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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{G}$\end{document} 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)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\omega (\theta )$\end{document} and G(θ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{G}(\theta )$\end{document} 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)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\omega (\theta )$\end{document} and G(θ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{G}(\theta )$\end{document}.


Introduction
The Chebyshev functional [1] for two integrable functions U and V on [v 1 , v 2 ] is defined by (1.1) The weighted Chebyshev functional (WCF in short) [1] for two integrable functions U and V on [v 1 , v 2 ] is defined by (1.2) where the function is positive and integrable on [v 1 , v 2 ]. Applications of (1.2) can be found in the field of probability and statistical problems. Also, applications of functional (1.2) can be found in the field of differential and integral equations. The reader may consult [2][3][4]. Dragomir [5] defined the following inequality for two differentiable functions U and V: where U , V ∈ L ∞ (v 1 , v 2 ) and the function is positive and integrable on [v 1 , v 2 ]. The researchers have studied the functionals (1.1) and (1.2) and established certain remarkable inequalities by employing different techniques. We refer the reader to [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. The existence and uniqueness of a miscible flow equation through porous media with a nonsingular fractional derivative, unified integral inequalities comprising pathway operators, certain results comprising the weighted Chebyshev functional using pathway fractional integrals and integral inequalities associated with Gauss hypergeometric function fractional integral operator can be found in [22][23][24][25].
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.

Preliminaries
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, for a function U(θ) are, respectively, given by and where is denoted by the well-known gamma function [42].

Definition 2.4
The one-sided R-L fractional integral I κ 0 of order κ > 0, for a function U(θ) is given by 43,44]) Let the function U be an integrable in X p (0, ∞) and suppose the function G is positive, increasing and monotone on [0, ∞) and having continuous derivative on [0, ∞) such that G(0) = 0. Then the generalized R-L left-and right-sided fractional integrals of a function U concerning another function G are, respectively, defined by and where κ ∈ C with (κ) > 0.
Remark 2.2 The following new weighted fractional integrals can be easily obtained: i. setting G(θ) = θ in Definition 2.6, we get the following weighted R-L fractional integral: ii. setting G(θ) = ln θ in Definition 2.6, then we get the following weighted Hadamard fractional integral operator: iii. setting G(θ) = θ η η , η > 0 in Definition 2.6, then we obtain the following weighted 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, ∞) and suppose the function G is positive, increasing and monotone on [0, ∞) and having continuous derivative on [0, ∞) such that G(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 1 Let the two function
and then integrating with respect to ϑ over (0, θ ) and applying (2.7), we have and then integrating with respect to ζ over (0, θ ), we have Also, on the other hand, we have Thus, we can write From (3.7), we estimate the following inequality: Hence, from (3.4) and (3.8), we get the desired proof.

Corollary 2 Let the two functions
Remark 3.1 If we put κ = μ in Theorem 2, then we obtain Theorem 1.
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 = 1 in Theorem 4, then we led to the following new result in terms of generalized fractional integral concerning another function G in the kernel. V (θ) ≤ M. Then, for all θ > 0, κ, μ > 0, the following fractional integral inequality holds: Then, for all θ > 0, κ > 0, the following weighted fractional integral inequality holds: (3.14) Proof By considering κ = μ in Theorem 3, we get the desired corollary.
Applying Theorem 4 for ω = 1, we obtain the following new result in terms of generalized fractional integral with respect to another function G in the kernel.
Applying Theorem 5 for ω = 1, we obtain the following new result in terms of generalized fractional integral with respect to another function G in the kernel.
By considering ω(θ ) = 1 in Theorem 6, 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.