Inequalities involving the mean and the standard deviation of nonnegative real numbers
- Oscar Rojo1Email author
https://doi.org/10.1155/JIA/2006/43465
© Oscar Rojo. 2006
Received: 22 December 2005
Accepted: 21 September 2006
Published: 8 November 2006
Abstract
Let
and
be the mean and the standard deviation of the components of the vector
, where
with
a positive integer. Here, we prove that if
, then
for
. The equality holds if and only if the
largest components of
are equal. It follows that
is a strictly increasing sequence converging to
, the largest component of
, except if the
largest components of
are equal. In this case,
for all
.
Keywords
Authors’ Affiliations
References
- Ciarlet PG: Introduction to Numerical Linear Algebra and Optimisation, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge; 1991.Google Scholar
- Rojo O, Rojo H: A decreasing sequence of upper bounds on the largest Laplacian eigenvalue of a graph. Linear Algebra and Its Applications 2004, 381: 97–116.MATHMathSciNetView ArticleGoogle Scholar
- Wolkowicz H, Styan GPH: Bounds for eigenvalues using traces. Linear Algebra and Its Applications 1980, 29: 471–506. 10.1016/0024-3795(80)90258-XMATHMathSciNetView ArticleGoogle Scholar
Copyright
This article is published under license to BioMed Central Ltd. 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.
