Comptes Rendus
no. 9-10
Complex Analysis
A conformal mapping example
[Un exemple dʼapplication conforme]

Présenté par : Gilles Pisier

Wayne Smith1 ; Alexander Volberg2
1 Department of Mathematics, University of Hawaii, Honolulu, HI 96822, USA
2 Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
Comptes Rendus. Mathématique, Volume 349 (2011) no. 9-10, pp. 511-514.

On construit un exemple de deux applications conformes φ et ψ du disque unité sur le même domaine telles que le rapport φ′/ψ′ soit borné et le rapport ψ′/φ′ non borné. On donne pour cela des expressions analytiques explicites pour φ, ψ.

An example is constructed of two Riemann maps φ and ψ of the unit disk onto the same domain such that φ′/ψ′ is bounded but not bounded away from zero. This is shown by producing explicit analytic expressions of φ and ψ.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2011.04.010
Wayne Smith 1 ; Alexander Volberg 2

1 Department of Mathematics, University of Hawaii, Honolulu, HI 96822, USA
2 Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA 
@article{CRMATH_2011__349_9-10_511_0,
     author = {Wayne Smith and Alexander Volberg},
     title = {A conformal mapping example},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {511--514},
     publisher = {Elsevier},
     volume = {349},
     number = {9-10},
     year = {2011},
     doi = {10.1016/j.crma.2011.04.010},
     language = {en},
}
TY  - JOUR
AU  - Wayne Smith
AU  - Alexander Volberg
TI  - A conformal mapping example
JO  - Comptes Rendus. Mathématique
PY  - 2011
SP  - 511
EP  - 514
VL  - 349
IS  - 9-10
PB  - Elsevier
DO  - 10.1016/j.crma.2011.04.010
LA  - en
ID  - CRMATH_2011__349_9-10_511_0
ER  - 
%0 Journal Article
%A Wayne Smith
%A Alexander Volberg
%T A conformal mapping example
%J Comptes Rendus. Mathématique
%D 2011
%P 511-514
%V 349
%N 9-10
%I Elsevier
%R 10.1016/j.crma.2011.04.010
%G en
%F CRMATH_2011__349_9-10_511_0
Wayne Smith; Alexander Volberg. A conformal mapping example. Comptes Rendus. Mathématique, Volume 349 (2011) no. 9-10, pp. 511-514. doi : 10.1016/j.crma.2011.04.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.04.010/

[1] K. Baranski; A. Volberg; A. Zdunik Brennanʼs conjecture and the Mandelbrot set, Int. Math. Res. Notices (IMRN) (1998), pp. 589-600

[2] J.E. Brennan The integrability of the derivative in conformal mapping, J. London Math. Soc. (2), Volume 18 (1978), pp. 261-272

[3] J. Garnett Bounded Analytic Functions, Academic Press, 1981

[4] H. Hedenmalm; S. Shimorin Weighted Bergman spaces and the integral means spectrum of conformal mappings, Duke Math. J., Volume 127 (2005) no. 2, pp. 341-393

[5] M.-R. Herman Are there critical points on the boundaries of singular domains, Comm. Math. Phys., Volume 99 (1985) no. 4, pp. 593-612

[6] V. Matache, W. Smith, Composition operators on a class of analytic function spaces related to Brennanʼs conjecture, Complex Anal. Oper. Theory, , in press. | DOI

[7] Ch. Pommerenke Boundary Behavior of Conformal Maps, Springer-Verlag, 1992

[8] J.T. Rogers Critical points on the boundaries of Siegel disks, Bull. Amer. Math. Soc., Volume 32 (1995) no. 3, pp. 317-321

Cité par Sources :

Commentaires - Politique