Suppose we have two annuli. The only way to have conformal map between them is to have same radius ratio.
Now lets take two annuli A(1,2) and A(1,8). I want to prove that conformal map between two these annuli cant happen using contradiction. So suppose i apply iteratively reflection principle. And eventually will get to 0. Since its bounded its a removable singularity.
Now how i conclude contradiction from here? What condition is not satisfied that stops this conformal map?