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 1/3 for polynomials of degree 3 and polynomials with real zeros.