Thread: Complex variables proof

This particular problem is giving me some trouble:

Suppose that f is an entire function and Re[f(z)] is less than or equal to c for all z. Show that f is constant.

Originally Posted by spoon737

This particular problem is giving me some trouble:

Suppose that f is an entire function and Re[f(z)] is less than or equal to c for all z. Show that f is constant.

This is a famous famous theorem.
In fact it gives a super short proof of the fundamental theorem of algebra. Here.

By the actual theorem is stated and proven. Here.