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I have the following homework question:

Let $u$ be a non-constant real-valued harmonic function in $\mathbb{C}$. Prove that $u^{-1}(c)$ is unbounded for every real number $c$

There is a hint that say we may use the result of the following exercise:

Prove that a positive harmonic function in $\mathbb{C}$ is non-constant

This exercise is taken from

Donald Sarason Notes on Complex Function Theory (page 89)

What I tried:

I don't really know how to start, but a reasonable beginning can be:

Assume by negation that there is $c\in\mathbb{R}$ s.t $$u^{-1}(c)\subseteq D(0,M)$$ where $$D(0,M)=\{z\in\mathbb{C}:\,|z|<M\}$$

We know that there is a harmonic conjugate $v$ s.t $$f=u+iv\in H(D(0,M))$$

I don't know how to move forward, or how to use the hint (since in this question $u$ need not be a positive function).

Can someone please direct me in the right direction ?

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  • $\begingroup$ Translate to get the case where $u^{-1}(0)$ is bounded. Show that $u$ has a fixed sign outside a certain radius. In case that sign is $+$, add a constant to get a positive harmonic function. $\endgroup$ – GEdgar May 29 '13 at 14:55
  • $\begingroup$ I don't understand the exercise you say is a hint. Why isn't the function $u(z)=1$ a counterexample? $\endgroup$ – Ted Shifrin May 29 '13 at 14:58
  • $\begingroup$ It's difficult to know what you know. Do you know Green's formulas and their consequences for harmonic functions? Do you know that if you had a compact level set, you could apply the maximum/minimum principle? $\endgroup$ – Ted Shifrin May 29 '13 at 15:16
  • $\begingroup$ @TedShifrin - that was a typo, sorry. I corrected it to mean constant instead of non-constant. I don't think I know "Green's formulas and their consequences for harmonic functions" but I do know the other things you mentioned $\endgroup$ – Belgi May 30 '13 at 1:26
  • $\begingroup$ @GEdgar - thanks for the hint! $\endgroup$ – Belgi May 30 '13 at 1:26
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I think you mean a positive harmonic function is constant. See Does there exist a harmonic function in the whold plane that is postive everywhere?

Also, you should be able to use the fact that $u^{-1}(c)$ is bounded to conclude that, by continuity, everything falls on one side of $c$. This can be manipulated into giving you a positive harmonic function.

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