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Gruss-type inequality by mean of a fractional integral

In this paper, using a fractional integral as proposed by Katugampola we establish a generalization of integral inequalities of Gruss-type. We prove two theorems associated with these inequalities and then immediately we enunciate and prove others inequalities associated with these fractional operator.


Introduction
In 1935, Gruss proved the following integral inequality [ The Gruss type inequality has some important applications. We mention: difference equations, integral arithmetic mean and h-integral arithmetic mean [2,3]. On the other hand, we also study the Gruss type inequality in spaces with intern product, consequently some applications for the Mellin transform of sequences and polynomials in Hilbert spaces [4].
In this sense, there are another important inequalities using integer order integrals among than we mention: Hölder's inequality, Jensen's inequality, Minkowski's inequality and reverse Minkowiski's inequality, [5,6,7,8,9,10,11]. For such inequalities, as well as for the study of functions, integrals and norms, the space of the p−integrable functions, L p (a, b), have a particular importance. However, in this work, we use the space of Lebesgue mensurable functions, which admits, as a particular case, the space L p (a, b).
The appearance of the fractional calculus allow several consequences, results and important theories in mathematics, physics, engineering, among others areas. Due to this fact, it was possible to define several fractional integrals, for example: Riemann-Liouville, Katugampola, Hadamard, Erdélyi-Kober, Liouville and Weyl types. There are other fractional integrals that can be found in [12]. Thus, with these fractional integrals some inequalities involving such formulations have been developed over the years, for example: Reverse Minkowiski, Hermite-Hadarmad, Ostrowski e Fejér inequalities, [8,13,14,15,16,17,18,19,20]. We also mention that, in the literature, there are generalizations for the Eq.(1.1) using, for example, fractional integrals of Riemann-Liouville, Hadamard and also the q−fractional integral, [21,22,23].
In this paper, using a fractional integral recently introduced [24], we propose a new generalization of Eq.(1.1), i.e., new Gruss-type inequalities that generalize inequalities obtained by means of Riemann-Liouville fractional integral [21].
The paper is organized as follow: In section 2, we present the Katugampola's fractional integral as well as the convenient space for such definition and the convenient parameters to recover the six fractional integrals as particular cases. In section 3, our main result, we present inequalities of Gruss-type. In section 4, we discuss other inequalities involving the Katugampola's fractional integral. Conclusions and future perspectives close the paper.

Preliminaries
In this section, we define the space X p c (a, b), where the Katumgampola's fractional integrals are defined. With a convenient choice of parameters, we recover six well-know fractional integrals, previously mentioned [12,24]. and In particular, when c = 1/p, the space X p c (a, b) coincides with the space L p (a, b). Definition 2. Let ϕ ∈ X p c (a, b), α > 0 and β, ρ, η, κ ∈ R. Then, the fractional integrals of a function ϕ, left-and right-sided, are defined by respectively, if the integrals exist.

Main results
We enunciate and prove the following lemma in order to use it in the first theorem which generalizes the inequalities of Gruss-type, Eq.(1.1). First of all, let α > 0, x > 0 and β, ρ, η, κ ∈ R. We define the following function to simplify the development and notation.
This result, was obtained in [21], considering the Riemann-Liouville fractional integral.
where ξ ∈ (0, x), x > 0, and integrating with respect to the variable ξ, from 0 to x, we obtain .
Using the Cauchy-Schwarz inequality associated with double integrals, we have which is Eq.(3.18).

Lemma 3. Let u be an integrable function on
Proof. Multiplying both sides of Eq.(3.4) by where ξ ∈ (0, x), x > 0, and integrating with respect to variable ξ, from 0 to x, we obtain
To prove the next theorem, we use Lemma 2 and Lemma 3, previously proved.

Other fractional integral inequalities
In this section, we present some integral inequalities involving the Katugampola's fractional operator. The results obtained were adapted from the paper Chinchane & Pachpatte [22], in which makes a brief approach with respect inequalities of Gruss-type, but in Hadamard sense.
A priori, it should be emphasize that, although the following results are adapted for the Katugampola's operator, the left-sided, Eq.(2.3), and that with convenient condition on the parameters of the fractional operator it is possible to obtain the Hadamard's operator. In this sense, the results presented here are, in fact, true for the Hadamard's operator when we admit a = 1 in Eq.(2.1).