International Journal of Mathematics and Mathematical Sciences
Volume 2012 (2012), Article ID 343191, 12 pages
http://dx.doi.org/10.1155/2012/343191
Research Article

Sharp Integral Inequalities Based on a General Four-Point Quadrature Formula via a Generalization of the Montgomery Identity

J. Pečarić1 and M. Ribičić Penava2

1Faculty of Textile Technology, University of Zagreb, Pierottijeva 6, 10000 Zagreb, Croatia
2Department of Mathematics, University of Osijek, Trg Ljudevita Gaja 6, 31 000 Osijek, Croatia

Received 28 March 2012; Accepted 10 July 2012

Academic Editor: Marianna Shubov

Copyright © 2012 J. Pečarić and M. Ribičić Penava. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider families of general four-point quadrature formulae using a generalization of the Montgomery identity via Taylor’s formula. The results are applied to obtain some sharp inequalities for functions whose derivatives belong to 𝐿 𝑝 spaces. Generalizations of Simpson’s 3/8 formula and the Lobatto four-point formula with related inequalities are considered as special cases.