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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 890404, 8 pages
http://dx.doi.org/10.1155/2013/890404
Research Article

Certain Subclasses of Multivalent Analytic Functions

1Department of Mathematics, Suqian College, Suqian 223800, China
2Department of Mathematics, Yangzhou University, Yangzhou 225002, China

Received 26 February 2013; Revised 17 May 2013; Accepted 29 June 2013

Academic Editor: Pedro M. Lima

Copyright © 2013 Yi-Ling Cang and Jin-Lin Liu. 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

Two new subclasses and of multivalent analytic functions are introduced. Distortion inequalities and inclusion relation for and are obtained. Some results of the partial sums of functions in these classes are also given.

1. Introduction

Throughout this paper, we assume that Let denote the class of functions of the form which are analytic in the unit disk .

For functions and analytic in , we say that is subordinate to in and write , if there exists a Schwarz function in such that

Let Then the Hadamard product (or convolution) of and is defined by The following lemma will be required in our investigation.

Lemma 1. Let defined by (2) satisfy Then where

Proof. For defined by (2), the function in (8) can be expressed as with In view of (1) and (9), we see that
Let inequality (6) hold. Then from (10) and (12) we deduce that Hence, by the maximum modulus theorem, we arrive at (7).

We now consider the following two subclasses of .

Definition 2. A function defined by (2) is said to be in the class if and only if it satisfies the coefficient inequality (6).

Definition 3. A function defined by (2) is said to be in the class if and only if it satisfies
It is obvious from Definitions 2 and 3 that
If we write then it is easy to verify that Thus we obtain the following inclusion relations: Therefore, by Lemma 1, we see that each function in the classes and is starlike with respect to -symmetric points. Analytic functions which are starlike with respect to symmetric points and related functions have been extensively investigated in [16]. Recently, several authors have obtained many important properties and characteristics of multivalent analytic functions (see, e.g., [711]).
The main object of this paper is to present some distortion inequalities of functions in the classes and which we have introduced here. In particular some results of inclusion relation and convolution of functions in these classes are also given. Further we derive several interesting results of the partial sums of functions in these classes.

2. Main Results

Theorem 4. Let and suppose that either(a) and , or(b) and .
 (i) If , then, for , The bounds in (19) are best possible for the function defined by  (ii) If , then, for , The bounds in (21) are best possible for the function defined by

Proof. Let . For and , we have , and so For and , we have and If either (a) or (b) is satisfied, then  (i) If then it follows from (23) to (25) that Hence we have for . (ii) If then (23) to (25) yield This leads to (21). The proof of the theorem is complete.

Theorem 5. Let  (i) If , then, for , Equalities in (33) are attained, for example, by  (ii) If , then, for , Equalities in (36) are attained, for example, by

Proof. Note that implies that  (i) For , it follows from (23), (24), and (38) that From this we can get (33). (ii) For , from (23), (24), and (38) we deduce that Hence we have (36). The proof of the theorem is complete.

Theorem 6. Let .(i)If , then, for , The bounds in (41) are sharp for the function defined by  (ii)  If , then, for , The bounds in (43) are sharp for the function defined by

Proof. Let . For and , we have , , and For and , we have , and so (i)  If then it follows from (45) and (46) that Hence we have for .(ii)  If then (45) and (46) yield This leads to (43). Thus we complete the proof.

Next, we generalize the inclusion relation which is mentioned in (18).

Theorem 7. If , then where

Proof. Since and , we see that
Let . In order to prove that , we need only to find the smallest such that for all , that is, that
For and , (56) is equivalent to Noting (1), a simple calculation shows that for all real and , and so the function is decreasing in . Therefore For and , (56) becomes
Consequently, by taking it follows from (55) to (60) that . The proof is complete.

Remark 8. If we take in Theorem 7, then from (1) we have . This shows that

Theorem 9. Let . Then where

Proof. For , from Lemma 1 we have (7), which is equivalent to or to Obviously If we put then, for , and, for ,
Now, making use of (65) to (69), we arrive at for , and . This gives the desired result (62). The proof of the theorem is complete.

Corollary 10. Let . Then where is the same as in Theorem 9.

Proof. Since if and only if it follows from Theorem 9 that Thus we complete the proof.

Finally, we derive certain results of the partial sums of functions in the classes and .

Let be given by (2) and define the partial sums and by For simplicity we use the notation defined by (16).

Theorem 11. Let and let either(a) and , or(b) and .Then, for , we have The bounds in (75) and (76) are best possible for each .

Proof. If either (a) or (b) is satisfied, then, for ,
Let . Then it follows from (77) that
If we put for and , then and we deduce from (78) that This implies that for , and so (75) holds true for .
Similarly, by setting it follows from (78) that Hence we have (76) for .
For , replacing (78) by and proceeding as the above, we see that (75) and (76) are also true.
Furthermore, taking the function defined by we have , Thus the proof of Theorem 11 is completed.

Remark 12. Replacing by , it follows from Theorem 11 that inequalities (75) and (76) are true. In Theorem 13 we improve the bounds in (75) and (76) for .

Theorem 13. Let and let either (a) or (b) in Theorem 11 be satisfied. Then, for , one has The bounds in (86) are sharp for the function defined by

Proof. In view of the assumptions of Theorem 13, it follows from (77) that If we put then (88) leads to . Hence we have (86). Sharpness can be verified easily.

Acknowledgment

The authors would like to express sincere thanks to the referees for careful reading and suggestions which helped them improve the paper.

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