I need to prove without using Picard's Little Theorem the following statement:
Let $f(z)$ an entire function such that $f(z) \notin \mathbb R$ for every $z \in \mathbb C$. Prove that $f$ is constant.
Do you have a way to do it?
Thanks
I need to prove without using Picard's Little Theorem the following statement:
Let $f(z)$ an entire function such that $f(z) \notin \mathbb R$ for every $z \in \mathbb C$. Prove that $f$ is constant.
Do you have a way to do it?
Thanks
Hint : The hypothesis implies that $Im f(z) \neq 0$ for all $z \in \mathbb{C}$, so $\mathbb{C}=\{Im f(z)>0\} \cup \{Im f(z) < 0\}.$
Use the fact that $\mathbb{C}$ is connected to deduce that either $Im f(z)<0$ for all $z$ for $Im f(z)>0$ or all $z$. Then, apply Liouville's theorem to either $g(z):=e^{if(z)}$ or $g(z):=e^{-if(z)}$.