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Inequalities involving the mean and the standard deviation of nonnegative real numbers
Journal of Inequalities and Applicationsvolume 2006, Article number: 43465 (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
.
References
- 1.
Ciarlet PG: Introduction to Numerical Linear Algebra and Optimisation, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge; 1991.
- 2.
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.
- 3.
Wolkowicz H, Styan GPH: Bounds for eigenvalues using traces. Linear Algebra and Its Applications 1980, 29: 471–506. 10.1016/0024-3795(80)90258-X
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Keywords
- Standard Deviation
- Positive Integer
- Real Number
- Large Component
- Nonnegative Real Number
