Bound on difference of roots between two “close” polynomials

Question

Let $p(z) = z^n + a_{n-1}z^{n-1}+\dots+a_0$ and $q(z) = z^n + b_{n-1}z^{n-1}+\dots+b_0$ be two complex monic polynomials with $|a_i - b_i| < \epsilon < 1$. Furthermore, assume that all the coefficients of both polynomials are bounded in modulus by some fixed number $K$. Then I believe that the Hausdorff distance between the roots of $p$ and $q$ considered as sets in the complex plane is atmost $C(K,n)\times \epsilon^{1/n}$ where $C(K,n)$ is a constant that depends on $K$ and $n$. Is this true?