Hermite–Hadamard and Pachpatte Type Inequalities for Coordinated Preinvex Fuzzy-Interval-Valued Functions Pertaining to a Fuzzy-Interval Double Integral Operator

Many authors have recently examined the relationship between symmetry and generalized convexity. Generalized convexity and symmetry have become a new area of study in the field of inequalities as a result of this close relationship. In this article, we introduce the idea of preinvex fuzzyinterval-valued functions (preinvex F·I-V·F) on coordinates in a rectangle drawn on a plane and show that these functions have Hermite–Hadamard-type inclusions. We also develop Hermite–Hadamardtype inclusions for the combination of two coordinated preinvex functions with interval values. The weighted Hermite–Hadamard-type inclusions for products of coordinated convex interval-valued functions discussed in a recent publication by Khan et al. in 2022 served as the inspiration for our conclusions. Our proven results expand and generalize several previous findings made in the body of literature. Additionally, we offer appropriate examples to corroborate our theoretical main findings.

On the other hand, Archimedes' calculation of the circumference of a circle can be linked to the theory of interval analysis, which has a lengthy history. However, due to a lack of applications to other sciences, it was forgotten for a very long time. Burkill [17] developed several fundamental interval function features in 1924. Kolmogorov's [18] generalization of Burkill's findings from single-valued functions to multi-valued functions came shortly after. Of course, throughout the following 20 years, numerous additional outstanding achievements were also obtained. Please take notice that Moore was the first to realize how interval analysis might be used to calculate the error boundaries of computer numerical solutions. The theoretical and applied research on interval analysis has received a lot of attention and has produced useful discoveries during the past 50 years since Moore [19] published the first monograph on the subject in 1966. In more recent years, Nikodem et al. [20] and, particularly, Budak et al. [21], Chalco-Cano et al. [22,23], Costa et al. [24][25][26], Román-Flores et al. [27,28], Flores-Franuli et al. [29], and Zhao et al. [30][31][32][33] have expanded various well-known inequalities. For more information, see  and the references are therein.
We introduce the coordinated preinvex functions in fuzzy interval-valued settings, which are inspired by Dragomir [34], Latif and Dragomir [44], and Khan et al. [37,38]. We also talk about how coordinated fuzzy-interval preinvexity and preinvexity relate to one another. The key findings of this study are new fuzzy-interval versions of Hermite-Hadamard-type inequalities that we develop with the help of newly defined coordinated fuzzy-interval preinvexity. Finally, we provide some examples to highlight our key findings. The current findings can also be seen as instruments for further study into topics like inequalities for fuzzy-interval-valued functions, fuzzy interval optimization, and generalized convexity.

