Suppose $\hat{p}/\hat{q}$ and $p/q$ are two rational functions where $p,q,\hat{p},\hat{q}$ are of degree $n$. Suppose they satisfy that $|p(z)/q(z) - \hat{p}(z)/\hat{q}(z)| < \epsilon$ for any $z$ on unit circle.
Moreover, we assume that $p$ has all its roots inside unit circle and $q$ has all its roots outside unit circle, and that $C \ge |q(z)| \ge c > 0$, $C \ge |\hat{q}(z)| \ge c > 0$for any $z$ on unit circle.
The question is whether we can prove the following statement: suppose $\hat{p},\hat{q},p,q$ satisfy conditions above, then for any $z$ on unit circle, $|q(z)- \hat{q}(z)| < \epsilon' = \epsilon^{c_1}c^{c_2}C^{c_3}n^{c_4}$ for postive constant $c_1$ and constant $c_2,c_3,c_4$.
(In general I am wondering what conditions on $p,q$ can make the statement true. As you may notice, all the conditions assumed above attempt to rule out the possibility that $p$ cancel some factor with $q$ in some way. The lower bound and upper bound on $q,\hat{q}$ may help us remove the denominator. Other similar assumptions on $p$ and $q$ would be also acceptable. )
(PS: for my problem, it suffices to prove that $\Re(q(z)/\hat{q}(z)) > 0$ for all $z$ on unit circle. I am not sure whether this is really easier).
Thanks!