Analytic onto maps from D to D

Question

We just characterized using the Schwarz Lemma the conformal self maps of the open unit disk. I am now trying to characterize the holomorphic onto maps from $\mathbb{D}$ onto $\mathbb{D}$. As a natural starting point, I tried constructing various surjective maps that failed to be one-to-one, but am having trouble finding any! Can someone give me a push in the right direction?

Answer 1

Such functions ($\,\,f:D\to D$ holomorphic and onto, but not 1-1) are:

1. $f_n(z)=\lambda z^n$, $\lvert \lambda\rvert=1$.

2. $g_{n,a}(z)=\lambda\left(\dfrac{z-a}{1-\overline{a}z}\right)^{\!n},\,$ $\lvert a\rvert<1,\,$ $\lvert \lambda\rvert=1$.