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Journal of Complex Analysis
Volume 2013 (2013), Article ID 509717, 7 pages
http://dx.doi.org/10.1155/2013/509717
Research Article

Class of Multivalent Analytic Functions Defined by a Linear Operator

Department of Mathematics, Faculty of Science, Al al-Bayt University, P.O. Box 130095, Mafraq, Jordan

Received 14 August 2012; Accepted 27 September 2012

Academic Editor: Bao Qin Li

Copyright © 2013 B. A. Frasin. 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 in (Prajapat, 2012), we introduce the class of analytic and -valent functions in the open unit disk . Furthermore, we obtain some sufficient conditions for starlikeness and close-to-convexity and some angular properties 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 analytic and -valent in the open unit disk and . In particular, we set . A function is said to be in the class of -valently starlike of order in if and only if it satisfies the inequality

Furthermore, a function is said to be in the class of -valently close-to-convex of order in if and only if it satisfies the inequality

In particular, we write and , where and are the usual subclasses of consisting of functions which are starlike and close-to-convex in , respectively.

In [1], Prajapat define a generalized multiplier transformation operator as follows:

We see that for , we have where and . It is readily verified from (5) that

We observe that the operator generalize several previously studied familiar operators, and we will show some of the interesting particular cases as follows:(i) (see [2]);(ii) (see [3, 4]);(iii) (see [57]);(iv) (see [8, 9]);(v) (see [10]);(vi) (see [11]);(vii) (see [12]);(viii) (see [13]).(For other generalizations of the operator , see [1]).

Making use of the above operator , we introduce the class of analytic 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 (7) implies that

We note that , the class which has been introduced and studied by the author in [14]. Also, we have , . The class is the class which has been introduced and studied by Frasin and Darus [15] (see also [16, 17]).

In this paper, we obtain some sufficient conditions and some angular properties 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 , one has 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 and for all . If there exist two points ,    such that for ,  , and for , then we have where

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

Unless otherwise mentioned, we shall assume in the remainder of this paper that Making use of Lemma 2, we first prove the following.

Theorem 5. If satisfies for some   , then .

Proof. Define the function by Then is analytic in and . It follows from (15) and the identities (6) and (1.6) that Suppose that there exists such that Then from Lemma 2, we have (9). Therefore, letting , we obtain that Which contradicts our assumption (14). Therefore we have in . Finally, we have that is, . This completes the proof of the theorem.

Putting and in Theorem 5, we obtain the following.

Corollary 6. If satisfies for some , then .

Putting and in Theorem 5, one obtains the following.

Corollary 7. If satisfies for some    , then .

Letting in Corollary 7, one has

Corollary 8. If satisfies for some , then . In particular, if satisfies then is close-to-convex in .

Putting , , and in Theorem 5, one obtains the following.

Corollary 9. If satisfies for some , then .

Putting in Corollary 9, one easily obtains the following result due to Owa [21].

Corollary 10. If satisfies for some , then . In particular, if satisfies then is starlike in .

Remark 11. We note that the results obtained by the author [14, Theorem 2.1, Corollaries 2.2–2.5] are not corrects. The correct results are given by Corollaries 6, 7, and 9.

Next we prove the following.

Theorem 12. If satisfies then , where .

Proof . Define the function by Then, we see that is analytic in . A computation shows that where For all real satisfying , we have Let . Then and for all real and , . By using Lemma 3, we have , that is, .

Putting and in Theorem 12, we have the following.

Corollary 13 (see [14]). If satisfies then , where . In particular, if satisfies then is close-to-convex in .

Putting ,  , and in Theorem 12, one has the following.

Corollary 14 (see [14]). If satisfies then , where . In particular, if satisfies then is starlike in .

3. Argument Properties

Finally, we prove the following.

Theorem 15. Suppose that for and . If satisfies for , then

Proof. Define the function by Then we see that analytic in , , and for all . It follows from (39) that
Suppose that there exists two points such that the condition (10) is satisfied, then by Lemma 4, we obtain (11) under the constraint (12). Therefore, we have which contradict the assumption of the theorem. This completes the proof.

Putting and in Theorem 15, one has the following.

Corollary 16 (see [14]). Suppose that for and . If satisfies for , then In particular, if satisfies for , then Putting , , and in Theorem 15, one has the following.

Corollary 17 (see [14]). Suppose that for and . If satisfies for , then In particular, if satisfies for , then

Remark 18. Taking different choices of , and in the above theorems, we obtain some sufficient conditions for starlikeness and close-to-convexity and some angular properties for functions belonging to new classes defined by the previously operators mentioned in Section 1.

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