  The Golden-Thompson trace inequality which states that $Tr\, e^{H+K} \leq
Tr\, e^H e^K$ has proved to be very useful in quantum statistical mechanics.
Golden used it to show that the classical free energy is less than the quantum
one. Here we make this G-T inequality more explicit by proving that for some
operators, notably the operators of interest in quantum mechanics, $H=\Delta$
or $H= -\sqrt{-\Delta +m}$ and $K=$ potential, $Tr\, e^{H+(1-u)K}e^{uK}$ is a
monotone increasing function of the parameter $u$ for $0\leq u \leq 1$. Our
proof utilizes an inequality of Ando, Hiai and Okubo (AHO): $Tr\,
X^sY^tX^{1-s}Y^{1-t} \leq Tr\, XY$ for positive operators X,Y and for
$\tfrac{1}{2} \leq s,\,t \leq 1 $ and $s+t \leq \tfrac{3}{2}$. The obvious
conjecture that this inequality should hold up to $s+t\leq 1$, was proved false
by Plevnik. We give a different proof of AHO and also give more counterexamples
in the $\tfrac{3}{2}, 1$ range. More importantly we show that the inequality
conjectured in AHO does indeed hold in this range if $X,Y$ have a certain
positivity property -- one which does hold for quantum mechanical operators,
thus enabling us to prove our G-T monotonicity theorem.
