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International Journal of Mathematics and Mathematical Sciences
Volume 2012 (2012), Article ID 309465, 9 pages
Research Article

Coefficient Bounds for Certain Subclasses of Analytic Functions Defined by Komatu Integral Operator

School of Civil Aviation College, Kocaeli University, Arslanbey Campus, 41285 Izmit-Kocaeli, Turkey

Received 23 March 2012; Accepted 11 July 2012

Academic Editor: Yuri Latushkin

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.


We determine the coeffcient bounds for functions in certain subclasses of analytic functions of complex order, which are introduced here by means of a certain non-homogeneous Cauchy–Euler type differential equation of order m. Relevant connections of some of the results obtained with those in earlier works are also provided.

1. Introduction, Definitions and Preliminaries

Let =(,) be the set of real numbers, let be the set of complex numbers, ={1,2,3,}=0{0}(1.1) be the set of positive integers and ={1}={2,3,4,}.(1.2)

Let 𝒜 denote the class of functions of the form 𝑓(𝑧)=𝑧+𝑛=2𝑎𝑛𝑧𝑛(1.3) which are analytic in the unit disk: 𝕌={𝑧|𝑧|<1}.(1.4)

Recently, Komatu [1] introduced a certain integral operator 𝐿𝛿𝑎 defined by 𝐿𝛿𝑎𝑎𝑓(𝑧)=𝛿Γ(𝛿)10𝑡𝑎21log𝑡𝛿1𝑓(𝑧𝑡)𝑑𝑡,𝑧𝕌;𝑎>0;𝛿0;𝑓(𝑧)𝒜.(1.5)

Thus, if 𝑓𝒜 is of the form (1.3), then it is easily seen from (1.5) that (see [1]) 𝐿𝛿𝑎𝑓(𝑧)=𝑧+𝑛=2𝑎𝑎+𝑛1𝛿𝑎𝑛𝑧𝑛,𝑎>0;𝛿0.(1.6)

Using the relation (1.6), it is easily verfied that 𝑧𝐿𝑎𝛿+1𝑓(𝑧)=𝑎𝐿𝛿𝑎𝑓(𝑧)(𝑎1)𝐿𝑎𝛿+1𝐿𝑓(𝑧),𝛿𝑎𝑧𝑓(𝐿𝑧)=𝑧𝛿𝑎𝑓(𝑧).(1.7)

We note that:(i)for 𝑎=1 and 𝛿=𝑘 (𝑘 is any integer), the multiplier transformation 𝐿𝑘1𝑓(𝑧)=𝐼𝑘𝑓(𝑧) was studied by Flett [2] and Sălageăn [3];(ii)for 𝑎=1 and 𝛿=𝑘 (𝑘0), the differential operator 𝐿1𝑘𝑓(𝑧)=𝐷𝑘𝑓(𝑧) was studied by Sălageăn [3];(iii)for 𝑎=2 and 𝛿=𝑘 (𝑘 is any integer), the operator 𝐿𝑘2𝑓(𝑧)=𝐿𝑘𝑓(𝑧) was studied by Uralegaddi and Somanatha [4];(iv)for 𝑎=2, the multiplier transformation 𝐿𝛿2𝑓(𝑧)=𝐼𝛿𝑓(𝑧) was studied by Jung et al. [5].

Using the operator 𝐿𝛿𝑎, we now introduce the following classes.

Definition 1.1. One says that a function 𝑓𝒜 is in the class 𝒮𝑎,𝛿(𝑏,𝛽) if 1Re1+𝑏𝑧𝐿𝛿𝑎𝑓(𝑧)𝐿𝛿𝑎𝑓(𝑧)1>𝛽,(1.8) where 𝑧𝕌;𝑎>0;𝛿0;0𝛽<1;𝑏{0}.

Definition 1.2. One says that a function 𝑓𝒜 is in the class 𝒞𝑎,𝛿(𝑏,𝛽) if 1Re1+𝑏𝑧𝐿𝛿𝑎𝑓(𝑧)𝐿𝛿𝑎𝑓(𝑧)>𝛽,(1.9) where 𝑧𝕌;𝑎>0;𝛿0;0𝛽<1;𝑏{0}.
Note that 𝑓𝒞𝑎,𝛿(𝑏,𝛽)𝑧𝑓𝒮𝑎,𝛿(𝑏,𝛽).(1.10) In particular, the classes 𝒮𝑎,𝛿(𝑏,0)𝒮𝑎,𝛿(𝑏),𝒞𝑎,𝛿(𝑏,0)𝒞𝑎,𝛿(𝑏)(1.11) introduced by Bulut [6].
Making use of the Komatu integral operator 𝐿𝛿𝑎, we now introduce each of the following subclasses of analytic functions.

