Convolution Properties of a Subclass of Analytic Univalent Functions

Saurabh Porwal Department of Mathematics, UIET, CSJM University, Kanpur 208024, India Correspondence should be addressed to Saurabh Porwal; saurabhjcb@rediffmail.com 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. The main objective of the present paper is to investigate some interesting properties on convolution and generalized convolution of functions for the classes R(n, α) and R(n, α). Our results improve the results of previous authors.


Introduction
Let denote the class of functions of the form ( Now for 0 ≤ < 1, ∈ = {1, 2, 3, . . .}, 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).
and for , ∈ , such that ≥ , then The Hadamard product of two functions ( ) of the form (1) and ( ) is of the form 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.

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 0 ≤ < 1 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 0 ≤ < 1 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 → 1 − , we must have which is the required condition.
Several authors such as [2][3][4][5][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][3][4][5][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.
0 of nonnegative numbers is said to be a convex null sequence if → 0 as → ∞ and

convex null sequence. Then the function
is analytic in and Re 1 ( ) > 0, ∈ . Lemma 4 is due to Fejér [7]. The assertion of Lemma 5 readily follows by using the Herglotz representation for ( ).
Thus, the proof of Theorem 8 is established.
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 = 1, 2, . . . , ; then the convolution 1 * 2 * ⋅ ⋅ ⋅ belongs to the class (∑ =1 , ), Proof. The proof of the above theorem is much akin to that of Theorem 8. Hence, we omit the details involved.
In our next result we improve the result of Theorem 11 for the case when and are any real numbers such that > 0, > 1.
Remark 13. Here we give some open problems for the readers.
(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).