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Let $f$ be an entire function such that $ |f(z)| \to \infty$ as $|z| \to \infty$. Prove that $f$ is a polynomial.

user61496
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2 Answers2

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Consider the holomorphic function on $\mathbb{C}^*$ $$ g(z)=f\left(\frac{1}{z}\right). $$ So $$ \lim_{z\rightarrow 0}\;|g(z)|=+\infty. $$ Obviously, $0$ is not a removable singularity.

Now by the Casorati-Weierstrass theorem, $0$ is not an essential singularity either. Indeed, take $r>0$ such that $|g(z)|\geq 1$ for all $0<|z|<r$. Then $g(z\;;\;0<|z|<r\})$ is certainly not dense in $\mathbb{C}$.

Edit: As pointed out by @mrf, it is actually easy to see that this implies $0$ is a pole for $g$. It suffices to use Riemann's theorem on removable singularities adequately. Indeed $h(z)=1/g(z)$ is holomorphic and bounded in some $\{z\;;\;0<|z|<r\}$ with limit $0$ at $0$, so $0$ is a removable singularity of $h$ and a zero of the holomorphic extension of $h$. Factor $h(z)=z^nk(z)$ with $k$ holomorphic and $k(0)\neq 0$. The result follows. And this is indeed more elementary than Casorati-Weierstrass. So keep the Casorati-Weierstrass argument for weaker assumptions such as the ones I mention below.

So $0$ must be a pole. Hence the Laurent series of $g$ at $0$ is of the form $$ g(z)=\frac{a_n}{z^n}+\ldots+\frac{a_1}{z}+a_0. $$ Note that there are no positive powers of $z$ because this comes from the power series of $f$.

The result follows clearly by going back to $f$.

Edit: Note the same proof works if we simply assume that $f$ is entire and there exist $\alpha>0$ and $M>0$ such that $$ |f(z)|\geq \alpha\qquad \forall |z|\geq M. $$

Of course, there could be even weaker assumptions like there exists $M>0$ such that $f(\{z\;;\; |z|>M\})$ is not dense in $\mathbb{C}$ and the proof would still work.

Julien
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  • This is a good answer +1, but is it possible to do with more elementary tools than Casorati-Weierstrass? –  Feb 23 '13 at 13:02
  • @user58512 Thanks. Maybe, since the assumption is much stronger than what is really needed to apply Casorati-Weierstrass. Let's wait and see. – Julien Feb 23 '13 at 13:17
  • Since $|g|\to \infty$ as $z\to 0$, we see that $0$ is a pole directly (without Weierstrass-Casorati). – mrf Feb 23 '13 at 13:22
  • @Seirios Thanks for the edit, you made me understand that it was possible to do this. After six months...it was high time. – Julien Feb 23 '13 at 13:23
  • Often this property is used as the very definition of pole, but even with other plausible definitions, it's straight-forward to see that an isolated singularity is a pole iff $|f| \to\infty$ at the point in question. – mrf Feb 23 '13 at 13:28
  • @mrf: This basically shifts the burden to Riemann's theorem on removable singularities, which, perhaps, is slightly more elementary than Liouville and Casorati-Weierstrass. – Martin Feb 23 '13 at 13:32
  • @mrf Oh boy, you're right. This is easy. – Julien Feb 23 '13 at 13:52
  • @julien: I agree that we can write $g(z)= \sum\limits_{k=-n}^{\infty} c_kz^k$, so $f(z)= \sum\limits_{k=-n}^{\infty} c_kz^{-k}$, but how do you know the behaviour of this series for $z \to 0$ to deduce that $c_k=0$ if $k>0$? – Seirios Mar 03 '13 at 08:35
  • @Seirios $f$ entire so $f(z)=\sum_{n\geq 0} c_nz^n$, so $g(z)=\sum_{n\geq 0} c_nz^{-n}$. $0$ is a pole for $g$ so $g(z)=c_nz^{-n}+\ldots+c_0$. So $f(z)=c_nz^n+\ldots+c_0$. – Julien Mar 03 '13 at 12:44
  • @Seirios These hold for all $z\neq 0$. Now if $f$ is entire, it is continuous at $0$, so the last equality holds also at $z=0$. – Julien Mar 03 '13 at 12:54
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Hint: The set of zeroes of $f$ is bounded and discrete, hence finite and this allows us to find a polynomial $p$ such that $f(z)=p(z)g(z)$ for some entire function $g\colon \mathbb C\to\mathbb C^\times$. What scenarios can happen with $g(z)$ as $z\to\infty$?

Ittay Weiss
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Hagen von Eitzen
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    If a set of points is bounded and discrete, can you then conclude is finite? Any sequence that converges is bounded and discrete, and infinite, like $\sum_0^\infty \frac{1}{2^k}$ – MyUserIsThis Feb 23 '13 at 11:31
  • @GitGud Yes thanks. – MyUserIsThis Feb 23 '13 at 11:33
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    @MyUserIsThis: Some people take discrete to mean closed and discrete for which the statement that boundedness implies finiteness is true. The set of zeroes of $f$ is clearly closed. – Martin Feb 23 '13 at 13:33
  • Hm, upon rereading I think that whoever upvoted me here should be ashamed :9 – Hagen von Eitzen Feb 23 '13 at 15:25