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Journal of Complex Analysis
Volume 2013 (2013), Article ID 629394, 5 pages
A New Subclass of Meromorphic Close-to-Convex Functions
1Department of Mathematics, Government Engineering College, Bikaner 334001, India
2Department of Mathematics, Government Dungar College, Bikaner 334001, India
Received 31 August 2012; Revised 13 November 2012; Accepted 13 November 2012
Academic Editor: Alfonso Montes-Rodriguez
Copyright © 2013 Amit Soni and Shashi Kant. 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.
A new subclass (, , ) of meromorphic close-to-convex functions, defined by means of subordination, is investigated. Some results such as inclusion relationship, coefficient inequality, convolution property, and distortion property for this class are derived. The results obtained here are extension of earlier known work.
Let denote the class of functions of the form which are analytic in the punctured open unit disk . For any two analytic function and in , we say that is subordinate to in , written as , if there exists a Schwarz function such that , for . In particular, if is univalent in , then is subordinate to , if and only if . If is given by (1) and is given by then the Hadamard product (or convolution) of the function and is defined by A function is said to be in the class of meromorphic starlike functions of order if it satisfies the inequality Moreover, a function is said to be in the class of meromorphic close to convex function if it satisfies the condition Recently, Wang et al.  introduced and discussed the class of meromorphic functions which satisfies the inequality where .
The class is very close to the interesting subclass of close-to-convex function for analytic function introduced and studied by Gao and Zhou . Many classes related to the class have been further studied by some authors. Especially, Wang et al. [3, 4], Kowalczyk and Leś-Bomba , Xu et al. , Şeker , and Cho et al.  introduced generalization of and obtained some properties for analytic functions in each class.
More recently, by means of subordination, Sim and Kwon  discussed a subclass of the class . A function is said be in the class if it satisfies the following subordination relation: where and . Here, the assumption that is meromorphic starlike function of order makes the function meromorphic starlike. So, instead of in (6) and (7), we can consider, because if , then is also a meromorphic starlike function, which motivates us to define a new subclass of meromorphic close-to-convex functions as follows.
Definition 1. A function is said to be in the class if there exists , such that The class provides a generalization of the classes given by Sim and Kwon  and Wang et al. . The transformation involved in the class is analytic and convex univalent in . Moreover,
In this paper, we aim at proving results such as inclusion relationship, coefficient bounds, and distortion theorem for the class .
2. Properties of Meromorphic Starlike Functions
In beginning, we prove the following result of meromorphic starlike functions.
Theorem 2. Suppose that and with . Then,
Proof. Let . By definition, we know that Next, we suppose that Then, we easily find that It follows that By noting that, which implies that , it completes the proof of Theorem 2.
For and , Theorem 2 gives the following corollary.
Corollary 3. If and , then .
Lemma 4 (see ). Let . Then,
Theorem 5. Let . Then,
Proof. Suppose that . Then, Since , by Lemma 4, we have It implies that , which completes the proof.
3. Coefficient Estimates
In this section,we obtain the coefficient estimates of functions belonging to the class . For this, we require the following Lemmas.
Lemma 6 (see ). Let be analytic in and be analytic and convex in . If , then
Lemma 7. Let and be given by (2). Then,
Proof. According to Corollary 3, we have and if then it is well known that Substituting the series expressions of and in (23) and comparing the coefficients of both sides of this equation, we get On substituting the value from (25) in (24), we get the result.
Theorem 8. Let . If then
Proof. Suppose that . Then, we know that
If we set it follows that In view of Lemma 6, we know that By substituting the series expression of functions ,, and in (29), we obtain Since is univalent in , it is well known that On comparing the coefficient of in both sides of (32), we have Substituting the values from (24) and (31) to (34), we have This evidently completes the proof of Theorem 8.
Theorem 10. Let . If , then
Proof. Suppose that . Then, we know that where is given by (23). It is easy to see that the condition (43) can be written as Note that By substituting (45) into (44), we get the desired assertion (42) of Theorem 10.
4. Distortion Theorem
Lemma 11 (see ). Let . Then,
Theorem 12. Let. If , then
Proof. Suppose that . By definition, we have the transformation which maps in the closed disk This implies that Thus, by virtue of Lemma 11, we readily get the desired result.
Remark. Taking , and , we obtain the results such as inclusion relation, coefficient inequality, convolution property, and distortion theorem derived by Wang et al. .
The authors would like to express their thanks to the referee for careful reading and suggestions made for the improvement of the paper.
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