I'm interested to bound/approximate the the inverse of the regularized upper incomplete gamma function $Q^{-1}(a,z)$, where $Q(a,z) = \frac{\int_z^\infty t^{a-1} e^{-t} \mathrm{d} t}{\Gamma(a)} $.
I found in here functions.wolfram.com, that the asymptotic behavior $Q^{-1}(a,z)$ is
$$Q^{-1}(a,z) \propto W_{-1} \left( - \frac{z^{1/(a-1)} \Gamma(a)^{1/(a-1)}}{a-1} \right)$$ as $z \to 0$, where (I guess that) $W_{-1} (\cdot)$ is the Lambert W function.
Is there other (more insightful) approximation? If not, can you please show me how to obtain this (or a reference)?
Any clues and suggestions are much appreciated.