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

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

Recommended by Wing-Sum Cheung

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.