The Hadamard and Generalized Fractional Integral Fuzzy-Number-Valued Operators for Mappings of One and Two Variables, and Their Related Fuzzy Number Inequalities

Abstract

In this study, we introduce new versions of fuzzy fractional integral operators for both one- and two-variable cases. Using these operators, several Hermite–Hadamard-type ($H$-type) inclusions are established for fuzzy-number-valued convex functions ($F$·$N\cdot V$-functions) and $F$·$N\cdot V$-coordinated convex functions. These results are obtained by employing $F$·$N\cdot V$-weighted functions within the framework of the newly defined Hadamard and generalized fractional integrals in one- and two-dimensional settings. The use of generalized fractional integral operators provides a unified approach that encompasses a wide class of classical and modern fractional integrals, including the fuzzy Riemann–Liouville and Hadamard types. This unified setting enables the derivation of more comprehensive and flexible inequality results in the fuzzy-number context. The inclusions obtained in this work significantly extend and generalize several known $H\cdot H$-type inequalities previously established for real-valued and interval-valued functions ($I\cdot V$-functions). Furthermore, the proposed results yield a variety of meaningful special cases by specifying suitable kernel functions and parameters of the generalized fractional integrals. In particular, we derive new weighted $H\cdot H$-type inclusions involving logarithmic functions in the fuzzy-number framework. These findings underscore the effectiveness of generalized fractional integrals in capturing nonlocal behavior and uncertainty, and they provide new tools for further investigations in fuzzy analysis, fractional calculus, and generalized convexity.

1. Introduction

One of the most fundamental results in the theory of convex functions is the $H\cdot H$-inequality, originally introduced by Charles Hermite and Jacques Hadamard; see [1,2]. This inequality has attracted considerable attention due to its elegant structure, clear geometrical interpretation, and wide range of applications in mathematical analysis, optimization, and related fields. Specifically, let $I\subset R$ be an interval and let $\mathcal{o},\mathcal{l}\in I$ with $\mathcal{o}<\mathcal{l}$. If $\mathfrak{S}:I\subset R\to [0,1]$ is a convex function, then the $H\cdot H$-inequality states that the value of the function at the midpoint of the interval is bounded above and below by the integral mean of the function and the arithmetic mean of its endpoint values, namely,

$$\mathfrak{S}({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}})\le {\displaystyle \frac{1}{\mathcal{l}-\mathcal{o}}}{\int }_{\mathcal{o}}^{\mathcal{l}}\mathfrak{S}(\mathcal{k})d\mathcal{k}\le {\displaystyle \frac{\mathfrak{S}(\mathcal{o})+\mathfrak{S}(\mathcal{l})}{2}}.$$

(1)

This inequality provides a useful estimate for the integral average of a convex function and serves as a powerful analytical tool. Owing to its significance, numerous generalizations and extensions have been developed, including variants for different classes of convexity, multidimensional settings, and integral operators. These developments continue to play an important role in advancing both theoretical results and practical applications. When the function $\mathfrak{S}$ is concave, the $H\cdot H$-inequality holds with the reverse ordering of inequalities. Over the years, this classical result has motivated extensive research, and numerous authors have established various forms and extensions of $H\cdot H$-type inequalities under different assumptions. It is worth noting that the $H\cdot H$-inequality can be derived directly from Jensen’s inequality, and thus may be viewed as a refined manifestation of the concept of convexity.

In recent decades, renewed interest in the $H\cdot H$-inequality for convex functions has led to a broad spectrum of refinements, improvements, and generalizations. These include extensions to different classes of convexity, alternative integral operators, and multidimensional settings, as documented in a growing body of literature (see, for example, [3,4]). Such developments highlight the continuing relevance of this inequality in modern mathematical analysis.

A significant role in these investigations is played by interval analysis, which provides a systematic framework for handling uncertainty in mathematical modeling and computer-assisted computations. Although the origins of interval-based reasoning can be traced back [5,6] to early mathematical works, such as those of Archimedes on the quadrature of the circle, interval analysis as a formal and structured theory did not gain substantial attention until the mid-twentieth century. Since then, it has become an important tool in both theoretical and applied mathematics, particularly in the study of inequalities, numerical analysis, and uncertainty quantification.

In recent years, interval analysis has become an important mathematical tool for dealing with uncertainties and rounding errors in numerical computations. It provides a reliable framework for representing imprecise quantities by intervals and for performing rigorous numerical calculations. Over the past decades, significant theoretical developments and practical applications of interval analysis have been reported in the literature. Fundamental concepts and computational techniques of interval arithmetic and interval computations were systematically studied in classical works on interval methods and their applications [7,8]. Furthermore, interval analysis has been successfully applied to global optimization problems and the reliable solution of nonlinear systems of equations [9]. Its applications have also been extended to various engineering and scientific problems, where interval methods provide guaranteed bounds for uncertain parameters and measurements [10]. In addition, interval arithmetic plays a crucial role in computer arithmetic and reliable numerical computation, ensuring accurate error control in scientific computing [11]. The development of extended interval spaces has further enriched the theoretical framework of interval analysis and opened new directions for mathematical modeling and analysis [12]. These advances have naturally led to the broader acceptance of absolute inequalities and interval-based approaches in modern mathematical analysis. As a natural generalization of $I\cdot V$-analysis, fuzzy-number theory provides a more flexible and expressive framework for describing uncertainty by incorporating degrees of membership rather than sharp bounds. Consequently, the study of inequalities for fuzzy-number-valued and set-valued functions has attracted growing interest. In this context, a $H\cdot H$-type inequality for set-valued functions, which generalizes $I\cdot V$-functions, was established by Sadowska, see [13]. Subsequently, several authors extended these ideas to fuzzy-number-valued functions, thereby enriching the theory and widening its range of applications.

Moreover, the introduction of generalized fractional integral operators has played a crucial role in the advancement of $H\cdot H$-type inequalities within the fuzzy setting. In particular, new $H\cdot H$-type inequalities involving fractional integrals in relation to another function were obtained by Jleli and Samet [14], providing a unified approach that encompasses several classical fractional operators. Furthermore, fractional integrals of one function in relation to another for $I\cdot V$-functions were first presented by Tunç [15], laying the groundwork for subsequent extensions to fuzzy-number-valued functions. These developments strongly motivate the present study, which focuses on establishing $H\cdot H$-type inequalities for coordinated convex fuzzy-number-valued functions involving generalized fractional integrals.

A unified framework that encompasses both the Riemann–Liouville and Hadamard fractional integrals was introduced through the novel fractional integral proposed by Katugampola. This generalization has proved to be highly effective in deriving new analytical results and has stimulated further developments in fractional inequality theory. Using generalized fractional integrals and classical integrals, Zhao et al. [16] and Budak and Agarwal [17] established $H\cdot H$-type inequalities for coordinated convex functions, thereby unifying and extending several important fractional integral operators, including the Riemann–Liouville, Hadamard, and Katugampola fractional integrals. Subsequently, $H\cdot H$-inequalities involving Riemann–Liouville fractional integrals [18] were investigated for $I\cdot V$-functions by Budak and coauthors [19], further highlighting the applicability of fractional operators in interval analysis. In a related direction, $I$.$V$-left- and -right-sided generalized fractional double integrals were introduced by Kara and collaborators [20], providing a broader setting for studying inequalities involving multivariable $I\cdot V$-functions. The concept of $I\cdot V$-convexity has also been extensively explored in the literature. In addition to classical $I\cdot V$-convex functions, several generalized convexity notions—such as $I\cdot V$-$L\cdot R$-convex functions—have been examined to obtain sharper and more flexible inequality results. On the other hand, the theory of fuzzy sets was first introduced by Zadeh to deal with uncertainty and vagueness in mathematical models [21]. Since then, fuzzy set theory has become an important tool in various branches of mathematics and applied sciences. In particular, the concept of metric spaces of fuzzy sets has been extensively studied, providing a fundamental framework for analyzing fuzzy-valued functions and their applications [22]. Moreover, the study of fuzzy differential equations has attracted considerable attention due to its significant role in modeling dynamic systems under uncertainty. Several researchers have contributed to the development of this field, including the pioneering work on fuzzy differential equations and their solutions [23,24]. Furthermore, the concept of fuzzy quasilinear spaces and their applications has provided a useful structure for studying fuzzy-valued mappings and related mathematical problems [25]. In addition, the theory of fuzzy integrals for set-valued and fuzzy mappings has been developed to extend classical integration concepts to fuzzy environments [26]. These developments have laid the foundation for further research in fuzzy analysis and its applications in various mathematical models. These studies form an important foundation for extending $H\cdot H$-type inequalities to fuzzy-number-valued functions—see [27]—and coordinated convexity frameworks involving fuzzy fractional integrals [28], which constitute the main focus of the present work. In recent years, several researchers have introduced and investigated different notions of $L\cdot R$-convexity for $I\cdot V$-functions [29], thereby broadening the classical concept of $I\cdot V$-convexity (see, for example, [30,31]). Within this generalized framework, a variety of $H\cdot H$-type inequalities have been established for $L\cdot R$-$I\cdot V$-convex functions, demonstrating the flexibility and effectiveness of $L\cdot R$-convexity in inequality analysis.

Following the extension of the $H\cdot H$-inequality proposed by Zabandan [32] in 2020, further developments were achieved for fractional and logarithmic integral operators. These results significantly enriched the theory by incorporating nonlocal operators and logarithmic kernels. In a related direction, important $H\cdot H$-type inequalities for coordinated convex functions were derived by Sarıkaya and Kılıçer [33], highlighting the role of coordinated convexity in multivariable settings. Moreover, several authors have extended $H\cdot H$-type inequalities to the setting of Riemann–Liouville fractional integrals combined with logarithmic integral operators. These extensions have further deepened the understanding of fractional inequalities and have opened new avenues for research in $I\cdot V$ and fuzzy-number-valued analysis. Such advances provide strong motivation for the present study, which aims to develop new $H\cdot H$-type inequalities for coordinated convex fuzzy-number-valued functions involving generalized fractional integrals. The following inequality is known as fuzzy-fractional $H\cdot H$-type inequality, such that, as discussed in [34],

$$\tilde{\mathfrak{S}}({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}}){\le }_F{\displaystyle \frac{\mathsf{\Gamma }(⋌+1)}{2[\mathcal{l}-\mathcal{o}{]}^{⋌}}}\odot \left\{{\mathcal{J}}_{{\mathcal{o}}^{+}}^{⋌}\tilde{\mathfrak{S}}(\mathcal{l})\oplus {\mathcal{J}}_{{\mathcal{l}}^{-}}^{⋌}\tilde{\mathfrak{S}}(\mathcal{o})\right\}{\le }_F{\displaystyle \frac{\tilde{\mathfrak{S}}(\mathcal{o})+\tilde{\mathfrak{S}}(\mathcal{l})}{2}}.$$

(2)

where $\tilde{\mathfrak{S}}$ is a fuzzy-number-valued convex function over $\left[\mathcal{o},\mathcal{l}\right].$

The principal aim of this paper is to establish several $H\cdot H$-type inclusions for $I\cdot V$-convex functions and $I\cdot V$-coordinated convex functions by employing $I\cdot V$-weighted functions. The results obtained in this study not only unify but also extend earlier findings reported in the literature, particularly those presented in [27,28].

The paper is organized as follows. Section 2 recalls the fundamental definitions, properties, and preliminary results related to $I\cdot V$-functions that are essential for the subsequent analysis. In Section 3, we present a concise overview of fractional integrals for $I\cdot V$-functions involving one and two variables. Section 4 is devoted to the derivation of weighted $H\cdot H$-type inclusions for $I\cdot V$-convex functions, where several known results are recovered as special cases. Finally, in the context of coordinated convex functions, we establish new and significant $H\cdot H$-type inclusions, thereby further enriching the existing theory.

2. Preliminary Concepts

Let $\tilde{A}$ be a fuzzy set in a nonempty set $R$ with the membership function

$${\eta }_{\tilde{A}}:R\to [0,1]$$

(3)

For any $\mu \in (0, 1]$, the $\mu$-cut (or $\mu$-level set) of $\tilde{A}$ is defined as the crisp set

$${\left[\tilde{A}\right]}_{\mu }=\left\{\mathcal{k}\in R| \tilde{\eta }(\mathcal{k})\ge \mu \right\}$$

(4)

Additionally, the 0-cut of $\tilde{A}$ is defined as

$${\left[\tilde{A}\right]}_0=\overline{\left\{\mathcal{k}\in R| \tilde{\eta }(\mathcal{k})\ge 0\right\}}$$

(5)

where the bar denotes the closure of the set.

The family of $\mu$-cuts $\left\{{\left[\tilde{A}\right]}_{\mu }| \mu \in [0, 1]\right\}$ provides a complete characterization of the fuzzy set $\tilde{A}$ and plays a fundamental role in fuzzy analysis, particularly in the definition of fuzzy numbers, fuzzy arithmetic, and fuzzy-valued integrals.

Note that the following four conditions are standard and essential for characterizing a fuzzy number. Normality ensures the existence of at least one element with full membership, convexity guarantees that intermediate values are represented consistently, upper semi-continuity ensures well-behaved membership functions, and compact support guarantees boundedness. Together, these conditions ensure that a fuzzy number is mathematically well-defined and suitable for analysis and applications.

Definition 1

([21]). A fuzzy number ($F$·$N$) is a normal, convex, upper semi-continuous fuzzy set on $R$ with compact support, whose μ-cuts are nonempty closed bounded intervals for all $\mu \in (0, 1]$. The set of all $F$·$N$s is denoted by ${\mathbb{F}}_0$.

Theorem 1

([22]). A fuzzy set is if $\tilde{\eta }:R\to [0,1]$ is a $F$·$N$ if, and only if, there exist two real-valued functions.

$${\eta }_{*}, {\eta }^{*}:[0,1]\to R$$

such that, for every  $\mu \in (0, 1]$, the $\mu$-cut of $\tilde{\eta }$ is given by

$$\left[{\eta }_{*}(\mu ),{\eta }^{*}(\mu ) \right].$$

Moreover, the endpoint functions ${\eta }_{*}(\mu ),{\eta }^{*}(\mu )$ satisfy the following conditions:

• ${\eta }_{*}(\mu )$ is bounded, non-decreasing, and left-continuous on $(0, 1]$;
• ${\eta }^{*}(\mu )$ is bounded, non-increasing, and left-continuous on $(0, 1]$;
• ${\eta }_{*}(\mu )\le {\eta }^{*}(\mu )$ for all $\mu \in (0, 1]$.

Conversely, any pair of functions $({\eta }_{*}(\mu ),{\eta }^{*}(\mu ) )$ satisfying conditions (1)–(3) uniquely determines a $F$·$N$ “$\tilde{\eta }$“.

Let $\tilde{\eta },õ\in {\mathbb{F}}_0$ represented parametrically $\left\{({\eta }_{*}(\mu ),{\eta }^{*}(\mu )):\mu \in \left[0, 1\right]\right\}$ and $\left\{({\mathrm{o}}_{*}(\mu ),{\mathrm{o}}^{*}(\mu )):\mu \in \left[0, 1\right]\right\}$, respectively. We say that $\tilde{\eta }{\le }_{\mathbb{F}}õ$ if for all $\mu \in (0, 1]$, ${\eta }^{*}(\mu )\le {\mathrm{o}}^{*}(\mu )$, and ${\eta }_{*}(\mu )\le {\mathrm{o}}_{*}(\mu )$ or $\left[{\eta }_{*}(\mu ),{\eta }^{*}(\mu ) \right]{\le }_I\left[{\mathrm{o}}_{*}(\mu ),{\mathrm{o}}^{*}(\mu )\right]$, where ${\le }_I$ is a partial-ordered relation between the intervals. If $\tilde{\eta }{\le }_{\mathbb{F}}õ,$ then there exists $\mu \in (0, 1]$ such that ${\eta }^{*}(\mu )<{\mathrm{o}}^{*}(\mu )$ or ${\eta }_{*}(\mu )\le {\mathrm{o}}_{*}(\mu )$. The fuzzy numbers $\tilde{\eta }$ and $õ$ are said to be comparable in relation to the partial order ${\le }_{\mathbb{F}}$ if either $\tilde{\eta }{\le }_{\mathbb{F}}õ$ or $\tilde{\eta }{\ge }_{\mathbb{F}}õ$ holds. Otherwise, they are termed non-comparable. The set ${\mathbb{F}}_0$ forms a partially ordered set under the relation ${\le }_{\mathbb{F}}$. In what follows, we may equivalently write $\tilde{\eta }{\le }_{\mathbb{F}}õ$ instead of $õ{\ge }_{\mathbb{F}}\tilde{\eta }$ whenever convenient.

Let $\tilde{\eta },õ\in {\mathbb{F}}_0$. If there exists a fuzzy number $\tilde{\delta }\in {\mathbb{F}}_0$ such that

$$\tilde{\eta }=õ\oplus \tilde{\delta },$$

(6)

where $\oplus$ denotes fuzzy addition, then the generalized Hukuhara ($\mathcal{g}\mathcal{H}$) difference between $\tilde{\eta }$ and õ is said to exist. In that sense, $\tilde{\delta }$ is called the $\mathcal{g}\mathcal{H}$-difference of $\tilde{\eta }$ and $õ$, and it is denoted by

$$\tilde{\eta }{\ominus }_{\mathcal{g}\mathcal{H}}õ,$$

(7)

as defined in ref. [25].

If the $\mathcal{g}\mathcal{H}$-difference exists, then for each $\mu \in (0, 1]$ the endpoint functions satisfy

$${(\tilde{\delta })}^{*}(\mu )={(\tilde{\eta }{\ominus }_{\mathcal{g}\mathcal{H}}\mathrm{õ})}^{*}(\mu )={\eta }^{*}(\mu )-{\mathrm{o}}^{*}(\mu ),{(\tilde{\delta })}_{*}(\mu )={(\tilde{\eta }{\ominus }_{\mathcal{g}\mathcal{H}}\mathrm{õ})}_{*}(\mu )={\eta }_{*}(\mu )-{\mathrm{o}}_{*}(\mu ),$$

and

$$\tilde{\eta }{\ominus }_{\mathcal{g}\mathcal{H}}õ=\tilde{\delta }\iff \{\begin{array}{c}\tilde{\delta }=\tilde{\eta }{\ominus }_{\mathcal{g}\mathcal{H}}õ\\ or \tilde{\eta }=õ\oplus (-1)\odot \tilde{\delta }\end{array}$$

(8)

Having established these fundamental properties of fuzzy numbers under addition and scalar multiplication, we now proceed to the case where $0<\mathfrak{e}\in R$ and $\tilde{\eta },õ\in {\mathbb{F}}_0$; then, $\mathfrak{e}\odot \tilde{\eta }$ and $\tilde{\eta }\oplus õ$ are defined as

$$\tilde{\eta }\oplus õ=\left\{({\eta }_{*}(\mu )+{\mathrm{o}}_{*}(\mu ),{\eta }^{*}(\mu )+{\mathrm{o}}^{*}(\mu )):\mu \in \left[0, 1\right]\right\},$$

(9)

$$\mathfrak{e}\odot \tilde{\eta }=\left\{(\mathfrak{e}{\eta }_{*}(\mu ),{\mathfrak{e}\eta }^{*}(\mu )):\mu \in \left[0, 1\right]\right\}.$$

(10)

Remark 1.

It is evident that ${\mathbb{F}}_0$ is closed under “$\odot$” scalar addition, and the features of ${\mathbb{F}}_0$ that were previously defined match those that result from the traditional extension concept. Furthermore, we obtain

$$\tilde{\eta }\oplus \mathfrak{e}=\left\{({\eta }_{*}(\mu )+\mathfrak{e},{\eta }^{*}(\mu )+\mathfrak{e}):\mu \in \left[0, 1\right]\right\}.$$

(11)

for every scalar number $\mathfrak{e}\in R$. A full metric space is generally understood to be the space ${\mathbb{F}}_0$ equipped with the supremum metric, which is written as $(\tilde{\eta },õ)=\underset{0\le \mu \le 1}{\mathit{sup}}H({\left[\tilde{\eta }\right]}^{\mu }, {\left[õ\right]}^{\mu })$) (see, e.g., [26]).

Definition 2

([26]). Let $\tilde{\mathfrak{S}}:\left[\mathcal{o},\mathcal{l}\right]\to {\mathbb{F}}_0$ be a $F$·$N\cdot V$-mapping. Then, $\mathbb{D}$-continuouity of $\tilde{\mathfrak{S}}$ at a point ${\mathcal{k}}_0$ over fuzzy supremum metric $\mathbb{D}$ if

$$\underset{\mathcal{k}\to {\mathcal{k}}_0}{\lim }\mathbb{D}(\tilde{\mathfrak{S}}(\mathcal{k}),\tilde{\mathfrak{S}}({\mathcal{k}}_0))=0.$$

(12)

Equivalently, $\tilde{\mathfrak{S}}$ is D-continuous at ${\mathcal{k}}_0$ if, and only if, for every $\mu \in (0, 1]$, the endpoint functions

$${\mathfrak{S}}_{*}(\cdot ;\mu ),{\mathfrak{S}}^{*}(\cdot ;\mu ):\left[\mathcal{o},\mathcal{l}\right]\to R$$

(13)

are continuous at ${\mathcal{k}}_0$, uniformly in relation to $\mu$.

The mapping $\tilde{\mathfrak{S}}$ is called $\mathbb{D}$-continuous on $\left[\mathcal{o},\mathcal{l}\right]$ if it is $\mathbb{D}$-continuous at every point of $\left[\mathcal{o},\mathcal{l}\right].$

Remark 2.

$\mathbb{D}$-continuity in relation to the fuzzy supremum metric guarantees uniform level-wise continuity of the μ-cut endpoints and is particularly suitable for studying $F$·$N\cdot V$-integrals, fractional operators, and $H\cdot H$-type inequalities.

Definition 3

([27]). A mapping$\tilde{\mathfrak{S}}:\left[\mathcal{o},\mathcal{l}\right]\to {\mathbb{F}}_0$ is called $F$·$N\cdot V$-mapping if, for every $\mathcal{k}\in [\mathcal{o},\mathcal{l}]$, $\tilde{\mathfrak{S}}(\mathcal{k})$ is a $F$·$N$. Equivalently, for every $\mathcal{k}\in [\mathcal{o},\mathcal{l}]$ and $\mu \in \left[0, 1\right],$the $\mu$-cut of $\tilde{\mathfrak{S}}(\mathcal{k})$ is a closed and bounded interval given by

$${\left[\tilde{\mathfrak{S}}(\mathcal{k})\right]}^{\mu }={\mathfrak{S}}_{\mu }(\mathcal{k})=\left[{\mathfrak{S}}_{*}(\mathcal{k}; \mu ), {\mathfrak{S}}^{*}(\mathcal{k}; \mu )\right],$$

(14)

where ${\mathfrak{S}}_{*}(\mathcal{k}; \mu )\le {\mathfrak{S}}^{*}(\mathcal{k}; \mu ).$

Definition 4

([27]). Let $\tilde{\mathfrak{S}}:[\mathcal{o},\mathcal{l}]\subset R\to {\mathbb{F}}_0$ be a $F$·$N\cdot V$ mapping such that, for each $\mu \in (0, 1],$ the endpoints mapping

$${\mathfrak{S}}_{*}(\cdot ; \mu ),{\mathfrak{S}}^{*}(\cdot ; \mu ):\left[\mathcal{o},\mathcal{l}\right]\to R,$$

are integrable on$\left[\mathcal{o},\mathcal{l}\right].$

The $F$·$N\cdot V$ Aumann integral of $\tilde{\mathfrak{S}}$ over $\left[\mathcal{o},\mathcal{l}\right]$ is defined as the $F$·$N$:

$$(FA){\int }_{\mathcal{o}}^{\mathcal{l}}\tilde{\mathfrak{S}}(\mathcal{k})d\mathcal{k},$$

(15)

whose $\mu$-cut is given by

$${\left[(FA){\int }_{\mathcal{o}}^{\mathcal{l}}\tilde{\mathfrak{S}}(\mathcal{k})d\mathcal{k}\right]}^{\begin{array}{c} \\ \mu \end{array}}=(IA){\int }_{\mathcal{o}}^{\mathcal{l}}{\mathfrak{S}}_{\mu }(\mathcal{k})d\mathcal{k}=\left[{\int }_{\mathcal{o}}^{\mathcal{l}}{\mathfrak{S}}_{*}(\mathcal{k};\mu )d\mathcal{k},{\int }_{\mathcal{o}}^{\mathcal{l}}{\mathfrak{S}}^{*}(\mathcal{k};\mu )d\mathcal{k}\right].$$

(16)

where $FA$ and $IA$ represent $F$·$N\cdot V$-Aumann and interval Aumann integrals, respectively.

The $F$·$N\cdot V$-Aumann integral preserves the $F$·$N\cdot V$-structure and plays a fundamental role in fuzzy analysis, particularly in the study of f $F$·$N\cdot V$-inequalities, $F$·$N\cdot V$-fractional integrals, and $F$·$N\cdot V$ $H\cdot H$-type results.

3. Fuzzy-Number-Valued Generalized Fractional Integral and Convexity

This section consists of the classical definition of fuzzy convex function and newly defined $\mathsf{F}$·$\mathsf{N}\cdot \mathit{V}$-generalized fractional integrals. Additionally, some extended versions of $\mathsf{H}\cdot \mathsf{H}$-type inequality are also obtained.

Definition 5.

A mapping $\tilde{\mathfrak{S}}:\Delta =[\mathcal{o},\mathcal{l}]\to {\mathbb{F}}_0^{+}$ is said to be a $F$·$N\cdot V$-convex mapping if the following inequality holds:

$$\tilde{\mathfrak{S}}(\mathcal{v}\mathcal{k}+(1-\mathcal{v})\mathcal{s}){\le }_F\mathcal{v}\odot \tilde{\mathfrak{S}}(\mathcal{k})\oplus (1-\mathcal{v})\odot \tilde{\mathfrak{S}}(\mathcal{s})$$

for all $(\mathcal{k},\mathcal{s})\in [\mathcal{o},\mathcal{l}],$ and $\mathcal{v}\in [0, 1].$

Definition 6.

