$-Godunova–Levin convex $F$·$N$·$M$s are established in this section. Additionally, several instances are provided to support the theory produced in this study’s application.

Theorem 6.

Let $\tilde{\mathcal{P}}:\left[\theta , \lambda \right]\to {\mathbb{E}}_C$ be an $ħ$-Godunova–Levin convex $F$·$N$·$M$ with non-negative real-valued function $ħ:[0, 1]\to {\mathfrak{N}}^{+}$ and $ħ\left(\frac{1}{2}\right)\ne 0$, and $\mathit{\text{Ι-V-M}}$s are classified according to their $\mathcal{ı}$-levels such that ${\mathcal{P}}_{\mathcal{ı}}:\left[\theta , \lambda \right]\subset \mathfrak{N}\to {\mathcal{X}}_I^{+}$ are given by ${\mathcal{P}}_{\mathcal{ı}}\left(\mathsf{б}\right)=\left[{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right), {\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\right]$ for all $\mathsf{б}\in \left[\theta , \lambda \right]$ and for all $\mathcal{ı}\in \left[0, 1\right]$. If $\tilde{\mathcal{P}}\in {\mathcal{F}\mathcal{A}}_{\left(\left[\theta , \lambda \right], \mathcal{ı}\right)}$, then

$$\frac{ħ\left(\frac{1}{2}\right)}{2}\odot \tilde{\mathcal{P}}\left(\frac{\theta +\lambda }{2}\right){\le }_{\mathbb{F}}\frac{1}{\lambda -\theta } \odot \left(FA\right){\int }_{\theta }^{\lambda }\tilde{\mathcal{P}}\left(\mathsf{б}\right)d\mathsf{б}{\le }_{\mathbb{F}}\left[\tilde{\mathcal{P}}\left(\theta \right)\oplus \tilde{\mathcal{P}}\left(\lambda \right)\right]\odot {\int }_0^1\frac{1}{ħ\left(\kappa \right)}d\kappa .$$

(24)

If $\tilde{\mathcal{P}}$ is an $ħ$-Godunova–Levin concave $F$·$N$·$M$, then (24) is reversed.

$$\frac{ħ\left(\frac{1}{2}\right)}{2}\odot \tilde{\mathcal{P}}\left(\frac{\theta +\lambda }{2}\right){\ge }_{\mathbb{F}}\frac{1}{\lambda -\theta }\odot \left(FA\right){\int }_{\theta }^{\lambda }\tilde{\mathcal{P}}\left(\mathsf{б}\right)d\mathsf{б}{\ge }_{\mathbb{F}}\left[\tilde{\mathcal{P}}\left(\theta \right)\oplus \tilde{\mathcal{P}}\left(\lambda \right)\right]\odot {\int }_0^1\frac{1}{ħ\left(\kappa \right)}d\kappa .$$

(25)

Proof. 

Let $\tilde{\mathcal{P}}:\left[\theta , \lambda \right]\to {\mathbb{E}}_C$ be a $ħ$-Godunova–Levin convex $F$·$N$·$M$. Then, for $a,b\in \left[\theta , \lambda \right]$, we have

$$\tilde{\mathcal{P}}\left(\kappa a+\left(1-\kappa \right)b\right){\le }_{\mathbb{F}}\frac{\tilde{\mathcal{P}}\left(a\right)}{ħ\left(\kappa \right)}\oplus \frac{\tilde{\mathcal{P}}\left(b\right)}{ħ\left(1-\kappa \right)},$$

If $\kappa =\frac{1}{2}$, then we have

$$\tilde{\mathcal{P}}\left(\frac{a+b}{2}\right){\le }_{\mathbb{F}}\frac{\tilde{\mathcal{P}}\left(a\right)}{ħ\left(\frac{1}{2}\right)}\oplus \frac{\tilde{\mathcal{P}}\left(b\right)}{ħ\left(\frac{1}{2}\right)}.$$

Let $a=\kappa \theta +\left(1-\kappa \right)\lambda$ and $b=\left(1-\kappa \right)\theta +\kappa \lambda$. Then, from the above inequality, we have

$$ħ\left(\frac{1}{2}\right)\odot \tilde{\mathcal{P}}\left(\frac{\theta +\lambda }{2}\right){\le }_{\mathbb{F}}\tilde{\mathcal{P}}\left(\kappa \theta +\left(1-\kappa \right)\lambda \right)\oplus \tilde{\mathcal{P}}\left(\left(1-\kappa \right)\theta +\kappa \lambda \right).$$

Therefore, for every $\mathcal{ı}\in [0, 1]$, we have

$$\begin{array}{c}ħ\left(\frac{1}{2}\right){\mathcal{P}}_{*}\left(\frac{\theta +\lambda }{2}, \mathcal{ı}\right)\le {\mathcal{P}}_{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda , \mathcal{ı}\right)+{\mathcal{P}}_{*}\left(\left(1-\kappa \right)\theta +\kappa \lambda , \mathcal{ı}\right),\\ ħ\left(\frac{1}{2}\right){\mathcal{P}}^{*}\left(\frac{\theta +\lambda }{2}, \mathcal{ı}\right)\le {\mathcal{P}}^{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda , \mathcal{ı}\right)+{\mathcal{P}}^{*}\left(\left(1-\kappa \right)\theta +\kappa \lambda ,\mathcal{ı}\right).\end{array}$$

Then,

$$\begin{array}{c}ħ\left(\frac{1}{2}\right){\int }_0^1{\mathcal{P}}_{*}\left(\frac{\theta +\lambda }{2}, \mathcal{ı}\right)d\kappa \le {\int }_0^1{\mathcal{P}}_{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda ,\mathcal{ı}\right)d\kappa +{\int }_0^1{\mathcal{P}}_{*}\left(\left(1-\kappa \right)\theta +\kappa \lambda , \mathcal{ı}\right)d\kappa ,\\ ħ\left(\frac{1}{2}\right){\int }_0^1{\mathcal{P}}^{*}\left(\frac{\theta +\lambda }{2},\mathcal{ı}\right)d\kappa \le {\int }_0^1{\mathcal{P}}^{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda , \mathcal{ı}\right)d\kappa +{\int }_0^1{\mathcal{P}}^{*}\left(\left(1-\kappa \right)\theta +\kappa \lambda ,\mathcal{ı}\right)d\kappa .\end{array}$$

It follows that

$$\begin{array}{c}ħ\left(\frac{1}{2}\right){\mathcal{P}}_{*}\left(\frac{\theta +\lambda }{2}, \mathcal{ı}\right)\le \frac{2}{\lambda -\theta }{\int }_{\theta }^{\lambda }{\mathcal{P}}_{*}\left(\mathsf{б}, \mathcal{ı}\right)d\mathsf{б},\\ ħ\left(\frac{1}{2}\right){\mathcal{P}}^{*}\left(\frac{\theta +\lambda }{2}, \mathcal{ı}\right)\le \frac{2}{\lambda -\theta }{\int }_{\theta }^{\lambda }{\mathcal{P}}^{*}\left(\mathsf{б}, \mathcal{ı}\right)d\mathsf{б}.\end{array}$$

That is

$$ħ\left(\frac{1}{2}\right)\left[{\mathcal{P}}_{*}\left(\frac{\theta +\lambda }{2}, \mathcal{ı}\right), {\mathcal{P}}^{*}\left(\frac{\theta +\lambda }{2}, \mathcal{ı}\right)\right]{\le }_I\frac{2}{\lambda -\theta }\left[{\int }_{\theta }^{\lambda }{\mathcal{P}}_{*}\left(\mathsf{б}, \mathcal{ı}\right)d\mathsf{б}, {\int }_{\theta }^{\lambda }{\mathcal{P}}^{*}\left(\mathsf{б}, \mathcal{ı}\right)d\mathsf{б}\right].$$

Thus,

$$\frac{ħ\left(\frac{1}{2}\right)}{2}\odot \tilde{\mathcal{P}}\left(\frac{\theta +\lambda }{2}\right){\le }_{\mathbb{F}}\frac{1}{\lambda -\theta }\odot \left(FA\right){\int }_{\theta }^{\lambda }\tilde{\mathcal{P}}\left(\mathsf{б}\right)d\mathsf{б}.$$

(26)

In a similar way as above, we have

$$\frac{1}{\lambda -\theta }\odot \left(FA\right){\int }_{\theta }^{\lambda }\tilde{\mathcal{P}}\left(\mathsf{б}\right)d\mathsf{б}{\le }_{\mathbb{F}}\left[\tilde{\mathcal{P}}\left(\theta \right)\oplus \tilde{\mathcal{P}}\left(\lambda \right)\right]\odot {\int }_0^1\frac{1}{ħ\left(\kappa \right)}d\kappa .$$

(27)

Combining (26) and (27), we have

$$\frac{ħ\left(\frac{1}{2}\right)}{2}\odot \tilde{\mathcal{P}}\left(\frac{\theta +\lambda }{2}\right){\le }_{\mathbb{F}}\frac{1}{\lambda -\theta }\odot \left(FA\right){\int }_{\theta }^{\lambda }\tilde{\mathcal{P}}\left(\mathsf{б}\right)d\mathsf{б}{\le }_{\mathbb{F}}\left[\tilde{\mathcal{P}}\left(\theta \right)\oplus \tilde{\mathcal{P}}\left(\lambda \right)\right]\odot {\int }_0^1\frac{1}{ħ\left(\kappa \right)}d\kappa .$$

Hence, the required result is obtained. □

Note that, by using same steps, Formula (25) can be proved with the help of an $ħ$-Godunova–Levin concave $F$·$N$·$M$.

Remark 5.

If $ħ\left(\kappa \right)=\frac{1}{{\kappa }^s}$, then Theorem 6 simplifies to the outcome for the $s$-convex $F$·$N$·$M$, which is also a new one:

$$2^{s-1}\odot \tilde{\mathcal{P}}\left(\frac{\theta +\lambda }{2}\right){\supseteq }_{\mathbb{F}}\frac{1}{\lambda -\theta }\odot \left(FA\right){\int }_{\theta }^{\lambda }\tilde{\mathcal{P}}\left(\mathsf{б}\right)d\mathsf{б}{\supseteq }_{\mathbb{F}}\frac{1}{s+1}\odot \left[\tilde{\mathcal{P}}\left(\theta \right)\oplus \tilde{\mathcal{P}}\left(\lambda \right)\right].$$

(28)

If $ħ\left(\kappa \right)=\frac{1}{\kappa }$, then Theorem 6 simplifies to the outcome for a convex $F$·$N$·$M,$ which is also a new one:

$$\tilde{\mathcal{P}}\left(\frac{\theta +\lambda }{2}\right){\supseteq }_{\mathbb{F}}\frac{1}{\lambda -\theta }\odot \left(FA\right){\int }_{\theta }^{\lambda }\tilde{\mathcal{P}}\left(\mathsf{б}\right)d\mathsf{б}{\supseteq }_{\mathbb{F}}\frac{\tilde{\mathcal{P}}\left(\theta \right)\oplus \tilde{\mathcal{P}}\left(\lambda \right)}{2}.$$

(29)

If $ħ\left(\kappa \right)\equiv 1$, then Theorem 6 simplifies to the outcome for a $P$-convex $F$·$N$·$M$, which is also a new one:

$$\frac{1}{2}\tilde{\mathcal{P}}\left(\frac{\theta +\lambda }{2}\right){\supseteq }_{\mathbb{F}}\frac{1}{\lambda -\theta }\odot \left(FA\right){\int }_{\theta }^{\lambda }\tilde{\mathcal{P}}\left(\mathsf{б}\right)d\mathsf{б}{\supseteq }_{\mathbb{F}}\tilde{\mathcal{P}}\left(\theta \right)\oplus \tilde{\mathcal{P}}\left(\lambda \right).$$

(30)

If ${\mathcal{P}}_{*}\left(\mathsf{б}, \mathcal{ı}\right)={\mathcal{P}}^{*}\left(\mathsf{б}, \mathcal{ı}\right)$ with $\mathcal{ı}=1$, then Theorem 6 simplifies to the outcome for the classical $ħ$-convex function; see [49]:

$$\frac{1}{2ħ\left(\frac{1}{2}\right)}\mathcal{P}\left(\frac{\theta +\lambda }{2}\right)\supseteq \frac{1}{\lambda -\theta }\left(IA\right){\int }_{\theta }^{\lambda }\mathcal{P}\left(\mathsf{б}\right)d\mathsf{б}\supseteq \left[\mathcal{P}\left(\theta \right)+\mathcal{P}\left(\lambda \right)\right]{\int }_0^1ħ\left(\kappa \right)d\kappa .$$

(31)

If ${\mathcal{P}}_{*}\left(\mathsf{б}, \mathcal{ı}\right)\ne {\mathcal{P}}^{*}\left(\mathsf{б}, \mathcal{ı}\right)$ with $\mathcal{ı}=1$ and $ħ\left(\kappa \right)=\frac{1}{\kappa }$, then Theorem 6 simplifies to the outcome for the classical convex function; see [49]:

$$\mathcal{P}\left(\frac{\theta +\lambda }{2}\right)\supseteq \frac{1}{\lambda -\theta }\left(IA\right){\int }_{\theta }^{\lambda }\mathcal{P}\left(\mathsf{б}\right)d\mathsf{б}\supseteq \frac{\mathcal{P}\left(\theta \right)+\mathcal{P}\left(\lambda \right)}{2}.$$

(32)

If  ${\mathcal{P}}_{*}\left(\mathsf{б}, \mathcal{ı}\right)={\mathcal{P}}^{*}\left(\mathsf{б}, \mathcal{ı}\right)$ with $\mathcal{ı}=1$, then Theorem 6 simplifies to the outcome for the classical $ħ$-convex function; see [50]:

$$\frac{ħ\left(\frac{1}{2}\right)}{2}\mathcal{P}\left(\frac{\theta +\lambda }{2}\right)\le \frac{1}{\lambda -\theta }{\int }_{\theta }^{\lambda }\mathcal{P}\left(\mathsf{б}\right)d\mathsf{б}\le \left[\mathcal{P}\left(\theta \right)+\mathcal{P}\left(\lambda \right)\right]{\int }_0^1\frac{1}{ħ\left(\kappa \right)}d\kappa .$$

(33)

If ${\mathcal{P}}_{*}\left(\mathsf{б}, \mathcal{ı}\right)={\mathcal{P}}^{*}\left(\mathsf{б}, \mathcal{ı}\right)$ with $\mathcal{ı}=1$ and $ħ\left(\kappa \right)=\frac{1}{{\kappa }^s}$, then Theorem 6 simplifies to the outcome for the classical $s$-convex function; see [50]:

$$2^{s-1}\mathcal{P}\left(\frac{\theta +\lambda }{2}\right)\le \frac{1}{\lambda -\theta }{\int }_{\theta }^{\lambda }\mathcal{P}\left(\mathsf{б}\right)d\mathsf{б}\le \frac{1}{s+1}\left[\mathcal{P}\left(\theta \right)+\mathcal{P}\left(\lambda \right)\right].$$

(34)

If ${\mathcal{P}}_{*}\left(\mathsf{б}, \mathcal{ı}\right)={\mathcal{P}}^{*}\left(\mathsf{б}, \mathcal{ı}\right)$ with $\mathcal{ı}=1$ and $ħ\left(\kappa \right)=\frac{1}{\kappa }$, then Theorem 6 simplifies to the outcome for the classical convex function:

$$\mathcal{P}\left(\frac{\theta +\lambda }{2}\right)\le \frac{1}{\lambda -\theta }{\int }_{\theta }^{\lambda }\mathcal{P}\left(\mathsf{б}\right)d\mathsf{б}\le \frac{\mathcal{P}\left(\theta \right)+\mathcal{P}\left(\lambda \right)}{2}.$$

(35)

If ${\mathcal{P}}_{*}\left(\mathsf{б}, \mathcal{ı}\right)={\mathcal{P}}^{*}\left(\mathsf{б}, \mathcal{ı}\right)$ with $\mathcal{ı}=1$ and $ħ\left(\kappa \right)\equiv 1$, then Theorem 6 simplifies to the outcome for the classical $P$-convex function:

$$\frac{1}{2}\mathcal{P}\left(\frac{\theta +\lambda }{2}\right)\le \frac{1}{\lambda -\theta }{\int }_{\theta }^{\lambda }\mathcal{P}\left(\mathsf{б}\right)d\mathsf{б}\le \mathcal{P}\left(\theta \right)+\mathcal{P}\left(\lambda \right).$$

(36)

Example 3.

