Journal of Inequalities and Applications
Volume 2006 (2006), Article ID 43465, 15 pages
doi:10.1155/JIA/2006/43465
Inequalities involving the mean and the standard deviation of nonnegative real numbers
Departamento de Matemáticas, Universidad Católica del Norte, Antofagasta 1280, Casilla, Chile
Received 22 December 2005; Revised 18 August 2006; Accepted 21 September 2006
Copyright © 2006 Oscar Rojo. 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
m(y)=∑j=1nyj/n
and s(y)=m(y2)−m2(y)
be the mean and the standard deviation of the components of the vector y=(y1,y2,…,yn−1,yn), where yq= (y1q,y2q,…,yn−1q,ynq) with q a positive integer. Here, we prove that if y≥0,
then m(y2p)+(1/n−1)s(y2p)≤m(y2p+1)+(1/n−1)s(y2p+1)
for p=0,1,2,…. The equality holds if and only if the (n−1) largest components of y are equal. It follows that (l2p(y))p=0∞, l2p(y)=(m(y2p)+(1/n−1)s(y2p))2−p,
is a strictly increasing sequence converging to y1, the largest
component of y, except if the (n−1) largest components of y are equal. In this case, l2p(y)=y1 for all p.