The Open Mapping Theorem
The Open Mapping Theorem states:
Letbe a function analytic and non – constant on a regionand letbe an open subset ofThenis open. Furthermore, ifis a region, thenis also a region.
The non – constant condition is required because ifis constant thenis a singleton set which is certainly not open. Many functions do not map open sets to open sets. The functionmapsto the first quadrantwhich is not an open set.
The Open Mapping Theorem provides a quick way of proving that there are no analytic functions which map certain regions to certain other regions. There is no analytic function which mapsto
Letbe a non – constant entire function which mapstothen by the open mapping theorem,is open, butand sosince no open disc with centrelies in
Sinceno such functionexists.
It is easily proved that ifis analytic and non – constant on a regionthenis also a region, sinceis a connected open set. The Open Mapping Theorem shows thatis open, and sinceis analytic onis also connected sois a region.