Abstract
We consider trace functions (A, B) -> Tr[(A(q/2) B-P A(q/2))(s)] where A and B are positive n x n matrices and ask when these functions are convex or concave. We also consider operator convexity/concavity of A(q/2) B-P A(q/2) and convexity/concavity of the closely related trace functional Tr[A(q/2)B(P)A(q/2)C(r)]. The concavity questions are completely resolved, thereby settling cases left open by Hiai; the convexity questions are settled in many cases. As a consequence, the Audenaert Datta Renyi entropy conjectures are proved for some cases. (C) 2015 Elsevier Inc. All rights reserved.