Certain subclasses of bi-univalent functions associated with the Hohlov operator
Abstract
In this paper, we introduce and investigate two new subclasses of the function class of bi-univalent functions defined in the open unit disk, which are associated with the Hohlovoperator, that is, a familiar special case of the widely- (and extensively-) investigated Dziok-Srivastava linear operator. Furthermore, we find estimates on the Taylor-Maclaurin coefficients|a_2| and |a_3| for functions in these new subclasses. Several (known or new) consequences of theresults are also pointed out.About this article
References
[1] D.A. Brannan and J.G. Clunie (Editors), Aspects of Contemporary Complex Analysis, Academic Press, London, 1980.
[2] D. A. Brannan, J. Clunie and W. E. Kirwan, Coefficient estimates for a class of star-like functions, Canad. J. Math. 22 (1970), 476–485.
[3] D. A. Brannan and T. S. Taha, On some classes of bi-univalent functions, Studia Univ. Babe¸s-Bolyai Math. 31 (2) (1986), 70–77.
[4] J. Dziok and H. M. Srivastava, Classes of analytic functions associated with the generalized hypergeometric function, Appl. Math. Comput. 103 (1999), 1–13.
[5] J. Dziok and H. M. Srivastava, Certain subclasses of analytic functions associated with the generalized hypergeometric function, Intergral Transforms Spec. Funct. 14 (2003), 7–18.
View more references (12)
[6] B. A. Frasin and M. K. Aouf, New subclasses of bi-univalent functions, Appl. Math. Lett. 24 (2011), 1569–1573.
[7] T. Hayami and S. Owa, Coefficient bounds for bi-univalent functions, Pan Amer. Math. J. 22 (4) (2012), 15–26.
[8] Yu. E. Hohlov, Convolution operators that preserve univalent functions, Ukrain. Mat. Zh. 37 (1985), 220–226.
[9] Yu. E. Hohlov, Hadamard convolutions, hypergeometric functions and linear operators in the class of univalent functions, Dokl. Akad. Nauk Ukrain. SSR Ser. A 1984 (7) (1984), 25–27.
[10] M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc. 18 (1967), 63–68.
[11] X.-F. Li and A.-P. Wang, Two new subclasses of bi-univalent functions, Internat. Math. Forum 7 (2012), 1495–1504.
[12] E. Netanyahu, The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in z < 1, Arch. Rational Mech. Anal. 32 (1969), 100–112.
[13] C. Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht, G¨ottingen, 1975.
[14] H. M. Srivastava, A. K. Mishra and P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math.Lett. 23 (2010), 1188–1192.
[15] T. S. Taha, Topics in Univalent Function Theory, Ph.D. Thesis, University of London, 1981.
[16] Q.-H. Xu, Y.-C. Gui and H. M. Srivastava, Coefficient estimates for a certain subclass of analytic and bi-univalent functions, Appl. Math. Lett. 25 (2012), 990–994.
[17] Q.-H. Xu, H.-G. Xiao and H. M. Srivastava, A certain general subclass of analytic and bi-univalent functions and associated coefficient estimate problems, Appl. Math. Comput. 218 (2012), 11461–11465.