What is a simple way to prove that two annuli $A_1 = {z: r_1 < |z| < R_1}$ and $A_2 = {z: r_2 < |z| < R_2}$ are conformally equivalent if and only if $R_1/r_1 = R_2/r_2$, using standard results of complex analysis?
I'm not looking for a concise proof so much as one that uses elementary concepts.