subclasses of analytic functions associated with generalised...

International Scholarly Research NetworkISRN Mathematical AnalysisVolume 2012, Article ID 632429, 9 pagesdoi:10.5402/2012/632429

Research ArticleSubclasses of Analytic Functions Associated withGeneralised Multiplier Transformations

Rashidah Omar1, 2 and Suzeini Abdul Halim2

1 Faculty of Computer and Mathematical Sciences, MARA University of Technology,40450 Shah Alam, Selangor, Malaysia

2 Institute of Mathematical Sciences, Faculty of Science, University of Malaya,50603 Kuala Lumpur, Malaysia

Correspondence should be addressed to Rashidah Omar, [email protected]

Received 20 January 2012; Accepted 25 March 2012

Academic Editors: O. Miyagaki and W. Yu

Copyright q 2012 R. Omar and S. Abdul Halim. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

New subclasses of analytic functions in the open unit disc are introduced which are defined usinggeneralised multiplier transformations. Inclusion theorems are investigated for functions to be inthe classes. Furthermore, generalised Bernardi-Libera-Livington integral operator is shown to bepreserved for these classes.

1. Introduction

Let A denote the class of functions f normalised by f�z� � z �∑∞

n�2 anzn in the open unit

disk D :� {z ∈ C : |z| < 1}. Also let S�, C, and K denote, respectively, the subclasses ofA consisting of functions which are starlike, convex, and close to convex in D. An analyticfunction f is subordinate to an analytic function g, written f�z� ≺ g�z� �z ∈ D� if there existsan analytic function w in D such that w�0� � 0 and |w�z�| < 1 for |z| < 1 and f�z� � g�w�z��.In particular, if g is univalent in D, then f�z� ≺ g�z� is equivalent to f�0� � g�0� and f�D� ⊂g�D�. The convolution of two analytic functions ϕ�z� �

∑∞n�2 anz

n and ψ�z� �∑∞

n�0 bnzn is

defined by ϕ�z� ∗ ψ�z� � ∑∞n�0 anbnzn � ψ�z� ∗ ϕ�z�.For any real numbers k and λwhere k ≥ 0, λ ≥ 0, c ≥ 0, Cǎtaş �1� defined the multiplier

transformations I�k, λ, c�f�z� by the following series:

I�k, λ, c�f�z� � z �∞∑

n�2

[1 � λ�n − 1� � c

1 � c

]kanz

n. �1.1�

2 ISRN Mathematical Analysis

Recently, some properties of functions using the multiplier transformations have beenstudied in �2–6�. Using the convolution, we extend the multiplier transformation in �1.1�to be a unified operator. The approach used is similar to Noor’s �7�, only we generalise andextend to include powers and uses the multiplier Cǎtaş as basis instead of the Ruscheweyhoperator.

Set the function

fk,c�z� � z �∞∑

n�2

[1 � c

1 � λ�n − 1� � c]kzn �k, λ ∈ R, k ≥ 0, λ ≥ 0, c ≥ 0�, �1.2�

and note that, for λ � 1, fk,c�z� is the generalised polylogarithm functions discussed in �8�. Anew function fμ

k,c�z� is defined in terms of the Hadamard product �or convolution� as follows:

fk,c�z� ∗ fμk,c�z� �z

�1 − z�μ(μ > 0

). �1.3�

Motivated by �9–11� and analogous to �1.1�, the following operator is introduced:

Ikc(λ, μ

)f�z� � fμk,c ∗ f�z�

� z �∞∑

n�2

(μ)n−1

�n − 1�![1 � λ�n − 1� � c

1 � c

]kanz

n.�1.4�

The operator Ikc �λ, μ�f unifies other previously defined operators. For examples,

�i� Ikc �λ, 1�f is the I1�δ, λ, l�f given in �1�,

�ii� Ikc �1, 1�f is the Ikc f given in �12�,

also, for any integer k,

�iii� Ik0 �λ, 1�f�z� ≡ Dkλf�z� given in �13�,�iv� Ik0 �1, 1�f�z� ≡ Dkf�z� given in �14�,�v� Ik1 �1, 1�f�z� ≡ Ikf�z� given in �15�.

