Research Article | Open Access

A. G. Alamoush, M. Darus, "On Certain Class of Non-Bazilevič Functions of Order Defined by a Differential Subordination", *International Journal of Differential Equations*, vol. 2014, Article ID 458090, 6 pages, 2014. https://doi.org/10.1155/2014/458090

# On Certain Class of Non-Bazilevič Functions of Order Defined by a Differential Subordination

**Academic Editor:**Salim Messaoudi

#### Abstract

We introduce a new subclass of Non-Bazilevič functions of order . Some subordination relations and inequality properties are discussed. The results obtained generalize the related work of some authors. In addition, some other new results are also obtained.

#### 1. Introduction

Let denote the class of the functions of the form which are analytic in the open unit disk . Let and be analytic in . Then we say that the function is subordinate to in if there exists an analytic function in such that and , denoted or . If is univalent in , then the subordination is equivalent to and .

Assume that , a function , is in if and only if The class was introduced by Obradović [1] recently. This class of functions was said to be of Non-Bazilevič type. To this date, this class was studied in a direction of finding necessary conditions over that embeds this class into the class of univalent functions or its subclasses which is still an open problem.

Assume that , , , , and , we consider the following subclass of : where all the powers are principal values, and we apply this agreement to get the following definition.

*Definition 1. *Let denote the class of functions in satisfying the inequality
where , , , and .

The classes and were studied by Wang et al. [2].

In the present paper, similarly we define the following class of analytic functions.

*Definition 2. *Let denote the class of functions in satisfying the inequality
where , , , , , and . All the powers in (5) are principal values.

We say that the function in this class is Non-Bazilevič functions of type .

*Definition 3. *Let if and only if and it satisfies
where , , , , and .

In particular, if , it reduces to the class studied in [2].

If , , , , and , then the class reduces to the class of non-Bazilevi functions. If , and , then the class reduces to the class of non-Bazilevič functions of order . Tuneski and Darus studied the Fekete-Szegö problem of the class [3]. Other works related to Bazilevič and non-Bazilevič can be found in ([4–9]).

In the present paper, we will discuss the subordination relations and inequality properties of the class . The results presented here generalize and improve some known results, and some other new results are obtained.

#### 2. Some Lemmas

Lemma 4 (see [10]). *Let be analytic in and be analytic and convex in , . If
**
where and , then
**
and is the best dominant for the differential subordination (7).*

Lemma 5 (see [11]). *Let ; then
*

Lemma 6 (see [12]). *Let be analytic and convex in , , . If
**
then
*

Lemma 7 (see [13]). *Let be analytic in and analytic and convex in . If , then , for .*

Lemma 8. *Let , , , , , , and . Then if and only if
**
where
*

*Proof. *Let
Then, by taking the derivatives of both sides of (14) and through simple calculation, we have
since , we have

#### 3. Main Results

Theorem 9. *Let , , , , , , and . If , then
*

*Proof. *First let ; then is analytic in . Now, suppose that ; by Lemma 8, we know that
It is obvious that is analytic and convex in , . Since , , , and ; therefore, it follows from Lemma 4 that

Corollary 10. *Let , , , , and . If satisfies
**
then
**
or equivalent to
*

Corollary 11. *Let , , , , and ; then
*

Theorem 12. *Let , , , , and ; then
*

*Proof. *Suppose that we have , and
Since , therefore it follows from Lemma 5 that
that is . So Theorem 12 is proved when .

When , then we can see from Corollary 11 that ; then But It is obvious that is analytic and convex in . So we obtain from Lemma 6 and differential subordinations (26) and (27) that that is, . Thus we have

Corollary 13. *Let , and ; then
*

Theorem 14. *Let , and . If , then
*

*Proof. *Suppose that ; then from Theorem 9 we know that
Therefore, from the definition of the subordination, we have

Corollary 15. *Let , and . If , then
*

Corollary 16. *Let , and . If ; then
**
then
*

Corollary 17. *Let , and . If , then
**
and inequality (38) is sharp, with the extremal function defined by
*

*Proof. *Suppose that ; from Theorem 9 we know
Therefore, from the definition of the subordination and , we have that
It is obvious that inequality (38) is sharp, with the extremal function given by (39).

Corollary 18. *Let , and . If , then
**
and inequality (42) is equivalent to
**
The inequality (42) is sharp, with the extremal function defined by
*

Corollary 19. *Let , , and . If , then
**
and inequality (45) is sharp, with the extremal function given by (39).*

*Proof. *Applying similar method as in Corollary 17, we get the result.

Corollary 20. *Let , , and. If satisfies
**
then
**
and inequality (47) is equivalent to
**
and inequality (47) is sharp, with the extremal function defined by equality (44).*

If , then (see [2, 12]). So we have the following.

Corollary 21. *Let , , and. If , then
**
and inequality (49) is sharp, with the extremal function defined by equality (39).*

*Proof. *From Theorem 9 we have
Since , we have
Thus, from inequality (38), we can get inequality (49). It is obvious that inequality (49) is sharp, with the extremal function defined by equality (39).

Corollary 22. *Let , and . If , then
**
and inequality (52) is sharp, with the extremal function defined by equality (39).*

*Proof. *Applying similar method as in Corollary 21, we get the required result.

*Remark 23. *From Corollaries 21 and 22, we can generalize the corresponding results and some other special classes of analytic functions.

Corollary 24. *Let , , , and ; if , then one has
**
and inequality (53) is sharp, with the extremal function defined by equality (39).*

*Proof. *Suppose that ; then we have
It follows from Lemma 7 that
Thus, we can get (53). Notice that
we obtain that the inequality (53) is sharp.

*Remark 25. *Setting , and in Corollary 24 we get the results obtained by [14].

#### Conflict of Interests

The authors declare that they have no conflict of interests.

#### Authors’ Contribution

Both authors read and approved the final paper.

#### Acknowledgments

The authors would like to acknowledge and appreciate the financial support received from Universiti Kebangsaan Malaysia under the Grant AP-2013-009. The authors also would like to thank the referees for the comments and suggestions to improve the paper.

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#### Copyright

Copyright © 2014 A. G. Alamoush and M. Darus. 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.