Properties for certain subclasses of meromorphic functions defined by a multiplier transformation
© Cho and Yoon; licensee Springer 2013
Received: 27 September 2012
Accepted: 26 January 2013
Published: 1 March 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 ().
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 .
which are analytic in the punctured open unit disk .
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.
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]).
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]
and is the best dominant of (1.5).
Lemma 1.2 [, Theorem 2.4, p.54]
where denotes the convex hull of .
Lemma 1.3 [, Lemma 5, p.656]
where ϕ is given by (1.4).
2 Inclusion relations
Therefore , and so we complete the proof of Theorem 2.1. □
which completes the proof of Theorem 2.2. □
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. □
since h is convex univalent in . This shows that .
which implies that . Hence the bound cannot be increased when (). □
3 Convolution properties
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. □
is also in the class .
In view of (3.1) and (3.2), an application of Theorem 3.1 leads to . □
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.
is convex univalent in
4 Integral operators
Therefore, by Lemma 1.1, we conclude that Theorem 4.2 holds true as stated. □
Therefore we conclude that the bound cannot be increased for each c (). □
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).
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