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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.06359 |
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| _version_ | 1866909775621521408 |
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| author | Arora, Vibhuti Chen, Jiaolong Kumar, Shankey Li, Qianyun |
| author_facet | Arora, Vibhuti Chen, Jiaolong Kumar, Shankey Li, Qianyun |
| contents | The aim of this paper is twofold.
First, we obtain a Schwarz-Pick type lemma for the $α$-harmonic mapping $u=P_α[ϕ]$,
where $ϕ\in L^{p}(\mathbb{S}^{n-1},\mathbb{R} )$ and $p\in[1,\infty]$.
We get an explicit form of the sharp function $\mathbf{C}_{α, q}(x)$ in the inequality
$|\nabla u(x)| \leq \mathbf{C}_{α, q}(x)\|ϕ\|_{L^p(\mathbb{S}^{n-1}, \mathbb{ R} )}$.
Second, we prove a Landau type theorem for $u=P_α[ϕ]$,
where $ϕ\in L^{\infty}(\mathbb{S}^{n-1},\mathbb{R}^{n})$.
These results generalize and extend the corresponding
results due to Kalaj (Complex Anal. Oper. Theory, 2024) and
Khalfallah et al. (Mediterr. J. Math.,
2021). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_06359 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Schwarz-Pick type lemma and Landau type theorem for $α$-harmonic mappings Arora, Vibhuti Chen, Jiaolong Kumar, Shankey Li, Qianyun Analysis of PDEs Primary 31B05, Secondary 42B30 The aim of this paper is twofold. First, we obtain a Schwarz-Pick type lemma for the $α$-harmonic mapping $u=P_α[ϕ]$, where $ϕ\in L^{p}(\mathbb{S}^{n-1},\mathbb{R} )$ and $p\in[1,\infty]$. We get an explicit form of the sharp function $\mathbf{C}_{α, q}(x)$ in the inequality $|\nabla u(x)| \leq \mathbf{C}_{α, q}(x)\|ϕ\|_{L^p(\mathbb{S}^{n-1}, \mathbb{ R} )}$. Second, we prove a Landau type theorem for $u=P_α[ϕ]$, where $ϕ\in L^{\infty}(\mathbb{S}^{n-1},\mathbb{R}^{n})$. These results generalize and extend the corresponding results due to Kalaj (Complex Anal. Oper. Theory, 2024) and Khalfallah et al. (Mediterr. J. Math., 2021). |
| title | Schwarz-Pick type lemma and Landau type theorem for $α$-harmonic mappings |
| topic | Analysis of PDEs Primary 31B05, Secondary 42B30 |
| url | https://arxiv.org/abs/2509.06359 |