On the zeros of a polynomial and its derivatives

If $p(z)$ is univariate polynomial with complex coefficients having all its zeros inside the closed unit disk, then the Gauss-Lucas theorem states that all zeros of $p'(z)$ lie in the same disk. We study the following question: what is the maximum distance from the arithmetic mean of all zeros of $p(z)$ to a nearest zero of ${p'(z)}$? We obtain bounds for this distance depending on degree. We also show that this distance is equal to $\frac{1}{3}$ for polynomials of degree 3 and polynomials with real zeros.