Let $D$ be the unit disk and $f \in \text {Aut} (D).$ Then can we always say that $f(\partial D) \subseteq \partial D\ $?
What I know is that if $f \in \text {Aut} (D)$ with $f(a) = 0$ for some $a \in D$ then $f = c\ \varphi_a,$ where $c \in S^1 = \partial D$ and $\varphi_a : \mathbb C_{\infty} \longrightarrow \mathbb C_{\infty}$ is the Möbius transformation given by $$\varphi_a (z) = \frac {z - a} {1 - \overline {a} z}$$ Now let $z \in \partial D.$ Then $|z| = |\overline {z}| = 1.$ So we have $$\begin{align*} |\varphi_a (z)| & = \left | \frac {z - a} {1 - \overline {a} z} \right | \\ & = \left |\frac {\overline {z} (z - a)} {\overline {z} - \overline {a}} \right | \\ & = |\overline {z}| \left |\frac {z - a} {\overline {z - a}} \right | \\ & = 1. \end{align*}$$ This shows that $\varphi_a (\partial D) \subseteq \partial D.$ Hence $f(\partial D) \subseteq \partial D,$ since $f = c\ \varphi_a$ with $|c| = 1.$
Is my reasoning correct at all? Could anyone please check it?
Thanks for your time.