Let $\mathfrak{g}:\left[\mathcal{o},\mathcal{l}\right]\to \mathbb{R}$ be an increasing and positive monotone mapping on $(\mathcal{o},\mathcal{l})$ having a continuous derivative ${\mathfrak{g}}^{\prime }(\mathcal{k})$ on $(\mathcal{o},\mathcal{l})$ and $\tilde{\mathfrak{S}}\in I{R}_{(\left[\mathcal{o},\mathcal{l}\right])}$. The $F$·$N\cdot V$-left-sided $({\mathbb{J}}_{{\mathcal{o}}^{+};\mathfrak{g}}^{⋌} \tilde{\mathfrak{S}}(\mathcal{k}))$ and right-sided $({\mathbb{J}}_{{\mathcal{l}}^{-};\mathfrak{g}}^{⋌} \tilde{\mathfrak{S}}(\mathcal{k}))$ fractional integrals of $\tilde{\mathfrak{S}}$ in relation to the mapping $\mathfrak{g}$ on $\left[\mathcal{o},\mathcal{l}\right]$ of order $⋌>0$ are described by:

$${\mathbb{J}}_{{\mathcal{o}}^{+};\mathfrak{g}}^{⋌} \tilde{\mathfrak{S}}(\mathcal{k})={\displaystyle \frac{1}{\mathsf{\Gamma }(⋌)}}(FA){\int }_{\mathcal{o}}^{\mathcal{k}}{\displaystyle \frac{{\mathfrak{g}}^{\prime }(\mathcal{v})}{{\left[\mathfrak{g}(\mathcal{k})-\mathfrak{g}(\mathcal{v})\right]}^{1-⋌}}}\tilde{\mathfrak{S}}(\mathcal{v})d\mathcal{v},\mathcal{k}>\mathcal{o}$$

(17)

and

$${\mathbb{J}}_{{\mathcal{l}}^{-};\mathfrak{g}}^{⋌} \tilde{\mathfrak{S}}(\mathcal{k})={\displaystyle \frac{1}{\mathsf{\Gamma }(⋌)}}(FA){\int }_{\mathcal{k}}^{\mathcal{l}}{\displaystyle \frac{{\mathfrak{g}}^{\prime }(\mathcal{v})}{{\left[\mathfrak{g}(\mathcal{v})-\mathfrak{g}(\mathcal{k})\right]}^{1-⋌}}}\tilde{\mathfrak{S}}(\mathcal{v})d\mathcal{v},\mathcal{k}<\mathcal{l},$$

(18)

respectively.

Corollary 1.

If $\tilde{\mathfrak{S}}:[\mathcal{o},\mathcal{l}]\to {\mathbb{F}}_0$ is an $F$·$N\cdot V$-mapping such that from $\mu$-cuts, here, we produce the collection of $I\cdot V$-mappings ${\mathfrak{S}}_{\mu }:[\mathcal{o},\mathcal{l}]\to {\mathbb{R}}_{\mathcal{I}}$ such that ${\left[\tilde{\mathfrak{S}}(\mathcal{v})\right]}^{\mu }={\mathfrak{S}}_{\mu }(\mathcal{v})=[\underset{\_}{\mathfrak{S}}(\mathcal{v};\mu ),\stackrel{\mathfrak{‾}}{\mathfrak{S}}(\mathcal{v};\mu )]$, for all $\mu \in [0,1]$ and $\mathfrak{g}:[\mathcal{o},\mathcal{l}]\to \mathbb{R}$ be an increasing and positive function on $(\mathcal{o},\mathcal{l}]$, having a continuous derivative ${\mathfrak{g}}^{\prime }(\mathcal{k})$ on $(\mathcal{o},\mathcal{l})$, then we obtain the following relation

$${\mathcal{J}}_{\mathcal{o}+;\mathfrak{g}}^{⋌}{\left[\tilde{\mathfrak{S}}(\mathcal{k})\right]}^{\mu }={\mathcal{J}}_{\mathcal{o}+;\mathfrak{g}}^{⋌}{\mathfrak{S}}_{\mu }(\mathcal{k})=\left[{\mathcal{I}}_{\mathcal{o}+;\mathfrak{g}}^{⋌}\underset{\_}{\mathfrak{S}}(\mathcal{k};\mu ),{\mathcal{I}}_{\mathcal{o}+;\mathfrak{g}}^{⋌}\stackrel{\mathfrak{‾}}{\mathfrak{S}}(\mathcal{k};\mu )\right]$$

(19)

and

$${\mathcal{J}}_{\mathcal{l}-;\mathfrak{g}}^{⋌}{\left[\tilde{\mathfrak{S}}(\mathcal{k})\right]}^{\mu }={\mathcal{J}}_{\mathcal{l}-;\mathfrak{g}}^{⋌}{\mathfrak{S}}_{\mu }(\mathcal{k})=\left[{\mathcal{I}}_{\mathcal{l}-;\mathfrak{g}}^{⋌}\underset{\_}{\mathfrak{S}}(\mathcal{k};\mu ),{\mathcal{I}}_{\mathcal{l}-;\mathfrak{g}}^{⋌}\stackrel{\mathfrak{‾}}{\mathfrak{S}}(\mathcal{k};\mu )\right].$$

(20)

Proof. 

The result follows directly from Theorem 1 together with the stated properties of the function $\mathfrak{g}$. □

We now proceed to the proof of the first theorem. Throughout the subsequent analysis, the following notation will be used:

$$\begin{array}{c}\tilde{\mathcal{T}}(\mathcal{k})=\tilde{\mathfrak{S}}(\mathcal{o}+\mathcal{l}-\mathcal{k}),\mathcal{k}\in [\mathcal{o},\mathcal{l}]\\ \tilde{\Phi }(\mathcal{k})=\tilde{\mathfrak{S}}(\mathcal{k})\oplus \tilde{\mathcal{T}}(\mathcal{k}).\end{array}$$

(21)

Theorem 2.

Let $\mathfrak{g}:[\mathcal{o},\mathcal{l}]\to \mathbb{R}$ be positive increasing mappings on $(\mathcal{o},\mathcal{l}]$ with continuous derivatives such that ${\mathfrak{g}}^{\prime }(\mathcal{k})$ on $(\mathcal{o},\mathcal{l})$. If $\tilde{\mathfrak{S}}:[\mathcal{o},\mathcal{l}]\to {\mathbb{F}}_0^{+}$ is a $F$·$N\cdot V$-coordinated convex mapping on $[\mathcal{o},\mathcal{l}]$ such that from $\mu$-cuts here, we produce the collection of $I\cdot V$-mappings ${\mathfrak{S}}_{\mu }:[\mathcal{o},\mathcal{l}]\to {\mathbb{R}}_{\mathrm{I}}^{+}$ such that ${\left[\tilde{\mathfrak{S}}(\mathcal{v})\right]}^{\mu }={\mathfrak{S}}_{\mu }(\mathcal{v})=[\underset{\_}{\mathfrak{S}}(\mathcal{v};\mu ),\stackrel{\mathfrak{‾}}{\mathfrak{S}}(\mathcal{v};\mu )]$, for all $\mu \in [0,1]$, then for $⋌ > 0$, the following $H\cdot H$-type inequality holds in relation to $\mathfrak{g}$:

$$\begin{array}{c} \tilde{\mathfrak{S}}({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}}){\le }_F{\displaystyle \frac{\mathsf{\Gamma }(⋌+1)}{4[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o}){]}^{⋌}}}\odot \left\{{\mathcal{J}}_{{\mathcal{o}}^{+};\mathfrak{g}}^{⋌}\tilde{\Phi }(\mathcal{l})\oplus {\mathcal{J}}_{{\mathcal{l}}^{-};\mathfrak{g}}^{⋌}\tilde{\Phi }(\mathcal{o})\right\}{\le }_F{\displaystyle \frac{\tilde{\mathfrak{S}}(\mathcal{o})+\tilde{\mathfrak{S}}(\mathcal{l})}{2}}. \end{array}$$

(22)

Proof. 

Let $u=\mathcal{o}\mathcal{r}+(1-\mathcal{r})\mathcal{l}$ and $v=(1-\mathcal{r})\mathcal{o}+\mathcal{r}\mathcal{l}$ for $\mathcal{r}\in [0,1]$. Convexity of $\tilde{\mathfrak{S}}$ allows us to write

$$\tilde{\mathfrak{S}}({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}})=\tilde{\mathfrak{S}}({\displaystyle \frac{u+v}{2}}){\le }_F{\displaystyle \frac{1}{2}}\odot \tilde{\mathfrak{S}}(u)+{\displaystyle \frac{1}{2}}\odot \tilde{\mathfrak{S}}(v).$$

That is, for all $\mu$-cuts,

$$\begin{array}{c} {\mathfrak{S}}_{\mu }({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}}){\le }_I{\displaystyle \frac{1}{2}}{\mathfrak{S}}_{\mu }(\mathcal{o}\mathcal{r}+(1-\mathcal{r})\mathcal{l})+{\displaystyle \frac{1}{2}}{\mathfrak{S}}_{\mu }((1-\mathcal{r})\mathcal{o}+\mathcal{r}\mathcal{l})\end{array}$$

(23)

Both sides of (23) are multiplied by

$${\displaystyle \frac{(\mathcal{l}-\mathcal{o})}{\mathsf{\Gamma }(⋌)}}{\displaystyle \frac{{\mathfrak{g}}^{\prime }(\mathcal{r}\mathcal{l}+(1-\mathcal{r})\mathcal{o})}{[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{r}\mathcal{l}+(1-\mathcal{r})\mathcal{o}){]}^{1-⋌}}}$$

and then integrating over $[0,1]$ in relation to $\mathcal{r}$, we have

$$\begin{array}{c} {\mathfrak{S}}_{\mu }({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}}){\displaystyle \frac{[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o}){]}^{⋌}}{\mathsf{\Gamma }(⋌+1)}} {\le }_I{\displaystyle \frac{1}{2}}{\mathcal{J}}_{\mathcal{o}+;\mathfrak{g}}\mathcal{T}(\mathcal{l})+{\displaystyle \frac{1}{2}}{\mathcal{J}}_{{\mathcal{o}}^{+};\mathfrak{g}}^{⋌}{\mathfrak{S}}_{\mu }(\mathcal{l})\\ ={\displaystyle \frac{1}{2}}{\mathcal{J}}_{{\mathcal{o}}^{+};\mathfrak{g}}^{⋌}{\Phi }_{\mu }(\mathcal{l})\\ {\displaystyle \frac{(\mathcal{l}-\mathcal{o})}{\mathsf{\Gamma }(⋌)}}{\displaystyle \frac{{\mathfrak{g}}^{\prime }(\mathcal{r}\mathcal{l}+(1-\mathcal{r})\mathcal{o})}{[\mathfrak{g}(\mathcal{r}\mathcal{l}+(1-\mathcal{r})\mathcal{o})-\mathfrak{g}(\mathcal{o}){]}^{1-⋌}}}\end{array}$$

(24)

and then integrating over $[0,1]$ in relation to $\mathcal{r}$, we get

$$\begin{array}{c}{\mathfrak{S}}_{\mu }({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}}){\displaystyle \frac{[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o}){]}^{⋌}}{\mathsf{\Gamma }(⋌+1)}}{\le }_I{\displaystyle \frac{1}{2}}{\mathcal{J}}_{{\mathcal{l}}^{-};\mathfrak{g}}^{⋌}\mathcal{T}(\mathcal{o})+{\displaystyle \frac{1}{2}}{\mathcal{J}}_{{\mathcal{l}}^{-};\mathfrak{g}}^{⋌}{\mathfrak{S}}_{\mu }(\mathcal{o})\\ ={\displaystyle \frac{1}{2}}{\mathcal{J}}_{{\mathcal{l}}^{-};\mathfrak{g}}^{⋌}{\Phi }_{\mu }(\mathcal{o})\end{array}$$

(25)

The inequalities (24) and (25) are added side by side to get

$${\mathfrak{S}}_{\mu }({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}}){\le }_I{\displaystyle \frac{\mathsf{\Gamma }(⋌+1)}{4[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o}){]}^{⋌}}}\left\{{\mathcal{J}}_{{\mathcal{o}}^{+};\mathfrak{g}}^{⋌}{\Phi }_{\mu }(\mathcal{l})+{\mathcal{J}}_{{\mathcal{l}}^{-};\mathfrak{g}}^{⋌}{\Phi }_{\mu }(\mathcal{o})\right\}$$

As a result, relation (22)’s first section is finished.

Nevertheless, due to ${\mathfrak{S}}_{\mu }$’s convexity, we may write

$$\begin{array}{c}{\mathfrak{S}}_{\mu }(\mathcal{o}\mathcal{r}+(1-\mathcal{r})\mathcal{l})+{\mathfrak{S}}_{\mu }((1-\mathcal{r})\mathcal{o}+\mathcal{r}\mathcal{l}){\le }_I{\mathfrak{S}}_{\mu }(\mathcal{o})+{\mathfrak{S}}_{\mu }(\mathcal{l})\end{array}$$

(26)

Both sides of (26) are multiplied by

$${\displaystyle \frac{(\mathcal{l}-\mathcal{o})}{\mathsf{\Gamma }(⋌)}}{\displaystyle \frac{{\mathfrak{g}}^{\prime }(\mathcal{r}\mathcal{l}+(1-\mathcal{r})\mathcal{o})}{[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{r}\mathcal{l}+(1-\mathcal{r})\mathcal{o}){]}^{1-⋌}}}$$

and then integrating over $[0,1]$ in relation to $\mathcal{r}$, we get

$$\begin{array}{c}{\mathcal{J}}_{{\mathcal{o}}^{+};\mathfrak{g}}^{⋌}\mathcal{T}(\mathcal{l})+{\mathcal{J}}_{{\mathcal{o}}^{+};\mathfrak{g}}^{⋌}{\mathfrak{S}}_{\mu }(\mathcal{l})={\mathcal{J}}_{{\mathcal{o}}^{+};\mathfrak{g}}^{⋌}{\Phi }_{\mu }(\mathcal{l}){\le }_I{\displaystyle \frac{[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o}){]}^{⋌}}{\mathsf{\Gamma }(⋌+1)}}\{{\mathfrak{S}}_{\mu }(\mathcal{o})+{\mathfrak{S}}_{\mu }(\mathcal{l})\}\end{array}$$

(27)

Similarly, both sides of (27) are multiplied by

$${\displaystyle \frac{(\mathcal{l}-\mathcal{o})}{\mathsf{\Gamma }(⋌)}}{\displaystyle \frac{{\mathfrak{g}}^{\prime }(\mathcal{r}\mathcal{l}+(1-\mathcal{r})\mathcal{o})}{[\mathfrak{g}(\mathcal{r}\mathcal{l}+(1-\mathcal{r})\mathcal{o})-\mathfrak{g}(\mathcal{o}){]}^{1-⋌}}}$$

and then integrating over $[0,1]$ in relation to $\mathcal{r}$, we have

$$\begin{array}{c}{\mathcal{J}}_{{\mathcal{l}}^{-};\mathfrak{g}}^{⋌}\mathcal{T}(\mathcal{o})+{\mathcal{J}}_{{\mathcal{l}}^{-};\mathfrak{g}}^{⋌}{\mathfrak{S}}_{\mu }(\mathcal{o})={\mathcal{J}}_{{\mathcal{l}}^{-};\mathfrak{g}}^{⋌}{\Phi }_{\mu }(\mathcal{o}){\le }_I{\displaystyle \frac{[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o}){]}^{⋌}}{\mathsf{\Gamma }(⋌+1)}}\{{\mathfrak{S}}_{\mu }(\mathcal{o})+{\mathfrak{S}}_{\mu }(\mathcal{l})\}\end{array}$$

(28)

Summing the relations (27) and (28),

$${\mathcal{J}}_{{\mathcal{l}}^{-};\mathfrak{g}}^{⋌}{\Phi }_{\mu }(\mathcal{o})+{\mathcal{J}}_{{\mathcal{o}}^{+};\mathfrak{g}}^{⋌}{\Phi }_{\mu }(\mathcal{l}){\le }_I{\displaystyle \frac{2[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o}){]}^{⋌}}{\mathsf{\Gamma }(⋌+1)}}\{{\mathfrak{S}}_{\mu }(\mathcal{o})+{\mathfrak{S}}_{\mu }(\mathcal{l})\}$$

The proof is finished in this manner. □

Remark 3.

(i) The inequality (2) is obtained by taking $\mathfrak{g}(\mathcal{v})=\mathcal{v}$ in (22). (ii) The $F$·$N\cdot V$-left-sided and right-sided Riemann–Liouville fractional integrals are obtained by taking $\mathfrak{g}(\mathcal{v})=\mathcal{v}$ in (17) and (18), respectively. (iii) We obtain the $FA$-integral (15) of the $\tilde{\mathfrak{S}}$ function if we take $\mathfrak{g}(\mathcal{v})=\mathcal{v}$ and $⋌=1$ in (17) and (18).

Theorem 3.

Let $\mathfrak{g}:[\mathcal{o},\mathcal{l}]\to \mathbb{R}$ be positive increasing mappings on $(\mathcal{o},\mathcal{l}]$ with continuous derivatives such that ${\mathfrak{g}}^{\prime }(\mathcal{k})$ on $(\mathcal{o},\mathcal{l})$. If $\tilde{\mathfrak{S}}:[\mathcal{o},\mathcal{l}]\to {\mathbb{F}}_0^{+}$ is a $F$·$N\cdot V$-coordinated convex mapping on $[\mathcal{o},\mathcal{l}]$ such that from $\mu$-cuts here, we produce the collection of $I\cdot V$-mappings ${\mathfrak{S}}_{\mu }:[\mathcal{o},\mathcal{l}]\to {\mathbb{R}}_{\mathrm{I}}^{+}$ such that ${\left[\tilde{\mathfrak{S}}(\mathcal{v})\right]}^{\mu }={\mathfrak{S}}_{\mu }(\mathcal{v})=[\underset{\_}{\mathfrak{S}}(\mathcal{v};\mu ),\stackrel{\mathfrak{‾}}{\mathfrak{S}}(\mathcal{v};\mu )]$, for all $\mu \in [0,1]$, then for $⋌ > 0$, the following $H\cdot H$-type inequality holds in relation to $\mathfrak{g}$:

$$\begin{array}{c}{\displaystyle \frac{{\mathrm{Ǩ}}_{\mathfrak{g}}^{⋌}}{\mathsf{\Gamma }(⋌+1)}}\odot \tilde{\mathfrak{S}}({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}})\\ {\le }_F{\displaystyle \frac{1}{2}}\odot \left\{{\mathcal{J}}_{{({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}})}^{-};\mathfrak{g}}^{⋌}\tilde{\Phi }(\mathcal{o})\oplus {\mathcal{J}}_{{({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}})}^{+};\mathfrak{g}}^{⋌}\tilde{\Phi }(\mathcal{l})\right\}\\ {\le }_F{\displaystyle \frac{{\mathrm{Ǩ}}_{\mathfrak{g}}^{⋌}}{\mathsf{\Gamma }(⋌+1)}}\odot ({\displaystyle \frac{\tilde{\mathfrak{S}}(\mathcal{o})\oplus \tilde{\mathfrak{S}}(\mathcal{l})}{2}})\end{array}$$

(29)

where

$${\mathrm{Ǩ}}_{\mathfrak{g}}^{⋌}={\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}})\right]}^{⋌}\oplus {\left[\mathfrak{g}({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}$$

Proof. 

Convexity of $\tilde{\mathfrak{S}}$ allows us to write

$$\tilde{\mathfrak{S}}({\displaystyle \frac{\mathcal{k}+\mathcal{s}}{2}}){\le }_F{\displaystyle \frac{\tilde{\mathfrak{S}}(\mathcal{k})\oplus \tilde{\mathfrak{S}}(\mathcal{s})}{2}}$$

for $\mathcal{k},\mathcal{s}\in [\mathcal{o},\mathcal{l}]$. If we take $\mathcal{k}={\displaystyle \frac{\mathcal{r}}{2}}\mathcal{o}+{\displaystyle \frac{2-\mathcal{r}}{2}}\mathcal{l}$ and $\mathcal{s}={\displaystyle \frac{2-\mathcal{r}}{2}}\mathcal{o}+{\displaystyle \frac{\mathcal{r}}{2}}\mathcal{l}$ for $\mathcal{r}\in [0,1]$, we obtain

$$\begin{array}{c}{\mathfrak{S}}_{\mu }({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}}){\le }_I{\displaystyle \frac{1}{2}}{\mathfrak{S}}_{\mu }({\displaystyle \frac{\mathcal{r}}{2}}\mathcal{o}+{\displaystyle \frac{2-\mathcal{r}}{2}}\mathcal{l})+{\displaystyle \frac{1}{2}}{\mathfrak{S}}_{\mu }({\displaystyle \frac{2-\mathcal{r}}{2}}\mathcal{o}+{\displaystyle \frac{\mathcal{r}}{2}}\mathcal{l}){\le }_I{\displaystyle \frac{{\mathfrak{S}}_{\mu }(\mathcal{o})+{\mathfrak{S}}_{\mu }(\mathcal{l})}{2}}.\end{array}$$

(30)

Both sides of (30) are multiplied by

$${\displaystyle \frac{(\mathcal{l}-\mathcal{o})}{2\mathsf{\Gamma }(⋌)}}{\displaystyle \frac{{\mathfrak{g}}^{\prime }(\frac{\mathcal{r}}{2}\mathcal{o}+\frac{2-\mathcal{r}}{2}\mathcal{l})}{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\frac{\mathcal{r}}{2}\mathcal{o}+\frac{2-\mathcal{r}}{2}\mathcal{l})\right]}^{1-⋌}}}$$

and then integrating over $[0,1]$ in relation to $\mathcal{r}$, we get

$$\begin{array}{ll}& \left\{{\displaystyle \frac{(\mathcal{l}-\mathcal{o})}{2\mathsf{\Gamma }(⋌)}}{\displaystyle {\int }_0^1} {\displaystyle \frac{{\mathfrak{g}}^{\prime }(\frac{\mathcal{r}}{2}\mathcal{o}+\frac{2-\mathcal{r}}{2}\mathcal{l})}{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\frac{\mathcal{r}}{2}\mathcal{o}+\frac{2-\mathcal{r}}{2}\mathcal{l})\right]}^{1-⋌}}}d\mathcal{r}\right\}{\mathfrak{S}}_{\mu }({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}})\\ {\le }_I& {\displaystyle \frac{(\mathcal{l}-\mathcal{o})}{4\mathsf{\Gamma }(⋌)}}(IA){\displaystyle {\int }_0^1} {\displaystyle \frac{{\mathfrak{g}}^{\prime }(\frac{\mathcal{r}}{2}\mathcal{o}+\frac{2-\mathcal{r}}{2}\mathcal{l})}{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\frac{\mathcal{r}}{2}\mathcal{o}+\frac{2-\mathcal{r}}{2}\mathcal{l})\right]}^{1-⋌}}}\left[{\mathfrak{S}}_{\mu }({\displaystyle \frac{\mathcal{r}}{2}}\mathcal{o}+{\displaystyle \frac{2-\mathcal{r}}{2}}\mathcal{l})+{\mathfrak{S}}_{\mu }({\displaystyle \frac{2-\mathcal{r}}{2}}\mathcal{o}+{\displaystyle \frac{\mathcal{r}}{2}}\mathcal{l})\right]d\mathcal{r}\\ {\le }_I& \left\{{\displaystyle \frac{(\mathcal{l}-\mathcal{o})}{\mathsf{\Gamma }(⋌)}}{\displaystyle {\int }_0^1} {\displaystyle \frac{{\mathfrak{g}}^{\prime }(\frac{\mathcal{r}}{2}\mathcal{o}+\frac{2-\mathcal{r}}{2}\mathcal{l})}{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\frac{\mathcal{r}}{2}\mathcal{o}+\frac{2-\mathcal{r}}{2}\mathcal{l})\right]}^{1-⋌}}}d\mathcal{r}\right\}{\displaystyle \frac{{\mathfrak{S}}_{\mu }(\mathcal{o})+{\mathfrak{S}}_{\mu }(\mathcal{l})}{4}}\end{array}$$

By change in variable $\mathcal{v}={\displaystyle \frac{\mathcal{r}}{2}}\mathcal{o}+{\displaystyle \frac{2-\mathcal{r}}{2}}\mathcal{l}$, we have

$$\begin{array}{c}\left\{{\displaystyle \frac{1}{\mathsf{\Gamma }(⋌)}}{\displaystyle {\int }_{{\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}}}^{\mathcal{l}}} {\displaystyle \frac{{\mathfrak{g}}^{\prime }(\mathcal{v})}{[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{v}){]}^{1-⋌}}}d\mathcal{v}\right\}{\mathfrak{S}}_{\mu }({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}})\\ {\le }_I{\displaystyle \frac{1}{2\mathsf{\Gamma }(⋌)}}(IA){\displaystyle {\int }_{{\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}}}^{\mathcal{l}}} {\displaystyle \frac{{\mathfrak{g}}^{\prime }(\mathcal{v})}{[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{v}){]}^{1-⋌}}}[{\mathfrak{S}}_{\mu }(v)+T(v)]dv\\ {\le }_I\left\{{\displaystyle \frac{1}{\mathsf{\Gamma }(⋌)}}{\displaystyle {\int }_{{\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}}}^{\mathcal{l}}} {\displaystyle \frac{{\mathfrak{g}}^{\prime }(\mathcal{v})}{[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{v}){]}^{1-⋌}}}d\mathcal{v}\right\}{\displaystyle \frac{{\mathfrak{S}}_{\mu }(\mathcal{o})+{\mathfrak{S}}_{\mu }(\mathcal{l})}{2}}\end{array}$$

(31)

It is clear that

$$\begin{array}{c}{\displaystyle {\int }_{{\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}}}^{\mathcal{l}}} {\displaystyle \frac{{\mathfrak{g}}^{\prime }(\mathcal{v})}{[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{v}){]}^{1-⋌}}}dv={\displaystyle \frac{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\frac{\mathcal{o}+\mathcal{l}}{2})\right]}^{⋌}}{⋌}}\end{array}$$

(32)

Inequality (32) and Definition 6 are used to get

$$\begin{array}{c}{\displaystyle \frac{1}{\mathsf{\Gamma }(⋌+1)}}{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}})\right]}^{⋌}{\mathfrak{S}}_{\mu }({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}})\\ {\le }_I{\displaystyle \frac{1}{2}}{\mathcal{J}}_{({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}}){+}_{;\mathfrak{g}}}^{⋌}{\Phi }_{\mu }(\mathcal{l})\\ {\le }_I{\displaystyle \frac{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\frac{\mathcal{o}+\mathcal{l}}{2})\right]}^{⋌}}{\mathsf{\Gamma }(⋌+1)}}{\displaystyle \frac{{\mathfrak{S}}_{\mu }(\mathcal{o})+{\mathfrak{S}}_{\mu }(\mathcal{l})}{2}}\end{array}$$

