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International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 528570, 10 pages
http://dx.doi.org/10.1155/2012/528570
Research Article

Neighborhoods of Certain Multivalently Analytic Functions

Civil Aviation College, Kocaeli University, Arslanbey Campus, 41285 İzmit-Kocaeli, Turkey

Received 28 March 2012; Accepted 16 May 2012

Academic Editor: Hernando Quevedo

Copyright © 2012 Serap Bulut. 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

We introduce and investigate two new general subclasses of multivalently analytic functions of complex order by making use of the familiar convolution structure of analytic functions. Among the various results obtained here for each of these function classes, we derive the coefficient bounds, distortion inequalities, and other interesting properties and characteristics for functions belonging to the classes introduced here.

1. Introduction and Definitions

Let =(,) be the set of real numbers, and let be the set of complex numbers, ={1,2,3,}=0{0},={1}={2,3,4,}.(1.1)

Let 𝒯𝑝 denote the class of functions of the form 𝑓(𝑧)=𝑧𝑝𝑗=𝑘𝑎𝑗𝑧𝑗𝑝<𝑘;𝑎𝑗,0(𝑗𝑘);𝑘,𝑝(1.2) which are analytic and 𝑝-valent in the open unit disk 𝕌={𝑧𝑧,|𝑧|<1}.(1.3)

Denote by 𝑓𝑔 the Hadamard product (or convolution) of the functions 𝑓 and 𝑔, that is, if 𝑓 is given by (1.2) and 𝑔 is given by 𝑔(𝑧)=𝑧𝑝+𝑗=𝑘𝑏𝑗𝑧𝑗𝑝<𝑘;𝑏𝑗,0(𝑗𝑘);𝑘,𝑝(1.4) then (𝑓𝑔)(𝑧)=𝑧𝑝𝑗=𝑘𝑎𝑗𝑏𝑗𝑧𝑗=(𝑔𝑓)(𝑧).(1.5)

In [1], the author defined the following general class.

Definition 1.1. Let the function 𝑓𝒯𝑝. Then we say that 𝑓 is in the class 𝒮𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛) if it satisfies the condition |||||1𝑏𝑧𝑛𝜆,𝜇𝑔(𝑚+𝑛)(𝑧)𝜆,𝜇𝑔(𝑚)(𝑧)(𝑝𝑚)𝑛|||||<𝛽,𝑚+𝑛<𝑝<𝑘;𝑝,𝑛;𝑚0;𝑏{0};0𝜇𝜆1;0<𝛽1;𝑧𝕌(1.6) where 𝜆,𝜇(𝑧)=𝜆𝜇𝑧2𝑓(𝑧)+(𝜆𝜇)𝑧𝑓(𝑧)+(1𝜆+𝜇)𝑓(𝑧),(1.7)𝑔 is given by (1.4), and (𝜈)𝑛 denotes the falling factorial defined as follows: (𝜈)0𝜈0,=1=(𝜈)𝑛𝜈𝑛=𝜈(𝜈1)(𝜈𝑛+1)=𝑛!(𝑛).(1.8)

Various special cases of the class 𝒮𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛) were considered by many earlier researchers on this topic of geometric function theory. For example, 𝒮𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛) reduces to the function class(i)𝒮𝑛𝑔(𝑝,𝜆,𝑏,𝛽) for 𝑚=0, 𝑛=1, and 𝜇=0, studied by Mostafa and Aouf [2],(ii)𝒮𝑔(𝑝,𝑘,𝑏,𝑚,𝑛) for 𝜆=𝜇=0, and 𝛽=1, studied by Srivastava et al. [3],(iii)𝒮𝑔(𝑝,𝑛,𝑏,𝑚) for 𝑛=1, 𝜆=𝜇=0, and 𝛽=1, studied by Prajapat et al. [4],(iv)𝒮𝑛,𝑝(𝑔;𝜆,𝜇,𝛼) for 𝑚=0, 𝑛=1, 𝛽=1, and 𝑏=𝑝(1𝛼)(0𝛼<1), studied by Srivastava and Bulut [5],(v)𝒯𝒮𝑔(𝑝,𝑚,𝛼) for 𝑚=0, 𝑛=1, 𝜆=𝜇=0, 𝛽=1, and 𝑏=𝑝(1𝛼)(0𝛼<1), studied by Ali et al. [6].

