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

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.

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References

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    Ciarlet PG: Introduction to Numerical Linear Algebra and Optimisation, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge; 1991.

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    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.

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    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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Correspondence to Oscar Rojo.

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Keywords

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