The following relations are easily derived using the following definition:

�1 � c�Ik�1c(λ, μ

)f�z� � �1 − λ � c�Ikc

(λ, μ

)f�z� � λz

[Ikc(λ, μ

)f�z�

]′, �1.5�

μIkc(λ, μ � 1

)f�z� � z

[Ikc(λ, μ

)f�z�

]′�(μ − 1)Ikc

(λ, μ

)f�z�. �1.6�

LetN be the class of all analytic and univalent functions φ inD and for which φ�D� is convexwith φ�0� � 1 and Re{φ�z�} > 0 for z ∈ D. For φ, ψ ∈ N, Ma and Minda �16� studied

ISRN Mathematical Analysis 3

the subclasses S��φ�, C�φ�, and K�φ, ψ� of the class A. These classes are defined using theprinciple of subordination as follows:

S�(φ):�

{

f : f ∈ A, zf′�z�

f�z�≺ φ�z� in D

}

,

C(φ):�

{

f : f ∈ A, 1 � zf′′�z�

f ′�z�≺ φ�z� in D

}

,

K(φ, ψ

):�

{

f : f ∈ A, ∃g ∈ S�(φ) such that zf′�z�

g�z�≺ ψ�z� in D

}

.

�1.7�

Obviously, we have the following relationships for special choices φ and ψ:

S�(1 � z1 − z

)

� S�, C(1 � z1 − z

)

� C, K(1 � z1 − z ,

1 � z1 − z

)

� K. �1.8�

Using the generalisedmultiplier transformations Ikc �λ, μ�f , new classes Skc �λ, μ;φ�,C

kc �λ, μ;φ�

and Kkc �λ, μ;φ, ψ� are introduced and defined below

Skc(λ, μ;φ

):�

{f ∈ A : Ikc

(λ, μ

)f�z� ∈ S�(φ)

},

Ckc(λ, μ;φ

):�

{f ∈ A : Ikc

(λ, μ

)f�z� ∈ C(φ)

},

Kkc(λ, μ;φ, ψ

):�

{f ∈ A : Ikc

(λ, μ

)f�z� ∈ K(φ, ψ)

}.

�1.9�

It can be shown easily that

f�z� ∈ Ckc(λ, μ;φ

) ⇐⇒ zf ′�z� ∈ Skc(λ, μ;φ

). �1.10�

Janowski �17� introduced class S��A,B� � S��1 �Az�/�1 � Bz�� and in particular for φ�z� ��1 �Az�/�1 � Bz�, we set

Skc

(

λ, μ;1 �Az1 � Bz

)

� S�k,c[μ;A,B

]�−1 ≤ B < A ≤ 1�. �1.11�

In �18�, the authors studied the inclusion properties for classes defined using Dziok-Srivastava operator. This paper investigates the similar properties for analytic functions inthe classes defined by the generalised multiplier transformations Ikc �λ, μ�f . Furthermore,applications of other families of integral operators are considered involving these classes.

2. Inclusion Properties Involving IKc �λ, μ�f

In proving our results, the following lemmas are needed.


Lemma 2.1 �see �19��. Let φ be convex univalent inD, with φ�0� � 1 andRe�κφ�z��η� > 0 �κ, η ∈C�. If p is analytic inD with p�0� � 1, then

p�z� �zp′�z�

κp�z� � η≺ φ�z� �⇒ p�z� ≺ φ�z�. �2.1�

Lemma 2.2 �see �20��. Let φ be convex univalent in D and ω be analytic in D with Re{ω�z�} ≥ 0.If p is analytic inD and p�0� � φ�0�, then

p�z� �ω�z�zp′�z� ≺ φ�z� �⇒ p�z� ≺ φ�z�. �2.2�

Theorem 2.3. For any real numbers k and λ where k ≥ 0, λ ≥ 0 and c ≥ 0.Let φ ∈N and Re{φ�z� � �1 − λ � c�/λ} > 0, then Sk�1c �λ, μ;φ� ⊂ Skc �λ, μ;φ� �μ > 0�.

