Trying to find a concave function defined on the positive reals, satisfying some inequalities, I came up with the following relation
$f\left(x\right) = 1 - \left(1 - f\left(x+1\right)\right)^{\frac{x}{x+1}}$
where $x \geq 0$. The only progress I could make was figuring that $f(0) = 0$, and if I postulate some value for $f(1)$ I can in principle calculate $f(n)$, where $n$ is a positive integer. Any hints for further progress are appreciated.