We consider $ħ\left(\kappa \right)=\frac{1}{\kappa }$ for $\kappa \in \left[0, 1\right]$, and the $F$·$N$·$M$ $\tilde{\mathcal{P}}:\left[\theta , \lambda \right]=[2, 3]\to {\mathbb{E}}_C$ is defined by,

$$\tilde{\mathcal{P}}\left(\mathsf{б}\right)\left(\theta \right)=\left\{\begin{array}{c}\frac{\theta -2+{\mathsf{б}}^{\frac{1}{2}}}{1-{\mathsf{б}}^{\frac{1}{2}}} \theta \in \left[2-{\mathsf{б}}^{\frac{1}{2}}, 3\right]\\ \frac{ 2+{\mathsf{б}}^{\frac{1}{2}}-\theta }{{\mathsf{б}}^{\frac{1}{2}}-1} \theta \in (3, 2+{\mathsf{б}}^{\frac{1}{2}}]\\ 0 \mathrm{otherwise}.\end{array}\right.$$

Then, for each $\mathcal{ı}\in \left[0,1\right],$ we have ${\mathcal{P}}_{\mathcal{ı}}\left(\mathrm{б}\right)=\left[\left(1-\mathcal{ı}\right)\left(2-{\mathrm{б}}^{\frac{1}{2}}\right)+3\mathcal{ı},\left(1-\mathcal{ı}\right)\left(2+{\mathrm{б}}^{\frac{1}{2}}\right)+3\mathcal{ı}\right]$. Since left and right end-point mappings, ${\mathcal{P}}_{\mathrm{*}}\left(\mathrm{б},\mathcal{ı}\right)=\left(1-\mathcal{ı}\right)\left(2-{\mathrm{б}}^{\frac{1}{2}}\right)+3\mathcal{ı},$ ${\mathcal{P}}^{\mathrm{*}}\left(\mathrm{б},\mathcal{ı}\right)=\left(1-\mathcal{ı}\right)\left(2+{\mathrm{б}}^{\frac{1}{2}}\right)+3\mathcal{ı}$, are $ħ$-Godunova–Levin convex mappings for each $\mathcal{ı}\in [0,1]$, $\tilde{\mathcal{P}}\left(\mathrm{б}\right)$ is an $ħ$-Godunova–Levin convex $F$·$N$·$M$. We clearly see that $\mathcal{P}\in L\left(\left[\theta ,\lambda \right],{\mathbb{E}}_C\right)$. Now, we compute the following:

$$\begin{gathered} \frac{ħ\left(\frac{1}{2}\right)}{2}{\mathcal{P}}_{*}\left(\frac{\theta +\lambda }{2}, \mathcal{ı}\right)\le \frac{1}{\lambda -\theta }{\int }_{\theta }^{\lambda }{\mathcal{P}}_{*}\left(\mathsf{б}, \mathcal{ı}\right)d\mathsf{б}\le \left[{\mathcal{P}}_{*}\left(\theta , \mathcal{ı}\right)+{\mathcal{P}}_{*}\left(\lambda , \mathcal{ı}\right)\right]{\int }_0^1\frac{1}{ħ\left(\kappa \right)}d\kappa . \\ \frac{ħ\left(\frac{1}{2}\right)}{2}{\mathcal{P}}_{*}\left(\frac{\theta +\lambda }{2}, \mathcal{ı}\right)={\mathcal{P}}_{*}\left(\frac{5}{2}, \mathcal{ı}\right)=\frac{4-\sqrt{10}}{2}\left(1-\mathcal{ı}\right)+3\mathcal{ı}, \\ \frac{1}{\lambda -\theta }{\int }_{\theta }^{\lambda }{\mathcal{P}}_{*}\left(\mathsf{б}, \mathcal{ı}\right)d\mathsf{б}={\int }_2^3\left(\left(1-\mathcal{ı}\right)\left(2-{\mathsf{б}}^{\frac{1}{2}}\right)+3\mathcal{ı}\right)d\mathsf{б}=\frac{\left(6+4\sqrt{2}-6\sqrt{3}\right)}{3}\left(1-\mathcal{ı}\right)+3\mathcal{ı}, \\ \left[{\mathcal{P}}_{*}\left(\theta , \mathcal{ı}\right)+{\mathcal{P}}_{*}\left(\lambda , \mathcal{ı}\right)\right]{\int }_0^1\frac{1}{ħ\left(\kappa \right)}d\kappa =\frac{\left(4-\sqrt{2}-\sqrt{3}\right)}{2}\left(1-\mathcal{ı}\right)+3\mathcal{ı}, \end{gathered}$$

for all $\mathcal{ı}\in \left[0, 1\right].$ This means

$$\frac{4-\sqrt{10}}{2}\left(1-\mathcal{ı}\right)+3\mathcal{ı}\le \frac{\left(6+4\sqrt{2}-6\sqrt{3}\right)}{3}\left(1-\mathcal{ı}\right)+3\mathcal{ı}\le \frac{\left(4-\sqrt{2}-\sqrt{3}\right)}{2}\left(1-\mathcal{ı}\right)+3\mathcal{ı}.$$

Similarly, it can be easily shown that

$$\frac{ħ\left(\frac{1}{2}\right)}{2}{\mathcal{P}}^{*}\left(\frac{\theta +\lambda }{2}, \mathcal{ı}\right)\ge \frac{1}{\lambda -\theta } {\int }_{\theta }^{\lambda }{\mathcal{P}}^{*}\left(\mathsf{б}, \mathcal{ı}\right)d\mathsf{б}\ge \left[{\mathcal{P}}^{*}\left(\theta , \mathcal{ı}\right)+{\mathcal{P}}^{*}\left(\lambda , \mathcal{ı}\right)\right]{\int }_0^1\frac{1}{ħ\left(\kappa \right)}d\kappa .$$

for all $\mathcal{ı}\in \left[0, 1\right],$ such that

$$\begin{gathered} \frac{ħ\left(\frac{1}{2}\right)}{2}{\mathcal{P}}^{*}\left(\frac{\theta +\lambda }{2}, \mathcal{ı}\right)={\mathcal{P}}^{*}\left(\frac{5}{2}, \mathcal{ı}\right)=\frac{4+\sqrt{10}}{2}\left(1-\mathcal{ı}\right)+3\mathcal{ı}, \\ \frac{1}{\lambda -\theta }{\int }_{\theta }^{\lambda }{\mathcal{P}}^{*}\left(\mathsf{б}, \mathcal{ı}\right)d\mathsf{б}={\int }_2^3\left(\left(1-\mathcal{ı}\right)\left(2+{\mathsf{б}}^{\frac{1}{2}}\right)+3\mathcal{ı}\right)d\mathsf{б}=\frac{\left(6-4\sqrt{2}+6\sqrt{3}\right)}{3}\left(1-\mathcal{ı}\right)+3\mathcal{ı}, \\ \left[{\mathcal{P}}^{*}\left(\theta , \mathcal{ı}\right)+{\mathcal{P}}^{*}\left(\lambda , \mathcal{ı}\right)\right]{\int }_0^1\frac{1}{ħ\left(\kappa \right)}d\kappa =\frac{\left(4+\sqrt{2}+\sqrt{3}\right)}{2}\left(1-\mathcal{ı}\right)+3\mathcal{ı}. \end{gathered}$$

From which, we have

$$\frac{4+\sqrt{10}}{2}\left(1-\mathcal{ı}\right)+3\mathcal{ı}\le \frac{\left(6-4\sqrt{2}+6\sqrt{3}\right)}{3}\left(1-\mathcal{ı}\right)+3\mathcal{ı}\le \frac{\left(4+\sqrt{2}+\sqrt{3}\right)}{2}\left(1-\mathcal{ı}\right)+3\mathcal{ı},$$

that is

$$\begin{gathered} \left[\frac{4-\sqrt{10}}{2}\left(1-\mathcal{ı}\right)+3\mathcal{ı},\frac{4+\sqrt{10}}{2}\left(1-\mathcal{ı}\right)+3\mathcal{ı}\right]{\le }_I\left[\frac{\left(6+4\sqrt{2}-6\sqrt{3}\right)}{3}\left(1-\mathcal{ı}\right)+3\mathcal{ı},\frac{\left(6-4\sqrt{2}+6\sqrt{3}\right)}{3}\left(1-\mathcal{ı}\right)+3\mathcal{ı}\right] \\ {\le }_I\left[\frac{\left(4-\sqrt{2}-\sqrt{3}\right)}{2}\left(1-\mathcal{ı}\right),\frac{\left(4+\sqrt{2}+\sqrt{3}\right)}{2}\left(1-\mathcal{ı}\right)+3\mathcal{ı},\right] \end{gathered}$$

for all $\mathcal{ı}\in \left[0,1\right].$

Hence,

$$\frac{ħ\left(\frac{1}{2}\right)}{2}\odot \tilde{\mathcal{P}}\left(\frac{\theta +\lambda }{2}\right){\le }_{\mathbb{F}}\frac{1}{\lambda -\theta }\odot \left(FA\right){\int }_{\theta }^{\lambda }\tilde{\mathcal{P}}\left(\mathsf{б}\right)d\mathsf{б}{\le }_{\mathbb{F}}\left[\tilde{\mathcal{P}}\left(\theta \right)\oplus \tilde{\mathcal{P}}\left(\lambda \right)\right]\odot {\int }_0^1\frac{1}{ħ\left(\kappa \right)}d\kappa .$$

Theorem 7.

Let $\tilde{\mathcal{P}}:\left[\theta , \lambda \right]\to {\mathbb{E}}_C$ be an $ħ$-Godunova–Levin convex $F$·$N$·$M$ with non-negative real-valued function $ħ:[0, 1]\to {\mathfrak{N}}^{+}$ and $ħ\left(\frac{1}{2}\right)\ne 0,$ and $\mathit{\text{Ι-V-M}}$s are classified according to their $\mathcal{ı}$-levels such that ${\mathcal{P}}_{\mathcal{ı}}:\left[\theta , \lambda \right]\subset \mathfrak{N}\to {\mathcal{X}}_I^{+}$ are given by ${\mathcal{P}}_{\mathcal{ı}}\left(\mathsf{б}\right)=\left[{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right), {\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\right]$ for all $\mathsf{б}\in \left[\theta , \lambda \right]$ and for all $\mathcal{ı}\in \left[0, 1\right]$. If $\tilde{\mathcal{P}}\in {\mathcal{F}\mathcal{A}}_{\left(\left[\theta , \lambda \right], \mathcal{ı}\right)}$, then

$$\begin{gathered} \frac{{\left[ħ\left(\frac{1}{2}\right)\right]}^2}{4}\odot \tilde{\mathcal{P}}\left(\frac{\theta +\lambda }{2}\right){\le }_{\mathbb{F}} {⪧}_2{\le }_{\mathbb{F}}\frac{1}{\lambda -\theta }\odot \left(FA\right){\int }_{\theta }^{\lambda }\tilde{\mathcal{P}}\left(\mathsf{б}\right)d\mathsf{б} \\ {\le }_{\mathbb{F}}{⪧}_1{\le }_{\mathbb{F}}\left[\tilde{\mathcal{P}}\left(\theta \right)\oplus \tilde{\mathcal{P}}\left(\lambda \right)\right]\odot \left[\frac{1}{2}+\frac{1}{ħ\left(\frac{1}{2}\right)}\right]{\int }_0^1ħ\left(\kappa \right)d\kappa , \end{gathered}$$

(37)

If $\tilde{\mathcal{P}}$ is an $ħ$-Godunova–Levin concave $F$·$N$·$M$, then (37) is reversed.

$$\begin{gathered} \frac{{\left[ħ\left(\frac{1}{2}\right)\right]}^2}{4}\odot \tilde{\mathcal{P}}\left(\frac{\theta +\lambda }{2}\right){\ge }_{\mathbb{F}} {⪧}_2{\ge }_{\mathbb{F}}\frac{1}{\lambda -\theta }\odot \left(FA\right){\int }_{\theta }^{\lambda }\tilde{\mathcal{P}}\left(\mathsf{б}\right)d\mathsf{б} \\ {\ge }_{\mathbb{F}}{⪧}_1{\ge }_{\mathbb{F}}\left[\tilde{\mathcal{P}}\left(\theta \right)\oplus \tilde{\mathcal{P}}\left(\lambda \right)\right]\odot \left[\frac{1}{2}+\frac{1}{ħ\left(\frac{1}{2}\right)}\right]{\int }_0^1ħ\left(\kappa \right)d\kappa , \end{gathered}$$

where

$$\begin{gathered} {⪧}_1=\left[\frac{\tilde{\mathcal{P}}\left(\theta \right)\oplus \tilde{\mathcal{P}}\left(\lambda \right)}{2}\oplus \tilde{\mathcal{P}}\left(\frac{\theta +\lambda }{2}\right)\right]\odot {\int }_0^1\frac{1}{ħ\left(\kappa \right)}d\kappa , \\ {⪧}_2=\frac{\left[ħ\left(\frac{1}{2}\right)\right]}{4}\odot \left[\tilde{\mathcal{P}}\left(\frac{3\theta +\lambda }{4}\right)\oplus \tilde{\mathcal{P}}\left(\frac{\theta +3\lambda }{4}\right)\right], \end{gathered}$$

and ${⪧}_1=\left[{{⪧}_1}_{*}, {{⪧}_1}^{*}\right]$, ${⪧}_2=\left[{{⪧}_2}_{*}, {{⪧}_2}^{*}\right].$

Proof. 

Taking $\left[\theta , \frac{\theta +\lambda }{2}\right],$ we have

$$ħ\left(\frac{1}{2}\right)\odot \tilde{\mathcal{P}}\left(\frac{\kappa \theta +\left(1-\kappa \right)\frac{\theta +\lambda }{2}}{2}+\frac{(1-\kappa )\theta +\kappa \frac{\theta +\lambda }{2}}{2}\right){\le }_{\mathbb{F}}\tilde{\mathcal{P}}\left(\kappa \theta +\left(1-\kappa \right)\frac{\theta +\lambda }{2} \right)\oplus \tilde{\mathcal{P}}\left(\left(1-\kappa \right)\theta +\kappa \frac{\theta +\lambda }{2}\right).$$

Therefore, for every $\mathcal{ı}\in [0, 1]$, we have

$$\begin{array}{c}ħ\left(\frac{1}{2}\right){\mathcal{P}}_{*}\left(\frac{\kappa \theta +\left(1-\kappa \right)\frac{\theta +\lambda }{2}}{2}+\frac{\left(1-\kappa \right)\theta +\kappa \frac{\theta +\lambda }{2}}{2}, \mathcal{ı}\right)\le {\mathcal{P}}_{*}\left(\kappa \theta +\left(1-\kappa \right)\frac{\theta +\lambda }{2}, \mathcal{ı}\right)+{\mathcal{P}}_{*}\left(\left(1-\kappa \right)\theta +\kappa \frac{\theta +\lambda }{2}, \mathcal{ı}\right),\\ ħ\left(\frac{1}{2}\right){\mathcal{P}}^{*}\left(\frac{\kappa \theta +\left(1-\kappa \right)\frac{\theta +\lambda }{2}}{2}+\frac{\left(1-\kappa \right)\theta +\kappa \frac{\theta +\lambda }{2}}{2}, \mathcal{ı}\right)\le {\mathcal{P}}^{*}\left(\kappa \theta +\left(1-\kappa \right)\frac{\theta +\lambda }{2}, \mathcal{ı}\right)+{\mathcal{P}}^{*}\left(\left(1-\kappa \right)\theta +\kappa \frac{\theta +\lambda }{2},\mathcal{ı}\right).\end{array}$$

In consequence, we obtain

$$\begin{array}{c}\frac{ħ\left(\frac{1}{2}\right)}{4}{\mathcal{P}}_{*}\left(\frac{3\theta +\lambda }{4}, \mathcal{ı}\right)\le \frac{1}{\lambda -\theta }{\int }_{\theta }^{\frac{\theta +\lambda }{2}}{\mathcal{P}}_{*}\left(\mathsf{б}, \mathcal{ı}\right)d\mathsf{б},\\ \frac{ħ\left(\frac{1}{2}\right)}{4}{\mathcal{P}}^{*}\left(\frac{3\theta +\lambda }{4}, \mathcal{ı}\right)\le \frac{1}{\lambda -\theta }{\int }_{\theta }^{\frac{\theta +\lambda }{2}}{\mathcal{P}}^{*}\left(\mathsf{б}, \mathcal{ı}\right)d\mathsf{б}.\end{array}$$

That is

$$\frac{ħ\left(\frac{1}{2}\right)}{4}\left[{\mathcal{P}}_{*}\left(\frac{3\theta +\lambda }{4}, \mathcal{ı}\right), {\mathcal{P}}^{*}\left(\frac{3\theta +\lambda }{4}, \mathcal{ı}\right)\right]{\le }_I\frac{1}{\lambda -\theta }\left[{\int }_{\theta }^{\frac{\theta +\lambda }{2}}{\mathcal{P}}_{*}\left(\mathsf{б}, \mathcal{ı}\right)d\mathsf{б}, {\int }_{\theta }^{\frac{\theta +\lambda }{2}}{\mathcal{P}}^{*}\left(\mathsf{б}, \mathcal{ı}\right)d\mathsf{б}\right].$$

It follows that

$$\frac{ħ\left(\frac{1}{2}\right)}{4}\odot \tilde{\mathcal{P}}\left(\frac{3\theta +\lambda }{4}\right){\le }_{\mathbb{F}}\frac{1}{\lambda -\theta }\odot {\int }_{\theta }^{\frac{\theta +\lambda }{2}}\tilde{\mathcal{P}}\left(\mathsf{б}\right)d\mathsf{б}.$$

(38)

