On quasiconformal extension of harmonic mappings with nonzero pole
Let $\Sigma _H^k(p)$ be the class of sense-preserving univalent harmonic mappings defined on the open unit disk $\mathbb{D}$ of the complex plane with a simple pole at $z=p \in (0,1)$ that have $k$-quasiconformal extensions ($0\le k<1$) to the extended complex plane. We first derive a sufficient...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Académie des sciences
2025-05-01
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| Series: | Comptes Rendus. Mathématique |
| Subjects: | |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.686/ |
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| Summary: | Let $\Sigma _H^k(p)$ be the class of sense-preserving univalent harmonic mappings defined on the open unit disk $\mathbb{D}$ of the complex plane with a simple pole at $z=p \in (0,1)$ that have $k$-quasiconformal extensions ($0\le k<1$) to the extended complex plane. We first derive a sufficient condition for harmonic mappings defined on $\mathbb{D}$ with pole at $z=p\in (0,1)$ to belong in the class $\Sigma _H^k(p)$. As a consequence of this, we derive a convolution result involving functions in $\smash{\Sigma _H^{k_i}(p)}$, $0\le k_i<1$ for $i=1,2$. We also consider harmonic mappings with a nonzero pole defined on a linearly connected domain $\Omega \subset \mathbb{D}$ and prove criteria for univalence and quasiconformal extensions for such mappings. |
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| ISSN: | 1778-3569 |