Proving bounded entire functions I'm having trouble with the following question, it was from an old final exam.
We didn't actually discuss bounding much in class, and the textbook isn't very elaborate on this topic, so I'd appreciate any help for this question.

Find all entire functions $f(z)$ that obey "there is an integer $n$ such that $$|f(z)| < |z|^n + 1,\quad \forall {z \in C, |z| > 100}.$$

So since $f$ is entire, we either use Cauchy or power series representation? But I'm not too sure how to begin.
 A: This is actually called Extended Liouvilles Theorem or generalized Liouvilles Theorem, Which Says: If $f$ is entire and There exist constant $A,B$ such that $|f(z)|<A+B|z|^k$ for some $k\ge 0$, Then $f$ is a polynomial of degree atmost $k$.
$n=0$ we invoke Liouville, Now apply Induction, consider $$g(z)=\frac{f(z)-f(0)}{z}, \text{for }z\neq 0 \text { and } f'(0)\text{ at } z=0$$ $g$ is entire and by the hypothesis of $f$, $|g(z)|<1+|z|^{n-1}.$
Hence $g$ is a polynomial of degree atmost $n-1$ and $f$ is a polynomial of degree atmost $n$
A: Hints: Prove that if $n=0$, then $f$ is a constant (use Liouville's Theorem). Prove that if $n=1$, then $f$ is linear. The trick here is to show the first derivative of $f$ is constant by using Cauchy's formula for the first derivative in terms of an integral. You can use standard tricks to evaluate that integral and show it must be bounded, so the first derivative is bounded. Keep going. 
A: Check that all polynomials have this property.  Then bound the coefficients of such a function $f$ around $z=0$ using a Cauchy integral over a circle of radius $R$ and let $R \to \infty$.  
