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.
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I found this http://math.stackexchange.com/questions/262355/polynomial-bounded-real-part-of-an-entire-function?rq=1 – Seth Apr 13 '14 at 15:37
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You can also argue with the Poisson integral, like here. – Daniel Fischer Apr 13 '14 at 15:46