Preliminaries
Let R I be the space of all closed and bounded intervals of R and Q ∈ R I be defined by outstanding achievements were also obtained. Please take notice that Moore was the first to realize how interval analysis might be used to calculate the error boundaries of computer numerical solutions. The theoretical and applied research on interval analysis has received a lot of attention and has produced useful discoveries during the past 50 years since Moore [19] published the first monograph on the subject in 1966. In more recent years, Nikodem et al. [20] and, particularly, Budak et al. [21], Chalco-Cano et al. [22,23], Costa et al. [24][25][26], Román-Flores et al. [27,28], Flores-Franuli et al. [29], and Zhao et al. [30][31][32][33] have expanded various well-known inequalities. For more information, see  and the references are therein. We introduce the coordinated preinvex functions in fuzzy interval-valued settings, which are inspired by Dragomir [34], Latif and Dragomir [44], and Khan et al. [37,38]. We also talk about how coordinated fuzzy-interval preinvexity and preinvexity relate to one another. The key findings of this study are new fuzzy-interval versions of Hermite-Hadamard-type inequalities that we develop with the help of newly defined coordinated fuzzyinterval preinvexity. Finally, we provide some examples to highlight our key findings. The current findings can also be seen as instruments for further study into topics like inequalities for fuzzy-interval-valued functions, fuzzy interval optimization, and generalized convexity.
Let ∈ ℝ and ⋅ be defined by  It can be easily seen that "⊇ " looks like "up and down" on the real line ℝ, so we call " ⊇ " as "up and down" (or "UD" order, in short) [40]. outstanding achievements were also obtained. Please take notice that Moore was the f to realize how interval analysis might be used to calculate the error boundaries of c puter numerical solutions. The theoretical and applied research on interval analysis received a lot of attention and has produced useful discoveries during the past 50 ye since Moore [19] published the first monograph on the subject in 1966. In more rec years, Nikodem et al. [20] and, particularly, Budak et al. [21], Chalco-Cano et al. [22,Costa et al. [24][25][26], Román-Flores et al. [27,28], Flores-Franuli et al. [29], and Zhao e [30][31][32][33] have expanded various well-known inequalities. For more information, see [ 69] and the references are therein. We introduce the coordinated preinvex functions in fuzzy interval-valued settin which are inspired by Dragomir [34], Latif and Dragomir [44], and Khan et al. [37,38]. also talk about how coordinated fuzzy-interval preinvexity and preinvexity relate to another. The key findings of this study are new fuzzy-interval versions of Hermite-Ha mard-type inequalities that we develop with the help of newly defined coordinated fuz interval preinvexity. Finally, we provide some examples to highlight our key findin The current findings can also be seen as instruments for further study into topics like qualities for fuzzy-interval-valued functions, fuzzy interval optimization, and generali convexity.
Let ∈ ℝ and ⋅ be defined by looks like "up and down" on the real line ℝ, so we call " ⊇ " as "up and down" (or "UD" or in short) [40].
If Q * = Q * , then Q is said to be degenerate. In this article, all intervals will be nondegenerate intervals. If Q * ≥ 0, then [Q * , Q * ] is called a positive interval. The set of all positive intervals is denoted by R + I and defined as R + I = {[Q * , Q * ] : [Q * , Q * ] ∈ R I and Q * ≥ 0}. Let λ ∈ R and λ · Q be defined by Then, the Minkowski difference Z − Q, addition Q + Z, and Q × Z for Q, Z ∈ R I are defined by [Z * , Z * ] × [Q * , Q * ] = [min{Z * Q * , Z * Q * , Z * Q * , Z * Q * }, max{Z * Q * , Z * Q * , Z * Q * , Z * Q * }] (4) It can be easily seen that "⊇ I " looks like "up and down" on the real line R, so we call "⊇ I " as "up and down" (or "UD" order, in short) [40]. Z * ≤ Q * , Z * ≤ Q * or Z * ≤ Q * , Z * < Q * it is a partial interval order relation. The relation [Z * , Z * ] ≤ I [Q * , Q * ] is coincident to [Z * , Z * ] ≤ [Q * , Q * ] on R I . It can be easily seen that "≤ I " looks like "left and right" on the real line R, so we call "≤ I " as "left and right" (or "LR" order, in short) [39,40].
It is a familiar fact that (R I , d H ) is a complete metric space [42][43][44].
Definition 1 ([40,41]). A fuzzy subset L of R is distinguished by mapping ψ : R → [0, 1] called the membership mapping of L. That is, a fuzzy subset L of R is mapping ψ : R → [0, 1] . So, for further study, we have chosen this notation.We appoint F to denote the set ofall fuzzy subsets of R. Let ψ ∈ F. Then, ψ is known as a fuzzy number or fuzzy interval if the following properties are satisfied by ψ: (1) ψ should be normal if there exists received a lot of attention and has produced useful discoveries during the past 50 years since Moore [19] published the first monograph on the subject in 1966. In more recent years, Nikodem et al. [20] and, particularly, Budak et al. [21], Chalco-Cano et al. [22,23], Costa et al. [24][25][26], Román-Flores et al. [27,28], Flores-Franuli et al. [29], and Zhao et al. [30][31][32][33] have expanded various well-known inequalities. For more information, see  and the references are therein.
We introduce the coordinated preinvex functions in fuzzy interval-valued settings, which are inspired by Dragomir [34], Latif and Dragomir [44], and Khan et al. [37,38]. We also talk about how coordinated fuzzy-interval preinvexity and preinvexity relate to one another. The key findings of this study are new fuzzy-interval versions of Hermite-Hadamard-type inequalities that we develop with the help of newly defined coordinated fuzzyinterval preinvexity. Finally, we provide some examples to highlight our key findings. The current findings can also be seen as instruments for further study into topics like inequalities for fuzzy-interval-valued functions, fuzzy interval optimization, and generalized convexity.

Preliminaries
Let ℝ be the space of all closed and bounded intervals of ℝ and ∈ ℝ be defined by If * = * , then is said to be degenerate. In this article, all intervals will be nondegenerate intervals. If * ≥ 0, then [ * , * ] is called a positive interval. The set of all positive intervals is denoted by ℝ and defined as ℝ = [ * , * ]: [ * , * ] ∈ ℝ and * ≥ 0 .

