International Journal of Nonlinear Analysis and Applications

Coefficient bounds for a generalized subclass of bounded turning functions associated with Sigmoid function

Document Type : Research Paper

Authors

1 Department of Mathematics, Khalsa College, Amritsar-143001, Punjab, India

2 Department of Mathematics, G.N.D.U. College, Chungh-143304, Tarn-Taran(Punjab), India

Abstract

In this paper, we introduce a subclass of analytic functions associated with the Sigmoid function and determine the upper bounds for various coefficient functionals such as  Fekete-Szego functional, second Hankel determinant, Zalcman functional and third Hankel determinant. Also, the concept is extended to two-fold and three-fold symmetric functions. The results proved earlier, follow as special cases of the results of this paper.

Keywords

[1] S. Altinkaya and S. Yalcin, Third Hankel determinant for Bazilevic functions, Adv. Math., Sci. J. 5 (2016), no. 2, 91–96.
[2] M. Arif, M. Raza, H. Tang, S. Hussain and H. Khan, Hankel determinant of order three for familiar subsets of analytic functions related with sine function, Open Math. 17 (2019), 1615–1630.
[3] K.O. Babalola, On H3(1) Hankel determinant for some classes of univalent functions, Inequal. Th. Appl. 6 (2010), 1–7.
[4] R. Bucur, D. Breaz and L. Georgescu, Third Hankel determinant for a class of analytic functions with respect to symmetric points, Acta Univ. Apulensis 42 (2015), 79–86.
[5] C. Caratheodory, Uber den variabilitatsbereich der fourier’schen konstanten von positiven harmonishen funktionen, Rend. Circ. Mat. Palermo 32 (1911), 193–217.
[6] R. Ehrenborg, The Hankel determinant of exponential polynomials, Amer. Math. Month. 107 (2000), 557–560.
[7] M. Fekete and G. Szego, Eine Bemer Kung uber ungerade schlichte Functionen, J. Lond. Math. Soc. 8 (1933), 85–89.
[8] P. Goel and S.S. Kumar, Certain class of starlike functions associated with modified Sigmoid function, Bull. Malays. Math. Sci. Soc. 43 (2020), 957–991.
[9] T. Hayami and S. Owa, Generalized Hankel determinant for certain classes, Int. J. Math. Anal. 4 (2010), 2573–2585.
[10] A. Janteng, S.A. Halim and M. Darus, Hankel determinant for starlike and convex functions, Int. J. Math. Anal. 1 (2007), 619–625.
[11] O.A.F. Joseph, B.O. Moses and M.O. Oluwayemi, Certain new classes of analytic functions defined by using Sigmoid function, Adv. Math.: Sci. J. (2016), no. 11, 83–89.
[12] S.R. Keogh and E.P. Merkes, A coefficient inequality for certain subclasses of analytic functions, Proc. Amer. Math. Soc. 20 (1969), 8–12.
[13] M.G. Khan, N.E. Cho, T.G. Shaba, B. Ahmed and W.K. Mashwani, Coefficient functionals for a class of bounded turning functions related to modified Sigmoid function, AIMS Math. 7 (2021), no. 2, 3133–3149.
[14] M.G. Khan, B. Ahmed, G.M. Moorthy, R. Chinram and W.K. Mashwani, Applications of modified Sigmoid function to a class of starlike functions, J. Funct. Spaces 2020 (2020), Article ID. 8844814.
[15] J.W. Layman, The Hankel transform and some of its properties, J. Integer Seq. 4 (2001), 1–11.
[16] R.J. Libera and E.J. Zlotkiewiez, Early coefficients of the inverse of a regular convex function, Proc. Amer. Math. Soc. 85(1982), 225–230.
[17] R.J. Libera and E.J. Zlotkiewiez, Coefficient bounds for the inverse of a function with derivative in P, Proc. Amer. Math. Soc. 87 (1983), 251–257.
[18] E. Lindelof, Memoire sur certaines inegualites dans la theorie des fonctions monog`enes et sur quelques proprietes nouvelles de ces fonctions dans la voisinage d’un point singulier essentiel, Acta. Soc. Sci. Fennica 35 (1909), no. 7, 1–35.
[19] W. Ma, Generalized Zalcman conjecture for starlike and typically real functions, J. Math. Anal. Appl. 234 (1999), 328–329.
[20] W. Ma and D. Minda, A unified treatment of some special classes of univalent functions, Proc. Conf. Complex Anal.: Int. Press Inc: Somerville, MA, USA, 1992.
[21] T.H. MacGregor, Functions whose derivative has a positive real part, Trans. Amer. Math. Soc. 104 (1962), 532–537.
[22] T.H. MacGregor, The radius of univalence of certain analytic functions, Proc. Amer. Math. Soc. 14 (1963), 514–520.
[23] B.S. Mehrok and G. Singh, Estimate of second Hankel determinant for certain classes of analytic functions, Scient. Magna 8 (2012), no. 3, 85–94.
[24] G. Murugusundramurthi and N. Magesh, Coefficient inequalities for certain classes of analytic functions associated with Hankel determinant, Bull. Math. Anal. Appl. 1 (2009), no. 3, 85–89.
[25] J.W. Noonan and D.K. Thomas, On the second Hankel determinant of a really mean p-valent functions, Trans. Amer. Math. Soc. 223 (1976), no. 2, 337–346.
[26] K.I. Noor, Hankel determinant problem for the class of functions with bounded boundary rotation, Rev. Roumain. Math. Pures Appl. 28 (1983), no. 8, 731–739.
[27] Ch. Pommerenke, Univalent functions, Math. Lehrbucher, vandenhoeck and Ruprecht, Gottingen, 1975.
[28] C. Ramachandran and K. Dhanalakshmi, The Fekete-Szego problem for a subclass of analytic functions related to Sigmoid function, Int. J. Pure Appl. Math. 111 (2017), no. 3, 389–398.
[29] M. O. Reade, On close-to-convex univalent functions, Michigan Math. J. 3 (1955-56), 59–62.
[30] G. Shanmugam, B. Adolf Stephen and K.O. Babalola, Third Hankel determinant for α starlike functions, Gulf J. Math. 2 (2014), no. 2, 107–113.
[31] G. Singh, Hankel determinant for a new subclass of analytic functions, Scient. Magna, 8 (2012), no. 4, 61–65.
[32] G. Singh and G. Singh, On third Hankel determinant for a subclass of analytic functions, Open Sci. J. Math. Appl. 3 (2015), no. 6, 172–175.
[33] G. Singh, G. Singh and G. Singh, Certain subclasses of multivalent functions defined with generalized Salagean operator and related to Sigmoid function and lemniscate of Bernoulli, J. Fract. Calc. Appl. 13 (2022), no. 1, 65–81.
International Journal of Nonlinear Analysis and Applications
Volume 14, Issue 11
November 2023
Pages 1-10
  • Receive Date: 11 January 2023
  • Revise Date: 19 February 2023
  • Accept Date: 24 February 2023