# Estimates of initial coefficients for certain subclasses of bi-univalent functions involving quasi-subordination

Volume 10, Issue 3, pp 1004--1011 Publication Date: March 20, 2017       Article History
• 867 Views

### Authors

Obaid Algahtani - Department of Mathematics, College of Science, King Saud University, P. O. Box 231428, Riyadh 11321, Saudi Arabia.

### Abstract

The object of the present paper is to introduce and investigate new subclasses of the function class $\Sigma$ of bi-univalent functions defined in the open unit disk U, involving quasi subordination. The coefficients estimate $|a_2|$ and $|a_3|$ for functions in these new subclasses are also obtained.

### Keywords

• Univalent functions
• bi-univalent functions
• quasi-subordination
• subordination.

•  30C45
•  30C50

### References

• [1] D. A. Brannan, J. Clunie, W. E. Kirwan, Coefficient estimates for a class of star-like functions, Canad. J. Math., 22 (1970), 476–485.

• [2] D. A. Brannan, T. S. Taha, On some classes of bi-univalent functions, Studia Univ. Babe-Bolyai Math., 31 (1986), 70–77.

• [3] M. Çağlar, H. Orhan, N. Yağmur, Coefficient bounds for new subclasses of bi-univalent functions, Filomat, 27 (2013), 1165–1171.

• [4] E. Deniz, Certain subclasses of bi-univalent functions satisfying subordinate conditions, J. Class. Anal., 2 (2013), 49–60.

• [5] P. L. Duren, Univalent functions, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, New York (1983)

• [6] S. P. Goyal, P. Goswami, Estimate for initial Maclaurin coefficients of bi-univalent functions for a class defined by fractional derivatives, J. Egyptian Math. Soc., 20 (2012), 179–182.

• [7] M. Haji Mohd, M. Darus, Fekete-Szegő problems for quasi-subordination classes, Abstr. Appl. Anal., 2012 (2012 ), 14 pages.

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

• [9] Z. Nehari, Conformal Mapping, Reprinting of the 1952 edition, Dover, New York, NY, USA (1975)

• [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] F. Y. Ren, S. Owa, S. Fukui , Some inequalities on quasi-subordinate functions, Bull. Austral. Math. Soc. , 43 (1991), 317–329.

• [12] M. S. Robertson, Quasi-subordination and coefficient conjectures, Bull. Amer. Math. Soc., 76 (1970), 1–9.

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

• [14] Q.-H. Xu, H.-G. Xiao, 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.