Chebyshev Type Inequalities for the Riemann-Liouville Variable-Order Fractional Integral Operator

This paper presents Chebyshev Type inequalities for the Riemann-Liouville variable-order fractional integral operator using two synchronous functions on the set of real numbers. It is the first result of its kind in the current literature using variable-order Riemann-Liouville fractional integral operator. Some special cases for the result obtained in the paper are discussed.


Introduction
Fractional integral inequalities are very important in establishing the uniqueness of solutions for certain fractional differential equations and integral equations. They also provide upper and lower bounds for the solutions of fractional boundary value problems. Generally speaking, without inequalities, the advance of differential and integral equations would not be at its present stage. For more works on fractional integral (or differential) inequalities, one may refere the book [1](and the references therein) and the papers [2], [3], [4], [5], [6](and the references therein). In 2009, using the Riemann-Liouville constant-order fractional integral operator, Belarbi and Dahmani [2] established some new integral inequalities for the Chebyshev functional [7] in the case of two synchronous functions. In 2010, Dahmani [10] used the Riemann-Liouville fractional integral to present recent results on fractional integral inequalities. By considering the extended Chebyshev functional in the case of synchronous functions, he established two main results. The first one deals with some inequalities using one fractional parameter. The second result concerns others inequalities using two fractional parameters. In 2011, Dahmani, Mechouar and Braham [3]using the Riemann-Liouville fractional integral operator they established some integral results related to Chebyshev's functional in the case of differentiable functions whose derivatives belong to the space L p ([0, ∞[).
The following three Theorems are established by Belarbi and Dahmani(see [2]). In our main result they are special cases.

Preliminaries
Throughout this paper, we use the following definitions.
Definition 1. Two functions f and g are said to be synchronous on Definition 2. Given (z) > 0, we define the gamma function, Γ(z), as In the following definition of Riemann-Liouville variable-order fractional integral we used the notation RL stands for Riemann-Liouville.

Main result
In this section, we introduce one inequality for two synchronous functions on R. This inequality is for Riemann-Liouville two variable-order of fractional integral operator which is defined as (4). From this inequality, four important inequality produced as a corollary by assuming monotonocity and differentiability of the two synchronous functions.
Theorem 4. Let f and g be two synchronous functions on R. Then for all a, c ∈ R, t > a, s > c, and α, β : Proof. Since f and g are synchronous on R, then for all x, y ∈ R, we've Now multiply inequality (6) by (t−x) α(t,x)−1 /Γ(α(t, x)) and integrate from a to t with respect to x, that is, which means Now multiply inequality (8) by (s − y) β(s,y)−1 /Γ(β(s, y)) and integrate from c to s with respect to y, that is, which means     Proof. Define h(t) = g(t) − m 1 t, h(s) = g(s) − m 2 s. As we can see h is differentiable and it is increasing on R. It is also easy to verify that f and h are synchronous on R. From inequality (5) replace g(t) and g(s) by h(t) = g(t) − m 1 t and h(s) = g(s) − m 2 s respectively. Then from Theorem (4), we've