Re: Minimum modulus of a polynomial

From: Jon Slaughter (Jon_Slaughter_at_Hotmai.com)
Date: 10/08/04 

Date: Fri, 08 Oct 2004 05:01:52 -0000

> I have a polynomial with positive coefficients (decreasing at least
> geometrically, so I know all zeros are outside |z|=1).
>
> Does someone have a reference to what I believe is true, that the
> minimum modulus on |z|=1 is at z=-1?
>
> Thanks,
>
> Tim Norfolk
>
> --irascible since 1957
>

It looks as if your right for an odd degree polynomial, but for even degree, it takes on two minimum's not at z = -1. for odd, it seems right.

for example:

P(z) = 1 + 1/2*z + 1/4*z^2

then P(z) = 0 gives -1 +- I*sqrt(3), which is outsize |z|=1,

but |P(e^(I*2*Pi*0.5)| = 0.75, while for 0.35, you get ~0.65

so, unless I'm making a big boo boo somewhere ;/