Definition 1.3. One denotes by 𝒮𝑎,𝛿(𝜆,𝑏,𝐴,𝐵) the class of functions 𝑓𝒜 satisfying 11+𝑏𝑧𝐿𝜆𝑧𝛿𝑎𝑓(𝑧)+(1𝜆)𝐿𝛿𝑎𝑓(𝑧)𝐿𝜆𝑧𝛿𝑎𝑓(𝑧)+(1𝜆)𝐿𝛿𝑎𝑓(𝑧)11+𝐴𝑧1+𝐵𝑧,(1.12) where 𝑧𝕌;𝑎>0;𝛿0;1𝐵<𝐴1;0𝜆1;𝑏{0}.

Definition 1.4. A function 𝑓𝒜 is said to be in the class 𝑎,𝛿(𝜆,𝑏,𝐴,𝐵,𝑚;𝑢) if it satisfies the following non-homogenous Cauchy-Euler differential equation: 𝑧𝑚𝑑𝑚𝑤𝑑𝑧𝑚+𝑚1(𝑢+𝑚1)𝑧𝑚1𝑑𝑚1𝑤𝑑𝑧𝑚1𝑚𝑚𝑤++𝑚1𝑗=0(𝑢+𝑗)=𝑔(𝑧)𝑚1𝑗=0(𝑢+𝑗+1)𝑤=𝑓(𝑧)𝒜;𝑔𝒮𝑎,𝛿(𝜆,𝑏,𝐴,𝐵);𝑚.;𝑢(1,)(1.13)

Remark 1.5. If we set 𝛿=0 in the classes 𝒮𝑎,𝛿(𝜆,𝑏,𝐴,𝐵) and 𝑎,𝛿(𝜆,𝑏,𝐴,𝐵,𝑚;𝑢), then we have the classes 𝒮(𝜆,𝑏,𝐴,𝐵),𝒦(𝜆,𝑏,𝐴,𝐵,𝑚;𝑢)(1.14) introduced by Srivastava et al. [7], respectively.
If we take 𝐴=12𝛽  (0𝛽<1) and 𝐵=1 in the class 𝒮𝑎,𝛿(𝜆,𝑏,𝐴,𝐵), then we have a new class consisting of functions 𝑓𝒜 which satisfy the condition 1Re1+𝑏𝑧𝐿𝜆𝑧𝛿𝑎𝑓(𝑧)+(1𝜆)𝐿𝛿𝑎𝑓(𝑧)𝐿𝜆𝑧𝛿𝑎𝑓(𝑧)+(1𝜆)𝐿𝛿𝑎𝑓(𝑧)1>𝛽,𝑧𝕌.(1.15) We denote this class by 𝒮𝑎,𝛿(𝜆,𝑏,𝛽). Also we denote by 𝑎,𝛿(𝜆,𝑏,𝛽,𝑚;𝑢) for corresponding class to 𝑎,𝛿(𝜆,𝑏,12𝛽,1,𝑚;𝑢).
Note that taking 𝜆=0 and 𝜆=1 for the class 𝒮𝑎,𝛿(𝜆,𝑏,𝛽), we have the classes 𝒮𝑎,𝛿(𝑏,𝛽) and 𝒞𝑎,𝛿(𝑏,𝛽), respectively. In particular, the classes 𝒮𝑎,0(𝜆,𝑏,𝛽)𝒮𝒞(𝑏,𝜆,𝛽),𝑎,0(𝜆,𝑏,𝛽,2;𝑢)(𝑏,𝜆,𝛽;𝑢)(1.16) are studied by Altıntaş et al. [8].
In this work, by using the principle of subordination, we obtain coefficient bounds for functions in the subclasses 𝒮𝑎,𝛿(𝜆,𝑏,𝐴,𝐵),𝑎,𝛿(𝜆,𝑏,𝐴,𝐵,𝑚;𝑢)(1.17) of analytic functions of complex order, which we have introduced here. Our results would unify and extend the corresponding results obtained earlier by Robertson [9], Nasr and Aouf [10], Altıntaş et al. [8] and Srivastava et al. [7].
In our investigation, we will make use of the principle of subordination between analytic functions, which is explained in Definition 1.6 below (see [11]).