Definition 1.2. Let the function 𝑓𝒯𝑝. Then we say that 𝑓 is in the class 𝒦𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛;𝑞,𝑢) if it satisfies the following nonhomogenous Cauchy-Euler differential equation (see, e.g., [7, page 1360, Equation (9)] and [5, page 6512, Equation (1.9)]): 𝑧𝑞𝑑𝑞𝑤𝑑𝑧𝑞+𝑞1(𝑢+𝑞1)𝑧𝑞1𝑑𝑞1𝑤𝑑𝑧𝑞1𝑞𝑞𝑤++𝑞1𝜀=0(𝑢+𝜀)=(𝑧)𝑞1𝜀=0(𝑢+𝜀+𝑝),(1.9) where 𝑤=𝑓(𝑧)𝒯𝑝;𝒮𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛);𝑞,𝑢(𝑝,).(1.10)

Setting 𝑚=0, 𝑛=1, 𝜇=0, and 𝑞=2 in Definition 1.2, we have the special class introduced by Mostafa and Aouf [2].

Following the works of Goodman [8] and Ruscheweyh [9] (see also [10, 11]), Altıntaş [12] defined the 𝛿-neighborhood of a function 𝑓𝒯(𝑝) by 𝒩𝛿𝑘(𝑓)=𝒯𝑝(𝑧)=𝑧𝑝𝑗=𝑘𝑐𝑗𝑧𝑗,𝑗=𝑘𝑗||𝑎𝑗𝑐𝑗||.𝛿(1.11)

It follows from the definition (1.11) that if 𝑒(𝑧)=𝑧𝑝(𝑝),(1.12) then 𝒩𝛿𝑘(𝑒)=𝒯𝑝(𝑧)=𝑧𝑝𝑗=𝑘𝑐𝑗𝑧𝑗,𝑗=𝑘𝑗||𝑐𝑗||.𝛿(1.13)

The main object of this paper is to investigate the various properties and characteristics of functions belonging to the above-defined classes 𝒮𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛),𝒦𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛;𝑞,𝑢).(1.14) Apart from deriving coefficient bounds and distortion inequalities for each of these classes, we establish several inclusion relationships involving the 𝛿-neighborhoods of functions belonging to the general classes which are introduced above.

2. Coefficient Bounds and Distortion Theorems

Lemma 2.1 (see [1]). Let the function 𝑓𝒯𝑝 be given by (1.2). Then 𝑓 is in the class 𝒮𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛) if and only if 𝑗=𝑘(𝑗)𝑚(𝑗𝑚)𝑛(𝑝𝑚)𝑛||𝑏||+𝛽𝜓(𝑗)𝑎𝑗𝑏𝑗||𝑏||(𝛽𝑝)𝑚𝜓(𝑝),𝑚+𝑛<𝑝<𝑘;𝑝,𝑛;𝑚0,;𝑏{0};0<𝛽1;𝑧𝕌(2.1) where 𝜓(𝑠)=(𝑠1)(𝜆𝜇𝑠+𝜆𝜇)+1(0𝜇𝜆1).(2.2)

Remark 2.2. If we set 𝑚=0, 𝑛=1, and 𝜇=0 in Lemma 2.1, then we have [2, Lemma 1].

