Subclasses of Bi-Univalent Functions of Complex Order Based on Subordination Conditions Involving Wright Hypergeometric Functions
DOI:
https://doi.org/10.5614/j.math.fund.sci.2015.47.1.5Keywords:
analytic functions, bi-univalent functions, bi-starlike and bi-convex functions, Dziok-Srivastava operator, Gaussian hypergeometric function, univalent functions.Abstract
In this paper, we introduce and investigate a new subclass of bi-univalent functionsΣ of complex order dened in the open unit disk, which are associated with hypergeometric functions and satisfying subordinate conditions. Furthermore, we nd estimates on the Taylor-Maclaurin coecients| a2 | and| a3 | for functions in the new subclass. Several (known or new) consequences of the results are also pointed out.References
Ma, W.C. & Minda, D., A Unified Treatment of Some Special Classes of Functions, in: Proceedings of the Conference on Complex Analysis, Tianjin, 1992, pp. 157-169, Conf. Proc. Lecture Notes Anal. 1. Int. Press, Cambridge, MA, 1994.
Srivastava, H.M., Mishra, A.K. & Gochhayat, P., Certain Subclasses of Analytic and Bi-Univalent Functions, Appl. Math. Lett., 23, pp. 1188-1192, 2010.
Hayami, T. & Owa, S., Coefficient Bounds for Bi-Univalent Functions, Pan Amer. Math. J., 22(4), pp. 15-26, 2012.
Frasin, B.A. & Aouf, M.K., New Subclasses of Bi-Univalent Functions, Appl. Math. Lett., 24, pp. 1569-1573, 2011.
Li, X.-F. & Wang, A.-P., Two New Subclasses of Bi-Univalent Functions, Internat. Math. Forum, 7, pp. 1495-1504, 2012.
Xu, Q.-H., Gui, Y.-C. & Srivastava, H.M., Coefficient Estimates for A Certain Subclass of Analytic and Bi-Univalent Functions, Appl. Math. Lett., 25, pp. 990-994, 2012.
Xu, Q.-H., Xiao, H.-G. & Srivastava, H.M., A Certain General Subclass of Analytic and Bi-Univalent Functions and Associated Coefficient Estimate Problems, Appl. Math. Comput., 218, pp.11461-11465, 2012.
Srivastava, H.M., Some Fox's-Wright Generalized Hypergeometric Functions and Associated Families of Convolution Operators, Applicable Analysis and Discrete Mathematics, 1, pp. 56-71, 2007.
Dziok, J. & Raina, Families of Analytic Functions Associated With The Wright Generalized Hypergeometric Function, Demonstratio Math., 37(3), pp. 533-542, 2004.
Wright, E.M., The Asymptotic Expansion of the Generalized Hypergeometric Function, Proc. London. Math. Soc., 46(3), pp. 89-408, 1946.
Dziok, J. & Srivastava, H.M., Certain Subclasses of Analytic Functions Associated with the Generalized Hypergeometric Function, Intergral Transform Spec. Funct., 14, pp. 7-18, 2003.
Bernardi, S.D., Convex and Starlike Univalent Functions, Trans. Amer. Math. Soc., 135, pp. 429-446, 1969.
Carlson, B.C. & Shaffer, S.B., Starlike and Prestarlike Hypergeometric Functions, SIAM J. Math. Anal., 15, pp. 737-745, 1984.
Libera, R.J., Some Classes of Regular Univalent Functions, Proc. Amer. Math. Soc., 16, pp. 755-758, 1965.
Livingston, A.E., On the Radius of Univalence of Certain Analytic Functions, Proc. Amer. Math. Soc., 17, pp. 352-357, 1966.
Ruscheweyh, St., New Criteria for Univalent Functions, Proc. Amer. Math. Soc., 49, pp. 109-115, 1975.
Srivastava, H.M. & Owa, S., Some Characterization and Distortion Theorems Involving Fractional Calculus, Generalized Hypergeometric Functions, Hadamard Products, Linear Operators and Certain Subclasses of Analytic Functions, Nagoya Math. J., pp. 106, 1-28, 1987.
Deniz, E., Certain Subclasses of Bi-Univalent Functions Satisfying Subordinate Conditions, Journal of Classical Analysis, 2(1), pp. 49-60, 2013.
Ali, R.M., Leo, S.K., Ravichandran, V. & Supramaniam, S., Coefficient Estimates for Bi-Univalent Ma-Minda Star-Like and Convex Functions, Appl. Math. Lett., 25, pp. 344-351, 2012.
Panigarhi, T. & Murugusundaramoorthy, G., Coefficient Bounds for Bi- Univalent Functions Analytic Functions Associated with Hohlov Operator, Proc. Jangjeon Math. Soc, 16(1), pp. 91-100, 2013.
Srivastava, H.M., Murugusundaramoorthy, G. & Magesh, N., Certain Subclasses of Bi-Univalent Functions Associated with The Hohlov Operator, Global Journal of Mathematical Analysis, 1(2), pp. 67-73, 2013.
Pommerenke, C., Univalent Functions, Vandenhoeck & Ruprecht, Gttingen, 1975.
Brannan, D.A. & Taha, T.S., On Some Classes of Bi-Univalent Functions, Studia Univ. Babes-Bolyai Math., 31(2) pp. 70-77, 1986.
