Some Generalized Fractional Integral Inequalities for Convex Functions with Applications

Dafang Zhao,Muhammad Aamir Ali,Thanin Sitthiwirattham,Chanon Promsakon

doi:10.3390/fractalfract6020094

Abstract

In this paper, we establish a generalized fractional integrals identity involving some parameters and differentiable functions. Then, we use the newly established identity and prove different generalized fractional integrals inequalities like midpoint inequalities, trapezoidal inequalities and Simpson’s inequalities for differentiable convex functions. Finally, we give some applications of newly established inequalities in the context of quadrature formulas.

Highlights

Some Generalized Fractional IntegralFractional calculus has grown in popularity and relevance over the last three decades, owing to its demonstrated applications in a wide range of seemingly disparate domains of science and engineering
Sarikaya and Ertugral [10] defined a new class of fractional integrals, called generalized fractional, and they used these integrals to prove a general version of Hermite–Hadamard type inequalities for convex functions
We proved some parameterized integral inequalities for differentiable convex functions via generalized fractional integrals

Summary

Introduction

Fractional calculus (that is, calculus of integrals and derivatives of any arbitrary real or complex order) has grown in popularity and relevance over the last three decades, owing to its demonstrated applications in a wide range of seemingly disparate domains of science and engineering. Because of the importance of fractional calculus, researchers have utilized it to establish various fractional integral inequalities that have been shown to be quite useful in approximation theory Inequalities such as Hermite–Hadamard, Simpson’s, midpoint, Ostrowski’s and trapezoidal inequalities are examples, and by using these inequalities, we can obtain the bounds of formulas used in numerical integration. Sarikaya and Yildrim [4] used Riemann–Liouville fractional integrals to prove some new Hermite–Hadamard type inequalities and midpoint type inequalities for differentiable convex functions. Sarikaya and Ertugral [10] defined a new class of fractional integrals, called generalized fractional, and they used these integrals to prove a general version of Hermite–Hadamard type inequalities for convex functions. Proved several variants of Ostrowski’s and Simpson’s type for differentiable convex functions via generalized fractional integrals.

Fractional Integrals and Related Inequalities

Key Equalities

Main Results

Applications to Quadrature Formulas

Conclusions