On Certain Class of Non-Bazilevič Functions of Order Defined by a Differential Subordination
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.
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  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. .
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 .
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 . 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 5 (see ). Let ; then
Lemma 6 (see ). Let be analytic and convex in , , . If then
Lemma 7 (see ). 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
Corollary 10. Let , , , , and . If satisfies then or equivalent to
Corollary 11. Let , , , , and ; then
Theorem 12. Let , , , , and ; then
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).
Proof. Applying similar method as in Corollary 17, we get the result.
Proof. Applying similar method as in Corollary 21, we get the required result.
Conflict of Interests
The authors declare that they have no conflict of interests.
Both authors read and approved the final paper.
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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