Comptes Rendus
no. 7-8
Complex analysis/Harmonic analysis
A characterization of Möbius transformations
[Une caractérisation des transformations de Möbius]

Présenté par : Gilles Pisier

Konstantin M. Dyakonov1
1 ICREA and Universitat de Barcelona, Departament de Matemàtica Aplicada i Anàlisi, Gran Via 585, E-08007 Barcelona, Spain
Comptes Rendus. Mathématique, Volume 352 (2014) no. 7-8, pp. 593-595.

Étant donnée une fonction intérieure θ, on démontre que sa dérivée θ′ est extérieure si et seulement si θ est une transformation de Möbius.

We prove that the derivative θ′ of an inner function θ is outer if and only if θ is a Möbius transformation. An alternative characterization involving a reverse Schwarz–Pick type estimate is also given.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2014.05.009
Konstantin M. Dyakonov 1

1 ICREA and Universitat de Barcelona, Departament de Matemàtica Aplicada i Anàlisi, Gran Via 585, E-08007 Barcelona, Spain 
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Konstantin M. Dyakonov. A characterization of Möbius transformations. Comptes Rendus. Mathématique, Volume 352 (2014) no. 7-8, pp. 593-595. doi : 10.1016/j.crma.2014.05.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.05.009/

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[2] C. Carathéodory Theory of Functions of a Complex Variable, Vol. II, Chelsea Publ. Co., New York, 1954

[3] K.M. Dyakonov Smooth functions and coinvariant subspaces of the shift operator, Algebra Anal., Volume 4 (1992) no. 5, pp. 117-147 (English transl. in St. Petersburg Math. J., 4, 1993, pp. 933-959)

[4] K.M. Dyakonov A reverse Schwarz–Pick inequality, Comput. Methods Funct. Theory, Volume 13 (2013), pp. 449-457

[5] J.B. Garnett Bounded Analytic Functions, Springer, New York, 2007

[6] J.L. Walsh Note on the location of zeros of the derivative of a rational function whose zeros and poles are symmetric in a circle, Bull. Amer. Math. Soc., Volume 45 (1939), pp. 462-470

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