On boundary angular derivatives of an analytic self-map of the unit disk

Vladimir Bolotnikov

doi:10.1016/j.crma.2009.01.018

Complex Analysis

On boundary angular derivatives of an analytic self-map of the unit disk
[Un problème aux limites à dérivées non normales pour une application analytique du disque sur lui-même]

Présenté par : Gilles Pisier

Vladimir Bolotnikov ¹

¹ Department of Mathematics, College of William and Mary, Williamsburg, VA 23187-8795, USA

Comptes Rendus. Mathématique, Volume 347 (2009) no. 5-6, pp. 227-230.

Résumé
Abstract

On se donne des nombres complexes $s_{0}, \dots, s_{N}$ , et on établit des conditions nécessaires et suffisantes d'existence d'une fonction analytique, définie sur le disque unité ouvert, bornée en module par un et admettant un développement asymptotique non tangentiel, en un point $t_{0}$ du bord, du type $f (z) = s_{0} + s_{1} (z - t_{0}) + \dots + s_{N} {(z - t_{0})}^{N} + o ({(z - t_{0})}^{N})$ . Ce critère peut être considéré commec l'analogue à la frontière du résultat classique de I. Schur.

Given complex numbers $s_{0}, \dots, s_{N}$ , we present necessary and sufficient conditions for the existence of a function f analytic and bounded by one in modulus on the open unit disk which admits the nontangential boundary asymptotic expansion $f (z) = s_{0} + s_{1} (z - t_{0}) + \dots + s_{N} {(z - t_{0})}^{N} + o ({(z - t_{0})}^{N})$ at a given point $t_{0}$ on the unit circle. This criterion can be considered as a boundary analog of the classical result of I. Schur.

Reçu le : 2008-11-12
Accepté le : 2009-01-08
Publié le : 2009-02-14

DOI : 10.1016/j.crma.2009.01.018

Affiliations des auteurs :

Vladimir Bolotnikov ¹

¹ Department of Mathematics, College of William and Mary, Williamsburg, VA 23187-8795, USA

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Vladimir Bolotnikov. On boundary angular derivatives of an analytic self-map of the unit disk. Comptes Rendus. Mathématique, Volume 347 (2009) no. 5-6, pp. 227-230. doi : 10.1016/j.crma.2009.01.018. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.01.018/

Bibliographie
Cité par

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