Faber polynomial coefficients of bi-subordinate functions

Samaneh G. Hamidi; Jay M. Jahangiri

doi:10.1016/j.crma.2016.01.013

Complex analysis

Faber polynomial coefficients of bi-subordinate functions
[Polynômes de Faber et coefficients des fonctions bi-subordonnées]

Présenté par : Jean-Pierre Demailly

Samaneh G. Hamidi ¹ ; Jay M. Jahangiri ²

¹ Department of Mathematics, Brigham Young University, Provo, UT 84604, USA
² Department of Mathematical Sciences, Kent State University, Burton, OH 44021, USA

Comptes Rendus. Mathématique, Volume 354 (2016) no. 4, pp. 365-370.

Résumé
Abstract

Une fonction est dite bi-univalente dans le disque unité ouvert $D$ si elle et son inverse sont univalentes dans $D$ . Dans le même ordre, une fonction est dite bi-subordonnée dans $D$ si elle et son inverse sont subordonnées à une fonction donnée dans $D$ . Le comportement des coefficients de telles fonctions est imprévisible et inconnu. Dans cette Note, nous utilisons les développements en polynômes de Faber afin d'établir une borne supérieure pour le $n^{e}$ ( $n \geq 3$ ) coefficient d'une fonction bi-subordonnée, lorsque les $n - 2$ précédents coefficients sont nuls. Nous donnons également des bornes plus précises pour les deux premiers coefficients de telles fonctions.

A function is said to be bi-univalent in the open unit disk $D$ if both the function and its inverse map are univalent in $D$ . By the same token, a function is said to be bi-subordinate in $D$ if both the function and its inverse map are subordinate to certain given function in $D$ . The behavior of the coefficients of such functions are unpredictable and unknown. In this paper, we use the Faber polynomial expansions to find upper bounds for the n-th ( $n \geq 3$ ) coefficients of classes of bi-subordinate functions subject to a gap series condition as well as determining bounds for the first two coefficients of such functions.

Reçu le : 2015-11-23
Accepté le : 2016-01-18
Publié le : 2016-03-03

DOI : 10.1016/j.crma.2016.01.013

Affiliations des auteurs :

Samaneh G. Hamidi ¹ ; Jay M. Jahangiri ²

¹ Department of Mathematics, Brigham Young University, Provo, UT 84604, USA
² Department of Mathematical Sciences, Kent State University, Burton, OH 44021, USA

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Samaneh G. Hamidi; Jay M. Jahangiri. Faber polynomial coefficients of bi-subordinate functions. Comptes Rendus. Mathématique, Volume 354 (2016) no. 4, pp. 365-370. doi : 10.1016/j.crma.2016.01.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.01.013/

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