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  • Research Article
  • Open Access

Inequalities involving the mean and the standard deviation of nonnegative real numbers

Journal of Inequalities and Applications20062006:43465

https://doi.org/10.1155/JIA/2006/43465

  • Received: 22 December 2005
  • Accepted: 21 September 2006
  • Published:

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

  • Standard Deviation
  • Positive Integer
  • Real Number
  • Large Component
  • Nonnegative Real Number

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Authors’ Affiliations

(1)
Departamento de Matemáticas, Universidad Católica del Norte, Antofagasta, Casilla, 1280, Chile

References

  1. Ciarlet PG: Introduction to Numerical Linear Algebra and Optimisation, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge; 1991.Google Scholar
  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.MATHMathSciNetView ArticleGoogle Scholar
  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-XMATHMathSciNetView ArticleGoogle Scholar

Copyright

© Oscar Rojo. 2006

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.

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