f is an entire function, and it satisfies f(C)⊆{z∈C∣Imz>0}. Show that f is constant.
I want to take advantage of the Liouville's Theorem, but I just can't figure out the relationship between its image part with its module.
Answer
Consider g(z)=1f(z)+i.
Clearly it is entire, and |g(z)|=1|f(z)+i|≤11=1.
Thus, g(z) is constant. Applying Liouville's theorem, g(z) is constant, implying f(z) is constant.
NOTE: This argument can be used to show that f(C) is dense in C by using a proof by contradiction.
No comments:
Post a Comment