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ISRN Mathematical Analysis
Volume 2012 (2012), Article ID 403028, 12 pages
Some Properties of Certain Subclasses of Analytic Functions with Complex Order
1School of Mathematics and Statistics, Anyang Normal University, Anyang, Henan 455002, China
2School of Econometrics and Management, Changsha University of Science and Technology, Changsha, Hunan 410114, China
3Department of Mathematics, Huaihua University, Huaihua, Hunan 418008, China
Received 2 November 2011; Accepted 30 November 2011
Academic Editor: G. Martin
Copyright © 2012 Zhi-Gang Wang 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.
The main purpose of this paper is to derive some coefficient inequalities and subordination properties for certain subclasses of analytic functions involving the Salagean operator. Relevant connections of the results presented here with those obtained in earlier works are also pointed out.
Let denote the class of functions of the form which are analytic in the open unit disk
For , we denote by and the usual subclasses of consisting of functions which are, respectively, starlike of order and convex of order in . Clearly, we know that
A function is said to be in the class if it satisfies the inequality for some . Also, a function is said to be in the class if and only if . The classes and were introduced and investigated recently by Owa and Srivastava  (see also Nishiwaki and Owa , Owa and Nishiwaki , and Srivastava and Attiya ).
Sălăgean  introduced the operator We note that
Given two functions , where is given by (1.1) and is defined by the Hadamard product (or convolution) is defined by
For two functions and , analytic in , we say that the function is subordinate to in , and write if there exists a Schwarz function , which is analytic in with such that Indeed, it is known that Furthermore, if the function is univalent in , then we have the following equivalence:
In recent years, Deng  (see also Kamali , Altintaş et al. , Srivastava et al. , and Xu et al. ) introduced and investigated the following subclass of involving the S Sălăgean lagean operator and obtained the coefficient bounds for this function class.
Definition 1.1. A function is said to be in the class if it satisfies the inequality
It is easy to see that the class includes the classes and as its special cases.
Now, motivated essentially by the above-mentioned function classes, we introduce the following subclass of of analytic functions.
Definition 1.2. A function is said to be in the class if it satisfies the inequality: where
It is also easy to see that the classes and are special cases of the class .
In this paper, we aim at proving some coefficient inequalities and subordination properties for the classes and . The results presented here would provide extensions of those given in earlier works. Several other new results are also obtained.
2. Coefficient Inequalities
In this section, we derive some coefficient inequalities for the classes and .
Theorem 2.1. Let If satisfies the coefficient inequality then .
Theorem 2.2. Let If satisfies the coefficient inequality then .
Proof. To prove , it suffices to show that We consider defined by Thus, for , we have It follows from (2.6) that , which implies that (2.7) holds, that is, . The proof of Theorem 2.2 is evidently completed.
To prove our next result, we need the following lemma.
Lemma 2.3. Let and . Suppose also that the sequence is defined by then
Proof. We make use of the principle of mathematical induction to prove the assertion (2.11) of Lemma 2.3. Indeed, from (2.10), we know that
which implies that (2.11) holds for .
We now suppose that (2.11) holds for , then Combining (2.10) and (2.13), we find that which shows that (2.11) holds for . The proof of Lemma 2.3 is evidently completed.
Theorem 2.4. Let , then
Proof. We first suppose that
Next, by setting
we easily find that . It follows from (2.18) that
We now find from (2.16), (2.18), and (2.19) that
By evaluating the coefficients of in both the sides of (2.20), we get
On the other hand, it is well known that
Combining (2.21) and (2.22), we easily get
Suppose that and . We define the sequence as follows: In order to prove that we use the principle of mathematical induction. By noting that thus, assuming that we find from (2.23) and (2.24) that Therefore, by the principle of mathematical induction, we have as desired.
By virtue of Lemma 2.3 and (2.24), we know that Combining (2.17), (2.29), and (2.30), we readily arrive at the coefficient estimates (2.15) asserted by Theorem 2.4.
Remark 2.6. We cannot show that the result of Theorem 2.4 is sharp. Indeed, if one can prove the sharpness of Theorem 2.4, the sharpness of the corresponding result obtained by Deng  follows easily.
3. Subordination Properties
A sequence of complex numbers is said to be a subordinating factor sequence if, whenever of the form (1.1) is analytic, univalent, and convex in , we have the subordination
To derive the subordination properties for the classes and , we need the following lemma.
Lemma 3.1 (see ). The sequence is a subordinating factor sequence if and only if
Theorem 3.2. If and , then for where, for convenience, The constant factor in the subordination result (3.4) cannot be replaced by a larger one.
Proof. Let and suppose that
where is defined by (3.7).
If is a subordinating factor sequence with , then the subordination result (3.4) holds. By Lemma 3.1, we know that this is equivalent to the inequality Since is an increasing function of , and using Theorem 2.1, we have This evidently proves the inequality (3.12), and hence also the subordination result (3.4), asserted by Theorem 3.2. The inequality (3.5) asserted by Theorem 3.2 follows from (3.4) by setting Finally, we consider the function defined by which belongs to the class . Thus, by (3.4), we know that Furthermore, it can be easily verified for the function given by (3.16) that We thus complete the proof of Theorem 3.2.
The proof of the following subordination result is much akin to that of Theorem 3.2. We, therefore, choose to omit the analogous details involved.
Corollary 3.3. If and , then for where, for convenience, The constant factor in the subordination result (3.19) cannot be replaced by a larger one.
The present investigation was supported by the National Natural Science Foundation under grants 11101053, 70971013, and 71171024, the Natural Science Foundation of Hunan Province under grant 09JJ1010, the Key Project of Chinese Ministry of Education under grant 211118, the Excellent Youth Foundation of Educational Committee of Hunan Province under grant 10B002, the Open Fund Project of Key Research Institute of Philosophies and Social Sciences in Hunan Universities under grant 11FEFM02, and the Key Project of Natural Science Foundation of Educational Committee of Henan Province under grant 12A110002 of China.
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