SUBCLASSES OF ANALYTIC FUNCTIONS DEFINED BY ...Subclasses of analytic functions defined by new...

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SUBCLASSES OF ANALYTIC FUNCTIONS DEFINED BY NEW GENERALISED DERIVATIVE OPERATOR (Subkelas Fungsi Analisis yang Ditakrif oleh Pengoperasi Terbitan Teritlak) ENTISAR EL-YAGUBI & MASLINA DARUS ABSTRACT A new generalised derivative operator 1 2 , , , nm D b λλ is introduced. This operator generalised some well-known operators studied earlier. New subclasses of analytic functions in the open unit disc which are defined using generalised derivative operator are introduced. Inclusion theorems are investigated. Furthermore, generalised Bernardi-Libera-Livington integral operator is shown to be preserved for these classes. Keywords: analytic functions; univalent functions; starlike functions; convex functions; close- to-convex functions; subordination; Hadamard product; integral operator ABSTRAK Pengoperasi terbitan baharu teritlak 1 2 , , , nm D b λλ diperkenalkan. Pengoperasi ini mengitlak beberapa pengoperasi terdahulu yang terkenal. Subkelas baharu fungsi analisis dalam cakera terbuka unit diperkenalkan yang ditakrif dengan menggunakan pengoperasi terbitan teritlak. Teorem rangkuman dikaji. Malah pengoperasi kamiran Bernardi-Libera-Livington ditunjukkan kekal untuk kelas tesebut. Kata kunci: fungsi analisis; fungsi univalen; fungsi bak bintang; fungsi cembung; fungsi hampir cembung; subordinasi; hasil darab Hadamard; pengoperasi kamiran 1. Introduction Let A denote the class of functions of the form 2 ( ) , k k k f z az = = (1.1) where a k is a complex number, which are analytic in the open unit disc U = {z :| z | < 1}. Also let S * ,C and K denote, respectively, the subclasses of A consisting of functions which are starlike, convex, and close to convex in U. An analytic function is subordinate to an analytic function , written f ( z ) g ( z ), (z U ) if there exists an analytic function w in U, such that w (0) = 0, |w (z) |<1 for |z|<1 and f(z) = g (w (z)). In particular, if is univalent in U, then f ( z ) g ( z ) is equivalent to f (0) = g (0) and f (U ) g (U). The convolution of two analytic functions () = 2 k k k z az = + and () = 2 k k k z bz = + is defined by Journal of Quality Measurement and Analysis Jurnal Pengukuran Kualiti dan Analisis JQMA 9(1) 2013, 47-56

Transcript of SUBCLASSES OF ANALYTIC FUNCTIONS DEFINED BY ...Subclasses of analytic functions defined by new...

Page 1: SUBCLASSES OF ANALYTIC FUNCTIONS DEFINED BY ...Subclasses of analytic functions defined by new generalised derivative operator 48 49 Srivastave derivative operator in the case 0,1,0,

SUBCLASSES OF ANALYTIC FUNCTIONS DEFINED BY NEW GENERALISED DERIVATIVE OPERATOR

(Subkelas Fungsi Analisis yang Ditakrif oleh Pengoperasi Terbitan Teritlak)

ENTISAR EL-YAGUBI & MASLINA DARUS

ABSTRACT

A new generalised derivative operator 1 2

,, ,n mD bλ λ   is introduced. This operator generalised some

well-known operators studied earlier. New subclasses of analytic functions in the open unit disc which are defined using generalised derivative operator are introduced. Inclusion theorems are investigated. Furthermore, generalised Bernardi-Libera-Livington integral operator is shown to be preserved for these classes.

Keywords: analytic functions; univalent functions; starlike functions; convex functions; close-to-convex functions; subordination; Hadamard product; integral operator

ABSTRAK

Pengoperasi terbitan baharu teritlak 1 2

,, ,n mD bλ λ   diperkenalkan. Pengoperasi ini mengitlak

beberapa pengoperasi terdahulu yang terkenal. Subkelas baharu fungsi analisis dalam cakera terbuka unit diperkenalkan yang ditakrif dengan menggunakan pengoperasi terbitan teritlak. Teorem rangkuman dikaji. Malah pengoperasi kamiran Bernardi-Libera-Livington ditunjukkan kekal untuk kelas tesebut.

