Interval Fej é r-Type Inequalities for Left and Right- λ -Preinvex Functions in Interval-Valued Settings

: For left and right λ -preinvex interval-valued functions (left and right λ -preinvex IVFs) in interval-valued Riemann operator settings, we create Hermite–Hadamard ( H - H ) type inequalities in the current study. Additionally, we create Hermite–Hadamard–Fej é r ( H - H -Fej é r)-type inequalities for preinvex functions of the left and right interval-valued type under some mild conditions. Moreover, some exceptional new and classical cases are also obtained. Some useful examples are also presented to prove the validity of the results.


Introduction
One of the most well-known inequalities in the theory of convex functions with a geometrical meaning and a wide range of applications is the traditional Hermite-Hadamard inequality. This disparity might be seen as a more sophisticated use of convexity. Recent years have seen resurgence in interest in the Hermite-Hadamard inequality for convex functions, leading to the study of several noteworthy improvements and extensions [1,2].
It is generally recognized how important set-valued analysis research is both theoretically and in terms of practical applications. Control theory and dynamical games have been the driving forces behind several developments in set-valued analysis. Since the beginning of the 1960s, mathematical programming and optimal control theory have been the driving forces behind these fields. A specific example is interval analysis, which was developed in an effort to address the interval uncertainty that frequently emerges in mathematical or computer models of some deterministic real-world processes, see [3][4][5][6][7]. In recent years, certain important inequalities for interval valued functions, including Hermite-Hadamard-and Ostrowski-type inequalities, have also been developed. By utilizing Hukuhara derivatives for interval valued functions, Chalco-Cano et al. developed Ostrowski-type inequalities for interval valued functions in [8,9]. The inequalities of Minkowski and Beckenbach for interval valued functions were established by Roman-Flores et al. in [10]. We direct readers to [11][12][13][14][15][16] for further results that are related to generalization of convex and interval-valued convex functions As additional references, Zhao et al. [17] introduced the idea of interval-valued coordinated convex functions; An et al. [18] introduced interval (h1, h2) convex functions; Nwaeze et al. [19] proved H-H inequality for n-polynomial convex interval-valued functions; and Tariboon et al. [20], Kalsoom et al. [21] and Ali et al. [22] refined this idea using quantum calculus. Recently, this concept was also generalized to convex fuzzy interval-valued functions by Khan et al. [23]. Interval-valued analysis has also been used in optimization in fuzzy environments [24][25][26][27][28][29][30].
We construct some new mappings in relation to Hermite-Hadamard-type inequalities and show new Hermite-Hadamard-Fejér-type inequalities that do actually give refinement inequalities in order to be motivated by the investigations undertaken in [14][15][16]28]. Some special cases which can be vied as applications of our main results are also discussed. Some non-trivial examples are also presented to discuss the validity of our main findings.

Preliminaries
We begin by recalling the basic notations and definitions. We define interval as We write len [B * , 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 The set of all positive intervals is denoted by K C + and defined as We will now look at some of the properties of intervals using arithmetic operations.
for all [B * , B * ], [G * , G * ] ∈ K C , it is an order relation. This relation is also known as left and right relation, see [27].
Moore [5] initially proposed the concept of Riemann integral for IVF, which is defined as follows: Then U is Riemann integrable (IR) over [ℴ, ς] if and only if, U * and U * both are Riemann integrable over [ℴ, ς] such that The collection of all Riemann-integrable interval-valued functions is denoted by IR ([ℴ, ς)]) . Definition 1. ( [29]). Let K be an invex set and λ : [0, 1] → R such that λ( ) > 0. Then IVF U : K → K + C is said to be left and right λ-preinvex on K with respect to if for all , ∈ K, σ ∈ [0, 1], where U ( ) ≥ p 0, : K × K → R. U is named as left and right λ-preconcave on K with respect to if inequality (7) is reversed. U is named as affine left and right λ-preinvex on K with respect to if Remark 2. The left and right λ-preinvex IVFs have some very nice properties similar to preinvex IVF: If U is left and right λ-preinvex IVF, then Y is also left and right λ-preinvex for Y ≥ 0. If U and ψ both are left and right λ -preinvex IVFs, then max(U ( ), ψ( )) is also left and right λ-preinvex IVF. Now we discuss some new special cases of λ-preinvex IVFs: (i) If λ(σ) = σ s , then left and right λ -preinvex IVF becomes left and right s-preinvex IVF, that is If ( , ) = − , then U is named as left and right s -convex IVF.
(ii) If λ(σ) = σ, then left and right λ -preinvex IVF becomes left and right preinvex IVF, that is If ( , ) = − , then U is named as left and right convex IVF. (iii) If λ(σ) ≡ 1, then left and right λ -preinvex IVF becomes left and right P IVF, that is If ( , ) = − , then U is named as left and right P IVF.

Main Results
We use the crucial assumption about bifunction : K × K → R that has been provided to demonstrate the main conclusions of this study.
If U is left and right λ-preconcave IVF, then (14) is reversed such that Proof. Let U : [ℴ, ℴ + (ς, ℴ)] → K + C be a left and right λ-preinvex IVF. Then, by hypothesis, we have Therefore, we have It follows that That is, In a similar way as above, we have Combining (16) and (17), we have which complete the proof.
Note that, inequality (14) is known as fuzzy-interval H-H inequality for left and right λ-preinvex IVF.
Hence, U ( ) is left and right λ-preinvex IVF with respect to (ς, ℴ) = ς − ℴ. Since U * ( ) = 2 2 and U * ( ) = 4 2 then, we compute the following Similarly, it can be easily shown that From which it follows that That is, Hence, and Theorem 3 is verified.
Proof. Using condition C, we can write By hypothesis, we have Integrating over [0, 1], we have Hence, Theorem 5 is demonstrated.

Conclusions
Over the past three decades, there has been an increase in interest in the field of convex mathematical inequalities. Novel findings are being added to the theory of inequalities as a result of the researchers' search for new generalizations of convex functions. A number of conclusions that hold for convex functions have been generalized in the current study using left and right λ-preinvex IVF. In this work, we developed several new mappings in order to obtain the innovative results. In addition to obtaining further modifications of the Hermite-Hadamard-and Fejér-type inequalities previously established for left and right λ-preinvex IVF, we have highlighted several intriguing characteristics of these mappings. The findings of this study, in our opinion, may serve as a source of motivation for mathematicians working in this area and for future researchers considering a career in this exciting area of mathematics.