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The sharp Lipschitz continuity problem with respect to the pseudo hyperbolic metric
Abstract
The main purpose of this paper is to address an open problem posed by Huang, Rasila, and Zhu in [Anal. Math. 48(2022), 1069-1080]. We first establish a sharp Lipschitz continuity result for locally univalent harmonic Bloch mappings with respect to the pseudo hyperbolic metric. This result is then applied to demonstrate that harmonic (K,K0)-quasiregular Bloch type mappings are Lipschitz continuous under the same metric, where K≥1 and K0≥0. Moreover, the estimates obtained are asymptotically sharp as K → 1+ and K0 → 0+. In particular, when K0 = 0, our theorem provides a solution to the aforementioned open problem in the asymptotically sharp sense.
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BibTeX
@misc{chen2026sharp,
title = {{The sharp Lipschitz continuity problem with respect to the pseudo hyperbolic metric}},
author = {Shaolin Chen and Hidetaka Hamada and Manzi Huang and Lei Jin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33140},
eprint = {2609.33140},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33140},
}
RIS
TY - UNPB
TI - The sharp Lipschitz continuity problem with respect to the pseudo hyperbolic metric
AU - Shaolin Chen
AU - Hidetaka Hamada
AU - Manzi Huang
AU - Lei Jin
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33140
UR - https://arxiv.org/abs/2609.33140
AB - The main purpose of this paper is to address an open problem posed by Huang, Rasila, and Zhu in [Anal. Math. 48(2022), 1069-1080]. We first establish a sharp Lipschitz continuity result for locally univalent harmonic Bloch mappings with respect to the pseudo hyperbolic metric. This result is then applied to demonstrate that harmonic (K,K_0)-quasiregular Bloch type mappings are Lipschitz continuous under the same metric, where K≥1 and K_0≥0. Moreover, the estimates obtained are asymptotically sharp as K → 1^+ and K_0 → 0^+. In particular, when K_0 = 0, our theorem provides a solution to the aforementioned open problem in the asymptotically sharp sense.
ER -