Some convexity and monotonicity results of trace functionals

In this paper, we prove the convexity of trace functionals $$(A,B,C)\mapsto \text{Tr}|B^{p}AC^{q}|^{s},$$ for parameters $(p,q,s)$ that are best possible. We also obtain the monotonicity under unital completely positive trace preserving maps of trace functionals of this type. As applications, we extend some results in \cite{HP12quasi,CFL16some} and resolve a conjecture in \cite{RZ14}. Other conjectures in \cite{RZ14} will also be discussed. We also show that some related trace functionals are not concave in general. Such concavity results were expected to hold in \cite{Chehade20} to derive equality conditions of data processing inequalities for $\alpha-z$ Renyi relative entropies.