I'm doing a course on complex analysis and while studying for my exam I found this problem:
Consider $p(z) = z^n + a_{n-1}z^{n−1} + · · · + a_0$ with $n ≥ 1$. I have to show that exist $z ∈ ∂\mathbb{D}$ such that $|p(z)| ≥ 1$, with $z ∈ \mathbb{C}$ and $\mathbb{D}$ is the unity disk.
The problem has also a hint which tells me to consider that $p(z)-z^n$ has at most $n-1$ zeros inside $\mathbb{D}$.
I'm familiarized with Cauchy local properties such as maximus module principle and analytic continuation principle.
My try was to supose that this $z$ didn't exist and then apply the maximus module principle to show that the maximum of de polynomial has to be in $∂\mathbb{D}$ to get to a contradiction, but I really don't find a way to continue. I also suppose that the hint given on the question suggests me to use the analytic continuation principle, but i don't know how to use it there.
I would be really grateful if someone could help me because I'm struggling there. Thanks in advance!
Best Answer
An easy solution without Rouche is to assume wlog $a_0 \ne 0$ as otherwise, we divide by the highest $z^k$ dividing $p$ and nothing changes as the problem here goes and take the $n$ inversion $q(z)=z^{n}\bar p(\frac{1}{\bar z})=\bar a_0z^n+\cdots+1$.
Since if $|w|=1, \frac{1}{\bar w}=w$, it follows that $|q(w)|=|p(w)|, |w|=1$ arbitrary.
But $q(0)=1$ so by maximum modulus there is $|w|=1, |p(w)|=|q(w)| \ge 1$. Done!