Define
 g(x) = (1+x) ln(1+x) - x.
One can check that g is strictly monotonically increasing for x>=0 by checking its derivative is ln(1+x). So g is invertible and its inverse is also strictly monotically increasing.


Is there an explicit closed form for its inverse?


With a page of calculations I can prove that
 (1/2) x/ln(1+sqrt(x)) <= g^{-1}(x) <= 2 x/ln(1+sqrt(x))
for all x>=0. Is this obvious? Can estimates like this be found in the literature?



Inverse of (1+x) ln(1+x) - x
Just to expand on Qiaochu's comment: let $W$ stand for the [Lambert W-function](http://en.wikipedia.org/wiki/Lambert_W_function), then if $g(x)=z$, we readily find that
$$
x=\exp\big(W((z-1)/e)+1\big)-1.
$$


