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I am aware of Liouville’s theorem from complex analysis that says that any holomorphic function that is bounded and entire must be constant. I want to use it to prove that any positive harmonic function $u: \mathbb{R}^2 \to \mathbb{R}$ must be constant. I found the question below which seems relevant.

On this question a commenter writes,

If you're just looking for a proof of Liouville's theorem for harmonic functions, then just use the fact that over a simply-connected domain, every harmonic function is equal to the real part of a holomorphic function.

It is said that this only works for dimension two, which is fine since that is specifically what I want. But I’m not sure how to relate the harmonic function being positive to the holomorphic function being bounded.

Specifically, if I write

$$f=u+iv$$

for $f$ holomorphic and $u$ as above, I’m not sure how to get any conditions on $v$, e.g. whether it is positive, bounded, etc. The only thing I know is it is harmonic.

How can I proceed to go from Liouville’s theorem from complex analysis to showing $u$ is constant?

Robin
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  • See https://math.stackexchange.com/q/229312/42969 – Martin R May 22 '23 at 14:29
  • My holomorphic function $u$ is not necessarily bounded, just positive @MartinR – Robin May 22 '23 at 14:34
  • The title of that question was misleading. The question and the answers apply to holomorphic functions whose real part is bounded above (or bounded below). I have now edited the title to make that clear. – Martin R May 22 '23 at 14:37
  • @MartinR Ah, I see now. Yes, this is what I need. Thank you. – Robin May 22 '23 at 14:40

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