We generalize Ostrowski inequality for higher order derivatives, by using a generalized Euler type identity. Some of the inequalities produced are sharp, namely attained by basic functions. The rest of the estimates are tight. We give applications to trapezoidal and mid-point rules. Estimates are given with respect to L∞ norm. c © 2006 Elsevier Ltd. All rights reserved.

This is the fifth and last in our series of notes concerning some classical inequalities such as the Ostrowski, Simpson, Iyengar, and Ostrowski–Grüss inequalities in R. In the last note, we propose an improvement of the Ostrowski–Grüss inequality which involves 3n knots where n = 1 is an arbitrary numbers. More precisely, suppose that fxkgk1⁄41 1⁄20;1 ; fykg n k1⁄41 1⁄20;1 , and fakg n k1⁄41 1⁄...

 (b− a)M, for all x ∈ [a, b] . The constant 14 is best possible in the sense that it cannot be replaced by a smaller constant. In [2], the author has proved the following Ostrowski type inequality. Theorem 2. Let f : [a, b] → R be continuous on [a, b] with a > 0 and differentiable on (a, b) . Let p ∈ R\ {0} and assume that Kp (f ) := sup u∈(a,b) { u |f ′ (u)| } < ∞. Then we have the inequality...

Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this article, we establish some companions of an Ostrowski-like type inequality for functions whose second derivatives in absolute value are convex and concave.

Inequalities are obtained for quadrature rules in terms of upper and lower bounds of the first derivative of the integrand. Bounds of Ostrowski type quadrature rules are obtained and the classical Iyengar inequality for the trapezoidal rule is recaptured as a special case. Applications to numerical integration are demonstrated.