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PreprintarXiv licencearXiv

A Counterexample to the Liu-Luo-Luo Conjecture on the Harmonic Landau Radius

  • Mikhail Borovikov
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Abstract

We disprove a conjecture of Liu, Luo, and Luo (2020) concerning the Landau radius for bounded planar harmonic mappings. We construct an explicit harmonic mapping FM satisfying the standard normalization conditions, which fails to be locally univalent strictly within the classical holomorphic Landau disk for all M > M* ≈ 4.451. By combining this counterexample with the lower bound established by Ponnusamy, Kalaj, and Vuorinen (2014), we demonstrate that the exact asymptotics of the harmonic Landau radius is π/8M.

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Cite

BibTeXRIS
BibTeX
@misc{borovikov2026counterexample,
  title = {{A Counterexample to the Liu-Luo-Luo Conjecture on the Harmonic Landau Radius}},
  author = {Mikhail Borovikov},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.39676},
  eprint = {2609.39676},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.39676},
}
RIS
TY  - UNPB
TI  - A Counterexample to the Liu-Luo-Luo Conjecture on the Harmonic Landau Radius
AU  - Mikhail Borovikov
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.39676
UR  - https://arxiv.org/abs/2609.39676
AB  - We disprove a conjecture of Liu, Luo, and Luo (2020) concerning the Landau radius for bounded planar harmonic mappings. We construct an explicit harmonic mapping F_M satisfying the standard normalization conditions, which fails to be locally univalent strictly within the classical holomorphic Landau disk for all M > M^* ≈ 4.451. By combining this counterexample with the lower bound established by Ponnusamy, Kalaj, and Vuorinen (2014), we demonstrate that the exact asymptotics of the harmonic Landau radius is π/8M.
ER  -