  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 trace
functionals of this type. As applications, we extend some results in
\cite{HP12quasi,CFL16some} and resolve a conjecture in \cite{RZ14} in the
matrix setting. 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 different problems.
