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The Golden-Thompson trace inequality which states that $Tr\, e^{H+K} \leqTr\, e^H e^K$ has proved to be very useful in quantum statistical mechanics.Golden used it to show that the classical free energy is less than the quantumone. Here we make this G-T inequality more explicit by proving that for someoperators, 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 amonotone increasing function of the parameter $u$ for $0\leq u \leq 1$. Ourproof 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 obviousconjecture that this inequality should hold up to $s+t\leq 1$, was proved falseby Plevnik. We give a different proof of AHO and also give more counterexamplesin the $\tfrac{3}{2}, 1$ range. More importantly we show that the inequalityconjectured in AHO does indeed hold in this range if $X,Y$ have a certainpositivity property -- one which does hold for quantum mechanical operators,thus enabling us to prove our G-T monotonicity theorem.

A trace inequality of Ando, Hiai and Okubo and a monotonicity property of the Golden-Thompson inequality