Journal of Inequalities and Applications
Volume 2010 (2010), Article ID 728251, 19 pages
doi:10.1155/2010/728251
Research Article

Positive Semidefinite Matrices, Exponential Convexity for Majorization, and Related Cauchy Means

M. Anwar,1 N. Latif,2 and J. Pečarić2,3

1Center for Advanced Mathematics and Physics, NUST Campus at College of Electrical & Mechanical Engineering, Peshawar Road, Rawalpindi 48000, Pakistan
2Abdus Salam School of Mathematical Sciences, GC University, Lahore 5400, Pakistan
3Faculty of Textile Technology, University of Zagreb, 1002 Zagreb, Croatia

Received 26 October 2009; Accepted 10 March 2010

Academic Editor: Panayiotis Siafarikas

Copyright © 2010 M. Anwar et al. 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 prove positive semidefiniteness of matrices generated by differences deduced from majorization-type results which implies exponential convexity and log-convexity of these differences and also obtain Lyapunov's and Dresher's inequalities for these differences. We introduce new Cauchy means and show that these means are monotone.