Recall that the Jordan curve $J\subset \mathbb{C}$ is called asymptotically conformal if $$\text{$\max_{z\in J(a,b)}\frac{|a-z|+|z-b|}{|a-b|}\to 1$ as $a,b\in J, |a-b|\to 0$}$$ where $J(a,b)$ is the smaller arc of $J$ between $a$ and $b.$
The following is Exercise 11.2 in the book titled "Boundary behaviour of conformal maps" by Christian Pommerenke.
Let $f$ map $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$ conformlly onto the inner domain of the asymptotically conformal curve $J$. Show that $f$ is $\alpha$-Holder continuous for each $\alpha<1.$
I think it is a very interesting result. Unfortunately, I have no idea to prove it. I would appreciate any solution or comments.