Kata kunci: fungsi analisis; fungsi univalen; fungsi bak bintang; fungsi cembung; fungsi hampir cembung; subordinasi; hasil darab Hadamard; pengoperasi kamiran

1. Introduction

Let A denote the class of functions of the form

2( ) ,k

kk

f z a z∞

=

= ∑ (1.1)

where ak is a complex number, which are analytic in the open unit disc U ={z ∈ :| z |<1}. Also let S*,C and K denote, respectively, the subclasses of A consisting of functions which are starlike, convex, and close to convex in U. An analytic function 𝑓 is subordinate to an analytic function 𝑔, written f (z) g(z), (z ∊ U ) if there exists an analytic function w in U, such that w (0) = 0, |w (z) |<1 for |z|<1 and f(z) = g (w (z)). In particular, if 𝑔 is univalent in U, then f (z) g(z) is equivalent to f (0) = g (0) and f (U ) ⊂ g (U).

The convolution of two analytic functions 𝜑(𝑧) =2

kk

kz a z

=

+∑  and 𝜓(𝑧) =2

kk

kz b z

=

+∑  is defined by

Journal of Quality Measurement and Analysis Jurnal Pengukuran Kualiti dan Analisis

JQMA 9(1) 2013, 47-56

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𝜑(𝑧) ∗ 𝜓(𝑧) = 2

kk k

kz a b z

=

+∑ = 𝜓(𝑧) ∗ 𝜑(𝑧). (1.2)

In order to derive our new generalised derivative operator, we define the analytic function

1 2

1 2, ,

2 2

1 ( )( 1)( ) ,1 ( 1)

mm k

bk

k bF z z zk bλ λ

λ λλ

=

⎡ ⎤+ + − += + ⎢ ⎥+ − +⎣ ⎦∑   (1.3)

where m,b∈ 0 ={0,1,2,…} and 2 1 0.λ λ≥ ≥ Now, we introduce the new generalised derivative

operator 1 2

,, ,n m

bDλ λ  as follows:

Definition 1.1. For ,f ∈Α the operator 1 2

,, ,n m

bDλ λ  is defined by 1 2

,, , : ,n mbDλ λ Α→Α

1 2 1 2

,, , , ,( ) ( ) ( ),n m m nb bD f z F z R f zλ λ λ λ= ∗   ,z U∈   (1.4)

where n,m,b∈ 0 = ∪{0}, 2 1 0,λ λ≥ ≥  and ( )nR f z denotes the Ruscheweyh derivative operator (Ruscheweyh 1975), given by

2( ) ( , ) ,n k

kk

R f z z C n k a z∞

=

= +∑ (n∈ 0 , z ∈U ),   (1.5)

where 1 1( , ) ( 1) (1) .k kC n k n − −= +  

If 𝑓 is given by (1.1), then we easily find from equality (1.4) that

1 2

, 1 2, ,

2 2

1 ( )( 1)( ) ( , ) ,1 ( 1)

mn m k

b kk

k bD f z z C n k a zk bλ λ

λ λλ

=

⎡ ⎤+ + − += + ⎢ ⎥+ − +⎣ ⎦∑ ( ),z U∈   (1.6)

where n,m,b∈ 0 = ∪{0}, 2 1 0,λ λ≥ ≥  and

1 1

1( , ) ( 1) (1) .k k

n kC n k n

n − −

+ −⎛ ⎞= = +⎜ ⎟⎝ ⎠

 

Note that, (n)k denotes the Pochhammer symbol (or the shifted factorial) defined by

(n)k =1 for k = 0,n∈ \{0}n(n+1)(n+ 2)(n+ k −1) for k ∈,n∈

⎧⎨⎪

⎩⎪ (1.7)

Remark 1.1. Special cases of the operator 1 2

,, ,n m

bDλ λ  include the Ruscheweyh derivative operator

in the case 1 2

,0, ,n

bDλ λ   (Ruscheweyh 1975), the Salagean derivative operator in the case 0,1,0,0m nD S≡

