In this paper, we focus on variational representations of some matrix symmetric norm functions that are related to the quantum R\'{ e}nyi relative entropy. Concretely, we obtain variational representations of the function (A,B)\mapsto \normmm{ (B^{ q/2}K^*A^pKB^{ q/2})^s} for symmetric norms by using the H\"{ o}lder inequality and Young inequality. These variational expressions enable us to make the proofs of the convexity/concavity of the trace function (A,B)\mapsto \tr (B^{ q/2}K^*A^pKB^{ q/2})^s more clear.
Categories: math.FA cs.IT math.IT