Journal of Inequalities and Applications
Volume 2008 (2008), Article ID 163202, 10 pages
Research Article

New Means of Cauchy's Type

Matloob Anwar1 and J. Pečarić1,2

1Abdus Salam School of Mathematical Sciences, GC University, Lahore Gulberg 54660, Pakistan
2Faculty of Textile Technology, University of Zagreb, 10000 Zagreb, Croatia

Received 30 December 2007; Accepted 7 April 2008

Academic Editor: Wing-Sum Cheung

Copyright © 2008 Matloob Anwar and J. Pečarić. 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.


We will introduce new means of Cauchy's type Mr,ls(f,μ) defined, for example, as Mr,ls(f,μ)=((l(ls)/r(rs))(Mrr(f,μ)Msr(f,μ)/Mll(f,μ)Msl(f,μ)))1/(rl), in the case when lrs, l,r0. We will show that this new Cauchy's mean is monotonic, that is, the following result. Theorem. Let t,r,u,v, such that tv, ru. Then for Mr,ls(f,μ), one has Mt,rsMv,us. We will also give some related comparison results.