Good approximation of the Lambert function for $(-\tfrac{1}{e},0)$

Question

I am looking for an $\epsilon > 0$:

$$\epsilon < \frac{1}{a}\left(1+R\,W_0(\beta)\right)$$

where

$$a = 1 + \frac{kv}{g}\,\sin{\alpha}, \hspace{20pt} R = \frac{v}{k}\,\cos{\alpha}, \hspace{20pt} \beta = -a \cdot e^{-a} \hspace{20pt} k,v,g \in \mathbb{R}_+, \hspace{20pt} \alpha \in (0,\pi/2)\,.$$

It therefore applies

$$R \in [0,\infty), \hspace{20pt} a \in [1,\infty), \hspace{20pt} \beta \in (-\tfrac{1}{e},0)\,.$$

I would like to estimate the Lambert function with a function $f(\beta) \leq W(\beta)$, but such that $\epsilon > 0$. This turns out to be very difficult. Do you have any ideas?

Answer 1

A simple approximation of Lambert function in this range is $$f(\beta)=\frac{2 \sqrt{2(1+ e \beta )}-3}{\sqrt{2 (1+ e\beta )}+3}$$ and for all this range $f(\beta)\leq W(\beta)$.

If you want better (but more complex approximations), let me know.