Smale’s mean value conjecture

Show that for all n ≥ 2, the optimal constant C_smale(n) equals 1 − 1/n in Smale’s mean value inequality: for any degree-n polynomial f and any z with f′(z) ≠ 0, there exists a critical point ξ of f′ such that |(f(z) − f(ξ))/(z − ξ)| ≤ C_smale(n) |f′(z)|.

Background

Smale’s mean value problem concerns sharp constants in an inequality relating function values, critical points, and derivatives for complex polynomials. While bounds are known, the exact constant remains conjectural.

Determining the exact constant would refine our understanding of polynomial behavior and derivative control at finite degree.

References

In Problem 1E, Smale conjectured that the lower bound was sharp, thus C_{\ref{smale}(n) = 1 - \frac{1}{n}.

— Mathematical exploration and discovery at scale (2511.02864 - Georgiev et al., 3 Nov 2025) in Subsection “Sendov’s conjecture and its variants” (Section 4.10)