Let's say we have $f_1$ and $f_2$, both strictly increasing and strictly concave on $[0,+\infty)$. $f_1(0)=f_2(0)=0$ and the difference $f_1-f_2$ is strictly positive and strictly increasing. That is, $f_1(x)>f_2(x)$ for $x>0$ and $f^\prime_1(x)>f^\prime_2(x)$ for $x>0$.
Can we prove the following intuitive result:
There exist $\phi$, strictly positive, strictly increasing and strictly concave, such that $f_2=\phi(f_1)$. We would have $\phi'<1$.
Thanks a lot !