Bounds for the univalent Bloch-Landau constant

Abstract

The univalent Bloch-Landau constant $\mathcal{U}$ is the largest number such that, for every univalent function $f$ on the unit disk, $f(\mathbb{D})$ contains a disk of radius $\mathcal{U} |f'(0)|$. We prove $0.612242\le\mathcal{U}<0.6562681$. 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 $2\times2$ 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 $\mathcal{U}\ge0.592$; 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 $\mathcal{U}\ge0.612242$ through a finite interval certificate, and this bound holds for every normalized univalent function with $a_2=0$ and $|a_3|\le1/3$. 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 $\mathcal{U}<0.6562681$ with a capacity estimate for almost critical arcs, improving the best published bound $0.6563937$.