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.
[Math] Liouville’s theorem for subharmonic functions
complex-analysis
Best Answer
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 $r>0$ $M(r)=\max \{ U(x): \|x\|=r\}.$