×

Certain subclasses of bi-close-to-convex functions associated with quasi-subordination. (English) Zbl 1474.30122

Summary: In the present investigation, we introduce certain new subclasses of the class of biunivalent functions in the open unit disc \(U = \{z : |z| < 1\}\) defined by quasi-subordination. We obtained estimates on the initial coefficients \(|a_2|\) and \(|a_3|\) for the functions in these subclasses. The results present in this paper would generalize and improve those in related works of several earlier authors.

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
30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination

References:

[1] Duren, P. L., Univalent Functions (1983), New York, NY, USA: Springer, New York, NY, USA · Zbl 0514.30001
[2] Robertson, M. S., Quasi-subordination and coefficient conjecture, Bulletin of the American Mathematical Society, 76, 1-9 (1970) · Zbl 0191.09101 · doi:10.1090/S0002-9904-1970-12356-4
[3] Altintas, O.; Owa, S., Mazorizations and quasi-subordinations for certain analytic functions, Proceedings of the Japan Academy, Series A, Mathematical Sciences, 68, 7, 181-185 (1992) · Zbl 0767.30014 · doi:10.3792/pjaa.68.181
[4] Lee, S. Y., Quasi-subordinate functions and coefficient conjectures, Journal of the Korean Mathematical Society, 12, 1, 43-50 (1975) · Zbl 0306.30013
[5] Magesh, N.; Balaji, V. K.; Yamini, J., Certain subclasses of bistarlike and biconvex functions based on quasi-subordination, Abstract and Applied Analysis, 2016 (2016) · Zbl 1470.30014 · doi:10.1155/2016/3102960
[6] Ren, F. Y.; Owa, S.; Fukui, S., Some inequalities on quasi-subordinate functions, Bulletin of the Australian Mathematical Society, 43, 2, 317-324 (1991) · Zbl 0712.30021 · doi:10.1017/S0004972700029117
[7] 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
[8] Brannan, D. A.; Taha, T. S.; Mazhar, S. M.; Hamoni, A.; Faour, N. S., On some classes of bi-univalent functions, See also Studia Univ. Babes-Bolyai Math., 31 (1986), No. 2, 70-77, Oxford, UK: Pergamon Press, Oxford, UK · Zbl 0614.30017
[9] Sakar, F. M.; Guney, H. O., Coefficient bounds for a new subclass of analytic bi-close-to-convex functions by making use of Faber polynomial expansion, Turkish Journal of Mathematics, 41, 4, 888-895 (2017) · Zbl 1424.30070 · doi:10.3906/mat-1605-117
[10] Seker, B.; Sumer Seker, S., On subclasses of bi-close-to-convex functions related to the odd-starlike functions, Palestine Journal of Mathematics, 6, 215-221 (2017) · Zbl 1374.30060
[11] Selvaraj, C.; Kumar, T. R. K., Bi-univalent coefficient estimates for certain subclasses of close-to-convex functions, International Journal of Mathematics and Its Applications, 3, 4-D, 69-74 (2015)
[12] Pommerenke, C., Univalent Functions (1975), Göttingen, Germany: Vandenhoeck & Ruprecht, Göttingen, Germany · Zbl 0298.30014
[13] Keogh, F. R.; Merkes, E. P., A coefficient inequality for certain classes of analytic functions, Proceedings of the American Mathematical Society, 20, 8-12 (1969) · Zbl 0165.09102 · doi:10.1090/S0002-9939-1969-0232926-9
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.