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New Subclasses of Multivalent Analytic Functions Associated with a Linear Operator
Making use of a linear operator, which is defined here by means of the Hadamard product (or convolution), we consider two subclasses and of multivalent analytic functions with negative coefficients in the open unit disk. Some modified Hadamard products, integral transforms, and the partial sums of functions belonging to these classes are studied.
Let denote the class of functions of the form which are analytic in the open unit disk .
For functions and analytic in , we say that is subordinate to in , written , if there exists an analytic function in such that Furthermore, if the function is univalent in , then
In terms of the Pochhammer symbol given by , we define the function by Corresponding to , we consider here a linear operator on by the following usual Hadamard product (or convolution): for given by (1). For , on was first defined by Carlson and Shaffer . Its differential-integral representation can be found in . We remark in passing that a much more general convolution operator than the operator was introduced by Dziok and Srivastava .
Let denote the subclass of consisting of functions of the form We now consider the following two subclasses of the class .
Definition 1. A function is said to be in the class if and only if where
Definition 2. A function is said to be in the class if and only if
For functions given by we denote by the modified Hadamard (or quasi-Hadamard) product of and ; that is,
The class with was introduced and studied earlier by Lee et al.  (and was further investigated by Aouf and Darwish , Aouf et al. , and Yaguchi et al. ). The class with was studied by Aouf  and Aouf et al. . Recently, Aouf  investigated the modified Hadamard products of several functions in the classes and for .
In the present paper, we prove a number of theorems involving the modified Hadamard products, integral transforms, and the partial sums of functions in the classes and . Some of our results are generalizations of the corresponding results in [4–9].
In proving our main results, we need the following lemmas.
Corollary 5. Let Then
Corollary 6. Let . Then if and only if , where , and , and
Corollary 7. Let be defined by (6). Then(i) is in the class if and only if (ii) is in the class if and only if
2. Modified Hadamard Products
Hereafter in this paper we assume that (8) is satisfied:
Theorem 8. Let and . Then where and The result is sharp; that is, cannot be decreased for each .
Proof. By (24) we have for . Let
Then Lemma 3 gives
Also, using Lemma 3,
if and only if
To prove the result of Theorem 8, it follows from (27) and (29) that we need to find the smallest such that that is, that Since we see that the right-hand side of (31) is a decreasing function of . Consequently, taking in (31), we have , where is given by (24).
Furthermore, by considering the functions defined by: we have and Noting that we conclude that cannot be decreased for each .
Theorem 9. Let and . Then where and The result is sharp for the functions defined by
Theorem 10. Let with and . Then where and The result is sharp for the functions defined by (33).
Proof. Obviously, for , , and . By applying Lemma 4, we know that if and only if Proceeding as in the proof of Theorem 8, we need to find the smallest such that Defining the function by we see that for and . Hence, the right-hand side of (43) is a decreasing function of . Thus, we arrive at , where is given by (40).
Sharpness can be verified easily.
By putting Theorem 10 reduces to the following.
Corollary 11. Let . Then , where , , and The result is sharp for the functions
Theorem 12. Let , and . Then one has (i) , where provided that . (ii) , where
Proof. It is clear that , , , ,
for . Let
Then Lemma 3 gives
if and only if
Also, by Lemma 3 and (52), we deduce that
To prove Theorem 12(i), it follows from (55) and (57) that we need to find the smallest such that for . This leads to the assertion of Theorem 12(i).
Analogously, we can prove Theorem 12(ii).
In the special case when Theorem 12 reduces to the following.
Corollary 15. Let . Then (i) , where provided that . (ii) , where
3. Integral Operator
Theorem 16. Let If , then the function defined by belongs to , where and The result is sharp; that is, the number cannot be decreased for each .
Proof. Note that . For
it follows from (68) that
To prove the result of Theorem 16, we need to find the smallest such that where we have used Lemma 3. In view of (67), it is easy to know that the right-hand side of (72) is a decreasing function of . Therefore, we conclude that where is given by (69).
Furthermore, it can easily be verified that the result is sharp, with the extremal function
If we let then Theorem 17 yields the following.
Corollary 18. Let . Then the function defined by (68) belongs to , where The number cannot be increased for each .
4. Partial Sums
In this section, we let be given by (6) and define the partial sums and by Also we make the notation simple by writing
Theorem 19. Let and . Then for , one has the following. The results are sharp for each .
Proof. Let and be given by (79). Then for , and so it follows from Lemma 3 that
If we put then and because of (82). Hence, we have for , which implies that (80) holds true for .
Similarly, by setting it follows from (82) that Therefore, we see that for , that is, that (81) holds for .
For , replacing (82) by and proceeding as the above, we know that (80) and (81) are also true.
Furthermore, by taking the function we find that , The proof of Theorem 19 is thus completed.
Corollary 20. Let and . Then we have The results are sharp for each .
Theorem 21. Let with and . Then one has The results are sharp for each .
Proof. Let , and be given by (79). Then it is easy to verify that
and hence we deduce from Lemma 3 that
Defining the function by it follows from (94) that This leads to the inequality (91) for .
Similarly, for the function defined by we deduce from (94) that This yields the inequality (92) for .
For , replacing (94) by we know that (91) and (92) are also true.
Furthermore, the bounds in (91) and (92) are the best possible for the function defined by (88).
Finally, Theorem 21 yields the following.
Corollary 22. Let with and . Then The results are sharp for each .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.