TITLE: Is a function that maps circles to circles necessarily a Möbius transformation?
QUESTION [5 upvotes]: I'm introducing myself to Complex analysis and Möbius transformations and I read that Möbius transformations map circles and lines to circles and lines.
Are there any other functions that are not Möbius transformations but they can map circles to circles?
If I know that $f(z)$ maps a circle to another circle, can I assume that $f(z)$ is a Möbius transformation?

REPLY [2 votes]: I presume you talk about analytic maps.  But even then you may take products of Möbius transformations which also maps $S^1=\{|z|=1\}$  to itself (1-1). Such transformations are called:
Blaschke products 
If you do not require 1-1 then you also have maps like $z\mapsto z^p$ and if you  require analyticity only in a neighborhood of $S^1$ there are many more.
On the other hand a map that always maps any circle or line to a circle or a line is either a Möbius transformation (whence meromorphic) or a Möbius transformation composed with complex conjugation. Perhaps this is more what you are after... (and a proof is not that difficult)