Some Integral Inequalities for Generalized Convex Fuzzy-Interval-Valued Functions via Fuzzy Riemann Integrals

In this study, we introduce the new concept of h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h$$\end{document}-convex fuzzy-interval-valued functions. Under the new concept, we present new versions of Hermite–Hadamard inequalities (H–H inequalities) are called fuzzy-interval Hermite–Hadamard type inequalities for h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h$$\end{document}-convex fuzzy-interval-valued functions (h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h$$\end{document}-convex FIVF) by means of fuzzy order relation. This fuzzy order relation is defined level wise through Kulisch–Miranker order relation defined on fuzzy-interval space. Fuzzy order relation and inclusion relation are two different concepts. With the help of fuzzy order relation, we also present some H–H type inequalities for the product of h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h$$\end{document}-convex FIVFs. Moreover, we have also established strong relationship between Hermite–Hadamard–Fej´er (H–H–Fej´er) type inequality and h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h$$\end{document}-convex FIVF. There are also some special cases presented that can be considered applications. There are useful examples provided to demonstrate the applicability of the concepts proposed in this study. This paper's thoughts and methodologies could serve as a springboard for more research in this field.


Introduction
The significance and supreme applications of convex functions are well known in different fields, especially in the study of integral inequalities, variational inequalities and optimization. Therefore, much attention has been given in studying and characterizing different directions of classical idea of convexity. Recently, many extensions and generalizations of convex functions have been studied. For more useful details, see [1-4, 6, 7, 14, 18-21, 25] and the references are therein. In classical approach, a real valued function Ψ ∶ K → ℝ is called convex if for all z, y ∈ K, ∈ [0, 1].
The concept of convexity with integral problem is an interesting area for research. Therefore, many inequalities have been introduced as applications of convex functions. Among those, the H-H inequality is an interesting outcome in convex analysis. The H-H inequality [16,17] for convex function Ψ ∶ K → ℝ on an interval K = [u, ] On the other hand, the concept of interval analysis was proposed and investigated by Moore [24], and Kulish and Miranker [23] to improve the reliability of the calculation results and automatically perform error analysis. It is a discipline in which an uncertain variable is represented by an interval of real numbers and real operations are replaced by interval operation. In last 5 decades, the concept of interval was used as a tool to handle uncertain problems. Recently, many authors have contributed their role in this theory and introduced new concepts. Zhao et al. [30] introduced h-convex interval-valued functions and proved that the following H-H type inequality for h-convex intervalvalued functions:
This study is organized as follows: Sect. 2 presents preliminary notions, new concepts and results in interval space, in the space of fuzzy-intervals and for h-convex FIVFs. Section 3 obtains fuzzy-interval H-H inequalities via h-convex FIVFs. In addition, some interesting examples are also given to verify our results. Section 4 gives conclusions.

Preliminaries
In this section, we recall some basic preliminary notions, definitions and results. With the help of these results, some new basic definitions and results are also discussed.
We b e g i n by r e c a l l i n g t h e b a s i c n o t ations and definitions. We define inter val as, * , * = z ∈ ℝ ∶ * ≤ z ≤ * and * , * ∈ ℝ , where * ≤ * . We write len * , * = * − * , If len * , * = 0 then, * , * is called degenerate. In this article, all intervals will be non-degenerate intervals. The collection of all closed and bounded intervals of ℝ is denoted and defined as K C = * , * ∶ * , * ∈ ℝand * ≤ * . If * ≥ 0 then, * , * is called positive interval. The set of all positive interval is denoted by K + C and defined as K + C = * , * ∶ * , * ∈ K C and * ≥ 0 . We'll now look at some of the properties of intervals using arithmetic operations. Let * , * , & * , & * ∈ K C and ∈ ℝ , then we have Page 3 of 15 158 Let ℝ be the set of real numbers. A mapping ∶ ℝ → [0, 1] called the membership function distinguishes a fuzzy subset set A of ℝ . This representation is found to be acceptable in this study. (ℝ) also stand for the collection of all fuzzy subsets of ℝ.
A real fuzzy interval is a fuzzy set in ℝ with the following properties: is upper semi continuous i.e., for given z ∈ ℝ, for every z ∈ ℝ there exist > 0 there exist > 0 such that The collection of all real fuzzy intervals is denoted by 0 . Let ∈ 0 be real fuzzy interval, if and only if, -levels [ ] is a nonempty compact convex set of ℝ . This is represented by from these definitions, we have where Thus a real fuzzy interval can be identified by a parametrized triples These two end point functions * ( ) and * ( ) are used to characterize a real fuzzy interval as a result.
it is partial order relation.
We will now look at some of the properties of fuzzy intervals using arithmetic operations. Let , Θ ∈ 0 and ∈ ℝ , then we have For ∈ 0 such that = Θ+ , we have the existence of the Hukuhara difference of and Θ , which we call the H-difference of and Θ , and denoted by −Θ . If H-difference exists, then Theorem 2.4. [13,26] The space 0 dealing with a supremum metric i.e., for , Θ ∈ 0 it is a complete metric space, where H denote the wellknown Hausdorff metric on space of intervals. Definition 2.5. [10] A fuzzy-interval-valued map Ψ ∶ K ⊂ ℝ → 0 is called FIVF. For each ∈ (0, 1], whose -levels define the family of IVFs Ψ ∶ K ⊂ ℝ → K C are given by Ψ (z) = Ψ * (z, ), Ψ * (z, ) for all z ∈ K. Here, for each ∈ (0, 1], the end point real functions Ψ * (., ), Ψ * (., ) ∶ K → ℝ are called lower and upper functions of Ψ.
We now discuss some special cases of h-convex FIVFs: FIVF, see [25], that is Note that, special cases (i) and (iii) are also new ones.

Main Results
In this section, we propose fuzzy-interval H-H inequalities for h-convex FIVFs. Furthermore, several examples are given to demonstrate the applicability of the theory produced in this research.

It follows that
That is Thus, In a similar way as above, we have Combining (16) and (17), we have Hence, the required result.
Proof. Take , + 2 , we have Therefore, for every ∈ [0, 1] , we have In consequence, we obtain That is It follows that In a similar way as above, we have Combining (18) and (19)   that is Hence, the proof has been completed.
Following results find the new versions of H-H inequalities for the product of two h-convex FIVFs.
Hence, the required result.  We now give HH-Fej´er inequalities for ℎ-convex FIVFs. Firstly, we obtain the second HH-Fej´er inequality for ℎ-convex FIVF.

Since
From (27) and (28), we have From which, we have that is Then we complete the proof.

Remark 3.2.
If h( ) = , then inequalities in Theorem 10 and 11 reduces for convex FIVFs which are also new one.

Conclusion
This study introduced the class of h-convex FIVFs and established some new H-H inequalities by means of fuzzy order relation on fuzzy-interval space. Moreover, we established strong relationship between H-H-Fej´er type inequality and h-convex FIVF. We provided relevant examples to demonstrate the application of the theory produced in this research.
To construct fuzzy-interval inequalities of FIVFs, we plan to use a variety of convex FIVFs. We hope that this notion will assist other authors in remunerating their contributions in other sectors of knowledge. In future, we try to explore this concept using different fuzzy fractional integral operators.