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/

[1] R.M. Ali Coefficients of the inverse of strongly starlike functions, Bull. Malays. Math. Sci. Soc. (2), Volume 26 (2003) no. 1, pp. 63-71

[2] R.M. Ali; V. Ravichandran; N. Seenivasagan Coefficient bounds for p-valent functions, Appl. Math. Comput., Volume 187 (2007) no. 1, pp. 35-46

[3] C.R. Leverenz Hermitian forms in function theory, Trans. Amer. Math. Soc., Volume 286 (1984) no. 2, pp. 675-688

[4] R.J. Libera; E.J. Złotkiewicz Early coefficients of the inverse of a regular convex function, Proc. Amer. Math. Soc., Volume 85 (1982) no. 2, pp. 225-230

[5] A.E. Livingston The coefficients of multivalent close-to-convex functions, Proc. Amer. Math. Soc., Volume 21 (1969), pp. 545-552

[6] W.C. Ma; D. Minda A unified treatment of some special classes of univalent functions, Tianjin, 1992 (Conf. Proc. Lecture Notes Anal., I), Int. Press, Cambridge, MA, USA (1992), pp. 157-169

[7] W.C. Ma; D. Minda Uniformly convex functions. II, Ann. Pol. Math., Volume 58 (1993) no. 3, pp. 275-285

[8] R. Mendiratta; S. Nagpal; V. Ravichandran A subclass of starlike functions associated with left-half of the lemniscate of Bernoulli, Int. J. Math., Volume 25 (2014) no. 9 (17 pp)

[9] J. Sokół Coefficient estimates in a class of strongly starlike functions, Kyungpook Math. J., Volume 49 (2009) no. 2, pp. 349-353

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