For every M>1, let f:D→C be harmonic and satisfy |f(z)|<M for all z∈D, together with the normalization f(0)=0, f_z(0)=1, and f_bar z(0)=0. Define r_0=frac1M+√M^2-1 and sigma_0=Mr_0^2. The Liu–Luo–Luo conjecture asserts that f is univalent on D_r_0=z∈C:|z|<r_0 and that f(D_r_0) contains the disk D_sigma_0=w∈C:|w|<sigma_0.

Liu–Luo–Luo conjecture on the harmonic Landau radius

For every $M>1$ , let $f:\mathbb{D}\to\mathbb{C}$ be harmonic and satisfy $|f(z)|<M$ for all $z\in\mathbb{D}$ , together with the normalization $f(0)=0$ , $f_z(0)=1$ , and $f_{\bar z}(0)=0$ . Define $r_0=\frac{1}{M+\sqrt{M^2-1}}$ and $\sigma_0=Mr_0^2$ . The Liu–Luo–Luo conjecture asserts that $f$ is univalent on $\mathbb{D}_{r_0}=\{z\in\mathbb{C}:|z|<r_0\}$ and that $f(\mathbb{D}_{r_0})$ contains the disk $\mathbb{D}_{\sigma_0}=\{w\in\mathbb{C}:|w|<\sigma_0\}$ .

References

Primary source

arXiv

Additional references

A Counterexample to the Liu-Luo-Luo Conjecture on the Harmonic Landau Radius — arXiv — Mikhail Borovikov

Posted 6 days ago · edited 6 days ago

Progress summary

Refreshed 6 days ago

Claimed progress

A preprint claims an explicit counterexample overturns the conjectured radius and establishes a different large-boundary asymptotic, but the claim has not been independently assessed.

The conjecture asserts that normalized bounded harmonic maps remain univalent on a specific disk, with the proposed radius and image radius sharp. The latest claim challenges this exact formula rather than merely proving another partial bound.

Known results

Liu (2022) recorded the conjectured radii $r_0=1/(M+\sqrt{M^2-1})$ and $\sigma_0=Mr_0^2$ , with a proposed extremal mapping; no proof was reported.

Counterexample claim by September 30, 2026

Mikhail Borovikov claims that an explicit construction loses local univalence inside the conjectured disk for $M>M^*\approx4.451$ . Combined with an earlier lower bound, the preprint claims the corrected asymptotic scale; both the construction and conclusion remain unrefereed and unverified.

Current status (as of October 2026): The conjectured exact radius remains unverified and is challenged by Borovikov's unrefereed counterexample claim; the claimed asymptotic formula is likewise unconfirmed.

Sources

No solutions have been posted yet.