Preliminaries
Let ℝ be the space of all closed and bounded intervals of ℝ and ∈ ℝ be fined by It can be easily seen that " looks like "up and down" on the real line ℝ, so we call " ⊇ " as "up and down" (or "UD" or in short) [40]. (2) ψ should be upper semi continuous on R if for given puter numerical solutions. The theoretical and applied research on interval ana received a lot of attention and has produced useful discoveries during the past since Moore [19] published the first monograph on the subject in 1966. In mo years, Nikodem et al. [20] and, particularly, Budak et al. [21], Chalco-Cano et a Costa et al. [24][25][26], Román-Flores et al. [27,28], Flores-Franuli et al. [29], and Z [30][31][32][33] have expanded various well-known inequalities. For more information 69] and the references are therein.
We introduce the coordinated preinvex functions in fuzzy interval-valued which are inspired by Dragomir [34], Latif and Dragomir [44], and Khan et al. [3 also talk about how coordinated fuzzy-interval preinvexity and preinvexity rela another. The key findings of this study are new fuzzy-interval versions of Hermi mard-type inequalities that we develop with the help of newly defined coordinat interval preinvexity. Finally, we provide some examples to highlight our key The current findings can also be seen as instruments for further study into topics qualities for fuzzy-interval-valued functions, fuzzy interval optimization, and gen convexity. on ℝ . It can be easily seen looks like "up and down" on the real line ℝ, so we call " ⊇ " as "up and down" (or "U in short) [40]. to realize how interval analysis might be used to calculate the error boundaries of computer numerical solutions. The theoretical and applied research on interval analysis has received a lot of attention and has produced useful discoveries during the past 50 years since Moore [19] published the first monograph on the subject in 1966. In more recent years, Nikodem et al. [20] and, particularly, Budak et al. [21], Chalco-Cano et al. [22,23], Costa et al. [24][25][26], Román-Flores et al. [27,28], Flores-Franuli et al. [29], and Zhao et al. [30][31][32][33] have expanded various well-known inequalities. For more information, see  and the references are therein. We introduce the coordinated preinvex functions in fuzzy interval-valued settings, which are inspired by Dragomir [34], Latif and Dragomir [44], and Khan et al. [37,38]. We also talk about how coordinated fuzzy-interval preinvexity and preinvexity relate to one another. The key findings of this study are new fuzzy-interval versions of Hermite-Hadamard-type inequalities that we develop with the help of newly defined coordinated fuzzyinterval preinvexity. Finally, we provide some examples to highlight our key findings. The current findings can also be seen as instruments for further study into topics like inequalities for fuzzy-interval-valued functions, fuzzy interval optimization, and generalized convexity.

Preliminaries
Let ℝ be the space of all closed and bounded intervals of ℝ and ∈ ℝ be defined by If * = * , then is said to be degenerate. In this article, all intervals will be nondegenerate intervals. If Then, the Minkowski difference − , addition + , and × for , ∈ ℝ are defined by to realize how interval analysis might be used to calculate the error boundari puter numerical solutions. The theoretical and applied research on interval an received a lot of attention and has produced useful discoveries during the pa since Moore [19] published the first monograph on the subject in 1966. In m years, Nikodem et al. [20] and, particularly, Budak et al. [21], Chalco-Cano et Costa et al. [24][25][26], Román-Flores et al. [27,28], Flores-Franuli et al. [29], and [30][31][32][33] have expanded various well-known inequalities. For more informatio 69] and the references are therein. We introduce the coordinated preinvex functions in fuzzy interval-value which are inspired by Dragomir [34], Latif and Dragomir [44], and Khan et al. [ also talk about how coordinated fuzzy-interval preinvexity and preinvexity re another. The key findings of this study are new fuzzy-interval versions of Herm mard-type inequalities that we develop with the help of newly defined coordina interval preinvexity. Finally, we provide some examples to highlight our ke The current findings can also be seen as instruments for further study into topi qualities for fuzzy-interval-valued functions, fuzzy interval optimization, and g convexity. if and only if

Preliminaries
outstanding achievements were also obtained. Please take notice that Moore w to realize how interval analysis might be used to calculate the error boundari puter numerical solutions. The theoretical and applied research on interval an received a lot of attention and has produced useful discoveries during the pa since Moore [19] published the first monograph on the subject in 1966. In m years, Nikodem et al. [20] and, particularly, Budak et al. [21], Chalco-Cano et Costa et al. [24][25][26], Román-Flores et al. [27,28], Flores-Franuli et al. [29], and [30][31][32][33] have expanded various well-known inequalities. For more informatio 69] and the references are therein.
We introduce the coordinated preinvex functions in fuzzy interval-value which are inspired by Dragomir [34], Latif and Dragomir [44], and Khan et al. [ also talk about how coordinated fuzzy-interval preinvexity and preinvexity re another. The key findings of this study are new fuzzy-interval versions of Herm mard-type inequalities that we develop with the help of newly defined coordina interval preinvexity. Finally, we provide some examples to highlight our key The current findings can also be seen as instruments for further study into topi qualities for fuzzy-interval-valued functions, fuzzy interval optimization, and g convexity.