In a similar way as above, we have

$$\frac{ħ\left(\frac{1}{2}\right)}{4}\odot \tilde{\mathcal{P}}\left(\frac{\theta +3\lambda }{4}\right){\le }_{\mathbb{F}}\frac{1}{\lambda -\theta }\odot {\int }_{\frac{\theta +\lambda }{2}}^{\lambda }\tilde{\mathcal{P}}\left(\mathsf{б}\right)d\mathsf{б}.$$

(39)

Combining (38) and (39), we have

$$\frac{ħ\left(\frac{1}{2}\right)}{4}\odot \left[\tilde{\mathcal{P}}\left(\frac{3\theta +\lambda }{4}\right)\oplus \tilde{\mathcal{P}}\left(\frac{\theta +3\lambda }{4}\right)\right]{\le }_{\mathbb{F}}\frac{1}{\lambda -\theta }\odot {\int }_{\theta }^{\lambda }\tilde{\mathcal{P}}\left(\mathsf{б}\right)d\mathsf{б}.$$

Now,

$$\frac{ħ\left(\frac{1}{2}\right)}{4}\odot \tilde{\mathcal{P}}\left(\frac{\theta +\lambda }{2}\right)=\frac{ħ\left(\frac{1}{2}\right)}{4}\odot \tilde{\mathcal{P}}\left(\frac{1}{2}.\frac{3\theta +\lambda }{4}+\frac{1}{2}.\frac{\theta +3\lambda }{4}\right).$$

Therefore, for every $\mathcal{ı}\in [0, 1]$, we have

$$\begin{gathered} \begin{array}{c}\frac{{\left[ħ\left(\frac{1}{2}\right)\right]}^2}{4}{\mathcal{P}}_{*}\left(\frac{\theta +\lambda }{2}, \mathcal{ı}\right)=\frac{{\left[ħ\left(\frac{1}{2}\right)\right]}^2}{4}{\mathcal{P}}_{*}\left(\frac{1}{2}.\frac{3\theta +\lambda }{4}+\frac{1}{2}.\frac{\theta +3\lambda }{4}, \mathcal{ı}\right),\\ \frac{{\left[ħ\left(\frac{1}{2}\right)\right]}^2}{4}{\mathcal{P}}^{*}\left(\frac{\theta +\lambda }{2}, \mathcal{ı}\right)=\frac{{\left[ħ\left(\frac{1}{2}\right)\right]}^2}{4}{\mathcal{P}}^{*}\left(\frac{1}{2}.\frac{3\theta +\lambda }{4}+\frac{1}{2}.\frac{\theta +3\lambda }{4}, \mathcal{ı}\right),\end{array} \\ \begin{array}{c}\le \frac{{\left[ħ\left(\frac{1}{2}\right)\right]}^2}{4}\left[\frac{{\mathcal{P}}_{*}\left(\frac{3\theta +\lambda }{4}, \mathcal{ı}\right)}{ħ\left(\frac{1}{2}\right)}+\frac{{\mathcal{P}}_{*}\left(\frac{\theta +3\lambda }{4}, \mathcal{ı}\right)}{ħ\left(\frac{1}{2}\right)}\right],\\ \le \frac{{\left[ħ\left(\frac{1}{2}\right)\right]}^2}{4}\left[\frac{{\mathcal{P}}^{*}\left(\frac{3\theta +\lambda }{4}, \mathcal{ı}\right)}{ħ\left(\frac{1}{2}\right)}+\frac{{\mathcal{P}}^{*}\left(\frac{\theta +3\lambda }{4}, \mathcal{ı}\right)}{ħ\left(\frac{1}{2}\right)}\right],\end{array} \\ \begin{array}{c}={{⪧}_2}_{*},\\ ={{⪧}_2}^{*},\end{array} \\ \begin{array}{c}\le \frac{1}{\lambda -\theta }{\int }_{\theta }^{\lambda }{\mathcal{P}}_{*}\left(\mathsf{б}, \mathcal{ı}\right)d\mathsf{б}, \\ \le \frac{1}{\lambda -\theta }{\int }_{\theta }^{\lambda }{\mathcal{P}}^{*}\left(\mathsf{б}, \mathcal{ı}\right)d\mathsf{б},\end{array} \\ \begin{array}{c}\le \left[\frac{{\mathcal{P}}_{*}\left(\theta , \mathcal{ı}\right)+{\mathcal{P}}_{*}\left(\lambda , \mathcal{ı}\right)}{2}+{\mathcal{P}}_{*}\left(\frac{\theta +\lambda }{2}, \mathcal{ı}\right)\right]{\int }_0^1ħ\left(\kappa \right)d\kappa , \\ \le \left[\frac{{\mathcal{P}}^{*}\left(\theta , \mathcal{ı}\right)+{\mathcal{P}}^{*}\left(\lambda , \mathcal{ı}\right)}{2}+{\mathcal{P}}^{*}\left(\frac{\theta +\lambda }{2}, \mathcal{ı}\right)\right]{\int }_0^1ħ\left(\kappa \right)d\kappa ,\end{array} \\ \begin{array}{c}={{⪧}_1}_{*},\\ ={{⪧}_1}^{*},\end{array} \\ \begin{array}{c}\le \left[\frac{{\mathcal{P}}_{*}\left(\theta , \mathcal{ı}\right)+{\mathcal{P}}_{*}\left(\lambda , \mathcal{ı}\right)}{2}+\frac{1}{ħ\left(\frac{1}{2}\right)}\left({\mathcal{P}}_{*}\left(\theta , \mathcal{ı}\right)+{\mathcal{P}}_{*}\left(\lambda , \mathcal{ı}\right)\right)\right]{\int }_0^1\frac{1}{ħ\left(\kappa \right)}d\kappa , \\ \le \left[\frac{{\mathcal{P}}^{*}\left(\theta , \mathcal{ı}\right)+{\mathcal{P}}^{*}\left(\lambda , \mathcal{ı}\right)}{2}+\frac{1}{ħ\left(\frac{1}{2}\right)}\left({\mathcal{P}}^{*}\left(\theta , \mathcal{ı}\right)+{\mathcal{P}}^{*}\left(\lambda , \mathcal{ı}\right)\right)\right]{\int }_0^1\frac{1}{ħ\left(\kappa \right)}d\kappa ,\end{array} \\ \begin{array}{c}=\left[{\mathcal{P}}_{*}\left(\theta , \mathcal{ı}\right)+{\mathcal{P}}_{*}\left(\lambda , \mathcal{ı}\right)\right]\left[\frac{1}{2}+\frac{1}{ħ\left(\frac{1}{2}\right)}\right]{\int }_0^1\frac{1}{ħ\left(\kappa \right)}d\kappa , \\ =\left[{\mathcal{P}}^{*}\left(\theta , \mathcal{ı}\right)+{\mathcal{P}}^{*}\left(\lambda , \mathcal{ı}\right)\right]\left[\frac{1}{2}+\frac{1}{ħ\left(\frac{1}{2}\right)}\right]{\int }_0^1\frac{1}{ħ\left(\kappa \right)}d\kappa ,\end{array} \end{gathered}$$

that is

$$\begin{gathered} \frac{\left[ħ\left(\frac{1}{2}\right)\right]}{4}\odot \tilde{\mathcal{P}}\left(\frac{\theta +\lambda }{2}\right){\le }_{\mathbb{F}}{⪧}_2{\le }_{\mathbb{F}}\frac{1}{\lambda -\theta }\odot \left(FA\right){\int }_{\theta }^{\lambda }\tilde{\mathcal{P}}\left(\mathsf{б}\right)d\mathsf{б} \\ {\le }_{\mathbb{F}}{⪧}_1{\le }_{\mathbb{F}}\left[\tilde{\mathcal{P}}\left(\theta \right)\oplus \tilde{\mathcal{P}}\left(\lambda \right)\right]\odot \left[\frac{1}{2}+\frac{1}{ħ\left(\frac{1}{2}\right)}\right]{\int }_0^1\frac{1}{ħ\left(\kappa \right)}d\kappa , \end{gathered}$$

hence, the result follows. □

Example 4.

We consider $ħ\left(\kappa \right)=\kappa ,$ for $\kappa \in \left[0, 1\right]$, and the $F$·$N$·$M$ $\tilde{\mathcal{P}}:\left[\theta , \lambda \right]=[2, 3]\to {\mathbb{E}}_C$ defined by, ${\mathcal{P}}_{\mathcal{ı}}\left(\mathsf{б}\right)=\left[\left(1-\mathcal{ı}\right)\left(2-{\mathsf{б}}^{\frac{1}{2}}\right)+3\mathcal{ı},\left(1-\mathcal{ı}\right)\left(2+{\mathsf{б}}^{\frac{1}{2}}\right)+3\mathcal{ı}\right],$ as in Example 3, then $\tilde{\mathcal{P}}\left(\mathsf{б}\right)$ is an $ħ$-Godunova–Levin convex $F$·$N$·$M$. We have ${\mathcal{P}}_{*}\left(\mathsf{б}, \mathcal{ı}\right)=\left(1-\mathcal{ı}\right)\left(2-{\mathsf{б}}^{\frac{1}{2}}\right)+3\mathcal{ı}$ and ${\mathcal{P}}^{*}\left(\mathsf{б}, \mathcal{ı}\right)=\left(1-\mathcal{ı}\right)\left(2+{\mathsf{б}}^{\frac{1}{2}}\right)+3\mathcal{ı}$. We now compute the following:

$$\begin{gathered} \begin{array}{c}\frac{{\left[ħ\left(\frac{1}{2}\right)\right]}^2}{4}{\mathcal{P}}_{*}\left(\frac{\theta +\lambda }{2}, \mathcal{ı}\right)={\mathcal{P}}_{*}\left(\frac{5}{2}, \mathcal{ı}\right)=\frac{4-\sqrt{10}}{2}\left(1-\mathcal{ı}\right)+3\mathcal{ı}, \\ \frac{{\left[ħ\left(\frac{1}{2}\right)\right]}^2}{4}{\mathcal{P}}^{*}\left(\frac{\theta +\lambda }{2}, \mathcal{ı}\right)={\mathcal{P}}^{*}\left(\frac{5}{2}, \mathcal{ı}\right)=\frac{4+\sqrt{10}}{2}\left(1-\mathcal{ı}\right)+3\mathcal{ı},\end{array} \\ \begin{array}{c}{{⪧}_2}_{*}=\frac{ħ\left(\frac{1}{2}\right)}{4}\left[{\mathcal{P}}_{*}\left(\frac{3\theta +\lambda }{4}, \mathcal{ı}\right)+{\mathcal{P}}_{*}\left(\frac{\theta +3\lambda }{4}, \mathcal{ı}\right)\right]=\frac{5-\sqrt{11}}{4}\left(1-\mathcal{ı}\right)+3\mathcal{ı}, \\ {{⪧}_2}^{*}=\frac{ħ\left(\frac{1}{2}\right)}{4}\left[{\mathcal{P}}^{*}\left(\frac{3\theta +\lambda }{4}, \mathcal{ı}\right)+{\mathcal{P}}^{*}\left(\frac{\theta +3\lambda }{4}, \mathcal{ı}\right)\right]=\frac{7+\sqrt{11}}{4}\left(1-\mathcal{ı}\right)+3\mathcal{ı}.\end{array} \\ \begin{array}{c}{{⪧}_1}_{*}=\left[\frac{{\mathcal{P}}_{*}\left(\theta , \mathcal{ı}\right)+{\mathcal{P}}_{*}\left(\lambda , \mathcal{ı}\right)}{2}+{\mathcal{P}}_{*}\left(\frac{\theta +\lambda }{2}, \mathcal{ı}\right)\right]{\int }_0^1\frac{1}{ħ\left(\kappa \right)}d\kappa =\frac{\left(8-\sqrt{2}-\sqrt{3}-\sqrt{10}\right)}{4}\left(1-\mathcal{ı}\right)+3\mathcal{ı}, \\ {{⪧}_1}^{*}=\left[\frac{{\mathcal{P}}^{*}\left(\theta , \mathcal{ı}\right)+{\mathcal{P}}^{*}\left(\lambda , \mathcal{ı}\right)}{2}+{\mathcal{P}}^{*}\left(\frac{\theta +\lambda }{2}, \mathcal{ı}\right)\right]{\int }_0^1\frac{1}{ħ\left(\kappa \right)}d\kappa =\frac{\left(8+\sqrt{2}+\sqrt{3}+\sqrt{10}\right)}{4}\left(1-\mathcal{ı}\right)+3\mathcal{ı},\end{array} \\ \begin{array}{c}\left[{\mathcal{P}}_{*}\left(\theta , \mathcal{ı}\right)+{\mathcal{P}}_{*}\left(\lambda , \mathcal{ı}\right)\right]\left[\frac{1}{2}+\frac{1}{ħ\left(\frac{1}{2}\right)}\right]{\int }_0^1\frac{1}{ħ\left(\kappa \right)}d\kappa =\frac{\left(4-\sqrt{2}-\sqrt{3}\right)}{2}\left(1-\mathcal{ı}\right)+3\mathcal{ı}, \\ \left[{\mathcal{P}}^{*}\left(\theta , \mathcal{ı}\right)+{\mathcal{P}}^{*}\left(\lambda , \mathcal{ı}\right)\right]\left[\frac{1}{2}+\frac{1}{ħ\left(\frac{1}{2}\right)}\right]{\int }_0^1\frac{1}{ħ\left(\kappa \right)}d\kappa =\frac{\left(4+\sqrt{2}+\sqrt{3}\right)}{2}\left(1-\mathcal{ı}\right)+3\mathcal{ı}.\end{array} \end{gathered}$$

Then, we obtain the following:

$$\begin{array}{c}\left(1-\mathcal{ı}\right)\frac{4-\sqrt{10}}{2}+3\mathcal{ı}\le \frac{5-\sqrt{11}}{4}\left(1-\mathcal{ı}\right)+3\mathcal{ı}\le \frac{\left(6+4\sqrt{2}-6\sqrt{3}\right)}{3}\left(1-\mathcal{ı}\right)+3\mathcal{ı}\\ \le \frac{\left(8-\sqrt{2}-\sqrt{3}-\sqrt{10}\right)}{4}\left(1-\mathcal{ı}\right)+3\mathcal{ı}\le \frac{\left(1-\mathcal{ı}\right)\left(4-\sqrt{2}-\sqrt{3}\right)}{2}+3\mathcal{ı}\\ \left(1-\mathcal{ı}\right)\frac{4+\sqrt{10}}{2}+3\mathcal{ı}\le \frac{7+\sqrt{11}}{4}\left(1-\mathcal{ı}\right)+3\mathcal{ı}\le \frac{\left(6-4\sqrt{2}+6\sqrt{3}\right)}{3}\left(1-\mathcal{ı}\right)+3\mathcal{ı}\\ \le \frac{\left(8+\sqrt{2}+\sqrt{3}+\sqrt{10}\right)}{4}\left(1-\mathcal{ı}\right)+3\mathcal{ı}\le \frac{\left(1-\mathcal{ı}\right)\left(4+\sqrt{2}+\sqrt{3}\right)}{2}+3\mathcal{ı}.\end{array}$$

Hence, Theorem 7 is verified.

The novel fuzzy Hermite–Hadamard inequalities for the product of two $ħ$-Godunova–Levin convex $F$·$N$·$M$s are found in the results.

Theorem 8.

Let $\tilde{\mathcal{P}},\tilde{\mathcal{J}} :\left[\theta , \lambda \right]\to {\mathbb{E}}_C$ be two $ħ$-Godunova–Levin convex $F$·$N$·$M$s with non-negative real-valued functions ${ħ}_1, {ħ}_2:[0, 1]\to {\mathfrak{N}}^{+}$ and ${ħ}_1\left(\frac{1}{2}\right){ħ}_2\left(\frac{1}{2}\right)\ne 0,$ and $\mathit{\text{Ι-V-M}}$s are classified according to their $\mathcal{ı}$-levels such that ${\mathcal{P}}_{\mathcal{ı}}, {\mathcal{J}}_{\mathcal{ı}}:\left[\theta , \lambda \right]\subset \mathfrak{N}\to {\mathcal{X}}_I^{+}$ are given by ${\mathcal{P}}_{\mathcal{ı}}\left(\mathsf{б}\right)=\left[{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right), {\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\right]$ and ${\mathcal{J}}_{\mathcal{ı}}\left(\mathsf{б}\right)=\left[{\mathcal{J}}_{*}\left(\mathsf{б},\mathcal{ı}\right), {\mathcal{J}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\right]$ for all $\mathsf{б}\in \left[\theta , \lambda \right]$ and for all $\mathcal{ı}\in \left[0, 1\right]$. If $\tilde{\mathcal{P}}, \tilde{\mathcal{J}}$, and $\tilde{\mathcal{P}}\otimes \tilde{\mathcal{J}}\in {\mathcal{F}\mathcal{A}}_{\left(\left[\theta , \lambda \right], \mathcal{ı}\right)}$, then

$$\frac{1}{\lambda -\theta }\odot \left(FA\right){\int }_{\theta }^{\lambda }\tilde{\mathcal{P}}\left(\mathsf{б}\right)\otimes \tilde{\mathcal{J}}\left(\mathsf{б}\right)d\mathsf{б}{\le }_{\mathbb{F}}\tilde{\mathcal{M}}\left(\theta ,\lambda \right)\odot {\int }_0^1\frac{1}{{ħ}_1\left(\kappa \right){ħ}_2\left(\kappa \right)}d\kappa \oplus \tilde{\mathcal{N}}\left(\theta ,\lambda \right)\odot {\int }_0^1\frac{1}{{ħ}_1\left(\kappa \right){ħ}_2\left(1-\kappa \right)}d\kappa ,$$

(40)

The (42) is reversed for an $ħ$-Godunova–Levin concave $F$·$N$·$M$.

