Inclusion relationships for certain classes of analytic functions involving the Choi-Saigo-Srivastava operator
© Cho and Yoon; licensee Springer 2013
Received: 27 November 2012
Accepted: 4 February 2013
Published: 1 March 2013
The purpose of the present paper is to investigate some inclusion properties of certain classes of analytic functions associated with a family of linear operators which are defined by means of the Hadamard product (or convolution). Some invariant properties under convolution are also considered for the classes presented here. The results presented here include several previous known results as their special cases.
which are analytic in the open unit disk . If f and g are analytic in , we say that f is subordinate to g, written or , if there exists an analytic function w in with and for such that . We denote by , and the subclasses of consisting of all analytic functions which are, respectively, starlike, convex and close-to-convex in (see, e.g., Srivastava and Owa ).
Let be a class of all functions ϕ which are analytic and univalent in and for which is convex with and for .
In particular, the operator (; ) was introduced by Choi et al.  who investigated (among other things) several inclusion properties involving various subclasses of analytic and univalent functions. For () and , we also note that the Choi-Saigo-Srivastava operator is the Noor integral operator of n th order of f studied by Liu  and Noor et al. [11–15].
Recently, Sokoł  extended the results given by Choi et al.  making use of some interesting proof techniques due to Ruscheweyh  and Ruscheweyh and Sheil-Small . In this paper, we investigate several inclusion properties of the classes , and . The integral-preserving properties in connection with the operator are also considered. Furthermore, relevant connections of the results presented here with those obtained in earlier works are pointed out.
2 Inclusion properties involving the operator
The following lemmas will be required in our investigation.
Therefore, equations (2.1), (2.2) and (2.3) follow from (2.5) immediately. □
Lemma 2.2 [, pp.60-61]
Let . If or , then the function defined by (2.4) belongs to the class .
Lemma 2.3 
where denotes the closed convex hull of .
At first, the inclusion relationship involving the class is contained in Theorem 2.1 below.
or, equivalently, , which completes the proof of Theorem 2.1. □
By using equations (1.4), (2.2) and (2.3), we have the following theorems.
Next, we prove the inclusion theorem involving the class .
which evidently proves Theorem 2.4. □
By using a similar method as in the proof of Theorem 2.4, we obtain the following two theorems.
Taking (; ) in Theorems 2.1-2.6, we have the following corollaries below.
To prove the theorems below, we need the following lemma.
Lemma 2.4 
Let . If and , then .
By using similar arguments to those used in the proof of Theorem 2.1, we conclude that (2.9) is subordinated to ϕ in and so . □
Finally, we give the inclusion properties involving the class .
Therefore we prove that .
Thus the proof of Theorem 2.7 is completed. □
The following results can be obtained by using the same techniques as in the proof of Theorem 2.7, and so we omit the detailed proofs involved.
For (), and , Theorems 2.1-2.2, Theorems 2.4-2.5 and Theorem 2.7 reduce the corresponding results obtained by Sokoł .
3 Inclusion properties involving various operators
The next theorem shows that the classes , and are invariant under convolution with convex functions.
- (ii)Let . Then, by (1.5), and hence from (i), . Since
- (iii)Let . Then there exists a function such that
we obtain (iii).
Remark 3.1 Letting (), and in Theorem 3.1, we have the corresponding results given by Sokoł .
Dedicated to Professor Hari M Srivastava.
The authors would like to express their thanks to the referees for some valuable comments 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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