(33)

Similarly, if we perform the operations that multiply (30)’s two sides by

$${\displaystyle \frac{(\mathcal{l}-\mathcal{o})}{2\mathsf{\Gamma }(⋌)}}{\displaystyle \frac{{\mathfrak{g}}^{\prime }(\frac{2-\mathcal{r}}{2}\mathcal{o}+\frac{\mathcal{r}}{2}\mathcal{l})}{{\left[\mathfrak{g}(\frac{2-\mathcal{r}}{2}\mathcal{o}+\frac{\mathcal{r}}{2}\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{1-⋌}}}$$

When we change the variable $\mathcal{v}={\displaystyle \frac{2-\mathcal{r}}{2}}\mathcal{o}+{\displaystyle \frac{\mathcal{r}}{2}}\mathcal{l}$ and integrate over $[0,1]$ with respect to $\mathcal{r}$, respectively, we get

$$\begin{array}{c}\left\{{\displaystyle \frac{1}{\mathsf{\Gamma }(⋌)}}{\displaystyle {\int }_{\mathcal{o}}^{{\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}}}} {\displaystyle \frac{{\mathfrak{g}}^{\prime }(\mathcal{v})}{[\mathfrak{g}(\mathcal{v})-\mathfrak{g}(\mathcal{o}){]}^{1-⋌}}}d\mathcal{v}\right\}{\mathfrak{S}}_{\mu }({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}})\\ {\le }_I{\displaystyle \frac{1}{2\mathsf{\Gamma }(⋌)}}(IA){\displaystyle {\int }_{\mathcal{o}}^{{\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}}}} {\displaystyle \frac{{\mathfrak{g}}^{\prime }(\mathcal{v})}{[\mathfrak{g}(\mathcal{v})-\mathfrak{g}(\mathcal{o}){]}^{1-⋌}}}[{\mathfrak{S}}_{\mu }(v)+\mathcal{T}(v)]dv\\ {\le }_I\left\{{\displaystyle \frac{1}{\mathsf{\Gamma }(⋌)}}{\displaystyle {\int }_{\mathcal{o}}^{{\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}}}} {\displaystyle \frac{{\mathfrak{g}}^{\prime }(\mathcal{v})}{[\mathfrak{g}(\mathcal{v})-\mathfrak{g}(\mathcal{o}){]}^{1-⋌}}}d\mathcal{v}\right\}{\displaystyle \frac{{\mathfrak{S}}_{\mu }(\mathcal{o})+{\mathfrak{S}}_{\mu }(\mathcal{l})}{2}}\end{array}$$

(34)

In relation (34), if we apply the following:

$${\int }_{\mathcal{o}}^{{\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}}} {\displaystyle \frac{{\mathfrak{g}}^{\prime }(\mathcal{v})}{[\mathfrak{g}(\mathcal{v})-\mathfrak{g}(\mathcal{o}){]}^{1-⋌}}}d\mathcal{v}={\displaystyle \frac{{\left[\mathfrak{g}(\frac{\mathcal{o}+\mathcal{l}}{2})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}}{⋌}}$$

we obtain

$$\begin{array}{c}{\displaystyle \frac{1}{\mathsf{\Gamma }(⋌+1)}}{\left[\mathfrak{g}({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}{\mathfrak{S}}_{\mu }({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}})\\ {\le }_I{\displaystyle \frac{1}{2}}{\mathcal{J}}_{{({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}})}^{-};\mathfrak{g}}^{⋌}{\Phi }_{\mu }(\mathcal{o})\\ {\le }_I{\displaystyle \frac{{\left[\mathfrak{g}(\frac{\mathcal{o}+\mathcal{l}}{2})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}}{\mathsf{\Gamma }(⋌+1)}}{\displaystyle \frac{{\mathfrak{S}}_{\mu }(\mathcal{o})+{\mathfrak{S}}_{\mu }(\mathcal{l})}{2}}\end{array}$$

(35)

Summing the relations (33) and (35),

$$\begin{array}{cc}& {\displaystyle \frac{{\mathrm{Ǩ}}_{\mathfrak{g}}^{⋌}}{\mathsf{\Gamma }(⋌+1)}}{\mathfrak{S}}_{\mu }({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}}){\le }_I{\displaystyle \frac{1}{2}}\left\{{\mathcal{J}}_{{({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}})}^{-};\mathfrak{g}}^{⋌}{\Phi }_{\mu }(\mathcal{o})+{\mathcal{J}}_{{({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}})}^{+};\mathfrak{g}}^{⋌}{\Phi }_{\mu }(\mathcal{l})\right\}\\ & {\le }_I{\displaystyle \frac{{\mathrm{Ǩ}}_{\mathfrak{g}}^{⋌}}{\mathsf{\Gamma }(⋌+1)}}{\displaystyle \frac{{\mathfrak{S}}_{\mu }(\mathcal{o})+{\mathfrak{S}}_{\mu }(\mathcal{l})}{2}}\end{array}$$

Thus, the proof is finished. □

Corollary 2.

Taking $\mathfrak{g}(\mathcal{v})=\mathcal{v}$ from (29) gives us

$$\begin{array}{c} \tilde{\mathfrak{S}}({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}}){\le }_F{\displaystyle \frac{2^{⋌-1}\mathsf{\Gamma }(⋌+1)}{(\mathcal{l}-\mathcal{o}{)}^{⋌}}}\odot \left\{{\mathcal{J}}_{{({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}})}^{-}}^{⋌}\tilde{\mathfrak{S}}(\mathcal{o})\oplus {\mathcal{J}}_{{({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}})}^{+}}^{⋌}\tilde{\mathfrak{S}}(\mathcal{l})\right\}{\le }_F{\displaystyle \frac{\tilde{\mathfrak{S}}(\mathcal{o})\oplus \tilde{\mathfrak{S}}(\mathcal{l})}{2}}\end{array}$$

(36)

4. Fuzzy-Number-Valued Double Integral, and Coordinated Convexity

Firstly, we recall the concept of $F$·$N\cdot V$-double integral given by Khan et al. [27] in a previous study.

Definition 7

([27]). Let $\Delta = \left[\mathcal{o},\mathcal{l}\right]\times \left[\mathsf{ı}, \mathsf{ȷ}\right]\subset R^2$ and let $\tilde{\mathfrak{S}}:[\mathcal{o},\mathcal{l}]\to {\mathbb{F}}_0$ be a $F$·$N\cdot V$-mapping. Assume that, for every $\mu \in (0, 1]$, the endpoint functions

$${\mathfrak{S}}_{*}((\cdot ,\cdot ); \mu ),{\mathfrak{S}}^{*}((\cdot ,\cdot ); \mu ):\left[\mathcal{o},\mathcal{l}\right]\times \left[\mathsf{ı}, \mathsf{ȷ}\right]\to R,$$

are integrable over $\Delta$.

The $F$·$N\cdot \mathit{V}$-double integral of $\tilde{\mathfrak{S}}$ over $\Delta$, in the sense of $F$·$N\cdot \mathit{V}$-Aumann, is defined as the $F$·$N$:

$$(FD){\iint }_{\mathsf{\Delta }}\tilde{\mathfrak{S}}(\mathcal{r},\mathcal{v})d\mathcal{r}d\mathcal{v},$$

(37)

whose $\mu$-cut is given by

$${\left[(FD){\iint }_{\mathsf{\Delta }}\tilde{\mathfrak{S}}(\mathcal{r},\mathcal{v})d\mathcal{r}d\mathcal{v} \right]}^{\begin{array}{c} \\ \mu \end{array}}=(ID){\iint }_{\mathsf{\Delta }}{\mathfrak{S}}_{\mu }(\mathcal{r},\mathcal{v})d\mathcal{r}d\mathcal{v}$$

$$=\left[{\iint }_{\mathsf{\Delta }}{\mathfrak{S}}_{*}((\mathcal{r},\mathcal{v}); \mu )d\mathcal{r}d\mathcal{v},{\iint }_{\mathsf{\Delta }}{\mathfrak{S}}^{*}((\mathcal{r},\mathcal{v}); \mu )d\mathcal{r}d\mathcal{v}\right].$$

The $F$·$N\cdot V$-double integral preserves the $F$·$N$-structure and generalizes the classical double integral to $F$·$N\cdot V$-mappings. It plays a fundamental role in the study of fuzzy partial differential equations, $F$·$N\cdot V$-fractional integrals, and $H\cdot H$-type inequalities for coordinated convex $F$·$N\cdot V$-mappings.

Definition 8.

Let $\tilde{\mathfrak{S}}\in {FA}_{(\left[\mathcal{o},\mathcal{l}\right]\times \left[\mathsf{ı},\mathsf{ȷ}\right])}.$ The Hadamard $F$·$N\cdot V$-fractional integrals ${\mathfrak{J}}_{\mathcal{o}+,\mathsf{ı}+}^{⋌,⋊}\tilde{\mathfrak{S}}, {\mathfrak{J}}_{\mathcal{o}+,\mathsf{ȷ}-}^{⋌,⋊}\tilde{\mathfrak{S}},$ ${\mathfrak{J}}_{\mathcal{l}-,\mathsf{ı}+}^{⋌,⋊}\tilde{\mathfrak{S}}, and {\mathfrak{J}}_{\mathcal{l}-,\mathsf{ȷ}-}^{⋌,⋊}\tilde{\mathfrak{S}}$ of order $⋌, ⋊ > 0$ with $\mathcal{o},\mathsf{ı} \ge 0$ are described by

$${\mathfrak{J}}_{\mathcal{o}+,\mathsf{ı}+}^{⋌,⋊}\tilde{\mathfrak{S}}(\mathcal{k},\mathcal{s})≔{\displaystyle \frac{1}{\mathsf{\Gamma }(⋌)\mathsf{\Gamma }(⋊)}}(FA){\int }_{\mathcal{o}}^{\mathcal{k}} {\int }_{\mathsf{ı}}^{\mathcal{s}}{(ln{\displaystyle \frac{\mathcal{k}}{\mathcal{v}}})}^{⋌-1}{(ln{\displaystyle \frac{\mathcal{s}}{\mathcal{r}}})}^{⋊-1}{\displaystyle \frac{\tilde{\mathfrak{S}}(\mathcal{v},\mathcal{r})}{\mathcal{v}\mathcal{r}}}d\mathcal{r}d\mathcal{v},\mathcal{k}>\mathcal{o},\mathcal{s}>\mathsf{ı},$$

(38)

$${\mathfrak{J}}_{\mathcal{o}+,\mathsf{ȷ}-}^{⋌,⋊}\tilde{\mathfrak{S}}(\mathcal{k},\mathcal{s})≔ {\displaystyle \frac{1}{\mathsf{\Gamma }(⋌)\mathsf{\Gamma }(⋊)}} (FA){\int }_{\mathcal{o}}^{\mathcal{k}} {\int }_{\mathcal{s}}^{\mathsf{ȷ}}{(ln{\displaystyle \frac{\mathcal{k}}{\mathcal{v}}})}^{⋌-1}{(ln{\displaystyle \frac{\mathcal{s}}{\mathcal{r}}})}^{⋊-1}{\displaystyle \frac{\tilde{\mathfrak{S}}(\mathcal{v},\mathcal{r})}{\mathcal{v}\mathcal{r}}}d\mathcal{r}d\mathcal{v},\mathcal{k}>\mathcal{o},\mathcal{s}<\mathsf{ȷ},$$

(39)

$${\mathfrak{J}}_{\mathcal{l}-,\mathsf{ı}+}^{⋌,⋊}\tilde{\mathfrak{S}}(\mathcal{k},\mathcal{s})≔ {\displaystyle \frac{1}{\mathsf{\Gamma }(⋌)\mathsf{\Gamma }(⋊)}} (FA){\int }_{\mathcal{k}}^{\mathcal{l}} {\int }_{\mathsf{ı}}^{\mathcal{s}}{(ln{\displaystyle \frac{\mathcal{v}}{\mathcal{k}}})}^{⋌-1}{(ln{\displaystyle \frac{\mathcal{s}}{\mathcal{r}}})}^{⋊-1}{\displaystyle \frac{\tilde{\mathfrak{S}}(\mathcal{v},\mathcal{r})}{\mathcal{v}\mathcal{r}}}d\mathcal{r}d\mathcal{v},\mathcal{k}<\mathcal{l},\mathcal{s}>\mathsf{ı},$$

(40)

and

$${\mathfrak{J}}_{\mathcal{l}-,\mathsf{ȷ}-}^{⋌,⋊}\tilde{\mathfrak{S}}(\mathcal{k},\mathcal{s})≔ {\displaystyle \frac{1}{\mathsf{\Gamma }(⋌)\mathsf{\Gamma }(⋊)}} (FA){\int }_{\mathcal{k}}^{\mathcal{l}} {\int }_{\mathcal{s}}^{\mathsf{ȷ}}{(ln{\displaystyle \frac{\mathcal{k}}{\mathcal{v}}})}^{⋌-1}{(ln{\displaystyle \frac{\mathcal{s}}{\mathcal{r}}})}^{⋊-1}{\displaystyle \frac{\tilde{\mathfrak{S}}(\mathcal{v},\mathcal{r})}{\mathcal{v}\mathcal{r}}}d\mathcal{r}d\mathcal{v},\mathcal{k}<\mathcal{l},\mathcal{s}<\mathsf{ȷ},$$

(41)

respectively.

Now, we give the following generalized $F$·$N\cdot V$-fractional integral operators.

Definition 9.

Let $\mathfrak{g} : \left[\mathcal{o},\mathcal{l}\right]\to \mathbb{R}$ be an increasing and positive monotone mapping on $(\mathcal{o},\mathcal{l}]$, having a continuous derivative ${\mathfrak{g}}^{\prime }(\mathcal{k})$ on $(\mathcal{o},\mathcal{l})$, and let $⋉ : \left[\mathsf{ı}, \mathsf{ȷ}\right]\to \mathbb{R}$ be an increasing and positive monotone mapping on $(\mathsf{ı}, \mathsf{ȷ}]$, having a continuous derivative ${⋉}^{\prime }(\mathcal{s})on (\mathsf{ı}, \mathsf{ȷ}) and \tilde{\mathfrak{S}}\in {FA}_{(\left[\mathcal{o},\mathcal{l}\right]\times \left[\mathsf{ı},\mathsf{ȷ}\right])}.$ The $F$· $N\cdot V$-left-sided and right-sided fractional integral operators for mappings of two variables are described by

$${\mathbb{J}}_{\mathcal{o}+,\mathsf{ı}+;\mathfrak{g},⋉}^{⋌,⋊}\tilde{\mathfrak{S}}(\mathcal{k},\mathcal{s}):={\displaystyle \frac{1}{\Gamma (⋌)\Gamma (⋊)}}(FA){\int }_{\mathcal{o}}^{\mathcal{k}} {\int }_{\mathsf{ı}}^{\mathcal{s}}{\displaystyle \frac{{\mathfrak{g}}^{\prime }(\mathcal{v})}{{\left[\mathfrak{g}(\mathcal{k})-\mathfrak{g}(\mathcal{v})\right]}^{1-⋌}}} {\displaystyle \frac{{⋉}^{\prime }(\mathcal{r})}{{\left[⋉(\mathcal{s})-⋉(\mathcal{r})\right]}^{1-⋊}}} \tilde{\mathfrak{S}}(\mathcal{v},\mathcal{r})d\mathcal{r}d\mathcal{v},\mathcal{k}>\mathcal{o},\mathcal{s}>\mathsf{ı},$$

(42)

$${\mathbb{J}}_{\mathcal{o}+,\mathsf{ȷ}-;\mathfrak{g},⋉}^{⋌,⋊}\tilde{\mathfrak{S}}(\mathcal{k},\mathcal{s}):={\displaystyle \frac{1}{\Gamma (⋌)\Gamma (⋊)}}(FA){\int }_{\mathcal{o}}^{\mathcal{k}} {\int }_{\mathcal{s}}^{\mathsf{ȷ}}{\displaystyle \frac{{\mathfrak{g}}^{\prime }(\mathcal{v})}{{\left[\mathfrak{g}(\mathcal{k})-\mathfrak{g}(\mathcal{v})\right]}^{1-⋌}}} {\displaystyle \frac{{⋉}^{\prime }(\mathcal{r})}{{\left[⋉(\mathcal{s})-⋉(\mathcal{r})\right]}^{1-⋊}}} \tilde{\mathfrak{S}}(\mathcal{v},\mathcal{r})d\mathcal{r}d\mathcal{v},\mathcal{k}>\mathcal{o},\mathcal{s}<\mathsf{ȷ},$$

(43)

$${\mathbb{J}}_{\mathcal{l}-,\mathsf{ı}+;\mathfrak{g},⋉}^{⋌,⋊}\tilde{\mathfrak{S}}(\mathcal{k},\mathcal{s}):={\displaystyle \frac{1}{\Gamma (⋌)\Gamma (⋊)}}(FA){\int }_{\mathcal{k}}^{\mathcal{l}} {\int }_{\mathsf{ı}}^{\mathcal{s}}{\displaystyle \frac{{\mathfrak{g}}^{\prime }(\mathcal{v})}{{\left[\mathfrak{g}(\mathcal{k})-\mathfrak{g}(\mathcal{v})\right]}^{1-⋌}}} {\displaystyle \frac{{⋉}^{\prime }(\mathcal{r})}{{\left[⋉(\mathcal{s})-⋉(\mathcal{r})\right]}^{1-⋊}}} \tilde{\mathfrak{S}}(\mathcal{v},\mathcal{r})d\mathcal{r}d\mathcal{v},\mathcal{k}<\mathcal{l},\mathcal{s}>\mathsf{ı},$$

(44)

and

$${\mathbb{J}}_{\mathcal{l}-,\mathsf{ȷ}-;\mathfrak{g},⋉}^{⋌,⋊}\tilde{\mathfrak{S}}(\mathcal{k},\mathcal{s}):={\displaystyle \frac{1}{\Gamma (⋌)\Gamma (⋊)}}(FA){\int }_{\mathcal{k}}^{\mathcal{l}} {\int }_{\mathcal{s}}^{\mathsf{ȷ}}{\displaystyle \frac{{\mathfrak{g}}^{\prime }(\mathcal{v})}{{\left[\mathfrak{g}(\mathcal{k})-\mathfrak{g}(\mathcal{v})\right]}^{1-⋌}}} {\displaystyle \frac{{⋉}^{\prime }(\mathcal{r})}{{\left[⋉(\mathcal{s})-⋉(\mathcal{r})\right]}^{1-⋊}}} \tilde{\mathfrak{S}}(\mathcal{v},\mathcal{r})d\mathcal{r}d\mathcal{v},\mathcal{k}<\mathcal{l},\mathcal{s}<\mathsf{ȷ},$$

(45)

for ⋌, ⋊ > 0.

Based on the definitions given above, we may provide the $F$·$N\cdot V$-integrals as follows:

$${\mathbb{J}}_{{\mathcal{o}}^{+};\mathfrak{g}}^{⋌} \tilde{\mathfrak{S}}(\mathcal{k},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}})≔{\displaystyle \frac{1}{\mathsf{\Gamma }(⋌)}}(FA){\int }_{\mathcal{o}}^{\mathcal{k}}{\displaystyle \frac{{\mathfrak{g}}^{\prime }(\mathcal{v})}{{\left[\mathfrak{g}(\mathcal{k})-\mathfrak{g}(\mathcal{v})\right]}^{1-⋌}}}\tilde{\mathfrak{S}}(\mathcal{v},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}})d\mathcal{v},\mathcal{k}>\mathcal{o}$$

(46)

$${\mathbb{J}}_{\mathcal{l}-;\mathfrak{g}}^{⋌} \tilde{\mathfrak{S}}(\mathcal{k},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}})≔{\displaystyle \frac{1}{\mathsf{\Gamma }(⋌)}}(FA){\int }_{\mathcal{k}}^{\mathcal{l}}{\displaystyle \frac{{\mathfrak{g}}^{\prime }(\mathcal{v})}{{\left[\mathfrak{g}(\mathcal{v})-\mathfrak{g}(\mathcal{k})\right]}^{1-⋌}}}\tilde{\mathfrak{S}}(\mathcal{v},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}})d\mathcal{v},\mathcal{k}<\mathcal{l}$$

(47)

$${\mathbb{J}}_{\mathsf{ı}+;⋉}^{⋊} \tilde{\mathfrak{S}}({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},\mathcal{s})≔{\displaystyle \frac{1}{\mathsf{\Gamma }(⋊)}}(FA){\int }_{\mathsf{ı}}^{\mathcal{s}}{\displaystyle \frac{{⋉}^{\prime }(\mathcal{v})}{{\left[⋉(\mathcal{s})-⋉(\mathcal{r})\right]}^{1-⋊}}}\tilde{\mathfrak{S}}({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},\mathcal{r})d\mathcal{r},\mathcal{s}>\mathsf{ı},$$

(48)

and

$${\mathbb{J}}_{\mathsf{ȷ}-;⋉}^{⋊} \tilde{\mathfrak{S}}({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},\mathcal{s})≔{\displaystyle \frac{1}{\mathsf{\Gamma }(⋊)}}(FA){\int }_{\mathsf{ı}}^{\mathcal{s}}{\displaystyle \frac{{⋉}^{\prime }(\mathcal{v})}{{\left[⋉(\mathcal{r})-⋉(\mathcal{s})\right]}^{1-⋊}}}\tilde{\mathfrak{S}}({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},\mathcal{r})d\mathcal{r},\mathcal{s}<\mathsf{ȷ}.$$

(49)

If we decide to $\mathfrak{g}(\mathcal{v})={\displaystyle \frac{{\mathcal{v}}^{\rho }}{\rho }} and ⋉(\mathcal{r})={\displaystyle \frac{{\mathcal{r}}^{\sigma }}{\sigma }} , \rho ,\sigma >0,$ the definition that follows is found in Definition 9.

Definition 10.

Let $\tilde{\mathfrak{S}} \in {FA}_{(\left[\mathcal{o},\mathcal{l}\right]\times \left[\mathsf{ı},\mathsf{ȷ}\right])}.$ The Katugampola fractional integrals for $F$·$N\cdot V$-mapping with two variables are described by

$${{}_{ }{}^{ \rho ,\sigma }\mathbb{I}}_{\mathcal{o}+,\mathsf{ı}+}^{⋌,⋊}\tilde{\mathfrak{S}}(\mathcal{k},\mathcal{s}):={\displaystyle \frac{ {\rho }^{1-⋌}{\sigma }^{1-⋊}}{\Gamma (⋌)\Gamma (⋊)}}(FA){\int }_{\mathcal{o}}^{\mathcal{k}} {\int }_{\mathsf{ı}}^{\mathcal{s}}{\displaystyle \frac{{\mathcal{v}}^{\rho -1}}{{\left[{\mathcal{k}}^{\rho }-{\mathcal{v}}^{\rho }\right]}^{1-⋌}}} {\displaystyle \frac{{\mathcal{r}}^{\sigma -1}}{{\left[{\mathcal{s}}^{\sigma }-{\mathcal{r}}^{\sigma }\right]}^{1-⋊}}} \tilde{\mathfrak{S}}(\mathcal{v},\mathcal{r})d\mathcal{r}d\mathcal{v},\mathcal{k}>\mathcal{o},\mathcal{s}>\mathsf{ı},$$

(50)

$${{}_{ }{}^{ \rho ,\sigma }\mathbb{I}}_{\mathcal{o}+,\mathsf{ȷ}-}^{⋌,⋊}\tilde{\mathfrak{S}}(\mathcal{k},\mathcal{s}):={\displaystyle \frac{ {\rho }^{1-⋌}{\sigma }^{1-⋊}}{\Gamma (⋌)\Gamma (⋊)}}(FA){\int }_{\mathcal{o}}^{\mathcal{k}} {\int }_{\mathcal{s}}^{\mathsf{ȷ}}{\displaystyle \frac{{\mathcal{v}}^{\rho -1}}{{\left[{\mathcal{k}}^{\rho }-{\mathcal{v}}^{\rho }\right]}^{1-⋌}}} {\displaystyle \frac{{\mathcal{r}}^{\sigma -1}}{{\left[{\mathcal{r}}^{\sigma }-{\mathcal{s}}^{\sigma }\right]}^{1-⋊}}} \tilde{\mathfrak{S}}(\mathcal{v},\mathcal{r})d\mathcal{r}d\mathcal{v},\mathcal{k}>\mathcal{o},\mathcal{s}<\mathsf{ȷ},$$

(51)

$${{}_{ }{}^{ \rho ,\sigma }\mathbb{I}}_{\mathcal{l}-,\mathsf{ı}+}^{⋌,⋊}\tilde{\mathfrak{S}}(\mathcal{k},\mathcal{s}):={\displaystyle \frac{ {\rho }^{1-⋌}{\sigma }^{1-⋊}}{\Gamma (⋌)\Gamma (⋊)}}(FA){\int }_{\mathcal{k}}^{\mathcal{l}} {\int }_{\mathsf{ı}}^{\mathcal{s}}{\displaystyle \frac{{\mathcal{v}}^{\rho -1}}{{\left[{\mathcal{v}}^{\rho }-{\mathcal{k}}^{\rho }\right]}^{1-⋌}}} {\displaystyle \frac{{\mathcal{r}}^{\sigma -1}}{{\left[{\mathcal{s}}^{\sigma }-{\mathcal{r}}^{\sigma }\right]}^{1-⋊}}} \tilde{\mathfrak{S}}(\mathcal{v},\mathcal{r})d\mathcal{r}d\mathcal{v},\mathcal{k}<\mathcal{l},\mathcal{s}<\mathsf{ı},$$

(52)

and

$${{}_{ }{}^{ \rho ,\sigma }\mathbb{I}}_{\mathcal{l}-,\mathsf{ȷ}-}^{⋌,⋊}\tilde{\mathfrak{S}}(\mathcal{k},\mathcal{s}):={\displaystyle \frac{ {\rho }^{1-⋌}{\sigma }^{1-⋊}}{\Gamma (⋌)\Gamma (⋊)}}(FA){\int }_{\mathcal{k}}^{\mathcal{l}} {\int }_{\mathcal{s}}^{\mathsf{ȷ}}{\displaystyle \frac{{\mathcal{v}}^{\rho -1}}{{\left[{\mathcal{v}}^{\rho }-{\mathcal{k}}^{\rho }\right]}^{1-⋌}}} {\displaystyle \frac{{\mathcal{r}}^{\sigma -1}}{{\left[{\mathcal{r}}^{\sigma }-{\mathcal{s}}^{\sigma }\right]}^{1-⋊}}} \tilde{\mathfrak{S}}(\mathcal{v},\mathcal{r})d\mathcal{r}d\mathcal{v},\mathcal{k}<\mathcal{l},\mathcal{s}<\mathsf{ȷ}.$$

(53)

Now, we recall the concept of $F$·$N\cdot V$-coordinated convex mappings that is given by Khan et al. [27,28] in a previous study as follows.

