Comptes Rendus
no. 6
Complex analysis
Bound for the fifth coefficient of certain starlike functions
[Borne pour le cinquième coefficient des fonctions étoilées]

Présenté par : the Editorial Board

V. Ravichandran1 ; Shelly Verma1
1 Department of Mathematics, University of Delhi, Delhi 110 007, India
Comptes Rendus. Mathématique, Volume 353 (2015) no. 6, pp. 505-510.

Nous appuyant sur une majoration de la valeur absolue d'un polynôme en les coefficients de fonctions de partie réelle positive, nous obtenons une majoration précise de la valeur absolue du cinquième coefficient d'une fonction analytique f normalisée, satisfaisant zf′(z)/f(z)≺φ(z), pour deux choix différents de φ. Notre preuve utilise une caractérisation des fonctions de partie réelle positive en termes de certaines formes hermitiennes semi-définies positives. Des inégalités bien connues pour ces fonctions de partie réelle positive résultent aussi sans difficulté de cette caractérisation.

For two different choices of φ, the sharp bound for the fifth coefficient of a normalized analytic function f satisfying zf′(z)/f(z)≺φ(z) is obtained by using a bound for a polynomial in the coefficients of functions with positive real part. Our proof uses a characterization of functions with positive real part in terms of certain positive semi-definite Hermitian form and certain well-known coefficient inequalities for functions with positive real part are shown to follow easily from this characterization.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2015.03.003
V. Ravichandran 1 ; Shelly Verma 1

1 Department of Mathematics, University of Delhi, Delhi 110 007, India 
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V. Ravichandran; Shelly Verma. Bound for the fifth coefficient of certain starlike functions. Comptes Rendus. Mathématique, Volume 353 (2015) no. 6, pp. 505-510. doi : 10.1016/j.crma.2015.03.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.03.003/

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