A function is said to be bi-univalent in the open unit disk if both the function and its inverse map are univalent in . By the same token, a function is said to be bi-subordinate in if both the function and its inverse map are subordinate to certain given function in . 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 () 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.
Une fonction est dite bi-univalente dans le disque unité ouvert si elle et son inverse sont univalentes dans . Dans le même ordre, une fonction est dite bi-subordonnée dans si elle et son inverse sont subordonnées à une fonction donnée dans . 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 () coefficient d'une fonction bi-subordonnée, lorsque les précédents coefficients sont nuls. Nous donnons également des bornes plus précises pour les deux premiers coefficients de telles fonctions.
Accepted:
Published online:
Samaneh G. Hamidi 1; Jay M. Jahangiri 2
@article{CRMATH_2016__354_4_365_0, author = {Samaneh G. Hamidi and Jay M. Jahangiri}, title = {Faber polynomial coefficients of bi-subordinate functions}, journal = {Comptes Rendus. Math\'ematique}, pages = {365--370}, publisher = {Elsevier}, volume = {354}, number = {4}, year = {2016}, doi = {10.1016/j.crma.2016.01.013}, language = {en}, }
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/
[1] Remarks on Faber polynomials, Int. Math. Forum, Volume 3 (2008) no. 9–12, pp. 449-456 (MR2386197)
[2] Differential calculus on the Faber polynomials, Bull. Sci. Math., Volume 130 (2006) no. 3, pp. 179-222 (MR2215663)
[3] An algebra of differential operators and generating functions on the set of univalent functions, Bull. Sci. Math., Volume 126 (2002) no. 5, pp. 343-367 MR1914725 (2004c:17048)
[4] Coefficient estimates for bi-univalent Ma–Minda starlike and convex functions, Appl. Math. Lett., Volume 25 (2012) no. 3, pp. 344-351 (MR2855984)
[5] Coefficient estimates for two new subclasses of bi-univalent functions with respect to symmetric points, J. Funct. Spaces (2015) (5 pp., MR3319198)
[6] Coefficient estimates for a certain subclass of analytic and bi-univalent functions, Acta Univ. Apulensis, Mat.-Inform., Volume 40 (2014), pp. 347-354 (MR3316514)
[7] Initial coefficient bounds for a general class of biunivalent functions, Int. J. Anal. (2014) (4 pp., MR3198331)
[8] Faber polynomial coefficient estimates for a comprehensive subclass of analytic bi-univalent functions, C. R. Acad. Sci. Paris, Ser. I, Volume 352 (2014) no. 6, pp. 479-484 (MR3210128)
[9] Certain subclasses of bi-univalent functions satisfying subordinate conditions, J. Class. Anal., Volume 2 (2013) no. 1, pp. 49-60 (MR3322242)
[10] Univalent Functions, Grundlehren der Mathematischen Wissenschaften, vol. 259, Springer, New York, 1983 (MR0708494)
[11] Über polynomische Entwickelungen, Math. Ann., Volume 57 (1903) no. 3, pp. 389-408 (MR1511216)
[12] New subclasses of bi-univalent functions, Appl. Math. Lett., Volume 24 (2011) no. 9, pp. 1569-1573 (MR2803711)
[13] Faber polynomial coefficient estimates for analytic bi-close-to-convex functions, C. R. Acad. Sci. Paris, Ser. I, Volume 352 (2014) no. 1, pp. 17-20 (MR3150761)
[14] Faber polynomial coefficient estimates for bi-univalent functions defined by subordinations, Bull. Iran. Math. Soc., Volume 41 (2015) no. 5, pp. 1103-1119
[15] On the coefficients of powers of a class of Bazilevic functions, Indian J. Pure Appl. Math., Volume 17 (1986) no. 9, pp. 1140-1144 (MR0864155)
[16] Coefficient estimates for certain classes of bi-univalent functions, Int. J. Math. Math. Sci. (2013) (4 pp., MR3100751)
[17] Coefficients of bi-univalent functions with positive real part derivatives, Bull. Malays. Math. Soc. (2), Volume 3 (2014), pp. 633-640 (MR3234504)
[18] Extremal problems for a family of functions with positive real part and for some related families, Ann. Pol. Math., Volume 23 (1970/1971), pp. 159-177 (MR0267103)
[19] Some extremal problems for certain families of analytic functions, I, Ann. Pol. Math., Volume 28 (1973), pp. 297-326 (MR0328059)
[20] Coefficient bounds for certain subclasses of bi-univalent functions, Int. Math. Forum, Volume 8 (2013), pp. 1337-1344 (MR3107010)
[21] Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett., Volume 23 (2010) no. 10, pp. 1188-1192 (MR2665593)
[22] Coefficient bounds for a certain class of analytic and bi-univalent functions, Filomat, Volume 29 (2015) no. 8, pp. 1839-1845
[23] On the Fekete–Szegö problem for classes of bi-univalent functions, Bull. Belg. Math. Soc. Simon Stevin, Volume 21 (2014) no. 1, pp. 169-178 (MR3178538)
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