International Journal of Nonlinear Analysis and Applications

Hankel determinant of a subclass of analytic and bi-univalent functions defined by means of subordination and q-differentiation

Document Type : Review articles

Authors

Department of Mathematics, University of Ilorin, PMB 1515, Ilorin, Nigeira

Abstract

In this present article, the $q$-derivative operator and the subordination principle are use to define a class of functions that are analytic and bi-univalent in the open unit disk. Our aim for this class is to obtain the upper bound for the second Hankel determinant for functions in this new subclass of analytic and bi-univalent functions.

Keywords

[1] A. Aral, V. Gupta and R.P. Agarwal, Applications of q-calculus in operator theory, Springer, New York, 2013.
[2] M.H. Annaby and Z.S. Mansour, q-Fractional calculus and equations, Springer Science+Business Media, New York, 2012.
[3] R.O. Ayinla and T.O. Opoola, The Fekete Szego functional and second Hankel determinant for a certain subclass of analytic functions, Appl. Math. 10 (2019), 1071–1078.
[4] E. Deniz, M. Kamali and S. Korkmaz, A certain subclass of bi-univalent functions associated with Bell numbers and q-Srivastava Attiya operator, AIMS Math. 5 (2020), 7259—7271.
[5] O.A. Fadipe-Joseph, W.T. Ademosu and G. Murugusundaramoorthy, Coefficient bounds for a new class of univalent functions involving Sˇalˇagean operator and the modified sigmoid function, Int. J. Nonlinear Anal. Appl. 9 (2018), 59–69.
[6] M.H. Golmohamadi, S. Najafzadeh and M.R. Foroutan, q-Analogue of Liu-Srivastava operator on meromorphic functions based on subordination, Int. J. Nonlinear Anal. Appl. 11 (2020), 219–228.
[7] A.W. Goodman, Univalent functions vol. I, Mariner Publishing Company Inc, Tampa, Florida, 1983.
[8] F.H. Jackson, On q-functions and a certain difference operator, Trans. Royal Soc. Edin. 46 (1908), 64–72.
[9] F.H. Jackson, On q-difference, Amer. J. Math. 32 (1910), 305–314.
[10] A.S. Juma, M.S. Abdulhussain and S.N. Al-khafaji, Certain subclass of p-valent meromorphic Bazilevic functions defined by fractional q-calculus operators, Int. J. Nonlinear Anal. Appl. 9 (2018), 223–230.
[11] A. Junod, Hankel determinants and orthogonal polynomials, Expo. Math. 21 (2003), 63–74.
[12] V. Kac and P. Cheung, Quantum calculus, Springer, New York, 2002.
[13] J.W. Layman, The Hankel transform and some of its properties, J. Integer Seq. 4 (2001), 1–11.
[14] A.O. Lasode and T.O. Opoola, On a generalized class of bi-univalent functions defined by subordination and q-derivative operator, Open J. Math. Anal. 5 (2021), 46–52.
[15] A.O. Lasode and T.O. Opoola, Fekete-Szeg¨o estimates and second Hankel determinant for a generalized subfamily of analytic functions defined by q-differential operator, Gulf J. Math. 11 (2021), no. 2, 36–43.
[16] M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc. 18 (1967), 63–68.
[17] R.J. Libera and E.J. Zlotkiewicz, Early coefficients of the inverse of a regular convex function, Proc. Amer. Math. Soc. 85 (1982), 225–230.
[18] R.J. Libera and E.J. Zlotkiewicz, Coefficient bounds for the inverse of a function with derivative in P, Proc. Amer. Math. Soc. 87 (1983), 251–257.
[19] W.C. Ma and D. Minda, A unified treatment of some special classes of univalent functions, Proc. Conf. Complex Anal., International Press, 1992, p. 157-–169.
[20] E.P. Mazi and T.O. Opoola, A newly defined subclass of bi-univalent functions satisfying subordinate conditions, Mathematica 84 (2019), 146–155.
[21] K.I. Noor, Hankel determinant problem for the class of functions with bounded boundary rotation, Rev. Roum. Math. Pures. Et. Appl. 28 (1983), 731–739.
[22] C. Pommerenke, On the coefficients and Hankel determinants of univalent functions, Proc. Lond. Math. Soc. 41 (1966), 111–122.
[23] C. Pommerenke, On the Hankel determinants of univalent functions, Mathematika 14 (1967), 108–112.
[24] S. Salehian and A. Motamednezhad, New subclasses of meromorphic bi-univalent functions by associated with subordinate, Int. J. Nonlinear Anal. Appl. 12 (2021), 61–74.
[25] H.M. Srivastava and D. Bansal, Coefficient estimates for a subclass of analytic and bi-univalent functions, J. Egyptian Math. Soc. 23 (2015), 242–246.
International Journal of Nonlinear Analysis and Applications
Volume 13, Issue 2
July 2022
Pages 3105-3114
  • Receive Date: 18 September 2021
  • Revise Date: 13 February 2022
  • Accept Date: 13 March 2022