Coefficient estimates for a subclass of analytic and bi-univalent functions

Document Type : Research Paper

Authors

1 Department of Mathematics‎, ‎Shahrood University Of Technology‎, ‎P.O‎. ‎Box 316-36155‎, ‎Shahrood‎, ‎Iran.

2 Department of Mathematics‎, ‎Mobarakeh Branch‎, ‎Islamic Azad University‎, ‎Mobarakeh‎, ‎P.O‎. ‎Box 84819-97817‎, ‎Isfahan‎, ‎Iran.

Abstract

In this paper, we introduce and investigate a subclass of analytic and bi-univalent functions in the open unit disk. Upper bounds for the second and third coefficients of functions in this subclass are founded. Our results, which are presented in this paper, generalize and improve those in related works of several earlier authors.

Keywords

Main Subjects


R. M. Ali, S. K. Lee, V. Ravichandran and S. Subramaniam, Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions, Appl. Math. Lett. 25 (2012), no. 3, 344--351.
D. A. Brannan and J. G. Clunie (Eds.), Aspects of contemporary complex analysis, Proceedings of the NATO Advanced Study Institute held at the University of Durham, Academic Press, New York-London, 1980.
P. L. Duren, Univalent Functions, Springer-Verlag, New York, Berlin, 1983.
T. Hayami and S. Owa, Coefficient bounds for bi-univalent functions, Panamer. Math. J. 22 (2012), no. 4 , 15--26.
B. A. Frasin and M. K. Aouf, New subclasses of bi-univalent functions, Appl. Math. Lett. 24 (2011), no. 9, 1569--1573.
B. A. Frasin, Coefficient bounds for certain classes of bi-univalent functions, Hacet. J. Math. Stat. 43 (2014), no. 3, 383--389.
C. Y. Gao and S. Q. Zhou, Certain subclass of starlike functions, Appl. Math. Comput. 187 (2007), no. 1, 176--182.
A. W. Kedzierawski, Some remarks on bi-univalent functions, Ann. Univ. Mariae Curie-Sklodowska Sect. A. 39 (1985) 77--81.
M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc. 18 (1967) 63--68.
H. M. Srivastava, A. K. Mishra and P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett. 23 (2010), no. 10, 1188--1192.
E. Netanyahu, The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in jzj < 1, Arch. Rational Mech. Anal. 32 (1969) 100--112.
D. L. Tan, Coefficient estimates for bi-univalent functions, Chinese Ann. Math. Ser. A. 5 (1984), no. 5, 559--568.
Q. H. Xu, Y. C. Gui and H. M. Srivastava, Coefficient estimates for a certain subclass of analytic and bi-univalent functions, Appl. Math. Lett. 25 (2012), no. 6, 990--994.
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), no. 23, 11461--11465.
D. G. Yang and J. L. Liu, A class of analytic functions with missing coefficients, Abstr. Appl. Anal. 2011, Article ID 456729, 16 pages.