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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?

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  • $\begingroup$ In arxiv.org/pdf/2004.01115.pdf the following bound is given for the range in question: $$ \sqrt {ex + 1} - 1 \le W_0 (x).$$ $\endgroup$
    – Gary
    Sep 13, 2020 at 11:22
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    $\begingroup$ @JCAA I need a function as a lower bound, not necessarily a constant. $\endgroup$
    – mahdi
    Sep 13, 2020 at 11:38

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$\require{begingroup} \begingroup$ $\def\e{\mathrm{e}}\def\W{\operatorname{W}}\def\Wp{\operatorname{W_0}}\def\Wm{\operatorname{W_{-1}}}\def\Catalan{\mathsf{Catalan}}$

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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  • $\begingroup$ Thanks! Do you know any other function as a lower bound which uses $log$ instead of square root? A function consisting of logarithms will be more helpful for me. $\endgroup$
    – mahdi
    Sep 14, 2020 at 10:30
  • $\begingroup$ @mahdi: Not at the moment. But, just in case, both $W_0(x)$ and $x$ (as well as $W_{-1}(x)$, not related to this question) can be transformed to parametric form, where $W_0(x(a))=\frac{a\ln a}{1-a}$ for $a\in[0,1],\ x(a)\in[-1/\mathrm e,0]$. $\endgroup$
    – g.kov
    Sep 14, 2020 at 10:47

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