(Salagean 1983), the generalised Salagean derivative operator introduced by Al-Oboudi in the

case 1 1

0,,0,0m mD Dλ λ≡ (Al-Oboudi 2004), the generalised Ruscheweyh derivative operator in the

case 1

1

,1,0,0n

nD D λλ ≡ (Al-Shaqsi & Darus 2009), the generalised Al-Shaqsi and Darus derivative

operator in the case , ,1,0,n m m bb nD D≡   (Darus & Al-Shaqsi 2008), the Uralegaddi and Somanatha

derivative operator in the case 0,1,0,1m mD D≡ (Uralegaddi & Somanatha 1992), the Cho and

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48 49

Srivastave derivative operator in the case 0,1,0,m mb bD D≡ (Cho & Srivastava 2003), the Eljamal

and Darus derivative operator in the case 0,1,0,m mb bD D≡ (Eljamal & Darus 2011), and the Cǎtas

derivative operator in the case 1 1

0,,0, ,m mb bD Dλ λ≡   (Cǎtas 2008).

To prove our results, we need the following equations throughout the paper:

1 2 1 2 1 2

, 1 , ,, , 1 2 , , 1 2 , ,(1 ) ( ) (1 ( ) ) ( ) ( ) ( ) ,n m n m n mb b bb D f z b D f z z D f zλ λ λ λ λ λλ λ λ λ+ ′⎡ ⎤+ = − + + + + ⎣ ⎦ (1.8)

1 2 1 2 1 2

1, , ,, , , , , ,( ) ( ) ( 1) ( ).n m n m n mb b bnD f z z D f z n D f zλ λ λ λ λ λ

+ ′⎡ ⎤= + −⎣ ⎦   (1.9)

Let N be the class of all analytic and univalent functions ø in U and for which ϕ (U) is convex with ϕ (0) =1 and Re{ϕ(z)}>0, for z∊U. For ϕ,ψ∊ N, Ma and Minda (1992) studied the subclasses

( ),S φ∗   ( ),C φ and ( , )K φ ψ of the class A. These classes are defined using the principle of subordination as follows:

S∗(φ) := f : f ∈A, z ′f (z)f (z)

φ(z) inU⎧⎨⎩

⎫⎬⎭,

C(φ) := f : f ∈A, 1+ z ′′f (z)′f (z)φ(z) inU

⎧⎨⎩

⎫⎬⎭, (1.10)

K (φ,ψ ) := f : f ∈A,∃ g ∈S∗(φ) such that z ′f (z)g(z)

ψ (z) in U⎧⎨⎩

⎫⎬⎭

.

Obviously, we have the following relationships for special choices of ϕ and ѱ

1 ,1zS Sz

∗ ∗+⎛ ⎞ =⎜ ⎟−⎝ ⎠   1 ,

1zC Cz

+⎛ ⎞ =⎜ ⎟−⎝ ⎠1 1, .1 1z zK Kz z

+ +⎛ ⎞ =⎜ ⎟− −⎝ ⎠   (1.11)

Using the generalised differential operator 1 2

,, , ,n mD fbλ λ new classes

1 2

,, , ( ),n m

bS λ λ φ  1 2

,, , ( )n m

bC λ λ φ   and

1 2

,, , ( , ),n m

bK λ λ φ ψ are introduced and defined as follows:

{ }1 2 1 2

, ,, , , ,( ) : : ( ) ( ) ,n m n mb bS f A D f z Sλ λ λ λφ φ∗= ∈ ∈  

{ }1 2 1 2

, ,, , , ,( ) : : ( ) ( ) ,n m n mb bC f A D f z Cλ λ λ λφ φ= ∈ ∈   (1.12)

{ }1 2 1 2

, ,, , , ,( , ) : : ( ) ( , ) .n m n mb bK f A D f z Kλ λ λ λφ ψ φ ψ= ∈ ∈  

It can be shown easily that

1 2 1 2

, ,, , , ,( ) ( ) ( ) ( ).n m n m

b bf z C zf z Sλ λ λ λφ φ′∈ ⇔ ∈ (1.13)

Janowski (1973) introduced the class [ , ]S A B∗ =   ((1 ) (1 )),S Az Bz∗ + +   and in particular for 𝜙 (z) = (1 + 𝐴z) / (1 + 𝐵𝑧), we set

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1 2 1 2

,, , , , , ,

1 [ , ],1

n mb n m b

AzS S A BBzλ λ λ λ

∗+⎛ ⎞ =⎜ ⎟+⎝ ⎠(1 1).A B≥ > ≥ − (1.14)

In (Omar and Halim 2012), the authors studied the inclusion properties for classes defined using Dziok-Srivastava operator. This paper investigates similar properties for analytic functions

in the classes defined by the generalized differential operator 1 2

, ., ,n mD fbλ λ   Furthermore,

applications of other families of integral operators are considered involving these classes.

