Tempered Fractional Integral Inequalities for Convex Functions

Certain new inequalities for convex functions by utilizing the tempered fractional integral are established in this paper. We also established some new results by employing the connections between the tempered fractional integral with the (R-L) fractional integral. Several special cases of the main result are also presented. The obtained results are more in a general form as it reduced certain existing results of Dahmani (2012) and Liu et al. (2009) by employing some particular values of the parameters.


Introduction
The domain of fractional calculus (FC) as engaged in derivatives and integrals of non-integer order. This area has a long history. The basis of it can be traced back to the letter between L'Hôpital and Leibniz in 1695 (See [1]). In the last three centuries, several mathematicians and physicists have devoted to the developments of the theories of fractional calculus [2][3][4][5][6][7][8][9][10][11][12][13]. Furthermore, fractional and fractal calculus applications are found in various fields [14][15][16][17][18]. In practical applications, certain various types of fractional operators such as Riemann-Liouville, Caputo, Riesz [11,12] and Hilfer [19] fractional operators are introduced. Freshly, the researchers have studied certain new fractional integral and derivative operators and their possible applications in various disciplines of sciences.
Khalil et al. [20] have introduced the notion of fractional conformable derivative (FCD) operators with some shortcomings. Abdeljawad [21] investigated the properties of the fractional conformable derivative operators. In [22], Jarad et al. introduced the fractional conformable integral and derivative operators. Anderson and Unless [23] developed the idea of conformable derivative by employing local proportional derivatives. Abdeljawad and Baleanu [24] investigated certain monotonicity results for fractional difference operators with discrete exponential kernels. Abdeljawad and Baleanu [25] have established fractional derivative operators with exponential kernel and their discrete versions. In [26], Atangana and Baleanu defined a new fractional derivative operator with the non-local and non-singular kernel. Caputo and Fabrizio [27] defined fractional derivative without a singular kernel. Certain properties of fractional derivative without a singular kernel can be found in the work of Losada and Nieto [28]. In [29], Jarad et al. defined generalized fractional derivatives generated by a class of local proportional derivatives.
On the other hand, fractional integral inequalities and its applications have also an essential role in the theory of differential equations and applied mathematics. A large number of several interesting integral inequalities are established by the researchers such as weighted Grüss type inequalities [30], Inequalities via R-L integrals [31], inequalities for extended gamma and confluent hypergeometric k-function [32], Gronwall inequalities involving k-fractional integral [33], inequalities involving generalized R-L integrals [34], the generalized R-L integrals with applications [35] and Grüss-type inequalities involving the generalized R-L integrals [36].
In [37], the following inequalities are presented where θ > 0 and v on [0, 1], which is the positive continuous function, such that In [38], the following inequalities are presented where µ > 0, ν > 0 and the positive continuous The following theorems are presented by Liu et al. [39]: Theorem 1. Let the two positive functions u and v be continuous functions on [a, b] such that u(θ) ≤ v(θ) for all θ ∈ [a, b]. Assume that the function u v is decreasing and the function u is increasing. Suppose that Ψ is a convex function with Ψ(0) = 0. Then the following inequality hold The applications of inequalities (1)-(3) can be found in the work of the various researchers. We refer the readers to [40][41][42][43][44].
Alzabut et al. [45] recently studied the Gronwall inequalities by considering generalized proportional fractional derivative operator. Rahman et al. [46] presented the Minkowski inequalities by employing proportional fractional integral. Dahmani [47] presented some classes of fractional integral inequalities by considering a family of n positive functions. Certainly, remarkable inequalities such as Hermite-Hadamard type [48], Chebyshev type [49][50][51], inequalities via generalized conformable integrals [52], Grüss type [53,54], fractional proportional inequalities and inequalities for convex functions [55], Hadamard proportional fractional integrals [56], bounds of proportional integrals with applications [57], inequalities for the weighted and the extended Chebyshev functionals [58], certain new inequalities for a class of n(n ∈ N) positive continuous and decreasing functions [59] and certain generalized fractional inequalities [60] are recently presented by utilizing several different kinds of fractional calculus approaches.

Main Results
Inequalities for convex functions by utilizing tempered fractional integral presented in this section.

Theorem 3. Let the two positive functions u and v be continuous on [a, b] and u(θ)
If the function u v is decreasing and the function u is increasing on [a, b]. Then for any convex function Ψ with Ψ(0) = 0. Then the following inequality holds for the tempered integral (6) where η ∈ C and (η) > 0.

Particular Cases
In [62], Li et al. gave the following connection of tempered fractional integral with the Riemann-Liouville fractional integral by a I η,ξ u(θ) = e −ξθ a I η e ξθ u(θ) .
By employing this connection (31) to Theorems 3 and 5, we get the following new results in term of Riemann-Liouville fractional integrals.
Similarly, we can get particular cases of Theorems 4 and 6.
The following Theorems are the particular results of Theorems 3 and 4 which can be obtained by setting η = 1 and θ = b in Theorems 7 and 8 respectively.

Conclusions
In this paper, we established certain inequalities for tempered fractional integrals via convex functions. We also established certain new particular results by employing the connections of tempered fractional integral with the Riemann-Liouville integral. The obtained results will reduce to the results given by Dahmani [65] by taking the parameter ξ = 0. Furthermore, by taking η = 1 and ξ = 0 the obtained inequalities will reduce to the results of Liu et al. ( [39], Theorem 9 and 10).