Exact value of the Bloch constants for planar harmonic mappings

For planar harmonic mappings of the unit disk, no Bloch theorem holds on the full normalized class, but well-defined Bloch constants exist for the class of open harmonic mappings normalized by $f_z(0)=1$ , $f_{\bar z}(0)=0$ (constant $B^O_{har}$ ) and for the $K$ -quasiregular harmonic classes (constants $B^K_{har}$ , $K\ge1$ , with $B^1_{har}$ equal to the classical analytic Bloch constant $B_{hol}$ ). All of these exact values are unknown. The current certified bounds are $B_{hol}/4\le B^O_{har}\le B_{hol}$ and $2B_{hol}/(K+1)\le B^K_{har}\le B_{hol}$ with $B_{hol}\in[0.4483,0.4719]$ , so the open-class constant is trapped in a factor-4 interval ( $0.112$ to $0.472$ ). The problem is to determine these constants exactly; it is likewise open whether the certified two-sided bounds can be strictly improved by an explicit sharpening of a certified factor (a lower bound $B^O_{har}\ge c\,B_{hol}$ with $c>1/4$ , respectively $B^K_{har}\ge c_K\,B_{hol}$ with $c_K>2/(K+1)$ , or a harmonic-specific upper bound below $B_{hol}$ ), and whether admitting a nonvanishing co-analytic part can shrink the guaranteed schlicht disk at all, i.e. whether $B^O_{har}=B_{hol}$ or $B^O_{har}<B_{hol}$ . The lineage of certified lower bounds (Chen–Gauthier–Hengartner's $0.0199$ , Grigoryan's $0.0263$ , Liu's $0.0277$ , Chen–Gauthier's $B_{hol}/4\approx0.112$ ) is established in the primary sources, and the exact values remain open in the literature through 2026.

Background

A planar harmonic mapping is a complex-valued harmonic function $f=u+iv$ on a planar domain; on the unit disk $\mathbb{D}$ it admits a canonical decomposition $f=h+\bar g$ with $h$ and $g$ holomorphic. Writing $f_z$ and $f_{\bar z}$ for the formal complex derivatives, the Jacobian is $J_f=|h'|^2-|g'|^2$ ; by Lewy's theorem $f$ is locally univalent exactly where $J_f\ne0$ , and $f$ is sense-preserving where $J_f>0$ . The extremal singular values of the differential are $\Lambda_f=|f_z|+|f_{\bar z}|$ and $\lambda_f=\big||f_z|-|f_{\bar z}|\big|$ , and $f$ is called $K$ -quasiregular ( $K\ge1$ ) if $\Lambda_f\le K\lambda_f$ everywhere.

Bloch's theorem states that a holomorphic $f$ on $\mathbb{D}$ with $f'(0)=1$ has an image containing a schlicht disk — a disk that is the one-to-one image of some subdomain of $\mathbb{D}$ — of universally bounded radius, and the (analytic) Bloch constant $B_{hol}$ is the largest admissible such bound. Its exact value is a famous open problem: classically $\sqrt{3}/4\le B_{hol}\le 0.4718617\ldots$ (lower bound Ahlfors, upper bound Ahlfors and Grunsky, who conjectured their value exact), with the lower bound improving only marginally for decades (Bonk 1990, $\sqrt3/4+10^{-14}$ ; Chen–Gauthier 1996, $\sqrt3/4+2\cdot10^{-4}$ ) until Wikström's 2026 computer-assisted preprint raised it to $\sqrt3/4+0.0153=0.4483\ldots$ (reference 9). The companion Landau theorem sharpens Bloch's existence statement for bounded holomorphic functions, giving an explicit univalence radius and covered-disk radius in terms of the bound $M$ on $|f|$ .

