There are well known bounds for $W\_0$, the "principal" real-valued branch of the Lambert-W function. For example, $W\_0(x)$ lies between $\log x - \log\log x$ and $\log x - \frac{1}{2}\log \log x$, and associated big-O asymptotics of $W\_0$ as $x\rightarrow\infty$.


What about bounds for $W\_{-1}(x)$, the other real-valued branch of the Lambert-W function as $x\rightarrow 0^-$? The limit is well-known to be $-\infty$, but how fast does $W\_{-1}(x)$ approach the limit? Are $O$ asymptotics known in terms of more elementary functions such as powers and logs?



Lambert $W_{-1}(x)$ as $x\rightarrow 0^-$: Asymptotic behavior
First, if $x \in ]-1/e,0[$, one has $1 < -W\_{-1}(x) \leq -\frac{1}{x}$ (it is easy to prove that these inequalities are equivalent to $-1/e < x \leq \frac{\exp(1/x)}{x}$, and that is true here). Then $0 < \ln(-W\_{-1}(x)) \leq -\ln(-x)$.


Since $W\_{-1}(x)\exp(W\_{-1}(x)) = x$, the above inequalities yield $1 < -W\_{-1}(x) = -\ln(-x)+\ln(-W\_{-1}(x)) \leq -2\ln(-x)$ (of course, one cas replace 2 by another suitable constant, if I only look for an asymptotic behavior, as we shall see). Finally, this gives $0 < \ln(-W\_{-1}(x)) < 2\ln(-\ln(-x))$. Combine everything here, and you already have $W\_{-1}(x) = \ln(-x) + O(\ln(-\ln(-x)))$ as $x \to 0^-$. One can do better, by reinjecting this estimate in the identity $W\_{-1}(x) = \ln(-x) - \ln(-W\_{-1}(x))$, and one obstains $W\_{-1}(x) = \ln(-x) - \ln(-\ln(-x)) + O\left(\frac{\ln(-\ln(-x))}{\ln(-x)}\right)$.


You may find some interets in reading "Asymptotic methods in analysis" by De Bruijn, the methods used to write $W\_0$ as an infinite sum of powers involving $\ln$ can be used with $W\_{-1}$, I guess.


