Let $u$ be a harmonic function on $\mathbb{R}^2\cong \mathbb{C}$ such that $Ref= u$ where $f$ is an entire function. If $|u(z)|\leq |z|^n$ for any $z\in\mathbb{C}$, then $f$ is a polynomial of degree no greater than $n$. How to prove this? I am quite confused. Thanks.
2026-03-30 12:38:46.1774874326
Real part of a holomorphic function is bounded by polynomial then the holomorphic function is a polynomial
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