For harmonic mappings the Bloch question must be reformulated: Chen, Gauthier and Hengartner (2000) exhibited normalized harmonic maps — depending on a parameter $k$ and with $df(0)$ the identity — whose images contain no schlicht disk of radius exceeding $\pi/(2k)$ , so no Bloch theorem exists for the full normalized harmonic class. The natural classes carrying the problem are instead: (i) open harmonic mappings — those mapping open sets to open sets; every sense-preserving harmonic map is open by Lewy's theorem — normalized by $f_z(0)=1$ , $f_{\bar z}(0)=0$ ; and (ii) $K$ -quasiregular harmonic mappings. For $b_f$ the supremum of radii of schlicht disks contained in $f(\mathbb{D})$ , the harmonic Bloch constants are $B^O_{har}=\inf\{b_f:\ f\ \text{open harmonic on}\ \mathbb{D},\ f_z(0)=1,\ f_{\bar z}(0)=0\}$ and $B^K_{har}=\inf\{b_f:\ f\ K\text{-quasiregular harmonic on}\ \mathbb{D},\ f_z(0)=1,\ |f_{\bar z}(0)|\le(K-1)/(K+1)\}.$ Chen–Gauthier–Hengartner proved the first quantitative bounds: $\beta^K_{har}\ge\pi/(8\sqrt2(1+2K))$ , where $\beta^K_{har}$ denotes the Bloch constant for $K$ -quasiregular maps under the alternative normalization $\lambda_f(0)=1$ , and $B^O_{har}\ge(\pi\sqrt2/16)(7-4\sqrt3)\approx0.0199$ . Grigoryan (2006) improved the open-class bound to $\ge0.0263$ via sharp coefficient estimates, and Liu (2009) to $\ge0.0277$ . Chen–Gauthier (2011) then changed the scale of the problem by relating the harmonic constants to the analytic one: $B^K_{har}\ge\frac{2}{K+1}\,B_{hol},\qquad B^O_{har}\ge\frac14\,B_{hol}\ge\frac{\sqrt3}{16}\approx0.108.$ Since holomorphic functions belong to both harmonic classes, $B^O_{har}\le B_{hol}$ and $B^K_{har}\le B_{hol}$ ; and $K=1$ quasiregularity forces $g\equiv0$ , so $B^1_{har}=B_{hol}$ exactly — the harmonic problem contains the classical analytic problem as its endpoint. No upper bound below $B_{hol}$ is known for $B^O_{har}$ , and whether harmonic generality shrinks the guaranteed schlicht disk at all is undecided.

Problem Statement

Determine the exact values of the Bloch constants for planar harmonic mappings — $B^O_{har}$ , over the class of open harmonic mappings of the unit disk (harmonic maps that take open sets to open sets) normalized by $f_z(0)=1$ and $f_{\bar z}(0)=0$ , and the family $B^K_{har}$ ( $K\ge1$ ), over the $K$ -quasiregular harmonic mappings (those satisfying $|f_z|+|f_{\bar z}|\le K\big||f_z|-|f_{\bar z}|\big|$ pointwise, where $f_z$ and $f_{\bar z}$ denote the complex partial derivatives of $f$ ) normalized by $f_z(0)=1$ and $|f_{\bar z}(0)|\le(K-1)/(K+1)$ , the family reducing at $K=1$ to the classical analytic Bloch constant $B_{hol}$ (the analogous constant for holomorphic maps with $f'(0)=1$ , currently known to satisfy $0.4483\le B_{hol}\le0.4719$ ). Here $b_f$ denotes the supremum of radii of schlicht disks (disks that are one-to-one images of subdomains of $\mathbb{D}$ , the unit disk) contained in $f(\mathbb{D})$ , and each constant is the infimum of $b_f$ over its class. The exact values of $B^O_{har}$ and of the whole family $B^K_{har}$ are unknown. It is likewise open whether the certified two-sided bounds $B_{hol}/4\le B^O_{har}\le B_{hol}$ and $2B_{hol}/(K+1)\le B^K_{har}\le B_{hol}$ admit a strict improvement of at least one certified factor — that is, a lower bound $B^O_{har}\ge c\,B_{hol}$ (respectively $B^K_{har}\ge c_K\,B_{hol}$ ) with an explicit $c>1/4$ (respectively $c_K>2/(K+1)$ ), or a harmonic-specific upper bound $B^O_{har}\le c\,B_{hol}$ with an explicit $c<1$ ; and whether admitting a nonvanishing co-analytic part (the $\bar g$ term in the canonical decomposition $f=h+\bar g$ with $h$ and $g$ holomorphic) can shrink the guaranteed schlicht disk at all, i.e. whether $B^O_{har}=B_{hol}$ or $B^O_{har}<B_{hol}$ . Any answer must be consistent with the counterexamples showing that no Bloch theorem holds for the full normalized harmonic class, so the constants cannot be posed over that larger class, and it must address the full open (respectively $K$ -quasiregular) class rather than a restricted family.

References

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