Lemma 2.3 (See[1]). Let the function 𝑓𝒯𝑝 given by (1.2) be in the class 𝒮𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛). Then, for 𝑏𝑗𝑏𝑘(𝑗𝑘), one has 𝑗=𝑘𝑎𝑗𝛽||𝑏||(𝑝)𝑚𝜓(𝑝)(𝑘)𝑚(𝑘𝑚)𝑛(𝑝𝑚)𝑛||𝑏||+𝛽𝜓(𝑘)𝑏𝑘,(2.3)𝑗=𝑘𝑗𝑎𝑗||𝑏||(𝑘𝑚)!𝛽(𝑝)𝑚𝜓(𝑝)(𝑘1)!(𝑘𝑚)𝑛(𝑝𝑚)𝑛||𝑏||+𝛽𝜓(𝑘)𝑏𝑘||𝑏||,𝑝>(2.4) where 𝜓 is defined by (2.2).

Remark 2.4. If we set 𝑚=0, 𝑛=1, and 𝜇=0 in Lemma 2.3, then we have [2, Lemma  2].

The distortion inequalities for functions in the class 𝒮𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛) are given by the following Theorem 2.5.

Theorem 2.5. Let a function 𝑓𝒯𝑝 be in the class 𝒮𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛). Then ||||𝑓(𝑧)|𝑧|𝑝+𝛽||𝑏||(𝑝)𝑚𝜓(𝑝)(𝑘)𝑚(𝑘𝑚)𝑛(𝑝𝑚)𝑛||𝑏||+𝛽𝜓(𝑘)𝑏𝑘|𝑧|𝑘,||||(2.5)𝑓(𝑧)|𝑧|𝑝𝛽||𝑏||(𝑝)𝑚𝜓(𝑝)(𝑘)𝑚(𝑘𝑚)𝑛(𝑝𝑚)𝑛||𝑏||+𝛽𝜓(𝑘)𝑏𝑘|𝑧|𝑘,(2.6) and in general ||𝑓(𝑟)||(𝑧)(𝑝)𝑟|𝑧|𝑝𝑟+𝛽||𝑏||(𝑝)𝑚(𝑘)𝑟𝜓(𝑝)(𝑘)𝑚(𝑘𝑚)𝑛(𝑝𝑚)𝑛||𝑏||+𝛽𝜓(𝑘)𝑏𝑘|𝑧|𝑘𝑟,||𝑓(𝑟)||(𝑧)(𝑝)𝑟|𝑧|𝑝𝑟𝛽||𝑏||(𝑝)𝑚(𝑘)𝑟𝜓(𝑝)(𝑘)𝑚(𝑘𝑚)𝑛(𝑝𝑚)𝑛||𝑏||+𝛽𝜓(𝑘)𝑏𝑘|𝑧|𝑘𝑟,𝑝>𝑟;𝑟0,;𝑧𝕌(2.7) where 𝜓 is defined by (2.2).

Proof. Suppose that 𝑓𝒮𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛). We find from the inequality (2.3) that ||||𝑓(𝑧)|𝑧|𝑝+|𝑧|𝑘𝑗=𝑘𝑎𝑗|𝑧|𝑝+𝛽||𝑏||(𝑝)𝑚𝜓(𝑝)(𝑘)𝑚(𝑘𝑚)𝑛(𝑝𝑚)𝑛||𝑏||+𝛽𝜓(𝑘)𝑏𝑘|𝑧|𝑘,(2.8) which is equivalent to (2.5) and ||||𝑓(𝑧)|𝑧|𝑝|𝑧|𝑘𝑗=𝑘𝑎𝑗|𝑧|𝑝𝛽||𝑏||(𝑝)𝑚𝜓(𝑝)(𝑘)𝑚(𝑘𝑚)𝑛(𝑝𝑚)𝑛||𝑏||+𝛽𝜓(𝑘)𝑏𝑘|𝑧|𝑘,(2.9) which is precisely the assertion (2.6).

If we set 𝑚=0, 𝑛=1, and 𝜇=0 in Theorem 2.5, then we get the following.