2. Inclusion Properties Involving 1 2

,, ,n mD fbλ λ  

To prove our results, we need the following lemmas:

Lemma 2.1 (see Eenigenburg et al. 1983). Let 𝜙 be convex univalent in U, with 𝜙(0)=1 and Re{ ( ) } 0,k zφ η+ >  (k,η ∈). If p is analytic in U with p(0)=1 then

p(z)+ z ′p (z)

kp(z)+ηφ(z)⇒ p(z) φ(z).

(2.1)

Lemma 2.2 (see Miller and Mocanu (1981)). Let 𝜙 be convex univalent in U and w be analytic in U with Re{ ( )} 0.w z ≥   If p is analytic in U and p(0)= 𝜙(0) then

p(z)+ w(z)z ′p (z) φ(z)⇒ p(z) φ(z). (2.2)

Theorem 2.3. For any real numbers ,m λ1 and 2 ,λ  where 0,m ≥   2 1 0λ λ≥ ≥  and 0.b ≥

Letφ ∈Ν   and 1 2 1 2Re{ ( ) (1 ( ) ) ( )} 0,z bφ λ λ λ λ+ − + + + > then 1 2 1 2

, 1 ,, , , ,( ) ( ),n m n m

b bS Sλ λ λ λφ φ+ ⊂   ( 0).n ≥

Proof. Let 1 2

, 1, , ( ),n m

bf S λ λ φ+∈ and set 1 2 1 2

, ,, , , ,( ) ( [ ( )] ) ( ( )),n m n m

b bp z z D f z D f zλ λ λ λ′= where p is analytic in U, with (0) 1.p =   Rearranging (1.8), we have

1 2 1 2

1 2 1 2

, 1 ,, , 1 2 , ,

1 2, ,, , , ,

(1 ) ( ) ( ) [ ( )](1 ( ) ) .

( ) ( )

n m n mb b

n m n mb b

b D f z z D f zb

D f z D f zλ λ λ λ

λ λ λ λ

λ λλ λ

+ ′+ += − + + +   (2.3)

Next, differentiating (2.3) logarithmically with respect to z and multiplying by z, we obtain

( )1 2 1 21 2 1 2

1 2 1 2 1 2 1 2

, ,, 1 ,, , , ,, , , ,

, 1 , , ,, , , , , , , , 1 2 1 2

( [ ( )] ) ( ( ))[ ( )] [ ( )]( ) ( ) ( [ ( )] ) ( ( )) (1 ( ) ) ( )

n m n mn m n mb bb b

n m n m n m n mb b b b

z z D f z D f zz D f z z D f zD f z D f z z D f z D f z b

λ λ λ λλ λ λ λ

λ λ λ λ λ λ λ λ λ λ λ λ

+

+

′′′ ′= +

′ + − + + + (2.4)

1 2 1 2

( )( ) .( ) (1 ( ) ) ( )

zp zp zp z bλ λ λ λ

′= +

+ − + + +  

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Since (z[Dλ1,λ2 ,bn,m f (z) ′] ) (Dλ1,λ2 ,b

n,m f (z)) φ(z) and applying Lemma 2.1, it follows that

𝑝 ≺ 𝜙. Thus1 2

,, , ( ).n m

bf S λ λ φ∈

Theorem 2.4. Let 1 2, , ,m λ λ ∈ °   where 0,m ≥   2 1 0λ λ≥ ≥  and 0.n ≥   Then

1 2 1 2

1, ,, , , ,( ) ( ),n m n m

b bS Sλ λ λ λφ φ+ ⊂   ( 0,b ≥   ).φ∈Ν

Proof. Let 1 2

1,, , ( ),n m

bf S λ λ φ+∈ and from (1.9), we obtain that

1 2 1 2

1 2 1 2

1, ,, , , ,, ,, , , ,

( ) [ ( )]( 1).

