The Minkowski's inequality by means of a generalized fractional integral

We use the definition of a fractional integral, recently proposed by Katugampola, to establish a generalization of the reverse Minkowski's inequality. We show two new theorems associated with this inequality, as well as state and show other inequalities related to this fractional operator.


Introduction
Studies involving integral inequalities are important in several areas of science: mathematics, physics, engineering, among others, in particular we mention: initial value problem, linear transformation stability, integral-differential equations, and impulse equations [1,2].
The space of p-integrable functions L p (a, b) play a relevant role in the study of inequalities involving integrals and sums. Further, it is possible to extend this space of p-integrable functions, to the space of the measurable Lebesgue functions, denoted by X p c (a, b), in which the space L p (a, b) is contained [3]. Thus, new results involving integral inequalities have been possible and consequently, some applications have been made [1,2]. We mention few of them, the inequalities of: Minkowski, Hlder, Hardy, Hermite-Hadamard, Jensen, among others [4,5,6,7,8,9,10].
Recently, Katugampola [23] proposed a fractional integral unifying other well known ones: Riemann-Liouville, Hadamard, Weyl, Liouville and Erdlyi-Kober. Motivated by this formulation, we present a generalization of the reverse Minkowski's inequality [24,25,26], using the fractional integral introduced by Katugampola. We point out that studies in this direction, involving fractional integrals, are growing in several branches of mathematics [18,27,28].
The work is organized as follows: In section 2, we present the definition of the fractional integral, as well as its particular cases. We present the main theorems involving the reverse Minkowski's inequality, as well as the suitable spaces for such definitions. In section 3, our main result, we propose the reverse Minkowski's inequality using the fractional integral. In section 4, we discuss other inequalities involving this fractional integral. Concluding remarks close the article.

Prelimiaries
In this section, we present the reverse Minkowski's inequality theorem associated with the classical Riemann integral and its respective generalization via Riemann-Liouville and Hadamard fractional integrals. In addition, we present the fractional integral introduced by Katugampola, and we conclude with a theorem in order to recover particular cases.
Erhan et al. [5] address the inequalities of Hermite-Hadamard and reverse Minkowski for two functions f and g by means of the classical Riemann integral. On the other hand, Lazhar [7] also proposed a work related to the inequality involving integrals, that is, Hardy's inequality and the reverse Minkowski's inequality. Two theorems below were motivation for the works performed so far, via the Riemann-Liouville and Hadamard integrals, involving the reverse Minkowski's inequality. and In particular, when c = 1/p the space X p c (a, b) coincides with the space L p (a, b) [3].
We present the definitions of the fractional integrals that will be useful in the development of the article: Riemann-Liouville fractional integral, Hadamard integral, Erdlyi-Kober integral, Katugampola integral, Weyl integral and Liouville integral.
be a finite interval on the real-axis R. The Riemann-Liouville fractional integrals (left-sided and right-sided) of order α ∈ C, Re(α) > 0, are defined by respectively [3,12]. Definition 3. Let (a, b) (0 ≤ a < b < ∞) be a finite or infinite interval on the half-axis R + . The Hadamard fractional integrals (left-sided and right-sided) of order α ∈ C, Re(α) > 0 of a real function f ∈ L p (a, b) are defined by respectively [3,12].
be a finite or infinite interval or halfaxis R + . Also let Re(α) > 0, σ > 0 and η ∈ C. The Erdlyi-Kober fractional integrals (left-sided and right-sided) of order α ∈ C of a real function f ∈ L p (a, b) are defined by respectively [3,12].
Zoubir [25] established the reverse Minkowski's inequality and another result that refers to the inequality via Riemann-Liouville fractional integral according to the following two theorems. .
In 2014, Chinchane et al. [26] and Sabrina et al. [30] also established the reverse Minkowski's inequality via Hadamard fractional integral as in two theorems below.
In 2014 Chinchane et al. [31] and recently Chinchane [32], established the reverse Minkowski's inequality via fractional integral of Saigo and the k-fractional integral, respectively.
In 2017, Katugampola [23] introduced a fractional integral that unifies the six fractional integrals above mentioned. Finally, we introduce this integral and with a theorem we study their respective particular cases.

Reverse Minkowski fractional integral inequality
In this section, our main contribution, we establish and prove the reverse Minkowski's inequality via generalized fractional integral Eq.(2.19) and a theorem that refers to the reverse Minkowski's inequality.
for m, M ∈ R * + and ∀t ∈ [a, x], then Proof. Using the condition which implies,

Other fractional integral inequalities
In this section we generalize the results discussed by Chinchane [32], Sulaiman [33] and Sroysang [34] on the reverse Minkowski's inequality via Riemann integral, using the fractional integral proposed by Katugampola [23].
Multiplying by f 1 p (t) both sides of Eq.(4.2), we can rewrite it as follows
Using Eq.(2.19) and Theorem 7 with the convenient conditions for each respective fractional integral, we have the previous theorems, that is, Theorem 10 to Theorem 15 introduced and demonstrated above, contain as particular cases, each result involving the following fractional integrals: Riemann-Liouville, Hadamard, Liouville, Weyl, Edrlyi-Kober, and Katugampola.

Concluding remarks
After a brief introduction to the fractional integral, proposed by Katugampola and fractional integrals in the sense of Riemann-Liouville and Hadamard, we generalize the reverse Minkowski's inequality obtaining, as a particular case, the inequality involving the fractional integral in the Riemann-Liouville sense and Hadamard sense [23]. We also show other inequalities using the Katugampola fractional integral. The application of this fractional integral can be used to generalize several inequalities, among them, we mention the Gruss-type inequality, recently introduced and proved [36]. A continuation of this work, with this formulation of fractional integral, consists in generalize the inequalities of Hermite-Hadamard and Hermite-Hadamard-Fjer. Moreover, we will discuss inequalities via M-fractional integral according to [37].