Where $\tilde{\mathcal{M}}\left(\theta ,\lambda \right)=\tilde{\mathcal{P}}\left(\theta \right)\otimes \tilde{\mathcal{J}}\left(\theta \right)\oplus \tilde{\mathcal{P}}\left(\lambda \right)\otimes \tilde{\mathcal{J}}\left(\lambda \right),$ $\tilde{\mathcal{N}}\left(\theta ,\lambda \right)=\tilde{\mathcal{P}}\left(\theta \right)\otimes \tilde{\mathcal{J}}\left(\lambda \right)\oplus \tilde{\mathcal{P}}\left(\lambda \right)\otimes \tilde{\mathcal{J}}\left(\theta \right),$ and ${\mathcal{M}}_{\mathcal{ı}}\left(\theta ,\lambda \right)=\left[{\mathcal{M}}_{\mathrm{*}}\left(\left(\theta ,\lambda \right),\mathcal{ı}\right),{\mathcal{M}}^{\mathrm{*}}\left(\left(\theta ,\lambda \right),\mathcal{ı}\right)\right]$ and ${\mathcal{N}}_{\mathcal{ı}}\left(\theta ,\lambda \right)=\left[{\mathcal{N}}_{\mathrm{*}}\left(\left(\theta ,\lambda \right),\mathcal{ı}\right),{\mathcal{N}}^{\mathrm{*}}\left(\left(\theta ,\lambda \right),\mathcal{ı}\right)\right].$

Proof. 

Let $\tilde{\mathcal{P}},\tilde{\mathcal{J}} :\left[\theta , \lambda \right]\to {\mathbb{E}}_C$ be two ${ħ}_1$-Godunova–Levin convex and ${ħ}_2$-Godunova–Levin convex $F$·$N$·$M$s. Then, we have

$$\begin{array}{c} \\ {\mathcal{P}}_{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda , \mathcal{ı}\right)\le \frac{{\mathcal{P}}_{*}\left(\mathsf{б}, \mathcal{ı}\right)}{{ħ}_1\left(\kappa \right)}+\frac{{\mathcal{P}}_{*}\left(s,\mathcal{ı}\right)}{{ħ}_1\left(1-\kappa \right)} \\ {\mathcal{P}}^{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda , \mathcal{ı}\right)\le \frac{{\mathcal{P}}^{*}\left(\mathsf{б}, \mathcal{ı}\right)}{{ħ}_1\left(\kappa \right)}+\frac{{\mathcal{P}}^{*}\left(s,\mathcal{ı}\right)}{{ħ}_1\left(1-\kappa \right)}, \end{array}$$

and

$$\begin{array}{c} \\ {\mathcal{J}}_{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda ,\mathcal{ı}\right)\le \frac{{\mathcal{J}}_{*}\left(\mathsf{б}, \mathcal{ı}\right)}{{ħ}_2\left(\kappa \right)}+\frac{{\mathcal{J}}_{*}\left(s,\mathcal{ı}\right)}{{ħ}_2\left(1-\kappa \right)}, \\ {\mathcal{J}}^{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda ,\mathcal{ı}\right)\le \frac{{\mathcal{J}}^{*}\left(\mathsf{б}, \mathcal{ı}\right)}{{ħ}_2\left(\kappa \right)}+\frac{{\mathcal{J}}^{*}\left(s,\mathcal{ı}\right)}{{ħ}_2\left(1-\kappa \right)}. \end{array}$$

From the definition of an $ħ$-Godunova–Levin convex $F$·$N$·$M$, it follows that $\tilde{\mathcal{P}}\left(x\right){\begin{array}{c}\begin{array}{c}\ge \end{array}\end{array}}_{\mathbb{F}}0$ and $\tilde{\mathcal{J}}\left(x\right){\begin{array}{c}\begin{array}{c}\ge \end{array}\end{array}}_{\mathbb{F}}0$, so

$$\begin{array}{c} \\ {\mathcal{P}}_{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda ,\mathcal{ı}\right){\mathcal{J}}_{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda ,\mathcal{ı}\right) \\ \le \left(\frac{{\mathcal{P}}_{*}\left(\mathsf{б}, \mathcal{ı}\right)}{{ħ}_1\left(\kappa \right)}+\frac{{\mathcal{P}}_{*}\left(s,\mathcal{ı}\right)}{{ħ}_1\left(1-\kappa \right)}\right)\left(\frac{{\mathcal{J}}_{*}\left(\mathsf{б}, \mathcal{ı}\right)}{{ħ}_2\left(\kappa \right)}+\frac{{\mathcal{J}}_{*}\left(s,\mathcal{ı}\right)}{{ħ}_2\left(1-\kappa \right)}\right) \\ ={\mathcal{P}}_{*}\left(\theta ,\mathcal{ı}\right){\mathcal{J}}_{*}\left(\theta ,\mathcal{ı}\right)\left[\frac{1}{{ħ}_1\left(\kappa \right){ħ}_2\left(\kappa \right)}\right]+{\mathcal{P}}_{*}\left(\lambda ,\mathcal{ı}\right){\mathcal{J}}_{*}\left(\lambda ,\mathcal{ı}\right)\left[\frac{1}{{ħ}_1\left(1-\kappa \right){ħ}_2\left(1-\kappa \right)}\right]\\ +{\mathcal{P}}_{*}\left(\theta ,\mathcal{ı}\right){\mathcal{J}}_{*}\left(\lambda ,\mathcal{ı}\right)\left[\frac{1}{{ħ}_1\left(\kappa \right){ħ}_2\left(1-\kappa \right)}\right]+{\mathcal{P}}_{*}\left(\lambda ,\mathcal{ı}\right){\mathcal{J}}_{*}\left(\theta ,\mathcal{ı}\right)\left[\frac{1}{{ħ}_1\left(1-\kappa \right){ħ}_2\left(\kappa \right)}\right],\\ {\mathcal{P}}^{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda ,\mathcal{ı}\right){\mathcal{J}}^{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda ,\mathcal{ı}\right) \\ \le \left(\frac{{\mathcal{P}}^{*}\left(\mathsf{б}, \mathcal{ı}\right)}{{ħ}_1\left(\kappa \right)}+\frac{{\mathcal{P}}^{*}\left(s,\mathcal{ı}\right)}{{ħ}_1\left(1-\kappa \right)}\right)\left(\begin{array}{c}\frac{{\mathcal{J}}^{*}\left(\mathsf{б}, \mathcal{ı}\right)}{{ħ}_2\left(\kappa \right)}+\frac{{\mathcal{J}}^{*}\left(s,\mathcal{ı}\right)}{{ħ}_2\left(1-\kappa \right)}\end{array}\right) \\ ={\mathcal{P}}^{*}\left(\theta ,\mathcal{ı}\right){\mathcal{J}}^{*}\left(\theta ,\mathcal{ı}\right)\left[\frac{1}{{ħ}_1\left(\kappa \right){ħ}_2\left(\kappa \right)}\right]+{\mathcal{P}}^{*}\left(\lambda ,\mathcal{ı}\right){\mathcal{J}}^{*}\left(\lambda ,\mathcal{ı}\right)\left[\frac{1}{{ħ}_1\left(1-\kappa \right){ħ}_2\left(1-\kappa \right)}\right]\\ +{\mathcal{P}}^{*}\left(\theta ,\mathcal{ı}\right){\mathcal{J}}^{*}\left(\lambda ,\mathcal{ı}\right)\left[\frac{1}{{ħ}_1\left(\kappa \right){ħ}_2\left(1-\kappa \right)}\right]+{\mathcal{P}}^{*}\left(\lambda ,\mathcal{ı}\right){\mathcal{J}}^{*}\left(\theta ,\mathcal{ı}\right)\left[\frac{1}{{ħ}_1\left(1-\kappa \right){ħ}_2\left(\kappa \right)}\right]. \end{array}$$

Integrating both sides of the above inequality over [0, 1] we obtain

$$\begin{array}{c} \\ {\int }_0^1{\mathcal{P}}_{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda ,\mathcal{ı}\right){\mathcal{J}}_{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda ,\mathcal{ı}\right)=\frac{1}{\lambda -\theta }{\int }_{\theta }^{\lambda }{\mathcal{P}}_{*}\left(x,\mathcal{ı}\right){\mathcal{J}}_{*}\left(x,\mathcal{ı}\right)dx\\ \le \left({\mathcal{P}}_{*}\left(\theta ,\mathcal{ı}\right){\mathcal{J}}_{*}\left(\theta ,\mathcal{ı}\right)+{\mathcal{P}}_{*}\left(\lambda ,\mathcal{ı}\right){\mathcal{J}}_{*}\left(\lambda ,\mathcal{ı}\right)\right){\int }_0^1\frac{1}{{ħ}_1\left(\kappa \right){ħ}_2\left(\kappa \right)}d\kappa \\ +\left({\mathcal{P}}_{*}\left(\theta ,\mathcal{ı}\right){\mathcal{J}}_{*}\left(\lambda ,\mathcal{ı}\right)+{\mathcal{P}}_{*}\left(\lambda ,\mathcal{ı}\right){\mathcal{J}}_{*}\left(\theta ,\mathcal{ı}\right)\right){\int }_0^1\frac{1}{{ħ}_1\left(\kappa \right){ħ}_2\left(1-\kappa \right)}d\kappa , \\ {\int }_0^1{\mathcal{P}}^{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda ,\mathcal{ı}\right){\mathcal{J}}^{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda ,\mathcal{ı}\right)=\frac{1}{\lambda -\theta } {\int }_{\theta }^{\lambda }{\mathcal{P}}^{*}\left(x,\mathcal{ı}\right){\mathcal{J}}^{*}\left(x,\mathcal{ı}\right)dx\\ \le \left({\mathcal{P}}^{*}\left(\theta ,\mathcal{ı}\right){\mathcal{J}}^{*}\left(\theta ,\mathcal{ı}\right)+{\mathcal{P}}^{*}\left(\lambda ,\mathcal{ı}\right){\mathcal{J}}^{*}\left(\lambda ,\mathcal{ı}\right)\right){\int }_0^1\frac{1}{{ħ}_1\left(\kappa \right){ħ}_2\left(\kappa \right)}d\kappa \\ +\left({\mathcal{P}}^{*}\left(\theta ,\mathcal{ı}\right){\mathcal{J}}^{*}\left(\lambda ,\mathcal{ı}\right)+{\mathcal{P}}^{*}\left(\lambda ,\mathcal{ı}\right){\mathcal{J}}^{*}\left(\theta ,\mathcal{ı}\right)\right){\int }_0^1\frac{1}{{ħ}_1\left(\kappa \right){ħ}_2\left(1-\kappa \right)}d\kappa .\end{array}$$

It follows that,

$$\begin{array}{c} \\ \frac{1}{\lambda -\theta }{\int }_{\theta }^{\lambda }{\mathcal{P}}_{*}\left(x,\mathcal{ı}\right){\mathcal{J}}_{*}\left(x,\mathcal{ı}\right)dx\le {\mathcal{M}}_{*}\left(\left(\theta ,\lambda \right),\mathcal{ı}\right){\int }_0^1\frac{1}{{ħ}_1\left(\kappa \right){ħ}_2\left(\kappa \right)}d\kappa \\ +{\mathcal{N}}_{*}\left(\left(\theta ,\lambda \right)\right),\mathcal{ı}{\int }_0^1\frac{1}{{ħ}_1\left(\kappa \right){ħ}_2\left(1-\kappa \right)}d\kappa , \\ \frac{1}{\lambda -\theta } {\int }_{\theta }^{\lambda }{\mathcal{P}}^{*}\left(x,\mathcal{ı}\right){\mathcal{J}}^{*}\left(x,\mathcal{ı}\right)dx\le {\mathcal{M}}^{*}\left(\left(\theta ,\lambda \right),\mathcal{ı}\right){\int }_0^1\frac{1}{{ħ}_1\left(\kappa \right){ħ}_2\left(\kappa \right)}d\kappa \\ +{\mathcal{N}}^{*}\left(\left(\theta ,\lambda \right),\mathcal{ı}\right){\int }_0^1\frac{1}{{ħ}_1\left(\kappa \right){ħ}_2\left(1-\kappa \right)}d\kappa ,\end{array}$$

that is

$$\begin{gathered} \frac{1}{\lambda -\theta }\left[{\int }_{\theta }^{\lambda }{\mathcal{P}}_{*}\left(x,\mathcal{ı}\right){\mathcal{J}}_{*}\left(x,\mathcal{ı}\right)dx, {\int }_{\theta }^{\lambda }{\mathcal{P}}^{*}\left(x,\mathcal{ı}\right){\mathcal{J}}^{*}\left(x,\mathcal{ı}\right)dx\right] \\ {\le }_I\left[{\mathcal{M}}_{*}\left(\left(\theta ,\lambda \right),\mathcal{ı}\right), {\mathcal{M}}^{*}\left(\left(\theta ,\lambda \right),\mathcal{ı}\right)\right]{\int }_0^1\frac{1}{{ħ}_1\left(\kappa \right){ħ}_2\left(\kappa \right)}d\kappa \\ +\left[{\mathcal{N}}_{*}\left(\left(\theta ,\lambda \right),\mathcal{ı}\right), {\mathcal{N}}^{*}\left(\left(\theta ,\lambda \right),\mathcal{ı}\right)\right]{\int }_0^1\frac{1}{{ħ}_1\left(\kappa \right){ħ}_2\left(1-\kappa \right)}d\kappa . \end{gathered}$$

Thus,

$$\begin{gathered} \frac{1}{\lambda -\theta }\odot \left(FA\right){\int }_{\theta }^{\lambda }\tilde{\mathcal{P}}\left(x\right)\otimes \tilde{\mathcal{J}}\left(x\right)dx \\ {\le }_{\mathbb{F}}\tilde{\mathcal{M}}\left(\theta ,\lambda \right)\odot {\int }_0^1\frac{1}{{ħ}_1\left(\kappa \right){ħ}_2\left(\kappa \right)}d\kappa \oplus \tilde{\mathcal{N}}\left(\theta ,\lambda \right)\odot {\int }_0^1\frac{1}{{ħ}_1\left(\kappa \right){ħ}_2\left(1-\kappa \right)}d\kappa , \end{gathered}$$

and the theorem has been established. □

Example 5.

