#### Abstract

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.

#### 1. Introduction

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 [1]. Its differential-integral representation can be found in [2]. We remark in passing that a much more general convolution operator than the operator was introduced by Dziok and Srivastava [3].

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. [4] (and was further investigated by Aouf and Darwish [5], Aouf et al. [6], and Yaguchi et al. [7]). The class with was studied by Aouf [8] and Aouf et al. [6]. Recently, Aouf [9] 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.

Lemma 3 (see [10, 11]). *A function defined by (6) is in the class if and only if
*

Lemma 4 (see [10, 11]). *A function defined by (6) is in the class if and only if
*

Making use of Lemmas 3 and 4, we can show the following two results.

Corollary 5. *Let
**
Then
*

Corollary 6. *Let . Then if and only if , where , and , and
*

If we let then Lemmas 3 and 4 reduce to the following result.

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
and hence
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 .

By using Lemma 4 instead of Lemma 3, the following theorem can be proved on the lines of the proof of Theorem 8. We omit the details involved.

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
and hence

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 13. *Let . Then* (i)* , where
* *provided that .* (ii)* , where
**Replacing Lemma 3 by Lemma 4 in the proof of Theorem 12, one can prove the following.*

Theorem 14. *Let ,
**
and . Then* (i)* , where
* *and , and are given as in Theorem 12, provided that .* (ii)* , where
**As a special case of Theorem 14, one has 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

With the aid of Lemma 4 (instead of Lemma 3) and using the same steps as in the proof of Theorem 16, we can prove the following.

Theorem 17. *Let (67) in Theorem 16 be satisfied. If , then the function defined by (68) belongs to , where is the same as in Theorem 16. The result is sharp for the 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
for .

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.

By virtue of Theorem 19 and Definition 2, we easily have the following.

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
for .

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.