( ) ( )

n m n mb b

n m n mb b

nD f z z D f zn

D f z D f zλ λ λ λ

λ λ λ λ

+ ′= + −   (2.5)

Making use of the differentiating (2.5) logarithmically with multiplying by z and setting

1 2 1 2

, ,, , , ,( ) ( [ ( )] ) ( ( )),n m n m

b bp z z D f z D f zλ λ λ λ′=we get the following:

z[Dλ1,λ2 ,bn+1,m f (z) ′]

Dλ1,λ2 ,bn+1,m f (z)

= p(z)+ z ′p (z)p(z)+ (n−1)

φ(z). (2.6)

Since 0.n ≥   and Re{ ( ) ( 1)} 0,z nφ + − > using Lemma 2.1, we conclude that 1 2

,, , ( ),n m

bf S λ λ φ∈

Corollary 2.5. Let 2 1 0,λ λ≥ ≥ 0,n ≥   and 1 1.A B≥ > ≥ −   Then

1 2 1 21, , , , , ,[ ; , ] [ ; , ]m b m bS n A B S n A Bλ λ λ λ∗ ∗+ ⊂   and

1 2 1 2, , , , , ,[ 1; , ] [ ; , ].m b m bS n A B S n A Bλ λ λ λ∗ ∗+ ⊂

Theorem 2.6. Let 2 1 0,λ λ≥ ≥ and 0.n ≥   Then 1 2 1 2

, 1 ,, , , ,( ) ( )n m n mb bC Cλ λ λ λφ φ+ ⊂ and

1 2 1 2

1, ,, , , ,( ) ( ).n m n mb bC Cλ λ λ λφ φ+ ⊂

Proof. Using (1.12) and Theorem 2.3, we observe that

1 2

, 1, ,( ) ( )n mbf z Cλ λ φ+∈  

1 2

, 1, ,( ) ( )n mbzf z S λ λ φ+′⇔ ∈

1 2

,, ,( ) ( )n mbzf z S λ λ φ′⇒ ∈

1 2

,, , ( ) ( )n mbD zf z Sλ λ φ∗′⇔ ∈

1 2

,, ,[ ( )] ( )n mbz D f z Sλ λ φ∗′⇔ ∈

1 2

,, , ( ) ( )n mbD f z Cλ λ φ⇔ ∈  

1 2

,, ,( ) ( ).n mbf z Cλ λ φ⇔ ∈ (2.7)

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To prove the second part of the theorem, we use similar steps and apply Theorem 2.4, the result is obtained.

Theorem 2.7. Let 2 1 0,λ λ≥ ≥ 0b ≥   and 1 2 1 2Re{ ( ) (1 ( ) ) ( )} 0.z bφ λ λ λ λ+ − + + + > Then

1 2 1 2

, 1 ,, , , ,( , ) ( , )n m n mb bK Kλ λ λ λφ ψ φ ψ+ ⊂   and

1 2 1 2

1, ,, , , ,( , ) ( , ),n m n mb bK Kλ λ λ λφ ψ φ ψ+ ⊂   ( , ) .Nφ ψ ∈

Proof. Let 1 2

, 1, , ( , ).n m

bf K λ λ φ ψ+∈   In view of the definition of the class 1 2

, 1, , ( , ),n m

bK λ λ φ ψ+   there is a function

1 2

, 1, , ( ),n m

bg S λ λ φ+∈ such that

1 2

1 2

, 1, ,, 1, ,

[ ( )]( )

( )

n mb

n mb

z D f zz

D g zλ λ

λ λ

ψ+

+

′p   (2.8)

Applying Theorem 2.3, then 1 2

,, , ( ),n m

bg S λ λ φ∈   and let q(z) = (z[Dλ1,λ2 ,b

n,m g(z) ′] ) (Dλ1,λ2 ,bn,m g(z)) φ(z).

Let the analytic function p with p(0)=1, as follows:

,, ,1 2,, ,1 2

[ ( )]( ) .

( )

n mb

n mb

z D f zp z

D g zλ λ

λ λ

′=   (2.9)

Thus, rearranging and differentiating (2.9), we have

1 2 1 2

1 2 1 2

, ,, , , ,, ,, , , ,

[ ( )] ( )[ ( )]( ).

