I want to find a lower bound for the branch $W_0(x)$ of Lambert $W$ function, for real values in range $-\frac{1}{e} \leq x \leq 0$. It is apparent that $-1$ is a lower bound for this function in the aforementioned range, but I need a slightly tighter lower bound.
Can anybody offer a better lower bound for this function using only elementary functions?
Best Answer
$\require{begingroup} \begingroup$ $\def\e{\mathrm{e}}\def\W{\operatorname{W}}\def\Wp{\operatorname{W_0}}\def\Wm{\operatorname{W_{-1}}}\def\Catalan{\mathsf{Catalan}}$
It looks that
\begin{align} f_1(x)&=\frac{1-\sqrt{1-(\e x)^2}}{\e x} \tag{1}\label{1} \end{align}
is slightly better lower bound for $\Wp(x)$ on $x\in[-\tfrac1\e,0]$, than \begin{align} f_2(x)&=\sqrt{\e x+1}-1 \tag{2}\label{2} \end{align}
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