We consider ${ħ}_1\left(\kappa \right)=\frac{1}{\kappa }, {ħ}_2\left(\kappa \right)=\frac{1}{\kappa },$ for $\kappa \in \left[0, 1\right]$, and the $F$·$N$·$M$s $\tilde{\mathcal{P}}, \tilde{\mathcal{J}}:\left[\theta , \lambda \right]=[0, 2]\to {\mathbb{E}}_C$ is defined by

$$\tilde{\mathcal{P}}\left(\mathsf{б}\right)\left(\sigma \right)=\left\{\begin{array}{c}\frac{\theta }{\mathsf{б}} \theta \in \left[0, \mathsf{б}\right]\\ \frac{ 2\mathsf{б}-\theta }{\mathsf{б}} \theta \in (\mathsf{б}, 2\mathsf{б}]\\ 0 \mathrm{otherwise},\end{array}\right.$$

$$\tilde{\mathcal{J}}\left(\mathsf{б}\right)\left(\sigma \right)=\left\{\begin{array}{c}\frac{\theta -\mathsf{б}}{2-\mathsf{б}} \theta \in \left[\mathsf{б}, 2\right]\\ \frac{ 8-e^{\mathsf{б}}-\theta }{8-e^{\mathsf{б}}-2} \theta \in (2, 8-e^{\mathsf{б}}]\\ 0 \mathrm{otherwise}.\end{array}\right.$$

Then, for each $\mathcal{ı}\in \left[0,1\right],$ we have ${\mathcal{P}}_{\mathcal{ı}}\left(\mathrm{б}\right)=\left[\mathcal{ı}\mathrm{б},(2-\mathcal{ı})\mathrm{б}\right]$ and ${\mathcal{J}}_{\mathcal{ı}}\left(\mathrm{б}\right)=\left[\left(1-\mathcal{ı}\right)\mathrm{б}+2\mathcal{ı},\left(1-\mathcal{ı}\right)\left(8-e^{\mathrm{б}}\right)+2\mathcal{ı}\right].$ The end-point functions ${\mathcal{P}}_{\mathrm{*}}\left(\mathrm{б},\mathcal{ı}\right)=\mathcal{ı}\mathrm{б},$ ${\mathcal{P}}^{\mathrm{*}}\left(\mathrm{б},\mathcal{ı}\right)=(2-\mathcal{ı})\mathrm{б}$ and ${\mathcal{J}}_{\mathrm{*}}\left(\mathrm{б},\mathcal{ı}\right)=\left(1-\mathcal{ı}\right)\mathrm{б}+2\mathcal{ı}$, ${\mathcal{J}}^{\mathrm{*}}\left(\mathrm{б},\mathcal{ı}\right)=\left(1-\mathcal{ı}\right)\left(8-e^{\mathrm{б}}\right)+2\mathcal{ı}$ are ${ħ}_1,{ħ}_2$-Godunova–Levin convex functions for each $\mathcal{ı}\in [0,1]$. Hence, $\tilde{\mathcal{P}},$ $\tilde{\mathcal{J}}$ both are ${ħ}_1,{ħ}_2$-Godunova–Levin convex $F$·$N$·$M$s. We now compute the following:

$$\begin{array}{c}\frac{1}{\lambda -\theta } {\int }_{\theta }^{\lambda }{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)\times {\mathcal{J}}_{*}\left(\mathsf{б},\mathcal{ı}\right)d\mathsf{б}=\frac{1}{2}\underset{0}{\overset{2}{\int }} \left(\mathcal{ı}\left(1-\mathcal{ı}\right){\mathsf{б}}^2+2{\mathcal{ı}}^2\mathsf{б}\right)d\mathsf{б}=\frac{2}{3}\mathcal{ı}\left(2+\mathcal{ı}\right) \\ \frac{1}{\lambda -\theta } {\int }_{\theta }^{\lambda }{\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\times {\mathcal{J}}^{*}\left(\mathsf{б},\mathcal{ı}\right)d\mathsf{б}=\frac{1}{2}\underset{0}{\overset{2}{\int }}\left(\left(1-\mathcal{ı}\right)\left(2-\mathcal{ı}\right)\mathsf{б}\left(8-e^{\mathsf{б}}\right)+2\mathcal{ı}\left(2-\mathcal{ı}\right)\mathsf{б}\right)d\mathsf{б}\approx \frac{\left(2-\mathcal{ı}\right)}{2}\left(\frac{1903}{250}-\frac{903}{250}\mathcal{ı}\right),\end{array}$$

$$\begin{array}{c}{\mathcal{M}}_{*}\left(\left(\theta ,\lambda \right), \mathcal{ı}\right){\int }_0^1\frac{1}{{ħ}_1\left(\kappa \right){ħ}_2\left(\kappa \right)}d\kappa =\frac{4\mathcal{ı}}{3} \\ {\mathcal{M}}^{*}\left(\left(\theta ,\lambda \right), \mathcal{ı}\right){\int }_0^1\frac{1}{{ħ}_1\left(\kappa \right){ħ}_2\left(\kappa \right)}d\kappa =\frac{2\left(2-\mathcal{ı}\right)\left[\left(1-\mathcal{ı}\right)\left(8-e^2\right)+2\mathcal{ı}\right]}{3},\end{array}$$

$$\begin{array}{c}{\mathcal{N}}_{*}\left(\left(\theta ,\lambda \right), \mathcal{ı}\right){\int }_0^1\frac{1}{{ħ}_1\left(\kappa \right){ħ}_2\left(1-\kappa \right)}d\kappa =\frac{2{\mathcal{ı}}^2}{3} \\ {\mathcal{N}}^{*}\left(\left(\theta ,\lambda \right), \mathcal{ı}\right){\int }_0^1\frac{1}{{ħ}_1\left(\kappa \right){ħ}_2\left(1-\kappa \right)}d\kappa =\frac{\left(2-\mathcal{ı}\right)\left(7-5\mathcal{ı}\right)}{3},\end{array}$$

for each $\mathcal{ı}\in \left[0, 1\right],$ which means

$$[\frac{2}{3}\mathcal{ı}\left(1+2\mathcal{ı}\right),\frac{\left(2-\mathcal{ı}\right)}{2}\left(\frac{1903}{250}-\frac{903}{250}\mathcal{ı}\right)]{\le }_I\frac{1}{3}\left[2\mathcal{ı}\left(2+\mathcal{ı}\right),\left(2-\mathcal{ı}\right)\left[2\left(1-\mathcal{ı}\right)\left(8-e^2\right)-\mathcal{ı}+7\right]\right]$$

Hence, Theorem 8 is demonstrated.

Theorem 9.

Let $\tilde{\mathcal{P}},\tilde{\mathcal{J}} :\left[\theta , \lambda \right]\to {\mathbb{E}}_C$ be two ${ħ}_1$-Godunova–Levin convex and ${ħ}_2$- Godunova–Levin convex $F$·$N$·$M$s with non-negative real-valued functions ${ħ}_1, {ħ}_2:[0, 1]\to {\mathfrak{N}}^{+}$, respectively, and ${ħ}_1\left(\frac{1}{2}\right){ħ}_2\left(\frac{1}{2}\right)\ne 0,$ respectively, and $\mathit{\text{Ι-V-M}}$s are classified according to their $\mathcal{ı}$-levels such that ${\mathcal{P}}_{\mathcal{ı}}, {\mathcal{J}}_{\mathcal{ı}}:\left[\theta , \lambda \right]\subset \mathfrak{N}\to {\mathcal{X}}_I^{+}$ are given, respectively, by ${\mathcal{P}}_{\mathcal{ı}}\left(\mathsf{б}\right)=\left[{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right), {\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\right]$ and ${\mathcal{J}}_{\mathcal{ı}}\left(\mathsf{б}\right)=\left[{\mathcal{J}}_{*}\left(\mathsf{б},\mathcal{ı}\right), {\mathcal{J}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\right]$ for all $\mathsf{б}\in \left[\theta , \lambda \right]$ and for all $\mathcal{ı}\in \left[0, 1\right]$. If $\tilde{\mathcal{P}}\otimes \tilde{\mathcal{J}}\in {\mathcal{F}\mathcal{A}}_{\left(\left[\theta , \lambda \right], \mathcal{ı}\right)}$, then

$$\begin{gathered} \frac{{ħ}_1\left(\frac{1}{2}\right){ħ}_2\left(\frac{1}{2}\right)}{2}\odot \tilde{\mathcal{P}}\left(\frac{\theta +\lambda }{2}\right)\otimes \tilde{\mathcal{J}}\left(\frac{\theta +\lambda }{2}\right){\le }_{\mathbb{F}}\frac{1}{\lambda -\theta }\odot \left(FA\right){\int }_{\theta }^{\lambda }\tilde{\mathcal{P}}\left(\mathsf{б}\right)\otimes \tilde{\mathcal{J}}\left(\mathsf{б}\right)d\mathsf{б} \\ \oplus \tilde{\mathcal{M}}\left(\theta ,\lambda \right)\odot {\int }_0^1\frac{1}{{ħ}_1\left(\kappa \right){ħ}_2\left(1-\kappa \right)}d\kappa \oplus \tilde{\mathcal{N}}\left(\theta ,\lambda \right)\odot {\int }_0^1\frac{1}{{ħ}_1\left(\kappa \right){ħ}_2\left(\kappa \right)}d\kappa , \end{gathered}$$

(41)

where $\tilde{\mathcal{M}}\left(\theta ,\lambda \right)=\tilde{\mathcal{P}}\left(\theta \right)\otimes \tilde{\mathcal{J}}\left(\theta \right)\oplus \tilde{\mathcal{P}}\left(\lambda \right)\otimes \tilde{\mathcal{J}}\left(\lambda \right),$ $\tilde{\mathcal{N}}\left(\theta ,\lambda \right)=\tilde{\mathcal{P}}\left(\theta \right)\otimes \tilde{\mathcal{J}}\left(\lambda \right)\oplus \tilde{\mathcal{P}}\left(\lambda \right)\otimes \tilde{\mathcal{J}}\left(\theta \right),$ and ${\mathcal{M}}_{\mathcal{ı}}\left(\theta ,\lambda \right)=\left[{\mathcal{M}}_{*}\left(\left(\theta ,\lambda \right), \mathcal{ı}\right), {\mathcal{M}}^{*}\left(\left(\theta ,\lambda \right), \mathcal{ı}\right)\right]$ and ${\mathcal{N}}_{\mathcal{ı}}\left(\theta ,\lambda \right)=\left[{\mathcal{N}}_{*}\left(\left(\theta ,\lambda \right), \mathcal{ı}\right), {\mathcal{N}}^{*}\left(\left(\theta ,\lambda \right), \mathcal{ı}\right)\right].$

Proof. 

By hypothesis, for each $\mathcal{ı}\in \left[0, 1\right],$ we have

$$\begin{gathered} \begin{array}{c}{\mathcal{P}}_{*}\left(\frac{\theta +\lambda }{2},\mathcal{ı}\right)\times {\mathcal{J}}_{*}\left(\frac{\theta +\lambda }{2},\mathcal{ı}\right)\le \frac{1}{{ħ}_1\left(\frac{1}{2}\right){ħ}_2\left(\frac{1}{2}\right)}\left[\begin{array}{c}{\mathcal{P}}_{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda , \mathcal{ı}\right)\times {\mathcal{J}}_{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda , \mathcal{ı}\right)\\ +{\mathcal{P}}_{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda , \mathcal{ı}\right)\times {\mathcal{J}}_{*}\left(\left(1-\kappa \right)\theta +\kappa \lambda , \mathcal{ı}\right)\end{array}\right] \\ +\frac{1}{{ħ}_1\left(\frac{1}{2}\right){ħ}_2\left(\frac{1}{2}\right)}\left[\begin{array}{c}{\mathcal{P}}_{*}\left(\left(1-\kappa \right)\theta +\kappa \lambda , \mathcal{ı}\right)\times {\mathcal{J}}_{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda , \mathcal{ı}\right)\\ +{\mathcal{P}}_{*}\left(\left(1-\kappa \right)\theta +\kappa \lambda , \mathcal{ı}\right)\times {\mathcal{J}}_{*}\left(\left(1-\kappa \right)\theta +\kappa \lambda , \mathcal{ı}\right)\end{array}\right], \\ {\mathcal{P}}^{*}\left(\frac{\theta +\lambda }{2},\mathcal{ı}\right)\times {\mathcal{J}}^{*}\left(\frac{\theta +\lambda }{2},\mathcal{ı}\right)\le \frac{1}{{ħ}_1\left(\frac{1}{2}\right){ħ}_2\left(\frac{1}{2}\right)}\left[\begin{array}{c}{\mathcal{P}}^{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda , \mathcal{ı}\right)\times {\mathcal{J}}^{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda , \mathcal{ı}\right)\\ +{\mathcal{P}}^{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda , \mathcal{ı}\right)\times {\mathcal{J}}^{*}\left(\left(1-\kappa \right)\theta +\kappa \lambda , \mathcal{ı}\right)\end{array}\right] \\ +\frac{1}{{ħ}_1\left(\frac{1}{2}\right){ħ}_2\left(\frac{1}{2}\right)}\left[\begin{array}{c}{\mathcal{P}}^{*}\left(\left(1-\kappa \right)\theta +\kappa \lambda , \mathcal{ı}\right)\times {\mathcal{J}}^{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda , \mathcal{ı}\right)\\ +{\mathcal{P}}^{*}\left(\left(1-\kappa \right)\theta +\kappa \lambda , \mathcal{ı}\right)\times {\mathcal{J}}^{*}\left(\left(1-\kappa \right)\theta +\kappa \lambda , \mathcal{ı}\right)\end{array}\right], \end{array} \\ \begin{array}{c}\le \frac{1}{{ħ}_1\left(\frac{1}{2}\right){ħ}_2\left(\frac{1}{2}\right)}\left[\begin{array}{c}{\mathcal{P}}_{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda , \mathcal{ı}\right){\times \mathcal{J}}_{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda , \mathcal{ı}\right)\\ +{\mathcal{P}}_{*}\left(\left(1-\kappa \right)\theta +\kappa \lambda , \mathcal{ı}\right)\times {\mathcal{J}}_{*}\left(\left(1-\kappa \right)\theta +\kappa \lambda , \mathcal{ı}\right)\end{array}\right] \\ +\frac{1}{{ħ}_1\left(\frac{1}{2}\right){ħ}_2\left(\frac{1}{2}\right)}\left[\begin{array}{c}\left({ħ}_1\left(\kappa \right){\mathcal{P}}_{*}\left(\theta , \mathcal{ı}\right)+{ħ}_1\left(1-\kappa \right){\mathcal{P}}_{*}\left(\lambda , \mathcal{ı}\right)\right)\\ \times \left({ħ}_2\left(1-\kappa \right){\mathcal{J}}_{*}\left(\theta , \mathcal{ı}\right)+{ħ}_2\left(\kappa \right){\mathcal{J}}_{*}\left(\lambda , \mathcal{ı}\right)\right)\\ +\left({{ħ}_1\left(1-\kappa \right)\mathcal{P}}_{*}\left(\theta , \mathcal{ı}\right)+{ħ}_1\left(\kappa \right){\mathcal{P}}_{*}\left(\lambda , \mathcal{ı}\right)\right)\\ \times \left({ħ}_2\left(\kappa \right){\mathcal{J}}_{*}\left(\theta , \mathcal{ı}\right)+{ħ}_2\left(1-\kappa \right){\mathcal{J}}_{*}\left(\lambda , \mathcal{ı}\right)\right)\end{array}\right], \\ \le \frac{1}{{ħ}_1\left(\frac{1}{2}\right){ħ}_2\left(\frac{1}{2}\right)}\left[\begin{array}{c}{\mathcal{P}}^{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda , \mathcal{ı}\right)\times {\mathcal{J}}^{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda , \mathcal{ı}\right)\\ +{\mathcal{P}}^{*}\left(\left(1-\kappa \right)\theta +\kappa \lambda , \mathcal{ı}\right)\times {\mathcal{J}}^{*}\left(\left(1-\kappa \right)\theta +\kappa \lambda , \mathcal{ı}\right)\end{array}\right] \\ +\frac{1}{{ħ}_1\left(\frac{1}{2}\right){ħ}_2\left(\frac{1}{2}\right)}\left[\begin{array}{c}\left({ħ}_1\left(\kappa \right){\mathcal{P}}^{*}\left(\theta , \mathcal{ı}\right)+{ħ}_1\left(1-\kappa \right){\mathcal{P}}^{*}\left(\lambda , \mathcal{ı}\right)\right)\\ \times \left({ħ}_2\left(1-\kappa \right){\mathcal{J}}^{*}\left(\theta , \mathcal{ı}\right)+{ħ}_2\left(\kappa \right){\mathcal{J}}^{*}\left(\lambda , \mathcal{ı}\right)\right)\\ +\left({ħ}_1\left(1-\kappa \right){\mathcal{P}}^{*}\left(\theta , \mathcal{ı}\right)+{ħ}_1\left(\kappa \right){\mathcal{P}}^{*}\left(\lambda , \mathcal{ı}\right)\right)\\ \times \left({ħ}_2\left(\kappa \right){\mathcal{J}}^{*}\left(\theta , \mathcal{ı}\right)+{ħ}_2\left(1-\kappa \right){\mathcal{J}}^{*}\left(\lambda , \mathcal{ı}\right)\right)\end{array}\right], \end{array} \\ \begin{array}{c}=\frac{1}{{ħ}_1\left(\frac{1}{2}\right){ħ}_2\left(\frac{1}{2}\right)}\left[\begin{array}{c}{\mathcal{P}}_{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda , \mathcal{ı}\right)\times {\mathcal{J}}_{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda , \mathcal{ı}\right)\\ +{\mathcal{P}}_{*}\left(\left(1-\kappa \right)\theta +\kappa \lambda , \mathcal{ı}\right){\times \mathcal{J}}_{*}\left(\left(1-\kappa \right)\theta +\kappa \lambda , \mathcal{ı}\right)\end{array}\right] \\ +\frac{2}{{ħ}_1\left(\frac{1}{2}\right){ħ}_2\left(\frac{1}{2}\right)}\left[\begin{array}{c}\left\{{ħ}_1\left(\kappa \right){ħ}_2\left(\kappa \right)+{ħ}_1\left(1-\kappa \right){ħ}_2\left(1-\kappa \right)\right\}{\mathcal{N}}_{*}\left(\left(\theta ,\lambda \right), \mathcal{ı}\right)\\ +\left\{{ħ}_1\left(\kappa \right){ħ}_2\left(1-\kappa \right)+{ħ}_1\left(1-\kappa \right){ħ}_2\left(\kappa \right)\right\}{\mathcal{M}}_{*}\left(\left(\theta ,\lambda \right), \mathcal{ı}\right)\end{array}\right], \\ =\frac{1}{{ħ}_1\left(\frac{1}{2}\right){ħ}_2\left(\frac{1}{2}\right)}\left[\begin{array}{c}{\mathcal{P}}^{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda , \mathcal{ı}\right)\times {\mathcal{J}}^{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda , \mathcal{ı}\right)\\ +{\mathcal{P}}^{*}\left(\left(1-\kappa \right)\theta +\kappa \lambda , \mathcal{ı}\right)\times {\mathcal{J}}^{*}\left(\left(1-\kappa \right)\theta +\kappa \lambda , \mathcal{ı}\right)\end{array}\right] \\ +\frac{2}{{ħ}_1\left(\frac{1}{2}\right){ħ}_2\left(\frac{1}{2}\right)}\left[\begin{array}{c}\left\{{ħ}_1\left(\kappa \right){ħ}_2\left(\kappa \right)+{ħ}_1\left(1-\kappa \right){ħ}_2\left(1-\kappa \right)\right\}{\mathcal{N}}^{*}\left(\left(\theta ,\lambda \right), \mathcal{ı}\right)\\ +\left\{{ħ}_1\left(\kappa \right){ħ}_2\left(1-\kappa \right)+{ħ}_1\left(1-\kappa \right){ħ}_2\left(\kappa \right)\right\}{\mathcal{M}}^{*}\left(\left(\theta ,\lambda \right), \mathcal{ı}\right)\end{array}\right],\end{array} \end{gathered}$$