Proof. Let f ∈ Sk�1c �λ, μ;φ�, and set p�z� � �z�Ikc �λ, μ�f�z��′�/�Ikc �λ, μ�f�z�� where p is

analytic inD with p�0� � 1. Rearranging �1.5�, we have

�1 � c�Ik�1c(λ, μ

)f�z�

Ikc(λ, μ

)f�z�

� �1 − λ � c� � λz[Ikc(λ, μ

)f�z�

]′

Ikc(λ, μ

)f�z�

. �2.3�

Next, differentiating �2.3� and multiplying by z gives

z[Ik�1c

(λ, μ

)f�z�

]′

Ik�1c(λ, μ

)f�z�

�z[Ikc(λ, μ

)f�z�

]′

Ikc(λ, μ

)f�z�

�z((z[Ikc(λ, μ

)f�z�

]′)/(Ikc(λ, μ

)f�z�

))′

(

z[Ikc(λ, μ

)f�z�

]′)/(Ikc(λ, μ

)f�z�

)� �1 − λ � c�/λ

� p�z� �zp′�z�

p�z� � �1 − λ � c�/λ.�2.4�

Since �z�Ik�1c �λ, μ�f�z��′�/�Ik�1c �λ, μ�f�z�� ≺ φ�z� and applying Lemma 2.1, it follows that

p ≺ φ. Thus f ∈ Skc �λ, μ;φ�.

Theorem 2.4. Let k, λ ∈ R, k ≥ 0, λ ≥ 0, and μ ≥ 1. Then Skc �λ, μ � 1;φ� ⊂ Skc �λ, μ;φ� �c ≥ 0;φ ∈N�.

Proof. Let f ∈ Skc �λ, μ � 1;φ�, and from �1.6�, we obtain that

μIkc(λ, μ � 1

)f�z�

Ikc(λ, μ

) �z[Ikc(λ, μ

)f�z�

]′

Ikc(λ, μ

) �(μ − 1). �2.5�


Making use of the differentiation on both sides in �2.5� and setting p�z� ��z�Ikc �λ, μ�f�z��

′�/�Ikc �λ, μ�f�z��, we get the following:

z[Ikc(λ, μ � 1

)f�z�

]′

Ikc(λ, μ � 1

)f�z�

� p�z� �zp′�z�

p�z� �(μ − 1) ≺ φ�z�. �2.6�

Since μ ≥ 1 and Re{φ�z��μ−1�} > 0, using Lemma 2.1, we conclude that f ∈ Skc �λ, μ;φ�.

Corollary 2.5. Let λ ≥ 0, μ ≥ 1, and −1 ≤ B < A ≤ 1. Then S�k�1,c�μ;A,B� ⊂ S�k,c�μ;A,B� andS�k,c�μ � 1;A,B� ⊂ S�

k,c�μ;A,B�.

Theorem 2.6. Let λ ≥ 0 and μ ≥ 1. Then Ck�1c �λ, μ;φ� ⊂ Ckc �λ, μ;φ� and Ckc �λ, μ � 1;φ� ⊂Ckc �λ, μ;φ�.

Proof. Using �1.10� and Theorem 2.3, we observe that

f�z� ∈ Ck�1c(λ, μ;φ

) ⇐⇒ zf ′�z� ∈ Sk�1c(λ, μ;φ

)

�⇒ zf ′�z� ∈ Skc(λ, μ;φ

)

⇐⇒ Ikc(λ, μ

)zf ′�z� ∈ S�(φ)

⇐⇒ z[Ikc(λ, μ

)f�z�

]′ ∈ S�(φ)

⇐⇒ Ikc(λ, μ

)f�z� ∈ C(φ)

⇐⇒ f ∈ Ckc(λ, μ;φ

).

�2.7�

To prove the second part of theorem, using the similar manner and applying Theorem 2.4,the result is obtained.

