Journal of Inequalities and Applications
Volume 2011 (2011), Article ID 562923, 16 pages
doi:10.1155/2011/562923
Research Article

Bessel and Grüss Type Inequalities in Inner Product Modules over Banach ∗ -Algebras

A. G. Ghazanfari1 and S. S. Dragomir2,3

1Department of Mathematics, Lorestan University, P.O. Box 465, Khoramabad, Iran
2School of Engineering and Science, Victoria University, P.O. Box 14428, Melbourne City, MC 8001, Australia
3School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3 Wits 2050, Johannesburg, South Africa

Received 11 January 2011; Accepted 1 March 2011

Academic Editor: Paolo E. Ricci

Copyright © 2011 A. G. Ghazanfari and S. S. Dragomir. 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 give an analogue of the Bessel inequality and we state a simple formulation of the Grüss type inequality in inner product 𝐶 ∗ -modules, which is a refinement of it. We obtain some further generalization of the Grüss type inequalities in inner product modules over proper 𝐻 ∗ -algebras and unital Banach ∗ -algebras for 𝐶 ∗ -seminorms and positive linear functionals.