Knockaert, Luc Best upper bounds based on the arithmetic-geometric mean inequality. (English) Zbl 1032.26017 Arch. Inequal. Appl. 1, No. 1, 85-90 (2003). 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. Reviewer: Peter S.Bullen (Vancouver) Cited in 1 Document MSC: 26D15 Inequalities for sums, series and integrals 15A42 Inequalities involving eigenvalues and eigenvectors 26E60 Means 60E15 Inequalities; stochastic orderings Keywords:eigenvalues; arithmetic-geometric mean inequality; Schwarz inequality; spectral number PDFBibTeX XMLCite \textit{L. Knockaert}, Arch. Inequal. Appl. 1, No. 1, 85--90 (2003; Zbl 1032.26017)