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ISRN Mathematical Analysis
Volume 2012 (2012), Article ID 403028, 12 pages
http://dx.doi.org/10.5402/2012/403028
Research Article

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.

Abstract

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.

1. Introduction

Let 𝒜 denote the class of functions of the form𝑓(𝑧)=𝑧+𝑗=2𝑎𝑗𝑧𝑗,(1.1) which are analytic in the open unit disk 𝕌={𝑧𝑧,|𝑧|<1}.(1.2)

For 0𝛼<1, we denote by 𝒮(𝛼) and 𝒦(𝛼) the usual subclasses of 𝒜 consisting of functions which are, respectively, starlike of order 𝛼 and convex of order 𝛼 in 𝕌. Clearly, we know that 𝑓𝒦(𝛼)𝑧𝑓𝒮(𝛼).(1.3)

A function 𝑓𝒜 is said to be in the class (𝛽) if it satisfies the inequality 𝑧𝑓(𝑧)𝑓(𝑧)<𝛽(𝑧𝕌),(1.4) for some 𝛽(𝛽>1). Also, a function 𝑓𝒜 is said to be in the class 𝒩(𝛽) if and only if 𝑧𝑓(𝛽). The classes (𝛽) and 𝒩(𝛽) were introduced and investigated recently by Owa and Srivastava [1] (see also Nishiwaki and Owa [2], Owa and Nishiwaki [3], and Srivastava and Attiya [4]).

Sălăgean [5] introduced the operator 𝐷0𝑓(𝑧)=𝑓(𝑧),𝐷1𝑓(𝑧)=𝐷𝑓(𝑧)=𝑧𝑓𝐷(𝑧),𝑛𝐷𝑓(𝑧)=𝐷𝑛1𝑓(𝑧)(𝑛={1,2,}).(1.5) We note that 𝐷𝑛𝑓(𝑧)=𝑧+𝑗=2𝑗𝑛𝑎𝑗𝑧𝑗𝑛0.={0}(1.6)

Given two functions 𝑓,𝑔𝐴, where 𝑓 is given by (1.1) and 𝑔 is defined by 𝑔(𝑧)=𝑧+𝑛=2𝑏𝑛𝑧𝑛,(1.7) the Hadamard product (or convolution) 𝑓𝑔 is defined by (𝑓𝑔)(𝑧)=𝑧+𝑛=2𝑎𝑛𝑏𝑛𝑧𝑛=(𝑔𝑓)(𝑧).(1.8)

For two functions 𝑓 and 𝑔, analytic in 𝕌, we say that the function 𝑓 is subordinate to 𝑔 in 𝕌, and write 𝑓(𝑧)𝑔(𝑧)(𝑧𝕌)(1.9) if there exists a Schwarz function 𝜔, which is analytic in 𝕌 with ||||𝜔(0)=0,𝜔(𝑧)<1(𝑧𝕌)(1.10) such that 𝑓(𝑧)=𝑔(𝜔(𝑧))(𝑧𝕌).(1.11) Indeed, it is known that 𝑓(𝑧)𝑔(𝑧),(𝑧𝕌)𝑓(0)=𝑔(0),𝑓(𝕌)𝑔(𝕌).(1.12) Furthermore, if the function 𝑔 is univalent in 𝕌, then we have the following equivalence: 𝑓(𝑧)𝑔(𝑧),(𝑧𝕌)𝑓(0)=𝑔(0),𝑓(𝕌)𝑔(𝕌).(1.13)

In recent years, Deng [6] (see also Kamali [7], Altintaş et al. [8], Srivastava et al. [9], and Xu et al. [10]) introduced and investigated the following subclass of 𝒜 involving the S Sălăgean lagean operator and obtained the coefficient bounds for this function class.

