# Inverse of the Schwartz-Christoffel map and the continuity

I have a question on the Schwartz-Christoffel 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-u^n)^{-2/n}\,du \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. That is, there exists $$C>0$$ such that $$\begin{equation} \| \phi(x)-\phi(y)\| \le C\|x-y\|^{1-2/n} \end{equation}$$ for any $$x,y \in \mathbb{D}$$. 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?

The inverse map satisfies $$|\phi^{-1}(z)-\phi^{-1}(a)|\leq c|z-a|^{n/(n-2)}$$ at any vertex $$a$$ (reciprocal to the exponent of the direct map). Since this exponent is $$>1$$ the inverse is just Lipschitz everywhere. You do not need any general theorems for this. At the vertex the inverse map behaves like a power. At all other points it is smooth (analytic).
• Thank you for your comment. If the inverse map $\phi^{-1}$ is $n/(n-2)$--Hölder continuous, it should be a constant function. Am I misunderstanding something? Feb 11, 2020 at 14:19
• Do you mean that $\phi^{-1}$ is a Lipschitz continuous function with respect to $\|\cdot\|$? Feb 11, 2020 at 16:51