On the weighted fractional Pólya–Szegö and Chebyshev-types integral inequalities concerning another function

The primary objective of this present paper is to establish certain new weighted fractional Pólya–Szegö and Chebyshev type integral inequalities by employing the generalized weighted fractional integral involving another function Ψ in the kernel. The inequalities presented in this paper cover some new inequalities involving all other type weighted fractional integrals by applying certain conditions on 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 Ψ(θ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Psi (\theta )$\end{document}. Also, the Pólya–Szegö and Chebyshev type integral inequalities for all other type fractional integrals, such as the Katugampola fractional integrals, generalized Riemann–Liouville fractional integral, conformable fractional integral, and Hadamard fractional integral, are the special cases of our main results 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 Ψ(θ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Psi (\theta )$\end{document}. Additionally, examples of constructing bounded functions are also presented in the paper.


Introduction
The field of integral inequalities plays an essential role in the diverse domain. The mathematicians have investigated that it is mainly a powerful tool for the improvement of both applied and pure mathematics. In [8], the authors established Grüss type integral inequalities by employing the classical fractional integrals. Certain new integral inequalities for the Riemann-Liouville (R-L) fractional integrals can be found in the work of Dahmani [6]. The inequalities involving an extension of the gamma function and confluent k-hypergeometric function were found in the work of Nisar et al. [26]. Nisar et al. [27] performed Gronwall inequalities with applications. Rahman et al. [42] gave certain inequalities for (k, ρ)-fractional integrals. Ostrowski type inequalities connecting local fractional integrals were found in [50]. Sarikaya et al. [51] developed generalized (k, s)-fractional integrals with applications. In [52], Set et al. introduced Grüss type inequalities by employing generalized k-fractional integrals. Recently, Nisar et al. [29] gave some new generalized fractional integral inequalities.
Very recently, the fractional conformable and proportional fractional integral operators have been given in [13,15]. Later on, Huang et al. [12] gave Hermite-Hadamard type inequalities by using fractional conformable integrals (FCI). Qi et al. [33] investigated Čebyšev type inequalities involving FCI. The Chebyshev type inequalities and certain Minkowski type inequalities are found in [25,30,43]. Nisar et al. [28] investigated some new inequalities for a class of n (n ∈ N) positive, continuous, and decreasing functions by employing FCI. Rahman et al. [41] introduced Grüss type inequalities for k-fractional conformable integrals. Some significant inequalities are given in [35-37, 39, 40, 46]. Very recently, Rahman et al. [38,44] presented fractional integral inequalities involving tempered fractional integrals. In [2], Abdeljawad et al. presented some new local fractional inequalities associated with generalized (s, m)-convex functions and applications. Qi et al. [34] proposed fractional integral versions of Hermite-Hadamard type inequality for generalized exponential convexity. In [3], Abdeljawad et al. presented new fractional integral inequalities for p-convexity within interval-valued functions. Zhou et al. [55] investigated some new inequalities by considering the generalized proportional Hadamard fractional integral operators. Rashid et al. [48] proposed some inequalities via generalized proportional fractional integrals. In [47], the authors presented reverse Minkowski's inequalities via generalized proportional fractional integrals. In [21], Mohammed and Abdeljawad proposed some modifications of fractional integral inequalities for convex functions. Abdeljawad et al. [1] presented modified conformable fractional integral inequalities of Hermite-Hadamard type with applications. Mohammed and Brevik [23] investigated a new version of Hermite-Hadamard for Riemann-Liouville fractional integrals. Mohammed and Abdeljawad [22] studied integral inequalities for generalized fractional integral with nonsingular kernels. Mohammed and Srikaya [24] proposed generalized fractional integral inequalities for twice differentiable functions.
In [31], Ntouyas et al. investigated some new Pólya-Szegö and Chebyshev type inequalities by considering the R-L fractional integrals.
This paper is composed as follows: In Sect. 2, we mention some basic definitions. Certain new Pólya-Szegö type inequalities for the weighted fractional integrals concerning another function are presented in Sect. 3. In Sect. 4, we present some new generalized Chebyshev type inequalities for the weighted fractional integrals concerning another function. In Sect. 5, certain new particular cases in terms of weighted fractional integrals are discussed. An example of constructing bounding functions is considered in Sect. 6. The concluding remarks are presented in Sect. 7.
Applying r = 0 on (2.2) gives When p → ∞, then Clearly, the space and in a similar way with the space

