Fractional Ostrowski Type Inequalities via Generalized Mittag–Leffler Function

If we study the theory of fractional differential equations then we notice the Mittag–Leffler function is very helpful in this theory. On the contrary, Ostrowski inequality is also very useful in numerical computations and error analysis of numerical quadrature rules. In this paper, Ostrowski inequalities with the help of generalized Mittag–Leffler function are established. In addition, bounds of fractional Hadamard inequalities are given as straightforward consequences of these inequalities.


Introduction
Exponential function plays a vital role in the theory of integer order differential equations. e symbol E α (z) is well known as the Mittag-Leffler function and it is a generalization of exponential function. It occurs in the solutions of fractional differential equations such as exponential function which exists in the solutions of differential equations. Due to its importance, Mittag-Leffler function is generalized by many mathematicians: For example, Wiman [1], Prabhakar [2], Shukla and Prajapati [3], Salim [4], Salim and Faraj [5], and Rahman et al. [6]. Mittag-Leffler function is also used in the formation of fractional integral operators. ese fractional integral operators provide generalizations of fractional differential equations and modeling of dynamic systems. Fractional integral operators also play a vital role in the advancement of classical mathematical inequalities. For example, Hadamard inequality, Ostrowski inequality, Gruss inequality, and many others have been presented for fractional integral and derivative operators, see [7][8][9][10][11][12][13][14][15][16]. e aim of this paper is to study well-known Ostrowski inequality for an integral operator which is directly associated with many fractional integral operators defined in near past.
Recently, Farid defined a unified integral operator in [17] (also see [18]). is unifies several kinds of fractional and conformable integrals in a compact formula and is given as follows.
en, for x ∈ [a, b] the left and right integral operators are defined by It can be noted that In the following, we state the Ostrowski inequality which is proved by Ostrowski [19] in 1938.
e Ostrowski inequality has been studied by many researchers to obtain its refinements, generalizations, and extensions. Also, their applications are analyzed for establishing the bounds of relations among special means and for estimations of numerical quadrature rules. For recent developments of Ostrowski inequality, we refer the reader to [8,9,11,[20][21][22][23][24][25][26] and references therein.
In Section 2, fractional version of Ostrowski inequalities with the help of Mittag-Leffler function has been established. e presented results may be useful in the study of fractional integral operators and their applications. Also, the error bounds of fractional Hadamard inequalities are presented in Section 3.

Main Results
First, we establish the following lemma for extended generalized Mittag-Leffler function.
Proof. We have After simple computation, one can obtain (11). Next, we give the generalized fractional Ostrowski type inequality containing extended generalized Mittag-Leffler function.  on (a, b), then for α, β ≥ 1, the following inequality for fractional integrals (6) and (7) holds: Proof. Let x ∈ [a, b], t ∈ [a, x], and α ≥ 1. en, the following inequality holds for the monotonically increasing function ψ 2 and the Mittag-Leffler function (1): From (14) and given condition of boundedness of ψ 1 ′ , one can have the following integral inequalities:

Mathematical Problems in Engineering
First, we consider inequality (15) as follows: erefore, (17) takes the following form after integrating by parts and using derivative property (11) and a simple computation: Similarly, adopting the same pattern from (16), one can obtain From (18) and (19), the following inequality is obtained: Now, on the contrary, we let x ∈ [a, b], t ∈ [x, b], and β ≥ 1. en, the following inequality holds for Mittag-Leffler function: From (21) and the condition of boundedness of ψ 1 ′ , one can have the following integral inequalities:

Mathematical Problems in Engineering
Following the same procedure as we did for (15) and (16), one can obtain from (22) and (23) the following modulus inequality: Inequalities (20) and (24) give (13) which is the required inequality.
Some comments on the abovementioned result are given as follows.