Definition 11.

A mapping $\tilde{\mathfrak{S}}:\Delta =[\mathcal{o},\mathcal{l}]\times [\mathsf{ı}, \mathsf{ȷ}]\to {\mathbb{F}}_0^{+}$ is said to be a $F$·$N\cdot V$-coordinated convex mapping if the following inequality holds:

$$\begin{gathered} \tilde{\mathfrak{S}}(\mathcal{v}\mathcal{k}+(1-\mathcal{v})\mathcal{s},\mathcal{r}u+(1-\mathcal{r})⋉) \\ {\le }_F\mathcal{v}\mathcal{r}\odot \tilde{\mathfrak{S}}(\mathcal{k},u)\oplus \mathcal{v}(1-\mathcal{r})\odot \tilde{\mathfrak{S}}(\mathcal{k},⋉)\oplus \mathcal{r}(1-\mathcal{v})\odot \tilde{\mathfrak{S}}(\mathcal{s},u)\oplus (1-\mathcal{r})(1-\mathcal{v})\odot \tilde{\mathfrak{S}}(\mathcal{s},⋉) \end{gathered}$$

for all $(\mathcal{k},u), (\mathcal{s},⋉)\in \Delta ,$ and $\mathcal{r},\mathcal{v} \in [0, 1].$

Lemma 1.

A mapping $\tilde{\mathfrak{S}} :\Delta =[\mathcal{o},\mathcal{l}]\times [\mathsf{ı}, \mathsf{ȷ}]\to {\mathbb{F}}_0^{+}$ is $F$· $N\cdot V$-convex on coordinates if and only if there exist two mappings ${\tilde{\mathfrak{S}}}_{\mathcal{k}} :[\mathsf{ı}, \mathsf{ȷ}]\to {\mathbb{F}}_0^{+}$, ${\tilde{\mathfrak{S}}}_{\mathcal{k}}(⋉)=\tilde{\mathfrak{S}}(\mathcal{k},⋉)$ and ${\tilde{\mathfrak{S}}}_{\mathcal{s}}:[\mathcal{o},\mathcal{l}]\to {\mathbb{F}}_0^{+}$, ${\tilde{\mathfrak{S}}}_{\mathcal{s}}(u)=\tilde{\mathfrak{S}}(\mathcal{s},u)$ that are $F$·$N\cdot V$-convex.

Proof. 

The proof of this lemma is followed immediately by the definition of $F$·$N\cdot V$-coordinated convex mapping. □

It is easy to prove that a $F$·$N\cdot V$-convex mapping is $F$·$N\cdot V$-coordinated convex, but the converse may not be true. For this, we can see the following example:

Example 1.

A $F$·$N\cdot V$-mapping $\tilde{\mathfrak{S}}:{[0, 1]}^2\to {\mathbb{F}}_0^{+}$ defined as $\tilde{\mathfrak{S}}(\mathcal{k},\mathcal{s})=[\mathcal{k}\mathcal{s}, (6 - e^{\mathcal{k}})(6 - e^{\mathcal{s}})]$ is $F$·$N\cdot V$-convex on coordinates, but it is not $F$·$N\cdot V$-convex on ${[0, 1]}^2$.

Note that the results obtained in the next section generalize and unify several existing $H\cdot H$-type inequalities. In the following section, we illustrate the applicability of our main theorems through special cases and related consequences.

Related Inequalities

In this section, the derived inequalities extend classical $H\cdot H$-results to the setting of $F$·$N\cdot V$-coordinated convex functions involving generalized fractional integrals, thereby providing a broader and more flexible analytical framework. Several important corollaries and special cases of the main results are discussed in the subsequent section.

Theorem 4.

Let $\mathfrak{g}:[\mathcal{o},\mathcal{l}]\to \mathbb{R}$ and $⋉ : [\mathsf{ı}, \mathsf{ȷ}]\to \mathbb{R}$ be positive increasing mappings on $(\mathcal{o},\mathcal{l}]$ and $(\mathsf{ı}, \mathsf{ȷ}]$ with continuous derivatives such that ${\mathfrak{g}}^{\prime }(\mathcal{k})$ on $(\mathcal{o},\mathcal{l})$ and ${⋉}^{\prime }(\mathcal{s})$ on $(\mathsf{ı}, \mathsf{ȷ})$, respectively. Let $\Delta =[\mathcal{o},\mathcal{l}]\times [\mathsf{ı}, \mathsf{ȷ}]$. If $\tilde{\mathfrak{S}}:\Delta \to {\mathbb{F}}_0^{+}$ is a $F$·$N\cdot V$-coordinated convex mapping on Δ such that from $\mu$-cuts here, we produce the collection of $I\cdot V$-mappings ${\mathfrak{S}}_{\mu }:\Delta \to {\mathbb{R}}_{\mathrm{I}}^{+}$ described by ${\left[\tilde{\mathfrak{S}}(\mathcal{k},\mathcal{s})\right]}^{\mu }={\mathfrak{S}}_{\mu }((\mathcal{k},\mathcal{s}))=[\underset{\_}{\mathfrak{S}}((\mathcal{k},\mathcal{s}); \mu ),\stackrel{\mathfrak{‾}}{\mathfrak{S}}((\mathcal{k},\mathcal{s}); \mu )]$, for all $\mu \in \left[0,1\right] and (\mathcal{k},\mathcal{s})\in \Delta$, then for $⋌, ⋊>0$, the following $H\cdot H$-type inequality holds:

$$\begin{gathered} \tilde{\mathfrak{S}}({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}}){\le }_F{\displaystyle \frac{\Gamma (⋌+1)\Gamma (⋊+1)}{16{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌} {\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}}\odot \left[{\mathbb{J}}_{\mathcal{o}+,\mathsf{ı}+;\mathfrak{g},⋉}^{⋌,⋊}\widetilde{\mathsf{Ϫ}}(\mathcal{l},\mathsf{ȷ})\oplus {\mathbb{J}}_{\mathcal{o}+,\mathsf{ȷ}-;\mathfrak{g},⋉}^{⋌,⋊}\widetilde{\mathsf{Ϫ}}(\mathcal{l},\mathsf{ı})\oplus {\mathbb{J}}_{\mathcal{l}-,\mathsf{ı}+;\mathfrak{g},⋉}^{⋌,⋊}\widetilde{\mathsf{Ϫ}}(\mathcal{o},\mathsf{ȷ})\oplus {\mathbb{J}}_{\mathcal{l}-,\mathsf{ȷ}-;\mathfrak{g},⋉}^{⋌,⋊}\widetilde{\mathsf{Ϫ}}(\mathcal{o},\mathsf{ı})\right] \\ {\le }_F{\displaystyle \frac{\tilde{\mathfrak{S}}(\mathcal{o},\mathsf{ı})\oplus \tilde{\mathfrak{S}}(\mathcal{o},\mathsf{ȷ})\oplus \tilde{\mathfrak{S}}(\mathcal{l},\mathsf{ı})\oplus \tilde{\mathfrak{S}}(\mathcal{l},\mathsf{ȷ})}{4}}, \end{gathered}$$

(54)

where

$$\widetilde{\mathsf{Ϫ}}1(\mathcal{k},\mathcal{s})=\tilde{\mathfrak{S}}(\mathcal{o}+\mathcal{l}-\mathcal{k},\mathcal{s}),$$

$$\widetilde{\mathsf{Ϫ}}2(\mathcal{k},\mathcal{s})=\tilde{\mathfrak{S}}(\mathcal{k},\mathsf{ı}+\mathsf{ȷ}-\mathcal{s}),$$

$$\widetilde{\mathsf{Ϫ}}3(\mathcal{k},\mathcal{s})=\tilde{\mathfrak{S}}(\mathcal{o}+\mathcal{l}-\mathcal{k},\mathsf{ı}+\mathsf{ȷ}-\mathcal{s}),$$

$$\tilde{\mathfrak{D}}(\mathcal{k},\mathcal{s})=\tilde{\mathfrak{S}}(\mathcal{k},\mathcal{s})\oplus \widetilde{\mathsf{Ϫ}}2(\mathcal{k},\mathcal{s})$$

$$\tilde{\mathcal{H}}(\mathcal{k},\mathcal{s})=\tilde{\mathfrak{S}}(\mathcal{k},\mathcal{s})\oplus \widetilde{\mathsf{Ϫ}}1(\mathcal{k},\mathcal{s})$$

$$\widetilde{\mathsf{Ϗ}}(\mathcal{k},\mathcal{s})=\widetilde{\mathsf{Ϫ}}1(\mathcal{k},\mathcal{s})\oplus \widetilde{\mathsf{Ϫ}}3(\mathcal{k},\mathcal{s})$$

$$\tilde{\mathcal{L}}(\mathcal{k},\mathcal{s})=\widetilde{\mathsf{Ϫ}}2(\mathcal{k},\mathcal{s})\oplus \widetilde{\mathsf{Ϫ}}3(\mathcal{k},\mathcal{s})$$

$$\widetilde{\mathsf{Ϫ}}(\mathcal{k},\mathcal{s})=\widetilde{\mathsf{Ϫ}}1(\mathcal{k},\mathcal{s})\oplus \widetilde{\mathsf{Ϫ}}2(\mathcal{k},\mathcal{s})\oplus \widetilde{\mathsf{Ϫ}}3(\mathcal{k},\mathcal{s})\oplus \tilde{\mathfrak{S}}(\mathcal{k},\mathcal{s})$$

$$={\displaystyle \frac{\tilde{\mathfrak{D}}(\mathcal{k},\mathcal{s})\oplus \tilde{\mathcal{H}}(\mathcal{k},\mathcal{s})\oplus \widetilde{\mathsf{Ϗ}}(\mathcal{k},\mathcal{s})\oplus \tilde{\mathcal{L}}(\mathcal{k},\mathcal{s})}{2}}$$

$for (\mathcal{k},\mathcal{s})\in [\mathcal{o},\mathcal{l}]\times [\mathsf{ı}, \mathsf{ȷ}]$, where for all $\mu$-cuts $\in \left[\mathrm{0,1}\right]$, we have ${\left[\widetilde{\mathsf{Ϫ}}1(\mathcal{k},\mathcal{s})\right]}^{\mu }={{\mathsf{Ϫ}}_1}_{\mu }(\mathcal{k},\mathcal{s})$, ${\left[\widetilde{\mathsf{Ϫ}}2(\mathcal{k},\mathcal{s})\right]}^{\mu }={{\mathsf{Ϫ}}_2}_{\mu }(\mathcal{k},\mathcal{s})$, ${\left[\widetilde{\mathsf{Ϫ}}3(\mathcal{k},\mathcal{s})\right]}^{\mu }={{\mathsf{Ϫ}}_3}_{\mu }(\mathcal{k},\mathcal{s})$, ${\left[\tilde{\mathfrak{D}}(\mathcal{k},\mathcal{s})\right]}^{\mu }={\mathfrak{D}}_{\mu }(\mathcal{k},\mathcal{s})$, ${\left[\tilde{\mathcal{H}}(\mathcal{k},\mathcal{s})\right]}^{\mu }={\mathcal{H}}_{\mu }(\mathcal{k},\mathcal{s})$, ${\left[\widetilde{\mathsf{Ϗ}}(\mathcal{k},\mathcal{s})\right]}^{\mu }={\mathsf{Ϗ}}_{\mu }(\mathcal{k},\mathcal{s})$, ${\left[\tilde{\mathcal{L}}(\mathcal{k},\mathcal{s})\right]}^{\mu }={\mathcal{L}}_{\mu }(\mathcal{k},\mathcal{s})$ and ${\left[\widetilde{\mathsf{Ϫ}}(\mathcal{k},\mathcal{s})\right]}^{\mu }={\mathsf{Ϫ}}_{\mu }(\mathcal{k},\mathcal{s}).$

Proof. 

Since F is a $F$·$N\cdot V$-coordinated convex mapping on Δ, we have

$$\tilde{\mathfrak{S}}({\displaystyle \frac{u+v}{2}},{\displaystyle \frac{p+q}{2}}){\le }_F{\displaystyle \frac{\tilde{\mathfrak{S}}(u, p)\oplus \tilde{\mathfrak{S}}(u, q)\oplus \tilde{\mathfrak{S}}(v, p)\oplus \tilde{\mathfrak{S}}(v, q)}{4}}$$

(55)

for $(u, p), (v, q)\in \Delta .$ Now, for $\mathcal{v},\mathcal{r}\in \left[0, 1\right],$ let $u=\mathcal{v}\mathcal{o}+(1 - \mathcal{v})\mathcal{l},v = (1 - \mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l},p=\mathsf{ı}\mathcal{r}+(1 - \mathcal{r})\mathsf{ȷ},$ and $q=(1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ}.$ Then, for all $\mu$-cuts $\in \left[0,1\right]$, we have

$$\begin{array}{ll}{\mathfrak{D}}_{\mu }({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}})& {\le }_I {\displaystyle \frac{1}{4}} {\mathfrak{D}}_{\mu }(\mathcal{v}\mathcal{o}+(1-\mathcal{v})\mathcal{l},\mathsf{ı}\mathcal{r}+(1-\mathcal{r})\mathsf{ȷ})\\ & +{\displaystyle \frac{1}{4}} {\mathfrak{S}}_{\mu }(\mathcal{v}\mathcal{o}+(1-\mathcal{v})\mathcal{l},(1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ})\\ & +{\displaystyle \frac{1}{4}}{\mathfrak{S}}_{\mu }((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l},\mathsf{ı}\mathcal{r}+(1-\mathcal{r})\mathsf{ȷ})+{\displaystyle \frac{1}{4}}{\mathfrak{S}}_{\mu }((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l},(1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ}).\end{array}$$

(56)

Both sides of (56) are multiplied by

$${\displaystyle \frac{(\mathcal{l}-\mathcal{o})(\mathsf{ȷ}-\mathsf{ı})}{\mathsf{\Gamma }(⋌)\mathsf{\Gamma }(⋊)}} {\displaystyle \frac{{\mathfrak{g}}^{\prime }((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l})}{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g} ((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l})\right]}^{1-⋌}}} {\displaystyle \frac{{⋉}^{\prime }((1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ})}{{\left[⋉(\mathsf{ȷ})-⋉((1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ})\right]}^{1-\mathsf{\beta }}}}$$

and integrating the resulting inequality in relation to $\mathcal{v}$, $\mathcal{r}$ over [0, 1] × [0, 1], we get

$$\begin{gathered} {\displaystyle \frac{(\mathcal{l}-\mathcal{o})(\mathsf{ȷ}-\mathsf{ı})}{\mathsf{\Gamma }(⋌)\mathsf{\Gamma }(⋊)}} {\mathfrak{S}}_{\mu }({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}}) (IA){\int }_0^1{\int }_0^1\left[{\displaystyle \frac{{\mathfrak{g}}^{\prime }((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l})}{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g} ((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l})\right]}^{1-⋌}}} {\displaystyle \frac{{⋉}^{\prime }((1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ})}{{\left[⋉(\mathsf{ȷ})-⋉((1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ})\right]}^{1-\mathsf{\beta }}}}\right]d\mathcal{r}d\mathcal{v} \\ {\le }_I {\displaystyle \frac{(\mathcal{l}-\mathcal{o})(\mathsf{ȷ}-\mathsf{ı})}{4\mathsf{\Gamma }(⋌)\mathsf{\Gamma }(⋊)}}(IA){\int }_0^1{\int }_0^1\left[{\displaystyle \frac{{\mathfrak{g}}^{\prime }((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l})}{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g} ((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l})\right]}^{1-⋌}}} {\displaystyle \frac{{⋉}^{\prime }((1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ})}{{\left[⋉(\mathsf{ȷ})-⋉((1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ})\right]}^{1-\mathsf{\beta }}}}{\mathfrak{S}}_{\mu }(\mathcal{v}\mathcal{o}+(1-\mathcal{v})\mathcal{l},\mathsf{ı}\mathcal{r}+(1-\mathcal{r})\mathsf{ȷ})\right]d\mathcal{r}d\mathcal{v} \\ +{\displaystyle \frac{(\mathcal{l}-\mathcal{o})(\mathsf{ȷ}-\mathsf{ı})}{4\mathsf{\Gamma }(⋌)\mathsf{\Gamma }(⋊)}}(IA){\int }_0^1{\int }_0^1\left[{\displaystyle \frac{{\mathfrak{g}}^{\prime }((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l})}{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g} ((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l})\right]}^{1-⋌}}} {\displaystyle \frac{{⋉}^{\prime }((1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ})}{{\left[⋉(\mathsf{ȷ})-⋉((1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ})\right]}^{1-\mathsf{\beta }}}}{\mathfrak{S}}_{\mu }(\mathcal{v}\mathcal{o}+(1-\mathcal{v})\mathcal{l},(1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ})\right] d\mathcal{r}d\mathcal{v} \\ +{\displaystyle \frac{(\mathcal{l}-\mathcal{o})(\mathsf{ȷ}-\mathsf{ı})}{4\mathsf{\Gamma }(⋌)\mathsf{\Gamma }(⋊)}}(IA){\int }_0^1{\int }_0^1\left[{\displaystyle \frac{{\mathfrak{g}}^{\prime }((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l})}{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g} ((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l})\right]}^{1-⋌}}} {\displaystyle \frac{{⋉}^{\prime }((1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ})}{{\left[⋉(\mathsf{ȷ})-⋉((1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ})\right]}^{1-\mathsf{\beta }}}}{\mathfrak{S}}_{\mu }((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l},\mathsf{ı}\mathcal{r}+(1-\mathcal{r})\mathsf{ȷ})\right] d\mathcal{r}d\mathcal{v} \\ +{\displaystyle \frac{(\mathcal{l}-\mathcal{o})(\mathsf{ȷ}-\mathsf{ı})}{4\mathsf{\Gamma }(⋌)\mathsf{\Gamma }(⋊)}}(IA){\int }_0^1{\int }_0^1\left[{\displaystyle \frac{{\mathfrak{g}}^{\prime }((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l})}{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g} ((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l})\right]}^{1-⋌}}} {\displaystyle \frac{{⋉}^{\prime }((1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ})}{{\left[⋉(\mathsf{ȷ})-⋉((1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ})\right]}^{1-\mathsf{\beta }}}}{\mathfrak{S}}_{\mu }((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l},(1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ})\right] d\mathcal{r}d\mathcal{v} \end{gathered}$$

By a simple calculation, we have

$$(IA){\int }_0^1{\int }_0^1\left[{\displaystyle \frac{{\mathfrak{g}}^{\prime }((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l})}{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g} ((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l})\right]}^{1-⋌}}} {\displaystyle \frac{{⋉}^{\prime }((1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ})}{{\left[⋉(\mathsf{ȷ})-⋉((1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ})\right]}^{1-\mathsf{\beta }}}}\right] d\mathcal{r}d\mathcal{v}={\displaystyle \frac{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}{\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}{⋌⋊(\mathcal{l}-\mathcal{o})(\mathsf{ȷ}-\mathsf{ı})}}$$

Using the change in variables $\tau = (1 - \mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l} and \eta = (1 - \mathcal{r})\mathsf{ı} + \mathcal{r}\mathsf{ȷ}$, we obtain

$$\begin{gathered} {\displaystyle \frac{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}{\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}{\mathsf{\Gamma }(⋌+1)\mathsf{\Gamma }(⋊+1)}} {\mathfrak{S}}_{\mu }({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}}){\le }_I{\displaystyle \frac{1}{4\mathsf{\Gamma }(⋌)\mathsf{\Gamma }(⋊)}}(IA){\int }_{\mathcal{o}}^{\mathcal{l}}{\int }_{\mathsf{ı}}^{\mathsf{ȷ}}{\displaystyle \frac{{\mathfrak{g}}^{\prime }(\tau )}{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g} (\tau )\right]}^{1-⋌}}} {\displaystyle \frac{{⋉}^{\prime }(\eta )}{{\left[⋉(\mathsf{ȷ})-⋉(\eta )\right]}^{1-\mathsf{\beta }}}}{\mathfrak{S}}_{\mu }(\mathcal{o}+\mathcal{l}-\tau , \mathsf{ı}+\mathsf{ȷ}-\eta )d\eta d\tau \\ +{\displaystyle \frac{1}{4\mathsf{\Gamma }(⋌)\mathsf{\Gamma }(⋊)}}(IA){\int }_{\mathcal{o}}^{\mathcal{l}}{\int }_{\mathsf{ı}}^{\mathsf{ȷ}}{\displaystyle \frac{{\mathfrak{g}}^{\prime }(\tau )}{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g} (\tau )\right]}^{1-⋌}}} {\displaystyle \frac{{⋉}^{\prime }(\eta )}{{\left[⋉(\mathsf{ȷ})-⋉(\eta )\right]}^{1-\mathsf{\beta }}}}{\mathfrak{S}}_{\mu }(\mathcal{o}+\mathcal{l}-\tau , \eta )d\eta d\tau \\ +{\displaystyle \frac{1}{4\mathsf{\Gamma }(⋌)\mathsf{\Gamma }(⋊)}}(IA){\int }_{\mathcal{o}}^{\mathcal{l}}{\int }_{\mathsf{ı}}^{\mathsf{ȷ}}{\displaystyle \frac{{\mathfrak{g}}^{\prime }(\tau )}{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g} (\tau )\right]}^{1-⋌}}} {\displaystyle \frac{{⋉}^{\prime }(\eta )}{{\left[⋉(\mathsf{ȷ})-⋉(\eta )\right]}^{1-\mathsf{\beta }}}}{\mathfrak{S}}_{\mu }(\tau , \mathsf{ı}+\mathsf{ȷ}-\eta )d\eta d\tau \\ +{\displaystyle \frac{1}{4\mathsf{\Gamma }(⋌)\mathsf{\Gamma }(⋊)}}(IA){\int }_{\mathcal{o}}^{\mathcal{l}}{\int }_{\mathsf{ı}}^{\mathsf{ȷ}}{\displaystyle \frac{{\mathfrak{g}}^{\prime }(\tau )}{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g} (\tau )\right]}^{1-⋌}}} {\displaystyle \frac{{⋉}^{\prime }(\eta )}{{\left[⋉(\mathsf{ȷ})-⋉(\eta )\right]}^{1-\mathsf{\beta }}}}{\mathfrak{S}}_{\mu }(\tau , \eta )d\eta d\tau \\ ={\displaystyle \frac{1}{4}}\left[{\mathbb{J}}_{\mathcal{o}+,\mathsf{ı}+;\mathfrak{g},⋉}^{⋌,⋊}{{\mathsf{Ϫ}}_3}_{\mu }(\mathcal{l},\mathsf{ȷ})+{\mathbb{J}}_{\mathcal{o}+,\mathsf{ı}+;\mathfrak{g},⋉}^{⋌,⋊}{{\mathsf{Ϫ}}_1}_{\mu }(\mathcal{l},\mathsf{ȷ})+{\mathbb{J}}_{\mathcal{o}+,\mathsf{ı}+;\mathfrak{g},⋉}^{⋌,⋊}{{\mathsf{Ϫ}}_2}_{\mu }(\mathcal{l},\mathsf{ȷ})+{\mathbb{J}}_{\mathcal{o}+,\mathsf{ı}+;\mathfrak{g},⋉}^{⋌,⋊}{\mathfrak{S}}_{\mu }(\mathcal{l},\mathsf{ȷ})\right] \\ ={\displaystyle \frac{1}{4}}{\mathbb{J}}_{\mathcal{o}+,\mathsf{ı}+;\mathfrak{g},⋉}^{⋌,⋊}{\mathsf{Ϫ}}_{\mu }(\mathcal{l},\mathsf{ȷ}) \end{gathered}$$

That is, we have

$${\displaystyle \frac{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}{\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}{\mathsf{\Gamma }(⋌+1)\mathsf{\Gamma }(⋊+1)}} {\mathfrak{S}}_{\mu }({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}}){\le }_I{\displaystyle \frac{1}{4}}{\mathbb{J}}_{\mathcal{o}+,\mathsf{ı}+;\mathfrak{g},⋉}^{⋌,⋊}{\mathsf{Ϫ}}_{\mu }(\mathcal{l},\mathsf{ȷ})$$

(57)

Similarly, both sides of (56) are multiplied by

$${\displaystyle \frac{(\mathcal{l}-\mathcal{o})(\mathsf{ȷ}-\mathsf{ı})}{\mathsf{\Gamma }(⋌)\mathsf{\Gamma }(⋊)}} {\displaystyle \frac{{\mathfrak{g}}^{\prime }((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l})}{{\left[\mathrm{g}(\mathrm{b})-\mathfrak{g} ((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l})\right]}^{1-⋌}}} {\displaystyle \frac{{⋉}^{\prime }((1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ})}{{\left[⋉((1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{1-\mathsf{\beta }}}}$$

and integrating the obtained inequality in relation to $\mathcal{v},\mathcal{r} over [0, 1] \times [0, 1],$ we obtain

$${\displaystyle \frac{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}{\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}{\mathsf{\Gamma }(⋌+1)\mathsf{\Gamma }(⋊+1)}}\odot \tilde{\mathfrak{S}}({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}}){\le }_F{\displaystyle \frac{1}{4}}\odot {\mathbb{J}}_{\mathcal{o}+,\mathsf{ȷ}-;\mathfrak{g},⋉}^{⋌,⋊}\widetilde{\mathsf{Ϫ}}(\mathcal{l},\mathsf{ı})$$

(58)

Moreover, both sides of (56) are multiplied by

$${\displaystyle \frac{(\mathcal{l}-\mathcal{o})(\mathsf{ȷ}-\mathsf{ı})}{\mathsf{\Gamma }(⋌)\mathsf{\Gamma }(⋊)}} {\displaystyle \frac{{\mathfrak{g}}^{\prime }((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l})}{{\left[\mathfrak{g} ((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{1-⋌}}} {\displaystyle \frac{{⋉}^{\prime }((1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ})}{{\left[⋉(\mathsf{ȷ})-⋉((1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ})\right]}^{1-\mathsf{\beta }}}}$$

and

$${\displaystyle \frac{(\mathcal{l}-\mathcal{o})(\mathsf{ȷ}-\mathsf{ı})}{\mathsf{\Gamma }(⋌)\mathsf{\Gamma }(⋊)}} {\displaystyle \frac{{\mathfrak{g}}^{\prime }((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l})}{{\left[\mathfrak{g} ((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{1-⋌}}} {\displaystyle \frac{{⋉}^{\prime }((1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ})}{{\left[⋉((1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{1-\mathsf{\beta }}}}$$

Then, integrating the established inequalities in relation to $\mathcal{v},\mathcal{r}$ over $[0, 1]\times [0, 1]$ and for all $\mu$-cuts $\in \left[0,1\right]$, we have the following inequalities:

$${\displaystyle \frac{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}{\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}{\mathsf{\Gamma }(⋌+1)\mathsf{\Gamma }(⋊+1)}} {\mathfrak{S}}_{\mu }({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}}){\le }_I{\displaystyle \frac{1}{4}}{\mathbb{J}}_{\mathcal{l}-,\mathsf{ı}+;\mathfrak{g},⋉}^{⋌,⋊}{\mathsf{Ϫ}}_{\mu }(\mathcal{o},\mathsf{ȷ})$$

(59)

and

$${\displaystyle \frac{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}{\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}{\mathsf{\Gamma }(⋌+1)\mathsf{\Gamma }(⋊+1)}} {\mathfrak{S}}_{\mu }({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}}){\le }_I{\displaystyle \frac{1}{4}}{\mathbb{J}}_{\mathcal{l}-,\mathsf{ȷ}-;\mathfrak{g},⋉}^{⋌,⋊}{\mathsf{Ϫ}}_{\mu }(\mathcal{o},\mathsf{ı})$$

(60)

respectively.

