SUBCLASSES OF MULTIVALENT FUNCTIONS OF COMPLEX ORDER ASSOCIATED WITH SIGMOID FUNCTION AND BERNOULLI LEMNISCATE

Abstract

In this present work, subclasses of multivalent functions of complex order associated with simple logistic sigmoid activation function and Bernoulli Lemniscate were investigated. Early few coefficient bounds, relevant connection to Fekete-Szego inequalities and second Hankel determinant for the two classes ML p, λ, b, Φm,n and GL p, λ, b, Φm,n were obtained. Our results are new in this direction and give birth to many corollaries.

Keywords

• Analytic function,Univalent function,Subordination,Coefficient bounds

References

1. Ali, R. M., Ravichandran, V. and Seenivasagan, N., (2007), Coefficient bounds for p−valent functions, Appl. Math. Comput., 187, pp. 35-46.
2. Ali, R. M., Ravichandran V. and Lee, K. S., (2009), Subclasses of Multivalent starlike and convex functions, Bull Belg. Math. Soc., 16, pp. 385-394.
3. Fadipe-Joseph, O. A., Oladipo, A. T. and Ezeafulukwe, U. A., (2013), Modified sigmoid function in univalent function theory, Int. J. Math. Sci. Eng. appl., 7, pp. 313-317.
4. Keogh, F. R and Merkes, E. P., (1969), Coefficient Inequality for certain class of analytic functions, Proc. Amer. Math. Soc., 20, pp. 8-12.
5. Murugusundaramoorthy, G. and Janani, T., (2015), Sigmoid function in the space of univalent λ−pseudo starlike functions, Int. J. Pure Appl. Math., 101, pp. 33-41.
6. Nasr, M. A. and Aouf, M. K., (1983), Radius of convexity for the class of starlike functions of complex order, Bull. Fac. Sci. Assiut. Univ. Sect. A, 12, pp. 153-159.
7. Noonman, J. W. and Thomas, D. K., (1976), On the second Hankel determinant of areally mean p−valent functions, Trans. Amer. Math. Soc., 223, pp. 337-346.
8. Oladipo, A. T., (2016), Coefficient inequality for subclass of analytic univalent functions related to simple logistic activation functions, stud. Univ. Babes-Bolyai Math., 61, pp. 45-52.

9. Olatunji, S. O., (2016), Sigmoid function in the space of univalent λ−pseudo starlike function with Sakaguchi type functions, J. Prog. Res. Math., 7, 1164-1172 (2016).
10. Olatunji, S. O., Dansu, E. J. and Abidemi, A., (2017), 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, pp. 97-105.
11. Pommerenke, C., (1975), Univalent functions, Gottingen: Vandenhoeck and Ruprecht.
12. Singh, G. and Singh, G., (2018), Subclasses of multivalent functions of complex order related to sigmoid function, J. Computer Math. Sci., 9, pp. 38-48.
13. Sokol, J. and Thomas, D. K., (2018), Further results on a class of starlike functions related to the Bernoulli Lemniscate, Houston J. Math., 44, pp. 83-95.
14. Wiatrowski, P., (1970), On the coefficients of some family of holomorphic functions, Zesyty Nauk. Uniw. Lodz Nauk. Mat.-Przyrod. (Ser. 2), 39, pp. 75-85.