Preliminaries
We appoint F I to denote the set ofall fuzzy intervals or fuzzy numbers of R.
Definition 2 ( [40,41]). Given ψ ∈ F I , the level sets or cut sets are given by ψ λ = Mathematics 2022, 10, x FOR PEER REVIEW outstanding achievements were also obtained. Please take no to realize how interval analysis might be used to calculate th puter numerical solutions. The theoretical and applied resea received a lot of attention and has produced useful discover since Moore [19] published the first monograph on the sub years, Nikodem et al. [20] and, particularly, Budak et al. [21] Costa et al. [24][25][26], Román-Flores et al. [27,28], Flores-Franu [30][31][32][33] have expanded various well-known inequalities. For 69] and the references are therein. We introduce the coordinated preinvex functions in fuz which are inspired by Dragomir [34], Latif and Dragomir [44] also talk about how coordinated fuzzy-interval preinvexity a another. The key findings of this study are new fuzzy-interval mard-type inequalities that we develop with the help of newly interval preinvexity. Finally, we provide some examples to The current findings can also be seen as instruments for furth qualities for fuzzy-interval-valued functions, fuzzy interval op convexity. outstanding achievements were also obtained. Pleas to realize how interval analysis might be used to ca puter numerical solutions. The theoretical and appl received a lot of attention and has produced useful since Moore [19] published the first monograph on years, Nikodem et al. [20] and, particularly, Budak Costa et al. [24][25][26], Román-Flores et al. [27,28], Flo [30][31][32][33] have expanded various well-known inequa 69] and the references are therein. We introduce the coordinated preinvex functio which are inspired by Dragomir [34], Latif and Drag also talk about how coordinated fuzzy-interval prei another. The key findings of this study are new fuzz mard-type inequalities that we develop with the help interval preinvexity. Finally, we provide some exa The current findings can also be seen as instruments qualities for fuzzy-interval-valued functions, fuzzy i convexity.

Preliminaries
Let ℝ be the space of all closed and bounde fined by is cal positive intervals is denoted by ℝ and defined as

.
Let ∈ ℝ and ⋅ be defined by in short) [40]. outstanding achievements were also obtained. Please take notice that Moore was the first to realize how interval analysis might be used to calculate the error boundaries of computer numerical solutions. The theoretical and applied research on interval analysis has received a lot of attention and has produced useful discoveries during the past 50 years since Moore [19] published the first monograph on the subject in 1966. In more recent years, Nikodem et al. [20] and, particularly, Budak et al. [21], Chalco-Cano et al. [22,23], Costa et al. [24][25][26], Román-Flores et al. [27,28], Flores-Franuli et al. [29], and Zhao et al. [30][31][32][33] have expanded various well-known inequalities. For more information, see  and the references are therein. We introduce the coordinated preinvex functions in fuzzy interval-valued settings, which are inspired by Dragomir [34], Latif and Dragomir [44], and Khan et al. [37,38]. We also talk about how coordinated fuzzy-interval preinvexity and preinvexity relate to one another. The key findings of this study are new fuzzy-interval versions of Hermite-Hadamard-type inequalities that we develop with the help of newly defined coordinated fuzzyinterval preinvexity. Finally, we provide some examples to highlight our key findings. The current findings can also be seen as instruments for further study into topics like inequalities for fuzzy-interval-valued functions, fuzzy interval optimization, and generalized convexity.

Preliminaries
Let ℝ be the space of all closed and bounded intervals of ℝ and ∈ ℝ be defined by If * = * , then is said to be degenerate. In this article, all intervals will be nondegenerate intervals. If Then, the Minkowski difference − , addition + , and × for , ∈ ℝ are defined by  outstanding achievements were also obtained. Please take notice that Moore was the first to realize how interval analysis might be used to calculate the error boundaries of computer numerical solutions. The theoretical and applied research on interval analysis has received a lot of attention and has produced useful discoveries during the past 50 years since Moore [19] published the first monograph on the subject in 1966. In more recent years, Nikodem et al. [20] and, particularly, Budak et al. [21], Chalco-Cano et al. [22,23], Costa et al. [24][25][26], Román-Flores et al. [27,28], Flores-Franuli et al. [29], and Zhao et al. [30][31][32][33] have expanded various well-known inequalities. For more information, see  and the references are therein. We introduce the coordinated preinvex functions in fuzzy interval-valued settings, which are inspired by Dragomir [34], Latif and Dragomir [44], and Khan et al. [37,38]. We also talk about how coordinated fuzzy-interval preinvexity and preinvexity relate to one another. The key findings of this study are new fuzzy-interval versions of Hermite-Hadamard-type inequalities that we develop with the help of newly defined coordinated fuzzyinterval preinvexity. Finally, we provide some examples to highlight our key findings. The current findings can also be seen as instruments for further study into topics like inequalities for fuzzy-interval-valued functions, fuzzy interval optimization, and generalized convexity.

