Hermite–Hadamard inequality for fractional integrals of Caputo–Fabrizio type and related inequalities

In this article, firstly, Hermite–Hadamard’s inequality is generalized via a fractional integral operator associated with the Caputo–Fabrizio fractional derivative. Then a new kernel is obtained and a new theorem valid for convex functions is proved for fractional order integrals. Also, some applications of our main findings are given.

Theorem 1 (See [22]) Let f : I → R be a convex function defined on the interval I of real numbers and a, b ∈ I with a < b. Then the following inequality holds: In the field of fractional analysis, many researchers have focused on defining new operators and modeling and implementing of the problems based on their features. The features that make the operators different from each other include 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. The distinctive feature of the Caputo-Fabrizio operator is that it has a non-singular kernel. With the help of the Caputo-Fabrizio operator, new studies have been made on many modeling problems and real-world problems. This is so because the definition of Caputo-Fabrizio is very effective in better describing heterogeneousness and systems with different scales with memory effects. The main basic feature of the Caputo-Fabrizio definition can be explained as a real power turned into the integer by the Laplace transformation, thus the exact solution can be easily found for various problems. Now, we will proceed by some necessary definitions and preliminary results which are used and referred throughout this paper.
where B(α) > 0 is a normalization function satisfying B(0) = B(1) = 1. For the right fractional derivative we have and the associated fractional integral is Fractional derivative and integral operators have recently been used to generalize existing kernels. The kernel which we will generalize with the help of a Caputo-Fabrizio fractional integral operator is proven by Dragomir and Agarwal.
In the following section, we will prove a theorem which is a variant of the Hermite-Hadamard inequality.

A generalization of Hermite-Hadamard inequality via the Caputo-Fabrizio fractional operator
, then the following double inequality holds: By multiplying both sides of (2) with α(b-a) 2B(α) and adding 2(1-α) So, the proof of the first inequality in (1) is completed by reorganizing the last inequality. For the proof of the second inequality in (1), if we use the right hand side of Hadamard inequality, we can write By making the same operations with (2) in (4), we have By reorganizing (5), the proof of the second inequality in (1) is completed.
, then we have the following inequality: , Proof Since f and g are convex functions on [a, b], we have Multiplying above inequalities both sides, we have Integrating (6) with respect to t over [0, 1], and making the change of variable, we obtain N(a, b).
Theorem 4 Let f , g : I ⊆ R → R be a convex function. If fg ∈ L ([a, b]), the set of integrable functions, then

where M(a, b) and N(a, b) are given in Theorem 3 and k ∈ [a, b], B(α) > 0 is a normalization function.
Proof Since f and g are convex functions on [a, b], for t = 1 2 , we have Multiplying the above inequalities at both sides, we have Integrating the above inequality with respect to t over [0, 1] and making the change of variable, one obtains N(a, b).

Some new results related with Caputo-Fabrizio fractional operator
In this section, firstly, we will generalize a lemma, then we will put forward a theorem with the help of the lemma.
where k ∈ [a, b] and B(α) > 0 is a normalization function.
Proof It is easy to see that By multiplying both sides with α(b-a) 2 2B(α) and subtracting 2(1-α) Thus, the proof is completed.
where k ∈ [a, b] and B(α) > 0 is a normalization function.
Proof By using Lemma 2, the properties of the absolute value and the convexity of |f | we have So the proof is completed.
Theorem 6 Let f : I ⊆ R → R be a differentiable positive mapping on I • and |f | q be convex on [a, b] where p > 1, p -1 + q -1 = 1, a, b ∈ I with a < b. If f ∈ L 1 [a, b] and α ∈ [0, 1], the following inequality holds: where k ∈ [a, b] and B(α) > 0 is a normalization function.
Proof By a similar argument to the proof of the previous theorem, but now using Lemma 2, the Hölder inequality and convexity of |f | q , we get So the proof is completed.

Application to special means
It is very important to give an application in terms of efficiency and usefulness of the results obtained. At the same time, the accuracy of the findings will be confirmed by the application to special means for real numbers a 1 , a 2 such that a 1 = a 2 : (1) The arithmetic mean , a 1 , a 2 ∈ R.
Now, using the results in Sect. 3, we have some applications to the special means of real numbers.
Proof In Theorem 5, if we set f (z) = e z with α = 1 and B(α) = B(1) = 1, then we obtain the result immediately.
Proof In Theorem 5, if we set f (z) = z n where n is an even number with α = 1 and B(α) = B(1) = 1, then we obtain the result immediately.