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Let $U\subseteq \mathbb{C}$ be an open set and let $u(z)=-\log(\mathrm{dist}(z,\partial U))$.

I need to show that $u$ is subharmonic on $U$?

$\partial U$ it means the boundary of $U$.

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    sorry about this i need to know how can i prove that$u$is subharmonic on U2011-03-04

2 Answers 2

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Hint: For fixed $x\in\partial U$, the function $-\log(|z-x|)$ is harmonic on $U$.

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    You ought to update your question and show us what you've done so far. For instance, have you shown that $u_x(z):=-\log(|z-x|)$ is a harmonic function?2011-03-09
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Suppose $U$ is connected open. Fix a $z\in U.$ Since $U$ is open, there is an $s>0$ such that $B(z,s)\subset U.$ Now, $\frac{1}{|\partial B(z,s)|}\int_{\partial B(z,s)}dist(w,\partial U) \leq dist(w,\partial U).$ Now, use Jensen inequality to finish the proof. Above argument shows that your function, say f(w) which is defined in the interior of U is subharmonic. You should also discuss, for instance, the riemann integrability of $dist(w,\partial U)$ on the boundary set $\partial B(z,r)$ if you want a complete solution.

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    can you show me pls2011-03-08