Some fuzzy-interval integral inequalities for harmonically convex fuzzy-interval-valued functions

: It is well-known fact that fuzzy interval-valued functions (F-I-V-Fs) are generalizations of interval-valued functions (I-V-Fs)


Introduction
Hermite and Hadamard's inequality [1,2] is one of the most well-known inequalities in convex function theory, with a geometrical interpretation and numerous applications. The ⋅͘ inequality is defined as follows for the convex function : → ℝ on an interval = [ , ]: for all , ∈ . If f is concave, the inequalities in (1) hold in the reversed direction. We should point out that Hermite-Hadamard inequality is a refinement of the concept of convexity, and it follows naturally from Jensen's inequality. In recent years, the Hermite-Hadamard inequality for convex functions has gotten a lot of attention, and a lot of improvements and generalizations have been examined; see [3][4][5][6][7][8][9][10][11][12] and the references therein.
Interval analysis, on the other hand, is a subset of set-valued analysis, which is the study of sets in the context of mathematical analysis and topology. It was created as a way to deal with interval uncertainty, which can be found in many mathematical or computer models of deterministic real-world phenomena. Archimedes' method, which is used to calculate the circumference of a circle, is a historical example of an interval enclosure. Moore, who is credited with being the first user of intervals in computer mathematics, published the first book on interval analysis in 1966, see [13]. Following the publication of his book, a number of scientists began to research the theory and applications of interval arithmetic. Interval analysis is now a useful technique in a variety of fields that are interested in ambiguous data because of its applications. Computer graphics, experimental and computational physics, error analysis, robotics, and many other fields have applications.
The goal of this study is to complete the fuzzy Riemann integrals for interval-valued functions and use these integrals to get the Hermite-Hadamard inequality. These integrals are also used to derive Hermite-Hadamard type inequalities for harmonically convex F-I-V-Fs.

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.
Moore [13] initially proposed the concept of Riemann integral for I-V-F, which is defined as follows: Let ℝ be the set of real numbers. A mapping ̃: ℝ → [0,1] called the membership function distinguishes a fuzzy subset set 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: 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 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.
for all , Proof. The demonstration of proof is similar to proof of Theorem 2.12, see [26].

Fuzzy-interval fractional Hermite-Hadamard inequalities
In this section, we will prove two types of inequalities. First one is . and their variant forms, and the second one is ⋅ Fejér inequalities for convex F-I-V-Fs where the integrands are F-I-V-Fs.  Thus, In a similar way as above, we have Combining (19) and (20), we have Hence, the required result.

It follows that
In a similar way as above, we have Combining (22) and (23), we have The theorem has been proved. First, we will purpose the following inequality linked with the right part of the classical − Fejér inequality for harmonically convex F-I-V-Fs through fuzzy order relation, which is said to be 2nd fuzzy − Fejér inequality.
From (32) and (33) Then we complete the proof.

Conclusions and future plan
Several novel conclusions in convex analysis and associated optimization theory can be obtained using this new class of functions known as harmonically convex F-I-V. The main findings include some new bounds with error estimations via fuzzy Riemann integrals. All of these papers aim to provide new estimations and optimal approaches. But, the main motivation of this paper is that we obtained new method by using fuzzy integrals for harmonically convex F-I-V-Fs calculus. The authors anticipate that this study may inspire more research in a variety of pure and applied sciences fields.