Suppose $f$ is analytic in $\mathbb D$ and $|f(z)|\rightarrow 1$ as $|z|\rightarrow 1^-$. Show that the number of solutions of $f(z)=\alpha$ is the same for all $\alpha$ in $\mathbb D$.
I guess try to express $f$ in the Blaschke Product form, but in the step to show that $f$ can be directly expressed as Blaschke Product form, I got stuck. Anyone have an idea? Thanks
Hint: take a circular contour just inside the disk, and use the Argument Principle.