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Let $f$ be entire in $\Bbb C$. If $Re(z)>0$ $\forall z \in \Bbb C$. Then prove that $f(z)$ is constant.

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If $Re(f(z))>M$ then taking $|\frac 1f|<\frac 1M$. Applying Liouville's theorem we are done. But if $Re(f)>0$ then can we conclude $|\frac 1f|<\infty$ & $\frac 1f$ is bounded so $f$ is constant. If it is so then comment otherwise give me some solution.

Ri-Li
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  • @Daniel Fischer: How do you show $g(z)$ is bounded? Taking $\frac 1f$? – Ri-Li Oct 20 '14 at 15:00
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    $w \mapsto \frac{w-1}{w+1}$ maps the right half-plane biholomorphically to the unit disk. It is easy to see that it maps the right half-plane into the unit disk, since $\lvert w-1\rvert < \lvert w+1\rvert$ if $\operatorname{Re} w > 0$. So $g$, the composition of that map with $f$ maps $\mathbb{C}$ into the unit disk, since $f$ maps it into the right half-plane. – Daniel Fischer Oct 20 '14 at 15:06

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