Companion to the Ostrowski–Grüss-Type Inequality of the Chebyshev Functional with an Application

Abstract

Recently, there have been many proven results of the Ostrowski–Grüss-type inequality regarding the error bounds for the Chebyshev functional when the functions or their derivatives belong to Lp spaces. In the existing literature, the main assumption in the weight-type results is that the derivative of the function is bounded by two constant functions. The aim of our paper is to extend those results in a way that the derivative of the function is bounded by two functions in Lp spaces. Furthermore, we give some new error estimations of the Chebyshev functional and applications to the one-point weight integral formulas.