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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 $\alpha$?

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, $c=\cdots$.

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, $c=\cdots$.

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 $\alpha \in (0,1]$, $\beta \in [1,2)$, 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 $\alpha$?

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, $c=\cdots$.

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 $\alpha$?

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, $c=\cdots$.

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 $\alpha \in (0,1]$, $\beta \in [1,2)$, 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 $\alpha$?

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 $\alpha \in (0,1]$, $\beta \in [1,2)$, 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 $\alpha$?

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, $c=\cdots$.