One of the most well-known concepts in the field of function theory is the Hermite-Hadamard inequality, which was found by C. Hermite and J. Hadamard (and described in sources such as [1], [2] p.137). This inequality has several real-world applications in addition to its geometric interpretation.

The Hermite-Hadamard inequalities have been established by numerous mathematicians. It's important to note that the Hermite-Hadamard inequality, which naturally follows from Jensen's inequality, can be seen as a development of the idea of convexity. Recently, there has been renewed interest in the Hermite-Hadamard inequality for convex functions, leading to a wide range of improvements and expansions that have been thoroughly investigated (see, for example, publications like [3,4,5,6,7,8]).

Interval analysis is a crucial topic since it is used in math and computer models as one method of addressing interval uncertainty. Even though this theory has a long history going back to Archimedes' calculation of a circle's circumference, significant research on the subject was not published until the 1950s. The first book [9] on interval analysis was published in 1966 by Ramon E. Moore, who is credited with developing interval calculus. After that, other academics studied the theory and uses of interval analysis.

Furthermore, by taking into account interval-valued functions in [10,11,12,13], well-known inequality types as Ostrowski, Minkowski, and Beckenbach, as well as some of their applications, were supplied. Additionally, Budak et al. in [14] developed a few inequalities utilizing interval-valued Riemann-Liouville fractional integrals. The definition of interval-valued harmonically convex functions was provided by Liu et al. in [15], and as a result, they are able to derive several Hermite-Hadamard type inequalities, including interval fractional integrals. The authors provided a fuzzy integral-based variation of Jensen's inequality for interval-valued functions in [16] and [17] and demonstrated several integral inequalities, [18,19,20,21,22,23,24,25,26,27,28,29]. In their proofs of Hermite-Hadamard type inequalities for set-valued functions in [30] and [31,32], Mitroi et al. made use of general forms of interval-valued convex functions. Rom'an Flores et al. found a few Gronwal type inequalities for interval-valued functions in [33]. Zhao et al. showed many kinds of integral inequalities for interval-valued functions in [34,35].

In [36], Jleli and Samet discovered brand-new Hermite-Hadamard type inequality involving fractional integrals with regard to a different function. Fractional integrals of a function with respect to another function were first introduced by Tunc in [37]. The Riemann-Liouville and Hadamard fractional integrals were generalized into a single form by Katugompala's novel fractional integration. Budak and Agarwal used generalized fractional integrals, which generalize some significant fractional integrals like the Riemann-Liouville fractional integrals, the Hadamard fractional integrals, and the Katugampola fractional integrals in [38], to establish the Hermite-Hadamard-type inequalities for co-ordinated convex function. Interval-valued left- and right-sided generalized fractional double integrals were defined by Kara et al. [39]. Numerous authors have concentrated on interval-valued functions in recent years. The authors of [40] introduced the idea of interval-valued general convex functions and used it to demonstrate a number of novel Hermite-Hadamard type inequalities. A fractional version of Hermite-Hadamard type inequalities for interval-valued harmonically convex functions was also provided by the authors in [41]. Researchers recently expanded the idea of interval-valued convexity and described various types of $UD$-convexity for interval-valued functions in [42,43,44,45,46]. For $UD$-fuzzy-number-valued convex functions, they also discovered a large number of Hermite-Hadmard type inequalities.

To express the collection of all positive fuzzy numbers over the real numbers, we introduce the notation ${\mathbb{F}}_0$ in the context of this article. The terms ${\mathcal{A}}_{[ʋ, ȡ]}$, ${\mathcal{I}\mathcal{A}}_{[ʋ, ȡ]},$ and ${\mathcal{F}\mathcal{A}}_{\left(\left[ʋ, ȡ\right]\right)}$ refer to the set of all $FNVM$ that are Riemann integrable real valued functions, Aumann’s integrable IV-Fs and fuzzy Aumann’s integrable on the interval $[ʋ, ȡ]$. The following theorem draws a link between functions that are integrable in the sense of Riemann ($\mathcal{A}$-integrable) and functions that are integrable in the sense of ${\mathcal{F}\mathcal{A}}_{ }$. Additionally, the sign "${\supseteq }_{\mathbb{F}}$" is used to denote the up and down ($UD$) fuzzy inclusion relationship for $\widetilde{Ɖ}$ and $\widetilde{₼}$ belonging to ${\mathbb{F}}_0$, where $\widetilde{₼}$ is thought of as a fuzzy subset of $\widetilde{Ɖ}$. If and only if for $ƺ$-levels, the conditions ${\left[\widetilde{Ɖ}\right]}^{ƺ}{\supseteq }_I{\left[\widetilde{₼}\right]}^{ƺ}$ is met, this $UD$-inclusion is true. Integral fuzzy inequalities generated from $FNVM$s have recently attracted the attention of several academics:

We provide the ideas of generalized fractional integrals for two-variable $FNVMs$ in order to demonstrate Hermite-Hadmard type inequalities for convex and coordinated convex functions, which are inspired by ongoing investigations. The main benefit of the newly established inequalities is that they can be converted into classical Hermite-Hadamard integral inequalities for coordinated $UD$-convex $FNVM$s as well as fuzzy Riemann-Liouville fractional Hermite-Hadamard, Hadamard, and Katugampola fractional Hermite-Hadamard inequalities without having to prove each one separately.

The format of this essay is as follows: A brief summary of the foundations of fuzzy-number-valued calculus and other relevant works in this area are presented in Section 2. In Section 3, we provide some generalized fractional integrals for $UD$-convex $FNVM$ with two variables. For $UD$-convex $FNVM$, we create a novel Hermite-Hadamard type inequality. Several Hermite-Hadamard type inequalities for coordinated $UD$-convex $FNVM$ are parented in Section 3. It is also taken into consideration how these findings compare to findings of a similar nature in the literature. Finally, Section 4 makes some suggestions for additional study.

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

These sets are known as $ƺ$-level sets or $ƺ$-cut sets of $\widetilde{Ɖ}$.

Remember the approaching notions, which are offered in the literature. If $\widetilde{Ɖ},\widetilde{₼}\in {\mathbb{F}}_0$ and $\mathcal{t}\in \mathfrak{R}$, then, for every $ƺ\in \left[0, 1\right],$ the arithmetic operations addition “$\oplus ”$, multiplication “$\otimes ”$, and scaler multiplication “$\odot ”$ are defined by

Equations (4) through (6) have immediate consequences for these outcomes.

is a complete metric space, where $\mathrm{H}$ indicates the well-known Hausdorff metric on the space of intervals.

$\forall$ $ƺ\in (0, 1].$ $\forall$ $ƺ\in \left(0, 1\right],$ ${\mathcal{F}\mathcal{A}}_{\left(\left[ʋ, ȡ\right], ƺ\right)}$ denotes the collection of all $FA$-integrable $FNVM$s over $[ʋ, ȡ]$.

where $ɤ>0$ and $\Gamma$ is the gamma function.

Recently, Allahviranloo et al. [49] introduced the fuzzy version of defined the fractional integral integrals such that:

and

respectively, where $\Gamma \left(\psi \right)={\int }_0^{\infty }{\mathtt{t}}^{\psi -1}{e}^{-\mathtt{t}}d\mathtt{t}$ is the Euler gamma function. The fuzzy left and right Riemann-Liouville fractional integral $\psi$ based on left and right end point functions can be defined, that is

where

and

The right Riemann-Liouville fractional integral, denoted by${\left[{\mathcal{I}}_{{ռ}^{-}}^{ɤ}\widetilde{Ɣ}\left(\psi \right)\right]}^{ƺ}$, can also be defined using the left and right end point functions.

