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

Serap Bulut School of Civil Aviation College, Kocaeli University, Arslanbey Campus, 41285 Izmit-Kocaeli, Turkey Correspondence should be addressed to Serap Bulut, serap.bulut@kocaeli.edu.tr Received 23 March 2012; Accepted 11 July 2012 Academic Editor: Yuri Latushkin Copyright q 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.


Introduction, Definitions and Preliminaries
Recently, Komatu 1 introduced a certain integral operator L δ a defined by Thus, if f ∈ A is of the form 1.3 , then it is easily seen from 1.5 that see 1 1.6 Using the relation 1.6 , it is easily verfied that

1.7
We note that: i for a 1 and δ k k is any integer , the multiplier transformation L k 1 f z I k f z was studied by Flett 2 and Sȃlageȃn 3 ; ii for a 1 and δ −k k ∈ N 0 , the differential operator L −k 1 f z D k f z was studied by Sȃlageȃn 3 ; iii for a 2 and δ k k is any integer , the operator L k 2 f z L k f z was studied by Uralegaddi and Somanatha 4 ; iv for a 2, the multiplier transformation L δ 2 f z I δ f z was studied by Jung et al.

.
Using the operator L δ a , we now introduce the following classes.

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Note that taking λ 0 and λ 1 for the class S a,δ λ, b, β , we have the classes S a,δ b, β and C a,δ b, β , respectively.In particular, the classes 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 if there exists a Schwarz function w z , analytic in U, with In particular, if the function g is univalent in U, 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 g given by
International Journal of Mathematics and Mathematical Sciences 5 be convex in U. Also let the function f given by 1.25

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.
Proof.Let the function f ∈ A be given by 1.3 .Define a function We note that the function h is of the form where, for convenience, From Definition 1.3 and 2.2 , we obtain that

International Journal of Mathematics and Mathematical Sciences
Let us set and define the function p z by Therefore, we have Hence, by Definition 1.6, we deduce that Note that p 0 g 0 1, p z ∈ g U , z ∈ U.

2.10
Also from 2.7 , we find

2.12
Since A 1 1, in view of 2.3 , 2.11 and 2.12 , we obtain for n ∈ N * .On the other hand, according to the Lemma 1.7, we obtain

2.14
By combining 2.14 and 2.13 , for n 2, 3, 4, we obtain respectively.Using the principle of mathematical induction, we obtain Now from 2.4 , it is clear that

2.17
This evidently completes the proof of Theorem 2.1.

2.18
Proof.Let the function f ∈ A be given by 1.3 .Also let Thus, by using Theorem 2.1, we obtain

2.21
This completes the proof of Theorem 2.2.
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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  For other related results, see also 9, 10 .

2 Let A denote the class of functions of the form f z z ∞ n 2 a n z n 1. 3 International
Let R−∞, ∞ be the set of real numbers, let C be the set of complex numbers,N : {1, 2, 3, . ..}N 0 \ {0}1.1 be the set of positive integers and N * : N \ {1} {2, 3, 4, . ..}. 1.Journal of Mathematics and Mathematical Sciences which are analytic in the unit disk: U {z ∈ C : |z| < 1}.1.4

International Journal of Mathematics and Mathematical Sciences 3 11 introduced by Bulut 6 .
Note thatf ∈ C a,δ b, β ⇔ zf ∈ S a,δ b, β .1.10Inparticular, the classes S a,δ b, 0 ≡ S a,δ b , C a,δ b, 0 ≡ C a,δ b 1.Making use of the Komatu integral operator L δ a , we now introduce each of the following subclasses of analytic functions.Definition 1.3.One denotes by S a,δ λ, b, A, B the class of functions f ∈ A satisfying 1 1 b

1 . 13 Remark 1 . 5 .
If we set δ 0 in the classes S a,δ λ, b, A, B and B 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 1 − 2β 0 ≤ β < 1 and B −1 in the class S a,δ λ, b, A, B , then we have a new class consisting of functions f ∈ A which satisfy the condition Re

Theorem 2 . 1 .
Let the function f ∈ A be defined by 1.3 .If the function f is in the class S a,δ λ, b, A, B , then

Theorem 2 . 2 .
Let the function f ∈ A be defined by 1.3 .If the function f is in the class B a,δ λ, b, A, B, m; u , then

3 . 2 .Corollary 3 . 3 .Remark 3 . 4 .Corollary 3 . 5 .− 1 a δ n− 2 j 0 j 2|b| 1 − β n − 1 ! 4 Corollary 3 . 6 .
Taking δ 0 in Corollary 3.1, we have 8, Theorem 1 .Let the function f ∈ A be defined by 1.3 .If the function f is in the class B a,δ λ, b, β, m; u , then |a n 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.Let the function f ∈ A be defined by 1.3 .If the function f is in the class S a,δ b, β , then |a n | ≤ a n , n ∈ N * .3.Let the function f ∈ A be defined by 1.3 .If the function f is in the class C a,δ b, β , then |a n of complex order, which we have introduced here.Our results would unify and extend the corresponding results obtained earlier byRobertson9 , Nasr and Aouf 10 , Altıntas ¸et al. 8 and Srivastava et al. 7 .
Let the function f ∈ A be defined by 1.3 .If the function f is in the class S a,δ λ, b, β , then