Let $\Delta(a, r)$ be the open disk of radius $r$ centered at the point $a$ in the complex plane, and $\operatorname{Aut}(\Delta(0, 1))$ be the set of linear fractional transformations that preserves the open unit disk, i.e. transformations of the form $z\mapsto e^{i\theta}(z-a)/(1-\bar az)$, where $a\in\Delta(0, 1)$ and $\theta\in\mathbb R$.
I want to show $\operatorname{Aut}(\Delta(0, 1))$ is equicontinuous on every compact subset of $\Delta(0, 1)$. Does it suffice to show it is equicontinuous on $\overline{\Delta(0,r)}$ for any $r<1$? I think it surely suffices to show that it is continuous on every closed disk contained in the unit disk, but I'm unsure about the former.
I want to show $\operatorname{Aut}(\Delta(0, 1))$ is equicontinuous on $\overline{\Delta(b,r)}$ which is contained in the unit disk. To do this, I evaluated $|f(z)-f(w)|$ for an arbitrary $f$ in $\operatorname{Aut}(\Delta(0, 1))$ and got $|f(z)-f(w)| \le |z-w|/(|1-\bar az||1-\bar aw|)$. But I can't go further. How do you get an upper bound for this?