The New Versions of Hermite–Hadamard Inequalities for Pre-invex Fuzzy-Interval-Valued Mappings via Fuzzy Riemann Integrals

In this study, we use the fuzzy order relation to show some novel variants of Hermite–Hadamard inequalities for pre-invex fuzzy-interval-valued mappings (F-I∙V-Ms), which we term fuzzy-interval Hermite–Hadamard inequalities and fuzzy-interval Hermite–Hadamard–Fejér inequalities. This fuzzy order relation is defined as the level of the fuzzy-interval space by the Kulisch–Miranker order relation. There are also some new exceptional instances mentioned. The theory proposed in this research is shown with practical examples that demonstrate its usefulness. This paper's approaches and methodologies might serve as a springboard for future study in this field.

As a novel non-probabilistic approach, interval analysis is a special instance of fuzzy-interval-valued analysis. There is no doubt that fuzzy-interval analysis is extremely important in both pure and practical research. One of the initial aims of the fuzzy-interval analysis process was to analyze the error estimations of finite state machines' numerical solutions. However, the fuzzy-interval analysis technique, which (2) has been used in mathematical models in engineering for over 50 years as one of the ways to solve interval uncertain structural systems, is a critical cornerstone. It is worth noting that applications in automatic error analysis, operation research, computer science, management sciences, artificial intelligence, control engineering, and decision sciences, see [30]. Furthermore, [31][32][33][34][35][36][37][38][39][40][41][42][43] has a number of applications in optimization theory relating to fuzzy interval-valued mappings. We refer interested readers to [43,44] and the bibliographies cited in them for recent developments in the field of interval-valued mappings. Moreover, Jensen's integral inequality for F-I•V-M was derived by Oseuna-Gomez et al. [13] and Costa et al. [45]. Costa and Floures used the same method to show Minkowski and Beckenbach's inequalities, with F-I•V-Ms as integrands.
This article is organized as follows: in the second section, we review and discuss the basic concepts of interval and fuzzy intervals, as well as a class of modified convex F-I•V-Ms known as pre-invex F-I•V-Ms. In the third section, we obtain fuzzy interval HH-inequalities and verify these inequalities with the help of examples by employing this class. Furthermore, pre-invex F-I•V-Ms introduce certain HH-Fejér inequalities. The final portion of this study concludes with findings and future plans.

Preliminaries
We will begin by introducing interval analysis theory, which will be used throughout this article.
Let K C be the collection of all closed and bounded intervals of ℝ that is The set of all positive interval is denoted by K C + and defined as We now discuss some properties of intervals under the arithmetic operations addition, multiplication and scalar multiplication. If * , * , ℭ * , ℭ * ∈ K C and ∈ ℝ , then arithmetic operations are defined by * , * + ℭ * , ℭ * = * + ℭ * , * +ℭ * , for all * , * , ℭ * , ℭ * ∈ K C , it is an order relation, see [18]. For given * , * , ℭ * , ℭ * ∈ K C , we say that * , The concept of Riemann integral for I•V-M first introduced by Moore [17] is defined as follows: it is partial order relation.

Definition 3 [8] Let K be an invex set. Then F-I•V-M
∶ K → is said to be:

Definition 4 [38] Let K be an invex set. Then F-I•V-M
∶ K → is said to be:

Remark 4
The pre-invex F-I•V-Ms have some very nice properties similar to convex F- In the case of ( , ) = − , we obtain the definition of convex F-I•V-M, see [37].

Main Results
Since ∶ K × K → ℝ is a bi-mapping, then we requiring following condition to prove the upcoming results: Condition C. Let K be an invex set with respect to . For any , ∈ K and ∈ [0, 1], From Condition C, it can be easily seen that when = 0, then ( , ) = 0 if and only if, = , for all , ∈ K . For more useful details and the applications of Condition C, see [36,[38][39][40][41][42].
(((for all ∈ K and for all Υ ∈ [0, 1]  In a similar way as above, we have Combining (17) and (18), we have This completes the proof. ( , ) = − . We now compute the following: .
where (   Proof Using condition C, we can write.

two pre-invex F-I•V-Ms along with family of I•V-Ms
By hypothesis, for each Υ ∈ [0, 1], we have   for each Υ ∈ [0, 1], that means Hence, Theorem 9 is verified. The next results, which are linked with the well-known Fejér-Hermite-Hadamard type inequalities, will be obtained using symmetric mappings of one variable forms.     From (26), we have that is hence Next, we construct first HH-Fejér inequality for pre-invex F-I•V-M, which generalizes first HH-Fejér inequalities for pre-invex mapping, see [16,43].
, This completes the proof.

Conclusion
In this work, some new HH-inequalities are established by means of fuzzy order relation on fuzzy-interval space for pre-invex F-I•V-Ms. Useful examples that verify the applicability of theory developed in this study are presented. In future, we intend to use various types of pre-invex F-I•V-Ms to construct fuzzy-interval inequalities of F-I•V-Ms. We hope that this concept will be helpful for other authors to play their roles in different fields of knowledge creation.
Author contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Funding No received any funding.
Availability of data and material Not applicable.

Conflict of interest
The authors declare that they have no competing interests.
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