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.
Soit $\Sigma _H^k(p)$ la classe des applications harmoniques injectives préservant l’orientation, définies sur le disque unité ouvert $\mathbb{D}$ du plan complexe ayant un pôle simple en $z=p \in (0,1)$ et qui ont une extension $k$-quasiconforme ($0\le k<1$) au plan complexe étendu. Nous décrivons d’abord une condition suffisante pour qu’une application harmonique appartienne à la classe $\Sigma _H^k(p)$. En conséquence, nous obtenons un résultat de convolution impliquant des applications dans $\smash{\Sigma _H^{k_i}(p)}$, $0\le k_i<1$ pour $i=1,2$. Nous considérons également des applications harmoniques avec un pôle non nul définies sur un domaine linéairement connecté $\Omega \subset \mathbb{D}$ et décrivons des critères d’univalence et d’extensions quasiconformes pour de telles applications.
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Keywords: Quasiconformal mappings, harmonic mappings, linearly connected domain, quasidisk
Mots-clés : Applications quasiconformes, applications harmoniques, domaine linéairement connecté, quasidisque
Bappaditya Bhowmik 1; Goutam Satpati 1
CC-BY 4.0
@article{CRMATH_2025__363_G5_445_0,
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},
}
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/
[1] Quasiconformal reflections, Acta Math., Volume 109 (1963), pp. 291-301 | DOI | MR | Zbl
[2] Loewner chain and quasiconformal extension of some classes of univalent functions, Complex Var. Elliptic Equ., Volume 65 (2020) no. 4, pp. 544-557 | DOI | MR | Zbl
[3] An area theorem for harmonic mappings with nonzero pole having quasiconformal extensions, Proc. Am. Math. Soc., Volume 152 (2024) no. 9, pp. 3881-3891 | DOI | MR | Zbl
[4] Quasiconformal extension of meromorphic functions with nonzero pole, Proc. Am. Math. Soc., Volume 144 (2016) no. 6, pp. 2593-2601 | DOI | MR
[5] Harmonic mappings in the plane, Cambridge Tracts in Mathematics, 156, Cambridge University Press, 2004, xii+212 pages | DOI | MR
[6] Univalent harmonic functions, Trans. Am. Math. Soc., Volume 299 (1987) no. 1, pp. 1-31 | DOI | MR
[7] Quasiconformal extension of harmonic mappings in the plane, Ann. Acad. Sci. Fenn., Math., Volume 38 (2013) no. 2, pp. 617-630 | DOI | MR | Zbl
[8] Explicit quasiconformal extensions and Löwner chains, Proc. Japan Acad., Ser. A, Volume 85 (2009) no. 8, pp. 108-111 | DOI | MR | Zbl
[9] Convolution and quasiconformal extension, Comment. Math. Helv., Volume 51 (1976) no. 1, pp. 99-104 | DOI | MR | Zbl
[10] Schlicht functions with a quasiconformal extension, Ann. Acad. Sci. Fenn., Ser. A I, Volume 500 (1971), 10 pages | MR | Zbl
[11] Univalent functions and Teichmüller spaces, Graduate Texts in Mathematics, 109, Springer, 1987, xii+257 pages | DOI | MR
[12] Quasiconformal mappings in the plane, Grundlehren der Mathematischen Wissenschaften, 126, Springer, 1973, viii+258 pages | DOI | MR
[13] Dirichlet’s principle, distortion and related problems for harmonic mappings, Publ. Inst. Math., Nouv. Sér., Volume 75(89) (2004), pp. 147-171 | DOI | MR | Zbl
[14] Boundary behaviour of conformal maps, Grundlehren der Mathematischen Wissenschaften, 299, Springer, 1992, x+300 pages | DOI | MR
[15] Quasiconformal extension of strongly spirallike functions, Comput. Methods Funct. Theory, Volume 12 (2012) no. 1, pp. 19-30 | DOI | MR | Zbl
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