Holomorphic function between $\{z\in \mathbb{C}: 1\leq |z|\leq 4\}$ and $\{z\in \mathbb{C}: 1\leq |z|\leq 2\}$

Question

Does there exist a holomorphic function $h$ that sends the set $\{z\in \mathbb{C}: 1\leq |z|\leq 4\}$ to the set $\{z\in \mathbb{C}: 1\leq |z|\leq 2\}$?

I tried proving it but I could not. Thanks for your help

Answer 1

It depends upon what other properties you want of the map? If you don't care about how the image is covered or what happens to boundaries then you may do it by abstract non-sense (applying the Riemann mapping theorem).

If, however, you want a genuine $d$-fold covering map ($h$ should be locally injective and each point in the image should have precisely $d$ preimages) then this is not possible, due to what is know as the modulus of an annulus. The modulus of an annulus is an invariant under bi-holomorphic maps. For a $d$-fold covering map the modulus is multiplied by $d$.

For an annulus $A_R=\{ 1 \leq |z| \leq R\}$ the modulus is $\log R$ and a $d$-fold covering map, like $z\mapsto z^d$ gives an image of modulus $d\log R$. One checks e.g. that $\log R^d = d \log R$.

So in particular, $A_2$ may be mapped to $A_4$ (by $z^2$, but you probably noticed this), whereas the converse is not possible.