Definition 1.1. A function 𝑓𝒜 is said to be in the class 𝒮𝑛(𝜆,𝛼,𝑏) if it satisfies the inequality 11+𝑏(1𝜆)𝐷𝑛+1𝑓(𝑧)+𝜆𝐷𝑛+2𝑓(𝑧)(1𝜆)𝐷𝑛𝑓(𝑧)+𝜆𝐷𝑛+1𝑓(𝑧)1>𝛼(𝑧𝕌),(1.14) where 𝑛0,𝑏{0},0𝛼<1,0𝜆1.(1.15)
It is easy to see that the class 𝒮𝑛(𝜆,𝛼,𝑏) includes the classes 𝒮(𝛼) and 𝒦(𝛼) as its special cases.
Now, motivated essentially by the above-mentioned function classes, we introduce the following subclass of 𝒜 of analytic functions.

Definition 1.2. A function 𝑓𝒜 is said to be in the class 𝑛(𝜆,𝛽,𝑏) if it satisfies the inequality: 11+𝑏(1𝜆)𝐷𝑛+1𝑓(𝑧)+𝜆𝐷𝑛+2𝑓(𝑧)(1𝜆)𝐷𝑛𝑓(𝑧)+𝜆𝐷𝑛+1𝑓(𝑧)1<𝛽(𝑧𝕌),(1.16) where 𝑛0,𝑏{0},𝛽>1,0𝜆1.(1.17)

It is also easy to see that the classes (𝛽) and 𝒩(𝛽) are special cases of the class 𝑛(𝜆,𝛽,𝑏).

In this paper, we aim at proving some coefficient inequalities and subordination properties for the classes 𝒮𝑛(𝜆,𝛽,𝑏) and 𝑛(𝜆,𝛽,𝑏). 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 𝒮𝑛(𝜆,𝛼,𝑏) and 𝑛(𝜆,𝛼,𝑏).

Theorem 2.1. Let 𝑛0,𝑏{0},0𝛼<1,0𝜆1.(2.1) If 𝑓𝒜 satisfies the coefficient inequality 𝑗=2(1𝜆)𝑗𝑛+𝜆𝑗𝑛+1||𝑏||(||𝑎𝑗1+1𝛼)𝑗||||𝑏||(1𝛼),(2.2) then 𝑓𝒮𝑛(𝜆,𝛼,𝑏).

Proof. To prove 𝑓𝒮𝑛(𝜆,𝛼,𝑏), it is sufficient to show that ||||(1𝜆)𝐷𝑛+1𝑓(𝑧)+𝜆𝐷𝑛+2𝑓(𝑧)(1𝜆)𝐷𝑛𝑓(𝑧)+𝜆𝐷𝑛+1𝑓||||<||𝑏||(𝑧)1(1𝛼)(𝑧𝕌).(2.3) By noting that ||||(1𝜆)𝐷𝑛+1𝑓(𝑧)+𝜆𝐷𝑛+2𝑓(𝑧)(1𝜆)𝐷𝑛𝑓(𝑧)+𝜆𝐷𝑛+1𝑓||||=|||||(𝑧)1𝑗=2(𝑗1𝜆)𝑛+1𝑗𝑛𝑗+𝜆𝑛+2𝑗𝑛+1𝑎𝑗𝑧𝑗11+𝑗=2(1𝜆)𝑗𝑛+𝜆𝑗𝑛+1𝑎𝑗𝑧𝑗1|||||𝑗=2𝑗(1𝜆)𝑛+1𝑗𝑛𝑗+𝜆𝑛+2𝑗𝑛+1||𝑎𝑗||1𝑗=2(1𝜆)𝑗𝑛+𝜆𝑗𝑛+1||𝑎𝑗||,(2.4) it follows from (2.2) that the above last expression is bounded by |𝑏|(1𝛼). This completes the proof of Theorem 2.1.

Theorem 2.2. Let 𝑛0,𝑏{0},𝛽>1,0𝜆1.(2.5) If 𝑓𝒜 satisfies the coefficient inequality 𝑗=2(1𝜆)𝑗𝑛+𝜆𝑗𝑛+1||||||||||𝑎𝑏1+𝑗+𝑗1(2𝛽1)𝑏𝑗||||𝑏||(2𝛽1),(2.6) then 𝑓𝑛(𝜆,𝛽).