Corollary 2.6. Let a function 𝑓𝒯𝑝 be in the class 𝒮𝑛𝑔(𝑝,𝜆,𝑏,𝛽). Then ||||𝑓(𝑧)|𝑧|𝑝+𝛽||𝑏||[]1+𝜆(𝑝1)||𝑏||[]𝑏𝑘𝑝+𝛽1+𝜆(𝑘1)𝑘|𝑧|𝑘,||||𝑓(𝑧)|𝑧|𝑝𝛽||𝑏||[]1+𝜆(𝑝1)||𝑏||[]𝑏𝑘𝑝+𝛽1+𝜆(𝑘1)𝑘|𝑧|𝑘,(2.10) and in general ||𝑓(𝑟)||(𝑧)𝑝!(𝑝𝑟)!|𝑧|𝑝𝑟+||𝑏||[]𝑘!𝛽1+𝜆(𝑝1)||𝑏||[]𝑏(𝑘𝑟)!𝑘𝑝+𝛽1+𝜆(𝑘1)𝑘|𝑧|𝑘𝑟,||𝑓(𝑟)||(𝑧)𝑝!(𝑝𝑟)!|𝑧|𝑝𝑟||𝑏||[]𝑘!𝛽1+𝜆(𝑝1)||𝑏||[]𝑏(𝑘𝑟)!𝑘𝑝+𝛽1+𝜆(𝑘1)𝑘|𝑧|𝑘𝑟,𝑝>𝑟;𝑟0,;𝑧𝕌(2.11) where 𝜓 is defined by (2.2).

The distortion inequalities for functions in the class 𝒦𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛;𝑞,𝑢) are given by Theorem 2.7 below.

Theorem 2.7. Let a function 𝑓𝒯𝑝 be in the class 𝒦𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛;𝑞,𝑢). Then ||||𝑓(𝑧)|𝑧|𝑝+𝛽||𝑏||(𝑝)𝑚𝜓(𝑝)𝑞1𝜀=0(𝑢+𝜀+𝑝)(𝑘)𝑚(𝑘𝑚)𝑛(𝑝𝑚)𝑛||𝑏||+𝛽𝜓(𝑘)(𝑞1)𝑞2𝜀=0(𝑢+𝜀+𝑘)𝑏𝑘|𝑧|𝑘,||||(2.12)𝑓(𝑧)|𝑧|𝑝𝛽||𝑏||(𝑝)𝑚𝜓(𝑝)𝑞1𝜀=0(𝑢+𝜀+𝑝)(𝑘)𝑚(𝑘𝑚)𝑛(𝑝𝑚)𝑛||𝑏||+𝛽𝜓(𝑘)(𝑞1)𝑞2𝜀=0(𝑢+𝜀+𝑘)𝑏𝑘|𝑧|𝑘,(2.13) and in general ||𝑓(𝑟)||(𝑧)(𝑝)𝑟|𝑧|𝑝𝑟+(𝑘)𝑟𝛽||𝑏||(𝑝)𝑚𝜓(𝑝)𝑞1𝜀=0(𝑢+𝜀+𝑝)(𝑘)𝑚(𝑘𝑚)𝑛(𝑝𝑚)𝑛||𝑏||+𝛽𝜓(𝑘)(𝑞1)𝑞2𝜀=0(𝑢+𝜀+𝑘)𝑏𝑘|𝑧|𝑘𝑟,||𝑓(𝑟)||(𝑧)(𝑝)𝑟|𝑧|𝑝𝑟(𝑘)𝑟𝛽||𝑏||(𝑝)𝑚𝜓(𝑝)𝑞1𝜀=0(𝑢+𝜀+𝑝)(𝑘)𝑚(𝑘𝑚)𝑛(𝑝𝑚)𝑛||𝑏||+𝛽𝜓(𝑘)(𝑞1)𝑞2𝜀=0(𝑢+𝜀+𝑘)𝑏𝑘|𝑧|𝑘𝑟,𝑝>𝑟;𝑟0,;𝑧𝕌(2.14) where 𝜓 is defined by (2.2).

