  We show that recent multivariate generalizations of the Araki-Lieb-Thirring
inequality and the Golden-Thompson inequality [Sutter, Berta, and Tomamichel,
Comm. Math. Phys. (2016)] for Schatten norms hold more generally for all
unitarily invariant norms and certain variations thereof. The main technical
contribution is a generalization of the concept of log-majorization which
allows us to treat majorization with regards to logarithmic integral averages
of vectors of singular values.
