SomeNewTempered Fractional Pólya-Szegö andChebyshev-Type Inequalities with Respect to Another Function

Department of Mathematics, Shaheed Benazir Bhutto University, Sheringal 18000, Upper Dir, Pakistan Department of Mathematics, College of Arts and Sciences, Prince Sattam Bin Abdulaziz University, Wadi Aldawaser 11991, Saudi Arabia Department of Mathematics and General Sciences, Prince Sultan University, Riyadh, KSA, Saudi Arabia Department of Medical Research, China Medical University, Taichung 40402, Taiwan Department of Computer Science and Information Engineering, Asia University, Taichung 40402, Taiwan Department of Mathematics, University of Sargodha, Sargodha, Pakistan


Introduction
e well-known Chebyshev functional [1] is defined by inequalities by utilizing functional (1).
In the last few decades, the researchers have considered that fractional integral inequalities are the most powerful tools for the development of both applied and pure mathematics. In [10], the authors presented some Grüss-type integral inequalities by considering fractional integrals. Some new integral inequalities in sense of Riemann-Liouville fractional integrals can be found in the work of Dahmani [11].
In [12], Sarikaya et al. gave the idea of generalized (k, s)-fractional integrals with applications. Set et al. [13] investigated some Grüss-type inequalities by considering generalized k-fractional integrals.
Very recently, the idea of fractional conformable and proportional fractional integral operators was proposed by Jarad et al. [14,15]. Later on, Huang et al. [16] presented generalized Hermite-Hadamard-type inequalities by considering generalized fractional conformable integrals. In [17], Qi et al. established Chebyshev-type inequalities for generalized fractional conformable integrals.
In [18], Ntouyas et al. investigated some new Pólya-Szegö-and Chebyshev-type inequalities by considering Riemann-Liouville fractional integrals. e tempered fractional integral was first studied by Buschman [19], but Li et al. [20] and Meerschaert et al. [21] have described the associated tempered fractional calculus more explicitly. Fernandez and Ustaoǧlu [22] investigated several analytic properties of the tempered fractional integral. In [23], Fahad et al. proposed the general form of the generalized tempered fractional integral concerning another function. In this paper, we investigate the said inequalities for the so-called tempered fractional integrals containing another function in the kernel. e structure of the paper as follows. In Section 2, some basic definitions are presented. Some new Pólya-Szegö-type for the so-called generalized tempered fractional integral in the sense of another function is presented in Section 3. In Section 4, we present some new generalized Chebyshev-type tempered fractional integral inequalities. In Section 5, certain new particular cases in terms of classical tempered fractional integrals are discussed. An example of constructing bounding functions is considered in Section 6. Finally, the concluding remarks are discussed in Section 7.

Preliminaries
In this section, we consider some well-known definitions and mathematical preliminaries.
Definition 1 (see [7]). Suppose that the functions , then the following inequality holds: where the constants B, A, C, D ∈ R and 1/4 is the sharp of inequality (6).

Definition 5.
e one-sided tempered fractional integral of order κ > 0, τ ≥ 0 is defined by Definition 6 (see [23]). Let the function f 1 be an integrable in the space X p Ψ (0, ∞) and assume that the function Ψ is positive, monotone, and increasing on [0, ∞[, and its derivative Ψ ′ is continuous on [0, ∞[ with Ψ(0) � 0. en, the left-sided generalized tempered fractional integral of the function f 1 concerning another function Ψ in the kernel is defined by where τ ≥ 0, κ, ∈ C with R(κ) > 0, and Γ(.) is the well-known gamma function.
In this article, we consider the following one-sided GTFintegral.

Definition 7.
Let the function f 1 be integrable in the space X p Ψ (0, ∞) and assume that the function Ψ is positive, monotone, and increasing on [0, ∞[, and its derivative Ψ ′ is continuous on [0, ∞[ with Ψ(0) � 0. en, the one-sided generalized tempered fractional integral of the function f 1 concerning another function Ψ in the kernel is defined by we define the following subintegrals for the generalized tempered integral Note that Remark 3. If we set Ψ(X) � X and τ � 0, then (18) will reduce to the subintegrals of Riemann-Liouville fractional integral defined by [18].

Pólya-Szegö-Type Tempered Fractional Integral Inequalities
In this section, we provide some new Pólya-Szegö-type tempered fractional integral inequalities for positive and integrable functions via tempered fractional integral (17) containing another function Ψ in the kernel.
en, for κ > 0, τ ≥ 0, and θ > 0, the following tempered fractional integral inequality holds: Proof. From the given hypothesis, we have Similarly, we have Taking product of (22) and (23), we get From (24), it can be written as Now, taking product of (25) with and integrating the resultant identity with respect to ϑ over (0, θ), we have With the aid of Definition 8, we can write By applying AM-GM inequality, i.e., It follows that which gives the desired assertion (21).

Particular Cases
e following new Pólya-Szegö-and Chebyshev-type inequalities for classical tempered fractional integral (14) can be easily established.

□
Similarly, we can derive particular result of eorem 2.

Applications
In this section, we define a way for constructing four bounded functions and then utilize them to present certain estimates of Chebyshev-type tempered fractional integral inequalities of two unknown functions.
Let the unit function Z(θ) be defined by and let the Heaviside unit step function Z a (θ) be defined by Z a (θ) � 1, θ > a, Suppose that the function U 1 is piecewise continuous function on [0, X] defined by