# Thread: Entire bounded function must be constant.

1. ## Entire bounded function must be constant.

I can kind of see whats going on, but not well enough to construct a logical progression towards a proof. Any ideas?

2. ## This feels like the solution.

This feels right, but I tend to get things wrong with my ignorance. Verify?

3. That basic idea (taking the exponential) is exactly what is needed, but some of the details are a bit dubious. You can use inequalities for real numbers, but not for complex numbers. It would be better to say $\exp(f(z)) = e^{u+iv} = e^ue^{iv}$ and therefore $|\exp(f(z))| = e^u$ (since $e^u>0$ and $|e^{iv}|=1$). Then apply Liouville's theorem.

4. Originally Posted by Opalg
That basic idea (taking the exponential) is exactly what is needed, but some of the details are a bit dubious. You can use inequalities for real numbers, but not for complex numbers. It would be better to say $\exp(f(z)) = e^{u+iv} = e^ue^{iv}$ and therefore $|\exp(f(z))| = e^u$ (since $e^u>0$ and $|e^{iv}|=1$). Then apply Liouville's theorem.
Of course. This makes a lot of sense to me. Thank you.