The metric increasing property of the exponential map is known to be
equivalent to the fact that the set of positive definite matrices is a
Riemannian manifold of nonpositive curvature. We show that this
property is an easy consequence of the logarithmic-geometric mean
inequality for positive numbers. Operator versions of this inequality
lead to a generalisation of the exponential metric increasing property
to all Schatten-von Neumann norms.