# Entire function such that $\lim_{ z \to \infty} \frac{f(z)}{z^N} = 0$ then $f$ is a polynomial of degree at most $N$.

I am tried to prove this:

Let a entire function $$f$$ such that $$\lim_{ z \to \infty} \frac{f(z)}{z^N} = 0.$$ Show that $$f$$ is a polynomial of degree at most $$N$$.

I find this results

Entire function bounded by a polynomial is a polynomial

So, I start my proof asuming that if $$f$$ is entire, it is equal to a power series centered at 0 with radius of convergence $$\infty$$, and its Taylor series is

$$f(z) = \sum_{n=0}^{\infty} \frac{f^{(N)}(0)}{N!}z^N$$

and I have the Cauchy formula says:

$$\frac{f^{n}(z_0)}{n!} = \frac{1}{2\pi i}\int_{|z-z_0|=r} \frac{f(z)}{(z-z_0)^{n+1}}dz$$

so if I consider $$z_0 = 0$$ and $$\frac{f^{n}(z_0)}{n!} {2\pi i} = \int_{|z|=r} \frac{f(z)}{z^{n+1}}dz$$

I tried to following the results of that, but I don't shure if I am in right way.

Can you help me to do it, please?

• For $n>N$, prove that $f^{(n)}(0)=0$. – Lord Shark the Unknown Apr 7 at 4:58
• Hint: show that $f^{(n)}(z)$ is a bounded entire function. Conclude using Liouville's. – Moya Apr 7 at 5:00
• There exists $R$ such that $|f(z)| <|z|^{N}$ for $|z| \geq R$. So $f$ is a polynomial of degree at most $N$ (by the result you have quoted). – Kavi Rama Murthy Apr 7 at 5:15

Hint following your idea: write $$f^{(n)}(0)$$ as line integral $$f^{(n)}(0) = \frac{n!}{2\pi i}\int_{|z|=r} \frac{f(z)}{z^{n+1}}\,dz$$ and apply the bound: $$\left|\int_{|z|=r}\frac{f(z)}{z^{n+1}}\,dz\right|\le \left(\sup_{|z|=r}\frac{f(z)}{z^N}\right)\frac1{r^{n+1-N}} \,2\pi r= \cdots$$ What happens when $$n$$ is large (how large?) enough?
• The integral is equals to 0 and $f^{(n)}(0) = 0$ but why I say that $f$ is a polynomial of grande at most N? It's because in the $n-th$ derivative is 0 and that's the limit? and where I use the hipotesis of my $lim_{z \to \infty} \frac{f(z)}{z^N} = 0$ – Ops Apr 7 at 17:10
• @Ops, the integral is $= 0$ for $n$ large enough. What is a Taylor series with coefficients $= 0$ for $n$ large enough? – Martín-Blas Pérez Pinilla Apr 7 at 17:12