Is there a harmonic function in the whole plane that is positive everywhere?

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.

Solution 1:

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

$$|h(z)| = e^{-\Re f(z)} < e^{0} < 1$$

Solution 2:

Continuing,a non-constant harmonic function is the real part of a non-constant entire function.so the real part must be positive.Little Picard Theorem: If a function is entire and non-constant, then the set of values that f(z) assumes is either the whole complex plane or the plane minus a single point. so we get a contradiction