Proof. Suppose that a function 𝑓𝒯𝑝 is given by (1.2), and also let the function 𝒮𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛) be occurring in the nonhomogenous Cauchy-Euler differential equation (1.9) with of course 𝑐𝑗0(𝑗𝑘).(2.15) Then we readily see from (1.9) that 𝑎𝑗=𝑞1𝜀=0(𝑢+𝜀+𝑝)𝑞1𝜀=0𝑐(𝑢+𝜀+𝑗)𝑗(𝑗𝑘),(2.16) so that 𝑓(𝑧)=𝑧𝑝𝑗=𝑘𝑎𝑗𝑧𝑗=𝑧𝑝𝑗=𝑘𝑞1𝜀=0(𝑢+𝜀+𝑝)𝑞1𝜀=0𝑐(𝑢+𝜀+𝑗)𝑗𝑧𝑗,||||(2.17)𝑓(𝑧)|𝑧|𝑝+|𝑧|𝑘𝑗=𝑘𝑞1𝜀=0(𝑢+𝜀+𝑝)𝑞1𝜀=0𝑐(𝑢+𝜀+𝑗)𝑗.(2.18) Moreover, since 𝒮𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛), the first assertion (2.3) of Lemma 2.3 yields the following inequality: 𝑐𝑗𝛽||𝑏||(𝑝)𝑚𝜓(𝑝)(𝑘)𝑚(𝑘𝑚)𝑛(𝑝𝑚)𝑛||𝑏||+𝛽𝜓(𝑘)𝑏𝑘,(2.19) and together with (2.19) and (2.18) it yields that ||||𝑓(𝑧)|𝑧|𝑝+𝛽||𝑏||(𝑝)𝑚𝜓(𝑝)𝑞1𝜀=0(𝑢+𝜀+𝑝)(𝑘)𝑚(𝑘𝑚)𝑛(𝑝𝑚)𝑛||𝑏||+𝛽𝜓(𝑘)𝑏𝑘|𝑧|𝑘𝑗=𝑘1𝑞1𝜀=0.(𝑢+𝜀+𝑗)(2.20) Finally, in view of the following sum: 𝑗=𝑘1𝑞1𝜀=0=(𝑗+𝑢+𝜀)𝑗=𝑘𝑞1𝜀=0(1)𝜀=1(𝑞1𝜀)!𝜀!(𝑗+𝑢+𝜀)(𝑞1)𝑞2𝜀=0,(𝑢+𝜀+𝑘)(𝑢{𝑘,𝑘1,𝑘2,}),(2.21) the assertion (2.12) of Theorem 2.7 follows at once from (2.20) together with (2.21). The assertion (2.13) can be proven by similarly applying (2.17), and (2.19)–(2.21).

Remark 2.8. If we set 𝑚=0, 𝑛=1, 𝜇=0, and 𝑞=2 in Theorem 2.7, then we have [2, Theorem 1].

3. Neighborhoods for the Classes 𝑆𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛) and 𝒦𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛;𝑞,𝑢)

In this section, we determine inclusion relations for the classes 𝒮𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛),𝒦𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛;𝑞,𝑢),(3.1) involving 𝛿-neighborhoods defined by (1.11) and (1.13).

Theorem 3.1 (see [1]). If 𝑏𝑗𝑏𝑘(𝑗𝑘) and ||𝑏||𝛿=(𝑘𝑚)!𝛽(𝑝)𝑚𝜓(𝑝)(𝑘1)!(𝑘𝑚)𝑛(𝑝𝑚)𝑛||𝑏||+𝛽𝜓(𝑘)𝑏𝑘||𝑏||,𝑝>(3.2) then 𝒮𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛)𝒩𝛿𝑘(𝑒),(3.3) where 𝑒 and 𝜓 are given by (1.12) and (2.2), respectively.

