Fractional Calculus for Convex Functions in Interval-Valued Settings and Inequalities

: In this paper, we discuss the Riemann–Liouville fractional integral operator for left and right convex interval-valued functions (left and right convex I · V-F ), as well as various related notions and concepts. First, the authors used the Riemann–Liouville fractional integral to prove Hermite– Hadamard type ( H – H type) inequality. Furthermore, H – H type inequalities for the product of two left and right convex I · V-Fs have been established. Finally, for left and right convex I · V-Fs , we found the Riemann–Liouville fractional integral Hermite–Hadamard type inequality ( H – H Fej é r type inequality). The ﬁndings of this research show that this methodology may be applied directly and is computationally simple and precise. convex functions and some supplementary interval analysis ﬁndings. Our results are a generalization of a number of previously published ﬁndings. In the future, we will use generalized interval and fuzzy Riemann–Liouville fractional operators to investigate this concept for generalized left and right convex I · V-Fs and F-I · V-Fs by using interval Katugampola fractional integrals and fuzzy Katugampola fractional integrals. For applications, see [53–56].


Introduction
Mathematical inequality, finance, engineering, statistics, and probability all use convex functions in some way. Convex and symmetric convex functions have strong relationships with inequalities. Because of their intriguing features in the mathematical sciences, there are expansive properties and strong links between the symmetric function and different fields of convexity, including convex functions, probability theory, and convex geometry on convex sets. Convex functions have a long and illustrious history in science, and they have been a hot focus of study for more than a century. Several researchers have proposed different convex function guesses, expansions, and variants. Many inequalities or equalities, such as the Ostrowski-type inequality, Hardy-type inequality, Opial-type inequality, Simpson inequality, Fejér-type inequality, and Cebysev-type inequalities, have been established using convex functions. Among these inequalities, the H-H inequality [1,2], on which many publications have been published, is likely the one that attracts the most attention from scholars. H-H inequality has been regarded as the most useful inequality in mathematical analysis since its discovery in 1883. It is also known as the conventional H-H Inequality equation. The expansions and generalizations of the H-H inequality have piqued the curiosity of a number of mathematicians. For various classes of convex functions and mappings, a number of mathematicians in the fields of pure and applied mathematics have worked to expand, generalize, counterpart, and enhance the H-H inequality (references [3][4][5][6][7][8][9][10][11][12][13] are a good place to start for interested readers).
Historically, Leibnitz and L'Hospital (1695) are credited with the invention of fractional calculus; however, Riemann, Liouville, and Grunwald-Letnikov, among others, made significant contributions to the field later on. The way that fractional operator speculation deciphers nature's existence in a grand and intentional fashion [14][15][16][17][18][19] has piqued the curiosity of researchers. By offering an enhanced form of an integral representation for the Appell k-series, Mubeen and Iqbal [20] have contributed to the present research.
Moreover, Khan et al. [21] exploited fuzzy order relations to introduce a new class of convex fuzzy-interval-valued functions (convex F-I·V-Fs), known as (h 1 , h 2 )-convex F-I·V-Fs, as well as a novel version of the H-H type inequality for (h 1 , h 2 )-convex F-I·V-Fs that incorporates the fuzzy interval Riemann integral. Khan et al. went a step further by providing new convex and extended convex I·V-F classes, as well as new fractional H-H type and H-H type inequalities for left and right (h 1 , h 2 )-preinvex I·V-F [22], left and right p-convex I·V-Fs [23], left and right log-h-convex I·V-Fs [24], and the references therein. For further analysis of the literature on the applications and properties of fuzzy Riemannian integrals, inequalities, and generalized convex fuzzy mappings, we refer the readers to cited works  and the references therein.
Motivated and inspired by the fascinating features of symmetry, convexity, and the fractional operator, we study the new H-H and related H-H type inequalities for left and right convex I·V-Fs, based upon the pseudo order relation and the Riemann-Liouville fractional integral operator.

Preliminaries
First, we offer some background information on interval-valued functions, the theory of convexity, interval-valued integration, and interval-valued fractional integration, which will be utilized throughout the article.
We offer some fundamental arithmetic regarding interval analysis in this paragraph, which will be quite useful throughout the article.
Let X I , X + I , X − I 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, respectively.

Definition 2.
[31] The I·V-F Y : K → X + I is named as the left and right convex-I·V-F on convex set K if the coming inequality, holds for all ω, z ∈ K and ν ∈ [0, 1] we have. If inequality (3) is reversed, then Y is named as the left and right concave on K. Y is affine, if and only if, it is both left and right convex and left and right concave.
for all ω ∈ K. Then, Y is a left and right convex I·V-F on K, if and only if, Y * (ω) and Y * (ω) both are convex functions.

Interval Fractional Hermite-Hadamard Inequalities
The major goal, and the main purpose of this section, is to develop a novel version of the H-H inequalities in the mode of interval-valued left and right convex functions.
If Y(ω) is a left and right concave I·V-F, then Proof. Let Y : [s, t] → X + I be a left and right convex I·V-F. Then, by hypothesis, we have: Therefore, we have Multiplying both sides by ν a−1 and integrating the obtained result, with respect to ν over (0, 1), we have Similar to the above, we have Combining (7) and (8), we have Hence, we achieve the required result.
, are left and right convex functions, then Y(ω) is a left and right convex I·V-F. We clearly see that Y ∈ L [s, t], X + I , and and Theorem 3 is verified.
The upcoming two results acquire the fractional inequalities for the product of left and right convex I·V-Fs.
Proof. Since Y, G are both left and right convex I·V-Fs, then we have From the definition of left and right convex I·V-Fs, it follows that 0 ≤ p Y(ω) and 0 ≤ p G(ω), so Analogously, we have Adding (9) and (10), we have Taking the multiplication of (11) by ν a−1 and integrating the obtained result, with respect to ν over (0, 1), we have Thus, and the theorem has been established.
Proof. Consider that Y, G : [s, t] → X + I are left and right convex I·V-Fs. Then, by hypoth- Taking the multiplication of (12) with ν a−1 and integrating over (0, 1), we get Hence, the required result is achieved.
The upcoming results discuss the H-H Fejér type inequality left and right convex I·V-F. Firstly, we achieve second H-H Fejér type inequality.
If Y is a concave I·V-F, then inequality (13) is reversed.
Proof. Let Y be a left and right convex I·V-F and ν a−1 C(νs + (1 − ν)t) ≥ 0. Then, we have and After adding (14) and (15), and integrating over [0, 1], we get then, from (16), we have That is, Hence, Now, we first obtain the H-H Fejér type inequality for the left and right convex I·V-F.
If Y is a concave I·V-F, then inequality (18) is reversed.