Sharp bounds for the principal branch of the Lambert W function?

Question

I'm looking for references for bounds on the principal $W_0$-branch of the Lambert W-function, specifically in the range $[ -\frac 1e, 0)$. I'm trying to work with the expression $W(-xe^{-x})$ with $x \in [1, \infty)$. This far I've found this paper: On certain inequalities involving the Lambert W function, but I would like something that is sharper than their theorem $3.5$ in the range $1 \leq x \leq 3$. As an added complication, I want to be able to symbolically write down the integral of an expression involving this approximation, so I'd like it to be as simple as possible. I realise I might be looking for unicorns, but does anyone know of any such bounds?

Answer 1

This is not a full answer.

I had to work a similar problem years ago and I used two different series expansions.

Built at $x=-\frac 1e$ $$W(x)=-1+p-\frac{p^2}{3}+\frac{11 p^3}{72}-\frac{43 p^4}{540}+\frac{769 p^5}{17280}-\frac{221 p^6}{8505}+\frac{680863 p^7}{43545600}-\frac{1963 p^8}{204120}+\frac{226287557 p^9}{37623398400}-\frac{5776369 p^{10}}{1515591000}+\frac{169709463197 p^{11}}{69528040243200}-\frac{1118511313 p^{12}}{709296588000}+O\left(p^{13}\right)$$ in which $p= \sqrt{(2(1+e x)}$

Built at $x=0$ $$W(x)=x-x^2+\frac{3 x^3}{2}-\frac{8 x^4}{3}+\frac{125 x^5}{24}-\frac{54 x^6}{5}+\frac{16807 x^7}{720}-\frac{16384 x^8}{315}+\frac{531441 x^9}{4480}-\frac{156250 x^{10}}{567}+\frac{2357947691 x^{11}}{3628800}-\frac{2985984 x^{12}}{1925}+O\left(x^{13}\right)$$ For sure, you can truncate or extend these series.

Numerical calculations show that the first one would be used for $-\frac1 {e} \leq x \leq -\frac1 {2e} $ and the second one for $-\frac1 {2e} \leq x \leq 0 $.

In practice, I transformed (for the same accuracy) these expansions into Padé approximants but it would not be of any use if you need to perform integrations.