Holomorphic function equal to polynomial. I have this question

Let $h$ an holomorphic function in $\mathbb{C}$ and suppose that 
  $$|h(z+7) -3|^2\leq |z-1|^9 + 42$$
  for all $z$ such that $|2z+63|>177$. Prove that $h$ is a polynomial.

How can I attack this problem? I can't see how because I wanted to use the Liouville theorem, but it isn't bounded in all $\mathbb{C}$. Could anyone give me an explanation?

The meromorphic function $g(z) = h(z+7)/(z-1)^9$ is bounded outside some disk, so its singularity at $\infty$ is removable.  Being a meromorphic function on the Riemann sphere, it is a rational function,
and so $h(z)$ is a rational function with no poles in $\mathbb C$, i.e. a polynomial.