Summing the inequalities (57) to (60), we get

$$\begin{gathered} {\mathfrak{S}}_{\mu }({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}}){\le }_I{\displaystyle \frac{\mathsf{\Gamma }(⋌+1)\mathsf{\Gamma }(⋊+1)}{{16 \left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}{\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}} \\ \times \left[{\mathbb{J}}_{\mathcal{o}+,\mathsf{ı}+;\mathfrak{g},⋉}^{⋌,⋊}{\mathsf{Ϫ}}_{\mu }(\mathcal{l},\mathsf{ȷ})+{\mathbb{J}}_{\mathcal{o}+,\mathsf{ȷ}-;\mathfrak{g},⋉}^{⋌,⋊}{\mathsf{Ϫ}}_{\mu }(\mathcal{l},\mathsf{ı})+{\mathbb{J}}_{\mathcal{l}-,\mathsf{ı}+;\mathfrak{g},⋉}^{⋌,⋊}{\mathsf{Ϫ}}_{\mu }(\mathcal{o},\mathsf{ȷ})+{\mathbb{J}}_{\mathcal{l}-,\mathsf{ȷ}-;\mathfrak{g},⋉}^{⋌,⋊}{\mathsf{Ϫ}}_{\mu }(\mathcal{o},\mathsf{ı})\right]. \end{gathered}$$

This completes the proof of the first inequality in (54).

For the proof of the second inequality in (54), since F is a coordinated convex, we have

$$\begin{gathered} {\mathfrak{S}}_{\mu }(\mathcal{v}\mathcal{o}+(1-\mathcal{v})\mathcal{l},\mathsf{ı}\mathcal{r}+(1-\mathcal{r})\mathsf{ȷ})+{\mathfrak{S}}_{\mu }(\mathcal{v}\mathcal{o}+(1-\mathcal{v})\mathcal{l},(1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ}) \\ +{\mathfrak{S}}_{\mu }((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l},\mathsf{ı}\mathcal{r}+(1-\mathcal{r})\mathsf{ȷ})+{\mathfrak{S}}_{\mu }((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l},(1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ}) \\ {\le }_I {\mathfrak{S}}_{\mu }(\mathcal{o},\mathsf{ı})+{\mathfrak{S}}_{\mu }(\mathcal{o},\mathsf{ȷ})+{\mathfrak{S}}_{\mu }(\mathcal{l},\mathsf{ı})+{\mathfrak{S}}_{\mu }(\mathcal{l},\mathsf{ȷ}). \end{gathered}$$

(61)

Both sides of (61) are multiplied by

$${\displaystyle \frac{(\mathcal{l}-\mathcal{o})(\mathsf{ȷ}-\mathsf{ı})}{\mathsf{\Gamma }(⋌)\mathsf{\Gamma }(⋊)}} {\displaystyle \frac{{\mathfrak{g}}^{\prime }((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l})}{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g} ((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l})\right]}^{1-⋌}}} {\displaystyle \frac{{⋉}^{\prime }((1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ})}{{\left[⋉(\mathsf{ȷ})-⋉((1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ})\right]}^{1-\mathsf{\beta }}}}$$

and integrating the resulting inequality in relation to $\mathcal{v},\mathcal{r} over [0, 1] \times [0, 1]$, we get

$${\displaystyle \frac{(\mathcal{l}-\mathcal{o})(\mathsf{ȷ}-\mathsf{ı})}{4\mathsf{\Gamma }(⋌)\mathsf{\Gamma }(⋊)}}(IA){\int }_0^1{\int }_0^1\left[\begin{array}{c}{\displaystyle \frac{{\mathfrak{g}}^{\prime }((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l})}{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g} ((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l})\right]}^{1-⋌}}} \\ {\displaystyle \frac{{⋉}^{\prime }((1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ})}{{\left[⋉(\mathsf{ȷ})-⋉((1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ})\right]}^{1-\mathsf{\beta }}}}\\ {\mathfrak{S}}_{\mu }(\mathcal{v}\mathcal{o}+(1-\mathcal{v})\mathcal{l},\mathsf{ı}\mathcal{r}+(1-\mathcal{r})\mathsf{ȷ})\end{array}\right]d\mathcal{r}d\mathcal{v}$$

$$+{\displaystyle \frac{(\mathcal{l}-\mathcal{o})(\mathsf{ȷ}-\mathsf{ı})}{4\mathsf{\Gamma }(⋌)\mathsf{\Gamma }(⋊)}}(IA){\int }_0^1{\int }_0^1\left[\begin{array}{c}{\displaystyle \frac{{\mathfrak{g}}^{\prime }((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l})}{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g} ((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l})\right]}^{1-⋌}}} \\ {\displaystyle \frac{{⋉}^{\prime }((1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ})}{{\left[⋉(\mathsf{ȷ})-⋉((1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ})\right]}^{1-\mathsf{\beta }}}}\\ {\mathfrak{S}}_{\mu }(\mathcal{v}\mathcal{o}+(1-\mathcal{v})\mathcal{l}, (1-r)\mathsf{ı}+r\mathsf{ȷ})\end{array}\right] d\mathcal{r}d\mathcal{v}$$

$$+{\displaystyle \frac{(\mathcal{l}-\mathcal{o})(\mathsf{ȷ}-\mathsf{ı})}{4\mathsf{\Gamma }(⋌)\mathsf{\Gamma }(⋊)}}(IA){\int }_0^1{\int }_0^1\left[\begin{array}{c}{\displaystyle \frac{{\mathit{v}}^{\prime }((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l})}{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g} ((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l})\right]}^{1-⋌}}} \\ {\displaystyle \frac{{⋉}^{\prime }((1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ})}{{\left[⋉(\mathsf{ȷ})-⋉((1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ})\right]}^{1-\mathsf{\beta }}}}\\ {\mathfrak{S}}_{\mu }((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l}, \mathsf{ı}r+(1-r)\mathsf{ȷ})\end{array}\right] d\mathcal{r}d\mathcal{v}$$

$$+{\displaystyle \frac{(\mathcal{l}-\mathcal{o})(\mathsf{ȷ}-\mathsf{ı})}{4\mathsf{\Gamma }(⋌)\mathsf{\Gamma }(⋊)}}(IA){\int }_0^1{\int }_0^1\left[\begin{array}{c}{\displaystyle \frac{{\mathfrak{g}}^{\prime }((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l})}{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g} ((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l})\right]}^{1-⋌}}}\\ {\displaystyle \frac{{⋉}^{\prime }((1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ})}{{\left[⋉(\mathsf{ȷ})-⋉((1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ})\right]}^{1-\mathsf{\beta }}}}\\ {\mathfrak{S}}_{\mu }((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l}, (1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ})\end{array}\right] d\mathcal{r}d\mathcal{v}$$

$${\le }_I{\displaystyle \frac{(\mathcal{l}-\mathcal{o})(\mathsf{ȷ}-\mathsf{ı})}{4\mathsf{\Gamma }(⋌)\mathsf{\Gamma }(⋊)}} \left[{\mathfrak{S}}_{\mu }(\mathcal{o},\mathsf{ı})+{\mathfrak{S}}_{\mu }(\mathcal{o},\mathsf{ȷ})+{\mathfrak{S}}_{\mu }(\mathcal{l},\mathsf{ı})+{\mathfrak{S}}_{\mu }(\mathcal{l},\mathsf{ȷ})\right]\times$$

$$(IA){\int }_0^1{\int }_0^1\left[{\displaystyle \frac{{\mathfrak{g}}^{\prime }((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l})}{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g} ((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l})\right]}^{1-⋌}}} {\displaystyle \frac{{⋉}^{\prime }((1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ})}{{\left[⋉(\mathsf{ȷ})-⋉((1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ})\right]}^{1-\mathsf{\beta }}}}\right] d\mathcal{r}d\mathcal{v}.$$

Then, we get

$${\mathbb{J}}_{\mathcal{o}+,\mathsf{ı}+;\mathfrak{g},⋉}^{⋌,⋊}{{\mathsf{Ϫ}}_3}_{\mu }(\mathcal{l},\mathsf{ȷ})+{\mathbb{J}}_{\mathcal{o}+,\mathsf{ı}+;\mathfrak{g},⋉}^{⋌,⋊}{{\mathsf{Ϫ}}_1}_{\mu }(\mathcal{l},\mathsf{ȷ})+{\mathbb{J}}_{\mathcal{o}+,\mathsf{ı}+;\mathfrak{g},⋉}^{⋌,⋊}{{\mathsf{Ϫ}}_2}_{\mu }(\mathcal{l},\mathsf{ȷ})+{\mathbb{J}}_{\mathcal{o}+,\mathsf{ı}+;\mathfrak{g},⋉}^{⋌,⋊}{\mathsf{Ϫ}}_{\mu }(\mathcal{l},\mathsf{ȷ})$$

$${\le }_I{\mathfrak{S}}_{\mu }(\mathcal{o},\mathsf{ı})+{\mathfrak{S}}_{\mu }(\mathcal{o},\mathsf{ȷ})+{\mathfrak{S}}_{\mu }(\mathcal{l},\mathsf{ı})+{\mathfrak{S}}_{\mu }(\mathcal{l},\mathsf{ȷ}){\displaystyle \frac{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}{\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}{\mathsf{\Gamma }(⋌+1)\mathsf{\Gamma }(⋊+1)}}$$

that is,

$${\displaystyle \frac{\mathsf{\Gamma }(⋌+1)\mathsf{\Gamma }(⋊+1)}{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}{\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}}{\mathbb{J}}_{\mathcal{o}+,\mathsf{ı}+;\mathfrak{g},⋉}^{⋌,⋊}{\mathsf{Ϫ}}_{\mu }(\mathcal{l},\mathsf{ȷ}){\le }_I{\mathfrak{S}}_{\mu }(\mathcal{o},\mathsf{ı})+{\mathfrak{S}}_{\mu }(\mathcal{o},\mathsf{ȷ})+{\mathfrak{S}}_{\mu }(\mathcal{l},\mathsf{ı})+{\mathfrak{S}}_{\mu }(\mathcal{l},\mathsf{ȷ}).$$

(62)

Similarly, both sides of (61) are multiplied by

$${\displaystyle \frac{(\mathcal{l}-\mathcal{o})(\mathsf{ȷ}-\mathsf{ı})}{\mathsf{\Gamma }(⋌)\mathsf{\Gamma }(⋊)}} {\displaystyle \frac{{\mathfrak{g}}^{\prime }((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l})}{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g} ((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l})\right]}^{1-⋌}}} {\displaystyle \frac{{⋉}^{\prime }((1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ})}{{\left[⋉((1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{1-\mathsf{\beta }}}}$$

$${\displaystyle \frac{(\mathcal{l}-\mathcal{o})(\mathsf{ȷ}-\mathsf{ı})}{\mathsf{\Gamma }(⋌)\mathsf{\Gamma }(⋊)}} {\displaystyle \frac{{\mathfrak{g}}^{\prime }((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l})}{{\left[\mathfrak{g} ((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{1-⋌}}} {\displaystyle \frac{{⋉}^{\prime }((1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ})}{{\left[⋉(\mathsf{ȷ})-⋉((1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ})\right]}^{1-\mathsf{\beta }}}}$$

and

$${\displaystyle \frac{(\mathcal{l}-\mathcal{o})(\mathsf{ȷ}-\mathsf{ı})}{\mathsf{\Gamma }(⋌)\mathsf{\Gamma }(⋊)}} {\displaystyle \frac{{\mathfrak{g}}^{\prime }((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l})}{{\left[\mathfrak{g} ((1-\mathcal{v})\mathcal{o}+\mathcal{v}\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{1-⋌}}} {\displaystyle \frac{{⋉}^{\prime }((1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ})}{{\left[⋉((1-\mathcal{r})\mathsf{ı}+\mathcal{r}\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{1-\mathsf{\beta }}}}$$

and integrating the resulting inequalities in relation to $\mathcal{v},\mathcal{r}$ over $[0, 1]\times [0, 1]$, we establish the following inequalities:

$${\displaystyle \frac{\mathsf{\Gamma }(⋌+1)\mathsf{\Gamma }(⋊+1)}{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}{\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}}{\mathbb{J}}_{\mathcal{o}+,\mathsf{ȷ}-;\mathfrak{g},⋉}^{⋌,⋊}{\mathsf{Ϫ}}_{\mu }(\mathcal{l},\mathsf{ı}){\le }_I{\mathfrak{S}}_{\mu }(\mathcal{o},\mathsf{ı})+{\mathfrak{S}}_{\mu }(\mathcal{o},\mathsf{ȷ})+{\mathfrak{S}}_{\mu }(\mathcal{l},\mathsf{ı})+{\mathfrak{S}}_{\mu }(\mathcal{l},\mathsf{ȷ}).$$

(63)

$${\displaystyle \frac{\mathsf{\Gamma }(⋌+1)\mathsf{\Gamma }(⋊+1)}{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}{\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}}{\mathbb{J}}_{\mathcal{l}-,\mathsf{ı}+;\mathfrak{g},⋉}^{⋌,⋊}{\mathsf{Ϫ}}_{\mu }(\mathcal{o},\mathsf{ȷ}){\le }_I{\mathfrak{S}}_{\mu }(\mathcal{o},\mathsf{ı})+{\mathfrak{S}}_{\mu }(\mathcal{o},\mathsf{ȷ})+{\mathfrak{S}}_{\mu }(\mathcal{l},\mathsf{ı})+{\mathfrak{S}}_{\mu }(\mathcal{l},\mathsf{ȷ}).$$

(64)

and

$${\displaystyle \frac{\mathsf{\Gamma }(⋌+1)\mathsf{\Gamma }(⋊+1)}{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}{\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}}{\mathbb{J}}_{\mathcal{l}-,\mathsf{ȷ}-;\mathfrak{g},⋉}^{⋌,⋊}{\mathsf{Ϫ}}_{\mu }(\mathcal{o},\mathsf{ı}){\le }_I{\mathfrak{S}}_{\mu }(\mathcal{o},\mathsf{ı})+{\mathfrak{S}}_{\mu }(\mathcal{o},\mathsf{ȷ})+{\mathfrak{S}}_{\mu }(\mathcal{l},\mathsf{ı})+{\mathfrak{S}}_{\mu }(\mathcal{l},\mathsf{ȷ}),$$

(65)

respectively.

By adding the inequalities (62) to (65), we have the following inequality.

$$\begin{gathered} {\displaystyle \frac{\mathsf{\Gamma }(⋌+1)\mathsf{\Gamma }(⋊+1)}{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}{\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}}\times \left[{\mathbb{J}}_{\mathcal{o}+,\mathsf{ı}+;\mathfrak{g},⋉}^{⋌,⋊}{\mathsf{Ϫ}}_{\mu }(\mathcal{l},\mathsf{ȷ})+{\mathbb{J}}_{\mathcal{o}+,\mathsf{ȷ}-;\mathfrak{g},⋉}^{⋌,⋊}{\mathsf{Ϫ}}_{\mu }(\mathcal{l},\mathsf{ı})+{\mathbb{J}}_{\mathcal{l}-,\mathsf{ı}+;\mathfrak{g},⋉}^{⋌,⋊}{\mathsf{Ϫ}}_{\mu }(\mathcal{o},\mathsf{ȷ})+{\mathbb{J}}_{\mathcal{l}-,\mathsf{ȷ}-;\mathfrak{g},⋉}^{⋌,⋊}{\mathsf{Ϫ}}_{\mu }(\mathcal{o},\mathsf{ı})\right] \\ {\le }_I 4[{\mathfrak{S}}_{\mu }(\mathcal{o},\mathsf{ı})+{\mathfrak{S}}_{\mu }(\mathcal{o},\mathsf{ȷ})+{\mathfrak{S}}_{\mu }(\mathcal{l},\mathsf{ı})+{\mathfrak{S}}_{\mu }(\mathcal{l},\mathsf{ȷ})]. \end{gathered}$$

(66)

If we divide both sides of inequality (66) by 16, then we have the second inequality in (54).

This completes the proof. □

Corollary 3.

Setting $\mathfrak{g}(\mathcal{v})=\mathcal{v} and ⋉(\mathcal{r})=\mathcal{r}$ in Theorem 4 yields the following inequalities for the Riemann–Liouville $F$·$N\cdot V$-fractional double integrals:

$$\begin{gathered} \tilde{\mathfrak{S}}({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}}){\le }_F{\displaystyle \frac{\mathsf{\Gamma }(⋌+1)\mathsf{\Gamma }(⋊+1)}{{4 (\mathcal{l}-\mathcal{o})}^{⋌}{(\mathsf{ȷ}-\mathsf{ı})}^{⋊}}}\odot \left[{\mathcal{J}}_{\mathcal{o}+,\mathsf{ı}+}^{⋌,⋊}\tilde{\mathfrak{S}}(\mathcal{l},\mathsf{ȷ})\oplus {\mathcal{J}}_{\mathcal{o}+,\mathsf{ȷ}-}^{⋌,⋊}\tilde{\mathfrak{S}}(\mathcal{l},\mathsf{ı})\oplus {\mathcal{J}}_{\mathcal{l}-,\mathsf{ı}+}^{⋌,⋊}\tilde{\mathfrak{S}}(\mathcal{o},\mathsf{ȷ})\oplus {\mathcal{J}}_{\mathcal{l}-,\mathsf{ȷ}-}^{⋌,⋊}\tilde{\mathfrak{S}}(\mathcal{o},\mathsf{ı})\right] \\ {\le }_F{\displaystyle \frac{\tilde{\mathfrak{S}}(\mathcal{o},\mathsf{ı})\oplus \tilde{\mathfrak{S}}(\mathcal{o},\mathsf{ȷ})\oplus \tilde{\mathfrak{S}}(\mathcal{l},\mathsf{ı})\oplus \tilde{\mathfrak{S}}(\mathcal{l},\mathsf{ȷ})}{4}}. \end{gathered}$$

Corollary 4.

Assuming the conditions of Theorem 4 hold, setting $\mathfrak{g}(\mathcal{v})=ln\mathcal{v}and ⋉(\mathcal{r})=ln \mathcal{r}$ yields the following inequalities for the Hadamard $F$·$N\cdot V$-fractional double integrals:

$$\begin{gathered} \tilde{\mathfrak{S}}({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}}){\le }_F{\displaystyle \frac{\mathsf{\Gamma }(⋌+1)\mathsf{\Gamma }(⋊+1)}{{16 \left[\ln \frac{\mathcal{l}}{\mathcal{o}}\right]}^{⋌}{\left[\ln \frac{\mathsf{ȷ}}{\mathsf{ı}}\right]}^{⋊}}}\odot \left[{\mathfrak{J}}_{\mathcal{o}+,\mathsf{ı}+}^{⋌,⋊}\widetilde{\mathsf{Ϫ}}(\mathcal{l},\mathsf{ȷ})\oplus {\mathfrak{J}}_{\mathcal{o}+,\mathsf{ȷ}-}^{⋌,⋊}\widetilde{\mathsf{Ϫ}}(\mathcal{l},\mathsf{ı})\oplus {\mathfrak{J}}_{\mathcal{l}-,\mathsf{ı}+}^{⋌,⋊}\widetilde{\mathsf{Ϫ}}(\mathcal{o},\mathsf{ȷ})\oplus {\mathfrak{J}}_{\mathcal{l}-,\mathsf{ȷ}-}^{⋌,⋊}\widetilde{\mathsf{Ϫ}}(\mathcal{o},\mathsf{ı})\right] \\ {\le }_F{\displaystyle \frac{\tilde{\mathfrak{S}}(\mathcal{o},\mathsf{ı})\oplus \tilde{\mathfrak{S}}(\mathcal{o},\mathsf{ȷ})\oplus \tilde{\mathfrak{S}}(\mathcal{l},\mathsf{ı})\oplus \tilde{\mathfrak{S}}(\mathcal{l},\mathsf{ȷ})}{4}}. \end{gathered}$$

Corollary 5.

Assuming the conditions of Theorem 4 hold, setting $\mathfrak{g}(\mathcal{v})={\displaystyle \frac{{\mathcal{v}}^{\rho }}{\rho }} and ⋉(\mathcal{r})={\displaystyle \frac{{\mathcal{r}}^{\sigma }}{\sigma }} , \rho ,\sigma >0$ yields the following inequalities for the Hadamard $F$·$N\cdot V$-Katugampola fractional double integrals:

$$\begin{gathered} \tilde{\mathfrak{S}}({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}}){\le }_F{\displaystyle \frac{\mathsf{\Gamma }(⋌+1)\mathsf{\Gamma }(⋊+1){\rho }^{⋌}{\sigma }^{⋊}}{{16 \left[{\mathcal{l}}^{\rho }-{\mathcal{o}}^{\rho }\right]}^{⋌}{\left[{\mathsf{ȷ}}^{\sigma }-{\mathsf{ı}}^{\sigma }\right]}^{⋊}}}\odot \left[{{}_{ }{}^{ \rho ,\sigma }\mathbb{I}}_{\mathcal{o}+,\mathsf{ı}+}^{⋌,⋊}\widetilde{\mathsf{Ϫ}}(\mathcal{l},\mathsf{ȷ})\oplus {{}_{ }{}^{ \rho ,\sigma }\mathbb{I}}_{\mathcal{o}+,\mathsf{ȷ}-}^{⋌,⋊}\widetilde{\mathsf{Ϫ}}(\mathcal{l},\mathsf{ı})\oplus {{}_{ }{}^{ \rho ,\sigma }\mathbb{I}}_{\mathcal{l}-,\mathsf{ı}+}^{⋌,⋊}\widetilde{\mathsf{Ϫ}}(\mathcal{o},\mathsf{ȷ})\oplus {{}_{ }{}^{ \rho ,\sigma }\mathbb{I}}_{\mathcal{l}-,\mathsf{ȷ}-}^{⋌,⋊}\widetilde{\mathsf{Ϫ}}(\mathcal{o},\mathsf{ı})\right] \\ {\le }_F{\displaystyle \frac{\tilde{\mathfrak{S}}(\mathcal{o},\mathsf{ı})\oplus \tilde{\mathfrak{S}}(\mathcal{o},\mathsf{ȷ})\oplus \tilde{\mathfrak{S}}(\mathcal{l},\mathsf{ı})\oplus \tilde{\mathfrak{S}}(\mathcal{l},\mathsf{ȷ})}{4}}. \end{gathered}$$

Theorem 5.

