10008113

The coefficients estimate problem for Taylor-Maclaurin series is still an open problem especially for a function in the subclass of bi-univalent functions. A function f ϵ A is said to be bi-univalent in the open unit disk D if both f and f-1 are univalent in D. The symbol A denotes the class of all analytic functions f in D and it is normalized by the conditions f(0) = f’(0) – 1=0. The class of bi-univalent is denoted by The subordination concept is used in determining second and third Taylor-Maclaurin coefficients. The upper bound for second and third coefficients is estimated for functions in the subclasses of bi-univalent functions which are subordinated to the function φ. An analytic function f is subordinate to an analytic function g if there is an analytic function w defined on D with w(0) = 0 and |w(z)| < 1 satisfying f(z) = g[w(z)]. In this paper, two subclasses of bi-univalent functions associated with Hohlov operator are introduced. The bound for second and third coefficients of functions in these subclasses is determined using subordination. The findings would generalize the previous related works of several earlier authors.

[1] Y. E. Hohlov, “Convolution operators that preserve univalent functions” Ukrain. Mat. Zh. vol. 37, pp 220-226, 1985.

[2] P. L. Duren, “Univalent Function”, Springer, New York, vol 259, 1983.

[3] M. Lewin, “On a coefficient problem for bi-univalent functions” Proc. Amer. Math. Soc. vol. 18, pp. 63-68, 1967.

[4] A. W. Kedzierawski, “Some remarks on bi-univalent functions” Annales Universitatis Mariae Curie-Skolodowska Sectio A, vol. 39, pp. 77-81, 1985.

[5] D. A. Brannan and T. S. Taha, “On some classes of bi-univalent functions” Studia Univ. babes-Bolyai Math, vol. 31, no. 2, pp. 70-77, 1986.

[6] W. C. Ma and D. Minda, “A unified treatment of some special classes of univalent functions” Conf. Proc. Lecture Note Anal. I, Int. Press, Cambridge, MA, 1994.

[7] S. Bulut, N. Magesh and C. Abirami. “A comprehensive class of analytic bi-univalent functions by means chebyshev polynomials” Journal of Fractional Calculus and Applications, vol. 8, no. 2, pp. 32-39, July 2017.

[8] H. M. Srivastava, S. S. Eker and R. M. Ali. “Coefficient bounds for a certain class of analytic and bi-univalent functions” Filomat, vol. 29, no. 8, pp. 1839-1845, 2015.

[9] S. K. Lee, V. Ravichandran and S. Supramaniam, “Initial coefficients of bi-univalent functions” Abstract and Applied Analysis, vol. 2014, Article ID 640856, 6 pages.

[10] G. Murugusundaramoothy, N. Magesh and V. Prameela, “Coefficient bounds for certain subclasses of bi-univalent function” Abstract and Applied Analysis, vol. 2013, Article ID 573017, 3 pages.

[11] E. Deniz, “Certain subclasses of bi-univalent functions satisfying subordinate conditions” Journal of Classical Analysis, vol. 2, no. 1, pp. 49-60, 2013.

[12] R. M. Ali, S. K. Lee, V. Ravichandran and S. Supramaniam, “Coefficient estimates for bi-univalent Ma-Minda starlike and convex” Applied Mathematics Letters, vol. 25, pp. 344-351, 2012.

[13] B. A. Frasin and M. K. Aouf, “New subclasses of bi-univalent functions” Applied Mathematics Letters, vol. 24, pp. 1569-1573, 2011.

[14] H. M. Srivastava, A. K. Mishra and P. Gochhayat, “Certain subclasses of analytic and bi-univalent functions” Applied Math. Letters, vol. 23, pp. 1188-1192, 2010.

[15] O. S. Babu, C. Selvaraj and G. Murugusundaramoorthy, “Subclasses of bi-univalent functions based on Hohlov operator” International Journal of Pure and Applied Mathematics, vol. 102, no. 3, pp. 473-482, 2015.

[16] J. Jothibasu, “certain subclasses of bi-univalent functions defined by Salagean operator” Eletronic Journal of Mathematical Analysis and Applications, vol. 3, no 1, pp. 150-157, 2015.

[17] Z. Peng, G. Murugusundaramoorthy and T.Janani, “Coefficient estimate of bi-univalent functions of complex order associated with Hohlov operator” Journal of Complex Analysis, vol. 2014, article ID 693908, 6 pages.

[18] C. Selvaraj and G. Thirupathi, “Coefficient bounds for a subclass of bi-univalent functions using differential operators” Ann. Acad. Rom. Sci. Ser. Math. Appl., vol. 6, no. 2, pp. 204-213, 2014.

[19] H. M. Srivastava, G. Murugusundaramoothy and N. Magesh, “Certain subclasses of bi-univalent functions associated with the Hohlov operator” Global Journal of Mathematical Analysis, vol. 1, no. 2, pp. 67-73, 2013.

[20] T. Panigrahi and G. Murugusundaramoorthy, “Coefficient bounds for bi-univalent analytic functions associated with Hohlov operator” Proceedings of the Jangjeon Math. Soc. Vol. 16, no. 1, pp. 91-100, 2013.

[21] M. K. Aouf, R. M. El-Ashwah and A. M. Abd-Eltawab, “New subclasses of bi-univalent functions involving Dziok-Srivastava operator” ISRN Mathematical Analysis, vol. 2013, article ID 387178, 5 pages.