Proof. To prove 𝑓𝑛(𝜆,𝛽,𝑏), it suffices to show that ||||11+𝑏(1𝜆)𝐷𝑛+1𝑓(𝑧)+𝜆𝐷𝑛+2𝑓(𝑧)(1𝜆)𝐷𝑛𝑓(𝑧)+𝜆𝐷𝑛+1𝑓||||<||||1(𝑧)11+𝑏(1𝜆)𝐷𝑛+1𝑓(𝑧)+𝜆𝐷𝑛+2𝑓(𝑧)(1𝜆)𝐷𝑛𝑓(𝑧)+𝜆𝐷𝑛+1||||.𝑓(𝑧)12𝛽(2.7) We consider 𝑀 defined by ||𝑀=(𝑏1)(1𝜆)𝐷𝑛𝑓(𝑧)+𝜆𝐷𝑛+1𝑓(𝑧)+(1𝜆)𝐷𝑛+1𝑓(𝑧)+𝜆𝐷𝑛+2||||𝑓(𝑧)(1𝜆)𝐷𝑛+1𝑓(𝑧)+𝜆𝐷𝑛+2𝑓[](𝑧)(2𝛽1)𝑏+1(1𝜆)𝐷𝑛𝑓(𝑧)+𝜆𝐷𝑛+1𝑓||=|||||(𝑧)𝑏𝑧+𝑗=2(𝑏1)(1𝜆)𝑗𝑛+𝜆𝑗𝑛+1+(1𝜆)𝑗𝑛+1+𝜆𝑗𝑛+2𝑎𝑗𝑧𝑗||||||||||𝑧+𝑗=2(1𝜆)𝑗𝑛+1+𝜆𝑗𝑛+2𝑎𝑗𝑧𝑗[](2𝛽1)𝑏+1𝑧+𝑗=2(1𝜆)𝑗𝑛+𝜆𝑗𝑛+1𝑎𝑗𝑧𝑗|||||.(2.8) Thus, for |𝑧|=𝑟<1, we have ||𝑏||𝑀𝑟+𝑗=2||||(𝑏11𝜆)𝑗𝑛+𝜆𝑗𝑛+1+(1𝜆)𝑗𝑛+1+𝜆𝑗𝑛+2||𝑎𝑗||𝑟𝑗||𝑏||(2𝛽1)𝑟𝑗=2||(1𝜆)𝑗𝑛+1+𝜆𝑗𝑛+2[](2𝛽1)𝑏+1(1𝜆)𝑗𝑛+𝜆𝑗𝑛+1||||𝑎𝑗||𝑟𝑗<𝑗=2||||𝑏1(1𝜆)𝑗𝑛+𝜆𝑗𝑛+1+(1𝜆)𝑗𝑛+1+𝜆𝑗𝑛+2+||(1𝜆)𝑗𝑛+1+𝜆𝑗𝑛+2[](2𝛽1)𝑏+1(1𝜆)𝑗𝑛+𝜆𝑗𝑛+1||||𝑎𝑗||||𝑏||2(𝛽1)𝑟.(2.9) It follows from (2.6) that 𝑀<0, which implies that (2.7) holds, that is, 𝑓𝑛(𝜆,𝛽,𝑏). The proof of Theorem 2.2 is evidently completed.

To prove our next result, we need the following lemma.

Lemma 2.3. Let 𝛽>1 and 𝑏{0}. Suppose also that the sequence {𝔹𝑗}𝑗=1 is defined by 𝔹1𝔹=1(𝑗=1),𝑗=2||𝑏||(𝛽1)𝑗1𝑗1𝑘=1𝔹𝑘(𝑗{1}),(2.10) then 𝔹𝑗=1(𝑗1)!𝑗2𝑘=02||𝑏||(𝛽1)+𝑘(𝑗{1}).(2.11)

