Hermite–Hadamard Type Inequalities for Interval-Valued Preinvex Functions via Fractional Integral Operators

In this article, the notion of interval-valued preinvex functions involving the Riemann–Liouville fractional integral is described. By applying this, some new refinements of the Hermite–Hadamard inequality for the fractional integral operator are presented. Some novel special cases of the presented results are discussed as well. Also, some examples are presented to validate our results. The established outcomes of our article may open another direction for different types of integral inequalities for fractional interval-valued functions, fuzzy interval-valued functions, and their associated optimization problems.


Introduction
The idea of convexity has a lot of recognition in the theory of inequality, and assumes an essential part in probability theory, operations research, finance, decision-making, and numerical analysis. As of late, several generalizations related to convex functions have been set up. The idea of integral inequality is a fascinating area for research within mathematical analysis. Some fundamental integral inequalities are being utilized as a tool for fostering the subjective properties of convexity. Moreover, fractional calculus deals with derivatives and integrations of any fractional order. Recently, studies on integral inequalities using fractional operators have become an interesting topic for several mathematicians. Sarikaya et al. [1] started the trend of applying Riemann-Liouville (R-L) fractional integral operator for the classical convex function to refine the well-known Hermite-Hadamard (H-H) inequality. It is seen that fractional calculus aims at establishing mathematical models.
In the last couple of years, several well-known inequalities like the Hermite-Hadamard inequality, the Ostrowski inequality, and the Minkowski inequality are presented with the help of interval analysis. Furthermore, recently, mathematicians have started establishing integral inequalities via interval-valued fractional operators. For example, Budak et al. [12] applied Riemann-Liouville fractional integral operator to study Hermite-Hadamard inequality and Pachpatte-type inequality as follows: Then, the H-H inequality is expressed as follows:  Consequently, Liu et al. [13] improved the Hermite-Hadamard inequality and presented the following refinements via fractional operator: In the same paper, the authors also introduced the concept of interval harmonically convex function, which reads as follows: They also presented a generalization of the Hermite-Hadamard inequality for this new class of interval harmonically convex function as follows: If Υ be an interval harmonically convex function and As further refinements, An et al. [14] introduced interval (h 1 , h 2 )− convex function, Nwaeze et al. [15] proved H − H inequality for n-polynomial convex interval-valued function, Zhao et al. [16] introduced the notion of interval-valued coordinated convex function, and Ali et al. [17], Kalsoom et al. [18], and Tariboon et al. [19] refined this concept via quantum calculus. Recently, this concept is also generalized to convex fuzzy interval-valued functions by Khan et al. [20]. Interval-valued analysis has also been used in decision-making methods with fuzzy environment [21][22][23], multi-criterion decision-making with single-valued neutrosophic soft aggregation operator [24], multi-criterion decision-making with SVTrN Dombi aggregation function [25], and decisionmaking process based on GRA approach [26]. The objective of this paper is to study fractional integral inequalities for interval-valued preinvex functions. Specifically, we establish Hermite-Hadamard inequality of differintegrals 1 + 1 2 ( 2 , 1 ) type via Riemann-Liouville fractional model. Furthermore, Pachpatte-type inequalities are also established for interval-valued preinvex functions using these fractional integrals. This paper is structured as follows: In Sect. 2, some basic notions about preinvexity and properties in the frame of interval-valued function are presented. Section 4 deals with presenting some novel H-H-and Pachpatte-type inequalities for interval-valued preinvex function involving Riemann-Liouville fractional integral operator. Finally, in Sect. 5, conclusions and future scope of this paper are hinted at.

Preliminaries
In this section, first, we present some pre-requisite knowledge about the interval-valued functions, the theory of preinvexity, interval-valued integration, and interval-valued fractional integration, which is used extensively throughout the paper.

Basic Properties of Interval-Valued Functions
Here, in this subsection, we present some basic arithmetic about interval analysis, which will be very helpful throughout the paper where and Let 1 be an interval, such that it does not contain 0. Then min = min 1 2 , 1 2 , 1 2 , 1 2 max = max 1 2 , 1 2 , 1 2 , 1 2 .