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

$$\begin{array}{c} \\ \frac{{ħ}_1\left(\frac{1}{2}\right){ħ}_2\left(\frac{1}{2}\right)}{2}{\mathcal{P}}_{*}\left(\frac{\theta +\lambda }{2},\mathcal{ı}\right)\times {\mathcal{J}}_{*}\left(\frac{\theta +\lambda }{2},\mathcal{ı}\right)\le \frac{1}{\lambda -\theta }\left(R\right){\int }_{\theta }^{\lambda }{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)\times {\mathcal{J}}_{*}\left(\mathsf{б},\mathcal{ı}\right)d\mathsf{б} \\ +{\mathcal{M}}_{*}\left(\left(\theta ,\lambda \right), \mathcal{ı}\right){\int }_0^1\frac{1}{{ħ}_1\left(\kappa \right){ħ}_2\left(1-\kappa \right)}d\kappa \\ +{\mathcal{N}}_{*}\left(\left(\theta ,\lambda \right), \mathcal{ı}\right){\int }_0^1\frac{1}{{ħ}_1\left(\kappa \right){ħ}_2\left(\kappa \right)}d\kappa , \\ \frac{{ħ}_1\left(\frac{1}{2}\right){ħ}_2\left(\frac{1}{2}\right)}{2}{\mathcal{P}}^{*}\left(\frac{\theta +\lambda }{2},\mathcal{ı}\right)\times {\mathcal{J}}^{*}\left(\frac{\theta +\lambda }{2},\mathcal{ı}\right)\le \frac{1}{\lambda -\theta }\left(R\right){\int }_{\theta }^{\lambda }{\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right){\times \mathcal{J}}^{*}\left(\mathsf{б},\mathcal{ı}\right)d\mathsf{б} \\ +{\mathcal{M}}^{*}\left(\left(\theta ,\lambda \right), \mathcal{ı}\right){\int }_0^1\frac{1}{{ħ}_1\left(\kappa \right){ħ}_2\left(1-\kappa \right)}d\kappa \\ +{\mathcal{N}}^{*}\left(\left(\theta ,\lambda \right), \mathcal{ı}\right){\int }_0^1\frac{1}{{ħ}_1\left(\kappa \right){ħ}_2\left(\kappa \right)}d\kappa , \end{array}$$

that is

$$\begin{gathered} \frac{{ħ}_1\left(\frac{1}{2}\right){ħ}_2\left(\frac{1}{2}\right)}{2}\odot \tilde{\mathcal{P}}\left(\frac{\theta +\lambda }{2}\right)\otimes \tilde{\mathcal{J}}\left(\frac{\theta +\lambda }{2}\right){\le }_{\mathbb{F}}\frac{1}{\lambda -\theta } \odot \left(FA\right){\int }_{\theta }^{\lambda }\tilde{\mathcal{P}}\left(\mathsf{б}\right)\otimes \tilde{\mathcal{J}}\left(\mathsf{б}\right)d\mathsf{б} \\ \oplus \tilde{\mathcal{M}}\left(\theta ,\lambda \right)\odot {\int }_0^1\frac{1}{{ħ}_1\left(\kappa \right){ħ}_2\left(1-\kappa \right)}d\kappa \oplus \tilde{\mathcal{N}}\left(\theta ,\lambda \right)\odot {\int }_0^1\frac{1}{{ħ}_1\left(\kappa \right){ħ}_2\left(\kappa \right)}d\kappa , \end{gathered}$$

Hence, the required result is obtained. □

Example 6.

We consider ${ħ}_1\left(\kappa \right)=\frac{1}{\kappa }, {ħ}_2\left(\kappa \right)=\frac{1}{\kappa },$ for $\kappa \in \left[0, 1\right]$, and the $F$·$N$·$M$s $\tilde{\mathcal{P}}, \tilde{\mathcal{J}}:\left[\theta , \lambda \right]=\left[0, 1\right]\to {\mathbb{E}}_C,$ as in Example 5. Then, for each $\mathcal{ı}\in \left[0, 1\right],$ we have ${\mathcal{P}}_{\mathcal{ı}}\left(\mathsf{б}\right)=\left[\mathcal{ı}\mathsf{б},(2-\mathcal{ı})\mathsf{б}\right]$ and ${\mathcal{J}}_{\mathcal{ı}}\left(\mathsf{б}\right)=\left[\left(1-\mathcal{ı}\right)\mathsf{б}+2\mathcal{ı},\left(1-\mathcal{ı}\right)\left(8-e^{\mathsf{б}}\right)+2\mathcal{ı}\right]$, and $\tilde{\mathcal{P}}\left(\mathsf{б}\right), \tilde{\mathcal{J}}\left(\mathsf{б}\right)$ are ${ħ}_1$-Godunova–Levin convex and ${ħ}_2$-Godunova–Levin convex $F$·$N$·$M$s, respectively. We have ${\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)=\mathcal{ı}\mathsf{б},$ ${\mathcal{P}}^{*}\left(\mathsf{б}, \mathcal{ı}\right)=(2-\mathcal{ı})\mathsf{б}$ and ${\mathcal{J}}_{*}\left(\mathsf{б},\mathcal{ı}\right)=\left(1-\mathcal{ı}\right)\mathsf{б}+2\mathcal{ı}$, ${\mathcal{J}}^{*}\left(\mathsf{б}, \mathcal{ı}\right)=\left(1-\mathcal{ı}\right)\left(8-e^{\mathsf{б}}\right)+2\mathcal{ı}$. We now compute the following:

$$\begin{array}{c}\frac{{ħ}_1\left(\frac{1}{2}\right){ħ}_2\left(\frac{1}{2}\right)}{2} {\mathcal{P}}_{*}\left(\frac{\theta +\lambda }{2},\mathcal{ı}\right)\times {\mathcal{J}}_{*}\left(\frac{\theta +\lambda }{2},\mathcal{ı}\right)=2\mathcal{ı}\left(1+\mathcal{ı}\right), \\ \frac{{ħ}_1\left(\frac{1}{2}\right){ħ}_2\left(\frac{1}{2}\right)}{2} {\mathcal{P}}^{*}\left(\frac{\theta +\lambda }{2},\mathcal{ı}\right)\times {\mathcal{J}}^{*}\left(\frac{\theta +\lambda }{2},\mathcal{ı}\right)=2\left[16-20\mathcal{ı}+6{\mathcal{ı}}^2+\left(2-3\mathcal{ı}+{\mathcal{ı}}^2\right)e\right].\end{array}$$

$$\begin{array}{c}\frac{1}{\lambda -\theta }{\int }_{\theta }^{\lambda }{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)\times {\mathcal{J}}_{*}\left(\mathsf{б},\mathcal{ı}\right)d\mathsf{б}=\frac{1}{2}\underset{0}{\overset{2}{\int }}\left(\mathcal{ı}\left(1-\mathcal{ı}\right){\mathsf{б}}^2+2{\mathcal{ı}}^2\mathsf{б}\right)d\mathsf{б}=\frac{4}{3}\mathcal{ı}\left(3-\mathcal{ı}\right), \\ \frac{1}{\lambda -\theta } {\int }_{\theta }^{\lambda }{\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\times {\mathcal{J}}^{*}\left(\mathsf{б},\mathcal{ı}\right)d\mathsf{б}=\frac{1}{2}\underset{0}{\overset{2}{\int }}\left(\left(1-\mathcal{ı}\right)\left(2-\mathcal{ı}\right)\mathsf{б}\left(8-e^{\mathsf{б}}\right)+2\mathcal{ı}\left(2-\mathcal{ı}\right)\mathsf{б}\right)d\mathsf{б}\\ \approx \frac{\left(2-\mathcal{ı}\right)}{2}\left(\frac{1903}{250}-\frac{903}{250}\mathcal{ı}\right).\end{array}$$

$$\begin{array}{c}{\mathcal{M}}_{*}\left(\left(\theta ,\lambda \right), \mathcal{ı}\right){\int }_0^1\frac{1}{{ħ}_1\left(\kappa \right){ħ}_2\left(1-\kappa \right)}d\kappa =\frac{2\mathcal{ı}}{3}, \\ {\mathcal{M}}^{*}\left(\left(\theta ,\lambda \right), \mathcal{ı}\right){\int }_0^1\frac{1}{{ħ}_1\left(\kappa \right){ħ}_2\left(1-\kappa \right)}d\kappa =\frac{\left(2-\mathcal{ı}\right)\left[\left(1-\mathcal{ı}\right)\left(8-e^2\right)+2\mathcal{ı}\right]}{3},\end{array}$$

$$\begin{array}{c}{\mathcal{N}}_{*}\left(\left(\theta ,\lambda \right), \mathcal{ı}\right){\int }_0^1\frac{1}{{ħ}_1\left(\kappa \right){ħ}_2\left(\kappa \right)}d\kappa =\frac{4{\mathcal{ı}}^2}{3}, \\ {\mathcal{N}}^{*}\left(\left(\theta ,\lambda \right), \mathcal{ı}\right){\int }_0^1\frac{1}{{ħ}_1\left(\kappa \right){ħ}_2\left(\kappa \right)}d\kappa =\frac{2\left(2-\mathcal{ı}\right)\left(7-5\mathcal{ı}\right)}{3},\end{array}$$

for each $\mathcal{ı}\in \left[0, 1\right],$ which means

$$\begin{gathered} 2\left[\mathcal{ı}\left(1+\mathcal{ı}\right),\left[16-20\mathcal{ı}+6{\mathcal{ı}}^2+\left(2-3\mathcal{ı}+{\mathcal{ı}}^2\right)e\right]\right]{\le }_I\left[\frac{2}{3}\mathcal{ı}\left(2+\mathcal{ı}\right),\frac{\left(2-\mathcal{ı}\right)}{2}\left(\frac{1903}{250}-\frac{903}{250}\mathcal{ı}\right)\right] \\ +\frac{1}{3}\left[2\mathcal{ı}\left(1+2\mathcal{ı}\right),\left(2-\mathcal{ı}\right)\left[\left(1-\mathcal{ı}\right)\left(8-e^2\right)-8\mathcal{ı}+14,\right]\right] \end{gathered}$$

hence, Theorem 9 is demonstrated.

The H-H Fejér inequalities for $ħ$-Godunova–Levin convex $F$·$N$·$M$s are now presented. The second H-H Fejér inequality for the $ħ$-Godunova–Levin convex $F$·$N$·$M$ is firstly obtained.

Theorem 10.

Let $\tilde{\mathcal{P}}:\left[\theta , \lambda \right]\to {\mathbb{E}}_C$ be an $ħ$-Godunova–Levin convex $F$·$N$·$M$ with $ħ:[0, 1]\to {\mathfrak{N}}^{+}$, and $\mathit{\text{Ι-V-M}}$s are classified according to their $\mathcal{ı}$-levels such that ${\mathcal{P}}_{\mathcal{ı}}:\left[\theta , \lambda \right]\subset \mathfrak{N}\to {\mathcal{X}}_I^{+}$ are given by ${\mathcal{P}}_{\mathcal{ı}}\left(\mathsf{б}\right)=\left[{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right), {\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\right]$ for all $\mathsf{б}\in \left[\theta , \lambda \right]$ and for all $\mathcal{ı}\in \left[0, 1\right]$. If $\tilde{\mathcal{P}}\in {\mathcal{F}\mathcal{R}}_{\left(\left[\theta , \lambda \right], \mathcal{ı}\right)}$ and $\mathfrak{B}:\left[\theta , \lambda \right]\to \mathfrak{N},\mathfrak{B}\left(\mathsf{б}\right)\ge 0,$ symmetric with respect to $\frac{\theta +\lambda }{2},$ then

$$\frac{1}{\lambda -\theta }\odot \left(FA\right){\int }_{\theta }^{\lambda }\mathcal{P}\left(\mathsf{б}\right)\odot \mathfrak{B}\left(\mathfrak{б}\right)d\mathsf{б}{\le }_{\mathbb{F}}\left[\mathcal{P}\left(\theta \right)\oplus \mathcal{P}\left(\lambda \right)\right]\odot {\int }_0^1\frac{\mathfrak{B}\left(\left(1-\kappa \right)\theta +\kappa \lambda \right)}{ħ\left(\kappa \right)}d\kappa .$$

(42)

Proof. 

Let $\tilde{\mathcal{P}}$ be an $ħ$-Godunova–Levin convex $F$·$N$·$M$. Then, for each $\mathcal{ı}\in \left[0, 1\right],$ we have

$$\begin{array}{c} \\ {\mathcal{P}}_{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda , \mathcal{ı}\right)B\left(\kappa \theta +\left(1-\kappa \right)\lambda \right) \\ \le \left(\frac{{\mathcal{P}}_{*}\left(\theta , \mathcal{ı}\right)}{ħ\left(\kappa \right)}+\frac{{\mathcal{P}}_{*}\left(\lambda , \mathcal{ı}\right)}{ħ\left(1-\kappa \right)}\right)B\left(\kappa \theta +\left(1-\kappa \right)\lambda \right),\\ {\mathcal{P}}^{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda , \mathcal{ı}\right)B\left(\kappa \theta +\left(1-\kappa \right)\lambda \right) \\ \le \left(\frac{{\mathcal{P}}^{*}\left(\theta , \mathcal{ı}\right)}{ħ\left(\kappa \right)}+\frac{{\mathcal{P}}^{*}\left(\lambda , \mathcal{ı}\right)}{ħ\left(1-\kappa \right)}\right)B\left(\kappa \theta +\left(1-\kappa \right)\lambda \right).\end{array}$$

(43)

and

$$\begin{array}{c} \\ {\mathcal{P}}_{*}\left(\left(1-\kappa \right)\theta +\kappa \lambda , \mathcal{ı}\right)B\left(\left(1-\kappa \right)\theta +\kappa \lambda \right) \\ \le \left(\frac{{\mathcal{P}}_{*}\left(\theta , \mathcal{ı}\right)}{ħ\left(1-\kappa \right)}+\frac{{\mathcal{P}}_{*}\left(\lambda , \mathcal{ı}\right)}{ħ\left(\kappa \right)}\right)B\left(\left(1-\kappa \right)\theta +\kappa \lambda \right),\\ {\mathcal{P}}^{*}\left(\left(1-\kappa \right)\theta +\kappa \lambda , \mathcal{ı}\right)B\left(\left(1-\kappa \right)\theta +\kappa \lambda \right) \\ \le \left(\frac{{\mathcal{P}}^{*}\left(\theta , \mathcal{ı}\right)}{ħ\left(1-\kappa \right)}+\frac{{\mathcal{P}}^{*}\left(\lambda , \mathcal{ı}\right)}{ħ\left(\kappa \right)}\right)B\left(\left(1-\kappa \right)\theta +\kappa \lambda \right).\end{array}$$

(44)

