Extended Liouville to Real Part of Entire Functions

Question

I want to prove if $f$ is entire function and $u(x,y)$ is real part of $f$ satisfying \begin{align*} |u(x,y)|\leq C|z|^n, z=x+iy \end{align*} then $f$ is polynomial with degree at most $n$. If $f(z)=\displaystyle \sum_{k=0}^{\infty}a_kz^k$, I tried to compute $a_k$ where $k>n$ using Cauchy's theorem. But I don't know how to replace $f$ with $u$ in the formula to get $a_k=0$ if $k>n$.