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ISRN Mathematical Analysis
Volume 2014 (2014), Article ID 190898, 4 pages
http://dx.doi.org/10.1155/2014/190898
Research Article

Convolution Properties of a Subclass of Analytic Univalent Functions

Department of Mathematics, UIET, CSJM University, Kanpur 208024, India

Received 26 September 2013; Accepted 19 November 2013; Published 2 February 2014

Academic Editors: K. Lurie and J.-L. Wu

Copyright © 2014 Saurabh Porwal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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 [26] 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 [26] 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. For this, we will require the following definition and lemmas.

Definition 3. A sequence of nonnegative numbers is said to be a convex null sequence if as and

Lemma 4. Let be a convex null sequence. Then the function is analytic in and , .

Lemma 5. Let be analytic in , , and in . For functions analytic in , the convolution function takes values in the convex hull of the image on under .

Lemma 4 is due to Fejér [7]. The assertion of Lemma 5 readily follows by using the Herglotz representation for .

Lemma 6. For , , let . Then

Proof. Let be given by (1). Since , hence, by definition which is equivalent to and hence, We observe that the sequence defined by and , , , is a convex null sequence; we have in view of Lemma 4 that Now and making use of (20), (21), and Lemma 5, we conclude that

Theorem 7. If and , where ,  ,  , then so does their Hadamard product (convolution)

Proof. To prove that we have to show that which is equivalent to or Since from (20) we have and since , from Lemma 6, we have or From (28), (30), and Lemma 5 we immediately have (27).
This establishes the proof of Theorem 7.

In our next result we improve the results of Theorem 7 for functions of the form (2).

Theorem 8. Let the functions , defined by (2), (8) be in the classes , , respectively, where , , and ; then defined by (9) is in the class , where .

Proof. Since , then, by Theorem 2, we have Similarly, since , we have Therefore, from (31), we have
Now, for the convolution function we have Thus, the proof of Theorem 8 is established.

Remark 9. From (4) and (5), we see that Thus, the result of Theorem 8 provides smaller class in comparison to the class given by Theorem 7.

Theorem 10. Let the functions defined as belong to the class for every ; then the convolution belongs to the class , where .

Proof. The proof of the above theorem is much akin to that of Theorem 8. Hence, we omit the details involved.

For any real numbers and , we define that the generalized convolution for functions and is of the form (2) and (8) as

In the special case, if we take , then we have

Theorem 11. If the functions and defined by (2) and (8) are in the classes and  , respectively, then where , and .

Proof. Since , by using Theorem 2, we have
From (40) we have Similarly for we have
Now
Since it suffices to show that if
For this we have to show that L.H.S. of (45) is bounded by which is equivalent to .

In our next result we improve the result of Theorem 11 for the case when and are any real numbers such that , .

Theorem 12. For , , , let and of the form (2) and (8) belong to the classes and , respectively; then

Proof. Since , by using Theorem 2, we have or ,  .
Equivalently, , , and since , we have or
Now Therefore, .

Remark 13. Here we give some open problems for the readers.
(1) Find and such that Theorem 12 holds.
(2) The result of Theorem 12 holds only for functions of the form (2); that is, the coefficients of expansion are negative. Therefore, it is natural to ask what is the analogue results for the function of the form (1).

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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