Sandwiched (quantum) $α$-Rényi divergence has been recently defined in the independent works of Wilde et al. (arXiv:1306.1586) and Müller-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 $α$-Rényi divergence satisfies the data processing inequality for all values of $α> 1$. Moreover we prove that $α$-Holevo information, a variant of Holevo information defined in terms of sandwiched $α$-Rényi divergence, is super-additive. Our results are based on Hölder's inequality, the Riesz-Thorin theorem and ideas from the theory of complex interpolation. We also employ Sion's minimax theorem.