If P a polynomial of degree n let M=Max |f(z)| if Z ∈ D(0,1)

Question

If P a polynomial of degree n let M=Max |f(z)| if Z ∈ D(0,1) prove if |z|>1 then P(z)<=M.|Z|^n

Maybe we can use the maximum modulus principle and consider ()=^n P(1/)

Answer 1

Note that $|p(z)| \leq M$ for $|z|=1$ by continuity. $q$ is a polynomial. Apply MMP to the unit disk. For $|z|=1$ we get $|q(z)| \leq M$ so MMP shows that $|q(z)| \leq M$ for $|z|<1$. This means $|p(z)| \leq M|z|^{n}$ for $|z|>1$.