Exact values of the classical Bloch constant $B$ and the Landau constant $L$

Bloch's theorem states that every function holomorphic in the unit disk and normalized by $f'(0)=1$ maps some subdomain univalently onto a Euclidean disk whose radius is bounded below by a universal constant. The optimal such constant is the Bloch constant $B$; the corresponding constant when the disk need only be covered (not univalently) is the Landau constant $L$. Both constants were introduced by Landau in 1929, and their exact values remain unknown despite nearly a century of work. The best current bounds are $0.4483127\ldots\le B<0.4718617\ldots$ and $0.51<L\le 0.5432590\ldots$, the lower bounds being due to Wikström (2026); the upper bound for $L$ is the value conjectured to be exact by Rademacher (1943), while the upper bound for $B$ lies only marginally below the Ahlfors–Grunsky value $0.4718617\ldots$ that was long conjectured to be exact (Zhang, 2026). The problem is to determine the exact values of $B$ and $L$; the exact value of the closely related univalent Bloch constant $U$ is likewise unknown.

Background

Let $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ be the unit disk and let $\mathcal{A}_1$ denote the class of functions holomorphic in $\mathbb{D}$ satisfying the normalization $f'(0)=1$. A map is called univalent (or schlicht) on a set if it is one-to-one there. Bloch's theorem (Bloch, 1925) asserts that for every $f\in\mathcal{A}_1$ there exists a subdomain of $\mathbb{D}$ on which $f$ is univalent and whose image is a Euclidean disk; such a disk in $f(\mathbb{D})$ is called a schlicht disk. Writing $B_f$ for the radius of the largest schlicht disk contained in the image $f(\mathbb{D})$, and $L_f$ for the radius of the largest disk of any kind contained in $f(\mathbb{D})$ (dropping the univalence requirement on the preimage), Landau (1929) defined the Bloch constant $B=\inf\{B_f:f\in\mathcal{A}_1\}$, the Landau constant $L=\inf\{L_f:f\in\mathcal{A}_1\}$, and the univalent Bloch constant $U=\inf\{B_f:f\in\mathcal{A}_1,\ f \text{ univalent in } \mathbb{D}\}$, and proved the first numerical estimates $0.39\le B\le L<0.555$ and $U>0.566$. These constants measure the universal covering capability of normalized holomorphic maps and are fundamental objects of geometric function theory; they satisfy the chain $B\le B_\ell\le L\le U$, where $B_\ell$ is the locally univalent Bloch constant obtained by restricting the infimum to functions with nonvanishing derivative in $\mathbb{D}$ — a constant introduced by Pommerenke (1970), who proved $B_\ell>1/2$.

The conjectured answers come from explicit extremal constructions. Ahlfors and Grunsky (1937) computed the function in $\mathcal{A}_1$ corresponding to the simply connected Riemann surface over the plane, without singular points, whose points above the vertices of the equilateral-triangle tiling are simple critical points; this construction yields the upper bound

$$B\le B'=\sqrt{\pi}\,2^{1/4}\,\frac{\Gamma(1/3)}{\Gamma(1/4)}\left(\frac{\Gamma(11/12)}{\Gamma(1/12)}\right)^{1/2}\approx 0.4718617,$$

where $\Gamma$ is Euler's Gamma function, and they conjectured that $B'$ is the exact value of $B$. That Ahlfors–Grunsky conjecture was later disproved: a recent construction exhibits normalized functions with $B_f<B'$, giving the strict upper bound $B<0.4718616521\ldots$ (Zhang, 2026). Rademacher (1943) analogously proved

$$L\le\frac{\Gamma(1/3)\Gamma(5/6)}{\Gamma(1/6)}\approx 0.5432590,$$

conjecturing sharpness. On the lower-bound side, Ahlfors (1938) extended the Schwarz lemma to conformal metrics $ds=p(z)|dz|$ on the disk — metrics admitting, at every point, a supporting metric whose Gaussian curvature is at most $-1$ — and obtained $B\ge\sqrt{3}/4$ and $L\ge 1/2$. Refining this method has produced only small gains relative to the gaps: Bonk (1990) first broke the $\sqrt{3}/4$ barrier with $B\ge\sqrt{3}/4+10^{-14}$, Chen and Gauthier (1996) reached $B>\sqrt{3}/4+2\times10^{-4}$, and a computer-assisted refinement of Bonk's method, whose numerical estimates are rigorously enclosed by interval arithmetic, currently gives $B\ge\sqrt{3}/4+0.0153=0.4483127\ldots$ (Wikström, 2026), still about $0.024$ short of the Ahlfors–Grunsky value; for the Landau constant the best lower bound is now $L>0.51$ (Wikström, 2026), superseding $L>1/2+2\times10^{-8}$ (Chen and Shiba, 2004, who also proved $B_\ell>1/2+2\times10^{-8}$). For the univalent Bloch constant the best current estimates are $0.5708858<U\le 0.6563937$ (Skinner, 2009; Carroll and Ortega-Cerdà, 2009).

The difficulty appears specific to holomorphic maps of the disk: the analogous constants for meromorphic functions on the whole plane have been determined exactly — $\pi/2$ for the locally univalent class (Minda, 1982) and $\arctan\sqrt{8}$ for the full meromorphic class (Bonk and Eremenko, 2000) — using negative-curvature and spherical geometry with no known counterpart in the analytic disk setting. A conjectural route to the classical constants, recounted by Faulhuber (2017), connects them through Baernstein's program to extremal problems for heat kernels on flat tori, but this program itself remains unresolved.

Problem Statement

Determine the exact values of the classical Bloch constant $B$ and Landau constant $L$ for the normalized analytic class $\mathcal{A}_1=\{f \text{ holomorphic in } \mathbb{D}:f'(0)=1\}$, where $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ is the unit disk and

$$B=\inf\{B_f:f\in\mathcal{A}_1\},\qquad L=\inf\{L_f:f\in\mathcal{A}_1\},$$

$B_f$ is the radius of the largest schlicht disk (a disk in $f(\mathbb{D})$ that is the univalent image of some subdomain of $\mathbb{D}$) and $L_f$ is the radius of the largest disk contained in $f(\mathbb{D})$. Concretely, the problem is to prove, for each of $B$ and $L$, a sharp two-sided equality — in particular to determine the exact value of $B$, now known to lie strictly below the Ahlfors–Grunsky value $B'=\sqrt{\pi}\,2^{1/4}\frac{\Gamma(1/3)}{\Gamma(1/4)}\sqrt{\frac{\Gamma(11/12)}{\Gamma(1/12)}}\approx 0.4718617$ (Zhang, 2026), and to decide whether $L$ equals Rademacher's value $\frac{\Gamma(1/3)\Gamma(5/6)}{\Gamma(1/6)}\approx 0.5432590$, or to identify the true values if Rademacher's conjecture fails. The two determinations are the joint objective — they concern the same extremal covering phenomenon, are linked by the inequality $B\le L$, and share the same conjectured structure and method barriers. The exact value of the univalent Bloch constant $U=\inf\{B_f:f\in\mathcal{A}_1 \text{ univalent}\}$, currently known only to satisfy $0.5708858<U\le 0.6563937$, is a natural companion question for the univalent subclass, recorded here as background context.

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