Finding the set of analytic functions whose image is a subset of a given set

Question

Let $A=${$z\in\mathbb{C}||z|=1$} and $B=${$z\in\mathbb{C}||z|<2$}. I want to find the the set of analytic functions such that $f(B)\subset A$. Is there a way to solve this? Hope someone could help me out. Thanks

EDIT: I also would like to look at the case $f(A)\subset B$

Answer 1

Hint: You should be able to see which constant functions satisfy the requirement. As for non-constant holomorphic functions, consider what $f(B)$ is topologically and whether it can be contained in $A$.