Document Type : Research Paper

Author

Department of Mathematics, Payame Noor University, Tehran, Iran.

Abstract

In this paper, we introduce and investigate two new subclasses of the functions class $\Sigma$ of bi-univalent functions defined in the open unit disk, which are associated with the Aghalary-Ebadian-Wang operator. We  estimate the coefficients $|a_{2} |$ and  $|a_{3} |$ for functions in these new subclasses. Several  consequences of the result are also pointed out.

Keywords

Main Subjects

###### ##### References
[1] R. Aghalary, A.  Ebadian, and Z.G. Wang, Subordination and superordination result involving certain convolution operators, Bull. Iranian Math. Soc. Vol. 36 No. 1 (2010), pp 137-147.

[2] D.A. Brannan and J.L. Clunie (Editors), Aspects of Contemporary Complex Analysis, Academic Press, London 1980.

[3] D.A. Brannan, J.L. Clunie, and W.E. Kirwan, Coefficient estimates for the class of star-like functions, Canad. J. Math. 22 (1970) 476-485.

[4] D.A. Brannan and T.S. Taha, On some classes of bi-univalent functions,  Studia Univ. „Babes-Bolyai", Math. 31 (2) (1986), 70-77.

[5] P.L. Duren, Univalent functions, Springer-Verlag, New York, 1983.

[6] B.A. Frasin and M.K. Aouf, New subclasses of bi-univalent function, Appl. Math. Lett. 24 (2011), 1569-1573.

[7] T. Hayami and S. Owa, Coefficient coefficient bounds for bi-univalent functions, Panamer. Math. J. 22 (4) (2012), 15-26.

[8] M. Lewin, On a coefficient problem for bi-univalent function, Proc. Amer. Math. Soc. 18 (1967), 63-68.

[9] X.F. Li and A.P. Wang, Two new subclasses of bi-univalent functions, Internat. Math. Forum, 7 (2012), 1495-1504.

[10] E. Netanyahu, The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in z<1, Arch. Rational Mech. Anal. 32 (1969) 100-112.
[11] H.M. Srivastava, G. Murugusundaramoorthy, and N. Magesh, Certain subclasses of bi-univalent functions associated with the Hoholv operator, Global J.  Math. Analysis, 1 (2) (2013) 67-73.

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

[13] T.S. Taha, Topic in Univalent Function Theory, Ph.D. Thesis, University of London 1981.

[14] Q.H. Xu, Y.C. Gui, and H.M. Srivastava, Coefficient estimates for a certain subclass of analytic and bi-univalent functions and associated coefficient estimate problems, Appl. Math. Lett. 25 (2012),990-994.

[15] Q.H. Xu, H.G. Xiao, and H.M. Srivastava, A certain general subclass of analytic and bi-univalent functions and associated coefficient estimate problems, Appl. Math. Comput. 218 (2012), 11461-11465.