Fractional Ostrowski Type Inequalities via Generalized Mittag–Leffler Function

Xinghua You,Kahkashan Maheen,Ghulam Farid

doi:10.1155/2020/4705632

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.

Highlights

Exponential function plays a vital role in the theory of integer order differential equations. e symbol Eα(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
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

Summary

Introduction

Exponential function plays a vital role in the theory of integer order differential equations. e symbol Eα(z) is well known as the Mittag–Leffler function and it is a generalization of exponential function. Mittag–Leffler function is used in the formation of fractional integral operators. Defined the extended generalized Mittag–Leffler function Ecμ,,δσ,,kl ,c(.; p) as follows. En, the extended generalized Mittag–Leffler function Ecμ,,δα,,kl ,c(t; p) is defined as. E corresponding left- and right-sided generalized fractional integrals εcμ,,δα,,kl,,ωc,a+ and εcμ,,δα,,kl,,ωc,b− are defined as follows. + b)/2)) a)2 􏼣(b a)M. e Ostrowski inequality has been studied by many researchers to obtain its refinements, generalizations, and extensions. E Ostrowski inequality has been studied by many researchers to obtain its refinements, generalizations, and extensions Their applications are analyzed for establishing the bounds of relations among special means and for estimations of numerical quadrature rules.

Main Results

Applications

Concluding Remarks