Definition 1.6. For two functions 𝑓 and 𝑔, analytic in 𝕌, one says that the function 𝑓(𝑧) is subordinate to 𝑔(𝑧) in 𝕌, and write 𝑓(𝑧)𝑔(𝑧),𝑧𝕌,(1.18) if there exists a Schwarz function 𝑤(𝑧), analytic in 𝕌, with ||||𝑤(0)=0,𝑤(𝑧)<1,𝑧𝕌,(1.19) such that 𝑓(𝑧)=𝑔(𝑤(𝑧)),𝑧𝕌.(1.20) In particular, if the function 𝑔 is univalent in 𝕌, the above subordination is equivalent to 𝑓(0)=𝑔(0),𝑓(𝕌)𝑔(𝕌).(1.21)
In order to prove our main results (Theorems 2.1 and 2.2 in Section 2), we first recall the following lemma due to Rogosinski [12].

Lemma 1.7. Let the function 𝑔 given by 𝑔(𝑧)=𝑘=1𝑏𝑘𝑧𝑘,𝑧𝕌,(1.22) be convex in 𝕌. Also let the function 𝑓 given by 𝑓(𝑧)=𝑘=1𝑎𝑘𝑧𝑘,𝑧𝕌,(1.23) be holomorphic in 𝕌. If 𝑓(𝑧)𝑔(𝑧),𝑧𝕌,(1.24) then ||𝑎𝑘||||𝑏1||,𝑘.(1.25)

2. The Main Results and Their Demonstration

We now state and prove each of our main results given by Theorems 2.1 and 2.2 below.

Theorem 2.1. Let the function 𝑓𝒜 be defined by (1.3). If the function 𝑓 is in the class 𝒮𝑎,𝛿(𝜆,𝑏,𝐴,𝐵), then ||𝑎𝑛||𝑎+𝑛1𝑎𝛿𝑛2𝑗=0||𝑏||𝑗+(𝐴𝐵)(𝑛1)!(1+𝜆(𝑛1)),𝑛.(2.1)

Proof. Let the function 𝑓𝒜 be given by (1.3). Define a function 𝐿(𝑧)=𝜆𝑧𝛿𝑎𝑓(𝑧)+(1𝜆)𝐿𝛿𝑎𝑓(𝑧),𝑧𝕌.(2.2) We note that the function is of the form (𝑧)=𝑧+𝑛=2𝐴𝑛𝑧𝑛,𝑧𝕌,(2.3) where, for convenience, 𝐴𝑛𝑎=(1+𝜆(𝑛1))𝑎+𝑛1𝛿𝑎𝑛,𝑛.(2.4) From Definition 1.3 and (2.2), we obtain that 11+𝑏𝑧(𝑧)(𝑧)11+𝐴𝑧1+𝐵𝑧,𝑧𝕌.(2.5) Let us set 𝑔(𝑧)=1+𝐴𝑧1+𝐵𝑧(2.6) and define the function 𝑝(𝑧) by 1𝑝(𝑧)=1+𝑏𝑧(𝑧)(𝑧)1,𝑧𝕌.(2.7) Therefore, we have 𝑝(𝑧)𝑔(𝑧),𝑧𝕌.(2.8) Hence, by Definition 1.6, we deduce that 𝑝(𝑧)=1+𝐴𝑤(𝑧)||||.1+𝐵𝑤(𝑧)𝑤(0)=0;𝑤(𝑧)<1(2.9) Note that 𝑝(0)=𝑔(0)=1,𝑝(𝑧)𝑔(𝕌),𝑧𝕌.(2.10) Also from (2.7), we find 𝑧[](𝑧)=1+𝑏(𝑝(𝑧)1)(𝑧).(2.11) Let 𝑝(𝑧)=1+𝑐1𝑧+𝑐2𝑧2+,𝑧𝕌.(2.12) Since 𝐴1=1, in view of (2.3), (2.11) and (2.12), we obtain (𝑛1)𝐴𝑛𝑐=𝑏𝑛1+𝑐𝑛2𝐴2++𝑐1𝐴𝑛1(2.13) for 𝑛. On the other hand, according to the Lemma 1.7, we obtain ||||𝑝(𝑚)(0)||||𝑚!𝐴𝐵,𝑚.(2.14) By combining (2.14) and (2.13), for 𝑛=2,3,4, we obtain ||𝐴2||||𝑏||||𝐴(𝐴𝐵),3||||𝑏||||𝑏||(𝐴𝐵)1+(𝐴𝐵),||𝐴2!4||||𝑏||||𝑏||||𝑏||(𝐴𝐵)1+(𝐴𝐵)2+(𝐴𝐵),3!(2.15) respectively. Using the principle of mathematical induction, we obtain ||𝐴𝑛||𝑛2𝑗=0||𝑏||𝑗+(𝐴𝐵)(𝑛1)!,𝑛.(2.16) Now from (2.4), it is clear that ||𝑎𝑛||𝑎+𝑛1𝑎𝛿𝑛2𝑗=0||𝑏||𝑗+(𝐴𝐵)(𝑛1)!(1+𝜆(𝑛1)),𝑛.(2.17) This evidently completes the proof of Theorem 2.1.