Preliminaries
Let ℝ be the space of all closed and bounded intervals of ℝ and ∈ ℝ be defined by If * = * , then is said to be degenerate. In this article, all intervals will be nondegenerate intervals. If Then, the Minkowski difference − , addition + , and × for , ∈ ℝ are defined by  looks like "up and down" on the real line ℝ, so we call " ⊇ " as "up and down" (or "UD" order, in short) [40]. Remember the approaching notions, which are offered in literature. If ψ, ∈ F I and λ ∈ R, then, for every λ ∈ [0, 1], the arithmetic operations are defined by These operations follow directly from the Equations (2)-(5), respectively.

Theorem 1 ([40]
). The space F I dealing with a supremum metric, i.e., for ψ, Is a complete metric space, where H denote the well-known Hausdorff metric on the space of intervals.
Condition 1 (see [46]). Let be an invex set with respect to .
The collection of all Riemann integrable real-valued functions and Riemann integrable I-V•F is denoted by ℛ [ , ] and ℛ [ , ] , respectively.
The collection of all Riemann integrable real-valued functions and Riemann integrable I-V·F is denoted by R [u, ν] and TR [u, ν] , respectively. outstanding achievements were also obtained. Please take notice to realize how interval analysis might be used to calculate the e puter numerical solutions. The theoretical and applied research received a lot of attention and has produced useful discoveries d since Moore [19] published the first monograph on the subject years, Nikodem et al. [20] and, particularly, Budak et al. [21], Ch Costa et al. [24][25][26], Román-Flores et al. [27,28], Flores-Franuli e [30][31][32][33] have expanded various well-known inequalities. For mo 69] and the references are therein.
We introduce the coordinated preinvex functions in fuzzy which are inspired by Dragomir [34], Latif and Dragomir [44], an also talk about how coordinated fuzzy-interval preinvexity and p another. The key findings of this study are new fuzzy-interval ver mard-type inequalities that we develop with the help of newly def interval preinvexity. Finally, we provide some examples to hig The current findings can also be seen as instruments for further s qualities for fuzzy-interval-valued functions, fuzzy interval optim convexity.
Let ∈ ℝ and ⋅ be defined by outstanding achievements were also obtained. Please take no to realize how interval analysis might be used to calculate th puter numerical solutions. The theoretical and applied resea received a lot of attention and has produced useful discover since Moore [19] published the first monograph on the sub years, Nikodem et al. [20] and, particularly, Budak et al. [21] Costa et al. [24][25][26], Román-Flores et al. [27,28], Flores-Franu [30][31][32][33] have expanded various well-known inequalities. For 69] and the references are therein. We introduce the coordinated preinvex functions in fuz which are inspired by Dragomir [34], Latif and Dragomir [44] also talk about how coordinated fuzzy-interval preinvexity a another. The key findings of this study are new fuzzy-interval mard-type inequalities that we develop with the help of newly interval preinvexity. Finally, we provide some examples to The current findings can also be seen as instruments for furth qualities for fuzzy-interval-valued functions, fuzzy interval op convexity.

Preliminaries
Let ℝ be the space of all closed and bounded interva fined by = [ * , * ] = ∈ ℝ| * ≤ ≤ * , ( outstanding achievements were also obtained. Please take notice that Moore was the first to realize how interval analysis might be used to calculate the error boundaries of computer numerical solutions. The theoretical and applied research on interval analysis has received a lot of attention and has produced useful discoveries during the past 50 years since Moore [19] published the first monograph on the subject in 1966. In more recent years, Nikodem et al. [20] and, particularly, Budak et al. [21], Chalco-Cano et al. [22,23], Costa et al. [24][25][26], Román-Flores et al. [27,28], Flores-Franuli et al. [29], and Zhao et al. [30][31][32][33] have expanded various well-known inequalities. For more information, see  and the references are therein. We introduce the coordinated preinvex functions in fuzzy interval-valued settings, which are inspired by Dragomir [34], Latif and Dragomir [44], and Khan et al. [37,38]. We also talk about how coordinated fuzzy-interval preinvexity and preinvexity relate to one another. The key findings of this study are new fuzzy-interval versions of Hermite-Hadamard-type inequalities that we develop with the help of newly defined coordinated fuzzyinterval preinvexity. Finally, we provide some examples to highlight our key findings. The current findings can also be seen as instruments for further study into topics like inequalities for fuzzy-interval-valued functions, fuzzy interval optimization, and generalized convexity.

