Let $p(z)=z^n+a_{n-1}z^{n-1}+...+a_0$, where $a_0,a_1,..a_{n-1}$ are complex numbers and $q(z)=1+a_{n-1}z+..+a_0z^n$. If $|p(z)|\le 1$ for all with $|z|\le 1$, Then
$(1) \space |q(z)| \le 1$ for all z with $|z|\le 1$
$(2) \space q(z)$ is a constant polynomial
$(3)\space p(z)=z^n$ for all complex numbers $z$
$(4)\space p(z)$ is a constant polynomial
The way I tried :-
We have $q(z)=z^np\big(\frac1z\big)$
On $|z|=1$ we have $z=e^{i\theta}, 0\le \theta \le 2\pi$.
By Maximum Modulus Theorem,
$\displaystyle\max_{|z|\le 1}|q(z)|=\max_{|z|=1}|q(z)|=\max_{0\le \theta \le 2\pi}|e^{in\theta}p(e^{-i\theta})|\le 1$
But $q(0)=1$ . So by Maximum Modulus Theorem , we have $q(z)=1$ i.e $a_{n-1}=a_{n-2}=...=a_0=0$
Hence $p(z)=z^n$. So correct options $(1),(2)$ and $(3)$
Is my work correct? Are there yet better approaches?
Thanks for your valuable time!!