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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,where $B$ and $C$ are any $n$-by-$n$ positive definite matrices, and $A$ is any$n$-by-$n$ matrix. We also obtain the monotonicity versions of tracefunctionals of this type. As applications, we extend some results in\cite{HP12quasi,CFL16some} and resolve a conjecture in \cite{RZ14} in thematrix setting. Other conjectures in \cite{RZ14} will also be discussed. Wealso show that some related trace functionals are not concave in general. Suchconcavity results were expected to hold in different problems.

Some convexity and monotonicity results of trace functionals