This is one of the past qualifying exam problems that I was working on.
I know that, when we let $z=x+iy$, ${|z|}^2=x^2+y^2$ is not harmonic. I do not know where to start to prove that there is no harmonic function that is positive everywhere.
Any help or ideas idea will be really appreciated.
Thank you in advance.
Best Answer
Let $f(z)$ be entire and positive. Consider
$$h(z) = e^{-f(z)}$$
If $\Re f(z) > 0$, we have $-\Re f(z) < 0$, so $h(z)$ is bounded and entire. What can you conclude about $h(z)$, and hence about $f(z)$?
To go into a little more detail, note that
$$|h(z)| = e^{-\Re f(z)} < e^{0} < 1$$