Bounded harmonic function is constant

Question

Can you please help me to prove that bounded harmonic function is constant?

Thanks a lot!

Answer 1

If $u$ is a harmonic function then there exists a conjugate function $v$ and an analytic function $f=u+iv$. It's easy to prove that $f$ is bounded too. Then you apply the Liouville's Theorem. $f$ is constant and then $u$ is constant too.

Answer 2

E. Nelson, "A Proof of Liouville's Theorem", Proc. Amer. Math. Soc. 12 (1961) 995

9 lines long. Not the shortest paper ever, but maximizes importance/length ...

http://www.jstor.org/stable/2034412

http://en.wikipedia.org/wiki/Harmonic_function#Liouville.27s_theorem

Edward Nelson's paper is freely and legally available here:

$\bullet\ $ pdf file,

$\bullet\ $ html page.