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## Lambert $W_{-1}(x)$ as $x\rightarrow 0^-$: Asymptotic behavior

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?

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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.