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 𝔹2||𝑏||1=2(𝛽1)=1!0𝑘=02||𝑏||,(𝛽1)+𝑘(2.12) which implies that (2.11) holds for 𝑗=2.
We now suppose that (2.11) holds for 𝑗=𝑚(𝑚2), then 𝔹𝑚=1(𝑚1)!𝑚2𝑘=02||𝑏||.(𝛽1)+𝑘(2.13) Combining (2.10) and (2.13), we find that 𝔹𝑚+1=2||𝑏||(𝛽1)𝑚𝑚𝑘=1𝔹𝑘=2||𝑏||(𝛽1)𝑚𝑚1𝑘=1𝔹𝑘+2||𝑏||(𝛽1)𝑚𝔹𝑚=2||𝑏||(𝛽1)𝑚𝑚12||𝑏||𝔹(𝛽1)𝑚+2||𝑏||(𝛽1)𝑚𝔹𝑚=2||𝑏||(𝛽1)+𝑚1𝑚𝔹𝑚=1𝑚!𝑚1𝑘=02||𝑏||,(𝛽1)+𝑘(2.14) which shows that (2.11) holds for 𝑗=𝑚+1. The proof of Lemma 2.3 is evidently completed.

Theorem 2.4. Let 𝑓𝑛(𝜆,𝛽,𝑏), then ||𝑎𝑗||1(𝑗1)!(1𝜆+𝜆𝑗)𝑗𝑛𝑗2𝑘=02||𝑏||(𝛽1)+𝑘(𝑗{1}).(2.15)

Proof. We first suppose that 𝐹(𝑧)=(1𝜆)𝐷𝑛𝑓(𝑧)+𝜆𝐷𝑛+1𝑓(𝑧)=𝑧+𝑗=2𝐵𝑗𝑧𝑗(𝑧𝕌;𝑓𝒜),(2.16) where 𝐵𝑗=𝑗𝑛(1𝜆+𝜆𝑗)𝑎𝑗.(2.17) Next, by setting (𝑧)=𝛽1(1/𝑏)𝑧𝐹(𝑧)/𝐹(𝑧)1𝛽1=1+1𝑧+2𝑧2+𝑧𝕌;𝑓𝑛,(𝜆,𝛽,𝑏)(2.18) we easily find that 𝒫. It follows from (2.18) that 𝑧𝐹[]𝐹(𝑧)=1+𝑏(𝛽1)(𝑧)𝑏(𝛽1)(𝑧)𝐹(𝑧).(2.19) We now find from (2.16), (2.18), and (2.19) that 𝑧+2𝐵2𝑧2++𝑗𝐵𝑗𝑧𝑗=[]+1+𝑏(𝛽1)𝑧+𝐵2𝑧2++𝐵𝑗𝑧𝑗+𝑏(𝛽1)1+1𝑧+2𝑧2++𝑗𝑧𝑗+𝑧+𝐵2𝑧2++𝐵𝑗𝑧𝑗.+(2.20) By evaluating the coefficients of 𝑧𝑗 in both the sides of (2.20), we get 𝑗𝐵𝑗=[]𝐵1+𝑏(𝛽1)𝑗𝑏(𝛽1)𝑗1+𝑗2𝐵2++1𝐵𝑗1+𝐵𝑗.(2.21) On the other hand, it is well known that ||𝑘||2(𝑘).(2.22) Combining (2.21) and (2.22), we easily get ||𝐵𝑗||2||𝑏||(𝛽1)𝑗1𝑗1𝑘=1||𝐵𝑘||𝐵1.=1;𝑗{1}(2.23)
Suppose that 𝛽>1 and 𝑏{0}. We define the sequence {𝔹𝑗}𝑗=1 as follows: 𝔹1𝔹=1(𝑗=1),𝑗=2||𝑏||(𝛽1)𝑗1𝑗1𝑘=1𝔹𝑘(𝑗{1}).(2.24) In order to prove that ||𝐵𝑗||𝔹𝑗(𝑗{1}),(2.25) we use the principle of mathematical induction. By noting that ||𝐵2||||𝑏||2(𝛽1),(2.26) thus, assuming that ||𝐵𝑚||𝔹𝑚(𝑚{2,3,,𝑗}),(2.27) we find from (2.23) and (2.24) that ||𝐵𝑗+1||2||𝑏||(𝛽1)𝑗𝑗𝑘=1||𝐵𝑘||2||𝑏||(𝛽1)𝑗𝑗𝑘=1𝔹𝑘=𝔹𝑗+1(𝑗).(2.28) Therefore, by the principle of mathematical induction, we have ||𝐵𝑗||𝔹𝑗(𝑗{1})(2.29) as desired.
By virtue of Lemma 2.3 and (2.24), we know that 𝔹𝑗=1(𝑗1)!𝑗2𝑘=02||𝑏||(𝛽1)+𝑘(𝑗{1}).(2.30) Combining (2.17), (2.29), and (2.30), we readily arrive at the coefficient estimates (2.15) asserted by Theorem 2.4.

