$1.$ Let $0<r<R$. Show that the open balls $B(z,r)$ and $B(z,R)$ are analytically isomorphic.
$2$. Show that the annuli $A(0;r_1,R_1)$ and $A(0; r_2, R_2)$ are analytically isomorphic iff $\frac {r_1}{R_1} = \frac {r_2}{R_2} $
I know that an analytic isomorphism is a bijective holomorphic map. How should I construct the maps in both the cases?
In both cases, the biholomorphic map can be given by multiplication by a constant.
Notice that for question (2) you also need to prove that in the case $\frac{r_1}{R_1}\neq\frac{r_2}{R_2}$ the annuli are not analytically isomorphic.