Proving that an entire, bounded function is a polynomial

Asked Nov 03 '18 at 21:35

Active Nov 03 '18 at 21:35

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I am trying to prove the following : Let f be entire and assume that $$|f(z)| \leq M|z|^n$$ for large z, for a constant M, and for some integer. Show that f is a polynomial of degree $\leq$ n

Here are my thoughts on the problem so far: Since we have the inequality above we know that f must be bounded by $M|z*|^n$ where z* = argmax(f) and n large enough to satisfy the inequality. Given that f is bounded and entire I think I can use Liouville's theorem which says that f should be constant. I am currently not sure how to go from f being constant to f being a polynomial, however.

asked Nov 03 '18 at 21:35

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Note that your reasoning cannot be correct: $f(z) = z^n$ satisfies the given inequality, but is not constant. – Martin R Nov 03 '18 at 21:41
This is a moot point, following Martin R's comment, but: constants are polynomials. ($c = c + 0z + 0z^2$, for example, if that helps.) – Jair Taylor Nov 03 '18 at 21:45
Use Cauchy estimate – Fakemistake Nov 03 '18 at 22:19

Proving that an entire, bounded function is a polynomial

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