GeometryVolume 2013 (2013), Article ID 497191, 6 pageshttp://dx.doi.org/10.1155/2013/497191

Research Article

Sufficient Conditions for Meromorphically -Valent Starlikeness and Close-to-Convexity

B. A. Frasin,¹ Tariq Al-Hawary,² and M. Darus²

¹Department of Mathematics, Faculty of Science, Al al-Bayt University, P.O. Box 130095, Mafraq, Jordan²School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia

Received 14 October 2012; Accepted 6 December 2012

Academic Editor: JinLin Liu

Copyright © 2013 B. A. Frasin et al. 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.

Making use of the linear operator defined by (Frasin 2012), we introduce the class of meromorphically -valent functions in the punctured unit disk . Furthermore, we obtain some sufficient conditions for starlikeness and close-to-convexity for functions belonging to this class. Several corollaries and consequences of the main results are also considered.

1. Introduction and Definitions

Let denote the class of functions of the form: which are -valent in the punctured unit disk . A function in is said to be meromorphically -valent starlike of order if and only if for some . We denote by the class of all meromorphically -valent starlike of order . Further, a function in is said to be meromorphically -valent convex of order if and only if for some . We denote by the class of all meromorphically -valent convex of order . A function belonging to is said to be meromorphically -valent close-to-convex of order if it satisfies for some . We denote by the subclass of consisting of functions which are meromorphically -valent close-to-convex of order in .

Many interesting families of analytic and multivalent functions were considered by earlier authors in Geometric Functions Theory (cf. e.g., [1–4]). Some subclasses of when were considered by (e.g.) Miller [5], Pommerenke [6], Clunie [7], Owa et al. [8], and Royster [9]. Furthermore, several subclasses of when were studied by (amongst others) Mogra et al. [10], Uralegaddi and Ganigi [11], Cho et al. [12], Aouf [13, 14], and Uralegaddi and Somanatha [15].

For a function in , Frasin [16] introduced and studied the following differential operator: and for

Note that for , we have the operator introduced and studied by Frasin and Darus [17].

It easily verified from (6) that Making use of the above operator , we now introduce a new class of meromorphically and -valent functions defined as follows.

Definition 1. A function is said to be a member of the class if and only if for some and for all .

Note that condition (8) implies that

Clearly, we have and .

In this paper, we obtain some sufficient conditions for functions belonging to the class . Several corollaries and consequences of the main results are also considered.

In order to derive our main results, we have to recall the following lemmas.

Lemma 2 (see [18]). Let be analytic in and such that . Then if attains its maximum value on circle at a point , we have where is a real number.

Lemma 3 (see [19]). Let be a set in the complex plane and suppose that is a mapping from × to which satisfies for , and for all real such that . If the function is analytic in such that for all , then .

Lemma 4 (see [20]). Let be analytic in with . If there exists a point such that then where and .

2. Sufficient Conditions for Meromorphically -Valent Starlikeness and Close-to-Convexity

Making use of Lemma 2, we first prove

Theorem 5. If satisfies for some   , and , then .

Proof. Define the function by Then is analytic in and . It follows from (14) and the identities (7) that Suppose that there exists such that then from Lemma 2, we have (10). Therefore, letting , with , we obtain that which contradicts our assumption (13). Therefore we have in . Finally, we have that is, . This completes the proof of the theorem.

Next we prove the following.

Theorem 6. If satisfies for some and ,  then

Proof. Define the function by Then, we see that is analytic in . Differentiating both sides of (20) with respect logarithmically, we get Using the identities (7) in (21), we find that From (20) and (22), we immediately get where For all real satisfying , we have Let . Then , and , for all real and , . By using Lemma 3, we have , that is, .

Finally, we prove the next theorem.

Theorem 7. If satisfies for some   ,  and ,  then .

Proof. Define the function by Then, we see that is analytic in with . From (22) it follows that and If there exists a point such that Then applying Lemma 4, we have where and . Thus, from (28) and (30) we get Therefore, we have This contradicts our assumption. Thus, we conclude that for all , that is,

3. Special Cases and Consequences

Among the various interesting and important consequences of Theorems 5–7, we mention now some of the corollaries relating to the classes , and , which are deducible from the main results.

Firstly, if we let ,  , and in Theorems 5–7, we get the following sufficient conditions for meromorphically -valent starlike functions.

Corollary 8. If satisfies for some and then .

Corollary 9. If satisfies for some and , then .

Corollary 10. If satisfies for some ,  then .

Setting and in Theorems 5–7, we get the following sufficient conditions for meromorphically -valent close-to-convex functions.

Corollary 11. If satisfies for some and ,  then .

Corollary 12. If satisfies for some , then .

