Estimates on Initial Coefficients of Certain Subclasses of Bi-Univalent Functions Associated with Al-Oboudi Differential Operator


  • Amol B. Patil
  • Uday H. Naik



Univalent Functions, Bi-Univalent Functions, Al-Oboudi Differential Operator, Salagean's Differential Operator


In the present investigation we introduce two subclasses ΔΣδ,μ [η, α, λ] and ΔΣδ,μ (η, β, λ) of the function class Σ of bi-univalent functions defined in the open unit disk. These subclasses are defined by using the Al-Oboudi differential operator, which is the generalized Salagean's differential operator. Also we find estimates on initial coeffcients |a2| and |a3| for the functions in these subclasses and consider some related subclasses in connection with these subclasses.


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Al-Oboudi, F. M., On univalent functions de ned by a generalized Salagean operator, Int. J. of Math. and Math. sci., vol. 2004, 27 (2004), 1429-1436.

Brannan, D. A., Clunie, J. G., Aspects of Contemporary Complex Analysis, (Proceedings of the NATO Advanced Study Institute held at the University of Durham, Durham; July 1-20, 1979), Academic Press, New York, London, 1980.

Brannan, D. A., Taha, T. S., On some classes of bi-univalent functions, in: Mazhar, S. M., Hamoui, A., Faour, N. S., (Eds.), Math. Anal. and Appl., Kuwait; February 18- 21, 1985, in: KFAS Proceedings Series, vol. 3, Pergamon Press, Elsevier Science Limited, Oxford, 1988, pp. 53-60. see also Studia Univ. Babes-Bolyai Math. 31 (2) (1986) 70-77.

Caglar, M., Orhan, H., Yagmur, N., Coecient bounds for new subclasses of bi-univalent functions, Filomat, 27 (7) (2013), 1165-1171.

Duren, P. L., Univalent Functions, Grundlehren der Mathematischen Wissenschaften 259, Springer, New York,1983.

Frasin, B. A., Aouf, M. K., New subclasses of bi-univalent functions, Appl. Math. Lett. 24 (2011), 1569-1573.

Lewin, M., On a coecient problem for bi-univalent functions, Proc. Amer. Math. Soc., 18 (1967), 63-68.

Netanyahu, E., The minimal distance of the image boundary from the origin and the second coecient of a univalent function in jzj < 1 , Arch. Rational Mech. Anal., 32 (1969), 100-112.

Pommerenke, Ch., Univalent functions, Vandenhoeck and Rupercht, Gottingen, 1975.

Porwal, S., Darus, M., On a new subclass of bi-univalent functions, J. of the Egyptian Math. Society, 21 (3)(2013), 190-193.

Salagean, G. S., Subclasses of univalent functions, in: Complex Analysis - Fifth Romanian Finish Seminar, Bucharest vol. 1, (1983), 362-372.

Srivastava, H. M., Mishra, A. K., Gochhayat, P., Certain subclasses of analytic and bi- univalent functions, Appl. Math. Lett. 23 (2010), 1188-1192.

Taha, T. S., Topics in Univalent Function Theory, Ph.D. Thesis, University of London, 1981.



How to Cite

Patil, A. B., & Naik, U. H. (2017). Estimates on Initial Coefficients of Certain Subclasses of Bi-Univalent Functions Associated with Al-Oboudi Differential Operator. The Journal of the Indian Mathematical Society, 84(1-2), 73–80.