Turkish Journal of Analysis and Number Theory
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Turkish Journal of Analysis and Number Theory. 2015, 3(3), 83-86
DOI: 10.12691/tjant-3-3-3
Open AccessArticle

Coefficient Estimates for Starlike and Convex Classes of -fold Symmetric Bi-univalent Functions

S. Sivasubramanian1 and P. Gurusamy2,

1Department of Mathematics, University College of Engineering Tindivanam, Anna University, Tindivanam, India

2Department of Mathematics, Velammal Engineering College, Surapet, Chennai, India

Pub. Date: September 14, 2015

Cite this paper:
S. Sivasubramanian and P. Gurusamy. Coefficient Estimates for Starlike and Convex Classes of -fold Symmetric Bi-univalent Functions. Turkish Journal of Analysis and Number Theory. 2015; 3(3):83-86. doi: 10.12691/tjant-3-3-3

Abstract

In an article of Pommerenke [10] he remarked that, for an -fold symmetric functions in the class , the well known lemma stated by Caratheodary for a one fold symmetric functions in still holds good. Exploiting this concept, we introduce certain new subclasses of the bi-univalent function class in which both and are -fold symmetric analytic with their derivatives in the class of analytic functions. Furthermore, for functions in each of the subclasses introduced in this paper, we obtain the coefficient bounds for and We remark here that the concept of -fold symmetric bi-univalent is not in the literature and the authors hope it will make the researchers interested in these type of investigations in the forseeable future. By the working procedure and the difficulty involved in these procedures, one can clearly conclude that there lies an unpredictability in finding the coefficients of a -fold symmetric bi-univalent functions.

Keywords:
analytic functions univalent functions bi-univalent functions -fold symmetric functions subordination

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References:

[1]  D. A. Brannan and J. G. Clunie, Aspects of contemporary complex analysis, Academic Press, London, 1980.
 
[2]  D. A. Brannan and T. S. Taha, On some classes of bi-univalent functions, Stud.Univ.Babes-Bolyai Math. 31(1986), 70-77.
 
[3]  P. L. Duren, Univalent functions, Springer-Verlag, New York, Berlin, Hiedelberg and Tokyo, 1983.
 
[4]  B. A. Frasin and M. K. Aouf, New subclasses of bi-univalent functions, Appl. Math. Lett. 24 (2011), 1569-1573.
 
[5]  J. Sokol, On a condition for α - starlikeness, J. Math. Anal. Appl. 352(2009), 696-701.
 
[6]  M . K . Aouf , J . Sokol and J . Dziok, On a subclass of strongly starlike functions, Appl. Math. Lett. 24 (2011), 27-32.
 
[7]  J . Sokol, A certain class of starlike functions, Comput. Math. Appl, 62(2011), 611-619.
 
[8]  M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc. 18 (1967), 63-68.
 
[9]  W. C. Ma, and D. Minda, A unified treatment of some special classes of unvalent functions, in Proceedings of the Conference on complex analysis. Tianjin (1992), 157-169.
 
[10]  Ch.Pommerenke, On the coefficients of close-to-convex functions, Michigan. Math. J. 9 (1962), 259-269.
 
[11]  H. M. Srivastava, A. K. Mishra and P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett. 23 (2010), 1188-1192.
 
[12]  H. M. Srivastava, S. Bulut, M. Cagler and N. Yagmur, Coefficient estimates for a general subclass of analytic and bi-univalent functions, Filomat. 27 (2013), 831-842.
 
[13]  Q.-H. Xu, H. M. Srivastava and Z. Li, A certain subclass of analytic and close-to-convex functions, Appl. Math. Lett. 24 (2011), 396-–401.
 
[14]  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), 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.
 
[16]  Zhigang Peng and Qiuqiu Han, On the Coefficients of several classes of bi-uivalent functions, Acta. Math. Sci. 34B(1) (2014), 228-240.
 
[17]  H. Tang, G-T. Deng and S-H. Li, Coefïcient estimates for new subclasses of Ma-Minda bi-univalent functions, J. Inequal. Appl. 2013, 2013:317, 1-10.