About this Journal Submit a Manuscript Table of Contents
Geometry
Volume 2013 (2013), Article ID 497191, 6 pages
http://dx.doi.org/10.1155/2013/497191
Research Article

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

1Department of Mathematics, Faculty of Science, Al al-Bayt University, P.O. Box 130095, Mafraq, Jordan
2School 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., [14]). 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 57, 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 57, 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 57, 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

  1. M. K. Aouf, “A generalization of meromorphic multivalent functions with positive coefficients,” Mathematica Japonica, vol. 35, no. 4, pp. 609–614, 1990. View at Zentralblatt MATH · View at MathSciNet
  2. S. R. Kulkarni, U. H. Naik, and H. M. Srivastava, “A certain class of meromorphically p-valent quasi-convex functions,” Panamerican Mathematical Journal, vol. 8, no. 1, pp. 57–64, 1998. View at MathSciNet
  3. H. M. Srivastava and S. Owa, Current Topics in Analytic Function Theory, World Scientific Publishing Co, River Edge, NJ, USA, 1992. View at MathSciNet
  4. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. J. Miller, “Convex meromorphic mappings and related functions,” Proceedings of the American Mathematical Society, vol. 25, pp. 220–228, 1970. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. Ch. Pommerenke, “On meromorphic starlike functions,” Pacific Journal of Mathematics, vol. 13, pp. 221–235, 1963. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. J. Clunie, “On meromorphic schlicht functions,” Journal of the London Mathematical Society, vol. 34, pp. 215–216, 1959. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. 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. View at Zentralblatt MATH · View at MathSciNet
  9. W. C. Royster, “Meromorphic starlike multivalent functions,” Transactions of the American Mathematical Society, vol. 107, pp. 300–308, 1963. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. 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. View at Zentralblatt MATH · View at MathSciNet
  12. 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. View at Zentralblatt MATH · View at MathSciNet
  13. M. K. Aouf, “On a class of meromorphic multivalent functions with positive coefficients,” Mathematica Japonica, vol. 35, no. 4, pp. 603–608, 1990. View at Zentralblatt MATH · View at MathSciNet
  14. 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. View at Zentralblatt MATH · View at MathSciNet
  15. 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. View at Publisher · View at Google Scholar · View at MathSciNet
  16. B. A. Frasin, “Differential operator of meromorphic p-valent functions,” Tamkang Journal of Mathematics, vol. 43, pp. 417–421, 2012.
  17. 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. View at Zentralblatt MATH · View at MathSciNet
  18. I. S. Jack, “Functions starlike and convex of order α,” Journal of the London Mathematical Society, vol. 3, pp. 469–474, 1971. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. M. Nunokawa, “On properties of non-Carathéodory functions,” Proceedings of the Japan Academy, Series A, vol. 68, no. 6, pp. 152–153, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. 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. View at Zentralblatt MATH · View at MathSciNet