Certain Grüss-type inequalities via tempered fractional integrals concerning another function

We study a generalized left sided tempered fractional (GTF)-integral concerning another function Ψ in the kernel. Then we investigate several kinds of inequalities such as Grüss-type and certain other related inequalities by utilizing the GTF-integral. Additionally, we present various special cases of the main result. By utilizing the connection between GTF-integral and Riemann–Liouville integral concerning another function Ψ in the kernel, certain distinct particular cases of the main result are also presented. Furthermore, certain other inequalities can be formed by applying various kinds of conditions on the function Ψ.

the idea of a conformable derivative by employing local proportional derivatives. In [9], Abdeljawad and Baleanu gave certain monotonicity results for fractional difference operators with discrete exponential kernels. In [10], Abdeljawad and Baleanu have defined a fractional derivative operator with exponential kernel and the discrete version. In [11], Atangana and Baleanu defined a new fractional derivative operator with the non-local and non-singular kernel. A fractional derivative without a singular kernel can be found in the work of Caputo and Fabrizio [12]. Certain properties of fractional derivatives without a singular kernel can be found in the work of Losada and Nieto [13]. In [14][15][16], the authors studied stability analysis and a numerical scheme for fractional Klein-Gordon equations, existence results in Banach space for a nonlinear impulsive system and results for mild solutions of fractional coupled hybrid boundary value problems.
A variety of such types of distinguished operators led researchers to establish new ideas and fractional integral inequalities by utilizing these new operators. In [17,18], Hasib et al. established various inequalities by using AB-fractional and Saigo fractional integral operators. Recently Alzabut et al. and Rahman et al. [19,20] studied generalized proportional derivatives and integral operators and established a certain Gronwall inequality and the Minkowski inequalities involving the said operators. Rahman et al. [21,22] presented fractional integral inequalities for a family of positive continuous and decreasing functions and inequalities for convex functions by employing proportional Hadamard fractional integrals. Recently, researchers presented several various remarkable inequalities with properties and applications for the fractional conformable integrals and proportional integrals. The interested reader may consult [23][24][25][26][27][28][29][30][31][32][33].

Preliminaries
In this section, we consider some well-known definitions and mathematical preliminaries.
where the constants B, A, C, D ∈ R and 1 4 is the sharp value of inequality (2.1).
If we consider r = 0, then (2.2) gives When p = ∞, then Note that the space The tempered fractional integral was first studied by Buschman [38], but Li et al. [39] and Meerschaert et al. [40] have described the associated tempered fractional calculus more explicitly. Fernandez and Ustaoǧlu [41] investigated several analytic properties of tempered fractional integrals. Definition 2.4 ([39, 40]) Suppose that [a, b] is a real interval and κ, ξ ∈ C with (κ) > 0 and (ξ ) ≥ 0, then the left sided tempered fractional integral is defined by Remark 2.1 Setting ξ = 0 in (2.3) yields the following Riemann-Liouville fractional integral: The tempered fractional integral (2.3) satisfies the following semigroup property: In [42], Fahad et al. defined the following general form of the generalized tempered fractional integral concerning another function. where ξ > 0, κ ∈ C with (κ) > 0 and Γ (·) is the well-known gamma function.
In this manuscript, we will consider the following one sided GTF-integral.
One can easily derive the following results.
will satisfy the following linearity property: The main goal of this manuscript is to establish certain inequalities such as Grüss-type and several other inequalities by utilizing the GTF-integral (2.1). Also, certain special and particular cases of the main result are presented.

Certain other inequalities via GTF-integral concerning another function
Certain other types of inequalities which involving generalized tempered fractional (GTF) integral (2.7) are presented in this section. and

Special cases
This section is devoted to certain special cases of the main result obtained in Sects. 3 and 4.
(I) Applying Theorem 3.1 for Ψ ( ) = , we attain the following result for a one sided tempered fractional integral.

Corollary 5.7
Suppose that the functions U and V are two positive functions defined on [0, ∞[ and let p 1 , q 1 > 1 be such that 1 p 1 + 1 q 1 = 1. Then, for > 0, the following inequalities hold: (II) Applying Theorem 4.1 for ξ = 1, we attain the following new result for a one sided generalized Riemann-Liouville fractional integral. If p 1 , q 1 > 1 are such that 1 p 1 + 1 q 1 = 1, then, for > 0, the following inequalities hold: In a similar way, we can obtain the special cases of Theorems 4.2 and 4.3 by applying similar procedures.

Particular cases
Here, we present certain new particular cases of our main result by employing the connection of GTF-integral (2.7) with the classical Riemann-Liouville expression containing another function in the kernel.
Li et al. [39] defined the connection of a tempered fractional integral (2.3) with the Riemann-Liouville fractional integral by Here, we propose the following connection of the GTF-integral (2.7) with the generalized Riemann-Liouville fractional integral as is the generalized Riemann-Liouville fractional integral concerning another function.
Applying the above connection (6.1) to Theorem 3.1, one can get the following new result in terms of the generalized Riemann-Liouville fractional integral in the sense of another function.
Then, for > 0 and κ, λ > 0, the following four inequalities hold: One can obtain the following new result of Theorem 4.1 in terms of the generalized Riemann-Liouville fractional integral in the sense of another function by utilizing (6.1).

Theorem 6.3
Suppose that the two positive functions U and V are defined on [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. If p 1 , q 1 > 1 is such that 1 p 1 + 1 q 1 = 1, then, for > 0, the following inequalities hold: Similarly, we can obtain particular new results of Theorems 4.2 and 4.3 in terms of the generalized Riemann-Liouville fractional integral in the sense of another function by utilizing (6.1). Also, one can easily obtain certain new results of Theorems presented in Sect. 6 by utilizing the special cases discussed in Remark 2.2.

Conclusion
In this paper, we presented various types of inequalities such as Grüss-type inequalities and certain other inequalities by employing a generalized tempered fractional (GTF)integral in the sense of another function Ψ . Furthermore, we have discussed several special cases by using Remark 2.2. Also, we proposed a connection between the GTF-integral with the classical Riemann-Liouville fractional integral and derived certain new results in terms of the Riemann-Liouville fractional integral concerning another function. One can easily obtain several other types of inequalities, such as Hadamard fractional integral inequalities and generalized fractional conformable inequalities by utilizing Remark 2.2. Moreover, certain new inequalities can be derived by utilizing the inequalities discussed in Sect. 6.