Let $\mathfrak{g}:[\mathcal{o},\mathcal{l}]\to \mathbb{R}$ and $⋉ : [\mathsf{ı}, \mathsf{ȷ}]\to \mathbb{R}$ be positive increasing mappings on $(\mathcal{o},\mathcal{l}]$ and $(\mathsf{ı}, \mathsf{ȷ}]$ with continuous derivatives such that ${\mathfrak{g}}^{\prime }(\mathcal{k})$ on $(\mathcal{o},\mathcal{l})$ and ${⋉}^{\prime }(\mathcal{s})$ on $(\mathsf{ı}, \mathsf{ȷ})$, respectively. Let $\Delta =[\mathcal{o},\mathcal{l}]\times [\mathsf{ı}, \mathsf{ȷ}]$. If $\tilde{\mathfrak{S}}:\Delta \to {\mathbb{F}}_0^{+}$ is a $F$·$N\cdot V$-coordinated convex mapping on Δ such that from $\mu$-cuts here, we produce the collection of $I\cdot V$-mappings ${\mathfrak{S}}_{\mu }:\Delta \to {\mathbb{R}}_{\mathrm{I}}^{+}$ described by ${\left[\tilde{\mathfrak{S}}(\mathcal{k},\mathcal{s})\right]}^{\mu }={\mathfrak{S}}_{\mu }((\mathcal{k},\mathcal{s}))=[\underset{\_}{\mathfrak{S}}((\mathcal{k},\mathcal{s}); \mu ),\stackrel{\mathfrak{‾}}{\mathfrak{S}}((\mathcal{k},\mathcal{s}); \mu )]$, for all $\mu \in \left[0,1\right] and (\mathcal{k},\mathcal{s})\in \Delta$, then, for $⋌, ⋊>0$, the following $H\cdot H$-type inequality holds:

$$\begin{gathered} \tilde{\mathfrak{S}}({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}}){\le }_F{\displaystyle \frac{\mathsf{\Gamma }(⋌+1)}{{8\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}}}\odot \left[{\mathbb{J}}_{\mathcal{o}+;\mathfrak{g}}^{⋌}\tilde{\mathcal{H}}(\mathcal{l},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}})\oplus {\mathbb{J}}_{\mathcal{l}-;\mathfrak{g}}^{⋌}\tilde{\mathcal{H}}(\mathcal{o},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}})\right] \\ \oplus {\displaystyle \frac{\mathsf{\Gamma }(⋊+1)}{{8\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}}\odot \left[{\mathbb{J}}_{\mathsf{ı}+;⋉}^{⋊}\tilde{\mathfrak{D}}({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}}, \mathsf{ȷ})\oplus {\mathbb{J}}_{\mathsf{ȷ}-;⋉}^{⋊}\tilde{\mathfrak{D}}({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}}, \mathsf{ı})\right] \\ {\le }_F{\displaystyle \frac{\mathsf{\Gamma }(⋌+1)\mathsf{\Gamma }(⋊+1)}{16{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}{\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}} \\ \odot \left[{\mathbb{J}}_{\mathcal{o}+,\mathsf{ı}+;\mathfrak{g},⋉}^{⋌,⋊}\widetilde{\mathsf{Ϫ}}(\mathcal{l},\mathsf{ȷ})\oplus {\mathbb{J}}_{\mathcal{o}+,\mathsf{ȷ}-;\mathfrak{g},⋉}^{⋌,⋊}\widetilde{\mathsf{Ϫ}}(\mathcal{l},\mathsf{ı})\oplus {\mathbb{J}}_{\mathcal{l}-,\mathsf{ı}+;\mathfrak{g},⋉}^{⋌,⋊}\widetilde{\mathsf{Ϫ}}(\mathcal{o},\mathsf{ȷ})\oplus {\mathbb{J}}_{\mathcal{l}-,\mathsf{ȷ}-;\mathfrak{g},⋉}^{⋌,⋊}\widetilde{\mathsf{Ϫ}}(\mathcal{o},\mathsf{ı})\right] \\ {\le }_F{\displaystyle \frac{\mathsf{\Gamma }(⋌+1)}{{16\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}}}\odot \left[{\mathbb{J}}_{\mathcal{o}+;\mathfrak{g}}^{⋌}\tilde{\mathcal{H}}(\mathcal{l},\mathsf{ı})\oplus {\mathbb{J}}_{\mathcal{o}+;\mathfrak{g}}^{⋌}\tilde{\mathcal{H}}(\mathcal{l},\mathsf{ȷ})\oplus {\mathbb{J}}_{\mathcal{l}-;\mathfrak{g}}^{⋌}\tilde{\mathcal{H}}(\mathcal{o},\mathsf{ı})\oplus {\mathbb{J}}_{\mathcal{l}-;\mathfrak{g}}^{⋌}\tilde{\mathcal{H}}(\mathcal{o},\mathsf{ȷ})\right] \\ \oplus {\displaystyle \frac{\mathsf{\Gamma }(⋊+1)}{{16\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}}\odot \left[{\mathbb{J}}_{\mathsf{ı}+;⋉}^{⋊}\tilde{\mathfrak{D}}(\mathcal{o},\mathsf{ȷ})\oplus {\mathbb{J}}_{\mathsf{ı}+;⋉}^{⋊}\tilde{\mathfrak{D}}(\mathcal{l},\mathsf{ȷ})\oplus {\mathbb{J}}_{\mathsf{ȷ}-;⋉}^{⋊}\tilde{\mathfrak{D}}(\mathcal{o},\mathsf{ı})\oplus {\mathbb{J}}_{\mathsf{ȷ}-;⋉}^{⋊}\tilde{\mathfrak{D}}(\mathcal{l},\mathsf{ı})\right] \\ {\le }_F{\displaystyle \frac{\tilde{\mathfrak{S}}(\mathcal{o},\mathsf{ı})\oplus \tilde{\mathfrak{S}}(\mathcal{o},\mathsf{ȷ})\oplus \tilde{\mathfrak{S}}(\mathcal{l},\mathsf{ı})\oplus \tilde{\mathfrak{S}}(\mathcal{l},\mathsf{ȷ})}{4}}, \end{gathered}$$

(67)

where

$$\widetilde{\mathsf{Ϫ}}1(\mathcal{k},\mathcal{s})=\tilde{\mathfrak{S}}(\mathcal{o}+\mathcal{l}-\mathcal{k},\mathcal{s}),$$

$$\widetilde{\mathsf{Ϫ}}2(\mathcal{k},\mathcal{s})=\tilde{\mathfrak{S}}(\mathcal{k},\mathsf{ı}+\mathsf{ȷ}-\mathcal{s}),$$

$$\widetilde{\mathsf{Ϫ}}3(\mathcal{k},\mathcal{s})=\tilde{\mathfrak{S}}(\mathcal{o}+\mathcal{l}-\mathcal{k},\mathsf{ı}+\mathsf{ȷ}-\mathcal{s}),$$

$$\tilde{\mathfrak{D}}(\mathcal{k},\mathcal{s})=\tilde{\mathfrak{S}}(\mathcal{k},\mathcal{s})\oplus \widetilde{\mathsf{Ϫ}}2(\mathcal{k},\mathcal{s})$$

$$\tilde{\mathcal{H}}(\mathcal{k},\mathcal{s})=\tilde{\mathfrak{S}}(\mathcal{k},\mathcal{s})\oplus \widetilde{\mathsf{Ϫ}}1(\mathcal{k},\mathcal{s})$$

$$\widetilde{\mathsf{Ϗ}}(\mathcal{k},\mathcal{s})=\widetilde{\mathsf{Ϫ}}1(\mathcal{k},\mathcal{s})\oplus \widetilde{\mathsf{Ϫ}}3(\mathcal{k},\mathcal{s})$$

$$\tilde{\mathcal{L}}(\mathcal{k},\mathcal{s})=\widetilde{\mathsf{Ϫ}}2(\mathcal{k},\mathcal{s})\oplus \widetilde{\mathsf{Ϫ}}3(\mathcal{k},\mathcal{s})$$

$$\widetilde{\mathsf{Ϫ}}(\mathcal{k},\mathcal{s})=\widetilde{\mathsf{Ϫ}}1(\mathcal{k},\mathcal{s})\oplus \widetilde{\mathsf{Ϫ}}2(\mathcal{k},\mathcal{s})\oplus \widetilde{\mathsf{Ϫ}}3(\mathcal{k},\mathcal{s})\oplus \tilde{\mathfrak{S}}(\mathcal{k},\mathcal{s})$$

$$={\displaystyle \frac{\tilde{\mathfrak{D}}(\mathcal{k},\mathcal{s})\oplus \tilde{\mathcal{H}}(\mathcal{k},\mathcal{s})\oplus \widetilde{\mathsf{Ϗ}}(\mathcal{k},\mathcal{s})\oplus \tilde{\mathcal{L}}(\mathcal{k},\mathcal{s})}{2}}$$

for $(\mathcal{k},\mathcal{s})\in [\mathcal{o},\mathcal{l}]\times [\mathsf{ı}, \mathsf{ȷ}].$ where for all $\mu$-cuts $\in \left[\mathrm{0,1}\right]$, we have ${\left[\widetilde{\mathsf{Ϫ}}1(\mathcal{k},\mathcal{s})\right]}^{\mu }={{\mathsf{Ϫ}}_1}_{\mu }(\mathcal{k},\mathcal{s})$, ${\left[\widetilde{\mathsf{Ϫ}}2(\mathcal{k},\mathcal{s})\right]}^{\mu }={{\mathsf{Ϫ}}_2}_{\mu }(\mathcal{k},\mathcal{s})$, ${\left[\widetilde{\mathsf{Ϫ}}3(\mathcal{k},\mathcal{s})\right]}^{\mu }={{\mathsf{Ϫ}}_3}_{\mu }(\mathcal{k},\mathcal{s})$, ${\left[\tilde{\mathfrak{D}}(\mathcal{k},\mathcal{s})\right]}^{\mu }={\mathfrak{D}}_{\mu }(\mathcal{k},\mathcal{s})$, ${\left[\tilde{\mathcal{H}}(\mathcal{k},\mathcal{s})\right]}^{\mu }={\mathcal{H}}_{\mu }(\mathcal{k},\mathcal{s})$, ${\left[\widetilde{\mathsf{Ϗ}}(\mathcal{k},\mathcal{s})\right]}^{\mu }={\mathsf{Ϗ}}_{\mu }(\mathcal{k},\mathcal{s})$, ${\left[\tilde{\mathcal{L}}(\mathcal{k},\mathcal{s})\right]}^{\mu }={\mathcal{L}}_{\mu }(\mathcal{k},\mathcal{s})$ and ${\left[\widetilde{\mathsf{Ϫ}}(\mathcal{k},\mathcal{s})\right]}^{\mu }={\mathsf{Ϫ}}_{\mu }(\mathcal{k},\mathcal{s})$.

Proof. 

Since $\tilde{\mathfrak{S}}$ is a $F$·$N\cdot V$-coordinated convex on $\Delta$, if we define the mapping $h_{\mathcal{k}}^1:\left[\mathsf{ı},\mathsf{ȷ}\right]\to {\mathbb{F}}_0^{+}, h_{\mathcal{k}}^1=\tilde{\mathfrak{S}}(\mathcal{k},\mathcal{s}),$ then $h_{\mathcal{k}}^1(\mathcal{s})$ is convex for all $\mathcal{k}\in [\mathcal{o},\mathcal{l}]$ and ${\tilde{\mathcal{H}}}_{\mathcal{k}}^1(\mathcal{s})=h_{\mathcal{k}}^1(\mathcal{s})\oplus \widetilde{h_{\mathcal{k}}^1}(\mathcal{s})=\tilde{\mathfrak{S}}(\mathcal{k},\mathcal{s})\oplus {\widetilde{\mathsf{Ϫ}}}_2(\mathcal{k},\mathcal{s})=\tilde{\mathfrak{D}}(\mathcal{k},\mathcal{s})$. If we apply the inequalities (22) for the convex mapping $h_{\mathcal{k}}^1(\mathcal{s})$, then we have

$$h_{\mathcal{k}}^1({\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}}){\le }_F{\displaystyle \frac{\mathsf{\Gamma }(⋊+1)}{{4\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}}\odot \left[{\mathbb{J}}_{\mathsf{ı}+;⋉}^{⋊}{\tilde{\mathcal{H}}}_{\mathcal{k}}^1(\mathsf{ȷ})\oplus {\mathbb{J}}_{\mathsf{ȷ}-;⋉}^{⋊}{\tilde{\mathcal{H}}}_{\mathcal{k}}^1(\mathsf{ı})\right]{\le }_F{\displaystyle \frac{h_{\mathcal{k}}^1(\mathsf{ı})\oplus h_{\mathcal{k}}^1(\mathsf{ȷ})}{2}},$$

Then, for all $\mu$-cuts $\in \left[0,1\right]$, we have

$$\begin{gathered} {\mathfrak{S}}_{\mu }(\mathcal{k},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}}){\le }_I{\displaystyle \frac{⋊}{{4\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}}\left[(IA){\int }_{\mathsf{ı}}^{\mathsf{ȷ}}{\displaystyle \frac{{⋉}^{\prime }(\mathcal{s})}{{\left[⋉(\mathsf{ȷ})-⋉(\mathcal{s})\right]}^{1-⋊}}}{\mathfrak{D}}_{\mu }(\mathcal{k},\mathcal{s})d\mathcal{s}+(IA){\int }_{\mathsf{ı}}^{\mathsf{ȷ}}{\displaystyle \frac{{⋉}^{\prime }(\mathcal{s})}{{\left[⋉(\mathcal{s})-⋉(\mathsf{ı})\right]}^{1-⋊}}}{\mathfrak{D}}_{\mu }(\mathcal{k},\mathcal{s})d\mathcal{s}\right] \\ {\le }_I{\displaystyle \frac{{\mathfrak{S}}_{\mu }(\mathcal{k},\mathsf{ı})+{\mathfrak{S}}_{\mu } (\mathcal{k},\mathsf{ȷ})}{2}} \end{gathered}$$

(68)

Both sides of (68) are multiplied by

$${\displaystyle \frac{⋌}{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}}} {\displaystyle \frac{{\mathrm{g}}^{\prime }(\mathcal{k})}{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{k})\right]}^{1-⋌}}}$$

and

$${\displaystyle \frac{⋌}{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}}} {\displaystyle \frac{{\mathrm{g}}^{\prime }(\mathcal{k})}{{\left[\mathfrak{g}(\mathcal{k})-\mathfrak{g}(\mathcal{o})\right]}^{1-⋌}}}$$

then by integrating the obtained results in relation to x from $\mathcal{o}$ to $\mathcal{l}$, we get

$$\begin{gathered} {\displaystyle \frac{ \Gamma (⋌+1)}{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}}}{\mathbb{J}}_{{\mathcal{o}}^{+};\mathfrak{g}}^{⋌}{\mathfrak{S}}_{\mu }(\mathcal{l},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}}) \\ {\le }_I{\displaystyle \frac{\mathsf{\Gamma }(⋌+1)\mathsf{\Gamma }(⋊+1)}{4{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}{\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}}\left[{\mathbb{J}}_{\mathcal{o}+,\mathsf{ı}+;\mathfrak{g},⋉}^{⋌,⋊}{\mathfrak{D}}_{\mu }(\mathcal{l},\mathsf{ȷ})+{\mathbb{J}}_{\mathcal{o}+,\mathsf{ȷ}-;\mathfrak{g},⋉}^{⋌,⋊}{\mathfrak{D}}_{\mu }(\mathcal{l},\mathsf{ı})\right] \\ {\le }_I{\displaystyle \frac{ \Gamma (⋌+1)}{2{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}}}\left[{\mathbb{J}}_{\mathcal{o}+;\mathfrak{g}}^{⋌}{\mathfrak{S}}_{\mu } (\mathcal{l},\mathsf{ı})+{\mathbb{J}}_{\mathcal{o}+;\mathfrak{g}}^{⋌}{\mathfrak{S}}_{\mu } (\mathcal{l},\mathsf{ȷ})\right] \end{gathered}$$

(69)

and

$$\begin{gathered} {\displaystyle \frac{ \Gamma (⋌+1)}{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}}}{\mathbb{J}}_{\mathcal{l}-;\mathfrak{g}}^{⋌}{\mathfrak{S}}_{\mu }(\mathcal{o},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}}) \\ {\le }_I{\displaystyle \frac{\mathsf{\Gamma }(⋌+1)\mathsf{\Gamma }(⋊+1)}{4{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}{\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}}\left[{\mathbb{J}}_{\mathcal{l}-,\mathsf{ı}+;\mathfrak{g},⋉}^{⋌,⋊}{\mathfrak{D}}_{\mu }(\mathcal{o},\mathsf{ȷ})+{\mathbb{J}}_{\mathcal{l}-,\mathsf{ȷ}-;\mathfrak{g},⋉}^{⋌,⋊}{\mathfrak{D}}_{\mu }(\mathcal{o},\mathsf{ı})\right] \\ {\le }_I{\displaystyle \frac{ \Gamma (⋌+1)}{2{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}}}\left[{\mathbb{J}}_{\mathcal{l}-;\mathfrak{g}}^{⋌}{\mathfrak{S}}_{\mu }(\mathcal{o},\mathsf{ı})+{\mathbb{J}}_{\mathcal{l}-;\mathfrak{g}}^{⋌}{\mathfrak{S}}_{\mu }(\mathcal{o},\mathsf{ȷ})\right] \end{gathered}$$

(70)

respectively.

On the other hand, since $\tilde{\mathfrak{S}}$ is a coordinated convex on Δ, if we define the mapping $h_{\mathcal{k}}^2 : \left[\mathsf{ı},\mathsf{ȷ}\right]\to {\mathbb{F}}_0^{+}, h_{\mathcal{k}}^2(\mathcal{s})={\widetilde{\mathsf{Ϫ}}}_1(\mathcal{k},\mathcal{s}),$ then $h_{\mathcal{k}}^2(\mathcal{s})$ is convex for all $\mathcal{k}\in [\mathcal{o},\mathcal{l}]$ and ${\tilde{\mathcal{H}}}_{\mathcal{k}}^2(\mathcal{s})=h_{\mathcal{k}}^2(\mathcal{s})\oplus \widetilde{h_{\mathcal{k}}^2}(\mathcal{s})={\widetilde{\mathsf{Ϫ}}}_1(\mathcal{k},\mathcal{s})\oplus {\widetilde{\mathsf{Ϫ}}}_3(\mathcal{k},\mathcal{s})=\widetilde{\mathsf{Ϗ}}(\mathcal{k},\mathcal{s})$. If we apply the inequalities (22) for the convex mapping $h_{\mathcal{k}}^2(\mathcal{s})$, then we have

$$h_{\mathcal{k}}^2({\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}}){\le }_F{\displaystyle \frac{\mathsf{\Gamma }(⋊+1)}{{4\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}}\odot \left[{\mathbb{J}}_{\mathsf{ı}+;⋉}^{⋊}{\tilde{\mathcal{H}}}_{\mathcal{k}}^2(\mathsf{ȷ})\oplus {\mathbb{J}}_{\mathsf{ȷ}-;⋉}^{⋊}{\tilde{\mathcal{H}}}_{\mathcal{k}}^2(\mathsf{ı})\right]{\le }_F{\displaystyle \frac{h_{\mathcal{k}}^2(\mathsf{ı})\oplus h_{\mathcal{k}}^2(\mathsf{ȷ})}{2}},$$

that is, for all $\mu$-cuts $\in \left[0,1\right]$, we have

$$\begin{gathered} {{\mathsf{Ϫ}}_1}_{\mu }(\mathcal{k},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}}) \\ {\le }_I{\displaystyle \frac{⋊}{{4\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}}\odot \left[(IA){\displaystyle \underset{\mathsf{ı}}{\overset{\mathsf{ȷ}}{\int }}}{\displaystyle \frac{{⋉}^{\prime }(\mathcal{s})}{{\left[⋉(\mathsf{ȷ})-⋉(\mathcal{s})\right]}^{1-⋊}}}{\mathsf{Ϗ}}_{\mu }(\mathcal{k},\mathcal{s})d\mathcal{s}+(IA){\displaystyle \underset{\mathsf{ı}}{\overset{\mathsf{ȷ}}{\int }}}{\displaystyle \frac{{⋉}^{\prime }(\mathcal{s})}{{\left[⋉(\mathcal{s})-⋉(\mathsf{ı})\right]}^{1-⋊}}}{\mathsf{Ϗ}}_{\mu }(\mathcal{k},\mathcal{s})d\mathcal{s}\right] \\ {\le }_I{\displaystyle \frac{{{\mathsf{Ϫ}}_1}_{\mu }(\mathcal{k},\mathsf{ı})+{{\mathsf{Ϫ}}_1}_{\mu } (\mathcal{k},\mathsf{ȷ})}{2}} \end{gathered}$$

(71)

Similarly, both sides of (71) are multiplied by

$${\displaystyle \frac{⋌}{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}}} {\displaystyle \frac{{\mathrm{g}}^{\prime }(\mathcal{k})}{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{k})\right]}^{1-⋌}}}$$

and

$${\displaystyle \frac{⋌}{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}}} {\displaystyle \frac{{\mathrm{g}}^{\prime }(\mathcal{k})}{{\left[\mathfrak{g}(\mathcal{k})-\mathfrak{g}(\mathcal{o})\right]}^{1-⋌}}}$$

then by integrating the obtained results in relation to x from $\mathcal{o}$ to $\mathcal{l}$, we get

$$\begin{gathered} {\displaystyle \frac{ \Gamma (⋌+1)}{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}}}{\mathbb{J}}_{{\mathcal{o}}^{+};\mathfrak{g}}^{⋌}{{\mathsf{Ϫ}}_1}_{\mu }(\mathcal{l},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}}) \\ {\le }_I{\displaystyle \frac{\mathsf{\Gamma }(⋌+1)\mathsf{\Gamma }(⋊+1)}{4{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}{\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}}\left[{\mathbb{J}}_{\mathcal{o}+,\mathsf{ı}+;\mathfrak{g},⋉}^{⋌,⋊}{\mathsf{Ϗ}}_{\mu }(\mathcal{l},\mathsf{ȷ})+{\mathbb{J}}_{\mathcal{o}+,\mathsf{ȷ}-;\mathfrak{g},⋉}^{⋌,⋊}{\mathsf{Ϗ}}_{\mu }(\mathcal{l},\mathsf{ı})\right] \\ {\le }_I{\displaystyle \frac{ \Gamma (⋌+1)}{2{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}}}\left[{\mathbb{J}}_{\mathcal{o}+;\mathfrak{g}}^{⋌}{{\mathsf{Ϫ}}_1}_{\mu }(\mathcal{l},\mathsf{ı})+{\mathbb{J}}_{\mathcal{o}+;\mathfrak{g}}^{⋌}{{\mathsf{Ϫ}}_1}_{\mu }(\mathcal{l},\mathsf{ȷ})\right] \end{gathered}$$

(72)

and

$$\begin{gathered} {\displaystyle \frac{ \Gamma (⋌+1)}{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}}}{\mathbb{J}}_{\mathcal{l}-;\mathfrak{g}}^{⋌}{{\mathsf{Ϫ}}_1}_{\mu }(\mathcal{o},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}}) \\ {\le }_I{\displaystyle \frac{\mathsf{\Gamma }(⋌+1)\mathsf{\Gamma }(⋊+1)}{4{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}{\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}}\left[{\mathbb{J}}_{\mathcal{l}-,\mathsf{ı}+;\mathfrak{g},⋉}^{⋌,⋊}{\mathsf{Ϗ}}_{\mu }(\mathcal{o},\mathsf{ȷ})+{\mathbb{J}}_{\mathcal{l}-,\mathsf{ȷ}-;\mathfrak{g},⋉}^{⋌,⋊}{\mathsf{Ϗ}}_{\mu }(\mathcal{o},\mathsf{ı})\right] \\ {\le }_I{\displaystyle \frac{ \Gamma (⋌+1)}{2{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}}}\left[{\mathbb{J}}_{\mathcal{l}-;\mathfrak{g}}^{⋌}{{\mathsf{Ϫ}}_1}_{\mu }(\mathcal{o},\mathsf{ı})+{\mathbb{J}}_{\mathcal{l}-;\mathfrak{g}}^{⋌}{{\mathsf{Ϫ}}_1}_{\mu }(\mathcal{o},\mathsf{ȷ})\right] \end{gathered}$$

(73)

respectively.

Moreover, if we define the mapping $h_{\mathcal{s}}^1 : \left[\mathcal{o},\mathcal{l}\right]\to {\mathbb{F}}_0^{+}, h_{\mathcal{s}}^1(\mathcal{k})=\tilde{\mathfrak{S}}(\mathcal{k},\mathcal{s}),$ then $h_{\mathcal{s}}^1(\mathcal{k})$ is convex for all $\mathcal{s}\in [\mathsf{ı}, \mathsf{ȷ}]$ and ${\tilde{\mathcal{H}}}_{\mathcal{s}}^1(\mathcal{k})=h_{\mathcal{s}}^1(\mathcal{k})\oplus \widetilde{h_{\mathcal{s}}^1}(\mathcal{k})=\tilde{\mathfrak{S}}(\mathcal{k},\mathcal{s})\oplus {{\mathsf{Ϫ}}_1}_{\mu }(\mathcal{k},\mathcal{s})=\tilde{\mathcal{H}}(\mathcal{k},\mathcal{s})$. If we apply the inequalities (22) for the convex mapping $h_{\mathcal{s}}^1(\mathcal{k})$, then we have

$$h_{\mathcal{s}}^1({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}}){\le }_F{\displaystyle \frac{\mathsf{\Gamma }(⋌+1)}{{4\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}}}\odot \left[{\mathbb{J}}_{\mathsf{ȷ}+;\mathfrak{g}}^{⋌}{\tilde{\mathcal{H}}}_{\mathcal{s}}^1(\mathcal{l})\oplus {\mathbb{J}}_{\mathcal{l}-;\mathfrak{g}}^{⋌}{\tilde{\mathcal{H}}}_{\mathcal{s}}^1(\mathcal{o})\right]{\le }_F{\displaystyle \frac{h_{\mathcal{s}}^1(\mathcal{o})\oplus h_{\mathcal{s}}^1(\mathcal{l})}{2}},$$

For all $\mu$-cuts $\in \left[0,1\right]$, we have

$$\begin{gathered} {\mathfrak{S}}_{\mu }( {\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},\mathcal{s}) \\ {\le }_I{\displaystyle \frac{⋌}{{4\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}}}\left[(IA){\int }_{\mathcal{o}}^{\mathcal{l}}{\displaystyle \frac{{\mathfrak{g}}^{\prime }(\mathcal{k})}{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{k})\right]}^{1-⋌}}}{\mathcal{H}}_{\mu }(\mathcal{k},\mathcal{s})d\mathcal{k}+(IA){\int }_{\mathcal{o}}^{\mathcal{l}}{\displaystyle \frac{{\mathfrak{g}}^{\prime }(\mathcal{k})}{{\left[\mathfrak{g}(\mathcal{k})-\mathfrak{g}(\mathcal{o})\right]}^{1-⋌}}}{\mathcal{H}}_{\mu }(\mathcal{k},\mathcal{s})d\mathcal{k}\right] \\ {\le }_I{\displaystyle \frac{{\mathfrak{S}}_{\mu }(\mathcal{o},\mathcal{s})+{\mathfrak{S}}_{\mu } (\mathcal{l},\mathcal{s})}{2}} \end{gathered}$$

(74)

Similarly, both sides of (74) are multiplied by

$${\displaystyle \frac{\mathsf{\beta }}{{\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{\mathsf{\beta }}}} {\displaystyle \frac{{\mathrm{g}}^{\prime }(\mathcal{s})}{{\left[⋉(\mathsf{ȷ})-⋉(\mathcal{s})\right]}^{1-\mathsf{\beta }}}}$$

and

$${\displaystyle \frac{\mathsf{\beta }}{{\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{\mathsf{\beta }}}} {\displaystyle \frac{{\mathrm{g}}^{\prime }(\mathcal{s})}{{\left[⋉(\mathcal{s})-⋉(\mathsf{ı})\right]}^{1-\mathsf{\beta }}}}$$

then integrating the established results in relation to y from $\mathsf{ı}$ to $\mathsf{ȷ}$, we obtain the following inequalities:

$$\begin{gathered} {\displaystyle \frac{ \Gamma (⋊ +1)}{{\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}}{\mathbb{J}}_{{\mathsf{ı}}^{+};⋉}^{⋊}{\mathfrak{S}}_{\mu }( {\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},\mathsf{ȷ}) \\ {\le }_I{\displaystyle \frac{\mathsf{\Gamma }(⋌+1)\mathsf{\Gamma }(⋊+1)}{4{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}{\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}}\left[{\mathbb{J}}_{\mathcal{o}+,\mathsf{ı}+;\mathfrak{g},⋉}^{⋌,⋊}{\mathcal{H}}_{\mu }(\mathcal{l},\mathsf{ȷ})+{\mathbb{J}}_{\mathcal{o}+,\mathsf{ȷ}-;\mathfrak{g},⋉}^{⋌,⋊}{\mathcal{H}}_{\mu }(\mathcal{o},\mathsf{ȷ})\right] \\ {\le }_I{\displaystyle \frac{ \Gamma (⋊ +1)}{2{\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}}\left[{\mathbb{J}}_{\mathsf{ı}+;⋉}^{⋊}{\mathfrak{S}}_{\mu } (\mathcal{o},\mathsf{ȷ})+{\mathbb{J}}_{\mathsf{ı}+;⋉}^{⋊}{\mathfrak{S}}_{\mu } (\mathcal{l},\mathsf{ȷ})\right] \end{gathered}$$

(75)

and

$$\begin{gathered} {\displaystyle \frac{ \Gamma (⋊+1)}{{\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}}{\mathbb{J}}_{\mathsf{ȷ}-;⋉}^{⋊}{\mathfrak{S}}_{\mu }( {\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},\mathsf{ı}) \\ {\le }_I{\displaystyle \frac{\mathsf{\Gamma }(⋌+1)\mathsf{\Gamma }(⋊+1)}{4{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}{\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}}\left[{\mathbb{J}}_{\mathcal{o}+,\mathsf{ȷ}-;\mathfrak{g},⋉}^{⋌,⋊}{\mathcal{H}}_{\mu }(\mathcal{l},\mathsf{ı})+{\mathbb{J}}_{\mathcal{l}-,\mathsf{ȷ}-;\mathfrak{g},⋉}^{⋌,⋊}{\mathcal{H}}_{\mu }(\mathcal{o},\mathsf{ı})\right] \\ {\le }_I{\displaystyle \frac{ \Gamma (⋊+1)}{2{\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}}\left[{\mathbb{J}}_{\mathsf{ȷ}-;⋉}^{⋊}{\mathfrak{S}}_{\mu } (\mathcal{o},\mathsf{ı})+{\mathbb{J}}_{\mathsf{ȷ}-;⋉}^{⋊}{\mathfrak{S}}_{\mu } (\mathcal{l},\mathsf{ı})\right] \end{gathered}$$

(76)

respectively.

