Some operator and trace function convexity theorems

We consider trace functions (A,B)↦Tr[(Aq/2BpAq/2)s](A,B)↦Tr[(Aq/2BpAq/2)s] where A and B   are positive n×nn×n matrices and ask when these functions are convex or concave. We also consider operator convexity/concavity of Aq/2BpAq/2Aq/2BpAq/2 and convexity/concavity of the closely related trace functional Tr[Aq/2BpAq/2Cr]Tr[Aq/2BpAq/2Cr]. 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 Rényi entropy conjectures are proved for some cases.