Ostrowski type inequalities and some selected quadrature formulae

TL;DR

This survey explores Ostrowski type inequalities and their applications in numerical integration, highlighting various inequalities and quadrature formulas, and emphasizing their historical development and error estimation techniques.

Contribution

It provides a comprehensive overview of Ostrowski inequalities, their modifications, and applications to quadrature formulas, including new variants and error bounds.

Findings

Several Ostrowski inequalities and their modifications are presented.

Connections between inequalities and quadrature error estimation are established.

Multiple specific inequalities and formulas are compared and analyzed.

Abstract

Some selected Ostrowski type inequalities and a connection with numerical integration are studied in this survey paper, which is dedicated to the memory of Professor D.S. Mitrinovic, who left us 25 years ago. His significant influence to the development of the theory of inequalities is briefly given in the first section of this paper. Beside some basic facts on quadrature formulas and an approach for estimating the error term using Ostrowski type inequalities and Peano kernel techniques, we give several examples of selected quadrature formulas and the corresponding inequalities, including the basic Ostrowski's inequality (1938), inequality of Milovanovic and Pecaric (1976) and its modifications, inequality of Dragomir, Cerone and Roumeliotis (2000), symmetric inequality of Guessab and Schmeisser (2002) and asymmetric inequality of Franjic (2009), as well as four point symmetric inequalites by Alomari (2012) and a variant with double internal nodes given by Liu and Park (2017).