If I have an equation of the form $a = b \cdot e^b$ then the Lambert W function tells me that $W(a) = b$.
But what if I instead have $a \geq b \cdot e^b$ ? Do I then have $W(a) \leq b$ or $W(a) \geq b$, or does it depend on the values of $a$ and $b$?
The nature of my problem makes me believe that $W(a) \leq b$ is the right answer, but I need someone to convince me that this is the case and if possible explain why. I can't seem to find anything online. If it matters, I'm working with the -1 branch of the Lambert W function.
(The actual problem can be found in this question that I posted earlier, but I didn't find it to be of importance to this question)