Let $f \in C^2(\mathbb{R}^2)$ be a non-constant function such that $f^4$ is bounded. Is $f$ harmonic?
If $f^4$ is bounded there exists $M$ such that $|f^4(x)| \leq M$. This means that $|f(x)| \leq M^{1/4}$, thus $f$ is bounded. Assume $f$ is harmonic then $f$ has to be constant by Liouville's theorem , contradiction $\Rightarrow$ $f$ is not harmonic
Question: Is my approach correct?