A Generalization of the Ostrowski–gruss Inequality

Abstract

A new generalization of the Ostrowski–Gruss inequality is introduced in three different cases for functions in L1[a, b] and L∞[a, b] spaces and its application is given for deriving error bounds of some quadrature rules.

A Main Class of Integral Inequalities With Applications

“…Following the papers of Masjed-Jamei and Dragomir [9,10,11,12,13], let us introduce the integral transform L K (. ; x) as follows.…”

An analogue of the Ostrowski inequality and applications

“…where w(x) is a positive function on [a, b], {x k } n k=0 and {w k } n k=0 are respectively nodes and weight coefficients and R n+1 ( f ) is the corresponding error [18]. Let Π d be the set of algebraic polynomials of degree at most d. The quadrature formula (14) has degree of exactness d if for every p ∈ Π d we have R n+1 (p) = 0. In addition, if R n+1 (p) 0 for some Π d+1 , formula (14) has precise degree of exactness d. The convergence order of quadrature formula (14) depends on the smoothness of the function f as well as on its degree of exactness.…”

“…Let Π d be the set of algebraic polynomials of degree at most d. The quadrature formula (14) has degree of exactness d if for every p ∈ Π d we have R n+1 (p) = 0. In addition, if R n+1 (p) 0 for some Π d+1 , formula (14) has precise degree of exactness d. The convergence order of quadrature formula (14) depends on the smoothness of the function f as well as on its degree of exactness. It is well known that for given n + 1 mutually different nodes {x k } n k=0 we can always achieve a degree of exactness d = n by interpolating at these nodes and integrating the interpolated polynomial instead of f .…”

“…For various generalizations, refinements and related Ostrowski type inequalities for functions of one or several variables one may refer to the monograph [7] and the references therein. See also [1,8,11,15] and [13,14,16,19] in this regard. Moreover, the Ostrowski inequality has an important role in numerical quadrature rules [9,12].…”

Generalized Ostrowski inequalities and computational integration

“…Also explicit error bounds for numerical quadrature and nonstandard quadrature formulas will be discussed. Also we recaptured many established results from articles [12], [13], [14] and [21].…”