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Best upper bounds based on the arithmetic-geometric mean inequality. (English) Zbl 1032.26017

In this paper the following inequality is proved: if \(0< \lambda_1\leq\cdots\leq \lambda_n\) are the eigenvalues of a positive definite matrix then \[ {{ \lambda_n}\over{ \lambda_1}}\leq \tau\Bigl({{(\lambda_1+ \cdots +\lambda_n)^n}\over{n^n(\lambda_1\lambda_2 \cdots \lambda_n)}}\Bigr), \] where \(\tau(x) = 2x-1 + \sqrt{(2x-1)^2-1}\); there is equality in the case \(n=2\). Applications are made to condition numbers of matrices, standard deviations and Schwarz’s inequality.

MSC:

26D15 Inequalities for sums, series and integrals
15A42 Inequalities involving eigenvalues and eigenvectors
26E60 Means
60E15 Inequalities; stochastic orderings
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