Fractional Integral Inequalities of Gruss Type via Generalized Mittag-Leffler Function

Ghulam Farid ,Vishnu Narayan Mishra ,Saqib Mehmood ,Atiq Ur Rehman

doi:10.28924/2291-8639-17-2019-548

Abstract

We use generalized fractional integral operator containing the generalized Mittag-Leffler function to establish some new integral inequalities of Gr¨uss type. A cluster of fractional integral inequalities have been identified by setting particular values to parameters involved in the Mittag-Leffler special function. Presented results contain several fractional integral inequalities which reflects their importance.

Highlights

Gruss inequality; Generalized fractional integral operator; Mittag-Leffler function
In 1935, Gruss [5] proved the following inequality a2a2 − a1 a1 f1(t)f2(t)dt − (M − m)(N − n) ≤a2 − a1 a1 f1(t)dt a2 − a1f2(t)dt a1 where f and g are two integrable functions on [a,b] and satisfying the following conditions (1.1)m ≤ f1(x) ≤ M, n ≤ f2(x) ≤ N m, M, n, N ∈ R, x ∈ [a, b].In the literature inequality (1.1) is well known as the Gruss inequality
For example selecting p = 0, fractional integral inequalities for fractional integral operators defined by Salim and Faraj in [12], selecting l = δ = 1, fractional integral inequalities for fractional integral operators defined by Rahman et al in [11], selecting p = 0 and l = δ = 1, fractional integral inequalities for fractional integral operators defined by Shukla and Prajapati in [13] and see [14], selecting p = 0 and l = δ = k = 1, fractional integral inequalities for fractional integral operators defined by Prabhakar in [10], selecting p = ω = 0 fractional integral inequalities for Riemann-Liouville fractional integrals

Summary

Introduction

Gruss inequality; Generalized fractional integral operator; Mittag-Leffler function. Our interest in this paper is to give some generalized fractional integral inequalities of Gruss type by use of generalized fractional integral operators due to the Mittag-Leffler function. In the following we define an extended generalized Mittag-Leffler function Eμγ,,αδ,,kl,c(t; p) as fallows: Definition 1.1.

Results

Conclusion