International Journal of Differential Equations

Volume 2014 (2014), Article ID 458090, 6 pages

http://dx.doi.org/10.1155/2014/458090

Research Article

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

A. G. Alamoush and M. Darus

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia (UKM), 43600 Bangi, Selangor, Malaysia

Received 29 April 2014; Revised 27 June 2014; Accepted 2 July 2014; Published 17 July 2014

Academic Editor: Salim Messaoudi

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.

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.

References

M. Obradović, “A class of univalent functions,” Hokkaido Mathematical Journal, vol. 27, no. 2, pp. 329–335, 1998.

Z. Wang, C. Gao, and M. Liao, “On certain generalized class of non-Bazilevič functions,” Acta Mathematica Academiae Paedagogicae Nyíregyháziensis, vol. 21, no. 2, pp. 147–154, 2005.

N. Tuneski and M. Darus, “Fekete-Szegö functional for non-Bazilevic functions,” Acta Mathematica Academiae Paedagogicae Nyí regyháziensis, vol. 18, pp. 63–65, 2002.

A. A. Amer and M. Darus, “Distortion theorem for certain class of Bazilevic functions,” International Journal of Mathematical Analysis, vol. 6, no. 9–12, pp. 591–597, 2012.

R. W. Ibrahim and M. Darus, “Mixed class of functions of non-Bazilevic type and bounded turning,” Far East Journal of Mathematical Sciences, vol. 67, no. 1, pp. 141–152, 2012.

R. W. Ibrahim and M. Darus, “Argument estimate for non-Bazilevic type and bounded turning functions,” Far East Journal of Mathematical Sciences, vol. 68, no. 2, pp. 175–183, 2012.

R. W. Ibrahim, M. Darus, and N. Tuneski, “On subordination for classes of non-Bazilevic type,” Annales Universitatis Mariae Curie-Sklodowska Lublin-Polonia A, vol. 64, no. 2, pp. 49–60, 2010.

M. Darus and R. W. Ibrahim, “On subclasses of uniformly Bazilevic type functions involving generalised differential and integral operators,” Far East Journal of Mathematical Sciences (FJMS), vol. 33, no. 3, pp. 401–411, 2009.

T. N. Shanmugam, S. Sivasubramanian, M. Darus, and S. Kavitha, “On sandwich theorems for certain subclasses of non-Bazilevic functions involving Cho-Kim transformation,” Complex Variables and Elliptic Equations, vol. 52, no. 10-11, pp. 1017–1028, 2007.

S. S. Miller and P. T. Mocanu, “Differential subordinations and univalent functions,” The Michigan Mathematical Journal, vol. 28, no. 2, pp. 157–172, 1981.

M. S. Liu, “On a subclass of -valent close-to-convex functions of order and type ,” Journal of Mathematical Study, vol. 30, no. 1, pp. 102–104, 1997.

M.-S. Liu, “On certain class of analytic functions defined by differential subordination,” Acta Mathematica Scientia B, vol. 22, no. 3, pp. 388–392, 2002.

W. Rogosinski, “On the coefficients of subordination functions,” Proceedings of the London Mathematical Society 2, vol. 48, pp. 48–82, 1943.

R. Singh, “On Bazilevic functions,” Proceedings of the American Mathematical Society, vol. 38, pp. 261–271, 1973.