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We suppose that $f$ is entire such that $|f(z)| \geq 1/(1+|z|)$ for all $ z \in \mathbb{C}$. Show that $f$ is constant.

My thoughts so far: since $|f(z)| \geq 1/(1+|z|)$, $f$ has no zeroes, so we can divide by $f$, giving $1/|f(z)| \leq 1 + |z|$. Setting $g(z) = 1/f(z)$, it now suffices to show that $g$ is constant. Using Cauchy Estimates we can show that $g'$ is bounded, hence constant by Liouville's Theorem. But this is the furthest I can get. Any hint from here would be much appreciated.

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What you have done already shows that $g(z)=az+b$. But $g(z)=1/f(z)$ can never be zero, so $a=0$.

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