Simpson’s Second-Type Inequalities for Co-Ordinated Convex Functions and Applications for Cubature Formulas

Sabah Iftikhar,Hijaz Ahmad,Samet Erden,Muhammad Aamir Ali,Jamel Baili

doi:10.3390/fractalfract6010033

Abstract

Inequality theory has attracted considerable attention from scientists because it can be used in many fields. In particular, Hermite–Hadamard and Simpson inequalities based on convex functions have become a cornerstone in pure and applied mathematics. We deal with Simpson’s second-type inequalities based on coordinated convex functions in this work. In this paper, we first introduce Simpson’s second-type integral inequalities for two-variable functions whose second-order partial derivatives in modulus are convex on the coordinates. In addition, similar results are acquired by considering that powers of the absolute value of second-order partial derivatives of these two-variable functions are convex on the coordinates. Finally, some applications for Simpson’s 3/8 cubature formula are given.

Highlights

Introduction and PreliminariesSimpson’s rules (Thomas Simpson 1710–1761) are well-known methods in numerical analysis for the purpose of numerical integration and the numerical approximation of definite integrals
Two famous Simpson rules are known in the literature, and one of them is the following estimation known as Simpson’s second-type (Simpson’s 38 ) inequality
Afterwards, Hermite–Hadamard, Simpson- and Newton-type inequalities for harmonically convex mappings have been observed by some researchers

Summary

Introduction

Introduction and PreliminariesSimpson’s rules (Thomas Simpson 1710–1761) are well-known methods in numerical analysis for the purpose of numerical integration and the numerical approximation of definite integrals. In [2], Sarikaya et al gave some inequalities of Simpson’s type based on s-convexity and their applications for special means of real numbers. Gao and Shi obtained new inequalities of Newton’s type for functions whose absolute values of second derivatives are convex in [4]. Afterwards, Hermite–Hadamard-, Simpson- and Newton-type inequalities for harmonically convex mappings have been observed by some researchers.

Results

Conclusion