Exact value of the univalent Bloch–Landau constant $U$

The univalent Bloch–Landau constant $U$, introduced by Landau in 1929, is the largest constant such that the image of every univalent (conformal) map of the unit disk contains a schlicht disk of radius $U|f'(0)|$; equivalently, $U$ is the infimum over normalized univalent functions of the inradius of the image domain. Determining $U$ exactly is a classical open problem of geometric function theory: the best known bounds, $0.5708858 < U \le 0.6563937$ (Skinner 2009; Carroll–Ortega-Cerdà 2009), have stood since 2009, and no conjectured exact value is established in the literature.

Background

Bloch's theorem (1920s) states that for every function $f$ analytic in the unit disk $\mathbb{D}=\{z:|z|<1\}$ with $f'(0)=1$ the image contains a schlicht disk — a disk on which $f$ is one-to-one, because $f$ maps some subdomain of $\mathbb{D}$ univalently onto it. The classical Bloch constant $B$ is the largest constant such that every such $f$ has a schlicht disk of radius at least $B$, and the Landau constant $L$ is the analogous constant for disks centered at the origin; both are also open. In the same 1929 paper Landau introduced a third constant of the family by restricting to functions that are themselves univalent (one-to-one and holomorphic) in $\mathbb{D}$. For univalent $f$ every disk contained in the image is automatically schlicht, so the relevant quantity is the inradius $R_f(\mathbb{D})$ of the image domain — the supremum of radii of Euclidean disks contained in $f(\mathbb{D})$. The univalent Bloch–Landau constant (also called the schlicht Bloch constant; denoted $U$ by Landau, $\mathfrak{A}$ by Robinson, and $K$ in Finch's survey) is $U=\inf\Big\{\frac{R_f(\mathbb{D})}{|f'(0)|}:\ f \text{ univalent in } \mathbb{D},\ f(0)=0\Big\},$ equivalently $U=\inf_{f\in S} B_f$ where $S$ is the normalized univalent class ($f(0)=0$, $f'(0)=1$) and $B_f$ is the radius of the largest schlicht disk in $f(\mathbb{D})$. The four constants satisfy $B \le B_\ell \le L \le U$, where $B_\ell$ is the locally univalent Bloch constant (the same infimum over analytic $f$ with $f'(0)=1$ and $f'\ne 0$ in $\mathbb{D}$). The Koebe one-quarter theorem (the image of a normalized univalent function contains the disk of radius $1/4$ about $f(0)$) gives $U \ge 1/4$; Landau proved $U>0.566$. Lower bounds have crept up through Reich ($0.569$), Jenkins ($0.5705$), Toppila ($0.5708$), Zhang and Jenkins ($0.57088$), Xiong ($0.570884$), to Skinner ($0.5708858$), who optimized a pipeline going back to Landau: reduce to the Bloch class $\{f\in S:\|f\|=1\}$, where $\|f\|=\sup_{z\in\mathbb{D}}(1-|z|^2)|f'(z)|$ is the Bloch seminorm, and convert pointwise growth estimates — obtained from coefficient inequalities for univalent functions such as the Fekete–Szegö inequality (a sharp bound on $|a_3-\lambda a_2^2|$ for the Taylor coefficients of $f\in S$) — into guaranteed omitted-value and hence schlicht-disk bounds, with numerical optimization over auxiliary parameters. These estimates are delicate and numerically sensitive, and the gains have been confined to the fourth through sixth decimals. Upper bounds come from explicit domains. Since $U$ is an infimum, any simply connected domain $D_0$ of inradius $1$ that is the image of $\mathbb{D}$ under a conformal map $f$ with $f(0)=0$ yields $U \le 1/|f'(0)|$. Robinson (1935) obtained $U<0.658$; a proof that extremal domains — domains realizing the infimum — exist was first written down explicitly in his follow-up paper (1936). Goodman (1945) used a disk with radial slit arcs removed to get $U<0.65647$; Beller and Hummel (1985) refined the same construction to $U<0.6564155$; and Carroll and Ortega-Cerdà (2009) obtained the current best $U\le 0.6563937$ by trimming the slit arcs so that the domain is harmonically symmetric in each removed arc with respect to the origin — harmonic symmetry meaning that the Poisson kernels associated with the two prime ends bounded by an internal boundary arc take equal values at the origin — exploiting Fedorov's explicit solution of the Pólya–Chebotarev problem (finding the continuum of minimal logarithmic capacity containing a given finite point set) for four symmetrically placed points, together with conformal welding (constructing a conformal map by gluing copies of the disk along boundary arcs). The theoretical guidance for these constructions is Jenkins' criterion, extended by Carroll: an extremal domain contains a disk of radius exactly $U$ (an extremal disk), and it is harmonically symmetric in each internal boundary arc that does not touch the extremal disk. These extremality conditions are necessary but not known to be sufficient, and no conjectured exact value of $U$ is established in the literature.

