Let $p(z)$ be a polynomial not identically zero on $\Bbb C$ with degree n. Suppose that $|p(z)|\le|z|^n$ for any z satisfies $|z|\le1$. Then there exists a complex number $c\in \Bbb C$ such that $p(z)$ is in the form of $p(z)=cz^n$ for any $z$ on $\Bbb C$.
I already know that, if an entire function f is bounded by a polynomial with degree n on $\Bbb C$, for example $kz^n$, then the function is an polynomial with the degree under n in $\Bbb C$ by estimating the Taylor expansion coefficient of f. These two problem are similiar, but I cannot find the connection. Any help would be apprecicated!