# An extension of the Golden Thompson inequality

For three symmetric positive-semidefinite matrices $A,B,C$ I am trying to figure out if the following inequality holds, at least in some cases:

$$tr(Ae^{B+C})≤tr(Ae^Be^C)$$ Note that if $A=I$ then this is the Golden Thompson inequality:

$$tr(e^{B+C})≤tr(e^Be^C)$$

I wonder if the first inequality is true or if adding certain assumptions (such as $A≼I$ i.e. $I−A$ is positive semidefinite) will suffice to prove it. I have tried searching for similar inequalities with no luck.

The difficulties with generalizations of the Golden-Thompson inequality to three matrices arise because the trace of a product of three positive symmetric matrices is in general not positive; unlike the trace of the product of two positive symmetric matrices, which is positive: ${\rm tr}\,e^B e^C={\rm tr}\,XX^t=\sum_{n,m}X_{nm}^2>0$, with $X=e^{B/2}e^{C/2}$ and $X^t$ the transpose of $X$.
So while ${\rm tr}\,Ae^{B+C}$ is a positive number, ${\rm tr}\,Ae^Be^C$ can be negative.