A bound for Smale's mean value conjecture for complex polynomials

Smale's mean value conjecture is an inequality that relates the locations of critcal points and critical avalues of a polynomial p to the value and derivative of p at some given non-critical point. Using known estimatates for the logarithmic capacity of a connected set in the plane containing three given points, we give a new bound for the constant in Smale's inequality in terms of the degree d of p. The bound improves previous results when d > 8.