Journal of Complex Analysis

Journal of Complex Analysis / 2013 / Article

Research Article | Open Access

Volume 2013 |Article ID 851318 | https://doi.org/10.1155/2013/851318

A. O. Mostafa, M. K. Aouf, A. Shamandy, E. A. Adwan, "Certain Subclasses of -Valent Meromorphic Functions Associated with a New Operator", Journal of Complex Analysis, vol. 2013, Article ID 851318, 4 pages, 2013. https://doi.org/10.1155/2013/851318

Certain Subclasses of -Valent Meromorphic Functions Associated with a New Operator

Academic Editor: J. Dziok
Received21 Dec 2012
Accepted10 Mar 2013
Published03 Apr 2013

Abstract

We introduce two classes of -valent meromorphic functions associated with a new operator and derive several interesting results for these classes.

1. Introduction

Let denote the class of functions of the form which are analytic and -valent in the punctured unit disc . Let be the class of functions analytic in satisfying the properties and where , , and . The class was introduced by Padmanabhan and Parvatham [1]. For , the class was introduced by Pinchuk [2]. Also we note that , where is the class of functions with positive real part greater than and , where is the class of functions with positive real part. From (2), we have if and only if there exists such that It is known that the class is a convex set (see [3]).

For functions given by (1) and given by the Hadamard product (or convolution) of and is defined by

Aqlan et al. [4] defined the operator by

Mostafa [5] used Aqlan et al. operator and defined the following linear operator as follows.

First put and let be defined by Then Using (7) and (9), we have where denotes the Pochhammer symbol given by It is readily verified from (10) that (see [5])

It is noticed that by putting in (10), we obtain the operator

Now, by using the linear operator , we introduce classes of -valent Bazilevic functions of as follows.

Definition 1. A function is said to be in the class if it satisfies the following condition:

Definition 2. A function is said to be in the class if it satisfies the following condition:

In this paper, we investigate several properties of the classes and .

2. Main Results

Unless otherwise mentioned, we assume throughout this paper that , , , , , and all powers are understood as principle values.

To prove our results we need the following lemma.

Lemma 3 (see [6]). Let , and be a complex-valued function satisfying the conditions:(i) is continuous in a domain .(ii) and .(iii) whenever and .
If is analytic in such that and for , then in .

Employing the technique used by Noor and Muhammad [7] and Aouf and Seoudy [8] for multivalent functions, we prove the following theorems.

Theorem 4. If , then where is given by

Proof. Setting where are analytic in with , and is given by (3). Differentiating both sides of (19) with respect to and using (12) in the resulting equation, we obtain which implies that We form the functional by choosing , , that is, Clearly, the first two conditions of Lemma 3 are satisfied. Now, we verify the condition (iii) as follows: where We note that if and only if , . From given by (18), we have , , and . Therefore, applying Lemma 3, we have and consequently for . This completes the proof of Theorem 4.

Similarly, we can prove the following theorem for the class .

Theorem 5. If , then where is given by

Theorem 6. If , then where is given by

Proof. Let and where are analytic in with and is given by (3). Differentiating both sides of (29) with respect to and using (12) in the resulting equation, we obtain which implies that We form the functional by choosing , , that is, Clearly, the conditions (i) and (ii) of Lemma 3 are satisfied. Now, we verify the condition (iii) as follows: where We note that if and only if , . From as given by (28), we have , , and . Therefore, applying Lemma 3, we have and consequently for . This completes the proof of Theorem 6.

Similarly, we can prove the following theorem for the class .

Theorem 7. If , then where is given by

Remark 8. Putting , in Theorems 4 and 6, we obtain the corresponding results for the operator defined in (14).

Acknowledgment

The authors would like to thank the referees of the paper for their helpful suggestions.

References

  1. K. S. Padmanabhan and R. Parvatham, “Properties of a class of functions with bounded boundary rotation,” Annales Polonici Mathematici, vol. 31, no. 3, pp. 311–323, 1975/76. View at: Google Scholar | MathSciNet
  2. B. Pinchuk, “Functions of bounded boundary rotation,” Israel Journal of Mathematics, vol. 10, pp. 6–16, 1971. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  3. K. I. Noor, “On subclasses of close-to-convex functions of higher order,” International Journal of Mathematics and Mathematical Sciences, vol. 15, no. 2, pp. 279–289, 1992. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  4. E. Aqlan, J. M. Jahangiri, and S. R. Kulkarni, “Certain integral operators applied to meromorphic p-valent functions,” Journal of Natural Geometry, vol. 24, no. 1-2, pp. 111–120, 2003. View at: Google Scholar | MathSciNet
  5. A. Mostafa, “Inclusion results for certain subclasses of p-valent meromorphic functions associated with a new operator,” Journal of Inequalities and Applications, vol. 2012, p. 169, 2012. View at: Publisher Site | Google Scholar | MathSciNet
  6. S. S. Miller and P. T. Mocanu, “Second-order differential inequalities in the complex plane,” Journal of Mathematical Analysis and Applications, vol. 65, no. 2, pp. 289–305, 1978. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  7. K. I. Noor and A. Muhammad, “Some properties of the subclass of p-valent Bazilevic functions,” Acta Universitatis Apulensis, no. 17, pp. 189–197, 2009. View at: Google Scholar | MathSciNet
  8. M. K. Aouf and T. M. Seoudy, “Some properties of certain subclasses of p-valent Bazilevic functions associated with the generalized operator,” Applied Mathematics Letters, vol. 24, no. 11, pp. 1953–1958, 2011. View at: Publisher Site | Google Scholar | MathSciNet

Copyright © 2013 A. O. Mostafa 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.


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