Let $f$ be an entire function and suppose there is a constant $M$, and an integer $n \geq 1 $ such that $|f(z)| \leq M|z|^n $ for large $|z|$. Show that $f$ is a polynomial of degree $ \leq n$.
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Okay, I edited the title – odnerpmocon Mar 30 '15 at 22:43
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Hints without spoilers: Liouville's theorem or Cauchy's estimate – bashfuloctopus Mar 30 '15 at 22:47
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Hint Let $g(z)=f(z)-f(0)-\frac{f'(0)}{1!}z-..-\frac{f^{(n-1)}(0)}{(n-1)!}z^n$.
Prove that $\frac{g(z)}{z^n}$ has a removable discontinuity at $0$, and its continuation is entire and bounded.
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1@BijanDatta Well, it is easy to prove that $g/z^n$ is differentiable at $z \neq 0$, and $g/z^n$ has a Taylor series at $z=0$, thus it is differentiable.. – N. S. Mar 07 '19 at 23:25