Second Hankel Determinant of Bi-univalent Functions


  • Kiu Shu Huey Faculty of Science and Natural Resources, Universiti Malaysia Sabah, Malaysia
  • Aini Janteng Faculty of Science and Natural Resources, Universiti Malaysia Sabah, Malaysia
  • Jaludin Janteng Labuan Faculty of International Finance, Universiti Malaysia Sabah, Malaysia
  • Andy Liew Pik Hern Faculty of Science and Natural Resources, Universiti Malaysia Sabah, Malaysia



Analytic functions, Bi-univalent functions, Second Hankel determinant, Subordination


Let  be the class of functions which are analytic in the open unit disk and having the form . Denote  to be the class for all functions in  that are univalent in . Then, let  denote the class of bi-univalent functions in . In this paper, we obtain the second Hankel determinant for certain classes of analytic bi-univalent function which are defined by subordinations in the open unit disk . In particular, we determine the initial coefficients  and  and obtained the upper bound for the functional  of functions  in the classes of analytic bi-univalent function which are defined by subordinations in .


-[1] Aldweby, H. & Darus, M. (2017). On Fekete-Szegö problems for certain subclasses defined by q-derivative. Journal of Function Spaces, 2017(3).

Aldweby, H. & Darus, M. (2016). Coefficient estimates of classes of q-starlike and q-convex functions. Advanced Studies in Contemporary Mathematics, 26(1), 21-26.

Brannan, D. A. & Clunie, J. (1979). Aspects of contemporary complex analysis. In: Proceedings of the NATO Advanced Study Institute Held at University of Durham. Academic, New York.

Brown, J. W. & Churcill, R. V. (2009). Complex variables and applications. 8th edition. McGraw-Hill Companies, New York.

Choo, C. P. & Janteng, A. (2013). Estimate on the second hankel functional for a subclass of close-to-convex functions with respect to symmetric points. International Journal of Mathematical Analysis, 7(13-16), 781-788.

Duren, P. L. (1983). Univalent Functions, Grundlehren der Mathematischen Wissenschaften. Vol. 259. Springer, New York.

Fekete, M. & Szegö, G. (1933). Eine Bemerkung Uber Ungerade Schlichte Funktionen. J. Lond. Math. Soc., 8, 85-89.

Frasin, B. A. & Aouf, M. K. (2011). New subclasses of bi-univalent functions. Appl. Math. Lett., 24, 1569-1573.

Grenander, U. & Szegö, G. (1958). Toeplitz forms and their applications. University of California Press, Berkeley & Los Angeles.

Halim, S. A., Janteng, A. & Darus, M. (2006). Classes with negative coefficients and starlike with respect to other points II. Tamkang Journal of Mathematics, 37(4), 345-354.

Jackson, F. H. (1909). On q- functions and a certain difference operator. Transaction of the Royal Society of Edinburgh, 46, 253-281.

Jahangiri, J. M. & Hamidi, S. G. (2013). Coefficient estimates for certain classes of bi-univalent functions. Int. J. Math. Math. Sci., 4.

Janteng, A. & Halim, S. A. (2009). A subclass of quasi-convex functions with respect to symmetric points, Applied Mathematical Sciences, 3(12), 551-556.

Kanas, S., Adegani, E. A. & Zireh, A. (2017). An unified approach to second Hankel determinant of bi-subordinate functions. Mediterranean Journal of Mathematics, 14(6), 233-244.

Kodzron, M. J. (2007). A guide to some of the complex analysis necessary for understanding the Schramm-Loewner evolution (SLE). The Basic Theory of Univalent Function, University of Regina, Canada.

Lewin, M. (1967). On a coefficient problem for bi-univalent functions. Proc. Am. Math. Soc., 18, 63-68.

Li, X. F. & Wang, A. P. (2012). Two new subclasses of bi-univalent functions. Int. Math. Forum, 7, 1495-1504.

Liew, A. P. H., Janteng, A. & Omar, R. (2020). Hankel determinant H2(3) for certain subclasses of univalent functions. Mathematics and Statistics, 8(5), 566-569.

Ma, W. C. & Minda, D. (1994). A unified treatment of some special classes of univalent functions. Conf. Proc. Lecture Note Anal. I. Int. Press, Cambridge, MA, 157-169.

Mustafa, N., Murugusundaramoorthy, G. & Janani, T. (2018). Second Hankel determinant for a certain subclass of bi-univalent functions. Mediterranean Journal of Mathematics, 15(3), 1-17.

Noonan, J. W. & Thomas, D. K. (1976). On the second Hankel determinant of areally mean p-valent functions. Trans. Am. Math. Soc., 223, 337-346.

Pommerenke, C. (1975). Univalent Functions. Vandenhoeck and Rupercht, Götingen.

Srivastava, H. M., Mishra, A. K. & Gochhayat, P. (2010). Certain subclasses of analytic and biunivalent functions, Appl. Math. Lett., 23, 1188-1192.

Zireh, A. & Analouei Adegani, E. (2016). Coefficient estimates for a subclass of analytic and bi-univalent functions. Bull. Iran. Math. Soc., 42, 881-889.