math.CV — Complex Variables

Smale's mean value constant (1981) exceeds 1.1559

Contributed by Dave Goel

We prove that Smale's mean value constant exceeds $1.1559837$, improving the bound $1.1018$ of Jatar and Ng. Smale proved in 1981 that for every complex polynomial $P$ of degree at least $2$ and every point $z$ with $P'(z)\neq0$, some critical point $\zeta$ of $P$ satisfies $|P(z)-P(\zeta)|\le K\,|z-\zeta|\,|P'(z)|$ with $K=4$, and he conjectured that $K=1$ suffices. The best constant $K^*$ is called Smale's mean value constant. Smale's conjecture is false: a proof that $K^*>1$ appeared in 2026. Jatar and Ng showed that every pair $(W,B)$ of a normalized univalent map $W$ of the unit disk and a finite Blaschke product $B$ with $B(0)>0$, such that $(WB)'$ has no zero in the disk, gives $K^*\ge1/B(0)$. We construct such a pair explicitly, with $B$ of degree two, from a polynomial of degree $18$ with Gaussian rational coefficients, and we verify the required properties by a computation in exact arithmetic. Since the best constant $K(d)$ for degree $d$ is nondecreasing in $d$ (Crane), $K(d)$ also exceeds $1.1559837$ for all large $d$. All theorems, lemmas and corollaries of this paper, and the results of Jatar and Ng that it uses, are also proved in the Lean 4 proof assistant and checked with the Lean comparator.