If p(z) is univariate polynomial with complex coefficients having all
its zeros inside the closed unit disk, then the Gauss-Lucas theorem st
ates that all zeros of p'(z) lie in the same disk. We study the follow
ing 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 th
is distance depending on degree. We also show that this distance is eq
ual to 1/3 for polynomials of degree 3 and polynomials with real zeros
.