Theorem 2.7. Let λ ≥ 0, c ≥ 0 and Re{�1 − λ � c�/λ} > 0.Then Kk�1c �λ, μ;φ, ψ� ⊂ Kkc �λ, μ;φ, ψ� and Kkc �λ, μ � 1;φ, ψ� ⊂ Kkc �λ, μ;φ, ψ��φ, ψ ∈N�.

Proof. Let f ∈ Kk�1c �λ, μ;φ, ψ�. In view of the definition of the class Kk�1c �λ, μ;φ, ψ�, there is afunction g ∈ Sk�1c �λ, μ;φ� such that

z[Ik�1c

(λ, μ

)f�z�

]′

Ik�1c(λ, μ

)g�z�

≺ ψ�z�. �2.8�

Applying Theorem 2.3, then g ∈ Skc �λ, μ;φ� and let q�z� � �z�Ikc �λ, μ�g�z��′�/�Ikc �λ, μ�g�z�� ≺

φ�z�.Let the analytic function p with p�0� � 1 as

p�z� �z[Ikc(λ, μ

)f�z�

]′

Ikc(λ, μ

)g�z�

. �2.9�


Thus, rearranging and differentiating �2.9�, we have

[Ikc(λ, μ

)zf ′�z�

]′

Ikc(λ, μ

)g�z�

�p�z�

[Ikc(λ, μ

)g�z�

]′

Ikc(λ, μ

)g�z�

� p′�z�. �2.10�

Making use �1.5�, �2.9�, �2.10�, and q�z�, we obtain that

z[Ik�1c

(λ, μ

)f�z�

]′

Ik�1c(λ, μ

)g�z�

�

[Ik�1c

(λ, μ

)zf ′�z�

]

Ik�1c(λ, μ

)g�z�

��1 − λ � c�Ikc

(λ, μ

)zf ′�z� � λz

[Ikc(λ, μ

)zf ′�z�

]′

�1 − λ � c�Ikc(λ, μ

)g�z� � λz

[Ikc(λ, μ

)g�z�

]′

�

(�1 − λ � c�Ikc

(λ, μ

)zf ′�z�

)/(Ikc(λ, μ

)g�z�

)�(λz

[Ikc(λ, μ

)zf ′�z�

]′)/(Ikc(λ, μ

)g�z�

)

�1 − λ � c� �(

λz[Ikc(λ, μ

)g�z�

]′)/(Ikc(λ, μ

)g�z�

)

��1 − λ � c�p�z� � λ[p�z�q�z� � p′�z�]

�1 − λ � c� � λq�z�

� p�z� �zp′�z�

q�z� � �1 − λ � c�/λ ≺ ψ�z�.�2.11�

Since q�z� ≺ φ�z� and Re{�1−λ�c�/λ} > 0, then Re{q�z��1−λ�c�/λ} > 0. Using Lemma 2.2,we conclude that p�z� ≺ ψ�z� and thus f ∈ Kkc �λ, μ;φ, ψ�. By using similar manner and �1.6�,we obtain the second result.

In summary, using subordination technique inclusion properties has been establishedfor certain analytic functions defined via the generalised multiplier transformation.

3. Inclusion Properties Involving Fcf

In this section, we determine properties of generalised Bernardi-Libera-Livington integraloperator defined by �21–24�

Fc[f�z�

]�c � 1zc

∫z

0tc−1f�t�dt �c > −1,Re c ≥ 0�

� z �∞∑

n�2

c � 1n � c

anzn

�3.1�


and satisfies the following:

cIkc(λ, μ

)Fc[f�z�

]� z

[Ikc(λ, μ

)Fc[f�z�

]]′� �c � 1�Ikc

(λ, μ

)f�z�. �3.2�

Theorem 3.1. If f ∈ Skc �λ, μ;φ�, then Fcf ∈ Skc �λ, μ;φ�.

