New Hermite–Hadamard-type inequalities for -convex fuzzy-interval-valued functions

In this paper, we introduce the non-convex interval-valued functions for fuzzy-interval-valued functions, which are called -convex fuzzy-interval-valued functions, by means of fuzzy order relation. This fuzzy order relation is defined level-wise through Kulisch–Miranker order relation given on the interval space. By using the -convexity concept, we present fuzzy-interval Hermite–Hadamard inequalities for fuzzy-interval-valued functions. Several exceptional cases are debated, which can be viewed as useful applications. Interesting examples that verify the applicability of the theory developed in this study are presented. The results of this paper can be considered as extensions of previously established results.


Introduction
The following integral inequality is known in the literature as the Hermite-Hadamard inequality [16,17]: where F : K → R is a convex function on the interval K = [u, ϑ] with u <ϑ. So the concept of convexity in an integral problem is an interesting area for research. Therefore, much attention has been given to studying and characterizing different directions of classical convexity. Recently, many extensions and generalizations Hermite-Hadamard inequality for generalized convex functions have been established. For more useful details, see [1, 3-5, 7, 10, 19-23, 28] and the references therein. On the other hand, the theory of interval analysis fell in to oblivion for a long time because of lack of applications in other sciences. The concept of interval analysis was proposed and investigated by Moore [26] and Kulish and Miranker [25]. For the first time it was used in numerical analysis to determine the error bounds of numerical solutions of a finite state machine. For fundamental details and applications, we refer the readers to the papers [14,27,32,33] and the references therein. Inspired by the above literature, in 2018, Zhao et al. introduced h-convex interval-valued functions and proved the Hermite-Hadamard-type inequality for h-convex interval-valued functions [34]. As a step forward, An et al. [2] presented the class of (h 1 , h 2 )-convex interval-valued functions and established the following interval-valued Hermite-Hadamard-type inequality for such functions: We refer to the readers for further analysis of the literature on the applications and properties of generalized convex functions and Hermite-Hadamard integral inequalities to [6,8,9,13,15,17,24,30,31] and the references therein.
There are some integrals to deal with fuzzy-interval-valued functions, where the integrands are fuzzy-interval-valued functions. For instance, Oseuna-Gomez et al. [29] and Costa et al. [11] constructed Jensen's integral inequality for fuzzy-interval-valued functions. By using the same approach, Costa and Roman-Flores also presented Minkowski and Beckenbach's inequalities, where the integrands are fuzzy-interval-valued functions. Motivated by [11,12,29] and [34], we generalize integral inequality (2) by constructing fuzzy-interval integral inequality for convex fuzzy-interval-valued functions, where the integrands are convex fuzzy-interval-valued functions.
This study is organized as follows: Sect. 2 presents preliminaries and results in the interval space, in the space of fuzzy-intervals, and for fuzzy integrals. Section 3 introduces the new classes of (h 1 , h 2 )-convex fuzzy-interval-valued functions and investigates their properties. Section 4 obtains fuzzy-interval Hermite-Hadamard inequalities via (h 1 , h 2 )convex fuzzy-interval-valued functions. In addition, some interesting examples are also given to verify our results. Section 4 gives conclusions and directions for future works.

Preliminaries
Let K C be the collection of all closed and bounded intervals of R, that is, K C = {[ω * , ω * ] : ω * , ω * ∈ R and ω * ≤ ω * }. If ω * ≥ 0, then [ω * , ω * ] is called a positive interval. The set of all positive intervals is denoted by K + C and defined as K We now discuss some properties of intervals under the arithmetic operations of addition, multiplication, and scalar multiplication. If [μ * , μ * ], [ω * , ω * ] ∈ K C , and ρ ∈ R, then these arithmetic operations are defined by it is an order relation, see [25]. For given The concept of Riemann integral for interval-valued functions first introduced by Moore [26] and is defined as follows: The collections of all Riemann integrable real-valued functions and Riemann integrable interval-valued functions are denoted by R [c,d] and IR [c,d] , respectively.
Let R be the set of real numbers. A fuzzy subset set A of R is distinguished by a function ϕ : R → [0, 1] called the membership function. In this study this depiction is approved. Moreover, the collection of all fuzzy subsets of R is denoted by F(R).
A real fuzzy-interval ϕ is a fuzzy set in R with the following properties: (1) ϕ is normal, i.e., there exists x ∈ R such that ϕ(x) = 1; (2) ϕ is upper semicontinuous, i.e., for every x ∈ R and ε > 0 there exists δ > 0 such that The collection of all real fuzzy-intervals is denoted by F C (R).

Proposition 1 ([12])
Let ϕ, φ ∈ F C (R). Then the relation " " given on F C (R) by it is a partial order relation. Now we discuss some properties of real fuzzy-intervals under addition, scalar multiplication, multiplication, and division. If ϕ, φ ∈ F C (R) and ρ ∈ R, then these arithmetic operations are defined by If ψ ∈ F C (R) is such that ϕ = φ+ ψ, then we have the existence of Hukuhara difference of ϕ and φ, and we say that ψ is the H-difference of ϕ and φ, which is denoted by ϕ-φ . If the H-difference exists, then Remark 2 Obviously, F C (R) is closed under addition and nonnegative scalar multiplication. And the above-defined properties on F C (R) are equivalent to those derived from the usual extension principle. Furthermore, for each scalar number ρ ∈ R, Theorem 4 ( [18,29]) The space F C (R) equipped with a supremum metric, i.e., for ψ, φ ∈ F C (R) it is a complete metric space, where H denote the well-known Hausdorff metric on the space of intervals.
From the above literature review, the following results can be concluded; see [12,18,24,26].
. Note that if both end point functions are Lebesgue-integrable, then F is a fuzzy Annum integrable function; see [18,24,26].

(h 1 , h 2 )-Convex fuzzy-interval-valued functions
In this section, we put forward the definitions of (h 1 , h 2 )-convex fuzzy-interval-valued functions and investigate their basic properties.

Hermite-Hadamard-type inequalities for fuzzy-interval-valued functions
Proof Let F : [u, ϑ] → F C (R) be an (h 1 , h 2 )-convex fuzzy-interval-valued function. Then, by hypothesis, we have Therefore, for every γ ∈ [0, 1], we have It follows that That is, In a similar way as above, we have Combining (16) and (17), we have Hence, the required result follows.