Some trace inequalities for exponential and logarithmic functions.

Consider a function $F(X,Y)$ of pairs of positive matrices with values in the positive matrices such that whenever $X$ and $Y$ commute $F(X,Y)= X^pY^q.$ Our first main result gives conditions on $F$ such that ${\rm Tr}[ X \log (F(Z,Y))] \leq {\rm Tr}[X(p\log X + q \log Y)]$ for all $X,Y,Z$ such that ${\rm Tr} Z = {\rm Tr} X$. (Note that $Z$ is absent from the right side of the inequality.) We give several examples of functions $F$ to which the theorem applies. Our theorem allows us to give simple proofs of the well known logarithmic inequalities of Hiai and Petz and several new generalizations of them which involve three variables $X,Y,Z$ instead of just $X,Y$ alone. The investigation of these logarithmic inequalities is closely connected with three quantum relative entropy functionals: The standard Umegaki quantum relative entropy $D(X||Y) = {\rm Tr} [X(\log X-\log Y])$, and two others, the Donald relative entropy $D_D(X||Y)$, and the Belavkin-Stasewski relative entropy $D_{BS}(X||Y)$. They are known to satisfy $D_D(X||Y) \leq D(X||Y)\leq D_{BS}(X||Y)$. We prove that the Donald relative entropy provides the sharp upper bound, independent of $Z$, on ${\rm Tr}[ X \log (F(Z,Y))]$ in a number of cases in which $(Z,Y)$ is homogeneous of degree $1$ in $Z$ and $-1$ in $Y$. We also investigate the Legendre transforms in $X$ of $D_D(X||Y)$ and $D_{BS}(X||Y)$, and show how our results for these lead to new refinements of the Golden-Thompson inequality.