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?

Let $f(z) = h(z+7)-3$. Outside of the bounded domain $|2z+63| \le 177$, $|f(z)|$ is $O(|z|^9)$. By the well-known estimate derived from the Cauchy integral formula for the $n$th derivative of a holomorphic function, any entire function which is bounded in absolute value by a polynomial in $|z|$ for $|z|$ sufficiently large has a derivative which is identically zero-- in other words, is a polynomial. Therefore $f$ is a polynomial, and equivalently $h$ is a polynomial. 
