Research Article | Open Access
Saurabh Porwal, "Convolution Properties of a Subclass of Analytic Univalent Functions", International Scholarly Research Notices, vol. 2014, Article ID 190898, 4 pages, 2014. https://doi.org/10.1155/2014/190898
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.
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
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 .
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. 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 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
Since from (20) we have
and since , from Lemma 6, we have
From (28), (30), and Lemma 5 we immediately have (27).
This establishes the proof of 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.
In the special case, if we take , then we have
Proof. Since , by using Theorem 2, we have
From (40) we have Similarly for we have
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 , .
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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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.