Let $A_{R,r}=\{z\in\mathbb C: r\lt|z|<R\}$. Prove that there can not be a function $q \in O(A_{R,r})$ such that $q^2(z)=z$
$z=a\times e^{\phi i}, \; r\lt a \lt R, \phi \in [0,2\pi]$
and the square root would lool like this.
$q(z)=\sqrt{a}\times e^{\phi i/2}$
I don't understand why this can not be a square root for every $z \in A_{R,r}$.