Pólya–Szegö Integral Inequalities Using the Caputo–Fabrizio Approach

: In this article, we establish some of the Pólya–Szegö and Minkowsky-type fractional integral inequalities by considering the Caputo–Fabrizio fractional integral. Moreover, we give some special cases of Pólya–Szegö inequalities.


Introduction
Mathematical integral inequalities plays a very important role in classical differential and integral equations, which have many applications in many fields.
Recently, many researchers in several fields have found different results about some known fractional calculus and applications by means of the Riemann-Liouville [5][6][7][8][9][10][11], k-Riemann Liouville [12,13], Caputo [5,12,14], Hadamard [15,16], Marichev-Saigo-Maeda [17], Saigo [18][19][20], Katugamapola [21], Atangana-Baleanu [22] and some other fractional integral operators. Many mathematicians have worked on the Pólya-Szegö inequalities using various fractional integral operators in recent years (see [23][24][25][26]). Caputo and Fabrizio [27,28] obtained new fractional derivatives and integrals without a singular kernel, which apply to time and spatial fractional derivatives. In [29], the authors defined the weighted Caputo-Fabrizio fractional derivative and studied related linear and nonlinear fractional differential equations.  [33,34] proved some new integral inequalities by using generalized fractional integral operators and some classical inequalities for integrable functions and their applications to the Zipf-Mandelbrot law. Motivated by the above work, the main objective of this article is to establish some new results for the Pólya-Szegö inequality and some other inequalities using the Caputo-Fabrizio fractional integrals. 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 Pólya-Szegö and Minkowskytype fractional inequalities by considering the Caputo-Fabrizio fractional operator. Finally, conclusion are given in Section 4.

Preliminaries
First, the definitions of the Caputo-Fabrizio fractional integrals are reviewed.
Definition 1 ([28,31,35]). Let α ∈ R such that 0 < α ≤ 1. The Caputo-Fabrizio fractional integral of order α of a function f is defined by For α = 1, it is reduced to This integral operator will be at the center of our main results.

Fractional Pólya-Szegö Inequality
In this section, we investigate some new fractional Pólya-Szegö inequalities by considering the Caputo-Fabrizio integral operator. Theorem 1. Let h 1 and h 2 be two integrable functions on [0, ∞). Assume that there exist four positive integrable functions P 1 , P 2 , R 1 and R 2 on [0, ∞) such that Then for x > 0 and α > 0, the following inequality holds: Proof. To prove (7), since η ∈ (0, x) and x > 0, we have Multiplying (8) and (9), we have Integrating (11) with respect to η from 0 to x, we obtain By considering inequality it follows that which gives the required inequality (7).
Hereafter, we present some special cases of the above theorem.

Proposition 1.
Let h 1 and h 2 be two integrable functions on [0, ∞) such that Then for x > 0 and α > 0, the following inequality holds:
Then for x > 0 and α, β > 0, we have Now, we establish the Minkowsky-type inequality using the Caputo-Fabrizio integral operator.

Theorem 4. Let h 1 and h 2 be two integrable functions on
Then for all α > 0, we have Proof. Since, Taking the c 1 th power of both sides and multiplying the resulting inequality by integrating (24) with respect to η from 0 to x, we get On the other hand, Using the Young inequality, we obtain Multiplying (28) by 1 α e −( 1−α α )(x−η) , then integrating the resulting inequality with respect to η from 0 to x, we get and from the equations (26), (27) and (29), we obtain . (30) Now, using the inequality (x + y) m ≤ 2 m−1 (x m + y m ), m > 1, x, y ≥ 0, we have and Inserting (31), (32) in (30) we get the required inequality (22). This completes the proof.

Conclusions
Nchama et al. [35] investigated some integral inequalities by considering the Caputo-Fabrizio fractional integral operator. In [28], Caputo and Fabrizio introduced a new fractional differential and integral operator. In the above work, we have applied the Caputo-Fabrizo fractional integral operator to establish some Pólya-Szegö and Minkowsky-type fractional integral inequalities. With the help of this study, we have established more general inequalities than in the classical cases due to the nonsingularity of the kernel. We believe that the Caputo-Fabrizio fractional integral is a formalism due to its nonsingularity of the kernel, which may provide an alternative way to solve many problems. The obtained fractional integral inequalities are very general and can be specialized to discover numerous interesting fractional integral inequalities. The inequalities investigated in this paper bring some contributions to the fields of fractional calculus and Caputo-Fabrizio fractional integral operator. These inequalities should lead to some applications for determining bounds and uniqueness of solutions in fractional differential equations.
Author Contributions: A.B.N., V.L.C., S.K.P. and C.C. equally contribute to the manuscript. All authors have read and agreed to the published version of the manuscript.