and

where$\tilde{\mathcal{M}}\left(\mathrm{ʋ},\mathrm{ȡ}\right)=\widetilde{\mathrm{Ɣ}}\left(\mathrm{ʋ}\right)\otimes \tilde{\mathcal{J}}\left(\mathrm{ʋ}\right)\oplus \widetilde{\mathrm{Ɣ}}\left(\mathrm{ȡ}\right)\otimes \tilde{\mathcal{J}}\left(\mathrm{ȡ}\right),$ $\tilde{\mathcal{N}}\left(\mathrm{ʋ},\mathrm{ȡ}\right)=\widetilde{\mathrm{Ɣ}}\left(\mathrm{ʋ}\right)\otimes \tilde{\mathcal{J}}\left(\mathrm{ȡ}\right)\oplus \widetilde{\mathrm{Ɣ}}\left(\mathrm{ȡ}\right)\otimes \tilde{\mathcal{J}}\left(\mathrm{ʋ}\right),$${\mathcal{M}}_{\mathrm{ƺ}}\left(\mathrm{ʋ},\mathrm{ȡ}\right)=\left[{\mathcal{M}}_{\mathrm{*}}\left(\left(\mathrm{ʋ},\mathrm{ȡ}\right),\mathrm{ƺ}\right),{\mathcal{M}}^{\mathrm{*}}\left(\left(\mathrm{ʋ},\mathrm{ȡ}\right),\mathrm{ƺ}\right)\right]$, and ${\mathcal{N}}_{\mathrm{ƺ}}\left(\mathrm{ʋ},\mathrm{ȡ}\right)=\left[{\mathcal{N}}_{\mathrm{*}}\left(\left(\mathrm{ʋ},\mathrm{ȡ}\right),\mathrm{ƺ}\right),{\mathcal{N}}^{\mathrm{*}}\left(\left(\mathrm{ʋ},\mathrm{ȡ}\right),\mathrm{ƺ}\right)\right].$

Interval and fuzzy Aumann's type integrals are defined as follows for coordinated $IVM$ $Ɣ\left(⍬,\psi \right)$ and coordinated $FNVM$ $\widetilde{Ɣ}\left(⍬,\psi \right)$:

for all $\mathrm{ƺ}\in \left[0,1\right].$

The family of all $FD$-integrable of $FNVM$s over coordinates and $D$-integrable functions over coordinates are denoted by ${\mathcal{F}\mathfrak{O}}_{∆}$ and ${\mathfrak{O}}_{\left(∆, ƺ\right)},$ for all $ƺ\in \left[0, 1\right].$

Here is the main definition of fuzzy Riemann-Liouville fractional integral on the coordinates of the function $\widetilde{Ɣ}\left(⍬,\psi \right)$ by:

Here is the newly defined concept of coordinated convexity over fuzzy number space in the codomain via $UD$-relation given by:

for all$\left(ծ, ռ\right), \left(ʋ,ȡ\right)\in ∆,$and $\epsilon ,s\in \left[0, 1\right],$ where $\widetilde{Ɣ}\left(⍬\right){\ge }_{\mathbb{F}}\tilde{0}.$ If inequality (23) is reversed, then $\widetilde{Ɣ}$ is referred to be coordinate concave $FNVM$ on $∆$.

for all $\left(⍬,\psi \right)\in ∆$ and for all $ƺ\in \left[0, 1\right]$. Then, $\widetilde{Ɣ}$ is coordinated $UD$-convex $FNVM$ on $∆,$ if and only if, for all $ƺ\in \left[0, 1\right],$ ${Ɣ}_{*}\left(\left(⍬,\psi \right), ƺ\right)$ and ${Ɣ}^{*}\left(\left(⍬,\psi \right), ƺ\right)$ are coordinated convex and concave functions, respectively.

Then, for each $ƺ\in \left[0, 1\right],$ we have ${Ɣ}_{ƺ}\left(⍬\right)=\left[\left(1-ƺ\right)⍬\psi +5ƺ,\left(1-ƺ\right)\left(6+e^{⍬}\right)\left(6+e^{\psi }\right)+5ƺ\right]$. Since endpoint functions ${Ɣ}_{*}\left(\left(⍬,\psi \right),ƺ\right),$ ${Ɣ}^{*}\left(\left(⍬,\psi \right),ƺ\right)$ are coordinate concave functions for each $ƺ\in \left[0, 1\right]$. Hence, $\widetilde{Ɣ}\left(⍬,\psi \right)$ is coordinate $UD$-convex $FNVM$.

