Riemann–Liouville Fractional Integral Inequalities for Generalized Harmonically Convex Fuzzy-Interval-Valued Functions

The framework of fuzzy-interval-valued functions (FIVFs) is a generalization of interval-valued functions (IVF) and single-valued functions. To discuss convexity with these kinds of functions, in this article, we introduce and investigate the harmonically h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf{h}$$\end{document}-convexity for FIVFs through fuzzy-order relation (FOR). Using this class of harmonically h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf{h}$$\end{document}-convex FIVFs (H-h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{H}-\mathsf{h}$$\end{document}-convex FIVFs), we prove some Hermite–Hadamard (H⋅H) and Hermite–Hadamard–Fejér (H⋅H Fejér) type inequalities via fuzzy interval Riemann–Liouville fractional integral (FI Riemann–Liouville fractional integral). The concepts and techniques of this paper are refinements and generalizations of many results which are proved in the literature.


Introduction
Fractional calculus dates back to the seventeenth century, when Leibniz and Marquis de l'Hospital began a conversation about semi-derivatives. Many well-known mathematicians were inspired by this subject to investigate modern views of the area. The theory of fractional calculus grew greatly in the late nineteenth century. Math, physics, viscoelasticity, rheology, chemistry, and statistical physics, as well as electrical and mechanical engineering, are now covered.
On the other hand, Costa [11] just uncovered Jensen's type inequality in FIVF. Costa and Roman-Flores [12,13] looked at the characteristics of several types of inequalities in the context of FIVF and IVF. Roman-Flores et al. [14] established Gronwall inequality for IVFs. Furthermore, Chalco-Cano et al. [15,16] employed the generalized Hukuhara derivative to demonstrate Ostrowski-type inequalities for IVFs, as well as numerical integration applications in IVF. Nikodem et al. [17] and Matkowski and Nikodem [18] proposed new versions of Jensen's inequality for strongly convex and convex functions.
Zhao et al. [19,20] were employed IVFs to generate Chebyshev, Jensen's, and HH type inequalities. Zhang et al. [21] recently employed a pseudo order relation to extend Jensen's inequalities for set-valued and fuzzy-set-valued functions and develop a novel form of Jensen's inequalities. Budek [22] subsequently established an interval-valued fractional Riemann-Liouville HH inequality for convex IVF using an inclusion relation. For further detail, see [23][24][25][26][27] and the references therein.
Recently, Khan et al. [28] used FOR to construct a new class of convex FIVFs which is known as ( 1 , 2 )-convex FIVFs, as well as some new versions of the H⋅H type inequality for ( 1 , 2 )-convex FIVFs that incorporates the FI Riemann integral. Khan et al. went a step further by providing new convex and extended convex FIVF classes, as well as new fractional H⋅H type and H⋅H type inequalities for convex FIVF [29], -convex FIVF [30], ( 1 , 2 )-preinvex FIVF [31], log-s-convex FIVFs in the second sense [32], harmonically convex FIVFs [33], coordinated convex FIVFs [34] and the references therein. We suggest readers to  and the references therein for more study of literature on the applications and properties of FI, as well as inequalities and extended convex fuzzy mappings.
Motivated and inspired by ongoing research work, we have introduced the new generation of harmonic functions is known as ℋ − h-convex functions using FOR in Sect. 2. In Sect. 3, we have used FI fractional operators to derive new versions of Hermite-Hadamard inequalities with the help of this class. Furthermore, we have examined the study's special circumstances as applications. In the end, we have given conclusion and future plan.

Preliminaries
We will start by reviewing the fundamental notations and definitions.
The collection of all closed and bounded intervals of ℝ is denoted and defined as The set of all positive interval is denoted by K C + and defined as We will now look at some of the properties of intervals using arithmetic operations. Let * , * , * , * ∈ K C and ∈ ℝ , then we have where and For * , * , * , * ∈ K C , the inclusion ε ⊆ ε is defined by Remark 2.1. The relation ε ≤ I ε defined on K C by for all * , * , * , * ∈ K C , it is an order relation, see [35].
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. 0 also stands for the collection of all fuzzy subsets of ℝ.
The following FI Riemann-Liouville fractional integral operators were introduced by Allahviranloo et al. .
where and Similarly, the left and right end point functions can be used to define the right Riemann-Liouville fractional integral of .     Then, by (19), (5) and (6), we obtain Therefore, from (19), for each ∈ [0, 1] , left side of above inequality, we have Again, from (19), we obtain On fixing ( ) = , then from Definition 2.10, we obtain Definition 2.9.

Example 2.13. We consider the FIVFs
We shall develop a relationship between -convex FIVF and H − -convex FIVF in the next finding.
that is the proof the theorem has been completed.

Hermite-Hadamard Inequalities for Harmonically -Convex Fuzzy-Interval-Valued Functions
We shall prove two forms of inequalities in this section. The first is H⋅H and its variant forms, while the second is H⋅H Fejér inequalities for ℋ − -convex FIVFs with FIVFs as integrands. In the following, L [ , ], 0 denotes the family of Lebesgue measureable FIVFs.   Multiplying both sides by −1 and integrating the obtained result with respect to over (0, 1) , we have Let = (1− ) + and Z = +(1− ) . Then, we have

It follows that
That is In a similar way as above, we have Combining (23) and (24), we have that is Hence, the required result.
For the product of ℋ − -convex FIVFs, we now have some H⋅H inequalities. These inequalities are modifications of previously published inequalities [34,38,43].   (28) * Taking the result of multiplying (30) by −1 and integrating it with respect to over (0, 1), we get It follows that That is .    .
From which, we have that is As a result, the desired result has been achieved. Following result obtains the first FI fractional H ⋅ H Fejér inequality.