×

Initial coefficients of biunivalent functions. (English) Zbl 1474.30087

Summary: An analytic function \(f\) defined on the open unit disk is biunivalent if the function \(f\) and its inverse \(f^{- 1}\) are univalent in \(\mathbb{D}\). Estimates for the initial coefficients of biunivalent functions \(f\) are investigated when \(f\) and \(f^{- 1}\), respectively, belong to some subclasses of univalent functions. Some earlier results are shown to be special cases of our results.

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C50 Coefficient problems for univalent and multivalent functions of one complex variable
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Duren, P. L., Univalent Functions, 259 (1983), New York, NY, USA: Springer, New York, NY, USA · Zbl 0514.30001
[2] Lewin, M., On a coefficient problem for bi-univalent functions, Proceedings of the American Mathematical Society, 18, 63-68 (1967) · Zbl 0158.07802 · doi:10.1090/S0002-9939-1967-0206255-1
[3] Brannan, D. A.; Clunie, J.; Kirwan, W. E., Coefficient estimates for a class of star-like functions, Canadian Journal of Mathematics, 22, 476-485 (1970) · Zbl 0197.35602
[4] Netanyahu, E., The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in \(\left``|z\right``| < 1\), Archive for Rational Mechanics and Analysis, 32, 100-112 (1969) · Zbl 0186.39703
[5] Brannan, D. A.; Taha, T. S., On some classes of bi-univalent functions, Universitatis Babeş-Bolyai. Studia. Mathematica, 31, 2, 70-77 (1986) · Zbl 0614.30017
[6] Ali, R. M.; Lee, S. K.; Ravichandran, V.; Supramaniam, S., The Fekete-Szegő coefficient functional for transforms of analytic functions, Iranian Mathematical Society. Bulletin, 35, 2, article 276, 119-142 (2009) · Zbl 1193.30006
[7] Ali, R. M.; Ravichandran, V.; Seenivasagan, N., Coefficient bounds for \(p\)-valent functions, Applied Mathematics and Computation, 187, 1, 35-46 (2007) · Zbl 1113.30024 · doi:10.1016/j.amc.2006.08.100
[8] Frasin, B. A.; Aouf, M. K., New subclasses of bi-univalent functions, Applied Mathematics Letters, 24, 9, 1569-1573 (2011) · Zbl 1218.30024 · doi:10.1016/j.aml.2011.03.048
[9] Mishra, A. K.; Gochhayat, P., Fekete-Szegö problem for a class defined by an integral operator, Kodai Mathematical Journal, 33, 2, 310-328 (2010) · Zbl 1196.30013 · doi:10.2996/kmj/1278076345
[10] Shanmugam, T. N.; Ramachandran, C.; Ravichandran, V., Fekete-Szegő problem for subclasses of starlike functions with respect to symmetric points, Bulletin of the Korean Mathematical Society, 43, 3, 589-598 (2006) · Zbl 1118.30015 · doi:10.4134/BKMS.2006.43.3.589
[11] Srivastava, H. M., Some inequalities and other results associated with certain subclasses of univalent and bi-univalent analytic functions, Nonlinear Analysis. Nonlinear Analysis, Springer Series on Optimization and Its Applications, 68, 607-630 (2012), Berlin, Germany: Springer, Berlin, Germany · Zbl 1251.30028 · doi:10.1007/978-1-4614-3498-6_38
[12] Srivastava, H. M.; Mishra, A. K.; Gochhayat, P., Certain subclasses of analytic and bi-univalent functions, Applied Mathematics Letters, 23, 10, 1188-1192 (2010) · Zbl 1201.30020 · doi:10.1016/j.aml.2010.05.009
[13] Xu, Q.-H.; Xiao, H.-G.; Srivastava, H. M., A certain general subclass of analytic and bi-univalent functions and associated coefficient estimate problems, Applied Mathematics and Computation, 218, 23, 11461-11465 (2012) · Zbl 1284.30009 · doi:10.1016/j.amc.2012.05.034
[14] Xu, Q.-H.; Gui, Y.-C.; Srivastava, H. M., Coefficient estimates for a certain subclass of analytic and bi-univalent functions, Applied Mathematics Letters, 25, 6, 990-994 (2012) · Zbl 1244.30033 · doi:10.1016/j.aml.2011.11.013
[15] Murugusundaramoorthy, G.; Magesh, N.; Prameela, V., Coefficient bounds for certain subclasses of bi-univalent function, Abstract and Applied Analysis, 2013 (2013) · Zbl 1292.30005 · doi:10.1155/2013/573017
[16] Tang, H.; Deng, G.-T.; Li, S.-H., Coefficient estimates for new subclasses of Ma-Minda bi-univalent functions, Journal of Inequalities and Applications, 2013, article 317 (2013) · Zbl 1316.30018 · doi:10.1186/1029-242X-2013-317
[17] Hamidi, S. G.; Halim, S. A.; Jahangiri, J. M., Coefficent estimates for bi-univalent strongly starlike and Bazilevic functions, International Journal of Mathematics Research, 5, 1, 87-96 (2013)
[18] Bulut, S., Coefficient estimates for initial Taylor-Maclaurin coefficients for a subclass of analytic and bi-univalent functions defined by Al-Oboudi differential operator, The Scientific World Journal, 2013 (2013) · Zbl 1291.30065 · doi:10.1155/2013/171039
[19] Bulut, S., Coefficient estimates for a class of analytic and bi-univalent functions, Novi Sad Journal of Mathematics, 43, 2, 59-65 (2013) · Zbl 1349.30040
[20] Magesh, N.; Rosy, T.; Varma, S., Coefficient estimate problem for a new subclass of biunivalent functions, Journal of Complex Analysis, 2013 (2013) · Zbl 1310.30014 · doi:10.1155/2013/474231
[21] Srivastava, H. M.; Murugusundaramoorthy, G.; Magesh, N., On certain subclasses of bi-univalent functions associated with Hohlov operator, Global Journal of Mathematical Analysis, 1, 2, 67-73 (2013)
[22] Çağlar, M.; Orhan, H.; Yağmur, N., Coefficient bounds for new subclasses of bi-univalent functions, Filomat, 27, 7, 1165-1171 (2013) · Zbl 1324.30017 · doi:10.2298/FIL1307165C
[23] Srivastava, H. M.; Bulut, S.; Çağlar, M. C.; Yağmur, N., Coefficient estimates for a general subclass of analytic and bi-univalent functions, Filomat, 27, 5, 831-842 (2013) · Zbl 1432.30014 · doi:10.2298/FIL1305831S
[24] Kumar, S. S.; Kumar, V.; Ravichandran, V., Estimates for the initial coefficients of bi-univalent functions · Zbl 1308.30012
[25] Ma, W. C.; Minda, D., A unified treatment of some special classes of univalent functions, Proceedings of the Conference on Complex Analysis (Tianjin, 1992). Proceedings of the Conference on Complex Analysis (Tianjin, 1992), Conference Proceedings and Lecture Notes in Analysis, 157-169, Cambridge, Mass, USA: International Press, Cambridge, Mass, USA · Zbl 0823.30007
[26] Ali, R. M.; Lee, S. K.; Ravichandran, V.; Supramaniam, S., Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions, Applied Mathematics Letters, 25, 3, 344-351 (2012) · Zbl 1246.30018 · doi:10.1016/j.aml.2011.09.012
[27] Kędzierawski, A. W., Some remarks on bi-univalent functions, Annales Universitatis Mariae Curie-Skłodowska. Section A. Mathematica, 39, 1985, 77-81 (1988) · Zbl 0695.30014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.