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Inclusion relationships for certain classes of analytic functions involving the Choi-Saigo-Srivastava operator
Journal of Inequalities and Applications volume 2013, Article number: 83 (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.
Let denote the class of functions of the form
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 .
We now define the function by
where is the Pochhammer symbol (or the shifted factorial) defined (in terms of the gamma function) by
We also denote by the operator defined by
where the symbol (∗) stands for the Hadamard product (or convolution). Then it is easily observed from definitions (1.1) and (1.2) that
where the symbol denotes the familiar Ruscheweyh derivative  (also, see ) for . The operator , introduced and studied by Carlson-Shaffer , has been used widely on the space of analytic and univalent functions in (see also ). Corresponding to the function defined by (1.1), we also introduce a function given by
Analogous to , we now define the linear operator on as follows:
It is easily verified from definition (1.4) that
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].
By using the operator , we introduce the following classes of analytic functions for , , and :
We also note that
In particular, we set
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.
Lemma 2.1 Let , and () be defined by (1.3). Then, for , (),
Proof From definition (1.3), we know that
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 
Let and . Then, for every analytic function h in ,
where denotes the closed convex hull of .
At first, the inclusion relationship involving the class is contained in Theorem 2.1 below.
Theorem 2.1 Let , , and . If or , then
Proof Let . From the definition of , we have
where w is analytic in with () and . By using (1.4), (2.1) and (2.6), we get
It follows from (2.6) and Lemma 2.2 that and , respectively. Let us put . Then, by applying Lemma 2.3 to (2.7), we obtain
since s is convex univalent. Therefore, from the definition of subordination and (2.8), we have
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.
Theorem 2.2 Let , , , and . If or , then
Theorem 2.3 Let , , , and . If or , then
Next, we prove the inclusion theorem involving the class .
Theorem 2.4 Let , , and . If or , then
Proof Applying (1.5) and Theorem 2.1, we observe that
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.
Theorem 2.5 Let , , , and . If or , then
Theorem 2.6 Let , , , and . If or , then
Taking (; ) in Theorems 2.1-2.6, we have the following corollaries below.
Corollary 2.1 Let and let , and and . Then
Corollary 2.2 Let and let , and and . Then
Corollary 2.3 Let and let , and and . Then
To prove the theorems below, we need the following lemma.
Lemma 2.4 
Let . If and , then .
Proof Let . Then
where ω is an analytic function in with () and . Thus we have
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 .
Theorem 2.7 Let and , and let and . Then
We begin by proving that
Let . Then there exists a function such that
From (2.10), we obtain
where w is an analytic function in with () and . By virtue of (2.3), Lemma 2.2 and Lemma 2.4, we see that belongs to . Then, making use of (2.1), we have
Therefore we prove that .
For the second part, by using arguments similar to those detailed above with (2.2), we obtain
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.
Theorem 2.8 Let and , and let and . Then
Theorem 2.9 Let and , and let and . Then
Remark 2.1 (i) Taking (), (), and in Theorems 2.1-2.2, Theorems 2.4-2.5 and Theorem 2.7, we have the results obtained by Choi et al. , which extend the results earlier given by Noor et al. [12, 14] and Liu .
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.
Theorem 3.1 Let , , , , and let . Then
Proof (i) Let . Then we have
By using the same techniques as in the proof of Theorem 2.1, we obtain (i).
Let . Then, by (1.5), and hence from (i), . Since
we have (ii) applying (1.5) once again.
Let . Then there exists a function such that
where w is an analytic function in with () and . From Lemma 2.4, we have that . Since
we obtain (iii).
Now we consider the following operators defined by
Corollary 3.1 Let , , , , and let () be defined by (3.1) and (3.2). Then
Remark 3.1 Letting (), and in Theorem 3.1, we have the corresponding results given by Sokoł .
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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).
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. Inclusion relationships for certain classes of analytic functions involving the Choi-Saigo-Srivastava operator. J Inequal Appl 2013, 83 (2013). https://doi.org/10.1186/1029-242X-2013-83
- Hadamard product (or convolution)
- starlike function
- convex function
- close-to-convex function
- linear operator
- Choi-Saigo-Srivastava operator
- Carlson-Shaffer operator
- Ruscheweyh derivative operator