Preliminaries
Let ℝ be the space of all closed and bounded intervals of ℝ and ∈ ℝ be defined by Then, the Minkowski difference − , addition + , and × for , ∈ ℝ are defined by (4) )d 2022, 10, x FOR PEER REVIEW 2 of 31 outstanding achievements were also obtained. Please take notice that Moore was the first to realize how interval analysis might be used to calculate the error boundaries of computer numerical solutions. The theoretical and applied research on interval analysis has received a lot of attention and has produced useful discoveries during the past 50 years since Moore [19] published the first monograph on the subject in 1966. In more recent years, Nikodem et al. [20] and, particularly, Budak et al. [21], Chalco-Cano et al. [22,23], Costa et al. [24][25][26], Román-Flores et al. [27,28], Flores-Franuli et al. [29], and Zhao et al. [30][31][32][33] have expanded various well-known inequalities. For more information, see  and the references are therein. We introduce the coordinated preinvex functions in fuzzy interval-valued settings, which are inspired by Dragomir [34], Latif and Dragomir [44], and Khan et al. [37,38]. We also talk about how coordinated fuzzy-interval preinvexity and preinvexity relate to one another. The key findings of this study are new fuzzy-interval versions of Hermite-Hadamard-type inequalities that we develop with the help of newly defined coordinated fuzzyinterval preinvexity. Finally, we provide some examples to highlight our key findings. The current findings can also be seen as instruments for further study into topics like inequalities for fuzzy-interval-valued functions, fuzzy interval optimization, and generalized convexity.

Preliminaries
Let ℝ be the space of all closed and bounded intervals of ℝ and ∈ ℝ be defined by Then, the Minkowski difference − , addition + , and × for , ∈ ℝ are defined by outstanding achievements were also obtained. Please take notice that Moore was the first to realize how interval analysis might be used to calculate the error boundaries of computer numerical solutions. The theoretical and applied research on interval analysis has received a lot of attention and has produced useful discoveries during the past 50 years since Moore [19] published the first monograph on the subject in 1966. In more recent years, Nikodem et al. [20] and, particularly, Budak et al. [21], Chalco-Cano et al. [22,23], Costa et al. [24][25][26], Román-Flores et al. [27,28], Flores-Franuli et al. [29], and Zhao et al. [30][31][32][33] have expanded various well-known inequalities. For more information, see  and the references are therein. We introduce the coordinated preinvex functions in fuzzy interval-valued settings, which are inspired by Dragomir [34], Latif and Dragomir [44], and Khan et al. [37,38]. We also talk about how coordinated fuzzy-interval preinvexity and preinvexity relate to one another. The key findings of this study are new fuzzy-interval versions of Hermite-Hadamard-type inequalities that we develop with the help of newly defined coordinated fuzzyinterval preinvexity. Finally, we provide some examples to highlight our key findings. The current findings can also be seen as instruments for further study into topics like inequalities for fuzzy-interval-valued functions, fuzzy interval optimization, and generalized convexity.

Preliminaries
Let ℝ be the space of all closed and bounded intervals of ℝ and ∈ ℝ be defined by Then, the Minkowski difference − , addition + , and × for , ∈ ℝ are defined by (4) )d Mathematics 2022, 10, x FOR PEER REVIEW 2 of 31 outstanding achievements were also obtained. Please take notice that Moore was the first to realize how interval analysis might be used to calculate the error boundaries of computer numerical solutions. The theoretical and applied research on interval analysis has received a lot of attention and has produced useful discoveries during the past 50 years since Moore [19] published the first monograph on the subject in 1966. In more recent years, Nikodem et al. [20] and, particularly, Budak et al. [21], Chalco-Cano et al. [22,23], Costa et al. [24][25][26], Román-Flores et al. [27,28], Flores-Franuli et al. [29], and Zhao et al. [30][31][32][33] have expanded various well-known inequalities. For more information, see  and the references are therein. We introduce the coordinated preinvex functions in fuzzy interval-valued settings, which are inspired by Dragomir [34], Latif and Dragomir [44], and Khan et al. [37,38]. We also talk about how coordinated fuzzy-interval preinvexity and preinvexity relate to one another. The key findings of this study are new fuzzy-interval versions of Hermite-Hadamard-type inequalities that we develop with the help of newly defined coordinated fuzzyinterval preinvexity. Finally, we provide some examples to highlight our key findings. The current findings can also be seen as instruments for further study into topics like inequalities for fuzzy-interval-valued functions, fuzzy interval optimization, and generalized convexity.

