Big Theta of this modification of the secondary branch of the Lambert W function I am looking to find the big-$\Theta$ of $-W_{-1}(-\frac{a}{n})$ in terms of elementary functions where $a$ is a constant. Looking around and I find that this should be  $O(\log(n))$ and with maxima I found that the ratio seems to decrease towards $1$ as you increase $n$ implying that it is also $\Omega(\log(n))$

I get this from the fact that $W_{-1}(x) = -\ln(-x) - \ln(-(\ln(-x) - \ln(-(\ln(-x)-\cdots))))$

Is there any proof that this is correct or not?
Maple has
$$
W_{-1}(x)=
\ln  \left( -x \right) -\ln  \left( -\ln  \left( -x \right)  \right) +
{\frac {\ln  \left( -\ln  \left( -x \right)  \right) }{\ln  \left( -x
 \right) }}+{\frac {-\ln  \left( -\ln  \left( -x \right)  \right) +1/2
\, \left( \ln  \left( -\ln  \left( -x \right)  \right)  \right) ^{2}}{
 \left( \ln  \left( -x \right)  \right) ^{2}}}+O \left( -{\frac {
 \left( \ln  \left( -\ln  \left( -x \right)  \right)  \right) ^{3}}{
 \left( \ln  \left( -x \right)  \right) ^{3}}} \right) 
$$
as $x \to 0-$.  Since $W_{-1}(x) &lt; 0$ near $0$, at least my sign is better than yours.
Maple has
$$
W_{-1}(x)=
\ln  \left( -x \right) -\ln  \left( -\ln  \left( -x \right)  \right) +
{\frac {\ln  \left( -\ln  \left( -x \right)  \right) }{\ln  \left( -x
 \right) }}+{\frac {-\ln  \left( -\ln  \left( -x \right)  \right) +1/2
\, \left( \ln  \left( -\ln  \left( -x \right)  \right)  \right) ^{2}}{
 \left( \ln  \left( -x \right)  \right) ^{2}}}+O \left( -{\frac {
 \left( \ln  \left( -\ln  \left( -x \right)  \right)  \right) ^{3}}{
 \left( \ln  \left( -x \right)  \right) ^{3}}} \right) 
$$
as $x \to 0-$.  Since $W_{-1}(x) &lt; 0$ near $0$, at least my sign is better than yours.