$f$ also has to satisfy several conditions:
- $f: [0; +\infty) \rightarrow [0; +\infty)$
- $f(x) \ge x$ and $f(x) \ge 1$
- $f$ is strictly increasing
These conditions may not be used (this is not a problem from a MO), and I only care about all the values of $f$ that satisfy the functional equation.
I had found a family of $f$: $f(x) = (x^a+1)^{a^{-1}}$, but that solution is bad (for my purposes) and would like to find more $f$-es.
You may introduce more restrictions for $f$ (like differentiable, continuous, etc.), since I want analytic $f$-es.