Corollary 13. If satisfies for some , then .

Setting in Corollary 10, we have

Corollary 14. If satisfies for some , then . In particular, if satisfies then is meromorphically starlike of order .

Setting in Corollary 13, we have the following.

Corollary 15. If satisfies for some , then . In particular, if satisfies then is meromorphically close-to-convex of order .

Remark 16. (i) If we put in Corollaries 8 and 11, we get Corollaries 5 and 1, respectively, proved by Goyal and Prajapat [21].(ii) If we put and in Corollaries 9 and 12, we get Corollaries 8 and 4, respectively, proved by Goyal and Prajapat [21].

Acknowledgment

The work here was fully supported by LRGS/TD/2011/UKM/ICT/03/02.

References

M. K. Aouf, “A generalization of meromorphic multivalent functions with positive coefficients,” Mathematica Japonica, vol. 35, no. 4, pp. 609–614, 1990.

S. R. Kulkarni, U. H. Naik, and H. M. Srivastava, “A certain class of meromorphically -valent quasi-convex functions,” Panamerican Mathematical Journal, vol. 8, no. 1, pp. 57–64, 1998.

H. M. Srivastava and S. Owa, Current Topics in Analytic Function Theory, World Scientific Publishing Co, River Edge, NJ, USA, 1992.

H. M. Srivastava, H. M. Hossen, and M. K. Aouf, “A unified presentation of some classes of meromorphically multivalent functions,” Computers & Mathematics with Applications, vol. 38, no. 11-12, pp. 63–70, 1999.

J. Miller, “Convex meromorphic mappings and related functions,” Proceedings of the American Mathematical Society, vol. 25, pp. 220–228, 1970.

Ch. Pommerenke, “On meromorphic starlike functions,” Pacific Journal of Mathematics, vol. 13, pp. 221–235, 1963.

J. Clunie, “On meromorphic schlicht functions,” Journal of the London Mathematical Society, vol. 34, pp. 215–216, 1959.

S. Owa, H. E. Darwish, and M. K. Aouf, “Meromorphic multivalent functions with positive and fixed second coefficients,” Mathematica Japonica, vol. 46, no. 2, pp. 231–236, 1997.

W. C. Royster, “Meromorphic starlike multivalent functions,” Transactions of the American Mathematical Society, vol. 107, pp. 300–308, 1963.

M. L. Mogra, T. R. Reddy, and O. P. Juneja, “Meromorphic univalent functions with positive coefficients,” Bulletin of the Australian Mathematical Society, vol. 32, no. 2, pp. 161–176, 1985.

B. A. Uralegaddi and M. D. Ganigi, “A certain class of meromorphically starlike functions with positive coefficients,” Pure and Applied Mathematika Sciences, vol. 26, no. 1-2, pp. 75–81, 1987.

N. E. Cho, S. H. Lee, and S. Owa, “A class of meromorphic univalent functions with positive coefficients,” Kobe Journal of Mathematics, vol. 4, no. 1, pp. 43–50, 1987.

M. K. Aouf, “On a class of meromorphic multivalent functions with positive coefficients,” Mathematica Japonica, vol. 35, no. 4, pp. 603–608, 1990.

M. K. Aouf, “On a certain class of meromorphic univalent functions with positive coefficients,” Rendiconti di Matematica e delle sue Applicazioni Serie VII, vol. 11, no. 2, pp. 209–219, 1991.

B. A. Uralegaddi and C. Somanatha, “New criteria for meromorphic starlike univalent functions,” Bulletin of the Australian Mathematical Society, vol. 43, no. 1, pp. 137–140, 1991.

B. A. Frasin, “Differential operator of meromorphic p-valent functions,” Tamkang Journal of Mathematics, vol. 43, pp. 417–421, 2012.

B. A. Frasin and M. Darus, “On certain meromorphic functions with positive coefficients,” Southeast Asian Bulletin of Mathematics, vol. 28, no. 4, pp. 615–623, 2004.

I. S. Jack, “Functions starlike and convex of order ,” Journal of the London Mathematical Society, vol. 3, pp. 469–474, 1971.

S. S. Miller and P. T. Mocanu, “Differential subordinations and inequalities in the complex plane,” Journal of Differential Equations, vol. 67, no. 2, pp. 199–211, 1987.

M. Nunokawa, “On properties of non-Carathéodory functions,” Proceedings of the Japan Academy, Series A, vol. 68, no. 6, pp. 152–153, 1992.

S. P. Goyal and J. K. Prajapat, “A new class of meromorphic multivalent functions involving certain linear operator,” Tamsui Oxford Journal of Mathematical Sciences, vol. 25, no. 2, pp. 167–176, 2009.