I think so, yes. Suppose all the roots lie inside the unit disc. By compactness $|p(z)|$ takes a positive minimum $\delta$ on the contour (the unit circle). For $q(z)$ in the epsilon-neighbourhood, $q(z)-p(z)$ is a polynomial with epsilon coefficients, so we can take epsilon small enough so that on the contour of the region $|q(z)-p(z)|<|p(z)|$. <\delta<|p(z)|$. Then appeal to Rouché's theorem to learn that$p(z)$and$q(z)$have the same number of zeros in the region. 1 I think so, yes.$q(z)-p(z)$is a polynomial with epsilon coefficients, we can take epsilon small enough so that on the contour of the region$|q(z)-p(z)|<|p(z)|$. Then appeal to Rouché's theorem to learn that$p(z)$and$q(z)\$ have the same number of zeros in the region.