Dual mean value conjecture

Prove that every complex polynomial P of degree d≥2 satisfying P(0)=0 and P′(0)≠0 has a critical point b such that |P(b)/(bP′(0))|≥1/d, equivalently establish the conjectured lower bound D(P)=max_{b∈Z(P′)}|P(b)/(bP′(0))|≥1/d.

Background

The dual problem reverses the extremal direction of Smale’s mean value conjecture: instead of finding a critical point with a small normalized finite increment, it seeks one with a large normalized finite increment. The conjectured lower bound 1/d is sharp for the family P(z)=(z+1)^d−1.

The paper improves the unconditional lower bound from 1/d² to (d−1/2)^{1/d}/d², and obtains further bounds for critical points on the boundary of the maximal univalent lemniscate. These results do not establish the conjectured 1/d bound in general; the paper explicitly identifies degree 8 as the smallest degree for which the dual conjecture remains open.