On the weighted fractional integral inequalities for Chebyshev functionals

The goal of this present paper is to study some new inequalities for a class of diﬀerentiable functions connected with Chebyshev’s functionals by utilizing a fractional generalized weighted fractional integral involving another function G in the kernel. Also, we present weighted fractional integral inequalities for the weighted and extended Chebyshev’s functionals. One can easily investigate some new inequalities involving all other type weighted fractional integrals associated with Chebyshev’s functionals with certain choices of ω ( θ ) and G ( θ ) as discussed in the literature. Furthermore, the obtained weighted fractional integral inequalities will cover the inequalities for all other type fractional integrals such as Katugampola fractional integrals, generalized Riemann–Liouville fractional integrals, conformable fractional integrals and Hadamard fractional integrals associated with Chebyshev’s functionals with certain choices of ω ( θ ) and G ( θ ). MSC: 26A33; 26D10; 26D15;

In Sect. 2, we present some well-known definitions. Section 3 is devoted to the weighted fractional integral inequalities associated with Chebyshev's functionals (1.1)and(1.2)concerning another function G in the kernel. In Sect. 4, we give some new weighted fractional integral inequalities associated with weighted and extended Chebyshev's functionals (1.3) and (1.4) with respect to another function G in the kernel. Finally, we discuss concluding remarks in Sect. 5.

2P r e l i m i n a r i e s
In this section, we present the preliminaries and definitions.

Definition 2.3
The Riemann-Liouville (R-L) fractional integrals (left-and right-sided) I κ v 1 and I κ v 2 of order κ >0,forafunctionU (θ )are,respectively,givenby iii. setting G(θ)= θ η η , η > 0 in Definition 2.6,t h e nw eo b t a i nt h ef o l l o w i n gw e i g h t e d Katugampola fractional integral: Similarly, one can obtain other type of weighted fractional integrals.
In this paper, we consider the following one-sided generalized weighted fractional integral.

Definition 2.7
Let the function U be an integrable in the space X p U (0, ∞)andsupposethe function G is positive, increasing and monotone on [0, ∞) and having continuous derivative on [0, ∞)suchthatG(0) = 0. Then the one-sided generalized weighted fractional integral of the function 1 concerning another function G in the kernel is defined by

Weighted fractional integral inequalities associated with Chebyshev's functional
In this section, we present weighted fractional integral inequalities for a class of differentiable functions connected with Chebyshev's functional (1.1). Then, for all θ >0,κ >0, the following weighted fractional integral inequality holds:

Theorem 3 Let the two function
assume that the function G is increasing, positive and monotone on [0, ∞[ and having con- Then, for all θ >0,κ, μ >0,the following weighted fractional integral inequality holds: Proof Suppose that the functions U and V satisfy the hypothesis of Theorem 3.Then,for every ϑ, ζ ∈ [0, θ ]; u = v, θ > 0, there exists a constant c between ϑ and ζ such that Thus, for every ϑ, ζ ∈ [0, θ ], we have It follows that Therefore, we get Hence from (3.13), we get the desired inequality.
Setting ω =1inTheorem4, then we led to the following new result in terms of generalized fractional integral concerning another function G in the kernel.
Applying Theorem 5 for ω =1,weobtainthefollowingnewresultintermsofgeneralized fractional integral with respect to another function G in the kernel.

Weighted fractional integral inequalities associated with the weighted and the extended Chebyshev functionals
The following results presented in this section related to extended, weighted Chebyshev functionals.
By considering ω(θ )=1inTheorem6, we get the following new result in terms of generalized fractional integral concerning another function G in the kernel.

Concluding remarks
We presented some new weighted fractional integral inequalities for a class of differentiable functions connected with Chebyshev's, weighted Chebshev's and extended Chebyshev's functionals by utilizing weighted fractional integral operator recently introduced by Jarad et al. [45]. These inequalities are more general than the existing classical inequalities given in the literature. The special cases of our result can be found in [5,15,16,[48][49][50]. Also, one can easily obtain new fractional integral inequalities associated with Chebyshev"s, weighted Chebshev's and extended Chebyshev's functionals for another type of weighted and classical fractional integrals such as Katugampola, generalized Riemann-Liouville, classical Riemann-Liouville, generalized conformable and conformable fractional integrals with certain conditions on ω and G given in Remarks 2.2 and 2.3.