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I saw the following two theorems:

1) Let $f$ be an entire function such that $|f(z)|\leq k|z|^n$ for some $k>0$ and large $z$. Then $f$ is a polynomial of degree at most $n$.

2) Let $f$ be an entire function such that $|f(z)|\leq a+b|z|^n$ for some $b>0$, $a\geq 0$, and large $z$. Then $f$ is a polynomial of degree at most $n$.

Question: Can we prove (2) from (1)?

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  • $\begingroup$ I'm not sure if you can prove it from 1, but very similar logic applies to both. $\endgroup$ – Cameron Williams Feb 21 '15 at 5:27
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For $|z|>1$ you have $$a+b|z|^n < (a+b)|z|^n$$ so the answer is yes.

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Suppose $|f(z)| \le a + b|z|$ for $R \le |z|$. Then, for $R \le |z|$, $$ |f(z)| \le \frac{a}{R}R+b|z| \le \frac{a}{R}|z|+b|z| \le \left(\frac{a}{R}+b\right)|z|. $$

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