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Smale's mean value constant (1981) exceeds 1.1559, and other miscellanea

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

Contributed by Dave Goel ↗

Version 1 / Nov 11, 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; after the first version of this paper, the author proved that it equals 44. 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; K∗K^* is the best constant, called Smale's mean value constant. That conjecture was disproved in 2026. Jatar and Ng showed that K∗K^* equals the supremum of 1/B(0)1/B(0) over the admissible pairs (W,B)(W,B): WW is a normalised univalent map of the unit disk, BB is a finite Blaschke product with B(0)>0B(0)>0, and the derivative of the product WBWB has no zero in the disk. 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. Writing Bd\mathcal B_d for this supremum restricted to deg⁡B≤d\deg B\le d, we prove, partly with certified computation, that 1.1363375<B1<1.1778011.1363375<\mathcal B_1<1.177801, B2<100/61\mathcal B_2<100/61, and Bd≤4−0.00597/(dlog⁡d)\mathcal B_d\le4-0.00597/(d\log d) for every d≥2d\ge2, which improves the bound of Jatar and Ng for d≥452d\ge452. If a pair attains Bd\mathcal B_d with d≥1d\ge1, then WW does not extend holomorphically to any neighbourhood of the closed disk and is not rational. If 1/Bn(0)→41/B_n(0)\to4 along a sequence of admissible pairs (Wn,Bn)(W_n,B_n) (such sequences exist, since K∗=4K^*=4), then after rotation WnW_n tends to the Koebe function, as Jatar and Ng observed, and the zeros aa of BnB_n, weighted by log⁡(1/∣a∣)\log(1/|a|), have an explicit limit distribution. Every admissible pair (W,B)(W,B) with WW meromorphic on a neighbourhood of the closed disk has deg⁡B≤wt⁡(W)\deg B\le\operatorname{wt}(W), a weighted count of the points of the circle where ∣W∣|W| is stationary and arg⁡W\arg W decreases.

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.

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Version history

  1. v1 · 67 pagesSubmitted by Dave GoelInitial depositCurrentNov 11, 2026