Remark 2.5. Setting 𝜆=0, 𝑏=1, and 𝑛=0or1 in Theorem 2.4, we get the corresponding results obtained by Owa and Nishiwaki [3].

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 [6] follows easily.

3. Subordination Properties

In view of Theorems 2.1 and 2.2, we now introduce the following subclasses: 𝒮𝑛(𝜆,𝛼,𝑏)𝒮𝑛(𝜆,𝛼,𝑏),𝑛(𝜆,𝛽,𝑏)𝑛(𝜆,𝛽,𝑏),(3.1) which consist of functions 𝑓𝒜 whose Taylor-Maclaurin coefficients satisfy the inequalities (2.2) and (2.6), respectively.

A sequence {𝑏𝑗}𝑗=1 of complex numbers is said to be a subordinating factor sequence if, whenever 𝑓 of the form (1.1) is analytic, univalent, and convex in 𝕌, we have the subordination 𝑗=1𝑎𝑗𝑏𝑗𝑧𝑗𝑎𝑓(𝑧)1.=1;𝑧𝕌(3.2)

To derive the subordination properties for the classes 𝒮𝑛(𝜆,𝛼,𝑏) and 𝑛(𝜆,𝛼,𝑏), we need the following lemma.

Lemma 3.1 (see [11]). The sequence {𝑏𝑗}𝑗=1 is a subordinating factor sequence if and only if 1+2𝑗=1𝑏𝑗𝑧𝑗>0(𝑧𝕌).(3.3)

Theorem 3.2. If 𝒮𝑓𝑛(𝜆,𝛼,𝑏) and 𝑔𝒦(0), then ||𝑏||Φ(𝑛,𝜆,𝛼,𝑏)(𝑓𝑔)(𝑧)𝑔(𝑧),(3.4)(𝑓)>(1𝛼)+2𝑛||𝑏||(1+𝜆)1+(1𝛼)2𝑛||𝑏||(1+𝜆)1+(1𝛼),(3.5) for 0𝜆1,0𝛼<1,𝑏{0},𝑛0,(3.6) where, for convenience, 2Φ(𝑛,𝜆,𝛼,𝑏)=𝑛1||𝑏||(1+𝜆)1+(1𝛼)||𝑏||(1𝛼)+2𝑛||𝑏||.(1+𝜆)1+(1𝛼)(3.7) The constant factor 2𝑛1||𝑏||(1+𝜆)1+(1𝛼)||𝑏||(1𝛼)+2𝑛||𝑏||(1+𝜆)1+(1𝛼)(3.8) in the subordination result (3.4) cannot be replaced by a larger one.

