Estimation on initial coefficient bounds of generalized subclasses of bi-univalent functions

Document Type : Research Paper

Authors

1 Department of Mathematics, Arts, Commerce and Science College, Lanja-416701, India

2 Department of Mathematics, Willingdon College, Sangli-416415, India

3 Department of First Year Engineering, AISSMS College of Engineering, Pune-411001, India

Abstract

In the present investigation, we introduce the two subclasses $S^{\alpha}_{\Sigma}(\gamma, \rho, \lambda, \mu, \xi, \delta)$ and $S_{\Sigma}(\gamma, \rho, \lambda, \mu, \xi, \delta;\beta)$ of normalized analytic bi-univalent functions defined in the open unit disk and associated with the Ruscheweyh's operator. Further, we obtain bounds for the second and third Taylor-Maclaurin coefficients of the functions belong to these subclasses. We also provide relevant connections with earlier investigations of other researchers.

Keywords

[1] R.M. Ali, S.K. Lee, V. Ravichandran and S. Supramaniam, Coefficient estimates for bi-univalent Ma-Minda
starlike and convex functions, Appl. Math. Lett. 25 (2012), 344–351.
[2] D.A. Brannan and J.G. Clunie, Aspects of contemporary complex analysis, Proceedings of the NATO Advanced
Study Institute held at the University of Durham, Durham, Academic Press, New York and London, 1980.
[3] S. Bulut, Faber polynomial coefficient estimates for a subclass of analytic bi-univalent functions, Filomat 30
(2016), no. 6, 1567–1575.
[4] P.L. Duren, Univalent functions, Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, New York,
1983.
[5] B.A. Frasin, Coefficient bounds for certain classes of bi-univalent functions, Hact. J. Math. Stat. 43 (2014), no.
3, 383–389.
[6] B.A. Frasin and M.K. Aouf, New subclasses of bi-univalent functions, Appl. Math. Lett. 24 (2011), no. 9, 1569–
1573.
[7] M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc. 18 (1967), 63–68.
[8] N. Magesh and J. Yamini, Coefficient bounds for a certain subclasses of bi-univalent functions, Int. Math. Forum
8 (2013), no. 27, 1337–1344.
[9] G. Murugusundaramoorthy, N. Mangesh and V. Prameela, Coefficient bounds for certain subclasses of bi-univalent
functions, Abstr. Appl. Anal. 2013 (2013), Article ID 573017, 3 pages.
[10] U.H. Naik and A.B. Patil, On initial coefficient inequalities for certain new subclasses of bi-univalent functions,
J. Egyptian Math. Soc. 25 (2017), no. 3, 291–293.
[11] E. Netanyahu, The minimal distance of the image boundary for the origin and the second coefficient of a univalent
function in |z| < 1, Arch. Ration. Mech. Anal. 32 (1969), 100–112.
[12] A.B. Patil and U.H. Naik, Estimates on initial coefficients of certain subclasses of bi-univalent functions associated
with Al-Oboudi differential operator, J. Indian Math. Soc. 84 (2017), no. 1-2, 73–80.
[13] S. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc. 49 (1975), 109–115.
[14] S.A. Saleh, A.H. El-Qadeem and M.A. Mamon, Some estimation about Tayler-Maclaurin coefficients of generalized
subclasses of bi-univalent functions, Tbilisi Math. J. 13 (2020), no. 4, 23–32.
[15] H.M. Srivastava, S. Gaboury and F. Ghanim, Coefficient estimates for some general subclasses of analytic and
bi-univalent functions, Afr. Mat. 28 (2016), no. 5-6, 693–706.
[16] 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.
[17] H.M. Srivastava, G. Murugusundaramoorthy and N. Mangesh, Certain subclasses of bi-univalent functions associated with the Hohlov operator, Glob. J. Math. Anal. 1 (2013), no. 2, 67–73.[18] B. Srutha Keerthi and B. Raja, Coefficient inequality for certain new subclasses of analytic bi-univalent functions,
Abstr. Appl. Anal. 3 (2013), no. 1, 1–10.
[19] 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.
Volume 13, Issue 2
July 2022
Pages 989-997
  • Receive Date: 27 June 2021
  • Revise Date: 26 January 2022
  • Accept Date: 06 February 2022