International Journal of Mathematics and Mathematical Sciences 
Volume 2004 (2004), Issue 1, Pages 49-53
doi:10.1155/S0161171204208158

A lower bound for ratio of power means

Feng Qi,1 Bai-Ni Guo,1 and Lokenath Debnath2

 1Department of Applied Mathematics and Informatics, Jiaozuo Institute of Technology, Henan, Jiaozuo City 454000, China
2Department of Mathematics, University of Texas-Pan American, Edinburg, TX 78539, USA
Received 29 August 2002

Abstract

Let n and m be natural numbers. Suppose that {ai}i=1n+m is an increasing, logarithmically convex, and positive sequence. Denote the power mean Pn(r) for any given positive real number r by Pn(r)=((1/n)∑i=1nair)1/r. Then Pn(r)/Pn+m(r)≥an/an+m. The lower bound is the best possible.