Furthermore, if we define the mapping $h_{\mathcal{s}}^2 : \left[\mathcal{o},\mathcal{l}\right]\to {\mathbb{F}}_0^{+}, h_{\mathcal{s}}^2(\mathcal{k})={\widetilde{\mathsf{Ϫ}}}_2(\mathcal{k},\mathcal{s}),$ then $h_{\mathcal{s}}^2(\mathcal{k})$ is convex for all $\mathcal{s}\in [\mathsf{ı}, \mathsf{ȷ}]$ and ${\tilde{\mathcal{H}}}_{\mathcal{s}}^2(\mathcal{k})=h_{\mathcal{s}}^2(\mathcal{k})\oplus \widetilde{h_{\mathcal{s}}^2}(\mathcal{k})={\widetilde{\mathsf{Ϫ}}}_2(\mathcal{k},\mathcal{s})\oplus {\widetilde{\mathsf{Ϫ}}}_3(\mathcal{k},\mathcal{s})=\tilde{\mathcal{L}}(\mathcal{k},\mathcal{s})$. If we apply the inequalities (22) for the convex mapping $h_{\mathcal{s}}^2(\mathcal{k})$, then we have

$$h_{\mathcal{s}}^2({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}}){\le }_F{\displaystyle \frac{\mathsf{\Gamma }(⋌+1)}{{4\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}}}\odot \left[{\mathbb{J}}_{\mathcal{o}+;\mathfrak{g}}^{⋌}{\tilde{\mathcal{H}}}_{\mathcal{s}}^2(\mathcal{l})\oplus {\mathbb{J}}_{\mathcal{l}-;\mathfrak{g}}^{⋌}{\tilde{\mathcal{H}}}_{\mathcal{s}}^2(\mathcal{o})\right]{\le }_F{\displaystyle \frac{h_{\mathcal{s}}^2(\mathcal{o})\oplus h_{\mathcal{s}}^2(\mathcal{l})}{2}},$$

for all $\mu$-cuts $\in \left[0,1\right]$, we have

$$\begin{gathered} {{\mathsf{Ϫ}}_2}_{\mu }( {\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},\mathcal{s}){\le }_I{\displaystyle \frac{⋌}{{4\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}}}\left[(IA){\int }_{\mathcal{o}}^{\mathcal{l}}{\displaystyle \frac{{\mathfrak{g}}^{\prime }(\mathcal{k})}{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{k})\right]}^{1-⋌}}}{\mathcal{L}}_{\mu }(\mathcal{k},\mathcal{s})d\mathcal{k}+(IA){\int }_{\mathcal{o}}^{\mathcal{l}}{\displaystyle \frac{{\mathfrak{g}}^{\prime }(\mathcal{k})}{{\left[\mathfrak{g}(\mathcal{k})-\mathfrak{g}(\mathcal{o})\right]}^{1-⋌}}}{\mathcal{L}}_{\mu }(\mathcal{k},\mathcal{s})d\mathcal{k}\right] \\ {\le }_I{\displaystyle \frac{{{\mathsf{Ϫ}}_2}_{\mu }(\mathcal{o},\mathcal{s})+{{\mathsf{Ϫ}}_2}_{\mu }(\mathcal{l},\mathcal{s})}{2}} \end{gathered}$$

(77)

Similarly, both sides of (77) are multiplied by

$${\displaystyle \frac{\mathsf{\beta }}{{\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{\mathsf{\beta }}}} {\displaystyle \frac{{\mathrm{w}}^{\prime }(\mathcal{s})}{{\left[⋉(\mathsf{ȷ})-⋉(\mathcal{s})\right]}^{1-\mathsf{\beta }}}}$$

and

$${\displaystyle \frac{\mathsf{\beta }}{{ \left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{\mathsf{\beta }}}} {\displaystyle \frac{{\mathrm{w}}^{\prime }(\mathcal{s})}{{\left[⋉(\mathcal{s})-⋉(\mathsf{ı})\right]}^{1-\mathsf{\beta }}}}$$

then integrating the obtained results in relation to y from $\mathsf{ı}$ to $\mathsf{ȷ}$, we obtain the following inequalities:

$$\begin{gathered} {\displaystyle \frac{ \Gamma (⋊ +1)}{{\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}}{\mathbb{J}}_{{\mathsf{ı}}^{+};⋉}^{⋊}{{\mathsf{Ϫ}}_2}_{\mu }( {\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},\mathsf{ȷ}) \\ {\le }_I{\displaystyle \frac{\mathsf{\Gamma }(⋌+1)\mathsf{\Gamma }(⋊+1)}{4{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}{\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}}\left[{\mathbb{J}}_{\mathcal{o}+,\mathsf{ı}+;\mathfrak{g},⋉}^{⋌,⋊}{\mathcal{L}}_{\mu }(\mathcal{l},\mathsf{ȷ})+{\mathbb{J}}_{\mathcal{l}-,\mathsf{ı}+;\mathfrak{g},⋉}^{⋌,⋊}{\mathcal{L}}_{\mu }(\mathcal{o},\mathsf{ȷ})\right] \\ {\le }_I{\displaystyle \frac{ \Gamma (⋊ +1)}{2{\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}}\left[{\mathbb{J}}_{\mathsf{ı}+;⋉}^{⋊}{{\mathsf{Ϫ}}_2}_{\mu }(\mathcal{o},\mathsf{ȷ})+{\mathbb{J}}_{\mathsf{ı}+;⋉}^{⋊}{{\mathsf{Ϫ}}_2}_{\mu }(\mathcal{l},\mathsf{ȷ})\right] \end{gathered}$$

(78)

and

$$\begin{gathered} {\displaystyle \frac{ \Gamma (⋊+1)}{{\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}}{\mathbb{J}}_{\mathsf{ȷ}-;⋉}^{⋊}{{\mathsf{Ϫ}}_2}_{\mu }( {\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},\mathsf{ı}) \\ {\le }_I{\displaystyle \frac{\mathsf{\Gamma }(⋌+1)\mathsf{\Gamma }(⋊+1)}{4{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}{\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}}\left[{\mathbb{J}}_{\mathcal{o}+,\mathsf{ȷ}-;\mathfrak{g},⋉}^{⋌,⋊}{\mathcal{L}}_{\mu }(\mathcal{l},\mathsf{ı})+{\mathbb{J}}_{\mathcal{l}-,\mathsf{ȷ}-;\mathfrak{g},⋉}^{⋌,⋊}{\mathcal{L}}_{\mu }(\mathcal{o},\mathsf{ı})\right] \\ {\le }_I{\displaystyle \frac{ \Gamma (⋊+1)}{2{\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}}\left[{\mathbb{J}}_{\mathsf{ȷ}-;⋉}^{⋊}{{\mathsf{Ϫ}}_2}_{\mu }(\mathcal{o},\mathsf{ı})+{\mathbb{J}}_{\mathsf{ȷ}-;⋉}^{⋊}{{\mathsf{Ϫ}}_2}_{\mu }(\mathcal{l},\mathsf{ı})\right] \end{gathered}$$

(79)

respectively.

Summing the inequalities (69), (70), (72), (73), (75), (76), (78), and (79), we have the following inequalities:

$$\begin{gathered} {\displaystyle \frac{ \Gamma (⋌ + 1)}{{\left[\mathfrak{g}(\mathcal{l})- \mathfrak{g}(\mathcal{o})\right]}^{⋌}}}\left[\begin{array}{c}{\mathbb{J}}_{\mathcal{o}+;\mathfrak{g}}^{⋌}{\mathfrak{S}}_{\mu }(\mathcal{l},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}})+{\mathbb{J}}_{\mathcal{l}-;\mathfrak{g}}^{⋌}{\mathfrak{S}}_{\mu }(\mathcal{o},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}})\\ +{\mathbb{J}}_{\mathcal{o}+;\mathfrak{g}}^{⋌} {{\mathsf{Ϫ}}_1}_{\mu }(\mathcal{l},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}})+{\mathbb{J}}_{\mathcal{l}-;\mathfrak{g}}^{⋌}{{\mathsf{Ϫ}}_1}_{\mu }(\mathcal{o},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}})\end{array}\right] \\ +{\displaystyle \frac{ \Gamma (⋊ +1)}{{\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}}\left[\begin{array}{c}{\mathbb{J}}_{\mathsf{ı}+;⋉}^{⋊} {\mathfrak{S}}_{\mu }( {\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},\mathsf{ȷ})+{\mathbb{J}}_{\mathsf{ȷ}-;⋉}^{⋊}{\mathfrak{S}}_{\mu }( {\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},\mathsf{ı})\\ +{\mathbb{J}}_{\mathsf{ı}+;⋉}^{⋊}{{\mathsf{Ϫ}}_2}_{\mu }( {\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},\mathsf{ȷ})+{\mathbb{J}}_{\mathsf{ȷ}-;⋉}^{⋊}{{\mathsf{Ϫ}}_2}_{\mu }( {\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},\mathsf{ı})\end{array}\right] \\ {\le }_I{\displaystyle \frac{\mathsf{\Gamma }(⋌+1)\mathsf{\Gamma }(⋊+1)}{4{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}{\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}}\times \left[\begin{array}{c}{\mathbb{J}}_{\mathcal{o}+,\mathsf{ı}+;\mathfrak{g},⋉}^{⋌,⋊}{\mathfrak{D}}_{\mu }(\mathcal{l},\mathsf{ȷ})+{\mathbb{J}}_{\mathcal{o}+,\mathsf{ȷ}-;\mathfrak{g},⋉}^{⋌,⋊}{\mathfrak{D}}_{\mu }(\mathcal{l},\mathsf{ı})+{\mathbb{J}}_{\mathcal{l}-,\mathsf{ı}+;\mathfrak{g},⋉}^{⋌,⋊}{\mathfrak{D}}_{\mu }(\mathcal{o},\mathsf{ȷ})+{\mathbb{J}}_{\mathcal{l}-,\mathsf{ȷ}-;\mathfrak{g},⋉}^{⋌,⋊}{\mathfrak{D}}_{\mu }(\mathcal{o},\mathsf{ı})\\ +{\mathbb{J}}_{\mathcal{o}+,\mathsf{ı}+;\mathfrak{g},⋉}^{⋌,⋊}{\mathsf{Ϗ}}_{\mu }(\mathcal{l},\mathsf{ȷ})+{\mathbb{J}}_{\mathcal{o}+,\mathsf{ȷ}-;\mathfrak{g},⋉}^{⋌,⋊}{\mathsf{Ϗ}}_{\mu }(\mathcal{l},\mathsf{ı})+{\mathbb{J}}_{\mathcal{l}-,\mathsf{ı}+;\mathfrak{g},⋉}^{⋌,⋊}{\mathsf{Ϗ}}_{\mu }(\mathcal{o},\mathsf{ȷ})+{\mathbb{J}}_{\mathcal{l}-,\mathsf{ȷ}-;\mathfrak{g},⋉}^{⋌,⋊}{\mathsf{Ϗ}}_{\mu }(\mathcal{o},\mathsf{ı})\\ +{\mathbb{J}}_{\mathcal{o}+,\mathsf{ı}+;\mathfrak{g},⋉}^{⋌,⋊}{\mathcal{H}}_{\mu }(\mathcal{l},\mathsf{ȷ})+{\mathbb{J}}_{\mathcal{o}+,\mathsf{ȷ}-;\mathfrak{g},⋉}^{⋌,⋊}{\mathcal{H}}_{\mu }(\mathcal{o},\mathsf{ȷ})+{\mathbb{J}}_{\mathcal{o}+,\mathsf{ȷ}-;\mathfrak{g},⋉}^{⋌,⋊}{\mathcal{H}}_{\mu }(\mathcal{l},\mathsf{ı})+{\mathbb{J}}_{\mathcal{l}-,\mathsf{ȷ}-;\mathfrak{g},⋉}^{⋌,⋊}{\mathcal{H}}_{\mu }(\mathcal{o},\mathsf{ı})\\ +{\mathbb{J}}_{\mathcal{o}+,\mathsf{ı}+;\mathfrak{g},⋉}^{⋌,⋊}{\mathcal{L}}_{\mu }(\mathcal{l},\mathsf{ȷ})+{\mathbb{J}}_{\mathcal{o}+,\mathsf{ȷ}-;\mathfrak{g},⋉}^{⋌,⋊}{\mathcal{L}}_{\mu }(\mathcal{o},\mathsf{ȷ})+{\mathbb{J}}_{\mathcal{o}+,\mathsf{ȷ}-;\mathfrak{g},⋉}^{⋌,⋊}{\mathcal{L}}_{\mu }(\mathcal{l},\mathsf{ı})+{\mathbb{J}}_{\mathcal{l}-,\mathsf{ȷ}-;\mathfrak{g},⋉}^{⋌,⋊}{\mathcal{L}}_{\mu }(\mathcal{o},\mathsf{ı})\end{array}\right] \\ {\le }_I{\displaystyle \frac{ \Gamma (⋌ + 1)}{2{\left[\mathfrak{g}(\mathcal{l})- \mathfrak{g}(\mathcal{o})\right]}^{⋌}}}\left[\begin{array}{c}{\mathbb{J}}_{\mathcal{o}+;\mathfrak{g}}^{⋌}{\mathfrak{S}}_{\mu } (\mathcal{l},\mathsf{ı})+{\mathbb{J}}_{\mathcal{o}+;\mathfrak{g}}^{⋌}{\mathfrak{S}}_{\mu } (\mathcal{l},\mathsf{ȷ})+{\mathbb{J}}_{\mathcal{l}-;\mathfrak{g}}^{⋌}{\mathfrak{S}}_{\mu } (\mathcal{l},\mathsf{ı})+{\mathbb{J}}_{\mathcal{l}-;\mathfrak{g}}^{⋌}{\mathfrak{S}}_{\mu } (\mathcal{l},\mathsf{ȷ})\\ + {\mathbb{J}}_{\mathcal{o}+;\mathfrak{g}}^{⋌}{{\mathsf{Ϫ}}_1}_{\mu }(\mathcal{l},\mathsf{ı})+{\mathbb{J}}_{\mathcal{o}+;\mathfrak{g}}^{⋌}{{\mathsf{Ϫ}}_1}_{\mu }(\mathcal{l},\mathsf{ȷ})+{\mathbb{J}}_{\mathcal{l}-;\mathfrak{g}}^{⋌}{{\mathsf{Ϫ}}_1}_{\mu }(\mathcal{o},\mathsf{ı})+{\mathbb{J}}_{\mathcal{l}-;\mathfrak{g}}^{⋌}{{\mathsf{Ϫ}}_1}_{\mu }(\mathcal{o},\mathsf{ȷ})\end{array}\right] \\ +{\displaystyle \frac{ \Gamma (⋊ +1)}{{\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}}\left[\begin{array}{c}{\mathbb{J}}_{\mathsf{ı}+;⋉}^{⋊}{\mathfrak{S}}_{\mu } (\mathcal{o},\mathsf{ȷ})+{\mathbb{J}}_{\mathsf{ı}+;⋉}^{⋊}{\mathfrak{S}}_{\mu } (\mathcal{l},\mathsf{ȷ})+{\mathbb{J}}_{\mathsf{ȷ}-;⋉}^{⋊}{\mathfrak{S}}_{\mu } (\mathcal{o},\mathsf{ı})+{\mathbb{J}}_{\mathsf{ȷ}-;⋉}^{⋊}{\mathfrak{S}}_{\mu } (\mathcal{l},\mathsf{ı})\\ + {\mathbb{J}}_{\mathsf{ı}+;⋉}^{⋊}{{\mathsf{Ϫ}}_2}_{\mu }(\mathcal{o},\mathsf{ȷ})+{\mathbb{J}}_{\mathsf{ı}+;⋉}^{⋊}{{\mathsf{Ϫ}}_2}_{\mu }(\mathcal{l},\mathsf{ȷ})+{\mathbb{J}}_{\mathsf{ȷ}-;⋉}^{⋊}{{\mathsf{Ϫ}}_2}_{\mu }(o,\mathsf{ı}) +{\mathbb{J}}_{\mathsf{ȷ}-;⋉}^{⋊}{{\mathsf{Ϫ}}_2}_{\mu }(l, \mathsf{ı}) \end{array}\right] \end{gathered}$$

That is, we have

$$\begin{gathered} {\displaystyle \frac{ \Gamma (⋌+1)}{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}}}\left[{\mathbb{J}}_{\mathcal{o}+;\mathfrak{g}}^{⋌}{\mathcal{H}}_{\mu }(\mathcal{l},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}})+{\mathbb{J}}_{\mathcal{l}-;\mathfrak{g}}^{⋌}{\mathcal{H}}_{\mu }(\mathcal{o},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}})\right] \\ +{\displaystyle \frac{ \Gamma (⋊+1)}{{\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}}\left[{\mathbb{J}}_{\mathsf{ı}+;⋉}^{⋊}{\mathfrak{D}}_{\mu }( {\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},\mathsf{ȷ})+{\mathbb{J}}_{\mathsf{ȷ}-;⋉}^{⋊}{\mathfrak{D}}_{\mu }( {\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},\mathsf{ı})\right] \\ {\le }_I{\displaystyle \frac{\mathsf{\Gamma }(⋌+1)\mathsf{\Gamma }(⋊+1)}{2{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}{\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}}\times \left[\begin{array}{c}{\mathbb{J}}_{\mathcal{o}+,\mathsf{ı}+;\mathfrak{g},⋉}^{⋌,⋊}{\mathsf{Ϫ}}_{\mu }(\mathcal{l},\mathsf{ȷ})+{\mathbb{J}}_{\mathcal{o}+,\mathsf{ȷ}-;\mathfrak{g},⋉}^{⋌,⋊}{\mathsf{Ϫ}}_{\mu }(\mathcal{l},\mathsf{ı})\\ +{\mathbb{J}}_{\mathcal{l}-,\mathsf{ı}+;\mathfrak{g},⋉}^{⋌,⋊}{\mathsf{Ϫ}}_{\mu }(\mathcal{o},\mathsf{ȷ})+{\mathbb{J}}_{\mathcal{l}-,\mathsf{ȷ}-;\mathfrak{g},⋉}^{⋌,⋊}{\mathsf{Ϫ}}_{\mu }(\mathcal{o},\mathsf{ı})\end{array}\right] \\ {\le }_I{\displaystyle \frac{ \Gamma (⋌+1)}{{2\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}}}\left[{\mathbb{J}}_{\mathcal{o}+;\mathfrak{g}}^{⋌}{\mathcal{H}}_{\mu }(\mathcal{l},\mathsf{ı})+{\mathbb{J}}_{\mathcal{o}+;\mathfrak{g}}^{⋌}{\mathcal{H}}_{\mu }(\mathcal{l},\mathsf{ȷ})+{\mathbb{J}}_{\mathcal{l}-;\mathfrak{g}}^{⋌}{\mathcal{H}}_{\mu }(\mathcal{o},\mathsf{ı})+{\mathbb{J}}_{\mathcal{l}-;\mathfrak{g}}^{⋌}{\mathcal{H}}_{\mu }(\mathcal{o},\mathsf{ȷ})\right] \\ +2{\displaystyle \frac{ \Gamma (⋊+1)}{{\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}}\left[{\mathbb{J}}_{\mathsf{ı}+;⋉}^{⋊}{\mathfrak{D}}_{\mu }(\mathcal{o},\mathsf{ȷ})+{\mathbb{J}}_{\mathsf{ı}+;⋉}^{⋊}{\mathfrak{D}}_{\mu }(\mathcal{l},\mathsf{ȷ})+{\mathbb{J}}_{\mathsf{ȷ}-;⋉}^{⋊}{\mathfrak{D}}_{\mu }(\mathcal{o},\mathsf{ı})+{\mathbb{J}}_{\mathsf{ȷ}-;⋉}^{⋊}{\mathfrak{D}}_{\mu }(\mathcal{l},\mathsf{ı})\right] \end{gathered}$$

which completes the proof of the second and third inequalities in (67). On the other hand, from the first inequality in (22), we have

$${\mathfrak{S}}_{\mu }({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}}){\le }_I{\displaystyle \frac{⋌}{4{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}}}\left[\begin{array}{c}\underset{\mathcal{o}}{\overset{\mathcal{l}}{\int }}{\displaystyle \frac{{\mathfrak{g}}^{\prime }(\mathcal{k})}{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{k})\right]}^{⋌}}}\left[{\mathfrak{S}}_{\mu }(\mathcal{k})+{\mathfrak{S}}_{\mu }(\mathcal{o}+\mathcal{l}-\mathcal{k})\right]\mathsf{ȷ}k\\ +\underset{\mathcal{o}}{\overset{\mathcal{l}}{\int }}{\displaystyle \frac{{\mathfrak{g}}^{\prime }(\mathcal{k})}{{\left[\mathfrak{g}(\mathcal{k})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}}}\left[{\mathfrak{S}}_{\mu }(\mathcal{k})+{\mathfrak{S}}_{\mu }(\mathcal{o}+\mathcal{l}-\mathcal{k})\right]\mathsf{ȷ}k\end{array}\right]$$

(80)

Since F is the $F$·$N\cdot V$-coordinated convex on Δ, by using inequality (80), we obtain

$$\begin{gathered} {\mathfrak{S}}_{\mu }({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}}){\le }_I{\displaystyle \frac{⋌}{4{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}}}\left[\begin{array}{c}(IA)\underset{\mathcal{o}}{\overset{\mathcal{l}}{\int }}{\displaystyle \frac{{\mathfrak{g}}^{\prime }(\mathcal{k})}{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{k})\right]}^{⋌}}}\left[{\mathfrak{S}}_{\mu }(\mathcal{k},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}})+{\mathfrak{S}}_{\mu }(\mathcal{o}+\mathcal{l}-\mathcal{k},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}})\right]d\mathcal{k}\\ +(IA)\underset{\mathcal{o}}{\overset{\mathcal{l}}{\int }}{\displaystyle \frac{{\mathfrak{g}}^{\prime }(\mathcal{k})}{{\left[\mathfrak{g}(\mathcal{k})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}}}\left[{\mathfrak{S}}_{\mu }(\mathcal{k},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}})+{\mathfrak{S}}_{\mu }(\mathcal{o}+\mathcal{l}-\mathcal{k},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}})\right]d\mathcal{k}\end{array}\right] \\ ={\displaystyle \frac{ \Gamma (⋌+1)}{{4\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}}}\left[{\mathbb{J}}_{\mathcal{o}+;\mathfrak{g}}^{⋌}{\mathcal{H}}_{\mu }(\mathcal{l},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}})+{\mathbb{J}}_{\mathcal{l}-;\mathfrak{g}}^{⋌}{\mathcal{H}}_{\mu }(\mathcal{o},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}})\right], \end{gathered}$$

