Can one find a surjection $f:D\to{\bf C}$ such that $f$ is not a rational function?

Question

By considering a Mobius transformation, one can find a bijection between the open unit disc $D$ and the upper half plane $H$ and thus further give a surjection from $D$ to the whole complex plane ${\bf C}$. Can one find a surjection $f:D\to{\bf C}$ such that $f$ is not a rational function?

Answer 1

Yes. Since $\sin(z) : \mathbb C \to \mathbb C$ is surjective, the function $$f(z)= \sin(g(z))$$ where $g: D\to{\bf C}$ is your favorite (rational) surjection satisfies the requirements.

Answer 2

Consider $re^{it} \to \tan (\pi r/2)e^{it}.$