Polynomial Controlled by Degree Near 0

Question

If $f=\sum_{k=0}^{m} a_k x^k, g=\sum_{k=0}^{n} a_k x^k $ with$ n<m$. Then there exist $0<r<1$ such that $|g|<|f|$ on $C_r$.

I think this make sense as when |x|<1, |g| decreases faster. But I cannot write down the rigourous argument.