Proof. Let f ∈ Skc �λ, μ;φ�, then �z�Ikc �λ, μ�f�z��′�/�Ikc �λ, μ�f�z�� ≺ φ�z�. Taking the differenti-

ation on both sides of �3.2� and multiplying by z, we obtain

z[Ikc(λ, μ

)f�z�

]′

Ikc(λ, μ

)f�z�

�z[Ikc(λ, μ

)Fc[f�z�

]]′

Ikc(λ, μ

)Fc[f�z�

] �z((z[Ikc(λ, μ

)Fc[f�z�

]]′)/(Ikc(λ, μ

)Fc[f�z�

]))′

(

z[Ikc(λ, μ

)Fc[f�z�

]]′)/(Ikc(λ, μ

)Fc[f�z�

])� c

.

�3.3�

Setting p�z� � �z�Ikc �λ, μ�Fc�f�z��′�/�Ikc �λ, μ�Fc�f�z��, we have

z[Ikc(λ, μ

)f�z�

]′

Ikc(λ, μ

)f�z�

� p�z� �zp′�z�p�z� � c

. �3.4�

Lemma 2.1 implies �z�Ikc �λ, μ�Fc�f�z��′�/�Ikc �λ, μ�Fc�f�z�� ≺ φ�z�. Hence Fcf ∈ Skc �λ, μ;φ�.

Theorem 3.2. Let f ∈ Ckc �λ, μ;φ�, then Fcf ∈ Ckc �λ, μ;φ�.

Proof. By using �1.10� and Theorem 3.1, it follows that

f ∈ Ckc(λ, μ;φ

) ⇐⇒ zf ′�z� ∈ Skc(λ, μ;φ

)�⇒ Fc

[zf ′�z�

] ∈ Skc(λ, μ;φ

)

⇐⇒ z[Fc[f�z�

]]′ ∈ Skc(λ, μ;φ

) ⇐⇒ Fc[f�z�

] ∈ Ckc(λ, μ;φ

).

�3.5�

Theorem 3.3. Let φ, ψ ∈N, and f ∈ Kkc �λ, μ;φ, ψ�, then Fcf ∈ Kkc �λ, μ;φ, ψ�.

Proof. Let f ∈ Kkc �λ, μ;φ, ψ�, then there exists function g ∈ Skc �λ, μ;φ� such that�z�Ikc �λ, μ�f�z��

′�/�Ikc �λ, μ�g�z�� ≺ ψ�z�. Since g ∈ Skc �λ, μ;φ� therefore from Theorem 3.1,Fc�f�z�� ∈ Skc �λ, μ;φ�. Then let

q�z� �z[Ikc(λ, μ

)Fc[g�z�

]]′

Ikc(λ, μ

)Fc[g�z�

] ≺ φ�z�. �3.6�

Set

p�z� �z[Ikc(λ, μ

)Fc[f�z�

]]′

Ikc(λ, μ

)Fc[g�z�

] . �3.7�


By rearranging and differentiating �3.7�, we obtain that

[Ikc(λ, μ

)Fc[zf ′�z�

]]′

Ikc(λ, μ

)Fc[g�z�

] �p�z�

[Ikc(λ, μ

)Fc[g�z�

]]′

Ikc(λ, μ

)Fc[g�z�

] �

[Ikc(λ, μ

)Fc[g�z�

]]p′�z�

Ikc(λ, μ

)Fc[g�z�

] . �3.8�

Making use �3.2�, �3.7�, and �3.6�, it can be derived that

z[Ikc(λ, μ

)f�z�

]′

Ikc(λ, μ

)g�z�

� p�z� �zp′�z�c � q�z�

. �3.9�

Hence, applying Lemma 2.2, we conclude that p�z� ≺ ψ�z�, and it follows that Fc�f�z�� ∈Kkc �λ, μ;φ, ψ�.

For analytic functions in the classes defined by generalised multiplier transformations,the generalised Bernardi-Libera-Livington integral operator has been shown to be preservedin these classes.

4. Conclusion

Results involving functions defined using the generalised multiplier transformation, namely,inclusion properties and the Bernardi-Libera-Livington integral operator were obtained usingsubordination principles. In �18�, similar results were discussed for functions defined usingthe Dziok-Srivastava operator.

Acknowledgment

This research was supported by IPPP/UPGP/geran�RU/PPP�/PS207/2009A UniversityMalaya Grants 2009.

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