On quasi-subordination for bi-univalency involving generalized distribution series associated with remodelled $s$-sigmoid function

Document Type : Research Paper


1 Department of Mathematical Sciences, Federal University of Technology, Akure, Nigeria

2 Department of Pure and Applied Mathematics\\Ladoke Akintola University of Technology, P.M.B. 4000, Ogbomoso, Nigeria

3 Institute of Mathematics and Applications, Andharua, Bhubaneswar-751029, Odisha, India


In this paper, the authors investigate the bi-univalency of the generalized distribution series associated with quasi-subordination and remodelled $s$-sigmoid function. The early few coefficients are obtained to achieve our goal. The results obtained are new to the history of bi-univalency.


[1] A. Akg¨ul, On quasi subordination for analytic and bi-univalent function class, arXiv:1709.04503v1 [math.CV] (2017).
[2] A. Akg¨ul and L.-I. Cotˆirlˇa, Coefficient estimates for a family of starlike functions endowed with quasisubordination on conic domain, Symmetry 14 (2022), no. 3, 582.
[3] O. Altinas and S. Owa, Majorizations and quasi-subordinations for certain analytic functions, Proc. Japan Acad. A 68 (1992), no. 7, 181–185.
[4] U.A. Ezeafulukwe, M. Darus and O.A. Fadipe-Joseph, The q−analogue of sigmoid function in the space of univalent λ−pseudo starlike function, Int. J. Math. Comput. Sci. 15 (2020), no.2, 621–626.
[5] O.A. Fadipe-Joseph, A.T. Oladipo and U.A. Ezeafulukwe, Modified sigmoid function in univalent function theory, Int. J. Math. Sci. Eng. Appl. 7 (2013), no. 7, 313–317.
[6] V.B. Girgaonkar, S.B. Joshi and P.P. Yadav, Certain special subclasses of analytic function associated with biunivalent functions, Palestine J. Math. 6 (2017), no. 2, 617–623.
[7] P. Gurusamy, J. Sok´o land S. Sivasubramanian, The Fekete-Szeg¨o functional associated with kth root transformation using quasi-subordination, Comptes Rendus Math. 353 (2015), no. 7, 617–622.
[8] J.O. Hamzat, A.T. Oladipo and G. I. Oros, Bi-univalent problems involving certain new subclasses of generalized multiplier transform on analytic functions associated with modified sigmoid function, Symmetry 14 (2022), no. 7, 1479.
[9] F.R.Keogh and E.P. Merkes, A coefficient inequality for certain classes of analytic functions, Proc. Amer. Math. Soc. 20 (1969), 8–12.
[10] R.S. Khatu, U.H. Naik and A.B. Patil, Estimation on initial coefficient bounds of generalized subclasses of biunivalent functions, Int. J. Nonlinear Anal. Appl. 13 (2022), no. 2, 989–997.
[11] S.Y. Lee, Quasi-subordinate functions and coefficient conjectures, J. Korean Math. Soc. 12 (1975), no. 1, 43–50.
[12] P. Long, G. Murugusundaramoorthy, H. Tang and W. Wang, Subclasses of analytic and bi-univalent functions involving a generalized Mittag-Leffler function based on quasi-subordination, J. Math. Comput. Sci. 26 (2022), 379–394.
[13] W.C. Ma and D. Minda, A unified treatment of some special cases of univalent functions, Proc. Conf. Complex Anal., Tianjin, 1992, pp. 157–169.
[14] S.S. Miller and P.T. Mocanu, Differential Subordinations Theory and Applications, Series of Monographs and Text Books in Pure and Applied Mathematics, 225, Marcel Dekker, New York, 2000.
[15] M.H. Mohd and M. Darus, Fekete-Szeg¨o problems for Quasi-subordination classes, Abstr. Appl. Anal. 2012 (2012), Art. ID 192956, 14 pages.
[16] G. Murugusundaramoorthy and L.-I. Cotˆirlˇa, Bi-univalent functions of complex order defined by Hohlov operator associated with legendrae polynomial, AIMS Math. 7 (2022), no. 5, 8733–8750.
[17] G. Murugusundaramoorthy and T. Janani, Sigmoid function in the space of univalent λ-pseudo starlike functions, Int. J. Pure Appl. Math. Sci. 101 (2015), no. 1, 33–41.
[18] Z. Nehari, Conformal Mappings, McGraw-Hill, New York, 1952.
[19] A.T. Oladipo, Generalized distribution associated with univant functions in conical domain, Anal. Univer. Oradea Fasc Mat. Tom 26 (2019), no. 1, 163–169.
[20] A.T. Oladip, Analytic univalent function defined by generalized discrete probability distribution, Earthline J. Math. Sci. 5 (2021), no. 1, 169–178.
[21] S.O. Olatunji, Sigmoid function in the space of univalent λ-Pseudo starlike function with Sakaguchi type functions, J. Prog. Res. Math. 7 (2016), no. 4, 1164–1172.
[22] S.O. Olatunji, Fekete-Szeg¨o inequalities on certain subclasses of analytic functions defined byλ−pseudoq−difference operator associated with s−sigmoid function, Bol. Soc. Mat. Mexica. 28 (2022), no. 2, 1–15.
[23] S.O. Olatunji and S. Altinkaya, Generalized distribution associated with quasi-subordination in terms of error function and bell numbers, Jordan J. Math. Stat. 14 (2021), no. 1, 97–109.
[24] S.O. Olatunji and E.J. Dansu, Coefficient estimates for Bazileviˇc Ma-Minda functions in the space of sigmoid function, Malaya J. Mat. 4 (2016), no. 3, 505–512.
[25] S.O. Olatunji and P.T. Ajai, On subclasses of bi-univalent functions of Bazileviˇc type involving linear and Salageanoperator, Int. J. Pure Appl. Math. 92 (2014), no. 5, 645–656.
[26] S.O. Olatunji, E.J. Dansu and A. Abidemi, On a Sakaguchi type class of analytic functions associated with quasi-subordination in the space of modified sigmoid functions, Electronic J. Math. Anal. Appl. 5 (2017), no. 1, 97–105.
[27] S.O. Olatunji, A.M. Gbolagade, T. Anake and O.A. Fadipe-Joseph, Sigmoid function in the space of univalent function of Bazilevic type, Sci. Magna 97 (2013), no. 3, 43–51.
[28] G.I. Oros and L.-I. Cotˆirlˇa, Coefficient estimates and the Fekete-Szeg¨o problem for new classes of m-fold symmetric bi-univalent functions, Math. 10 (2022), no. 1, 129.
[29] T. Panigrahi, A certain new class of analytic functions associated with quasi-subordination in the space of modified sigmoid functions, Anal. Univer. Oradea Fasc Mat. Tom 25 (2018), no. 2, 77–83.
[30] S. Porwal, Generalized distribution and its geometric properties associated with univalent functions, J. Complex Annal. 2018 (2018), Art. ID 8654506.
[31] M.S. Robertson, Quasi-subordination and coefficient conjectures, Bull. Amer. Math. Soc. 76 (1970), 1–9.
[32] T. Panigrahi and R. K. Raina, Fekete-Szeg¨o coefficient functional for quasi-subordination class, Afr. Mat. 28 (2017), no. 5, 707–716.
Volume 14, Issue 1
January 2023
Pages 2213-2222
  • Receive Date: 05 September 2022
  • Revise Date: 06 December 2022
  • Accept Date: 09 December 2022