TITLE: The Koch snow flake, Holder exponents of conformal mappings
QUESTION [11 upvotes]: The Koch snow flake $K$ is a domain of $\mathbb{C}$, complex plane. Though, I do not state the precise definition, you can see the picture in wikipedia Koch snow flake.
The Koch snow flake $K$ is a quasidisk. Let $\mathbb{D}$ be an open unit disk and let $\phi:\mathbb{D} \to K$ be a conformal mapping. It is known that $\phi$ and the inverse map $\phi^{-1}$ are Holder continuous: there exist $\alpha \in (0,1]$, $\beta \in (0,1]$, and $L_1,L_2 \in (0,\infty)$ such that 
\begin{align*}
|\phi(z_1)-\phi(z_2)| &\le L_1 |z_1-z_2|^{\alpha},\quad z_1,z_2 \in \mathbb{D}, \\
|\phi^{-1}(w_1)-\phi^{-1}(w_2)| &\le L_2 |w_1-w_2|^{\beta},\quad w_1,w_2 \in K.
\end{align*}
My question
Is there a study for quantitative estimates on the Holder exponents $\alpha$ and $\beta$? 
I think that there is such study because the Koch snow flake is a famous fractal set.
ADD
By an argument by Benoît Kloeckner, $\alpha$ must be less than or equal to $\log3 / \log 4$. Is there a reasonable lower bound for $\alpha$ and $\beta$?
ADD2
Let $C$ be a closed Jordan curve. 
Lasley considers the following condition on $C$. 

Definition. Let $w_1$ and $w_3$ be points on $C$ and let $w_2$
   be on the arc of small diameter between $w_1$ and $w_3$. Then, $C$ is said to be a $c$-quasiconformal curve if there exist positive constants $c$ and $\delta$ such that
  \begin{align*}
\frac{|w_1-w_2|+|w_2-w_3|}{|w_1-w_3|} \le c
\end{align*}
  for any such $w_1,w_2,w_3$ with $|w_1-w_3| \le \delta$.

Lasley prove

Theorem. Suppose that $f$ maps $\mathbb{D}$ conformally onto the interior $\Omega$ of a $c$-quasiconformal curve $C$. Then, $f$ is
  $\alpha$-Holder continuous. Here,
\begin{align*} \alpha=\frac{2
 (\text{arcsin}(1/c))^2}{\pi^2 -\pi \text{arcsin}(1/c)}. 
\end{align*}

If $\Omega=K$, the Koch snowflake, $c=\cdots$.

REPLY [5 votes]: U.R Freiberg and M.R. Lancia, Energy Form on a Closed Fractal Curve (2004):
The Koch snow flake is the union of three Koch curves of Hausdorff dimension $D=\ln 4/\ln 3$ and Hölder exponent $\beta=\log 2/\log 3$.
See Proposition 2.2, attributed to a 1999 paper which I did not find online.
(For reference, Pietro Majer's argument for $\alpha=\log 3/\log 4$ is in this 2016 MO answer.)