# Solve the functional equation $f\left(x\right) = 1 - \left(1 - f\left(x+1\right)\right)^{\frac{x}{x+1}}$

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.

• The general solution is $f(x)=1-c(x)^x$ where $c:\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}$ is any function satisfying $c(x)=c(x+1)$ for all $x$. – Redundant Aunt May 7 at 13:27
• @Redundant Aunt Thanks for the answer. I now realise that a better formulation would have been in terms of function $g$ where $g(x) = 1-f(x)$. – dayandnightatom May 9 at 7:01
• @RedundantAunt post as an answer? – user574848 May 25 at 0:22