Interpolating the arithmetic–geometric mean inequality and its operator version
Copyright policy )Interpolating the arithmetic–geometric mean inequality and its operator version
The paper begins with a nice introduction to means and how inequalities between means often have a matricial unitarily-invariant norm version. While this happens in many known examples, \textit{F. Hiai} and \textit{H. Kosaki} [Indiana Univ. Math. J. 48, No. 3, 899--936 (1999; Zbl 0934.15023)] have shown a counterexample. In the paper under review, a ``more dramatic'' example is presented by considering a family of means that interpolate between the geometric and arithmetic mean, namely the Heron means \[ F_\alpha(a,b)=(1-\alpha)\sqrt{ab}+\alpha\,\frac{a+b}2,\;\;\;0\leq\alpha\leq1. \] The author shows that while \[ F_\alpha(a,b)\leq F_\beta(a,b) \] whenever \(\alpha,\beta\in[0,1]\) and \(\alpha\leq\beta\), these inequalities have a matricial unitarily-invariant norm version only when \(\beta\geq1/2\).
positive definite matrix, Numerical Analysis, operator inequalities, Algebra and Number Theory, Operator inequalities, inequalities for means, unitarily invariant norm, Unitarily invariant norm, geometric and arithmetic mean, Heron means, Positive definite matrix, Miscellaneous inequalities involving matrices, Norms of matrices, numerical range, applications of functional analysis to matrix theory, Discrete Mathematics and Combinatorics, Inequalities for means, Geometry and Topology, Means
positive definite matrix, Numerical Analysis, operator inequalities, Algebra and Number Theory, Operator inequalities, inequalities for means, unitarily invariant norm, Unitarily invariant norm, geometric and arithmetic mean, Heron means, Positive definite matrix, Miscellaneous inequalities involving matrices, Norms of matrices, numerical range, applications of functional analysis to matrix theory, Discrete Mathematics and Combinatorics, Inequalities for means, Geometry and Topology, Means
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