Hi guys I'm quite lost with a question which I've come across.
The question is:
Let $\ p(z) = c_nz^n+c_{n-1}z^{n-1}+...+c_1z+c_0 $ be a polynomial suppose $\ \lvert p(z) \rvert \le 1$ for all $\ z \in D(0,1), $ Show that $\lvert c_k \rvert \le 1$ for all $\ k = 0,1, ... n$
I'm quiet unsure as to what I should do. Any help would be aprreciated
$c_n = \frac {p^{(n)}(0)}{n!}$ (Taylor series)
$\frac {p^{(n)}(0)}{n!}= \frac 1{2\pi i}\int_{|z|=1} \frac {p(z)}{z^n} dz$ (Cauchy integral formula)
$|\int_{|z|=1} \frac {p(z)}{z^n} dz|\le 2\pi \max (|\frac {p(z)}{z^n}|)$
$|p(z)| \le 1, |z^n| = 1, \max (|\frac {p(z)}{z^n}|) \le 1$
$|c_n| \le 1$