Matrix Young Inequalities

Let p, q > 0 satisfy 1/p + 1/q = 1. We prove that for any pair A, B of n × n complex matrices there is a unitary matrix U, depending on A, B, such that$$U*\left| {AB*} \right|U \leqslant {\left| A \right|^p}/p + {\left| B \right|^q}/q.$$