Hermite–Jensen–Mercer-Type Inequalities via Caputo–Fabrizio Fractional Integral for h -Convex Function

: Integral inequalities involving many fractional integral operators are used to solve various fractional differential equations. In the present paper, we will generalize the Hermite–Jensen–Mercer-type inequalities for an h -convex function via a Caputo–Fabrizio fractional integral. We develop some novel Caputo–Fabrizio fractional integral inequalities. We also present Caputo–Fabrizio fractional integral identities for differentiable mapping, and these will be used to give estimates for some fractional Hermite–Jensen–Mercer-type inequalities. Some familiar results are recaptured as special cases of our results.


Introduction
Fractional calculus has undergone rapid development in both applied and pure mathematics because of its enormous use in image processing, physics, machine learning, networking, and other branches. For more on fractional calculus identities, see [1][2][3]. The fractional derivative has received rapid attention among experts from different branches of science. Most of the applied problems can not be modeled by classical derivations. The complications in real-world problems are addressed by fractional differential equations. The famous fractional integral contains Riemann-Liouville [4][5][6], Hadamard [6,7], Caputo-Fabrizio [8], and Katugampola [6], etc.
In this paper, we will restrict ourselves to the Caputo-Fabrizio fractional integral operator. In the current direction of fractional calculus, numerous analysts are characterizing new operators by various methods to cover most of the real-world problems. Usually, the operators are not the same as each other in terms of singularity and locality of kernels. The main aspect that makes Caputo-Fabrizio different from others is that it has a non-singular kernel, and it is useful to find exact solutions for various issues.
In the literature, for the Jensen inequality, several interesting studies are given. In [27], for a convex function, a variant of Jensen's inequality is proved by Mercer within the year 2003. Later, Matković et al. presented the Jensen-Mercer inequality for operators with applications in the year 2006 (see [28]).
The present paper is organized as follows. First, we write definitions and preliminary material associated with our present paper. In Section 2, we will present Hermite-Jensen-Mercer-type inequalities for a Caputo-Fabrizio fractional integral operator with the help of an h-convex function. In Section 3, we will develop new Lemmas and then present some results for an h-convex function via a Caputo-Fabrizio fractional integral operator. In Section 4, some more integral inequalities for h-convex functions are established making use of the Hölder-İşcan integral inequality for an improved power mean integral inequality, and at last, we will write concluding remarks to our present paper.
Throughout the paper, we need the following assumption: Let ξ : I = [υ, µ] → R be a positive function, 0 ≤ υ < µ and ξ ∈ L 1 [υ, µ]. Furthermore, consider h : (0, 1) → R is a non-negative function, h = 0 and I ⊆ R is an interval. Now, we begin with definitions and preliminary results, which will be used in this work. [39] The function ξ :

Definition 1. (Convex function)
Definition 4 ([8,41,42]). 1], then the definition of the left fractional derivative in the sense of Caputo and Fabrizio is defined as and the associated fractional integral is CF The right fractional derivative is defined as and the associated fractional integral is In [43,44], the Hölder-İşcan integral inequality and improved power-mean integral inequality is explained as follows.
Theorem 2. (Hölder-İşcan integral inequality) [43] Let ξ 1 and ξ 2 be real functions defined on [x 1 , x 2 ] and if |ξ 1 | q and |ξ 2 | q are integrable on [ x 1 , x 2 ]. If p > 1 and 1 p + 1 q = 1, then x 2 Theorem 3. (Improved power-mean integral inequality) [44] Let ξ 1 and ξ 2 be real functions defined on [x 1 , x 2 ] and if |ξ 1 |, |ξ 1 ||ξ 2 | q are integrable functions on [ x 1 , x 2 ]. Let q ≥ 1, then holds for all x 1 , The above inequality is integrated with respect to χ over [0, 1] and by change of variable technique, we can deduce Both sides of (3) multipled by Suitable rearrangement of (4) yields the first inequality of (2). By using h-convexity of ξ, we have Adding the above two inequalities and then by using the super additivity of function and Jensen-Mercer inequality yields that Integrating the inequality (5) with respect to χ over [0, 1] and by the change of variable technique, we can write 2 By making use of the same operations with (3) in (6), we have By suitable rearrangement of (7), we obtain inequality (2).
Theorem 5. Assume that ξ : Proof. By the Jensen-Mercer inequality, we have Both sides of the above inequality are multiplied by h(χ) and integrated with respect to χ over [0,1], and we obtain Now, we will use the right-hand side of the Hermite-Hadamard inequality for the h-convex function, and we obtain Both sides of (9) multiplying by and subtracting After suitable rearrangement, (10) yields the first inequality of (8).
For the second part of the inequality of (8), we will use the right-hand side of the Hermite-Hadamard integral inequality for the h-convex function, and we can write By using the same operations with (9) in (11), we have 1 0 h(χ)dχ to both sides of (12), we have which completes the proof. where and B(θ) > 0 is a normalization function.
Proof. Since ξ 1 and ξ 2 are h-convex functions on [x 1 , x 2 ] and making use of the Jensen-Mercer inequality, we have Multiplying both sides of the above inequalities, we can write Integrating the above inequality with respect to χ over [0,1] and then by the change of variable technique, we obtain 1 The above inequality is multipled by 2B(θ) , and adding 2(1−θ) Therefore, Thus, By suitable rearrangement, (14) yields required inequality (13).

Some Novel Results Related to the Caputo-Fabrizio Fractional Operator
In this section, we will present some new Lemmas, and then we develop some novel results for an h-convex function with the help of the Caputo-Fabrizio fractional integral operator. Lemma 1. Let ξ : I = [υ, µ] → R be a differentiable mapping on I • , where υ, µ ∈ I with υ < µ.
Proof. It is easy to see that With both sides of the above inequality multiplied by and subtracting After suitable rearrangements, we obtain the desired result.
where Proof. By making use of Lemma 3, the properties of the absolute value, the h-convexity of |ξ | and the Jensen-Mercer inequality yields This completes the proof.
Proof. From Lemma 3, Hölder's integral inequality, the h-convexity of |ξ | q and the Jensen-Mercer inequality yields that This completes the proof.
Next, we will prove the following theorems using the Hölder-İscan integral inequality and for improved power mean integral inequality, respectively. Theorem 9. Assume that ξ : I → R is a positive differentiable mapping on I • and |ξ | q is a h-convex function on [υ, µ], υ, µ ∈ I • with υ < µ for q ≥ 1, where υ, µ ∈ I with υ < µ. If Proof. Take q > 1, by using Lemma 3, the power mean inequality, the h-convexity of |ξ | q and the Jensen-Mercer inequality, and we have This completes the proof.
Proof. From Lemma 3, using the Hölder-İscan integral inequality, the h-convexity of |ξ | q and the Jensen-Mercer inequality yields This completes the proof.

Conclusions
In this note, we established the Hermite-Jensen-Mercer-type inequalities for an hconvex function in the Caputo-Fabrizio setting, and various Caputo-Fabrizio fractional integral inequalities are provided as well. We expect that this work will lead to the novel fractional integral research for Hermite-Hadamard inequalities. The remarks at the end of the results verify the generalization of the results. These results are new and set various interesting directions. In the future, we will prove the inequalities (2) and (8)