Lambert W function I wan to show that, for $x \in [-\frac{1}{\exp(1)},0)$, $W_0(x) + W_{-1}(x) \leq -2$. Can any body help or give a suggestion? It seems trivial from the graph of functions but I need a rigorous proof. 
Thanks in advance,
Reza

I found a simpler solution.
Simple algebraic manipulation shows that the parabola $e^{-1}(\frac{1}{2} x^2 + x -\frac{1}{2})$ is below the function $x e^x$ for $x \in (−1, 0)$ and above it for $x < −1$. Therefore, any line parallel to the $x$-axis with the $y$-intercept in the interval $y = [−\frac{1}{e}, 0)$, intersects the parabola at equal or larger $x$ coordinates than those corresponding to intersections with $xe^x$. From the
other side, the $x$ coordinate of the intersection points of these lines with $x e^x $ gives $W_0(y)$ and $W_{−1}(y)$. Consequently, for any $y \in [−\frac{1}{e}, 0)$, the sum $W_0(y) + W_{−1}(y)$ is less than or equal to the sum of the roots of $e^{-1} (\frac{1}{2} x^2 + x -\frac{1}{2}) = y$ which is always equal to $−2$. This completes the proof.
