A math analysis problem.
I know that it is an old question, but I came across it and I have an ansatz. First note that $$ (f\circ f)' = (f'\circ f)\cdot f' > (f\circ f\circ f)\cdot (f\circ f) $$ giving $$ f\circ f\circ f < \dfrac{(f\circ f)'}{f\circ f} = \big{(} \ln(\vert f\circ f\vert)\big{)}'. $$ That means that the question comes down to proving that $\vert f\circ f\vert$ is monotonely decreasing. In particular, $f\circ f$ needs to be either positive non-increasing or negative non-decreasing.