Journal of Inequalities and Applications
Volume 2008 (2008), Article ID 163202, 10 pages
doi:10.1155/2008/163202
Research Article
New Means of Cauchy's Type
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.
Abstract
We will introduce new means of Cauchy's type Mr,ls(f,μ) defined, for example, as Mr,ls(f,μ)=((l(l−s)/r(r−s))(Mrr(f,μ)−Msr(f,μ)/Mll(f,μ)−Msl(f,μ)))1/(r−l), in the case when l ≠ r ≠ s, l,r ≠ 0. We will show that this new Cauchy's mean is
monotonic, that is, the following result. Theorem. Let t,r,u,v∈ℝ, such that t≤v, r≤u. Then for Mr,ls(f,μ), one has Mt,rs≤Mv,us. We will also give some related comparison results.