The sharp Lipschitz continuity problem with respect to the pseudo hyperbolic metric

Shaolin Chen; Hidetaka Hamada; Manzi Huang; Lei Jin

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arXiv · 2609.33140

The sharp Lipschitz continuity problem with respect to the pseudo hyperbolic metric

Shaolin Chen·Hidetaka Hamada·Manzi Huang·Lei Jin

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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 $(K,K_0)$-quasiregular Bloch type mappings are Lipschitz continuous under the same metric, where $K\geq1$ and $K_{0}\geq0$. Moreover, the estimates obtained are asymptotically sharp as $K \to 1^+$ and $K_0 \to 0^+$. In particular, when $K_0 = 0$, our theorem provides a solution to the aforementioned open problem in the asymptotically sharp sense.

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BibTeX RIS

Shaolin Chen, Hidetaka Hamada, Manzi Huang, Lei Jin. 2026-09-27. The sharp Lipschitz continuity problem with respect to the pseudo hyperbolic metric. https://arxiv.org/abs/2609.33140

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