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 $W_{-1}(x) < 0$ near $0$, at least my sign is better than yours.