3
$\begingroup$

I'm having a bit of trouble on another problem, and I'm not sure where to start:

Show that for any polynomial $p(z)$ there is a $z$ with $|z|=1$ such that $|p(z)-1/z|\geq 1$.

Could anybody get me started with a tip or two? Thanks in advance.

1 Answers 1

0

You have to show that you can find $z$ of modulus $1$ such that $|zp(z)-1|\geq 1$.If it's not the the case, you apply Rouché's theorem to get a contradiction.

  • 0
    Well I need to clarify myself: so if that is not the case then then we have |zp(z)-1|<1\Rightarrow 0,but what does that lead to a contradiction from Rouches Theorem?2013-05-07