- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Journal of Mathematics

Volume 2013 (2013), Article ID 473530, 5 pages

http://dx.doi.org/10.1155/2013/473530

## Quasi-Hadamard Product of Certain *ω*-Starlike and *ω*-Convex Functions with respect to Symmetric Points

*ω*

*ω*

^{1}School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China^{2}School of Mathematics and Statistics, Chifeng University, Chifeng, Inner Mongolia 024000, China

Received 30 November 2012; Accepted 20 January 2013

Academic Editor: Herbert Homeier

Copyright © 2013 Huo Tang and Guan-Tie Deng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### References

- S. Kanas and F. Ronning, “Uniformly starlike and convex functions and other related classes of univalent functions,”
*Annales Universitatis Mariae Curie-Skłodowska A*, vol. 53, pp. 95–105, 1999. View at Zentralblatt MATH · View at MathSciNet - M. Acu and S. Owa, “On some subclasses of univalent functions,”
*Journal of Inequalities in Pure and Applied Mathematics*, vol. 6, no. 3, pp. 1–6, 2005. View at Zentralblatt MATH · View at MathSciNet - A. T. Oladipo, “On a subclasses of univalent functions,”
*Advances in Applied Mathematical Analysis*, vol. 4, no. 2, pp. 87–93, 2009. - A. T. Oladipo and D. Breaz, “On a class of family of Bazilevic functions,”
*Acta Universitatis Apulensis*, vol. 24, pp. 319–330, 2010. View at Zentralblatt MATH · View at MathSciNet - K. Sakaguchi, “On a certain univalent mapping,”
*Journal of the Mathematical Society of Japan*, vol. 11, pp. 72–75, 1959. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. T. Oladipo, “New subclasses of analytic functions with respect to other points,”
*International Journal of Pure and Applied Mathematics*, vol. 75, no. 1, pp. 1–12, 2012. - O. S. Olatunji and A. T. Oladipo, “On a new subfamilies of analytic and univalent functions with negative coefficient with respect to other points,”
*Bulletin of Mathematical Analysis and Applications*, vol. 3, no. 2, pp. 159–166, 2011. View at MathSciNet - R. M. Goel and B. C. Mehrok, “A subclass of starlike functions with respect to symmetric points,”
*Tamkang Journal of Mathematics*, vol. 13, no. 1, pp. 11–24, 1982. View at Zentralblatt MATH · View at MathSciNet - C. Selvaraj and N. Vasanthi, “Subclasses of analytic functions with respect to symmetric and conjugate points,”
*Tamkang Journal of Mathematics*, vol. 42, no. 1, pp. 87–94, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. Kumar, “Hadamard product of certain starlike functions,”
*Journal of Mathematical Analysis and Applications*, vol. 110, no. 2, pp. 425–428, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. Kumar, “Quasi-hadamard product of certain univalent functions,”
*Journal of Mathematical Analysis and Applications*, vol. 126, no. 1, pp. 70–77, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. E. Darwish, “The quasi-hadamard product of certain starlike and convex functions,”
*Applied Mathematics Letters*, vol. 20, no. 6, pp. 692–695, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. K. Aouf, “The quasi-hadamard product of certain analytic functions,”
*Applied Mathematics Letters*, vol. 21, no. 11, pp. 1184–1187, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet