A={$z\in\mathbb C:|z|>1$} and B={$z\in\mathbb C:|z|>0$}. Find an analytic surjection from A to B if it exists or prove why such a map does not exist.
I am able to prove that from B to A, no non constant analytic map exists because in all the three mutually exclusive and exhaustive cases of the origin being a removable singularity or a pole or an essential singularity, I get a contradiction. I am not able to use this result to conclude about analytic surjection from A to B. From A to B the inclusion is a non constant analytic function but I need a surjection. Please help.