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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.

mathstudent12
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1 Answers1

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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.

Davide Giraudo
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  • I see! It follows immediately from Rouché's theorem--haven't thought of using that before for some reason... – mathstudent12 Sep 24 '12 at 17:43
  • Well I need to clarify myself: so if that is not the case then then we have $|zp(z)-1|<1\Rightarrow 0<zp(z)<2$,but what does that lead to a contradiction from Rouches Theorem? – Myshkin May 07 '13 at 12:10