Suppose that an entire function $f(z)$ satisfies $\left|f(z)\right|\leq k\left|z\right|^n$ for sufficiently large $\left|z\right|$, where $n\in\mathbb{Z^+}$ and $k>0$ is constant. Show that $f$ is a polynomial of degree at most $n$.
Entire function bounded by a polynomial is a polynomial
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4Do you know Liouville's theorem? http://en.wikipedia.org/wiki/Liouville%27s_theorem_%28complex_analysis%29 – 2012-05-10
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0If $f$ is entire, then so are all of its derivatives. If an entire function is bounded, then it's constant. – 2015-11-17
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0No I meant the nth derivative is bounded in the plane. And I need to show that f is a polynomial of degree n. @GerryMyerson Thank you for the help. – 2015-11-17
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0@user, please take half a minute to work through the logical implications of my earlier comment. Everything you want is there. – 2015-11-17
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0I am sorry but I don't really see how the function being constant implies that f is a polynomial of degree n. @GerryMyerson – 2015-11-17
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0@user, did you take into account the first sentence of my earlier comment? – 2015-11-17
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0If what you mean is applied from the Liouville's Theorem(I googled it) I have no information of it @GerryMyerson Maybe that it is why I cannot really make any connection – 2015-11-17
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0@user, you can also get what you need from the answers posted to ron's question. – 2015-11-17
4 Answers
Since $f$ is entire, it is equal to a power series centered at zero with radius of convergence $\infty$, which must match its Taylor series there.
$$f(z)=\sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!}z^n$$
Since $|f(z)|\leq k|z|^m$, Cauchy's estimate gives
$$|f^{(n)}(0)|\leq \frac{n!k|z|^m}{R^n}$$ for all $|z|=R$. For $n>m$, letting $R\rightarrow\infty$, we see that $|f^{(n)}|=0$. It follows that $f$ is a polynomial of degree $\leq m$.
Hints:
- We have by Cauchy's integral formula that $$|f^{(d)}(0)|=\frac{d!}{2\pi R}\left|\int_{C(0,R)}\frac{f(z)}{z^{d+1}}dz\right|.$$
- What about $f^{(d)}(0)$ if $d\geq n+1$?
- Use the fact that $f$ is analytic at $0$ to get that $f(z)=\sum_{j=0}^n\frac{f^{(j)}(0)}{j!}z^j$ in a neighborhood of $0$.
- Show that the last formula is in fact true for all $z\in\Bbb C$.
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0shouldn't you have $\frac{d!}{2 \pi i}$? – 2018-05-19
A really dirty way to do this:
Theorem 1: Jensen's Formula Corollary
Suppose $f$ has order of growth $\rho$. Then there is a $C$ that for large enough $R$, $n(R) \le C R^{\rho} $ where $n(R)$ is the number of zeros whose magnitude is less than $R$.
Theorem 2: Hadamard Factorization Theorem
Suppose $f$ has order of growth $k \le \rho \lt k+1$ where $k$ is an integer. Then $f(z)$ can be written $z^m e^{g(z)} \prod_n E_k(z/a_n)$ where $E_k$ is the kth canonical Weirerstrass factor and $a_n$ is the nth zero of $f$ and $g(z)$ is a polynomial of degree $k$.
Then note that by assumption $f$ has zero order of growth. Theorem 1 it follows that $f$ has finitely many zeros. From Theorem 2 it follows that $f$ is a polynomial. Then we need to put in a tiny bit of work to show that the degree of this polynomial is the one we need. (Just argue about $|f(z)/z^n|$ as $z$ grows large)
Look at closed discs centered at the origin, use maximum modulus principle to show that the function obtains its maximum value on the boundary, show that if you take a larger disc, you obtain a higher value, and thus use Liouville's Theorem to get that $\lim_{|z| \to \infty} |f(z)| = \infty$. Then show that such a function is a polynomial.