Fractional Ostrowski Type Inequalities via Generalized Mittag–Leffler Function

Abstract

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.

1. Introduction

Exponential function plays a vital role in the theory of integer order differential equations. The symbol 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. These 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 [7–16]. The 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, in [7], Andrić et al. defined the extended generalized Mittag–Leffler function as follows.

Definition 1. Let , , and with , , and . Then, the extended generalized Mittag–Leffler function is defined aswhere is the generalized beta function defined as and is the Pochhammer symbol given by .
The corresponding left- and right-sided generalized fractional integrals and are defined as follows.

Definition 2 (see [7]). Let , , with , , and . Let and . Then, the generalized fractional integrals and are defined asRecently, Farid defined a unified integral operator in [17] (also see [18]). This unifies several kinds of fractional and conformable integrals in a compact formula and is given as follows.

Definition 3. Let , , be the functions such that be positive and , and be differentiable and strictly increasing. Also, let be an increasing function on and , , , , and . Then, for the left and right integral operators are defined bywhere .
The following definition can be deduced from Definition 3 (see [16]).

Definition 4. Let , , be the functions such that be positive and , and be differentiable and strictly increasing and , and . Then, for the left and right integral operators are defined byIt can be noted thatIn the following, we state the Ostrowski inequality which is proved by Ostrowski [19] in 1938.

Theorem 1. Let , where is an interval in , be a mapping differentiable in , the interior of and , . If for all , then for we have

The 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, 20–26] and references therein.

In Section 2, fractional version of Ostrowski inequalities with the help of Mittag–Leffler function has been established. The 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.

2. Main Results

First, we establish the following lemma for extended generalized Mittag–Leffler function.

Lemma 1. If with and , then

Proof. We haveAfter simple computation, one can obtain (11).
Next, we give the generalized fractional Ostrowski type inequality containing extended generalized Mittag–Leffler function.

Theorem 2. Let , where is an interval in , be a mapping differentiable in , the interior of and , . If is an integrable function, for all and be an increasing and positive function on , having continuous derivative on , then for , the following inequality for fractional integrals (6) and (7) holds:

Proof. Let , , and . Then, the following inequality holds for the monotonically increasing function and the Mittag–Leffler function (1):From (14) and given condition of boundedness of , one can have the following integral inequalities:First, we consider inequality (15) as follows:Therefore, (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 obtainFrom (18) and (19), the following inequality is obtained:Now, on the contrary, we let , , and . Then, the following inequality holds for Mittag–Leffler function:From (21) and the condition of boundedness of , one can have the following integral inequalities: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.
In the following, we give direct consequences of above theorem.

Corollary 1. If we put in (13), then we get the following fractional integral inequality:

Remark 1. (i)If we put in (13), then we obtain Theorem 5 in [9](ii)If we put and in (13), then we obtain Theorem 1 in [6](iii)If we put , , and in (13), then we obtain Ostrowski inequality (10)(iv)If we put in (25), then we obtain Corollary 1 in [9]The next result is a general form of fractional Ostrowski inequality containing generalized Mittag–Leffler function.

Theorem 3. Let , where is an interval in , be a mapping differentiable in , the interior of and , . If is integrable function and for all and be an increasing and positive function on , having continuous derivative on , then, for , the following inequalities for fractional integrals (6) and (7) hold:

Proof. The proof is similar to the proof of Theorem 2, just after comparing conditions on derivative of , so we left it for the reader.
Some comments on the abovementioned result are given as follows.

Remark 2. (i)If we put and in (26) and (27), then we obtain Theorem 1 in [6](ii)If we put in Theorem 3, then with some rearrangements we obtain Theorem 2(iii)If we put in (26) and (27), then we obtain Theorem 6 in [9]In the following, we have established a result related to fractional Ostrowski inequality containing generalized Mittag–Leffler function.

Theorem 4. Let , where is an interval in , be a mapping differentiable in , the interior of and , . If is integrable function, for all and and be an increasing and positive function on , then, for , the following inequality for fractional integrals (6) and (7) holds:

Proof. Let , , and . Then, the following inequality holds true for Mittag–Leffler function:From (29) and given condition of boundedness on , one can have the following integral inequalities:First, we consider inequality (30) as follows:Therefore, (32) takes the following form after integrating by parts and using derivative property (11) and a simple computationSimilarly, adopting the same pattern from (31), one can obtainFrom (33) and (34), the following inequality is obtained:Now, on the contrary, we let , , and . Then, the following inequality holds for Mittag–Leffler function:From (36) and given condition of boundedness of , one can have the following integral inequalities:Following the same procedure as we did for (30) and (31), one can obtain from (37) and (38) the following modulus inequality:Inequalities (35) and (39) give (28) which is required inequality.
Some direct consequences of the above theorem are given below.

Corollary 2. If we put in (28), then we get the following fractional integral inequality:

Remark 3. (i)If we put in (28), then we obtain Theorem 5 in [9](ii)If we put and in (28), then we obtain Theorem 1 in [6](iii)If we put in (40), then we obtain Corollary 2 in [9]

3. Applications

In this section, we just describe some applications of Theorem 4 and leave such applications of other results for the reader. By applying Theorem 4 at end points of the interval and adding the resulting inequalities, one obtains the error bounds of compact form of the fractional Hadamard inequality.

Theorem 5. Under the assumptions of Theorem 4, the following estimation of Hadamard inequality can be obtained:

Proof. By putting , , and in (40) then adding the resulting inequalities, we obtainMultiplying both sides of the above inequality by and using (8) and (9), inequality (41) can be obtained.

Remark 4. If in (41) is replaced by and by , then we get an error bound of the Hadamard inequality given in Theorem 1 in [20].

Theorem 6. Under the assumptions of Theorem 4, the following inequality can be obtained:

Proof. By putting for in (40), (43) can be obtained.

4. Concluding Remarks

We have established generalized fractional integral inequalities of Ostrowski type. By applying boundedness of a differentiable function and using properties of an extended generalized Mittag–Leffler function different generalized versions of Ostrowski type inequalities are analyzed. Also, some deductions from results of this paper are connected with already published results. Furthermore, all the results can be calculated for fractional integral operators defined in [2, 3, 5, 6, 27], and we left it for the reader.

Data Availability

No data were used to support the study.

Conflicts of Interest

The authors do not have any conflicts of interest.

Authors’ Contributions

All authors have equal contributions.

Acknowledgments

This research work was supported by the Higher Education Commission of Pakistan under NRPU 2016, Project no. 5421.