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Properties for certain subclasses of meromorphic functions defined by a multiplier transformation
Journal of Inequalities and Applications volume 2013, Article number: 46 (2013)
Some inclusion and convolution properties of certain subclasses of meromorphic functions associated with a family of multiplier transformations, which are defined by means of the Hadamard product (or convolution), are investigated. We also obtain closure properties for certain integral operators.
Let denote the class of analytic functions f in the open unit disk with the usual normalization . Let and denote the subclasses of consisting of starlike and convex functions of order α () and let and . If f and g are analytic in , we say that f is subordinate to g in , written as or , if there exists a Schwarz function w such that ().
A function is said to be prestarlike of order α in if
where denotes the familiar Hadamard product (or convolution) of two analytic functions f and g in . We denote this class by (see, for details, ). We note that and .
Let be the class of all functions h which are analytic and univalent in and for which is convex with .
Let ℳ denote the class of functions of the form
which are analytic in the punctured open unit disk .
For any , we denote the multiplier transformations of functions by
Obviously, we have
for all nonnegative integers s and t. The operators and are the multiplier transformations introduced and studied by Sarangi and Uraligaddi  and Uralegaddi and Somanatha [3, 4], respectively. Analogous to , we here define a new multiplier transformation as follows.
Let , , and let be such that
where is the Pochhammer symbol (or the shifted factorial) defined (in terms of the gamma function) by
We note that and . It is easily verified from (1.1) that
The definition (1.1) of the multiplier transformation is motivated essentially by the Choi-Saigo-Srivastava operator  for analytic functions, which includes the Noor integral operator studied by Liu  (also, see [7–9]).
We also define the function by
By using the operator , we introduce the following class of analytic functions for , , , and :
In the present paper, we derive some inclusion relations, convolution properties and integral preserving properties for the class .
The following lemmas will be required in our investigation.
Lemma 1.1 [, Lemma 2, p.192]
Let g be analytic in and h be analytic and convex univalent in U with . If
and is the best dominant of (1.5).
Lemma 1.2 [, Theorem 2.4, p.54]
Let and . Then for any analytic function F in ,
where denotes the convex hull of .
Lemma 1.3 [, Lemma 5, p.656]
Let . Then
where ϕ is given by (1.4).
2 Inclusion relations
Theorem 2.1 If , then
Then the function g is analytic in with . Differentiating both sides of (2.1), we have
Hence an application of Lemma 1.1 with yields
Since and h is convex univalent in U, it follows from (2.1), (2.2) and (2.3) that
Therefore , and so we complete the proof of Theorem 2.1. □
Theorem 2.2 If , then
Proof Let . Then
In view of Lemma 1.3, we see that the function has the Herglotz representation
where is a probability measure defined on the unit circle and
Since h is convex univalent in , it follows from (2.4) and (2.5) that
which completes the proof of Theorem 2.2. □
Theorem 2.3 If , then
Then from (1.4) and (2.6), we have
Differentiating both sides of (2.6) and using (1.4), we obtain
By a simple calculation with (2.7) and (2.8), we get
If , then it follows from (2.9) that
Hence an application of Lemma 1.1 yields
which shows that
Theorem 2.4 If and , then
Proof By using the same techniques as in the proof of Theorem 2.3 and (1.5), we have Theorem 2.4 and so we omit the detailed proof involved. □
Theorem 2.5 Let , and . If , where
then . The bound is sharp for the function
Then we have
Hence an application of Lemma 1.1 yields
If , where is given by (2.10), then from (2.13), we have
By using the Herglotz representation for ψ, it follows from (2.11) and (2.12) that
since h is convex univalent in . This shows that .
For and defined by
it is easy to verify that
Thus . Furthermore, for , we have
which implies that . Hence the bound cannot be increased when (). □
3 Convolution properties
Theorem 3.1 If and
Proof Let and . Then we have
The remaining part of the proof of Theorem 3.1 is similar to that of Theorem 2.2, and so we omit the details involved. □
Corollary 3.1 Let be given by (1.1). Then the function
is also in the class .
while, it is known  that
In view of (3.1) and (3.2), an application of Theorem 3.1 leads to . □
Theorem 3.2 If and
By using a similar method as in the proof of Theorem 3.1, we have
Since h is convex univalent in , it follows from (3.3) and Lemma 1.2 that Theorem 3.2 holds true. □
If we take and in Theorem 3.2, we have the following corollary.
Corollary 3.2 If and satisfies one of the following conditions:
is convex univalent in
4 Integral operators
Theorem 4.1 If , then the function F defined by
is in the class , where
Proof Let . Then from (4.1), we obtain
Define the function G by
Differentiating both sides of (4.3) with respect to z, we get
Furthermore, it follows from (4.2), (4.3) and (4.4) that
Since , from (4.5), we have
and so an application of Lemma 1.1 yields
Therefore we conclude that
Theorem 4.2 If and F are defined as in Theorem 4.1, if
then , where
Then G is analytic in with and
It follows from (4.2), (4.6), (4.7) and (4.8) that
Therefore, by Lemma 1.1, we conclude that Theorem 4.2 holds true as stated. □
Theorem 4.3 Let . If the function f is defined by
The bound σ is sharp for the function
Proof We note that for ,
Then from (4.9), we have
Next, we show that
where is given by (4.10). Letting
we see that
Then for (4.13) and (4.15), we have
This evidently gives (4.14), which is equivalent to
Let . Then, by using (4.12) and (4.16), an application of Theorem 3.1 yields
For h given by (4.11), we consider the function defined by
Then from (4.3), (4.5) and (4.17), we find that
Therefore we conclude that the bound cannot be increased for each c (). □
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The authors would like to express their thanks to the referees for valuable advice regarding a previous version of this paper. This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2012-0002619).
The authors declare that they have no competing interests.
All authors jointly worked on the results and they read and approved the final manuscript.
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Cho, N.E., Yoon, M. Properties for certain subclasses of meromorphic functions defined by a multiplier transformation. J Inequal Appl 2013, 46 (2013). https://doi.org/10.1186/1029-242X-2013-46
- meromorphic function
- starlike of order α
- convex of order α
- prestarlike of order α
- integral operator