Generalized k-Fractional Chebyshev-Type Inequalities via Mittag-Leffler Functions

Zhiqiang Zhang,Tao Yan,Sajid Mehmood,Chahn-Yong Jung,Ghulam Farid

doi:10.3390/axioms11020082

Abstract

Mathematical inequalities have gained importance and popularity due to the application of integral operators of different types. The present paper aims to give Chebyshev-type inequalities for generalized k-integral operators involving the Mittag-Leffler function in kernels. Several new results can be deduced for different integral operators, along with Riemann–Liouville fractional integrals by substituting convenient parameters. Moreover, the presented results generalize several already published inequalities.

Highlights

A large number of integral inequalities exist in the literature for different types of integral operators [1,2,3,4,5,6,7,8,9]
The results in this paper provide generalizations of various inequalities published in the literature of fractional integral inequalities
We give the definition of Riemann–Liouville integral operators, the classical Chebyshev inequality, Chebyshev inequalities for Riemann–Liouville integral operators, and definitions of generalized integral operators containing the Mittag-Leffler function

Summary

Introduction

Integral operators play a very important role in the field of mathematical inequalities.A large number of integral inequalities exist in the literature for different types of integral operators [1,2,3,4,5,6,7,8,9]. The Chebyshev inequality is studied extensively by using such extensions and generalizations (for details, see [3,5,10,11,12,13,14,15,16]). Inspired by this latest research, the aim of the present paper is to establish Chebyshevtype inequalities for generalized k-integral operators containing the Mittag-Leffler function in their kernels, which produce many well-known integral operators. We give the definition of Riemann–Liouville integral operators, the classical Chebyshev inequality, Chebyshev inequalities for Riemann–Liouville integral operators, and definitions of generalized integral operators containing the Mittag-Leffler function

Objectives

Results

Conclusion