Comptes Rendus
no. 2
Complex analysis
Yet another proof for a theorem of Landau on Koebe domains

Presented by: the Editorial Board

Manabu Ito1
1 10-20-101, Hirano-kita 1 chome, Hirano-ku, Osaka 547-0041, Japan
Comptes Rendus. Mathématique, Volume 352 (2014) no. 2, pp. 105-108.

The Koebe domain of a family of functions, holomorphic on the unit disk, is the largest domain that is contained in the image of the unit disk for every function of the family. In this note, we furnish a geometric proof of a classical theorem due to Landau on the Koebe domains for certain families of holomorphic functions. The method of proof involves our recently obtained results concerning estimates for hyperbolic metrics on subdomains.

Le domaine de Koebe dʼune famille de fonctions holomorphes sur le disque unité est le plus grand domaine qui est contenu dans lʼimage du disque par chaque fonction de la famille. Dans cette note, nous présentons une preuve géométrique dʼun théorème classique de Landau, relatif aux domaines de Koebe de certaines familles de fonctions holomorphes. La méthode de preuve met en jeu notre résultats récents concernant les estimations pour les métriques hyperboliques sur des sous-domaines.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2013.12.009

Manabu Ito 1

1 10-20-101, Hirano-kita 1 chome, Hirano-ku, Osaka 547-0041, Japan 
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Manabu Ito. Yet another proof for a theorem of Landau on Koebe domains. Comptes Rendus. Mathématique, Volume 352 (2014) no. 2, pp. 105-108. doi : 10.1016/j.crma.2013.12.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.12.009/

[1] A.W. Goodman Koebe domains for certain families, Proc. Amer. Math. Soc., Volume 74 (1979) no. 1, pp. 87-94

[2] M. Ito Upper estimates for hyperbolic metrics on subdomains, Arch. Math. (Basel), Volume 98 (2012) no. 4, pp. 383-387

[3] M. Ito Lower estimates for hyperbolic metrics on subdomains, Complex Var. Elliptic Equ. (2013) (in press) | DOI

[4] E. Landau Üer die Blochsche Konstante und zwei verwandte Weltkonstanten, Math. Z., Volume 30 (1929) no. 1, pp. 608-634 (in German)

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