Some Operator and Trace Function Convexity Theorems

We consider convex trace functions $\Phi_{p,q,s} = Trace[ (A^{q/2}B^p A^{q/2})^s]$ where $A$ and $B$ are positive $n\times n$ 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 $Trace[ A^{q/2}B^p A^{q/2} C^r]$. For concavity, these questions are completely settled, thereby settling cases left open by Hiai, while the convexity questions are settled in many cases. As a consequence, the Audenaert-Datta R\'enyi entropy conjectures are proved for some cases.