On the exponential metric increasing property

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Nonpositive curvature: A geometrical approach to Hilbert–Schmidt operators (2007, background): “Bhatia [6] for matrices, and his proof can be translated almost verbatim to the context of operator algebras with a trace, see [2].”
A Riemannian Framework for Tensor Computing (2005, background): “It is implicitly hidden under very general theorems on symmetric spaces in many diﬀerential geometry textbooks [Kobayashi and Nomizu, 1969, Helgason, 1978, Gamkrelidze, 1991] and sometimes considered as a well known result as in [Bhatia, 2003].”
Metric convexity of symmetric cones (2007, unknown): “It h as recently been discovered by Bhatia [6] that the non-positive curvature property of th e convex cone of positive definite matrices holds for metrics inherited from symmetric g au e functions.”
Norm inequalities related to the matrix geometric mean (2012, background): “(We remark here that it was shown in [10] that the generalised Golden Thompson inequality follows from a generalised exponential metric increasing property.”