[Borne pour le cinquième coefficient des fonctions étoilées]
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 , 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 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.
Accepté le :
Publié le :
@article{CRMATH_2015__353_6_505_0,
author = {Ravichandran, V. and Verma, Shelly},
title = {Bound for the fifth coefficient of certain starlike functions},
journal = {Comptes Rendus. Math\'ematique},
pages = {505--510},
publisher = {Elsevier},
volume = {353},
number = {6},
year = {2015},
doi = {10.1016/j.crma.2015.03.003},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2015.03.003/}
}
TY - JOUR AU - Ravichandran, V. AU - Verma, Shelly TI - Bound for the fifth coefficient of certain starlike functions JO - Comptes Rendus. Mathématique PY - 2015 SP - 505 EP - 510 VL - 353 IS - 6 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2015.03.003/ DO - 10.1016/j.crma.2015.03.003 LA - en ID - CRMATH_2015__353_6_505_0 ER -
%0 Journal Article %A Ravichandran, V. %A Verma, Shelly %T Bound for the fifth coefficient of certain starlike functions %J Comptes Rendus. Mathématique %D 2015 %P 505-510 %V 353 %N 6 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2015.03.003/ %R 10.1016/j.crma.2015.03.003 %G en %F CRMATH_2015__353_6_505_0
Ravichandran, V.; Verma, Shelly. Bound for the fifth coefficient of certain starlike functions. Comptes Rendus. Mathématique, Tome 353 (2015) no. 6, pp. 505-510. doi : 10.1016/j.crma.2015.03.003. http://www.numdam.org/articles/10.1016/j.crma.2015.03.003/
[1] Coefficients of the inverse of strongly starlike functions, Bull. Malays. Math. Sci. Soc. (2), Volume 26 (2003) no. 1, pp. 63-71
[2] Coefficient bounds for p-valent functions, Appl. Math. Comput., Volume 187 (2007) no. 1, pp. 35-46
[3] Hermitian forms in function theory, Trans. Amer. Math. Soc., Volume 286 (1984) no. 2, pp. 675-688
[4] Early coefficients of the inverse of a regular convex function, Proc. Amer. Math. Soc., Volume 85 (1982) no. 2, pp. 225-230
[5] The coefficients of multivalent close-to-convex functions, Proc. Amer. Math. Soc., Volume 21 (1969), pp. 545-552
[6] 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] Uniformly convex functions. II, Ann. Pol. Math., Volume 58 (1993) no. 3, pp. 275-285
[8] 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] Coefficient estimates in a class of strongly starlike functions, Kyungpook Math. J., Volume 49 (2009) no. 2, pp. 349-353
Cité par Sources :