Proof. Let 𝒮𝑓𝑛(𝜆,𝛼,𝑏) and suppose that 𝑔(𝑧)=𝑧+𝑗=2𝑐𝑗𝑧𝑗𝒦=𝒦(0),(3.9) then Φ(𝑛,𝜆,𝛼,𝑏)(𝑓𝑔)(𝑧)=Φ(𝑛,𝜆,𝛼,𝑏)𝑧+𝑗=2𝑎𝑗𝑐𝑗𝑧𝑗,(3.10) where Φ(𝑛,𝜆,𝛼,𝑏) is defined by (3.7).
If Φ(𝑛,𝜆,𝛼,𝑏)𝑎𝑗𝑗=1(3.11) is a subordinating factor sequence with 𝑎1=1, then the subordination result (3.4) holds. By Lemma 3.1, we know that this is equivalent to the inequality 1+𝑗=12𝑛||𝑏||(1+𝜆)1+(1𝛼)||𝑏||(1𝛼)+2𝑛||𝑏||𝑎(1+𝜆)1+(1𝛼)𝑗𝑧𝑗>0(𝑧𝕌).(3.12) Since 𝑗𝑛||𝑏||(1𝜆+𝜆𝑗)𝑗1+(1𝛼)𝑗2;0𝜆1;0𝛼<1;𝑏{0};𝑛0(3.13) is an increasing function of 𝑗, and using Theorem 2.1, we have 1+𝑗=12𝑛||𝑏||(1+𝜆)1+(1𝛼)||𝑏||(1𝛼)+2𝑛||𝑏||𝑎(1+𝜆)1+(1𝛼)𝑗𝑧𝑗2=1+𝑛(||𝑏||(1+𝜆)1+1𝛼)||𝑏||(1𝛼)+2𝑛||𝑏||𝑎(1+𝜆)1+(1𝛼)1𝑧+1||𝑏||(1𝛼)+2𝑛||𝑏||(1+𝜆)1+(1𝛼)𝑗=22𝑛||𝑏||𝑎(1+𝜆)1+(1𝛼)𝑗𝑧𝑗21𝑛||𝑏||(1+𝜆)1+(1𝛼)||𝑏||(1𝛼)+2𝑛||𝑏||𝑟1(1+𝜆)1+(1𝛼)||𝑏||(1𝛼)+2𝑛||𝑏||(1+𝜆)1+(1𝛼)𝑗=22𝑛||𝑏||||𝑎(1+𝜆)1+(1𝛼)𝑗||𝑟𝑗2>1𝑛||𝑏||(1+𝜆)1+(1𝛼)||𝑏||(1𝛼)+2𝑛||𝑏||||𝑏||(1+𝜆)1+(1𝛼)𝑟(1𝛼)||𝑏||(1𝛼)+2𝑛||𝑏||𝑟(1+𝜆)1+(1𝛼)=1𝑟>0(|𝑧|=𝑟<1).(3.14) 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 𝑧𝑔(𝑧)==1𝑧𝑗=1𝑧𝑗𝒦.(3.15) Finally, we consider the function 𝑓0 defined by 𝑓0||𝑏||(𝑧)=𝑧(1𝛼)2𝑛||𝑏||𝑧(1+𝜆)1+(1𝛼)2𝑛0,;0𝜆1;0𝛼<1;𝑏{0}(3.16) which belongs to the class 𝒮𝑛(𝜆,𝛼,𝑏). Thus, by (3.4), we know that Φ(𝑛,𝜆,𝛼,𝑏)𝑓0𝑧(𝑧)1𝑧(𝑧𝕌).(3.17) Furthermore, it can be easily verified for the function 𝑓0 given by (3.16) that min𝑧𝕌Φ(𝑛,𝜆,𝛼,𝑏)𝑓01(𝑧)=2.(3.18) 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 𝑓𝑛(𝜆,𝛼,𝑏) and 𝑔𝒦(0), then ||𝑏||(Ψ(𝑛,𝜆,𝛽,𝑏)(𝑓𝑔)(𝑧)𝑔(𝑧),(3.19)(𝑓)>𝛽1)+2𝑛1(||||||||1+𝜆)𝑏1+2+1(2𝛽1)𝑏2𝑛1||||||||(1+𝜆)𝑏1+2+1(2𝛽1)𝑏,(3.20) for 0𝜆1,𝛽>1,𝑏{0},𝑛0,(3.21) where, for convenience, 2Ψ(𝑛,𝜆,𝛽,𝑏)=𝑛2||||||||(1+𝜆)𝑏1+2+1(2𝛽1)𝑏||𝑏||(𝛽1)+2𝑛1||||||||(1+𝜆)𝑏1+2+1(2𝛽1)𝑏.(3.22) The constant factor 2𝑛2||||||||(1+𝜆)𝑏1+2+1(2𝛽1)𝑏||𝑏||(𝛽1)+2𝑛1||||||||(1+𝜆)𝑏1+2+1(2𝛽1)𝑏(3.23) in the subordination result (3.19) cannot be replaced by a larger one.

Remark 3.4. Putting 𝜆=0, 𝑏=1, and 𝑛=0or1 in Corollary 3.3, we get the corresponding results obtained by Srivastava and Attiya [4].

Acknowledgments

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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