I have a polynomial $p$ of degree $n$ satisfying $\lvert p(z) \lvert \leq c\ \ \forall z\in\partial B_1(0)$. (Isn't this true for any polynomial?) Show $\lvert p(z)\lvert \leq c \lvert z\lvert^n \ \ \forall z\in \mathbb{C}\backslash B_1(0)$.
The obvious attempt would be $|p(z)|=|p(\lvert z\lvert\frac{z}{\lvert z\lvert})|=|\sum_{i=0}^n a_i |z|^i (\frac{z}{\lvert z\lvert})^i|\leq |z|^n \sum_{i=0}^n |a_i (\frac{z}{\lvert z\lvert})^i|$ which doesn't lead anywhere.
I guess I have to apply some maximum principle but don't know how.