Some Fej é r-Type Inequalities for Generalized Interval-Valued Convex Functions

: The goal of this study is to create new variations of the well-known Hermite–Hadamard inequality ( HH -inequality) for preinvex interval-valued functions (preinvex I-V-F s). We develop several additional inequalities for the class of functions whose product is preinvex I-V-F s. The ﬁndings described here would be generalizations of those found in previous studies. Finally, we obtain the Hermite–Hadamard–Fej é r inequality with the support of preinvex interval-valued functions. Some new and classical special cases are also obtained. Moreover, some nontrivial examples are given to check the validity of our main results.


Introduction
Let R be the set of real numbers and J : K ⊆ R → R be a convex function. If o, q ∈ K with o ≤ q, then where K is a convex set, which is named Jensen's inequality [1]. The famous HH-inequality is then created by Hermite and Hadamard by adding the integral mean value of the convex function J to inequality (1). Let J : K ⊆ R → R be a convex function. If o, q ∈ K with o ≤ q, then If J is concave, the inequality (2) holds in the inverse fashion, see [2,3]. This inequality has several applications in numerical integration since it may be used to give a rough approximation of the integral mean on [o, q]. For details on the use and rising popularity of the HH-inequality, readers might refer to [4][5][6][7][8][9][10][11][12][13][14].
Research on fractional operator-type integral inequalities is becoming more and more popular, since fractional integral operators have several applications in a number of fields. Set et al. [40] used Raina's fractional integral operators to create new Hermite-Hadamard-Mercer inequalities. With a modified Mittag-Leffler kernel, Srivastava et al. [41] created the generalized left-side and right-side fractional integral operators, and they then used this large family of fractional integral operators to study the fascinating Chebyshev inequality. Sun established certain Hermite-Hadamard-type inequalities for extended h-convex functions and modified preinvex functions in refs. [28,42] using two local fractional integral operators with a Mittag-Leffler kernel. The two local fractional integral operators were then used by Xu et al. [43] to examine Hermite-Hadamard-Mercer for extended h-convex functions. For more information, see [44][45][46][47][48][49][50][51][52][53] and the references therein.
The goal of this study is to find certain HH-inclusions that are more generic. The study's general format consists of five sections, including an introduction. The remainder of the paper is organized as follows: Section 2 introduces certain types of integrals of real-valued functions and their accompanying HH-inequalities. In Section 2, we briefly summarize the idea of interval-valued functions. We discuss generalized integrals of interval-valued functions in Section 2, and provide some examples of these integrals. In Section 3, we use defined generalized integrals to show many HH-inclusions for intervalvalued convex functions and to validate the main results; we have provided some nontrivial examples. Finally, in Section 4, some findings and future study areas are explored.

Preliminaries
We offer some fundamental arithmetic regarding interval analysis in this paragraph, which will be quite useful throughout the article.
Let K C , K + C , K − C be the set of all closed intervals of R, the set of all closed positive intervals of R and the set of all closed negative intervals of R. Then, K C , K + C , and K − C are defined as K C = {[a * , a * ] : a * , a * ∈ R and a * ≤ a * }, (8) K + C = {[a * , a * ] : a * , a * ∈ K C and a * > 0}, (9) K − C = {[a * , a * ] : a * , a * ∈ K C and a * < 0}.
where J * , The collection of all Riemann-integrable real-valued functions and Riemann-integrable I-V-Fs is denoted by R [o,q] and TR [o,q] , respectively. Definition 1 ([72]). Let K be a convex set. Then I-V-F J : for all ω, κ ∈ K, ξ ∈ [0, 1]. J is named as concave on K if inequality (12) is reversed.

Definition 2 ([78]
). Let K be an invex set. Then, I-V-F J : K → K + C is named as preinvex on K with respect to w if for all ω, κ ∈ K, ξ ∈ [0, 1], where w : K × K → R. J is named as preincave on K with respect to w if inequality (13) is reversed. J is named as affine if J is both convex and concave.

Theorem 2.
Let K be an invex set and J : for all ω ∈ K. Then, J is preinvex I-V-F on K, if and only if, J * (ω) and J * (ω) are preinvex and preincave functions, respectively.
Proof. The proof of this result is similar to Theorem 6, see [63].

Main Results
In this section, we propose interval HH-inequalities for preinvex I-V-Fs. Moreover, some examples are presented that verify the applicability of theory developed in this study. Then,

It follows that
That is, In a similar way as above, we have Combining (17) and (18), we have This completes the proof.
We now compute the following: Similarly, it can be easily shown that such that From which, it follows that that is, Hence, Moreover, From the definition of left and right preinvex IV-F, it follows that 0 ≤ J(ω) and 0 ≤ A(ω), so Integrating both sides of the above inequality over [0,1], we obtain and the theorem has been established.
The following assumption is required to prove the next result regarding the bi-function w : K × K → R which is known as: Condition C [70]. Let K be an invex set with respect to w. For any o, q ∈ K and ξ ∈ [0, 1], Clearly for ξ = 0, we have w(q, o) = 0 if and only if, q = o, for all o, q ∈ K. For the applications of Condition C, see [5,8,42,69,70].

Proof. Using condition C, we can write
By hypothesis, we have Integrating over [0, 1], we have from which, we have We now compute the following: Hence, Theorem 5 is verified.
We now give HH-Fejér inequalities for preinvex I-V-Fs. Firstly, we obtain the second HH-Fejér inequality for preinvex I-V-F.
Proof. Using condition C, we can write Since J is a preinvex, we have By multiplying (28) by C(o + (1 − ξ)w(q, o)) = C(o + ξw(q, o)) and integrating it by ξ over [0, 1], we obtain From which, we have This completes the proof.

Conclusions
We constructed the new Hermite-integral Hadamard's inequality for preinvex intervalvalued functions in this study employing the interval integral operators with exponential kernel supplied by Moore in ref. [68]. In order to show the size relationship of the function values of the inequalities and to confirm the veracity of the findings, we offered four numerical examples. Our study of interval integral operator-type integral inequalities will broaden the practical application of Hermite-Hadamard-type inequalities, because integral operators are frequently used in engineering technology, such as mathematical models, and because different integral operators are suitable for different types of practical problems. We will study these inequalities further using various types of integral operators, because we are aware that integral operators are employed in many other fields. This will also provide a direction for our future research.

Conflicts of Interest:
The authors declare no conflict of interest.