Given $R>0 , M>0$ Let $f(z)$ be a entire function such as $|f(z)|\leq M|z|^m$ for evey z such as $|z|>R$. Show that $f(z)$ is a polynomial of degree m or lower.
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Do you know the Cauchy estimates? – Daniel Fischer Jun 10 '14 at 19:57
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thy, the hint said to see analytic part of $f(z)/z^m$ i think the Rouchè theorem solved this. – João Paulo Andrade Jun 10 '14 at 20:18
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This has been asked and answered many times, for example here – mrf Jun 10 '14 at 20:52
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Like Daniel Fischer pointed out, you can use the Cauchy estimate; since $f$ is entire, it converges to its Taylor series $$f(z_0)=\Sigma_{n=0}^{\infty}\frac {f^n(z_0)}{n!(z-z_0)^n}$$ at every $z_0$ in $\mathbb C$ . Use the Cauchy estimate on $|f^n(z_0)|$, together with your bound to show that $f^k(z_0)=0$ for all $n >k$ for some positive integer $k$. (you may need to translate by $z_0$ to be able to use $R$).
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