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
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
Copyright © 2004 Feng Qi et al. 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
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.