On Caputo–Fabrizio Fractional Integral Inequalities of Hermite–Hadamard Type for Modified h-Convex Functions

+e theory of convex functions plays an important role in engineering and applied mathematics. +e Caputo–Fabrizio fractional derivatives are one of the important notions of fractional calculus. +e aim of this paper is to present some properties of Caputo–Fabrizio fractional integral operator in the setting of h-convex function. We present some new Caputo–Fabrizio fractional estimates from Hermite–Hadamard-type inequalities. +e results of this paper can be considered as the generalization and extension of many existing results of inequalities and convex functions. Moreover, we also present some application of our results to special means of real numbers.

Many researchers in the last three decades are studying fractional calculus [8][9][10][11][12]. Some researchers deduced that it is essential to define new fractional derivatives with different singular or nonsingular kernels in order to provide more sufficient area to model more real-world problems in different fields of science and engineering [13][14][15][16][17][18][19].
In the present research, we will restrict ourselves to Caputo-Fabrizio fractional derivative. e features that make the operators different from each other comprise singularity and locality, while kernel expression of the operator is presented with functions such as the power law, the exponential function, or a Mittag-Leffler function. e unique feature of the Caputo-Fabrizio operator is that it has a nonsingular kernel. e main feature of the Caputo-Fabrizio operator can be described as a real power turned in to the integer by means of the Laplace transformation, and consequently, the exact solution can be easily found for several problems.
Fractional calculus plays a very significant role in the development of inequality theory. To study convex functions and its generalizations, the Hermite-Hadamard-type inequality is considered as one of the fundamental inequality is given as.
Theorem 1 (see [20]). Let ψ: J⊆R ⟶ R be a convex function and a 1 , b 1 ∈ J with a 1 < b 1 , then the following double inequality holds: interesting readers, we refer [24][25][26][27] to study about Hermite-Hadamard inequalities. e paper is organized as follows: first of all, we give some definitions and preliminary material related to our work. In Section 2, we will establish Hermite-Hadamardtype inequalities via Caputo-Fabrizio fractional integral operator for modified h-convex functions. Section 3 is devoted for some new inequalities via Caputo-Fabrizio fractional operator. At last, we give some application to special means and concluding remarks for our paper. Now, we start by some necessary definitions and preliminary results which will be used and in this paper.
In [28], Toader gave the concept of modified h-convex functions as follows.
Definition 2 (see [8,29,30]). Let ψ ∈ H 1 (a 1 , b 1 ), , then the left fractional derivative in the sense of Caputo and Fabrizio is given by (3) and the associated fractional integral is CF where B(σ) > 0 is a normalization function satisfying B(0) � B(1) � 1. e right fractional derivative is given as and the associated fractional integral is e following lemma is proven by Dragomir and Agarwal in [31].
Iscan gave a refinement of Hölder integral inequality in [33], which is given in the following theorem.

Generalization of Hermite-Hadamard Inequality via the Caputo-Fabrizio
Fractional Operator e following theorem is a variant of Hermite-Hadamard inequality for modified h-convex functions.
, then the following double inequality holds: where k ∈ [a 1 , b 1 ] and B(σ) > 0 is a normalization function.

Journal of Mathematics 3
After suitable rearrangement of (13), we get the required left-hand side of (11).
For the right-hand side, we will use the right-hand side of Hermite-Hadamard inequality for modified h-convex functions: By using the same operator with (12) in (14), we have CF After suitable rearrangement of (15), we get the required right-hand side of (11), which completes the proof. □ Remark 1. If we take h(α) � α, then inequality (11) reduces to the Hermite-Hadamard inequality for convex functions via Caputo-Fabrizio fractional operator [32].

Theorem 5. Let ψ and ϕ are modified h-convex functions on
, then we have the following inequality: where Proof. Since ψ and ϕ are convex on [a 1 , b 1 ], we have Multiplying both sides of (18) and (19), we have Integrating (20) with "α" over [0, 1], and using the change of variable technique, we obtain Multiplying both sides of (22) So, Journal of Mathematics 5 us, with suitable rearrangements, and the proof is completed.
Proof. Since ψ and ϕ are modified h-convex functions on J, then for α � (1/2), we have Multiplying the above inequalities at both sides, we have

Some New Results Related with Caputo-Fabrizio Fractional Operator
In this section, we establish some new inequalities for modified h-convex functions via Caputo-Fabrizio fractional operator.
Proof. Using Lemma 2 and the definition of modified h-convexity of |ψ ′ |, we get which completes the proof.
and σ ∈ [0, 1], then the following inequality holds: where k ∈ [a 1 , b 1 ], and B(σ) > 0 is a normalization function. Proof. Using Lemma 2, Hölder's integral inequality and modified h-convexity of |ψ ′ | q , we get  a 1 , b 1 ] and σ ∈ [0, 1], then the following inequality holds: where k ∈ [a 1 , b 1 ], and B(σ) > 0 is a normalization function. Proof. Assuming q > 1, using Lemma 2 and the power mean inequality and modified h-convexity of |ψ ′ | q , we get For q � 1, we use the estimates of eorem 7, which also follows step by step the above estimates. is completes the proof of theorem.
, then the following inequality holds:
Proof. Assuming q > 1 and using Lemma 2, improved power-mean integral inequality and modified h-convexity of |ψ ′ | q , we get For q � 1, we use the estimates of eorem 7 which also follows step by step the above estimates. is completes the proof of theorem. □ Corollary 2. If we take h(α) � α in inequality (41), we get the following inequality: Remark 8. Inequality (46) gives better results than inequality (39), and we have the following inequality: Proof. Using concavity of ψ: which completes the proof.

Application to Means
For two positive numbers a 1 > 0 and b 1 > 0, define ese means are, respectively, called the arithmetic and p-logarithmic means of two positive numbers a 1 and b 1 .

Conclusion
Hermite-Hadamard-type inequalities for modified h-convex functions via Caputo-Fabrizio integral operator are derived. Some new and interesting integral inequalities involving Caputo-Fabrizio fractional integral operator are also obtained for modified h-convex functions. Many existing results in literature become the particular cases for these results as mentioned in remarks.

Data Availability
All data required for this paper are included within the manuscript.

Conflicts of Interest
e authors declare no conflicts of interest.