I have a question on the Schwartz-Christffel formula. 

The map is a  conformal map from the unit disk to polygons. Let me give you a specific example. In fact, 
\begin{align*}
\phi(z)=\int_{0}^z (1-z^n)^{-2/n}\,dz
\end{align*}
maps $\mathbb{D}$ onto the interior of a regular polygon with $n$ sides. 

We can know the modulus of continuity of general conformal maps. The following result is well known.

>  Let $\mathbb{D} \subset \mathbb{C}$ be the unit disk centered at the
> origin. A conformal map $f$ defined on $\mathbb{D}$ is $\alpha$-Hölder
> continuous ($\alpha \in (0,1]$) if and only if there exists $L \in
(0,\infty)$ such that \begin{align*} |f'(z)| \le
 L(1-\|z\|^2)^{\alpha-1},\quad z \in \mathbb{D}, \end{align*} where we
> denote by $\|\cdot\|$ the Euclidean metric on $\mathbb{C}$.

According to this result, $\phi$ defined above is $(1-2/n)$-Hölder continuous. However, I do not know the Hölder continuity of $\phi^{-1}$. I think that the index should be bigger than $1-2/n$. Can we show this? What is the specific and optimal index?