On some applications of the Dziok-Srivastava operator

Carlson and Shaffer [B.C. Carlson, D.B. Shaffer, Starlike and prestarlike hypergeometric functions, SIAM J. Math. Anal. 15 (1984) 737-745] have introduced a linear operator associated with the Gaussian hypergeometric function which has been generalized by Dziok and Srivastava [J. Dziok, H.M. Srivastava, Classes of analytic functions associated with the generalized hypergeometric function, Appl. Math. Comput. 103 (1999) 1-13]. Certain classes of analytic functions defined by means of those operators have been considered in [J. Dziok, H.M. Srivastava, Certain subclasses of analytic functions associated with the generalized hypergeometric function, Integral Transform. Spec. Funct. 14 (2003) 7-18; J. Dziok, H.M. Srivastava, Some subclasses of analytic functions with fixed argument of coefficients associated with the generalized hypergeometric function, Adv. Stud. Contemp. Math. 5 (2) (2002) 115-125] and recently in [J. Dziok, On some applications of the Briot-Bouquet differential subordination, J. Math. Anal. Appl. 328 (2007) 295-301; J. Dziok, Some relations including various linear operators, Demonstratio Math. XL (2007) 77-84; J.-L. Liu, H.M. Srivastava, Certain properties of the Dziok-Srivastava operator, Appl. Math. Comput. 159 (2004) 485-493]. In the present paper, new results for a familiar class of multivalent functions have been obtained. We have used the methods of differential subordination and the properties of Hadamard product.