  Sandwiched (quantum) $\alpha$-R\'enyi divergence has been recently defined in
the independent works of Wilde et al. (arXiv:1306.1586) and M\"uller-Lennert et
al (arXiv:1306.3142v1). This new quantum divergence has already found
applications in quantum information theory. Here we further investigate
properties of this new quantum divergence. In particular we show that
sandwiched $\alpha$-R\'enyi divergence satisfies the data processing inequality
for all values of $\alpha> 1$. Moreover we prove that $\alpha$-Holevo
information, a variant of Holevo information defined in terms of sandwiched
$\alpha$-R\'enyi divergence, is super-additive. Our results are based on
H\"older's inequality, the Riesz-Thorin theorem and ideas from the theory of
complex interpolation. We also employ Sion's minimax theorem.
