Journal of Inequalities and Applications
Volume 2011 (2011), Article ID 350973, 19 pages
doi:10.1155/2011/350973
Research Article

Refinements of Results about Weighted Mixed Symmetric Means and Related Cauchy Means

László Horváth,1 Khuram Ali Khan,2,3 and J. Pečarić2,4

1Department of Mathematics, University of Pannonia, University Street 10, 8200 Veszprém, Hungary
2Abdus Salam School of Mathematical Sciences, GC University, 68-B, New Muslim Town, Lahore 54600, Pakistan
3Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan
4Faculty of Textile Technology, University of Zagreb, 10000 Zagreb, Croatia

Received 26 November 2010; Accepted 23 February 2011

Academic Editor: Michel Chipot

Copyright © 2011 László Horváth 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

A recent refinement of the classical discrete Jensen inequality is given by Horváth and Pečarić. In this paper, the corresponding weighted mixed symmetric means and Cauchy-type means are defined. We investigate the exponential convexity of some functions, study mean value theorems, and prove the monotonicity of the introduced means.