TITLE: A holomorphic bijection from the open unit disc to the complex plane
QUESTION [8 upvotes]: By Liouville's theorem, there is no non-constant holomorphic function from the complex plane to the unit disc. I wonder what the converse is like--surely there are holomorphic functions on the open unit disc into the complex plane, but are there any bijective ones?

REPLY [5 votes]: More generally: there is a holomorphic bijection from the open unit disk onto a region $U$ if and only if $U$ is simply connected and the complement of $U$ (in the Riemann sphere) has at least two points.