Certain New Subclassess of Analytic and Bi-univalent Functions

Main Article Content

C. R. Krishna, A. C. Chandrashekar, N. Ravikumar

Abstract

The paper presents two novel subclasses of the function class Σ, which consists of bi-univalent functions defined in the open unit disk D={ζ:|ζ|<1}. The authors investigate the properties of these new subclasses and provide estimates for the absolute values of the second, third, and fourth Taylor-Maclaurin coefficients r2, r3, and r4 for functions in these subclasses.

Article Details

References

  1. D.A. Brannan, T.S. Taha, On Some Classes of Bi-Univalent Functions, in: Mathematical Analysis and Its Applications, Elsevier, 1988: pp. 53–60. https://doi.org/10.1016/B978-0-08-031636-9.50012-7.
  2. B.A. Frasin, M.K. Aouf, New Subclasses of Bi-Univalent Functions, Appl. Math. Lett. 24 (2011), 1569–1573. https://doi.org/10.1016/j.aml.2011.03.048.
  3. T. Hayami, S. Owa, Coefficient Bounds for Bi-Univalent Functions, Pan Amer. Math. J. 22 (2012), 15–26.
  4. M. Lewin, On a Coefficient Problem for Bi-Univalent Functions, Proc. Amer. Math. Soc. 18 (1967), 63–68.
  5. X.F. Li, A.P. Wang, Two New Subclasses of Bi-Univalent Functions, Int. Math. Forum, 7 (2012), 1495-1504.
  6. 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. https://doi.org/10.1007/bf00247676.
  7. C. Pommerenke, Univalent Functions, Vandenhoeck and Ruprecht, (1975).
  8. N. Ravikumar, S. Latha, On Analytic Functions With Generalized Bounded Variation, Stud. Univ. Babes-Bolyai Math. 57 (2012), 43–52.
  9. H.M. Srivastava, A.K. Mishra, P. Gochhayat, Certain Subclasses of Analytic and Bi-Univalent Functions, Appl. Math. Lett. 23 (2010), 1188–1192. https://doi.org/10.1016/j.aml.2010.05.009.
  10. T.S. Taha, Topics in Univalent Function Theory, Ph.D. Thesis, University of London, (1981).
  11. Q.H. Xu, Y.C. Gui, H.M. Srivastava, Coefficient Estimates for a Certain Subclass of Analytic and Bi-Univalent Functions, Appl. Math. Lett. 25 (2012), 990–994. https://doi.org/10.1016/j.aml.2011.11.013.
  12. 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. https://doi.org/10.1016/j.amc.2012.05.034.