#### 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.