On Some Fractional Integral Inequalities Involving Caputo–Fabrizio Integral Operator

In this paper, we deal with the Caputo–Fabrizio fractional integral operator with a nonsingular kernel and establish some new integral inequalities for the Chebyshev functional in the case of synchronous function by employing the fractional integral. Moreover, several fractional integral inequalities for extended Chebyshev functional by considering the Caputo–Fabrizio fractional integral operator are discussed. In addition, we obtain fractional integral inequalities for three positive functions involving the same operator.

The main motivation of the Caputo-Fabrizio integral and derivative operator is that it is a general fractional integral and derivative. In addition, it has a non singular kernel which can be described as a real power turned into an integral by means of the Laplace transform. Consequently, an exact solution can be easily found for several problems. Nowadays, fractional integral and derivative play big role for modeling various phenomenon physics. However, in [23,24], Caputo and Fabrizio introduced new fractional derivatives and integrals without a singular kernel. Certain phenomena related to material heterogeneities cannot be well-modeled by considering the Riemann-Liouville and Caputo fractional derivatives due to the singular kernel. It stems from Caputo and Fabrizio's proposal of a new fractional integral involving the nonsingular kernel e −( 1−κ κ )(ξ−s) , 0 < κ < 1. Recently, many mathematicians in applied sciences are using the Caputo-Fabrizio fractional integral operator to model their problems. For more details, we refer to [25][26][27][28][29][30][31]. In [32], the authors presented the fundamental solutions to the Cauchy and Dirichlet problems based upon a heat conduction equation equipped with the Caputo-Fabrizio derivative, which is investigated on a line segment. The main advantage of the Caputo-Fabrizio integral operator is that the boundary condition of the fractional differential equations with Caputo-Fabrizio derivatives admits the same form as for the integer-order differential equations. In the literature, very little work has been conducted on fractional integral inequalities using Caputo and Caputo-Fabrizio integral operators. In [10,14,[16][17][18], the authors have established some new integral inequalities for the Chebyshev and extended Chebyshev functionals using different fractional operators. Recently, in [33], the authors have investigated several new estimations of the Hermite-Hadamard type inequality via generalized convex functions of the Raina type. In [34,35], the authors established fractional integral inequalities involving the Caputo-Fabrizio operator. From the above cited work, the main objective of this paper is to obtain some fractional integral inequalities for the functionals (1) and (2) by considering the Caputo-Fabrizio fractional integral operator. In addition, we establish some fractional integral inequalities for three positive and synchronous functions. The paper is organized into the following sections. Section 2 gives some basic definitions of fractional calculus. Section 3 is devoted to the proof of some fractional inequalities for Chebyshev functionals using the Caputo-Fabrizio fractional operator. Section 4 presents some inequalities involving the extended Chebyshev fractional in the case of synchronous function by employing the Caputo-Fabrizio fractional integral operator. Finally, concluding remarks are given in Section 5.

Preliminaries
Here, we provide some basic definitions of fractional calculus related to the Caputo-Fabrizio fractional integral operator. Definition 1 ([24,34]). Let κ ∈ R such that 0 < κ ≤ 1. The Caputo-Fabrizio fractional integral of order κ of a function φ is defined by For κ = 1, it is reduced to The above defintion may be extended to any κ > 0.
Definition 2 ([24,34]). Let κ, a ∈ R such that 0 < κ < 1. The Caputo-Fabrizio fractional derivative of order κ of a function φ is defined by In this study, the focus is put on the Caputo-Fabrizio fractional integral operator, aiming to demonstrate some new inequalities involving it.

Fractional Inequalities for Chebyshev Functional
Here, we obtain inequalities for the Chebyshev functional using the Caputo-Fabrizio fractional operator. Theorem 1. Let φ and ϕ be two synchronous functions on [0, ∞). Then for all ξ, κ > 0, we have From (6), we get By multiplying (7) by 1 κ e −( 1−κ κ )(ξ−µ) , which is positive, and then integrating the resulting identity with respect to µ from 0 to ξ, we have Hence, which implies that By multiplying (10) by 1 κ e −( 1−κ κ )(ξ−θ) , which is positive, and then integrating θ from 0 to ξ, we have Therefore It follows that This ends the proof of Theorem 1.

Theorem 2.
Let φ and ϕ be two synchronous functions on [0, ∞). Then, for all ξ, κ, λ > 0, we have Proof. To prove this theorem, first multiply the inequality (10) by 1 , which is positive. Then, by integrating the resulting identity with respect to θ over 0 to ξ, we obtain and this ends the proof of Theorem 2.

Fractional Inequalities for Extended Chebyshev Fractional
Here, we present some inequalities on extended Chebyshev fractional in the case of synchronous functions by employing the Caputo-Fabrizio fractional integral operator.
Proof. Since φ and ϕ are synchronous functions on [0, ∞), for all µ, θ ≥ 0, we have Owing to (21), we obtain By multiplying (22) by , which is positive, and then integrating with respect to µ from 0 to ξ, we have Consequently, By multiplying (24) by , which is positive, and then integrating with respect to θ from 0 to ξ, we have This completes the proof of the inequality (20). Now, we give our main result.
Proof. To prove this theorem, put u = p, v = q, and using Lemma 1, we get Now , multiplying both sides in (27) Again, by putting u = r, v = q, and using Lemma 1, we get By multiplying both sides of (29) by I κ 0,ξ [p(ξ)], we have With the same arguments as in the inequalities (29) and (30) Adding the inequalities (28), (30) and (31), we get the required inequality (26).
Proof. By multiplying both sides of (24) by v(θ) , which is positive, and then integrating with respect to θ from 0 to ξ, we have This completes the proof of Lemma 2.
Theorem 5. Let φ and ϕ be two integrable and synchronous functions on [0, ∞), and r, p, q: Then, for all ξ, κ, λ > 0, we have Proof. To prove this theorem, we put u = p, v = q and, by using Lemma 2, we get Now, multiplying both sides of (35) by I κ 0,ξ [r(ξ)], we obtain By putting u = r, v = q, and using Lemma 2, we get By multiplying both sides of (37) by I κ 0,ξ [p(ξ)], we have With the same argument as in the Equations (37) and (38), we obtain Adding the inequalities (36), (38) and (39), we get the inequality (34). Here, we give some fractional integral inequalities involving the Caputo-Fabrizio fractional integer operator.
Proof. From the condition (40), for any µ, θ ≥ 0, we have By multiplying both sides of the inequality (49) by 1 κ e −( 1−κ κ )(ξ−µ) , which is positive, and then integrating with respect to µ from 0 to ξ, we get With the same argument as in inequality (45), we obtain This completes the proof of inequality (48).

Concluding Remarks
In this paper, we studied the novel fractional integral inequalities for the Chebyshev and extended the Chebyshev functionals by considering the Caputo-Fabrizio fractional integral operator. In addition, we studied some inequalities for three positive functions using the same operator. The inequalities investigated in this paper make some contribution to the fields of fractional calculus and Caputo-Fabrizio fractional integral operators. In the future, we hope that inequalities presented in this paper can prove the existence and uniqueness of some ordinary differential equations, as well as initial and boundary value problems involving Caputo-Fabrizio fractional operators.