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Liouville's theorem states that a bounded entire function must be constant. A generalization of this is the Liouville theorem for subharmonic functions in the plane. I am looking for a proof of this assertion.

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This article gives a proof of the following version of Liouvilles theorem:

Liouville Theorem: If $U:\mathbb{R}^2\backslash {0} \rightarrow\mathbb{R}$ is a nonconstant harmonic function, then $$\liminf(M(r)/|\log r|)>0$$ either as $r\rightarrow 0$ or as $r\rightarrow \infty$, where for $r>0$ $M(r)=\max \{ U(x): \|x\|=r\}.$

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