Certain Families of Multivalent Analytic Functions Associated with Iterations of the Owa-Srivastava Fractional Differintegral Operator

By making use of a multivalent analogue of the Owa-Srivastava fractional differintegral operator and its iterations, certain new families of analytic functions are introduced. Several interesting properties of these function classes, such as convolution theorems, inclusion theorems, and class-preserving transforms, are studied. 1. Introduction Let denote the class of analytic functions in the open unit disk and let be the subclass of consisting of functions represented by the following Taylor-Maclaurin’s series: In a recent paper Patel and Mishra [1] studied several interesting mapping properties of the fractional differintegral operator: defined by where is given by (2). In the particular case and the fractional-differintegral operator was earlier introduced by Owa and Srivastava [2] (see also [3]) and this is popularly known as the Owa-Srivastava operator [4–6]. Moreover, for and , was investigated by Srivastava and Aouf [7] which was further extended to the range ,？？ by Srivastava and Mishra [8]. The following are some of the interesting particular cases of : Furthermore The -iterates of the operator are defined as follows: and, for , Similarly, for , represented by (2), let the operator be defined by the following: and, for , Very recently Srivastava et al. [6] considered the composition of the operators and and introduced the following operator: That is, for , given by (2), we know that The transformation includes, among many, the following two previously studied interesting operators as particular cases. (i)For , , , the fractional derivative operator was recently introduced and investigated by Al-Oboudi and Al-Amoudi [9, 10], in the context of functions represented by conical domains.(ii)For , , , , is the S？l？gean operator [11], which is, in fact, the -iterates of the popular Alexander’s differential transform [12]. We next recall the definition of subordination. Suppose that and in is univalent in . We say that is subordinate to in if and . Considering the function , it is readily checked that satisfies the conditions of the Schwarz lemma and In a broader sense the function is said to be subordinate to the function ( need not be univalent in ), written as if condition (15) holds for some Schwarz function (see [12] for details). We also need the following definition of Hadamard product (or convolution). For the functions and in , given by the following Taylor-Maclaurin’s series their Hadamard product (or convolution) is defined by It is easy to see that . The study of iterations of entire and meromorphic functions, as the number of iterations