Scalar Multiplication
where ∈ ℜ. Let ℜ I , ℜ + I and ℜ − I be the set of all closed intervals of ℜ , the set of all positive closed intervals of ℜ , and the set of all negative closed intervals of ℜ , respectively. We now discuss some algebraic properties of interval arithmetic.
1. Associative Property for Addition: . C a n c e l l a t i o n L aw : However, the distributive law does not always hold true.

Example 2.1
whereas Recently, many researchers have shown their interest in the interval analysis and established some innovative inequalities in different directions (see, for details, [27][28][29][30]).

The Concept of Preinvexity
In the year 1981, Hanson [31] introduced the concept of invex functions with respect to bifunction (⋅, ⋅) in mathematical programming. Soon after the publication of Hanson's work [31], Ben-Israel and Mond [32] explored the idea of invex sets and preinvex functions. The concept of preinvexity is more general than that of convexity.
In the year 1988, Weir and Mond [33] used the idea of invex sets to explore the concept of preinvexity.
The function −Υ is said to be preincave if and only if Υ is preinvex. We clearly see that if we choose then, as a result, we attain the classical convex function from the more general form of preinvex function.
We also need the following hypothesis regarding the function which was studied by Mohan and Neogy [34].
Condition C: Let ⊂ R n be an open invex subset with respect to ∶ × → ℜ . For any 1 , 2 ∈ and ∈ [0, 1] and In fact, for any 1 , 2 ∈ and 1 , 2 ∈ [0, 1] , we find from Condition C that In the year 2007, Noor [35] established and examined a new variant of H-H type inequality for preinvex functions, which is stated below as Theorem 2.1.
. Then, the following H-H type inequalities for preinvex functions holds true : For some detailed knowledge and perspectives about H-H inequality in the frame of Preinvexity, we encourage interested readers to (see, for example, [35,36] and the references cited therein).

H-H-Type Inequalities
The principal objective and the main aim of this section are to establish a novel version of the H-H-type inequalities in the mode of interval-valued preinvex functions. (3.2) This proves the first inequality asserted by Theorem 3.1.
For the proof of the second inequality in Theorem 3.1, we need the following results: and Adding the last two inequalities, we have Multiplying both sides of the above inequality by −1 ( > 0) and integrating over the closed interval [0, 1], we find that and if satisfies Condition C, then the following H − H type inequalities hold true : where and Proof Since Υ and are interval-valued preinvex functions, we have and Multiplying both the above inequalities, it is readily seen that Similarly, we can observe that and Consequently, we also have (3.9) . (3.10) (3.13) (3.14) w h e r e U( 1 , 2 ) and V( 1 , 2 )are defined as the above theorem.
Proof Since Υ and are interval-valued preinvex functions, we have Choosing Then, all the assumptions of theorem () are satisfied.

Conclusions
In this article, we investigated interval R-L fractional integrals, and using these fractional integrals, we have derived some new H-H-type inequalities and Pachpatte-type inequalities for interval-valued preinvex functions (see Theorem 2.1, Theorem 3.2, and Theorem 3.3). The discussed results and remarks can lead to some interesting inequalities if we choose different values of ( 2 , 1 ) and . The results presented will potentially motivate researchers to study analogous and more general integral inequalities for various other kinds of fractional integral operators.

Future Scopes
As a consequence of this article, the next step in the research direction is to study Jensen, Hermite-Hadamard, Ostrowski, Simpson, and Pachhpatte-type inequalities for interval-valued preinvex functions and fuzzy-interval-valued preinvex functions on quantum calculus, post-quantum calculus, time scales calculus, generalized fractional calculus, and coordinates. As interval analysis has a lot of applications in fields like computer graphics, robotics, error analysis, and computational mathematics, it can be applied to optimization engineering, convex optimization, industrial optimization, etc. And due to the advancement of fuzzy interval analysis, it will be very beneficial for artificial intelligence, decisionmaking methods, financial activities, etc.
Author Contributions Conceptualization: HMS and SKS; methodology, SKS, DB, and BK; writing-review and editing: HMS, SKS, and POM; project administration: POM; resources: BK. All authors have read and agreed to the final version of the manuscript.

Funding Not applicable.
Data Availability Not data were used to this study.

Conflict of Interest
The authors declare that there is no conflict of interest.
Ethical Approval Not applicable.

Consent for Publication Not applicable.
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