$\require{begingroup} \begingroup$ $\def\e{\mathrm{e}}\def\W{\operatorname{W}}\def\Wp{\operatorname{W_0}}\def\Wm{\operatorname{W_{-1}}}$
Let \begin{align} f_0(x)&=-\Wp(-\tfrac1x\,\exp(-\tfrac1x)) \tag{1}\label{1} ,\\ f_1(x)&=-\frac{1}{\Wm(-x\,\exp(-x))} \tag{2}\label{2} ,\quad x\in(0,1) . \end{align}
Then \begin{align} f_1(f_0(x))&=x \tag{3}\label{3} ,\\ f_0(f_1(x))&=x \tag{4}\label{4} . \end{align}
Note that for $x\in(0,1)$
\begin{align} \Wp(-\tfrac1x\,\exp(-\tfrac1x)) \ne -\tfrac1x \tag{5}\label{5} ,\\ \Wm(-x\,\exp(-x)) \ne -x \tag{6}\label{6} . \end{align}
Here is a graph:
The area under the curve $f_1(x)$ is
\begin{align} \int_0^1 f_1(x)\,dx &=\gamma \approx 0.5772156649 , \end{align}
hence the area bounded by $f_0(x)$ and $f_1(x)$ is $2\gamma-1$.
Question: is there known references where these mutually inverse functions were used?
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