International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 1-4, Pages 49-53
doi:10.1155/S0161171204208158
A lower bound for ratio of power means
Feng Qi1
, Bai-Ni Guo1
and Lokenath Debnath3
1Department of Applied Mathematics and Informatics, Jiaozuo Institute of Technology, Henan, Jiaozuo City 454000, China
3Department of Mathematics, University of Texas-Pan American, Edinburg, TX 78539, USA
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.