Can a Non-Surjective Smooth Harmonic Function on $\Bbb R^2$ Be Non-Constant?

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Here is this week's POTW:

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Prove that every smooth harmonic function from $\Bbb R^2$ to $\Bbb R$ that is not surjective must be constant.

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Alan receives honorable mention for providing a well thought-out, but incomplete, proof of the result. You can read my solution below.

Spoiler

Let $h : \Bbb R^2 \to \Bbb R$ be a smooth harmonic function that is not surjective. Let $f$ be an entire function whose real part is $h$. Since the image of $h$ misses at least one point $a\in \Bbb R$, the image of $f$ misses at least the line $\operatorname{Re}(z) = a$. By the Little Picard theorem, $f$ must be constant. Therefore, $h$ is constant.