#### Abstract

The main purpose of this paper is to derive some results associated with the quasi-Hadamard product of certain -starlike and -convex univalent analytic functions with respect to symmetric points.

#### 1. Introduction and Definitions

Let denote the class of functions of the form which are analytic and univalent in the open unit disk and normalized with , where is a fixed point in .

Throughout this paper, let the functions of the forms be regular and univalent in the unit disk .

Let be a fixed point in , and .

In [1], Kanas and Ronning defined the following classes of functions of -starlike and -convex, respectively, Recently, Acu and Owa [2], Oladipo [3], and Oladipo and Breaz [4] have studied the previous classes extensively.

Let be the subclass of consisting of functions given by (1) with and satisfying the condition These functions are called starlike with respect to symmetric points and were introduced by Sakaguchi [5].

Motivated by the previous classes , Oladipo [6] (see also [7]) defined the following classes of functions with respect to symmetric points.

Definition 1. (i) Let be the subclass of consisting of functions given by (1) satisfying the condition and is a fixed point in . These functions are called -starlike with respect to symmetric points.
(ii) Let be the subclass of consisting of functions given by (1) satisfying the condition and is a fixed point in . These functions are called -convex with respect to symmetric points.
Suppose that and are two analytic functions in . Then, we say that the function is subordinate to the function , and we write if there exists a Schwarz function with and such that
By applying the previous subordination definition, we define the following subclasses of and .

Definition 2. (i) Let be the subclass of consisting of functions given by (1) satisfying the condition where , , and is a fixed point in .
(ii) Let be the subclass of consisting of functions given by (1) satisfying the condition where , , and is a fixed point in .
For , we have which were introduced by Goel and Mehrok [8] and Selvaraj and Vasanthi [9], respectively.
Using (12) and (13), we can easily obtain the characterization properties for the classes and as follows.

Lemma 3. A function defined by (2) belongs to the class , if it satisfies the condition

Lemma 4. A function defined by (2) belongs to the class , if it satisfies the condition

Now, we introduce the following class of analytic functions in .

Definition 5. A function of the form (2), which is analytic in , belongs to the class , if it satisfies the condition where and is any fixed nonnegative real number.
We note that, for any nonnegative real number , the class is nonempty as the functions of the form where , , and , satisfy inequality (17).
Clearly, we have the following relationships:(i) and ;(ii);(iii).
Let us define the quasi- Hadamard product of the functions and by

Similarly, we can define the quasi-Hadamard product of more than two functions; for example, where the functions are given by (3).

In this paper, we derive certain results associated with the quasi-Hadamard product of functions in the classes , , and , which extend the results obtained by Kumar [10, 11], Darwish [12], and Aouf [13].

#### 2. Main Results

Unless otherwise mentioned, we will assume throughout the following results that , , is any fixed nonnegative real number, and is a fixed point in .

Theorem 6. Let the functions defined by (3) be in the class for every , and let the functions defined by (5) be in the class for every . Then, the quasi-Hadamard product belongs to the class .

Proof. Let ; then, It is sufficient to show that Since , by Lemma 4, we have for every . Thus, or for every . The right-hand expression of the last inequality is not greater than . Therefore, for every . Also, since , we find from (17) that for every . Hence, we obtain for every .
Using (26)–(28) for , , and , respectively, we have Thus, we have . This completes the proof of Theorem 6.

Upon setting in Theorem 6, we obtain the following result.

Corollary 7. Let the functions defined by (3) and the functions defined by (5) belong to the class for every and . Then, the quasi-Hadamard product belongs to the class .

Theorem 8. Let the functions defined by (3) be in the class for every , and let the functions defined by (5) be in the class for every . Then, the quasi-Hadamard product belongs to the class .

Proof. Suppose that is defined as (21). To prove the theorem, we need to show that Since , from (17), we have for every . Hence, we get for every . Further, since , by Lemma 3, we have for every . Whence, we obtain for every .
Using (32)–(34) for , , and , respectively, we get Therefore, we have . We complete the proof.

By taking in Theorem 8, we get the following result.

Corollary 9. Let the functions defined by (3) and the functions defined by (5) belong to the class for every and . Then, the quasi-Hadamard product belongs to the class .

By putting in Theorem 6, or in Theorem 8, we obtain the following result.

Corollary 10. Let the functions defined by (3) be in the class for every , and let the functions defined by (5) be in the class for every . Then, the quasi-Hadamard product belongs to the class .

Next, we discuss some applications of Theorems 6 and 8.

Taking into account the quasi-Hadamard product of functions only, in the proof of Theorem 6, and using (23) and (26) for and , respectively, we are led to the following.

Corollary 11. Let the functions defined by (3) belong to the class for every . Then, the quasi- Hadamard product belongs to the class .

Also, taking into account the quasi-Hadamard product of functions only, in the proof of Theorem 8, and using (33) and (34) for and , respectively, we are led to the following.

Corollary 12. Let the functions defined by (5) belong to the class for every . Then, the quasi-Hadamard product belongs to the class .

Remark 13. By taking in the previous results and making use of relationship (14), we obtain the corresponding results.

#### Acknowledgments

The present investigation is partly supported by the Natural Science Foundation of China under Grant 11271045, the Higher School Doctoral Foundation of China under Grant 20100003110004 and the Natural Science Foundation of Inner Mongolia of China under Grant 2010MS0117. The authors are thankful to the referees for their careful reading and making some helpful comments which have essentially improved the presentation of this paper.