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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?


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Do you know about the maximum principle for the real and imaginary parts of a harmonic function? Or Liouville's theorem and the Cauchy Riemann equations? – Jeff Dec 6 '11 at 15:32

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 $g(z):=e^{-if(z)}$.

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Or $g(z):=1/(i+f)$ or $g(z):=1/(f-i)$. – Jonas Meyer Dec 6 '11 at 15:56
Yes indeed, good idea. Thank you for the comment. – Kalim Dec 6 '11 at 16:57

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