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International Journal of Mathematics and Mathematical Sciences

Volume 2012 (2012), Article ID 309465, 9 pages

http://dx.doi.org/10.1155/2012/309465

## 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.

#### Abstract

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, be the set of positive integers and

Let denote the class of functions of the form which are analytic in the unit disk:

Recently, Komatu [1] introduced a certain integral operator defined by

Thus, if is of the form (1.3), then it is easily seen from (1.5) that (see [1])

Using the relation (1.6), it is easily verfied that

We note that:(i)for and ( is any integer), the multiplier transformation was studied by Flett [2] and Sălageăn [3];(ii)for and (), the differential operator was studied by Sălageăn [3];(iii)for and ( is any integer), the operator was studied by Uralegaddi and Somanatha [4];(iv)for , the multiplier transformation was studied by Jung et al. [5].

Using the operator , we now introduce the following classes.

*Definition 1.1. *One says that a function is in the class if
where .

*Definition 1.2. *One says that a function is in the class if
where .

Note that
In particular, the classes
introduced by Bulut [6].

Making use of the Komatu integral operator , we now introduce each of the following subclasses of analytic functions.

*Definition 1.3. *One denotes by the class of functions satisfying
where .

*Definition 1.4. *A function is said to be in the class if it satisfies the following non-homogenous Cauchy-Euler differential equation:

*Remark 1.5. *If we set in the classes and , then we have the classes
introduced by Srivastava et al. [7], respectively.

If we take and in the class , then we have a new class consisting of functions which satisfy the condition
We denote this class by . Also we denote by for corresponding class to .

Note that taking and for the class , we have the classes and , respectively. In particular, the classes
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
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 and , analytic in , one says that the function is subordinate to in , and write
if there exists a Schwarz function , analytic in , with
such that
In particular, if the function is univalent in , the above subordination is equivalent to

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 given by
**
be convex in . Also let the function given by
**
be holomorphic in . If
**
then
*

#### 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 be defined by (1.3). If the function is in the class , then
*

*Proof. *Let the function be given by (1.3). Define a function
We note that the function is of the form
where, for convenience,
From Definition 1.3 and (2.2), we obtain that
Let us set
and define the function by
Therefore, we have
Hence, by Definition 1.6, we deduce that
Note that
Also from (2.7), we find
Let
Since , in view of (2.3), (2.11) and (2.12), we obtain
for . On the other hand, according to the Lemma 1.7, we obtain
By combining (2.14) and (2.13), for , we obtain
respectively. Using the principle of mathematical induction, we obtain
Now from (2.4), it is clear that
This evidently completes the proof of Theorem 2.1.

Theorem 2.2. *Let the function be defined by (1.3). If the function is in the class , then
*

*Proof. *Let the function be given by (1.3). Also let
so that
Thus, by using Theorem 2.1, we obtain
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 which would obviously yield significant improvements over Theorems 2.1 and 2.2 (see Srivastava et al. [7]).

Setting and in Theorems 2.1 and 2.2, we have

Corollary 3.1. *Let the function be defined by (1.3). If the function is in the class , then
*

*Remark 3.2. *Taking in Corollary 3.1, we have [8, Theorem 1].

Corollary 3.3. *Let the function be defined by (1.3). If the function is in the class , then
*

*Remark 3.4. *Taking and in Corollary 3.3, we have [8, Theorem 2].

Letting and in Corollary 3.1, we get following corollaries, respectively.

Corollary 3.5. *Let the function be defined by (1.3). If the function is in the class , then
*

Corollary 3.6. *Let the function be defined by (1.3). If the function is in the class , then
*

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