Sorry I meant to ask if the first inequality is tight

sorry I meant whether the left-hand side is tight up to universal constant? Is that also obvious?

Ah.. yeah, just assume that $p$ and $\hat{p}$ has leading coefficient 1. This should be rather a minor thing though.

Thanks a lot for this example. I am wondering whether with the boundness property that I added (that is, $|p(z)|, |q(z)|$ have upper and lower bound on unit circle), the dependency of $\epsilon'$ on $\epsilon$ can be better. If you know of any tools for solving this kind of problems, it would also be great!

Sorry, yeah, $\hat{p}$ and $\hat{q}$ needs to also have degree $n$. Edited. Thanks!

@YemonChoi, I see. Sorry I am very new to this forum.. I guess all I should expect is that the question in my mind is indeed an open question and I didn't miss any common knowledge. Thanks!

@YemonChoi, alright .. You don't seem to like open-ended question..