Bounds for the univalent Bloch-Landau constant
Abstract
The univalent Bloch-Landau constant 𝒰 is the largest number such that, for every univalent function f on the unit disk, f(𝔻) contains a disk of radius 𝒰 |f'(0)|. We prove 0.612242≤𝒰<0.6562681. The previous lower bound, $0.57088586$, came from growth estimates, which control disks centered at f(0). Our improvement rests on a different mechanism. After Landau's normalization, a 2×2 compression of the Grunsky matrix constrains every omitted value through the same two coefficients, and choosing the center of the disk from these coefficients exploits this shared information. A short analytic proof based on this idea gives 𝒰≥0.592; it has been formalized in Lean, with the classical Grunsky and Fekete-Szegő inequalities as the only external inputs. Retaining complex quadratic forms of the matrix gives 𝒰≥0.612242 through a finite interval certificate, and this bound holds for every normalized univalent function with a2=0 and |a3|≤1/3. We also show that no argument based on disks centered at f(0) can give a bound above $0.6$. For the upper bound we construct an explicit domain, bounded by Goodman's first two slit generations, thirty analytic arcs and a circle. We certify 𝒰<0.6562681 with a capacity estimate for almost critical arcs, improving the best published bound $0.6563937$.
Related
- Asymptotic stability of Bloch wall for dispersive Landau-Lifshitz equation
- Upper Bounds and Fekete–Szegö Problem for Bi-Univalent Functions Involving Bazilevič and λ-Pseudo Functions Related to Lucas-Balancing Polynomials
- Asymptotic stability of a Bloch skyrmion for the Landau-Lifshitz-Gilbert equation
- Smale's mean value constant and admissible univalent Blaschke pairs
- Beyond the reduced Landau--Lifshitz--Bloch equation: Singular potential, global well-posedness, and micromagnetic limits
BibTeX
@misc{wikstrom2026bounds,
title = {{Bounds for the univalent Bloch-Landau constant}},
author = {Frank Wikström},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34580},
eprint = {2610.34580},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34580},
}
RIS
TY - UNPB
TI - Bounds for the univalent Bloch-Landau constant
AU - Frank Wikström
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34580
UR - https://arxiv.org/abs/2610.34580
AB - The univalent Bloch-Landau constant 𝒰 is the largest number such that, for every univalent function f on the unit disk, f(𝔻) contains a disk of radius 𝒰 |f'(0)|. We prove 0.612242≤𝒰<0.6562681. The previous lower bound, $0.57088586$, came from growth estimates, which control disks centered at f(0). Our improvement rests on a different mechanism. After Landau's normalization, a 2×2 compression of the Grunsky matrix constrains every omitted value through the same two coefficients, and choosing the center of the disk from these coefficients exploits this shared information. A short analytic proof based on this idea gives 𝒰≥0.592; it has been formalized in Lean, with the classical Grunsky and Fekete-Szegő inequalities as the only external inputs. Retaining complex quadratic forms of the matrix gives 𝒰≥0.612242 through a finite interval certificate, and this bound holds for every normalized univalent function with a_2=0 and |a_3|≤1/3. We also show that no argument based on disks centered at f(0) can give a bound above $0.6$. For the upper bound we construct an explicit domain, bounded by Goodman's first two slit generations, thirty analytic arcs and a circle. We certify 𝒰<0.6562681 with a capacity estimate for almost critical arcs, improving the best published bound $0.6563937$.
ER -