Preliminaries
Let ℝ be the space of all closed and bounded intervals of ℝ and ∈ ℝ be defined by Then, the Minkowski difference − , addition + , and × for , ∈ ℝ are defined by (4) outstanding achievements were also obtained. Please take notice that Moore wa to realize how interval analysis might be used to calculate the error boundarie puter numerical solutions. The theoretical and applied research on interval an received a lot of attention and has produced useful discoveries during the pas since Moore [19] published the first monograph on the subject in 1966. In mo years, Nikodem et al. [20] and, particularly, Budak et al. [21], Chalco-Cano et a Costa et al. [24][25][26], Román-Flores et al. [27,28], Flores-Franuli et al. [29], and Z [30][31][32][33] have expanded various well-known inequalities. For more information 69] and the references are therein. We introduce the coordinated preinvex functions in fuzzy interval-valued which are inspired by Dragomir [34], Latif and Dragomir [44], and Khan et al. [3 also talk about how coordinated fuzzy-interval preinvexity and preinvexity rel another. The key findings of this study are new fuzzy-interval versions of Hermi mard-type inequalities that we develop with the help of newly defined coordinat interval preinvexity. Finally, we provide some examples to highlight our key The current findings can also be seen as instruments for further study into topic qualities for fuzzy-interval-valued functions, fuzzy interval optimization, and ge convexity. outstanding achievements were also obtained. Please take notice that Moo to realize how interval analysis might be used to calculate the error boun puter numerical solutions. The theoretical and applied research on interv received a lot of attention and has produced useful discoveries during th since Moore [19] published the first monograph on the subject in 1966. years, Nikodem et al. [20] and, particularly, Budak et al. [21], Chalco-Can Costa et al. [24][25][26], Román-Flores et al. [27,28], Flores-Franuli et al. [29], [30][31][32][33] have expanded various well-known inequalities. For more inform 69] and the references are therein. We introduce the coordinated preinvex functions in fuzzy interval-v which are inspired by Dragomir [34], Latif and Dragomir [44], and Khan e also talk about how coordinated fuzzy-interval preinvexity and preinvex another. The key findings of this study are new fuzzy-interval versions of H mard-type inequalities that we develop with the help of newly defined coo interval preinvexity. Finally, we provide some examples to highlight ou The current findings can also be seen as instruments for further study into qualities for fuzzy-interval-valued functions, fuzzy interval optimization, a convexity. outstanding achievements were also obtained. Please take notice tha to realize how interval analysis might be used to calculate the erro puter numerical solutions. The theoretical and applied research on received a lot of attention and has produced useful discoveries dur since Moore [19] published the first monograph on the subject in years, Nikodem et al. [20] and, particularly, Budak et al. [21], Chalc Costa et al. [24][25][26], Román-Flores et al. [27,28], Flores-Franuli et al [30][31][32][33] have expanded various well-known inequalities. For more 69] and the references are therein. We introduce the coordinated preinvex functions in fuzzy int which are inspired by Dragomir [34], Latif and Dragomir [44], and K also talk about how coordinated fuzzy-interval preinvexity and pre another. The key findings of this study are new fuzzy-interval versio mard-type inequalities that we develop with the help of newly define interval preinvexity. Finally, we provide some examples to highli The current findings can also be seen as instruments for further stud qualities for fuzzy-interval-valued functions, fuzzy interval optimiza convexity. , λ) ∈ S(S λ ) , (19) where S(S λ ) = {S(., λ) → R : S(., λ) is integrable and S( Mathematics 2022, 10, x FOR PEER REVIEW outstanding achievements were also obtained. Please take notice that Moor to realize how interval analysis might be used to calculate the error bound puter numerical solutions. The theoretical and applied research on interva received a lot of attention and has produced useful discoveries during the since Moore [19] published the first monograph on the subject in 1966. In years, Nikodem et al. [20] and, particularly, Budak et al. [21], Chalco-Cano Costa et al. [24][25][26], Román-Flores et al. [27,28], Flores-Franuli et al. [29], a [30][31][32][33] have expanded various well-known inequalities. For more informa 69] and the references are therein. We introduce the coordinated preinvex functions in fuzzy interval-va which are inspired by Dragomir [34], Latif and Dragomir [44], and Khan et also talk about how coordinated fuzzy-interval preinvexity and preinvexity another. The key findings of this study are new fuzzy-interval versions of H mard-type inequalities that we develop with the help of newly defined coord interval preinvexity. Finally, we provide some examples to highlight our The current findings can also be seen as instruments for further study into t qualities for fuzzy-interval-valued functions, fuzzy interval optimization, an convexity. outstanding achievements were also obtained. Please take notic to realize how interval analysis might be used to calculate the puter numerical solutions. The theoretical and applied research received a lot of attention and has produced useful discoveries since Moore [19] published the first monograph on the subjec years, Nikodem et al. [20] and, particularly, Budak et al. [21], C Costa et al. [24][25][26], Román-Flores et al. [27,28], Flores-Franuli [30][31][32][33] have expanded various well-known inequalities. For m 69] and the references are therein. We introduce the coordinated preinvex functions in fuzzy which are inspired by Dragomir [34], Latif and Dragomir [44], a also talk about how coordinated fuzzy-interval preinvexity and another. The key findings of this study are new fuzzy-interval v mard-type inequalities that we develop with the help of newly d interval preinvexity. Finally, we provide some examples to hi The current findings can also be seen as instruments for further qualities for fuzzy-interval-valued functions, fuzzy interval opti convexity. outstanding achievements were also obtained. Please take notice that Moore was the to realize how interval analysis might be used to calculate the error boundaries of puter numerical solutions. The theoretical and applied research on interval analysi received a lot of attention and has produced useful discoveries during the past 50 y since Moore [19] published the first monograph on the subject in 1966. In more re years, Nikodem et al. [20] and, particularly, Budak et al. [21], Chalco-Cano et al. [22][23][24][25][26], Román-Flores et al. [27,28], Flores-Franuli et al. [29], and Zhao [30][31][32][33] have expanded various well-known inequalities. For more information, see 69] and the references are therein. We introduce the coordinated preinvex functions in fuzzy interval-valued sett which are inspired by Dragomir [34], Latif and Dragomir [44], and Khan et al. [37,38] also talk about how coordinated fuzzy-interval preinvexity and preinvexity relate to another. The key findings of this study are new fuzzy-interval versions of Hermite-H mard-type inequalities that we develop with the help of newly defined coordinated fu interval preinvexity. Finally, we provide some examples to highlight our key find The current findings can also be seen as instruments for further study into topics like qualities for fuzzy-interval-valued functions, fuzzy interval optimization, and genera convexity.

