In-betweenness: a geometric monotonicity property for operator means

We introduce the notions of in-betweenness and monotonicity with respect to a metric, for operator means. These notions can be seen as generalising their natural counterpart for scalar means, and as a relaxation of the notion of geodesity. We exhibit two classes of non-trivial means that are monotonic with respect to the Euclidean metric. We also show that all Kubo-Ando means are monotonic with respect to the trace metric, which is the natural metric for the geometric mean.Comment: 15 pages; a preliminary version has been presented at the June 2010 ILAS Conference in Pisa, Ital

A Singular Value Inequality for Heinz Means

We prove a matrix inequality for matrix monotone functions, and apply it to prove a singular value inequality for Heinz means recently conjectured by X. Zhan.Comment: 4 pages; Mistake in proof of Theorem 1 correcte

Trace inequalities for completely monotone functions and Bernstein functions

We prove a matrix trace inequality for completely monotone functions and for Bernstein functions. As special cases we obtain non-trivial trace inequalities for the power function x->x^q, which for certain values of q complement McCarthy's trace inequality and for others strenghten it.Comment: 16 page

Spectral radius of Hadamard product versus conventional product for non-negative matrices

We prove an inequality for the spectral radius of products of non-negative matrices conjectured by X. Zhan. We show that for all $n\times n$ non-negative matrices $A$ and $B$, $\rho(A\circ B)\le\rho((A\circ A)(B\circ B))^{1/2}\le\rho(AB)$, where $\circ$ represents the Hadamard product.Comment: 4.1 page