Smale's mean value constant and admissible univalent Blaschke pairs
Abstract
Let 𝒫 be the class of polynomials P of degree at least $2$ with P(0)=0, P'(0)=1, let S(P)=minP'(c)=0|P(c)/c| and let ℳ=supP∈𝒫S(P) be Smale's mean value constant. Smale proved that ℳ≤4, observed that 𝕄≥ 1, and asked whether ℳ=1. We show that ℳ coincides with ℬ=sup1/B(0), the supremum running over pairs (W,B) in which W is a normalised univalent map of the standard unit disk 𝔻 and B is a finite Blaschke product with B(0)>0 such that WB is locally univalent in 𝔻. Smale's question is therefore a problem in univalent function theory. We also introduce the degree-restricted constants ℬd for B with deg B ≤ d and show that 1.1018586788…=√(8081/6656) ≤ ℬ1 < √(7/5)=1.1832159566… This implies that ℳ>1.1018 and we also show that there are polynomials of arbitrarily large degree with S(P)>1.1018. The polynomial construction behind these results also settles Problems 1A and 1B of Smale's 1981 paper. For P∈𝒫 and an integer k≥2 let Sk(P)=minP'(c)=0|P(k)(0)/k!|1/(k−1)|P(c)|. Smale proved, by Löwner's coefficient estimate for inverses of univalent functions, that supP∈𝒫Sk(P)≤ Ck1/(k−1)<4, where Ck is the k-th Catalan number and he asked whether the constant $4$ (Problem 1A), or the constant 2=C2 for k=2 (Problem 1B), can be reduced. We prove that supP∈𝒫Sk(P)=Ck1/(k−1) for every k≥2. Hence the constants $4$ and $2$ cannot be reduced and the answers to both problems are negative.
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BibTeX
@misc{jatar2026smales,
title = {{Smale's mean value constant and admissible univalent Blaschke pairs}},
author = {Aneesh Jatar and Tuen Wai Ng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21472},
eprint = {2610.21472},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21472},
}
RIS
TY - UNPB
TI - Smale's mean value constant and admissible univalent Blaschke pairs
AU - Aneesh Jatar
AU - Tuen Wai Ng
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.21472
UR - https://arxiv.org/abs/2610.21472
AB - Let 𝒫 be the class of polynomials P of degree at least $2$ with P(0)=0, P'(0)=1, let S(P)=min_P'(c)=0|P(c)/c| and let ℳ=sup_P∈𝒫S(P) be Smale's mean value constant. Smale proved that ℳ≤4, observed that 𝕄≥ 1, and asked whether ℳ=1. We show that ℳ coincides with ℬ=sup1/B(0), the supremum running over pairs (W,B) in which W is a normalised univalent map of the standard unit disk 𝔻 and B is a finite Blaschke product with B(0)>0 such that WB is locally univalent in 𝔻. Smale's question is therefore a problem in univalent function theory. We also introduce the degree-restricted constants ℬ_d for B with deg B ≤ d and show that 1.1018586788…=√(8081/6656) ≤ ℬ_1 < √(7/5)=1.1832159566… This implies that ℳ>1.1018 and we also show that there are polynomials of arbitrarily large degree with S(P)>1.1018. The polynomial construction behind these results also settles Problems 1A and 1B of Smale's 1981 paper. For P∈𝒫 and an integer k≥2 let S_k(P)=min_P'(c)=0|P^(k)(0)/k!|^1/(k−1)|P(c)|. Smale proved, by Löwner's coefficient estimate for inverses of univalent functions, that sup_P∈𝒫S_k(P)≤ C_k^1/(k−1)<4, where C_k is the k-th Catalan number and he asked whether the constant $4$ (Problem 1A), or the constant 2=C_2 for k=2 (Problem 1B), can be reduced. We prove that sup_P∈𝒫S_k(P)=C_k^1/(k−1) for every k≥2. Hence the constants $4$ and $2$ cannot be reduced and the answers to both problems are negative.
ER -