Some Gruss-type Inequalities Using Generalized Katugampola Fractional Integral

The main objective of this paper is to obtain generalization of some Gruss-type inequalities in case of functional bounds by using a generalized Katugampola fractional integral.


INTRODUCTION:
In 1935, G. Grüss proved the renowned integral inequality [9] (see also [13]): In recent years, the inequalities are getting to play a very vital role in all mathematical fields, especially after the creation of fractional calculus which gave rise to several results and important theories in mathematics, engineering, physics, and other fields of science.
Dahmani et al. [4], in (2010), proved the following fractional version inequality 2 PRELIMINARIES: In this section, we give some definitions and properties available in literature that will be used in our paper, for more details (see [10], [11], [17]).
In particular, when c = 1/p, the space X p c (a, b) coincides with the space L p (a, b) .

Definition 2.2
The left-and right-sided fractional integrals of a function v where v ∈ X p c (a, b) , α > 0, and β, ρ, η, k ∈ R, are defined respectively by 2) if the integral exist.
To present and discuss our new results in this paper we use the left-sided fractional integrals, the right sided fractional can be proved similarly, also we consider a = 0, in (2.1), to obtain The above fractional integral has the following Composition (index) formulae For the convenience of establishing our results we define the following function as in [17]: let x > 0, α > 0, ρ, k, β, η ∈ R, then Proof. From the condition (3.1), for all τ ≥ 0, σ ≥ 0, we have Multiplying both sides of (3.3) by Integrating both sides of (3.5) with respect to σ over (0, x), we get
Now we give the lemma required for proving our next theorem Proof. For any τ, σ > 0, we have Multiplying both sides of (3.7) by where τ ∈ (0, x) and integrating over (0, x) with respect to the variable τ, we obtain

Now multiplying both sides of (3.8) by
where σ ∈ (0, x) and integrating over (0, x) with respect to the variable σ, we obtain . Which yields the required identity (3.6).
Our next result is on Gruss type inequality in case of functional bounds with same parameters Then for all x > 0 and α > 0, ρ > 0, β, η, k ∈ R, we have where T (ϕ, ψ, ω) as in [18], is defined by Proof. Define Multiplying both sides of (3.11) by where τ ∈ (0, x) and integrating over (0, x) with respect to the variable τ, we obtain Now multiplying both sides of (3.12) by where σ ∈ (0, x) and integrating the resulting identity over (0, x) with respect to the variable σ, we get Applying the Cauchy-Schwarz inequality to (3.13), we can write Combining the Inequalities (3.16), (3.17) with inequality (3.14), we obtain inequality (3.10).
Which is result given in [17].