  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.
