I encountered the following 2-part problem on a practice exam:
(a) Show that if $f:\Bbb C\to\Bbb C$ is entire and the real part of $f$ is always positive, then $f$ is constant. (b) Show that if $u:\Bbb R^2\to\Bbb R$ is a harmonic function with $u(x,y)>0$ for all $x,y\in\Bbb R$, then $u$ is constant.
Now, (a) was fairly simple. Putting $f=u+iv$, I took $g(z)=e^{-iz}$, so $|g(f(z))|=e^{-u(x,y)}
For (b), I wasn't certain what to do. According to the wikipedia article on harmonic conjugates, if the domain of a harmonic function is simply connected, then it admits a harmonic conjugate, and so (b) follows from (a), since the plane is of course simply connected. I had never seen this result before, so (obviously) didn't think to use it.
My question is this: Aside from proving that $u$ has a harmonic conjugate, I wonder if there are other ways that we can approach a proof of (b). My experience with harmonic analysis has been almost completely in the context of analytic functions. Any ideas?