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Smale's mean value constant (1981) exceeds 1.1559

A mix of human-written and AI-generated textHuman understanding: some partsmath.CV — Complex Variables

Contributed by Dave Goel ↗

Version 1 / Nov 10, 2026 / CC BY-SA 4.0

Abstract

We prove that Smale's mean value constant exceeds 1.15598371.1559837, improving the bound 1.10181.1018 of Jatar and Ng. Smale proved in 1981 that for every complex polynomial PP of degree at least 22 and every point zz with P′(z)≠0P'(z)\neq0, some critical point ζ\zeta of PP satisfies ∣P(z)−P(ζ)∣≤K ∣z−ζ∣ ∣P′(z)∣|P(z)-P(\zeta)|\le K\,|z-\zeta|\,|P'(z)| with K=4K=4, and he conjectured that K=1K=1 suffices. The best constant K∗K^* is called Smale's mean value constant. Smale's conjecture is false: a proof that K∗>1K^*>1 appeared in 2026. Jatar and Ng showed that every pair (W,B)(W,B) of a normalized univalent map WW of the unit disk and a finite Blaschke product BB with B(0)>0B(0)>0, such that (WB)′(WB)' has no zero in the disk, gives K∗≥1/B(0)K^*\ge1/B(0). We construct such a pair explicitly, with BB of degree two, from a polynomial of degree 1818 with Gaussian rational coefficients, and we verify the required properties by a computation in exact arithmetic. Since the best constant K(d)K(d) for degree dd is nondecreasing in dd (Crane), K(d)K(d) also exceeds 1.15598371.1559837 for all large dd. 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.

Provenance statement

This work was done overwhelmingly by AI models and supervised by the author. The models were Anthropic's Claude and OpenAI's ChatGPT. The author takes full responsibility for the paper.

References

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Version history

  1. v1 · 9 pagesSubmitted by Dave GoelInitial depositCurrentNov 10, 2026