Problem Statement

Determine the exact value of the univalent Bloch–Landau constant $U=\inf\Big\{\frac{R_f(\mathbb{D})}{|f'(0)|}:\ f \text{ univalent (one-to-one and holomorphic) in the unit disk } \mathbb{D},\ f(0)=0\Big\},$ where $R_f(\mathbb{D})$ is the inradius of the image domain $f(\mathbb{D})$, i.e. the supremum of radii of Euclidean disks contained in $f(\mathbb{D})$; equivalently, $U=\inf_{f\in S} B_f$ where $S$ is the class of normalized univalent functions ($f(0)=0$, $f'(0)=1$) and $B_f$ is the radius of the largest schlicht disk in $f(\mathbb{D})$. The problem is posed at its natural full generality: the whole class of univalent functions on the disk, with no restriction on degree, symmetry, or image geometry, and no restriction on the form of the answer. Currently $0.5708858<U\le 0.6563937$ (Skinner; Carroll–Ortega-Cerdà); determining $U$ means establishing the sharp lower bound $R_f(\mathbb{D})\ge U\,|f'(0)|$ for every univalent $f$ (the global lower bound), together with a matching extremal domain — a simply connected domain realized as $f(\mathbb{D})$ for a conformal $f$ with $f(0)=0$ — whose inradius-to-derivative ratio equals $U$ (the matching upper bound).

References

1. Edmund Landau, 'Über die Blochsche Konstante und zwei verwandte Weltkonstanten', Mathematische Zeitschrift 30 (1929) 608–634. DOI: 10.1007/BF01187791, https://doi.org/10.1007/BF01187791
2. R. M. Robinson, 'The Bloch constant 𝔄 for a schlicht function', Bulletin of the American Mathematical Society 41 (1935) 535–540. DOI: 10.1090/S0002-9904-1935-06138-5, https://doi.org/10.1090/S0002-9904-1935-06138-5
3. Raphael M. Robinson, 'Bloch functions', Duke Mathematical Journal 2 (1936) 453–459. DOI: 10.1215/s0012-7094-36-00237-5, https://doi.org/10.1215/s0012-7094-36-00237-5
4. Ruth E. Goodman, 'On the Bloch-Landau constant for schlicht functions', Bulletin of the American Mathematical Society 51 (1945) 234–239. DOI: 10.1090/S0002-9904-1945-08315-3, https://doi.org/10.1090/S0002-9904-1945-08315-3
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14. Tom Carroll, 'An Extension of Jenkin's Condition for Extremal Domains Associated with the Univalent Bloch-Landau Constant', Computational Methods and Function Theory 8 (2008) 159–165. DOI: 10.1007/BF03321679, https://doi.org/10.1007/BF03321679
15. Tom Carroll and Joaquim Ortega-Cerdà, 'The univalent Bloch-Landau constant, harmonic symmetry and conformal glueing', Journal de Mathématiques Pures et Appliquées 92 (2009) 396–406. DOI: 10.1016/j.matpur.2009.05.008, arXiv:0806.2282, https://arxiv.org/abs/0806.2282
16. Brian Skinner, 'The univalent Bloch constant problem', Complex Variables and Elliptic Equations 54 (2009) 951–955. DOI: 10.1080/17476930903197199, https://doi.org/10.1080/17476930903197199
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18. Steven Finch, 'Errata and Addenda to Mathematical Constants', arXiv:2001.00578 (v4, 2024), https://arxiv.org/abs/2001.00578 (§7.1 states $0.57088586 < U \le 0.6563937$)
19. Md Firoz Ali and Shaesta Azim, 'Bloch and Landau constants for meromorphic functions', arXiv:2603.05047 (2026), https://arxiv.org/abs/2603.05047