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May 28, 2018 at 19:06 comment added Fan Zheng @qingtang Alternatively, you can give yourself an "epsilon of room" and apply the maximum principle to $g-\epsilon\log|z-z_0|$ for some $g_0\notin\Omega$ and send $\epsilon\to0$.
Feb 19, 2016 at 22:35 comment added Kostya_I yep. If $\psi(z)$ is a conformal mapping from the upper half-plane to $\Omega$ that maps $\mathbb{R}$ to $\partial\Omega$, then $g(\psi(z))$ is a harmonic function in the upper half-plane which is bounded from above and vanishes on the real line. The only such function, up to scaling, is $-\mathfrak{Im}z$
Feb 19, 2016 at 18:06 comment added qingtang @Kostya_l Thanks. The domain is unbounded, the maximum principle might not be true in general, so can we really guarantee that $g(z)$ negative?
Feb 15, 2016 at 17:10 history edited Kostya_I CC BY-SA 3.0
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Feb 15, 2016 at 11:57 history edited Kostya_I CC BY-SA 3.0
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Feb 15, 2016 at 11:26 history answered Kostya_I CC BY-SA 3.0