After adding (45) and (46), and integrating over $\left[0, 1\right],$ we obtain

$$\begin{gathered} \begin{array}{c} \\ {\int }_0^1{\mathcal{P}}_{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda , \mathcal{ı}\right)\mathfrak{B}\left(\kappa \theta +\left(1-\kappa \right)\lambda \right)d\kappa +{\int }_0^1{\mathcal{P}}_{*}\left(\left(1-\kappa \right)\theta +\kappa \lambda , \mathcal{ı}\right)\mathfrak{B}\left(\left(1-\kappa \right)\theta +\kappa \lambda \right)d\kappa \\ \le {\int }_0^1\left[\begin{array}{c}{\mathcal{P}}_{*}\left(\theta , \mathcal{ı}\right)\left\{\frac{1}{ħ\left(\kappa \right)}\mathfrak{B}\left(\kappa \theta +\left(1-\kappa \right)\lambda \right)+\frac{1}{ħ\left(1-\kappa \right)}\mathfrak{B}\left(\left(1-\kappa \right)\theta +\kappa \lambda \right)\right\}\\ +{\mathcal{P}}_{*}\left(\lambda , \mathcal{ı}\right)\left\{\frac{1}{ħ\left(1-\kappa \right)}\mathfrak{B}\left(\kappa \theta +\left(1-\kappa \right)\lambda \right)+\frac{1}{ħ\left(\kappa \right)}\mathfrak{B}\left(\left(1-\kappa \right)\theta +\kappa \lambda \right)\right\}\end{array}\right]d\kappa ,\\ {\int }_0^1{\mathcal{P}}^{*}\left(\left(1-\kappa \right)\theta +\kappa \lambda , \mathcal{ı}\right)\mathfrak{B}\left(\left(1-\kappa \right)\theta +\kappa \lambda \right)d\kappa +{\int }_0^1{\mathcal{P}}^{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda , \mathcal{ı}\right)\mathfrak{B}\left(\kappa \theta +\left(1-\kappa \right)\lambda \right)d\kappa \\ \le {\int }_0^1\left[\begin{array}{c}{\mathcal{P}}^{*}\left(\theta , \mathcal{ı}\right)\left\{\frac{1}{ħ\left(\kappa \right)}\mathfrak{B}\left(\kappa \theta +\left(1-\kappa \right)\lambda \right)+\frac{1}{ħ\left(1-\kappa \right)}\mathfrak{B}\left(\left(1-\kappa \right)\theta +\kappa \lambda \right)\right\}\\ +{\mathcal{P}}^{*}\left(\lambda , \mathcal{ı}\right)\left\{\frac{1}{ħ\left(1-\kappa \right)}\mathfrak{B}\left(\kappa \theta +\left(1-\kappa \right)\lambda \right)+\frac{1}{ħ\left(\kappa \right)}\mathfrak{B}\left(\left(1-\kappa \right)\theta +\kappa \lambda \right)\right\}\end{array}\right]d\kappa ,\end{array} \\ \begin{array}{c} \\ =2{\mathcal{P}}_{*}\left(\theta , \mathcal{ı}\right){\int }_0^1\begin{array}{c}\frac{1}{ħ\left(\kappa \right)}B\left(\kappa \theta +\left(1-\kappa \right)\lambda \right)\end{array}d\kappa +2{\mathcal{P}}_{*}\left(\lambda , \mathcal{ı}\right){\int }_0^1\begin{array}{c}\frac{1}{ħ\left(\kappa \right)}B\left(\left(1-\kappa \right)\theta +\kappa \lambda \right)\end{array}d\kappa ,\\ =2{\mathcal{P}}^{*}\left(\theta , \mathcal{ı}\right){\int }_0^1\begin{array}{c}\frac{1}{ħ\left(\kappa \right)}B\left(\kappa \theta +\left(1-\kappa \right)\lambda \right)\end{array}d\kappa +2{\mathcal{P}}^{*}\left(\lambda , \mathcal{ı}\right){\int }_0^1\begin{array}{c}\frac{1}{ħ\left(\kappa \right)}B\left(\left(1-\kappa \right)\theta +\kappa \lambda \right)\end{array}d\kappa .\end{array} \end{gathered}$$

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

$$\begin{array}{c} \\ {\int }_0^1{\mathcal{P}}_{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda , \mathcal{ı}\right)\mathfrak{B}\left(\kappa \theta +\left(1-\kappa \right)\lambda \right)d\kappa \\ +{\int }_0^1{\mathcal{P}}_{*}\left(\left(1-\kappa \right)\theta +\kappa \lambda , \mathcal{ı}\right)\mathfrak{B}\left(\left(1-\kappa \right)\theta +\kappa \lambda \right)d\kappa \\ \le 2\left[{\mathcal{P}}_{*}\left(\theta , \mathcal{ı}\right)+{\mathcal{P}}_{*}\left(\lambda , \mathcal{ı}\right)\right]{\int }_0^1\begin{array}{c}\frac{1}{ħ\left(\kappa \right)}B\left(\left(1-\kappa \right)\theta +\kappa \lambda \right)\end{array}d\kappa ,\\ {\int }_0^1{\mathcal{P}}^{*}\left(\left(1-\kappa \right)\theta +\kappa \lambda , \mathcal{ı}\right)\mathfrak{B}\left(\left(1-\kappa \right)\theta +\kappa \lambda \right)d\kappa \\ +{\int }_0^1{\mathcal{P}}^{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda , \mathcal{ı}\right)\mathfrak{B}\left(\kappa \theta +\left(1-\kappa \right)\lambda \right)d\kappa \\ \le 2\left[{\mathcal{P}}^{*}\left(\theta , \mathcal{ı}\right)+{\mathcal{P}}^{*}\left(\lambda , \mathcal{ı}\right)\right]{\int }_0^1\begin{array}{c}\frac{1}{ħ\left(\kappa \right)}B\left(\left(1-\kappa \right)\theta +\kappa \lambda \right)\end{array}d\kappa .\end{array}$$

(45)

Since

$$\begin{array}{c} \\ {\int }_0^1{\mathcal{P}}_{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda , \mathcal{ı}\right)\mathfrak{B}\left(\kappa \theta +\left(1-\kappa \right)\lambda \right)d\kappa \\ ={\int }_0^1{\mathcal{P}}_{*}\left(\left(1-\kappa \right)\theta +\kappa \lambda , \mathcal{ı}\right)\mathfrak{B}\left(\left(1-\kappa \right)\theta +\kappa \lambda \right)d\kappa =\frac{1}{\lambda -\theta }{\int }_{\theta }^{\lambda }{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)\mathfrak{B}\left(\mathfrak{б}\right)d\mathsf{б} \\ {\int }_0^1{\mathcal{P}}^{*}\left(\left(1-\kappa \right)\theta +\kappa \lambda , \mathcal{ı}\right)\mathfrak{B}\left(\left(1-\kappa \right)\theta +\kappa \lambda \right)d\kappa \\ ={\int }_0^1{\mathcal{P}}^{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda , \mathcal{ı}\right)\mathfrak{B}\left(\kappa \theta +\left(1-\kappa \right)\lambda \right)d\kappa =\frac{1}{\lambda -\theta }{\int }_{\theta }^{\lambda }{\mathcal{P}}^{*}\left(\mathsf{б}, \mathcal{ı}\right)\mathfrak{B}\left(\mathfrak{б}\right)d\mathsf{б}\end{array}$$

(46)

Then, from (45) and (46), we have

$$\begin{array}{c} \\ \frac{1}{\lambda -\theta }{\int }_{\theta }^{\lambda }{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)\mathfrak{B}\left(\mathfrak{б}\right)d\mathsf{б}\le \left[{\mathcal{P}}_{*}\left(\theta , \mathcal{ı}\right)+{\mathcal{P}}_{*}\left(\lambda , \mathcal{ı}\right)\right]{\int }_0^1\begin{array}{c}\frac{1}{ħ\left(\kappa \right)}B\left(\left(1-\kappa \right)\theta +\kappa \lambda \right)\end{array}d\kappa , \\ \\ \frac{1}{\lambda -\theta }{\int }_{\theta }^{\lambda }{\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\mathfrak{B}\left(\mathfrak{б}\right)d\mathsf{б}\le \left[{\mathcal{P}}^{*}\left(\theta , \mathcal{ı}\right)+{\mathcal{P}}^{*}\left(\lambda , \mathcal{ı}\right)\right]{\int }_0^1\begin{array}{c}\frac{1}{ħ\left(\kappa \right)}B\left(\left(1-\kappa \right)\theta +\kappa \lambda \right)\end{array}d\kappa ,\end{array}$$

that is

$$\begin{gathered} \left[\frac{1}{\lambda -\theta }{\int }_{\theta }^{\lambda }{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)\mathfrak{B}\left(\mathsf{б}\right)d\mathsf{б}, \frac{1}{\lambda -\theta }{\int }_{\theta }^{\lambda }{\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\mathfrak{B}\left(\mathsf{б}\right)d\mathsf{б}\right] \\ {\le }_I\left[{\mathcal{P}}_{*}\left(\theta , \mathcal{ı}\right)+{\mathcal{P}}_{*}\left(\lambda , \mathcal{ı}\right), {\mathcal{P}}^{*}\left(\theta , \mathcal{ı}\right)+{\mathcal{P}}^{*}\left(\lambda , \mathcal{ı}\right)\right]{\int }_0^1\begin{array}{c}\frac{1}{ħ\left(\kappa \right)}B\left(\left(1-\kappa \right)\theta +\kappa \lambda \right)\end{array}d\kappa , \end{gathered}$$

hence

$$\frac{1}{\lambda -\theta }\odot \left(FA\right){\int }_{\theta }^{\lambda }\tilde{\mathcal{P}}\left(\mathsf{б}\right)\odot \mathfrak{B}\left(\mathfrak{б}\right)d\mathsf{б}{\le }_{\mathbb{F}}\left[\tilde{\mathcal{P}}\left(\theta \right)\oplus \tilde{\mathcal{P}}\left(\lambda \right)\right]\odot {\int }_0^1\frac{\mathfrak{B}\left(\left(1-\kappa \right)\theta +\kappa \lambda \right)}{ħ\left(\kappa \right)}d\kappa .$$

Now, generalizing the first H-H Fejér inequalities for classical Godunova–Levin convex functions, we build the first H-H Fejér inequality for an $ħ$-Godunova–Levin convex $F$·$N$·$M$. □

Theorem 11.

Let $\tilde{\mathcal{P}}:\left[\theta , \lambda \right]\to {\mathbb{E}}_C$ be an $ħ$-Godunova–Levin convex $F$·$N$·$M$ with $ħ:[0, 1]\to {\mathfrak{N}}^{+}$, and $\mathit{\text{Ι-V-M}}$s are classified according to their $\mathcal{ı}$-levels such that ${\mathcal{P}}_{\mathcal{ı}}:\left[\theta , \lambda \right]\subset \mathfrak{N}\to {\mathcal{X}}_I^{+}$ are given by ${\mathcal{P}}_{\mathcal{ı}}\left(\mathsf{б}\right)=\left[{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right), {\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\right]$ for all $\mathsf{б}\in \left[\theta , \lambda \right]$ and for all $\mathcal{ı}\in \left[0, 1\right]$. If $\tilde{\mathcal{P}}\in {\mathcal{F}\mathcal{R}}_{\left(\left[\theta , \lambda \right], \mathcal{ı}\right)}$ and $\mathfrak{B}:\left[\theta , \lambda \right]\to \mathfrak{N},\mathfrak{B}\left(\mathsf{б}\right)\ge 0,$ symmetric with respect to $\frac{\theta +\lambda }{2},$ and ${\int }_{\theta }^{\lambda }\mathfrak{B}\left(\mathfrak{б}\right)d\mathsf{б}>0$, then

$$\tilde{\mathcal{P}}\left(\frac{\theta +\lambda }{2}\right){\le }_{\mathbb{F}}\frac{2}{ħ\left(\frac{1}{2}\right){\int }_{\theta }^{\lambda }\mathfrak{B}\left(\mathfrak{б}\right)d\mathsf{б}}\odot \left(FA\right){\int }_{\theta }^{\lambda }\tilde{\mathcal{P}}\left(\mathsf{б}\right)\odot \mathfrak{B}\left(\mathfrak{б}\right)d\mathsf{б}.$$

(47)

Proof. 

Since $\tilde{\mathcal{P}}$ is an $ħ$-Godunova–Levin convex, then for $\mathcal{ı}\in \left[0, 1\right],$ we have

$$\begin{array}{c} \\ {\mathcal{P}}_{*}\left(\frac{\theta +\lambda }{2}, \mathcal{ı}\right)\le \frac{1}{ħ\left(\frac{1}{2}\right)}\left({\mathcal{P}}_{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda , \mathcal{ı}\right)+{\mathcal{P}}_{*}\left(\left(1-\kappa \right)\theta +\kappa \lambda , \mathcal{ı}\right)\right),\\ {\mathcal{P}}^{*}\left(\frac{\theta +\lambda }{2}, \mathcal{ı}\right)\le \frac{1}{ħ\left(\frac{1}{2}\right)}\left({\mathcal{P}}^{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda , \mathcal{ı}\right)+{\mathcal{P}}^{*}\left(\left(1-\kappa \right)\theta +\kappa \lambda , \mathcal{ı}\right)\right),\end{array}$$

(48)

Since $\mathfrak{B}\left(\kappa \theta +\left(1-\kappa \right)\lambda \right)=\mathfrak{B}\left(\left(1-\kappa \right)\theta +\kappa \lambda \right)$, then by multiplying (48) by $\mathfrak{B}\left(\left(1-\kappa \right)\theta +\kappa \lambda \right)$ and integrating it with respect to $\kappa$ over $\left[0, 1\right],$ we obtain

$$\begin{array}{c} \\ {\mathcal{P}}_{*}\left(\frac{\theta +\lambda }{2}, \mathcal{ı}\right){\int }_0^1\mathfrak{B}\left(\left(1-\kappa \right)\theta +\kappa \lambda \right)d\kappa \\ \le \frac{1}{ħ\left(\frac{1}{2}\right)}\left(\begin{array}{c}{\int }_0^1{\mathcal{P}}_{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda , \mathcal{ı}\right)\mathfrak{B}\left(\kappa \theta +\left(1-\kappa \right)\lambda \right)d\kappa \\ +{\int }_0^1{\mathcal{P}}_{*}\left(\left(1-\kappa \right)\theta +\kappa \lambda , \mathcal{ı}\right)\mathfrak{B}\left(\left(1-\kappa \right)\theta +\kappa \lambda \right)d\kappa \end{array}\right),\\ {\mathcal{P}}^{*}\left(\frac{\theta +\lambda }{2}, \mathcal{ı}\right){\int }_0^1\mathfrak{B}\left(\left(1-\kappa \right)\theta +\kappa \lambda \right)d\kappa \\ \le \frac{1}{ħ\left(\frac{1}{2}\right)}\left(\begin{array}{c}{\int }_0^1{\mathcal{P}}^{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda , \mathcal{ı}\right)\mathfrak{B}\left(\kappa \theta +\left(1-\kappa \right)\lambda \right)d\kappa \\ +{\int }_0^1{\mathcal{P}}^{*}\left(\left(1-\kappa \right)\theta +\kappa \lambda , \mathcal{ı}\right)\mathfrak{B}\left(\left(1-\kappa \right)\theta +\kappa \lambda \right)d\kappa \end{array}\right).\end{array}$$

(49)

Since

$$\begin{array}{c} \\ {\int }_0^1{\mathcal{P}}_{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda , \mathcal{ı}\right)\mathfrak{B}\left(\kappa \theta +\left(1-\kappa \right)\lambda \right)d\kappa \\ ={\int }_0^1{\mathcal{P}}_{*}\left(\left(1-\kappa \right)\theta +\kappa \lambda , \mathcal{ı}\right)\mathfrak{B}\left(\left(1-\kappa \right)\theta +\kappa \lambda \right)d\kappa \\ =\frac{1}{\lambda -\theta }{\int }_{\theta }^{\lambda }{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)\mathfrak{B}\left(\mathfrak{б}\right)d\mathsf{б} \\ \\ {\int }_0^1{\mathcal{P}}^{*}\left(\left(1-\kappa \right)\theta +\kappa \lambda , \mathcal{ı}\right)\mathfrak{B}\left(\left(1-\kappa \right)\theta +\kappa \lambda \right)d\kappa \\ ={\int }_0^1{\mathcal{P}}^{*}\left(\kappa \theta +\left(1-\kappa \right)\lambda , \mathcal{ı}\right)\mathfrak{B}\left(\kappa \theta +\left(1-\kappa \right)\lambda \right)d\kappa \\ =\frac{1}{\lambda -\theta }{\int }_{\theta }^{\lambda }{\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\mathfrak{B}\left(\mathfrak{б}\right)d\mathsf{б}. \\ \end{array}$$

(50)

And

$${\int }_0^1\mathfrak{B}\left(\left(1-\kappa \right)\theta +\kappa \lambda \right)d\kappa =\frac{1}{\lambda -\theta }{\int }_{\theta }^{\lambda }\mathfrak{B}\left(\mathfrak{б}\right)d\mathsf{б}$$

(51)

Then, from (50) and (51), (49) we have

$$\begin{array}{c} \\ {\mathcal{P}}_{*}\left(\frac{\theta +\lambda }{2}, \mathcal{ı}\right)\le \frac{2}{ħ\left(\frac{1}{2}\right){\int }_{\theta }^{\lambda }\mathfrak{B}\left(\mathsf{б}\right)d\mathsf{б}}{\int }_{\theta }^{\lambda }{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)\mathfrak{B}\left(\mathsf{б}\right)d\mathsf{б}, \\ {\mathcal{P}}^{*}\left(\frac{\theta +\lambda }{2}, \mathcal{ı}\right)\le \frac{2}{ħ\left(\frac{1}{2}\right){\int }_{\theta }^{\lambda }\mathfrak{B}\left(\mathsf{б}\right)d\mathsf{б}}{\int }_{\theta }^{\lambda }{\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\mathfrak{B}\left(\mathsf{б}\right)d\mathsf{б},\end{array}$$

from which, we have

$$\begin{array}{c} \\ \left[{\mathcal{P}}_{*}\left(\frac{\theta +\lambda }{2}, \mathcal{ı}\right), {\mathcal{P}}^{*}\left(\frac{\theta +\lambda }{2}, \mathcal{ı}\right)\right] \\ {\le }_I\frac{2}{ħ\left(\frac{1}{2}\right){\int }_{\theta }^{\lambda }\mathfrak{B}\left(\mathsf{б}\right)d\mathsf{б}}\left[{\int }_{\theta }^{\lambda }{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)\mathfrak{B}\left(\mathsf{б}\right)d\mathsf{б}, {\int }_{\theta }^{\lambda }{\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\mathfrak{B}\left(\mathsf{б}\right)d\mathsf{б}\right], \end{array}$$

that is

$$\tilde{\mathcal{P}}\left(\frac{\theta +\lambda }{2}\right){\le }_{\mathbb{F}}\frac{2}{ħ\left(\frac{1}{2}\right){\int }_{\theta }^{\lambda }\mathfrak{B}\left(\mathfrak{б}\right)d\mathsf{б}}\odot \left(FA\right){\int }_{\theta }^{\lambda }\tilde{\mathcal{P}}\left(\mathsf{б}\right)\odot \mathfrak{B}\left(\mathfrak{б}\right)d\mathsf{б}.$$

This completes the proof. □

Remark 6.

