Convolution Properties of a Subclass of Analytic Univalent Functions

The main objective of the present paper is to investigate some interesting properties on convolution and generalized convolution of functions for the classes and . Our results improve the results of previous authors. 1. Introduction Let denote the class of functions of the form which are analytic in the open unit disk and and satisfy the normalization condition . Let be the subclass of consisting of functions of the form (1) which are also univalent in . Further, denote the subclass of consisting of functions of the form Now for , , and , suppose that denotes the family of analytic univalent functions of the form (1) such that where stands for the Salagean operator introduced by Salagean in [1]. Further, let the subclass consist of functions in such that is of the form (2). Clearly, if , then and for , such that , then The Hadamard product of two functions of the form (1) and is of the form as and for the modified Hadamard product (quasi-convolution) of two functions of the form (2) and we define their convolution as In the present paper, we obtain a number of results on convolution and generalized convolution for the classes and . It is worthy to note that our results are quite new and not explored in the literature. 2. Main Results We first mention a sufficient condition for the function of the form (1) belonging to the class given by the following result which can be established easily. Theorem 1. Let the function be given by (1). Furthermore, let where and . Then . In the following theorem, it is proved that the condition (10) is also necessary for functions of the form (2). Theorem 2. Let be given by (2). Then , if and only if where and . Proof. The if part follows from Theorem 1, so we only need to prove the “only if” part of the theorem. To this end, for functions of the form (2), we notice that the condition is equivalent to The above required condition must hold for all values of in . Upon choosing the values of on the positive real axis and making , we must have which is the required condition. Several authors such as [2–6] studied the convolution properties for the functions with negative as well as positive coefficients only. Their results do not say anything for the function of the form (1). It is therefore natural to ask whether their results can be improved for function of the form (1). In our next theorem, we establish a result on convolution which improves the results of previous authors [2–6] to the case when is of the form (1). It is worth mentioning that the technique employed by us is entirely different from the previous authors.