Preliminaries
Let ℝ be the space of all closed and bounded intervals of ℝ and ∈ ℝ be fined by outstanding achievements were also obtained. Please take notice that Moore was to realize how interval analysis might be used to calculate the error boundaries puter numerical solutions. The theoretical and applied research on interval ana received a lot of attention and has produced useful discoveries during the past since Moore [19] published the first monograph on the subject in 1966. In mor years, Nikodem et al. [20] and, particularly, Budak et al. [21], Chalco-Cano et al Costa et al. [24][25][26], Román-Flores et al. [27,28], Flores-Franuli et al. [29], and Zh [30][31][32][33] have expanded various well-known inequalities. For more information, 69] and the references are therein. We introduce the coordinated preinvex functions in fuzzy interval-valued which are inspired by Dragomir [34], Latif and Dragomir [44], and Khan et al. [37 also talk about how coordinated fuzzy-interval preinvexity and preinvexity relat another. The key findings of this study are new fuzzy-interval versions of Hermit mard-type inequalities that we develop with the help of newly defined coordinate interval preinvexity. Finally, we provide some examples to highlight our key f The current findings can also be seen as instruments for further study into topics qualities for fuzzy-interval-valued functions, fuzzy interval optimization, and gen convexity. Note that, the Theorem 2 is also true for interval double integrals. The collection of all double integrable I-V·F is denoted TO ∆ , respectively.

Let
Note that, if end-point functions are Lebesgue-integrable, then S is a fuzzy double Aumann-integrable function over ∆.
Proof. From the Definition 9 of coordinated preinvex F·I-V·F, it can be easily proved.
From Lemma 1, we can easily note each preinvex F·I-V·F is a coordinated preinvex F·I-V·F. However, the converse is not true, see Example 1.
Remark 4. If one takes ϕ 1 (b, a) = b − a and ϕ 2 (ν, u) = ν − u, then S is known as aconvex F·I-V·F on coordinates if S satisfies the following inequality: which is valid defined by Khan et al. [38]. If one takes S * (x, ω) = S * (x, ω) with λ = 1, then S is known as a preinvex function on coordinates if S satisfies the following inequality which is defined by Latif and Dragomir [44].
If one takes S * (x, ω) = S * (x, ω) with λ = 1, then S is known as a convex function on coordinates if S satisfies the following inequality is valid, then S is named as IVFon coordinates, which is defined by Dragomir [34]. From Example 1, it can be easily seen that each coordinated preinvex F·I-V·F is not a preinvex F·I-V·F. Theorem 9. Let ∆ be a coordinated preinvex set, and let S : ∆ → F I be a F·I-V·F. Then, from λ-levels, we obtain the collection of I-V·Fs S λ : ∆ → R + I ⊂ R I are given by for all (x, ω) ∈ ∆ and for all λ ∈ [0, 1]. Then, S is a coordinated preinvex F·I-V·F on ∆, if and only if, for all λ ∈ [0, 1], S * ((x, ω), λ) and S * ((x, ω), λ) are coordinated preinvex functions.
Proof. The proof of Theorem 9 is similar to that of Theorem 8.
In the next results, to avoid confusion, we will not include the symbols (R), (IR), (FR), (ID), and (FD) before the integral sign.

Fuzzy-Interval Hermite-Hadamard Inequalities
In this section, we propose HHand HH-Fejér inequalities for coordinated preinvex F·I-V·Fs, and verify with the help of some nontrivial example.