From Theorem 10 and 11, we clearly see the following:

If $\mathfrak{B}\left(\mathsf{б}\right)=1$, then we acquire the inequality (24).

Let ${\mathcal{P}}_{*}(\mathsf{б}, \mathcal{ı})={\mathcal{P}}^{*}(\mathsf{б}, \mathcal{ı})$ with $\mathcal{ı}=1$ and $ħ\left(\kappa \right)=\frac{1}{\kappa }$. Then, from (42) and (47), we acquire the following inequality; see [48]:

$$\mathcal{P}\left(\frac{\theta +\lambda }{2}\right)\le \frac{1}{{\int }_{\theta }^{\lambda }\mathfrak{B}\left(\mathfrak{б}\right)d\mathsf{б}}(IA){\int }_{\theta }^{\lambda }\mathcal{P}\left(\mathsf{б}\right)\mathfrak{B}\left(\mathfrak{б}\right)d\mathsf{б}\le \frac{\mathcal{P}\left(\theta \right)+\mathcal{P}\left(\lambda \right)}{2}$$

(52)

If $\mathcal{P}$ is a lower Godunova–Levin convex $F$·$N$·$M$ on $[\theta ,\lambda ]$ and $ħ\left(\kappa \right)=\frac{1}{\kappa }$, then we derive the following subsequent inequality; see [41]:

$$\tilde{\mathcal{P}}\left(\frac{\theta +\lambda }{2}\right){\le }_{\mathbb{F}}\frac{1}{{\int }_{\theta }^{\lambda }\mathfrak{B}\left(\mathfrak{б}\right)d\mathsf{б}}\odot \left(FA\right){\int }_{\theta }^{\lambda }\tilde{\mathcal{P}}\left(\mathsf{б}\right)\odot \mathfrak{B}\left(\mathfrak{б}\right)d\mathsf{б}{\le }_{\mathbb{F}}\frac{\tilde{\mathcal{P}}\left(\theta \right)\oplus \tilde{\mathcal{P}}\left(\lambda \right)}{2}$$

(53)

If $\mathcal{P}$ is a lower Godunova–Levin convex $F$·$N$·$M$ on $[\theta ,\lambda ]$ with $\mathcal{ı}=1$ and $ħ\left(\kappa \right)=\frac{1}{\kappa }$, then from (42) and (47), we derive the following subsequent inequality; see [42]:

$$\mathcal{P}\left(\frac{\theta +\lambda }{2}\right){\le }_I\frac{1}{\lambda -\theta }\left(IA\right){\int }_{\theta }^{\lambda }\mathcal{P}\left(\mathsf{б}\right)d\mathsf{б}{\le }_I\frac{\mathcal{P}\left(\theta \right)+\mathcal{P}\left(\lambda \right)}{2}$$

(54)

If $\mathcal{P}$ is a lower Godunova–Levin convex $F$·$N$·$M$ on $[\theta ,\lambda ]$ with $\mathcal{ı}=1$ and $ħ\left(\kappa \right)=\frac{1}{\kappa }$, then from (42) and (47), we derive the following subsequent inequality; see [42]:

$$\mathcal{P}\left(\frac{\theta +\lambda }{2}\right){\le }_I\frac{1}{{\int }_{\theta }^{\lambda }\mathfrak{B}\left(\mathfrak{б}\right)d\mathsf{б}}\left(IA\right){\int }_{\theta }^{\lambda }\mathcal{P}\left(\mathsf{б}\right)\mathfrak{B}\left(\mathfrak{б}\right)d\mathsf{б}{\le }_I\frac{\mathcal{P}\left(\theta \right)+\mathcal{P}\left(\lambda \right)}{2}$$

(55)

Let $ħ\left(\kappa \right)=\frac{1}{\kappa }$ and ${\mathcal{P}}_{*}(\mathsf{б}, \mathcal{ı})={\mathcal{P}}^{*}(\mathsf{б}, \mathcal{ı})$ with $\mathcal{ı}=1$. Then, from (42) and (47), we obtain the following classical Fejér inequality:

$$\mathcal{P}\left(\frac{\theta +\lambda }{2}\right)\le \frac{1}{{\int }_{\theta }^{\lambda }\mathfrak{B}\left(\mathfrak{б}\right)d\mathsf{б}}{\int }_{\theta }^{\lambda }\mathcal{P}\left(\mathsf{б}\right)\mathfrak{B}\left(\mathfrak{б}\right)d\mathsf{б}\le \frac{\mathcal{P}\left(\theta \right)+\mathcal{P}\left(\lambda \right)}{2}$$

(56)

Example 7.

We consider $ħ\left(\kappa \right)=\frac{1}{\kappa },$ for $\kappa \in \left[0, 1\right]$, and the $F$·$N$·$M$ $\mathcal{P}:\left[0, 2\right]\to {\mathbb{E}}_C$ defined by

$$\mathcal{P}\left(\mathsf{б}\right)\left(\theta \right)=\left\{\begin{array}{c}\frac{\theta -2+{\mathsf{б}}^{\frac{1}{2}}}{\frac{3}{2}-2-{\mathsf{б}}^{\frac{1}{2}}} \theta \in \left[2-{\mathsf{б}}^{\frac{1}{2}}, \frac{3}{2}\right]\\ \frac{ 2+{\mathsf{б}}^{\frac{1}{2}}-\theta }{2+{\mathsf{б}}^{\frac{1}{2}}-\frac{3}{2}} \theta \in (\frac{3}{2}, 2+{\mathsf{б}}^{\frac{1}{2}}] \\ 0 \mathrm{otherwise},\end{array}\right.$$

Then, for each $\mathcal{ı}\in \left[0, 1\right],$ we have ${\mathcal{P}}_{\mathcal{ı}}\left(\mathsf{б}\right)=\left[\left(1-\mathcal{ı}\right)\left(2-{\mathsf{б}}^{\frac{1}{2}}\right)+\frac{3}{2}\mathcal{ı},\left(1-\mathcal{ı}\right)\left(2+{\mathsf{б}}^{\frac{1}{2}}\right)+\frac{3}{2}\mathcal{ı}\right]$. Since end-point mappings ${\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right),$ ${\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)$ are $ħ$-Godunova–Levin convex mappings for each $\mathcal{ı}\in [0, 1]$, $\mathcal{P}\left(\mathsf{б}\right)$ is an $ħ$-Godunova–Levin convex $F$·$N$·$M$. If

$$\mathfrak{B}\left(\mathsf{б}\right)=\left\{\begin{array}{c}\sqrt{\mathsf{б}}, \sigma \in \left[0, 1\right],\\ \sqrt{2-\mathsf{б}}, \sigma \in \left(1, 2\right], \end{array}\right.$$

then $\mathfrak{B}\left(2-\mathrm{б}\right)=\mathfrak{B}\left(\mathrm{б}\right)\ge 0$, for all $\mathrm{б}\in \left[0,2\right]$. ${\mathcal{P}}_{\mathrm{*}}\left(\mathrm{б},\mathcal{ı}\right)=\left(1-\mathcal{ı}\right)\left(2-{\mathrm{б}}^{\frac{1}{2}}\right)+\frac{3}{2}\mathcal{ı}$ and ${\mathcal{P}}^{\mathrm{*}}\left(\mathrm{б},\mathcal{ı}\right)=\left(1-\mathcal{ı}\right)\left(2+{\mathrm{б}}^{\frac{1}{2}}\right)+\frac{3}{2}\mathcal{ı}$. Now, we compute the following:

$$\begin{gathered} \begin{array}{c} \frac{1}{\lambda -\theta }{\int }_{\theta }^{\lambda }\left[{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)\right]\mathfrak{B}\left(\mathsf{б}\right)d\mathsf{б}=\frac{1}{2}{\int }_0^2\left[{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)\right]\mathfrak{B}\left(\mathsf{б}\right)d\mathsf{б}=\frac{1}{2}{\int }_0^1\left[{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)\right]\mathfrak{B}\left(\mathsf{б}\right)d\mathsf{б}+\frac{1}{2}{\int }_1^2{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)\mathfrak{B}\left(\mathsf{б}\right)d\mathsf{б},\\ \frac{1}{\lambda -\theta }{\int }_{\theta }^{\lambda }\left[{\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\right]\mathfrak{B}\left(\mathsf{б}\right)d\mathsf{б}=\frac{1}{2}{\int }_0^2\left[{\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\right]\mathfrak{B}\left(\mathsf{б}\right)d\mathsf{б}=\frac{1}{2}{\int }_0^1\left[{\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\right]\mathfrak{B}\left(\mathsf{б}\right)d\mathsf{б}+\frac{1}{2}{\int }_1^2{\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\mathfrak{B}\left(\mathsf{б}\right)d\mathsf{б},\end{array} \\ \begin{array}{c}=\frac{1}{2}{\int }_0^1\left[\left(1-\mathcal{ı}\right)\left(2-{\mathsf{б}}^{\frac{1}{2}}\right)+\frac{3}{2}\mathcal{ı}\right]\left(\sqrt{\mathsf{б}}\right)d\mathsf{б}+\frac{1}{2}{\int }_1^2\left[\left(1-\mathcal{ı}\right)\left(2-{\mathsf{б}}^{\frac{1}{2}}\right)+\frac{3}{2}\mathcal{ı}\right]\left(\sqrt{2-\mathsf{б}}\right)d\mathsf{б}\\ =\frac{\mathcal{ı}+5}{12}+\frac{1}{24}\left[3\pi \left(\mathcal{ı}-1\right)-4\left(\mathcal{ı}-4\right)\right], \\ =\frac{1}{2}{\int }_0^1\left[\left(1-\mathcal{ı}\right)\left(2+{\mathsf{б}}^{\frac{1}{2}}\right)+\frac{3}{2}\mathcal{ı}\right]\left(\sqrt{\mathsf{б}}\right)d\mathsf{б}+\frac{1}{2}{\int }_1^2\left[\left(1-\mathcal{ı}\right)\left(2+{\mathsf{б}}^{\frac{1}{2}}\right)+\frac{3}{2}\mathcal{ı}\right]\left(\sqrt{2-\mathsf{б}}\right)d\mathsf{б}\\ =\frac{1}{12}\left[11-5\mathcal{ı}\right]+\frac{1}{24}\left[-3\pi \mathcal{ı}-4\mathcal{ı}+3\pi +16\right]. \end{array} \end{gathered}$$

(57)

And

$$\begin{array}{c}\left[{\mathcal{P}}_{*}\left(\theta , \mathcal{ı}\right)+{\mathcal{P}}_{*}\left(\lambda , \mathcal{ı}\right)\right]{\int }_0^1\begin{array}{c}\frac{\mathfrak{B}\left(\left(1-\kappa \right)\theta +\kappa \lambda \right)}{ħ\left(\kappa \right)}\end{array}d\kappa \\ =\left[4\left(1-\mathcal{ı}\right)-\sqrt{2}\left(1-\mathcal{ı}\right)+3\mathcal{ı}\right]\left[{\int }_0^{\frac{1}{2}}\kappa \sqrt{2\kappa }d\kappa +{\int }_{\frac{1}{2}}^1\kappa \sqrt{2\left(1-\kappa \right)}d\kappa \right]\\ =\frac{1}{3}\left(4\left(1-\mathcal{ı}\right)-\sqrt{2}\left(1-\mathcal{ı}\right)+3\mathcal{ı}\right), \\ \left[{\mathcal{P}}^{*}\left(\theta , \mathcal{ı}\right)+{\mathcal{P}}^{*}\left(\lambda , \mathcal{ı}\right)\right]{\int }_0^1\frac{\mathfrak{B}\left(\left(1-\kappa \right)\theta +\kappa \lambda \right)}{ħ\left(\kappa \right)}d\kappa \\ =\left[4\left(1-\mathcal{ı}\right)+\sqrt{2}\left(1-\mathcal{ı}\right)+3\mathcal{ı}\right]\left[{\int }_0^{\frac{1}{2}}\kappa \sqrt{2\kappa }d\kappa +{\int }_{\frac{1}{2}}^1\kappa \sqrt{2\left(1-\kappa \right)}d\kappa \right]\\ =\frac{1}{3}\left(4\left(1-\mathcal{ı}\right)+\sqrt{2}\left(1-\mathcal{ı}\right)+3\mathcal{ı}\right). \end{array}$$

(58)

From (57) and (58), we have

$$\begin{gathered} \left[\frac{\mathcal{ı}+5}{12}+\frac{1}{24}\left[3\pi \left(\mathcal{ı}-1\right)-4\left(\mathcal{ı}-4\right)\right],\frac{1}{12}\left[11-5\mathcal{ı}\right]+\frac{1}{24}\left[-3\pi \mathcal{ı}-4\mathcal{ı}+3\pi +16\right]\right] \\ {\supseteq }_I\left[\frac{1}{3}\left(4\left(1-\mathcal{ı}\right)-\sqrt{2}\left(1-\mathcal{ı}\right)+3\mathcal{ı}\right), \frac{1}{3}\left(4\left(1-\mathcal{ı}\right)+\sqrt{2}\left(1-\mathcal{ı}\right)+3\mathcal{ı}\right)\right], \end{gathered}$$

for all $\mathcal{ı}\in \left[0,1\right].$

Hence, Theorem 10 is verified.

For Theorem 11, we have

$$\begin{array}{c}{\mathcal{P}}_{*}\left(\frac{\theta +\lambda }{2}, \mathcal{ı}\right)={\mathcal{P}}_{*}\left(1, \mathcal{ı}\right)=\frac{2+\mathcal{ı}}{2}, \\ {\mathcal{P}}^{*}\left(\frac{\theta +\lambda }{2}, \mathcal{ı}\right)={\mathcal{P}}^{*}\left(1, \mathcal{ı}\right)=\frac{3\left(2-\mathcal{ı}\right)}{2},\end{array}$$

(59)

$${\int }_{\theta }^{\lambda }\mathfrak{B}\left(\mathsf{б}\right)d\mathsf{б}={\int }_0^{\begin{array}{c} \\ 1\end{array}}\sqrt{\mathsf{б}}d\mathsf{б}+{\int }_1^2\sqrt{2-\mathsf{б}}d\mathsf{б}=\frac{4}{3},$$

$$\begin{array}{c} \\ \frac{2}{ħ\left(\frac{1}{2}\right){\int }_{\theta }^{\lambda }\mathfrak{B}\left(\mathsf{б}\right)d\mathsf{б}}{\int }_{\theta }^{\lambda }{\mathcal{P}}_{*}\left(\mathsf{б},\mathcal{ı}\right)\mathfrak{B}\left(\mathsf{б}\right)d\mathsf{б}=\frac{3}{4}\left[\frac{13-\mathcal{ı}}{6}+\frac{\pi \left(\mathcal{ı}-1\right)}{4}\right], \\ \frac{2}{ħ\left(\frac{1}{2}\right){\int }_{\theta }^{\lambda }\mathfrak{B}\left(\mathsf{б}\right)d\mathsf{б}}{\int }_{\theta }^{\lambda }{\mathcal{P}}^{*}\left(\mathsf{б},\mathcal{ı}\right)\mathfrak{B}\left(\mathsf{б}\right)d\mathsf{б}=\frac{3}{24}\left[\left[11-5\mathcal{ı}\right]+\frac{1}{2}\left[-3\pi \mathcal{ı}-4\mathcal{ı}+3\pi +16\right]\right].\\ \end{array}$$

(60)

From (59) and (60), we have

$$\left[\frac{2+\mathcal{ı}}{2}, \frac{3\left(2-\mathcal{ı}\right)}{2}\right]{\supseteq }_I\left[\frac{3}{4}\left[\frac{13-\mathcal{ı}}{6}+\frac{\pi \left(\mathcal{ı}-1\right)}{4}\right], \frac{3}{24}\left[\left[11-5\mathcal{ı}\right]+\frac{1}{2}\left[-3\pi \mathcal{ı}-4\mathcal{ı}+3\pi +16\right]\right]\right].$$

Hence, Theorem 11 has been verified.

7. Conclusions

The $ħ$-Godunova–Levin convex $F$·$N$·$M$ and fuzzy Aumann integral operators represent pivotal tools in addressing Hermite–Hadamard inequality problems. In this paper, we introduce a more comprehensive Godunova–Levin convex $F$·$N$·$M$, which extends several well-known classical convexities. The refinements of Milne, Ostrowski, and Hermite–Hadamard-type inequalities are introduced over $ħ$-Godunova–Levin convex $F$·$N$·$M$s. Moreover, some exceptional cases are also achieved. Nontrivial numerical examples are also given for the validation of main outcomes. Interested researchers can explore this article concept in fuzzy fractional calculus. In future, we will try to explore this concept for fuzzy fractional calculus and fuzzy q-fractional calculus.