Open Access

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

Journal of Inequalities and Applications20062006:43465

DOI: 10.1155/JIA/2006/43465

Received: 22 December 2005

Accepted: 21 September 2006

Published: 8 November 2006

Abstract

Let https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F43465/MediaObjects/13660_2005_Article_1597_IEq1_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F43465/MediaObjects/13660_2005_Article_1597_IEq2_HTML.gif be the mean and the standard deviation of the components of the vector https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F43465/MediaObjects/13660_2005_Article_1597_IEq3_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F43465/MediaObjects/13660_2005_Article_1597_IEq4_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F43465/MediaObjects/13660_2005_Article_1597_IEq5_HTML.gif a positive integer. Here, we prove that if https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F43465/MediaObjects/13660_2005_Article_1597_IEq6_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F43465/MediaObjects/13660_2005_Article_1597_IEq7_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F43465/MediaObjects/13660_2005_Article_1597_IEq8_HTML.gif . The equality holds if and only if the https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F43465/MediaObjects/13660_2005_Article_1597_IEq9_HTML.gif largest components of https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F43465/MediaObjects/13660_2005_Article_1597_IEq10_HTML.gif are equal. It follows that https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F43465/MediaObjects/13660_2005_Article_1597_IEq11_HTML.gif is a strictly increasing sequence converging to https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F43465/MediaObjects/13660_2005_Article_1597_IEq12_HTML.gif , the largest component of https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F43465/MediaObjects/13660_2005_Article_1597_IEq13_HTML.gif , except if the https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F43465/MediaObjects/13660_2005_Article_1597_IEq14_HTML.gif largest components of https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F43465/MediaObjects/13660_2005_Article_1597_IEq15_HTML.gif are equal. In this case, https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F43465/MediaObjects/13660_2005_Article_1597_IEq16_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F43465/MediaObjects/13660_2005_Article_1597_IEq17_HTML.gif .

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Authors’ Affiliations

(1)
Departamento de Matemáticas, Universidad Católica del Norte

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.