From Lemma 1 and Example 1, we can easily note that each $UD$-convex $FNVM$ is coordinated $UD$-convex $FNVM$. But the converse is not true.

Let one assumes that ${Ɣ}_{*}\left(\left(⍬,\psi \right),ƺ\right)\ne {Ɣ}^{*}\left(\left(⍬,\psi \right),ƺ\right)$ with $ƺ=1$ and ${Ɣ}_{*}\left(\left(⍬,\psi \right),ƺ\right)$ is affine function and ${Ɣ}^{*}\left(\left(⍬,\psi \right),ƺ\right)$ is a concave function. If the stated inequality here, see [32:]

is true.

for all $\left(⍬,\psi \right)\in ∆$ and for all $ƺ\in \left[0, 1\right]$. Then, $\widetilde{Ɣ}$ is coordinated left-$UD$-convex (concave) $FNVM$ on $∆,$ if and only if, for all $ƺ\in \left[0, 1\right],$ ${Ɣ}_{*}\left(\left(⍬,\psi \right), ƺ\right)$ and ${Ɣ}^{*}\left(\left(⍬,\psi \right), ƺ\right)$ are coordinated convex (concave) and affine functions on $∆$, respectively.

for all $\left(⍬,\psi \right)\in ∆$ and for all $ƺ\in \left[0, 1\right]$. Then, $\widetilde{Ɣ}$ is coordinated right-$UD$-convex (concave) $FNVM$ on $∆,$ if and only if, for all $ƺ\in \left[0, 1\right],$ ${Ɣ}_{*}\left(\left(⍬,\psi \right), ƺ\right)$ and ${Ɣ}^{*}\left(\left(⍬,\psi \right), ƺ\right)$ are coordinated affine and convex (concave) functions on $∆$, respectively.

for all $\left(⍬,\psi \right)\in ∆$ and for all $ƺ\in \left[0, 1\right]$. Then, $\widetilde{Ɣ}$ is coordinated $UD$-concave $FNVM$ on $∆,$ if and only if, for all $ƺ\in \left[0, 1\right],$ ${Ɣ}_{*}\left(\left(⍬,\psi \right), ƺ\right)$ and ${Ɣ}^{*}\left(\left(⍬,\psi \right), ƺ\right)$ are coordinated concave and convex functions, respectively.

Then, for each $ƺ\in \left[0, 1\right],$ we have ${Ɣ}_{ƺ}\left(⍬,\psi \right)=\left[\left(1-ƺ\right)\left(6-e^{⍬}\right)\left(6-e^{\psi }\right)+25ƺ,35\left(1-ƺ\right)⍬\psi +25ƺ\right]$. Since endpoint functions ${Ɣ}_{*}\left(\left(⍬,\psi \right),ƺ\right),$ ${Ɣ}^{*}\left(\left(⍬,\psi \right),ƺ\right)$ are coordinate concave and convex functions for each $ƺ\in \left[0, 1\right]$. Hence $\widetilde{Ɣ}\left(⍬,\psi \right)$ is coordinated $UD$-concave $FNVM$.

In the next results, to avoid confusion, we will not include the symbols $\left(R\right)$, $\left(IR\right)$, $\left(FR\right)$, $\left(ID\right)$, and $\left(FD\right)$ before the integral sign.

The main goal of this article is to develop a number of original fractional coordinated integral inequalities for the Hermite-Hadamard types using an coordinated $UD$-concave $FNVM$. We acquired the most recent estimates for mappings whose products are coordinated $UD$-concave $FNVM$s using the fuzzy fractional operators.

Here is first result of coordinated integral inequalities for the Hermite-Hadamard type using the fuzzy fractional operators via coordinated $UD$-concave $FNVM$s.

If $Ɣ\left(⍬\right)$ coordinated concave $FNVM$ then,