Asymptotic evaluation of a quantity

Question

Can we say that the following quantity (a recursion of logarithms):

$W_{-1}(x)=\ln \cfrac{-x}{-\ln \cfrac{-x}{-\ln \cfrac{-x}{...}}}$

is $\Theta(\ln x)$? i.e., asimptotically upper and lower bounded?

Answer 1

Your actual question seems to be about the asymptotics of $W_{-1}(x)$. I'm not sure how the expression for $W_{-1}(x)$ in your question is relevant.

Complete asymptotic expansions for the branches of $W$ were obtained in Corless et. al. "On the Lambert W Function" [PDF Link] and Jeffrey et. al. "Unwinding the branches of the Lambert W function" [HTML link].

In particular, taking only the largest term in the double series in equation (4.19) in the first paper,

\begin{align*} W_{-1}(z) &= \log(-x) - \log (-\log(-x)) + \Theta\!\left(\frac{\log(-\log(-x))}{\log(-x)}\right) \end{align*}

as $x \to 0^-$ with $x \in \mathbb R$.

Answer 2

At convergence, $$y=\ln\left(\frac{x}{y}\right),$$ or

$$x=ye^y$$ and $$y=W(x).$$