Assume $f:[0,1]\to [0,1]$ is an orientation-preserving homeomorphism, and $f$ is absolutely continuous on $[0,1]$, $\log f'\in H^{1/2}([0,1])$, that is to say, $$\int_{0}^{1}\int_{0}^{1}\frac{|\log f'(x)-\log f'(y)|^2}{|x-y|^2}dx dy<\infty,$$ my question is if there exists $M>0$ independent of $u,v$ such that
$$\frac{|u-v|}{|f^{-1}(u)-f^{-1}(v)|}|(f^{-1}(u))'|<M \quad a.e.?(*)$$
Basically, I think the conclusion does not hold, and I can show $(*)$ is bounded but the bound is not uniform, the main problem is that I have no idea how to use the condition $\log f'\in H^{1/2}([0,1])$, any help will be appreciated.