Definition 2.4 ([20, 49])
The one-sided R-L fractional integral of order κ > 0 is defined by 20,49]) Let the function 1 : [x 1 , x 2 ] → R be an integrable function, and assume that the function is increasing and positive monotone on (x 1 , x 2 ] and having continuous derivative on (x 1 , x 2 ). Then the left-and right-sided generalized Riemann-Liouville fractional integrals of a function 1 concerning another function are respectively defined by and where κ ∈ C with (κ) > 0.
Remark 2.1 The following new weighted fractional integrals can be easily obtained: i. Applying Definition 2.8 for (θ ) = θ , we get the following weighted R-L fractional integral: ii. Applying Definition 2.8 for (θ ) = θ , we get the following weighted Hadamard fractional integral operator: iii. Applying Definition 2.8 for (θ ) = θ η η , η > 0, we obtain the following weighted Katugampola fractional integral: Similarly, we can obtain another type of weighted fractional integrals.
In this paper, we analyze the subsequent one-sided generalized weighted fractional integral. Then the one-sided generalized weighted fractional integral of the function 1 with respect to another function in the kernel is given by Definition 2.8 For 0 = τ 0 < τ 1 < · · · < τ p < τ p+1 = τ , we define the following sub-integrals for generalized weighted integral: Remark 2.3 If we set (τ ) = τ and ω(θ ) = 1, then (2.10) will reduce to the sub-integrals of R-L fractional integral defined by [31].

Some weighted fractional Pólya-Szegö type integral inequalities
In this section, we present some new weighted fractional Pólya-Szegö type integral inequalities for positive and integrable functions by utilizing generalized weighted fractional integral (2.9) containing other function in the kernel.
Then, for κ > 0 and θ > 0, the following weighted fractional integral inequality holds: Proof Utilizing the given hypothesis, we have Similarly, we have The product of (3.3) and (3.4) yields From (3.5), it follows that and integrating the resultant identity with respect to ϑ over (0, θ ), we have Multiplying both sides of the above equation by ω -1 (θ ) and using Definition (2.9), we obtain Now, using the AM-GM inequality, i.e., p 1 + p 2 ≥ 2 √ p 1 p 2 , p 1 , p 2 R + , we get It follows that which gives the required result (3.2).

Lemma 3.3 Suppose that all the conditions of Lemma 3.1 hold, and assume that the function is increasing, positive, and monotone on [0, ∞[ and having continuous derivative on
[0, ∞[ with (0) = 0. Then, for κ, λ > 0 and θ > 0, the following weighted fractional integral inequality holds: Proof From the hypothesis given by (3.1), we have which in view of (2.9) yields (3.14) Similarly, one can obtain Hence, the product of (3.14) and (3.15) gives the desired assertion (3.13). Then, for κ, λ > 0 and θ > 0, the following weighted fractional integral inequality holds:

Theorem 4.2
Suppose that all the conditions of Theorem 4.1 are satisfied. Then, for κ > 0 and θ > 0, the following weighted fractional integral inequality holds: Proof Applying Theorem 4.1 for κ = λ, we get the desired assertion (4.9) of Theorem 4.2.
Similarly, one can derive the special case of Lemma 3.3. The following theorem represents the special case of Theorem 4.1 in terms of weighted classical fractional integral.  1). Then, for κ, λ > 0 and θ > 0, the following weighted fractional integral inequality holds: and Proof By employing Theorem 4.1 for (θ ) = θ , we get the desired Theorem 5.1.
By applying different choices given in Remark 2.1, some new inequalities can be obtained easily. Also, we can derive the particular cases of the main result by employing Remark 2.2.

Applications
Here, we define a way for constructing four bounded functions and then utilize them to present certain estimates of Chebyshev type weighted fractional integral inequalities of two unknown functions.

Concluding remarks
In this present investigation, we presented some new weighted fractional Pólya-Szegö and Chebyshev type integral inequalities by employing weighted fractional integral recently proposed by Jarad et al. [14]. It is worth mentioning that these inequalities cover the integral inequalities for the well-known fractional integral operators discussed in Remark 2.2.
In particular, if we take (θ ) = θ and ω(θ ) = 1, then the obtained inequalities reduce to the inequalities involving the R-L fractional integral established by Ntouyas et al. [31]. One can easily obtain Pólya-Szegö and Chebyshev type Hadamard fractional integral inequalities by applying (θ ) = ln θ and ω(θ ) = θ u . Also, one can easily derive the said Pólya-Szegö and Chebyshev type inequalities for other types of weighted fractional integrals such as Katugampola, generalized R-L, classical R-L, generalized conformable, and conformable fractional integrals by applying certain conditions on the function given in Remark 2.1.