Show that there is $z_0$ on the unit circle such that $|p(z_0)|>1$.

Question

Let $p(z)=z^n+a_{n-1}z^{n-1}+\dots+a_0$. Prove that if $p(z)\neq z^n$, then there is $z_0$ on the unit circle such that $|p(z_0)|>1$. Hint: Consider $q(z)=z^np(1/z)$.

I have really been messing around with $q(z)$, I have considered the case when $|z|=1$ (since that would put $z$ on the unit circle. I have done many other things, tried Triangle Inequality, Lower Triangle Inequality, etc. I just can't seem to find the required clever step to solve this.

I am only looking for a hint, please, as this is a homework problem. By hint, I mean a tad bit more of a hint than the one that is given in the problem. Thanks

score 3 · Accepted Answer · answered Mar 19 '15 at 15:25

3

Hint: Suppose $|p(z)|\leq 1$ for all $|z|=1.$ Note that $|p(z)|=|q(\bar{z})|$ for $|z|=1$ and $q(0)=1.$ Using Maximum modulus principle, you will get $q(z)$ a constant and $p(z)=z^n$ which is a contradiction.

answered Mar 19 '15 at 15:25

K_user

2,438

I thought that $q(0)$ was undefined. – nonremovable Mar 19 '15 at 17:51
Nevermind, I see it now. Thanks for the help! – nonremovable Mar 19 '15 at 18:09

Show that there is $z_0$ on the unit circle such that $|p(z_0)|>1$.

1 Answers1