Remark 3.2. If we set 𝑚=0, 𝑛=1, and 𝜇=0 in Theorem 3.1, then we have [2, Theorem 2].

Theorem 3.3. If 𝑏𝑗𝑏𝑘(𝑗𝑘) and ||𝑏||𝛿=(𝑘𝑚)!𝛽(𝑝)𝑚𝜓(𝑝)(𝑘1)!(𝑘𝑚)𝑛(𝑝𝑚)𝑛||𝑏||+𝛽𝜓(𝑘)𝑏𝑘1+𝑞1𝜀=0(𝑢+𝜀+𝑝)(𝑞1)𝑞2𝜀=0||𝑏||(𝑢+𝜀+𝑘)𝑝>,(3.4) then 𝒦𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛;𝑞,𝑢)𝒩𝛿𝑘(),(3.5) where and 𝜓 are given by (1.11) and (2.2), respectively.

Proof. Suppose that 𝑓𝒦𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛;𝑞,𝑢). Then, upon substituting from (2.16) into the following coefficient inequality: 𝑗=𝑘𝑗||𝑐𝑗𝑎𝑗||𝑗=𝑘𝑗𝑐𝑗+𝑗=𝑘𝑗𝑎𝑗𝑐𝑗0;𝑎𝑗,0(3.6) we obtain 𝑗=𝑘𝑗||𝑐𝑗𝑎𝑗||𝑗=𝑘𝑗𝑐𝑗+𝑗=𝑘𝑞1𝜀=0(𝑢+𝜀+𝑝)𝑞1𝜀=0(𝑢+𝜀+𝑗)𝑗𝑐𝑗.(3.7) Since 𝒮𝑔(𝑝,𝑘,𝜆,𝑏,𝛽,𝑚,𝑛), the assertion (2.4) of Lemma 2.3 yields 𝑗𝑐𝑗||𝑏||(𝑘𝑚)!𝛽(𝑝)𝑚𝜓(𝑝)(𝑘1)!(𝑘𝑚)𝑛(𝑝𝑚)𝑛||𝑏||+𝛽𝜓(𝑘)𝑏𝑘||𝑏||.𝑝>(3.8) Finally, by making use of (2.4) as well as (3.8) on the right-hand side of (3.7), we find that 𝑗=𝑘𝑗||𝑐𝑗𝑎𝑗||||𝑏||(𝑘𝑚)!𝛽(𝑝)𝑚𝜓(𝑝)(𝑘1)!(𝑘𝑚)𝑛(𝑝𝑚)𝑛||𝑏||+𝛽𝜓(𝑘)𝑏𝑘×1+𝑗=𝑘𝑞1𝜀=0(𝑢+𝜀+𝑝)𝑞1𝜀=0,(𝑢+𝜀+𝑗)(3.9) which, by virtue of the sum in (2.21), immediately yields 𝑗=𝑘𝑗||𝑐𝑗𝑎𝑗||||𝑏||(𝑘𝑚)!𝛽(𝑝)𝑚𝜓(𝑝)(𝑘1)!(𝑘𝑚)𝑛(𝑝𝑚)𝑛||𝑏||+𝛽𝜓(𝑘)𝑏𝑘×1+𝑞1𝜀=0(𝑢+𝜀+𝑝)(𝑞1)𝑞2𝜀=0(𝑢+𝜀+𝑘)=𝛿.(3.10) Thus, by applying the definition (1.11), we complete the proof of Theorem 3.3.

Remark 3.4. If we set 𝑚=0, 𝑛=1, 𝜇=0, and 𝑞=2 in Theorem 3.3, then we have [2, Theorem  3].

Acknowledgment

The present paper was supported by the Kocaeli University under Grant no. HD 2011/22.

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