Some New Simpson’s-Formula-Type Inequalities for Twice-Differentiable Convex Functions via Generalized Fractional Operators

Muhammad Aamir Ali,Jessada Tariboon,Suphawat Asawasamrit,Hasan Kara,Hüseyin Budak,Fatih Hezenci

doi:10.3390/sym13122249

Abstract

From the past to the present, various works have been dedicated to Simpson’s inequality for differentiable convex functions. Simpson-type inequalities for twice-differentiable functions have been the subject of some research. In this paper, we establish a new generalized fractional integral identity involving twice-differentiable functions, then we use this result to prove some new Simpson’s-formula-type inequalities for twice-differentiable convex functions. Furthermore, we examine a few special cases of newly established inequalities and obtain several new and old Simpson’s-formula-type inequalities. These types of analytic inequalities, as well as the methodologies for solving them, have applications in a wide range of fields where symmetry is crucial.

Highlights

Academic Editor: Nicusor MinculeteSimpson’s inequality is widely used in many areas of mathematics
For twice-differentiable functions, we have developed a generalized fractional version of the Simpson-type inequality in this paper
For k-Riemann–Liouville fractional integrals, we provided novel Simpson-type inequalities

Summary

Introduction

Simpson’s inequality is widely used in many areas of mathematics. For four times continuously differentiable functions, the classical Simpson’s inequality is expressed as follows: Received: 29 October 2021. In the papers [9,10], researchers extended the Simpson inequalities for differentiable functions to Riemann–Liouville fractional integrals. After giving the definition of the generalized fractional integral operators, we construct a new identity for twice-differentiable functions. Using this equality, we prove several Simpson-type inequalities for functions whose second derivatives are convex. The operators (1) and (2) reduce k to the k-Riemann–Liouville fractional integrals Jaα+,k f ( x ) and Jbα−,k f ( x ), respectively. Inspired by the ongoing studies, we give the generalized fractional version of the inequalities proved by Budak et al in [36] for twice-differentiable convex functions. The fundamental benefit of these inequalities is that they can be turned into classical integral inequalities of Simpson’s type [32], Riemann–Liouville fractional integral inequalities of Simpson’s type [36], and k-Riemann–Liouville fractional integral inequalities of Simpson’s type without having to prove each one separately

Simpson’s-Type Inequalities for Twice-Differentiable Functions

Findings

Conclusions