Variational representations related to quantum Renyi relative entropies

In this paper, we focus on variational representations of some matrix symmetric norm functions that are related to the quantum Renyi relative entropy. Concretely, we obtain variational representations of the function (A, B) bar right arrow vertical bar vertical bar vertical bar (B-q/2 K* A(p)KB(q/2))(s )vertical bar vertical bar vertical bar for symmetric norms by using the Holder inequality and Young inequality. These variational expressions enable us to make the proofs of the convexity/con- cavity of the trace function (A, B) bar right arrow Tr(B-q/2 K*A(p)KB(q/2))(s) more clear and give a new extension. (C) 2020 Elsevier Inc. All rights reserved.