I am trying to show that
$\frac{1}{2\left(1-e^x\right)}-\frac{1}{x}W_{-1}\left[\frac{x}{2\left(1-e^x\right)}\exp\left(\frac{x}{2\left(1-e^x\right)}\right)\right]\geq 1,$
for $x>0$, where $W_{-1}$ is the lower branch of the Lambert function. Do you have an idea? Thanks!