(81)

and similarly, we have

$$\begin{gathered} {\mathfrak{S}}_{\mu }({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}}){\le }_I{\displaystyle \frac{⋊}{4{\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}}\left[\begin{array}{c}(IA)\underset{\mathsf{ı}}{\overset{\mathsf{ȷ}}{\int }}{\displaystyle \frac{{⋉}^{\prime }(\mathcal{s})}{{\left[⋉(\mathsf{ȷ})-⋉(\mathcal{s})\right]}^{⋌}}}\left[{\mathfrak{S}}_{\mu }({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},\mathcal{s})+{\mathfrak{S}}_{\mu }({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},\mathsf{ı}+\mathsf{ȷ}-\mathcal{s})\right]d\mathcal{s}\\ +(IA)\underset{\mathsf{ı}}{\overset{\mathsf{ȷ}}{\int }}{\displaystyle \frac{{⋉}^{\prime }(\mathcal{s})}{{\left[⋉(\mathcal{s})-⋉(\mathsf{ı})\right]}^{⋌}}}\left[{\mathfrak{S}}_{\mu }({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},\mathcal{s})+{\mathfrak{S}}_{\mu }({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},\mathsf{ı}+\mathsf{ȷ}-\mathcal{s})\right]d\mathcal{s}\end{array}\right] \\ ={\displaystyle \frac{ \Gamma (⋊+1)}{{4\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}}\left[{\mathbb{J}}_{\mathsf{ı}+;⋉}^{⋊}{\mathfrak{D}}_{\mu }( {\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},\mathsf{ȷ})+{\mathbb{J}}_{\mathsf{ȷ}-;⋉}^{⋊}{\mathfrak{D}}_{\mu }( {\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},\mathsf{ı})\right] \end{gathered}$$

(82)

Combining the inequalities (81) and (82), we obtain the first inequality in (67). From the second inequality in (22), we have

$${\displaystyle \frac{⋌}{4{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}}}\left[\begin{array}{c}(IA)\underset{\mathcal{o}}{\overset{\mathcal{l}}{\int }}{\displaystyle \frac{{\mathfrak{g}}^{\prime }(\mathcal{k})}{{\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{k})\right]}^{⋌}}}\left[{\mathfrak{S}}_{\mu }(\mathcal{k})+{\mathfrak{S}}_{\mu }(\mathcal{o}+\mathcal{l}-\mathcal{k})\right]d\mathcal{k}\\ +(IA)\underset{\mathcal{o}}{\overset{\mathcal{l}}{\int }}{\displaystyle \frac{{\mathfrak{g}}^{\prime }(\mathcal{k})}{{\left[\mathfrak{g}(\mathcal{k})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}}}\left[{\mathfrak{S}}_{\mu }(\mathcal{k})+{\mathfrak{S}}_{\mu }(\mathcal{o}+\mathcal{l}-\mathcal{k})\right]d\mathcal{k}\end{array}\right]{\le }_I{\displaystyle \frac{{\mathfrak{S}}_{\mu }(\mathcal{o})+{\mathfrak{S}}_{\mu }(\mathcal{l})}{2}}$$

(83)

By using inequality (83), we obtain the following inequalities:

$${\displaystyle \frac{ \Gamma (⋌ + 1)}{{4\left[\mathfrak{g}(\mathcal{l})- \mathfrak{g}(\mathcal{o})\right]}^{⋌}}}\left[{\mathbb{J}}_{\mathcal{o}+;\mathfrak{g}}^{⋌}{\mathcal{H}}_{\mu }(\mathcal{l},\mathsf{ı})+{\mathbb{J}}_{\mathcal{l}-;\mathfrak{g}}^{⋌}{\mathcal{H}}_{\mu }(\mathcal{o},\mathsf{ı})\right]{\le }_I{\displaystyle \frac{{\mathfrak{S}}_{\mu }(\mathcal{o},\mathsf{ı})+{\mathfrak{S}}_{\mu }(\mathcal{l},\mathsf{ı})}{2}}$$

(84)

$${\displaystyle \frac{ \Gamma (⋌+1)}{{4\left[\mathfrak{g}(\mathcal{l})-\mathfrak{g}(\mathcal{o})\right]}^{⋌}}}\left[{\mathbb{J}}_{\mathcal{o}+;\mathfrak{g}}^{⋌}{\mathcal{H}}_{\mu }(\mathcal{l},\mathsf{ȷ})+{\mathbb{J}}_{\mathcal{l}-;\mathfrak{g}}^{⋌}{\mathcal{H}}_{\mu }(\mathcal{o},\mathsf{ȷ})\right]{\le }_I{\displaystyle \frac{{\mathfrak{S}}_{\mu }(\mathcal{o},\mathsf{ȷ})+{\mathfrak{S}}_{\mu }(\mathcal{l},\mathsf{ȷ})}{2}}$$

(85)

$${\displaystyle \frac{ \Gamma (⋊+1))}{{4\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}}\left[{\mathbb{J}}_{\mathsf{ı}+;⋉}^{⋊}{\mathfrak{D}}_{\mu }(\mathcal{o},\mathsf{ȷ})+{\mathbb{J}}_{\mathsf{ȷ}-;⋉}^{⋊}{\mathfrak{D}}_{\mu }(\mathcal{o},\mathsf{ı})\right]{\le }_I{\displaystyle \frac{{\mathfrak{S}}_{\mu }(\mathcal{o},\mathsf{ı})+{\mathfrak{S}}_{\mu }(\mathcal{o},\mathsf{ȷ})}{2}}$$

(86)

and

$${\displaystyle \frac{ \Gamma (⋊+1))}{{4\left[⋉(\mathsf{ȷ})-⋉(\mathsf{ı})\right]}^{⋊}}}\left[{\mathbb{J}}_{\mathsf{ı}+;⋉}^{⋊}{\mathfrak{D}}_{\mu }(\mathcal{l},\mathsf{ȷ})+{\mathbb{J}}_{\mathsf{ȷ}-;⋉}^{⋊}{\mathfrak{D}}_{\mu }(\mathcal{l},\mathsf{ı})\right]{\le }_I{\displaystyle \frac{{\mathfrak{S}}_{\mu }(\mathcal{l},\mathsf{ı})+{\mathfrak{S}}_{\mu }(\mathcal{l},\mathsf{ȷ})}{2}}$$

(87)

Combining the inequalities (84) to (87), we obtain the last inequality in (67). This completes the proof completely. □

Remark 4.

(i) Assuming the conditions of Theorem 5 hold, setting $\mathfrak{g}(\mathcal{v})=\mathcal{v},⋉(\mathcal{r})=\mathcal{r} and \mu =0$ yields the following inequalities for the Riemann–Liouville $F$·$N\cdot V$-fractional double integrals:

$$\begin{gathered} \mathfrak{S}({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}}) \\ {\le }_I{\displaystyle \frac{\mathsf{\Gamma }(⋌+1)}{4{(\mathcal{l}-\mathcal{o})}^{⋌}}}\left[{\mathcal{J}}_{\mathcal{o}+}^{⋌}\mathfrak{S}(\mathcal{l},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}})+{\mathcal{J}}_{\mathcal{l}-}^{⋌}\mathfrak{S}(\mathcal{o},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}})\right] \\ +{\displaystyle \frac{\mathsf{\Gamma }(⋊+1)}{4{(\mathsf{ȷ}-\mathsf{ı})}^{⋊}}}\left[{\mathcal{J}}_{\mathsf{ı}+}^{⋊}\mathfrak{S}({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},\mathsf{ȷ})+{\mathcal{J}}_{\mathsf{ȷ}-}^{⋊}\mathfrak{S}({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},\mathsf{ı})\right] \\ {\le }_I{\displaystyle \frac{\mathsf{\Gamma }(⋌+1)\mathsf{\Gamma }(⋊+1)}{4{(\mathcal{l}-\mathcal{o})}^{⋌}{(\mathsf{ȷ}-\mathsf{ı})}^{⋊}}}\left[\begin{array}{c}{\mathcal{J}}_{\mathcal{o}+,\mathsf{ı}+}^{⋌,⋊}\mathfrak{S}(\mathcal{l},\mathsf{ȷ})+{\mathcal{J}}_{\mathcal{o}+,\mathsf{ȷ}-}^{⋌,⋊}\mathfrak{S}(\mathcal{l},\mathsf{ı})\\ +{\mathcal{J}}_{\mathcal{l}-,\mathsf{ı}+}^{⋌,⋊}\mathfrak{S}(\mathcal{o},\mathsf{ȷ})+{\mathcal{J}}_{\mathcal{l}-,\mathsf{ȷ}-}^{⋌,⋊}\mathfrak{S}(\mathcal{o},\mathsf{ı})\end{array}\right] \\ {\le }_I{\displaystyle \frac{\mathsf{\Gamma }(⋌+1)}{8{(\mathcal{l}-\mathcal{o})}^{⋌}}}\left[{\mathcal{J}}_{\mathcal{o}+}^{⋌}\mathfrak{S}(\mathcal{l},\mathsf{ı})+{\mathcal{J}}_{\mathcal{o}+}^{⋌}\mathfrak{S}(\mathcal{l},\mathsf{ȷ})+{\mathcal{J}}_{\mathcal{l}-}^{⋌}\mathfrak{S}(\mathcal{o},\mathsf{ı})+{\mathcal{J}}_{\mathcal{l}-}^{⋌}\mathfrak{S}(\mathcal{o},\mathsf{ȷ})\right] \\ +{\displaystyle \frac{\mathsf{\Gamma }(⋊+1)}{8{(\mathsf{ȷ}-\mathsf{ı})}^{⋊}}}\left[{\mathcal{J}}_{\mathsf{ı}+}^{⋊}\mathfrak{S}(\mathcal{o},\mathsf{ȷ})+{\mathcal{J}}_{\mathsf{ȷ}-}^{⋊}\mathfrak{S}(\mathcal{l},\mathsf{ȷ})+{\mathcal{J}}_{\mathsf{ı}+}^{⋊}\mathfrak{S}(\mathcal{o},\mathsf{ı})+{\mathcal{J}}_{\mathsf{ȷ}-}^{⋊}\mathfrak{S}(\mathcal{l},\mathsf{ı})\right] \\ {\le }_I{\displaystyle \frac{\mathfrak{S}(\mathcal{o},\mathsf{ı})+\mathfrak{S}(\mathcal{o},\mathsf{ȷ})+\mathfrak{S}(\mathcal{l},\mathsf{ı})+\mathfrak{S}(\mathcal{l},\mathsf{ȷ})}{4}}. \end{gathered}$$

which are proved by Budak et al. [17] in a previous study.

(ii) Assuming the conditions of Theorem 5 hold, setting $\mathfrak{g}(\mathcal{v})=\mathcal{v},⋉(\mathcal{r})=\mathcal{r}, and \mu =0$ yields the following inequalities for the Riemann–Liouville $F$·$N\cdot V$-fractional double integrals:

$$\begin{gathered} \tilde{\mathfrak{S}}({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}}) \\ {\le }_F{\displaystyle \frac{\mathsf{\Gamma }(⋌+1)}{4{(\mathcal{l}-\mathcal{o})}^{⋌}}}\odot \left[{\mathcal{J}}_{\mathcal{o}+}^{⋌}\tilde{\mathfrak{S}}(\mathcal{l},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}})\oplus {\mathcal{J}}_{\mathcal{l}-}^{⋌}\tilde{\mathfrak{S}}(\mathcal{o},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}})\right] \\ \oplus {\displaystyle \frac{\mathsf{\Gamma }(⋊+1)}{4{(\mathsf{ȷ}-\mathsf{ı})}^{⋊}}}\odot \left[{\mathcal{J}}_{\mathsf{ı}+}^{⋊}\tilde{\mathfrak{S}}({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},\mathsf{ȷ})\oplus {\mathcal{J}}_{\mathsf{ȷ}-}^{⋊}\tilde{\mathfrak{S}}({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},\mathsf{ı})\right] \\ {\le }_F{\displaystyle \frac{\mathsf{\Gamma }(⋌+1)\mathsf{\Gamma }(⋊+1)}{4{(\mathcal{l}-\mathcal{o})}^{⋌}{(\mathsf{ȷ}-\mathsf{ı})}^{⋊}}}\odot \left[\begin{array}{c}{\mathcal{J}}_{\mathcal{o}+,\mathsf{ı}+}^{⋌,⋊}\tilde{\mathfrak{S}}(\mathcal{l},\mathsf{ȷ})\oplus {\mathcal{J}}_{\mathcal{o}+,\mathsf{ȷ}-}^{⋌,⋊}\tilde{\mathfrak{S}}(\mathcal{l},\mathsf{ı})\\ \oplus {\mathcal{J}}_{\mathcal{l}-,\mathsf{ı}+}^{⋌,⋊}\tilde{\mathfrak{S}}(\mathcal{o},\mathsf{ȷ})\oplus {\mathcal{J}}_{\mathcal{l}-,\mathsf{ȷ}-}^{⋌,⋊}\tilde{\mathfrak{S}}(\mathcal{o},\mathsf{ı})\end{array}\right] \\ {\le }_F{\displaystyle \frac{\mathsf{\Gamma }(⋌+1)}{8{(\mathcal{l}-\mathcal{o})}^{⋌}}}\odot \left[{\mathcal{J}}_{\mathcal{o}+}^{⋌}\tilde{\mathfrak{S}}(\mathcal{l},\mathsf{ı})\oplus {\mathcal{J}}_{\mathcal{o}+}^{⋌}\tilde{\mathfrak{S}}(\mathcal{l},\mathsf{ȷ})\oplus {\mathcal{J}}_{\mathcal{l}-}^{⋌}\tilde{\mathfrak{S}}(\mathcal{o},\mathsf{ı})\oplus {\mathcal{J}}_{\mathcal{l}-}^{⋌}\tilde{\mathfrak{S}}(\mathcal{o},\mathsf{ȷ})\right] \\ \oplus {\displaystyle \frac{\mathsf{\Gamma }(⋊+1)}{8{(\mathsf{ȷ}-\mathsf{ı})}^{⋊}}}\odot \left[{\mathcal{J}}_{\mathsf{ı}+}^{⋊}\tilde{\mathfrak{S}}(\mathcal{o},\mathsf{ȷ})\oplus {\mathcal{J}}_{\mathsf{ȷ}-}^{⋊}\tilde{\mathfrak{S}}(\mathcal{l},\mathsf{ȷ})\oplus {\mathcal{J}}_{\mathsf{ı}+}^{⋊}\tilde{\mathfrak{S}}(\mathcal{o},\mathsf{ı})\oplus {\mathcal{J}}_{\mathsf{ȷ}-}^{⋊}\tilde{\mathfrak{S}}(\mathcal{l},\mathsf{ı})\right] \\ {\le }_F{\displaystyle \frac{\tilde{\mathfrak{S}}(\mathcal{o},\mathsf{ı})\oplus \tilde{\mathfrak{S}}(\mathcal{o},\mathsf{ȷ})\oplus \tilde{\mathfrak{S}}(\mathcal{l},\mathsf{ı})\oplus \tilde{\mathfrak{S}}(\mathcal{l},\mathsf{ȷ})}{4}} \end{gathered}$$

Corollary 6.

Setting $\mathfrak{g}(\mathcal{v})=\ln \mathcal{v} and ⋉(\mathcal{r})=ln \mathcal{r}$ in Theorem 5 yields the following inequalities for the Hadamard $F$·$N\cdot V$-fractional double integrals:

$$\begin{gathered} \tilde{\mathfrak{S}}({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}}){\le }_F{\displaystyle \frac{\mathsf{\Gamma }(⋌+1)}{{8 \left[\ln \frac{\mathcal{l}}{\mathcal{o}}\right]}^{⋌}}}\odot \left[{\mathfrak{J}}_{\mathcal{o}+}^{⋌}\tilde{\mathcal{H}}(\mathcal{l},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}})\oplus {\mathfrak{J}}_{\mathcal{l}-}^{⋌}\tilde{\mathcal{H}}(\mathcal{o},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}})\right]\oplus {\displaystyle \frac{\mathsf{\Gamma }(⋊+1)}{8{\left[\ln \frac{\mathsf{ȷ}}{\mathsf{ı}}\right]}^{⋊}}}\odot \left[{\mathfrak{J}}_{\mathsf{ı}+}^{⋊}\tilde{\mathfrak{D}}( {\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},\mathsf{ȷ})\oplus {\mathfrak{J}}_{\mathsf{ȷ}-}^{⋊}\tilde{\mathfrak{D}}( {\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},\mathsf{ı})\right] \\ {\le }_F{\displaystyle \frac{\mathsf{\Gamma }(⋌+1)\mathsf{\Gamma }(⋊+1)}{{16 \left[\ln \frac{\mathcal{l}}{\mathcal{o}}\right]}^{⋌}{\left[\ln \frac{\mathsf{ȷ}}{\mathsf{ı}}\right]}^{⋊}}}\odot \left[{\mathfrak{J}}_{\mathcal{o}+,\mathsf{ı}+}^{⋌,⋊}\widetilde{\mathsf{Ϫ}}(\mathcal{l},\mathsf{ȷ})\oplus {\mathfrak{J}}_{\mathcal{o}+,\mathsf{ȷ}-}^{⋌,⋊}\widetilde{\mathsf{Ϫ}}(\mathcal{l},\mathsf{ı})\oplus {\mathfrak{J}}_{\mathcal{l}-,\mathsf{ı}+}^{⋌,⋊}\widetilde{\mathsf{Ϫ}}(\mathcal{o},\mathsf{ȷ})\oplus {\mathfrak{J}}_{\mathcal{l}-,\mathsf{ȷ}-}^{⋌,⋊}\widetilde{\mathsf{Ϫ}}(\mathcal{o},\mathsf{ı})\right] \\ {\le }_F{\displaystyle \frac{\mathsf{\Gamma }(⋌+1)}{{16\left[\ln \frac{\mathcal{l}}{\mathcal{o}}\right]}^{⋌}}}\odot \left[{\mathfrak{J}}_{\mathcal{o}+}^{⋌}\tilde{\mathcal{H}}(\mathcal{l},\mathsf{ı})\oplus {\mathfrak{J}}_{\mathcal{o}+}^{⋌}\tilde{\mathcal{H}}(\mathcal{l},\mathsf{ȷ})\oplus {\mathfrak{J}}_{\mathcal{l}-}^{⋌}\tilde{\mathcal{H}}(\mathcal{o},\mathsf{ı})\oplus {\mathfrak{J}}_{\mathcal{l}-}^{⋌}\tilde{\mathcal{H}}(\mathcal{o},\mathsf{ȷ})\right] \\ \oplus {\displaystyle \frac{\mathsf{\Gamma }(⋊+1)}{16{\left[\ln \frac{\mathsf{ȷ}}{\mathsf{ı}}\right]}^{⋊}}}\odot \left[{\mathfrak{J}}_{\mathsf{ı}+}^{⋊}\tilde{\mathfrak{D}}(\mathcal{o},\mathsf{ȷ})\oplus {\mathfrak{J}}_{\mathsf{ı}+}^{⋊}\tilde{\mathfrak{D}}(\mathcal{l},\mathsf{ȷ})\oplus {\mathfrak{J}}_{\mathsf{ȷ}-}^{⋊}\tilde{\mathfrak{D}}(\mathcal{o},\mathsf{ı})\oplus {\mathfrak{J}}_{\mathsf{ȷ}-}^{⋊}\tilde{\mathfrak{D}}(\mathcal{l},\mathsf{ı})\right] \\ {\le }_F{\displaystyle \frac{\tilde{\mathfrak{S}}(\mathcal{o},\mathsf{ı})\oplus \tilde{\mathfrak{S}}(\mathcal{o},\mathsf{ȷ})\oplus \tilde{\mathfrak{S}}(\mathcal{l},\mathsf{ı})\oplus \tilde{\mathfrak{S}}(\mathcal{l},\mathsf{ȷ})}{4}} \end{gathered}$$

Corollary 7.

Assuming the conditions of Theorem 5 hold, setting $\mathfrak{g}(\mathcal{v})={\displaystyle \frac{{\mathcal{v}}^{\rho }}{\rho }} and ⋉(\mathcal{r})={\displaystyle \frac{{\mathcal{r}}^{\sigma }}{\sigma }}$ yields the following inequalities for the $F$·$N\cdot V$-Katugampola fractional double integrals:

$$\begin{gathered} \tilde{\mathfrak{S}}({\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}}) \\ {\le }_F{\displaystyle \frac{\mathsf{\Gamma }(⋌+1){\rho }^{⋌}}{{{8 \left[{\mathcal{l}}^{\rho }-{\mathcal{o}}^{\rho }\right]}^{⋌}}^{⋌}}}\odot \left[{{}_{ }{}^{ \rho }\mathbb{I}}_{\mathcal{o}+}^{⋌}\tilde{\mathcal{H}}(\mathcal{l},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}})\oplus {{}_{ }{}^{ \rho }\mathbb{I}}_{\mathcal{l}-}^{⋌}\tilde{\mathcal{H}}(\mathcal{o},{\displaystyle \frac{\mathsf{ı}+\mathsf{ȷ}}{2}})\right]\oplus {\displaystyle \frac{\mathsf{\Gamma }(⋊+1){\sigma }^{⋊}}{8{\left[{\mathsf{ȷ}}^{\sigma }-{\mathsf{ı}}^{\sigma }\right]}^{⋊}}} \\ \odot \left[{{}_{ }{}^{ \sigma }\mathbb{I}}_{\mathsf{ı}+}^{⋊}\tilde{\mathfrak{D}}( {\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},\mathsf{ȷ})\oplus {{}_{ }{}^{ \sigma }\mathbb{I}}_{\mathsf{ȷ}-}^{⋊}\tilde{\mathfrak{D}}( {\displaystyle \frac{\mathcal{o}+\mathcal{l}}{2}},\mathsf{ı})\right] \\ {\le }_F{\displaystyle \frac{\mathsf{\Gamma }(⋌+1)\mathsf{\Gamma }(⋊+1){\rho }^{⋌}{\sigma }^{⋊}}{{16 \left[{\mathcal{l}}^{\rho }-{\mathcal{o}}^{\rho }\right]}^{⋌}{\left[{\mathsf{ȷ}}^{\sigma }-{\mathsf{ı}}^{\sigma }\right]}^{⋊}}}\odot \left[{{}_{ }{}^{ \rho ,\sigma }\mathbb{I}}_{\mathcal{o}+,\mathsf{ı}+}^{⋌,⋊}\widetilde{\mathsf{Ϫ}}(\mathcal{l},\mathsf{ȷ})\oplus {{}_{ }{}^{ \rho ,\sigma }\mathbb{I}}_{\mathcal{o}+,\mathsf{ȷ}-}^{⋌,⋊}\widetilde{\mathsf{Ϫ}}(\mathcal{l},\mathsf{ı})\oplus {{}_{ }{}^{ \rho ,\sigma }\mathbb{I}}_{\mathcal{l}-,\mathsf{ı}+}^{⋌,⋊}\widetilde{\mathsf{Ϫ}}(\mathcal{o},\mathsf{ȷ})\oplus {{}_{ }{}^{ \rho ,\sigma }\mathbb{I}}_{\mathcal{l}-,\mathsf{ȷ}-}^{⋌,⋊}\widetilde{\mathsf{Ϫ}}(\mathcal{o},\mathsf{ı})\right] \\ {\le }_F{\displaystyle \frac{\mathsf{\Gamma }(⋌+1){\rho }^{⋌}{\sigma }^{⋊}}{{16 \left[{\mathcal{l}}^{\rho }-{\mathcal{o}}^{\rho }\right]}^{⋌}}}\odot \left[{{}_{ }{}^{ \rho }\mathbb{I}}_{\mathcal{o}+}^{⋌}\tilde{\mathcal{H}}(\mathcal{l},\mathsf{ı})\oplus {{}_{ }{}^{ \rho }\mathbb{I}}_{\mathcal{o}+}^{⋌}\tilde{\mathcal{H}}(\mathcal{l},\mathsf{ȷ})\oplus {{}_{ }{}^{ \rho }\mathbb{I}}_{\mathcal{l}-}^{⋌}\tilde{\mathcal{H}}(\mathcal{o},\mathsf{ı})\oplus {{}_{ }{}^{ \rho }\mathbb{I}}_{\mathcal{l}-}^{⋌}\tilde{\mathcal{H}}(\mathcal{o},\mathsf{ȷ})\right] \\ \oplus {\displaystyle \frac{\mathsf{\Gamma }(⋊+1){\rho }^{⋌}{\sigma }^{⋊}}{16{\left[{\mathsf{ȷ}}^{\sigma }-{\mathsf{ı}}^{\sigma }\right]}^{⋊}}}\odot \left[{{}_{ }{}^{ \sigma }\mathbb{I}}_{\mathsf{ı}+}^{⋊}\tilde{\mathfrak{D}}(\mathcal{o},\mathsf{ȷ})\oplus {{}_{ }{}^{ \sigma }\mathbb{I}}_{\mathsf{ı}+}^{⋊}\tilde{\mathfrak{D}}(\mathcal{l},\mathsf{ȷ})\oplus {{}_{ }{}^{ \sigma }\mathbb{I}}_{\mathsf{ȷ}-}^{⋊}\tilde{\mathfrak{D}}(\mathcal{o},\mathsf{ı})\oplus {{}_{ }{}^{ \sigma }\mathbb{I}}_{\mathsf{ȷ}-}^{⋊}\tilde{\mathfrak{D}}(\mathcal{l},\mathsf{ı})\right] \\ {\le }_F{\displaystyle \frac{\tilde{\mathfrak{S}}(\mathcal{o},\mathsf{ı})\oplus \tilde{\mathfrak{S}}(\mathcal{o},\mathsf{ȷ})\oplus \tilde{\mathfrak{S}}(\mathcal{l},\mathsf{ı})\oplus \tilde{\mathfrak{S}}(\mathcal{l},\mathsf{ȷ})}{4}}. \end{gathered}$$

5. Conclusions

In this work, we have developed several weighted $H\cdot H$-type inequalities within the framework of $F$·$N\cdot V$-convex and $F$·$N\cdot V$-coordinated convex mappings. By employing differentiable $F$·$N\cdot V$-functions together with appropriate weighting schemes, we derived new inclusions that extend classical $H\cdot H$-inequalities to a more general fuzzy setting using different fuzzy fractional integrals. This approach provides a flexible and robust method for handling uncertainty and imprecision inherent in many real-world and mathematical problems. Furthermore, a number of meaningful special cases of the main results were examined, demonstrating that the established inequalities unify and generalize several well-known results involving Riemann–Liouville-type fractional integrals and logarithmic integral operators. These special cases highlight the effectiveness of generalized fractional integrals in the $F$·$N\cdot V$-framework and illustrate the broad applicability of the obtained results.

The findings presented in this paper contribute to the ongoing development of fuzzy analysis and fractional inequality theory. Future research may focus on extending the proposed inclusions to other generalized notions of convexity, such as strongly convex, s-convex, or preinvex $F$·$N\cdot V$-functions, as well as exploring additional classes of fractional operators. Such investigations may lead to further refinements and new applications of $H\cdot H$-type inequalities in fuzzy mathematics and related fields.