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Hi i need some hints and help with this problem.

Let $f\in\mathcal O(\mathbb C)$ and assume that $\Re f(z)\geq M$ for all $z\in\mathbb C$. Use Liouville´s theorem to prove that $f$ is constant function.

I am really stuck on this problem.

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Consider the function $g$ defined by $g(z)=\exp(-f(z))$. –  Mariano Suárez-Alvarez Sep 28 '11 at 19:53

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up vote 7 down vote accepted

Assuming $M>0$, we have that $\Re f(z)\geq M$ implies that $|f|\ge M$. In particular, $f$ is never zero. Then $1/f$ is entire and bounded and so constant by Liouville.

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@Davide: or you could consider the function $f+c$ for an appropriate real constant $c$. –  Mariano Suárez-Alvarez Sep 28 '11 at 20:18
    
@MarianoSuárez-Alvarez: yes, you're right. Hence we assume $M>0$ without loss of generality. –  Davide Giraudo Sep 28 '11 at 20:20
1  
More generally, this argument (after an appropriate shift) applies whenever there is a nonempty open subset of $\mathbb C$ that $f(z)$ is required to avoid. –  Robert Israel Sep 28 '11 at 23:35

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