Some fractional proportional integral inequalities

Gauhar Rahman,Aftab Khan,Thabet Abdeljawad,Kottakkaran Sooppy Nisar

doi:10.1186/s13660-019-2199-z

Abstract

In the last few years, various researchers studied the so-called conformable integrals and derivatives. Based on that notion some authors used modified conformable derivatives (proportional derivatives) to generate nonlocal fractional integrals and derivatives, called fractional proportional integrals and derivatives, which contain exponential functions in their kernels. Our aim in this paper is to establish some new integral inequalities by utilizing the fractional proportional-integral operators. In fact, certain new classes of integral inequalities for a class of n (nin mathbb{N}) positive continuous and decreasing functions on [a,b] are presented. The inequalities presented in this paper are more general than the existing classical inequalities.

Highlights

Fractional calculus is the generalization of derivatives and integrals of arbitrary noninteger order
Jarad et al [16] introduced Caputo and Riemann–Liouville generalized proportional fractional derivatives which comprise exponential functions in their kernels. They proved that the newly defined proportional fractional integrals possess a semi-group property and they provide an undeviating generalization to the existing Caputo and Riemann–Liouville fractional derivatives and integrals
The inequalities obtained in this paper are the generalization of integral inequalities involving the Riemann–Liouville fractional integral operators

Summary

Introduction

Fractional calculus is the generalization of derivatives and integrals of arbitrary noninteger order. This field has earned more recognition due to its applications in diverse domains [11, 18, 25]. In [17], the authors introduced the idea of fractional conformable derivative operators with a shortcoming that the new derivative operator does not tend to the original function when the order ρ → 0. In [4, 5, 7, 9, 19], some researchers defined new fractional derivative operators by using exponential and Mittag-Leffler functions in the kernels. In [16], Jarad et al introduced the left and right generalized proportional-integral operators which are respectively defined by aIβ,μf.

Iβb f

Consider a function

By multiplying both sides of

Multiplying both sides of