Sign of a quantity involving the Lambert function

Question

Is the following quantity

$$\frac{W(\ln x)}{x}\left(1-\frac{1}{1+W(\ln x)}\right)$$

positive for $x \geq 1$ ?

Thank you very much.

Answer 1

If you simply the expression you'll get:

$$\frac{(W(\ln x))^2}{x(1+W(\ln x))}$$

So the question becomes is $1+W(\ln x)$ positive for $x\geq1$.

As $x\geq1$ then $\ln x\geq0$ and so $W(\ln x)$ has only one solution (the upper branch) which is non-negative for all non-negative inputs. So $1+W(\ln x)\geq1$ so the expression is positive in the given domain.