Theorem 2.2. Let the function 𝑓𝒜 be defined by (1.3). If the function 𝑓 is in the class 𝑎,𝛿(𝜆,𝑏,𝐴,𝐵,𝑚;𝑢), then ||𝑎𝑛||𝑎+𝑛1𝑎𝛿𝑛2𝑗=0||𝑏||𝑗+(𝐴𝐵)(𝑛1)!(1+𝜆(𝑛1))𝑚1𝑗=0(𝑢+𝑗+1)𝑚1𝑗=0(𝑢+𝑗+𝑛),𝑛.(2.18)

Proof. Let the function 𝑓𝒜 be given by (1.3). Also let 𝑞(𝑧)=𝑧+𝑛=2𝐵𝑛𝑧𝑛𝒮𝑎,𝛿(𝜆,𝑏,𝐴,𝐵),(2.19) so that 𝑎𝑛=𝑚1𝑗=0(𝑢+𝑗+1)𝑚1𝑗=0𝐵(𝑢+𝑗+𝑛)𝑛,𝑛,𝑢(1,).(2.20) Thus, by using Theorem 2.1, we obtain ||𝑎𝑛||𝑎+𝑛1𝑎𝛿𝑛2𝑗=0||𝑏||𝑗+(𝐴𝐵)(𝑛1)!(1+𝜆(𝑛1))𝑚1𝑗=0(𝑢+𝑗+1)𝑚1𝑗=0(𝑢+𝑗+𝑛).(2.21) This completes the proof of Theorem 2.2.

3. Corollaries and Consequences

In this section, we apply our main results (Theorems 2.1 and 2.2) in order to deduce each of the following corollaries and consequences.

It is easy to see that ||𝑏||2||𝑏||𝑗+(𝐴𝐵)𝑗+(𝐴𝐵)1𝐵,𝑗,1𝐵<𝐴1,𝑏{0},(3.1) which would obviously yield significant improvements over Theorems 2.1 and 2.2 (see Srivastava et al. [7]).

Setting 𝐴=12𝛽  (0𝛽<1) and 𝐵=1 in Theorems 2.1 and 2.2, we have

Corollary 3.1. Let the function 𝑓𝒜 be defined by (1.3). If the function 𝑓 is in the class 𝒮𝑎,𝛿(𝜆,𝑏,𝛽), then ||𝑎𝑛||𝑎+𝑛1𝑎𝛿𝑛2𝑗=0||𝑏||𝑗+2(1𝛽)(𝑛1)!(1+𝜆(𝑛1)),𝑛.(3.2)

Remark 3.2. Taking 𝛿=0 in Corollary 3.1, we have [8, Theorem 1].

Corollary 3.3. Let the function 𝑓𝒜 be defined by (1.3). If the function 𝑓 is in the class 𝑎,𝛿(𝜆,𝑏,𝛽,𝑚;𝑢), then ||𝑎𝑛||𝑎+𝑛1𝑎𝛿𝑛2𝑗=0||𝑏||𝑗+2(1𝛽)(𝑛1)!(1+𝜆(𝑛1))𝑚1𝑗=0(𝑢+𝑗+1)𝑚1𝑗=0(𝑢+𝑗+𝑛),𝑛.(3.3)

Remark 3.4. Taking 𝛿=0 and 𝑚=2 in Corollary 3.3, we have [8, Theorem 2].

Letting 𝜆=0 and 𝜆=1 in Corollary 3.1, we get following corollaries, respectively.

Corollary 3.5. Let the function 𝑓𝒜 be defined by (1.3). If the function 𝑓 is in the class 𝒮𝑎,𝛿(𝑏,𝛽), then ||𝑎𝑛||𝑎+𝑛1𝑎𝛿𝑛2𝑗=0||𝑏||𝑗+2(1𝛽)(𝑛1)!,𝑛.(3.4)

Corollary 3.6. Let the function 𝑓𝒜 be defined by (1.3). If the function 𝑓 is in the class 𝒞𝑎,𝛿(𝑏,𝛽), then ||𝑎𝑛||𝑎+𝑛1𝑎𝛿𝑛2𝑗=0||𝑏||𝑗+2(1𝛽)𝑛!,𝑛.(3.5)

For other related results, see also [9, 10].


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