On quasiconformal extension of harmonic mappings with nonzero pole

Bappaditya Bhowmik; Goutam Satpati

doi:10.5802/crmath.686

Article de recherche - Analyse et géométrie complexes

On quasiconformal extension of harmonic mappings with nonzero pole
[Sur l’extension quasiconforme des applications harmoniques à pôle non nul]

Bappaditya Bhowmik ¹ ; Goutam Satpati ¹

¹ Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur – 721302, India

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 445-454.

Abstract (VO)
Résumé

Let $Σ_{H}^{k} (p)$ be the class of sense-preserving univalent harmonic mappings defined on the open unit disk $D$ of the complex plane with a simple pole at $z = p \in (0, 1)$ that have $k$ -quasiconformal extensions ( $0 \leq k < 1$ ) to the extended complex plane. We first derive a sufficient condition for harmonic mappings defined on $D$ with pole at $z = p \in (0, 1)$ to belong in the class $Σ_{H}^{k} (p)$ . As a consequence of this, we derive a convolution result involving functions in $Σ_{H}^{k_{i}} (p)$ , $0 \leq k_{i} < 1$ for $i = 1, 2$ . We also consider harmonic mappings with a nonzero pole defined on a linearly connected domain $Ω \subset D$ and prove criteria for univalence and quasiconformal extensions for such mappings.

Soit $Σ_{H}^{k} (p)$ la classe des applications harmoniques injectives préservant l’orientation, définies sur le disque unité ouvert $D$ du plan complexe ayant un pôle simple en $z = p \in (0, 1)$ et qui ont une extension $k$ -quasiconforme ( $0 \leq k < 1$ ) au plan complexe étendu. Nous décrivons d’abord une condition suffisante pour qu’une application harmonique appartienne à la classe $Σ_{H}^{k} (p)$ . En conséquence, nous obtenons un résultat de convolution impliquant des applications dans $Σ_{H}^{k_{i}} (p)$ , $0 \leq k_{i} < 1$ pour $i = 1, 2$ . Nous considérons également des applications harmoniques avec un pôle non nul définies sur un domaine linéairement connecté $Ω \subset D$ et décrivons des critères d’univalence et d’extensions quasiconformes pour de telles applications.

Reçu le : 2024-03-15
Révisé le : 2024-09-08
Accepté le : 2024-09-08
Publié le : 2025-05-26

DOI : 10.5802/crmath.686

Classification : 31A05, 30C62, 30C55
Keywords: Quasiconformal mappings, harmonic mappings, linearly connected domain, quasidisk
Mots-clés : Applications quasiconformes, applications harmoniques, domaine linéairement connecté, quasidisque

Affiliations des auteurs :

Bappaditya Bhowmik ¹ ; Goutam Satpati ¹

¹ Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur – 721302, India

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Bappaditya Bhowmik and Goutam Satpati},
     title = {On quasiconformal extension of harmonic mappings with nonzero pole},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {445--454},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {363},
     year = {2025},
     doi = {10.5802/crmath.686},
     language = {en},
}

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Bappaditya Bhowmik; Goutam Satpati. On quasiconformal extension of harmonic mappings with nonzero pole. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 445-454. doi : 10.5802/crmath.686. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.686/

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