科研之友
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摘要
We introduce a certain subclass of multivalent analytic functions by making use of the principle of subordination between these functions and Cˋtas operator. Such results as subordination and superordination properties, convolution properties, inclusion relationships, distortion theorems, inequality properties, and sufficient conditions for multivalent starlikeness are provide. The results presented here would provide extensions of those given in earlier works. Several other new results are also obtained. 1. Introduction Let denote the class of functions of the following form: which are analytic in the open unit disk . For simplicity, we write A function is said to be in the class of -valent starlike functions of order in if it satisfies the following inequality: Let be the class of analytic functions of the following form: Let , where is given by (1.1) and is defined by Then the Hadanard product (or convolution) of the functions and is defined by We consider the following multiplier transformations. Definition 1.1 (see [1]). Let . For , define the multiplier transformations on by the following infinite series: It is easily verified from (1.7), that It should be remarked that the class of multiplier transforms is a generalization of several other linear operators considered, in earlier investigations (see [2每12]). If is given by (1.1), then we have where In particular, we set For two functions and , analytic in , we say that the function is subordinate to in , and write if there exists a Schwarz function , which is analytic in with such that Indeed, it is known that Furthermore, if the function is univalent in , then we have the following equivalence: By making use of the linear operator and the above-mentioned principle of subordination between analytic functions, we introduce and investigate the following subclass of the class of -valent analytic functions. Definition 1.2. A function is said to be in the class if it satisfies the following subordination condition: where (and throughout this paper unless otherwise mentioned) the parameters , and are constrained as follows: For simplicity, we write Clearly, the class is a subclass of the familiar class of Bazileviˋ functions of type . If we set in the class , which was studied by Liu [13]. In particular, Zhu [14] determined the sufficient conditions such that . C tas [1, 5, 15], Cho and Srivastava [6], Cho and Kim [7], and Kumar et al. [10] obtained many interesting results associated with the multiplier operator. In the present paper, we aim at proving such results as subordination and superordination
