Kind of isolated singularity for analytic function between punctured disc and annulus

Question

If $S_1:=\{z: 0<\left| z\right|<R_1 \}$ and $S_2:=\{z: r<\left| z\right|<R_2 \}$, where $r, R_1, R_2 > 0$, and $\exists f:S_1\to S_2$ such that $f$ is analytic, then what kind of an isolated singularity is $0$ of $f$?

I'd appreciate some hints, as I'm not sure how to think about this question.

Answer 1

Hint: observe that $f$ is bounded on a punctured neighborhood of zero (namely $S_1$).