Let f(x) be a real-differentiable function with $f′(x)>0,f′′(x)>0 $ and
$$ f(f(x)) = \exp(x) $$
for all real $x$.
Tommy1729 adds the optimization condition
$$ max \int_{- \infty}^{\infty} \frac{dx}{\pi (1 + x^2 + f ' (x)^2 )} $$
Is this a uniqueness condition ? How to prove or disprove it ?
If not how many solutions are there ?
If there are infinitely many , are they countable ? Are they simply connected ?
What can we say about the solutions ?
Is there a similar looking optimization condition that is a uniqueness condition ?