( ) ( )

n m n mb b

n m n mb b

D zf z p z D g zp z

D g z D g zλ λ λ λ

λ λ λ λ

′ ′ ′′= + (2.10)

Making use of (1.8), (2.9), (2.10), and 𝑞 (z), we obtain that

1 2

1 2

, 1, ,, 1, ,

[ ( )]( )

n mb

n mb

z D f zD g z

λ λ

λ λ

+

+

′   1 2

1 2

, 1, ,, 1, ,

[ ( )]( )

n mb

n mb

D zf zD g zλ λ

λ λ

+

+

′=  

1 2 1 2

1 2 1 2

, ,1 2 , , 1 2 , ,

, ,1 2 , , 1 2 , ,

(1 ( ) ) ( ) ( ) [ ( )](1 ( ) ) ( ) ( ) [ ( )]

n m n mb b

n m n mb b

b D zf z z D zf zb D g z z D g z

λ λ λ λ

λ λ λ λ

λ λ λ λλ λ λ λ

′ ′ ′− + + + +=

′− + + + + (2.11)

( ) ( ) ( ) ( )( ) ( )

1 2 1 2 1 2 1 2

1 2 1 2

, , , ,1 2 , , , , 1 2 , , , ,

, ,1 2 1 2 , , , ,

(1 ( ) ) ( ) ( ) ( ) [ ( )] ( )

(1 ( ) ) ( ) [ ( )] ( )

n m n m n m n mb b b b

n m n mb b

b D zf z D g z z D zf z D g z

b z D g z D g zλ λ λ λ λ λ λ λ

λ λ λ λ

λ λ λ λ

λ λ λ λ

′ ′ ′− + + + +=

′− + + + +

1 2 1 2

1 2 1 2

(1 ( ) ) ( ) ( )[ ( ) ( ) ( )](1 ( ) ) ( ) ( )

b p z p z q z p zb q z

λ λ λ λλ λ λ λ

′− + + + + +=− + + + +

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= p(z)+ z ′p (z)q(z)+ (1− (λ1 + λ2 )+ b) (λ1 + λ2 )

ψ (z).

Since q(z) φ(z) and 1 2 1 2Re{(1 ( ) ) ( )} 0,bλ λ λ λ− + + + >   then

1 2 1 2Re{ ( ) (1 ( ) ) ( )} 0.q z bλ λ λ λ+ − + + + >

Using Lemma 2.2, we conclude that ( )p z  ≺ ( )zψ  and thus 1 2

,, , ( , ).n m

bf K λ λ φ ψ∈  

By using similar manner and (1.9), we obtain the second result.

In summary, by using subordination technique, inclusion properties have been established for certain analytic functions defined via the generalised differential operator.

3. Inclusion Properties Involving cF f

In this section, we determine properties of generalised Bernardi-Libera-Livington integral operator defined by (Bernardi 1969; Jung et al. 1993; Libera 1965; Livington 1966).

1

0

1[ ( )] ( )zc

c c

cF f z t f t dtz

−+= ∫

( ,f ∈Α 1).c > −

2

1 ,nnn

cz a zn c

=

+= ++∑ (3.1)

and satisfies the following:

1 2 1 2 1 2

, , ,, , , , , ,[ ( )] [ ( )] ( 1) ( ).n m n m n mb c b c bcD F f z z D F f z c D f zλ λ λ λ λ λ

′⎡ ⎤+ = +⎣ ⎦   (3.2)

Theorem 3.1. If 1 2

,, , ( ),n m

bf S λ λ φ∈ then 1 2

,, , ( ).n m

c bF f S λ λ φ∈

Proof. Let 1 2

,, , ( ),n m

bf S λ λ φ∈ then z[Dλ1,λ2 ,b

n,m f (z) ′]( ) Dλ1,λ2 ,bn,m f (z)( )φ(z).Taking the

differentiation on both sides of (3.2) and multiplying by 𝑧, we obtain

z Dλ1,λ2 ,bn,m f (z)⎡

⎣⎢⎤⎦⎥′

Dλ1,λ2 ,bn,m f (z)

=z Dλ1,λ2 ,b

n,m Fc[ f (z)]⎡⎣⎢

⎤⎦⎥′

Dλ1,λ2 ,bn,m Fc[ f (z)]

+

z z Dλ1,λ2 ,bn,m Fc[ f (z)]⎡

⎣⎢⎤⎦⎥′⎛

⎝⎜

⎠⎟ Dλ1,λ2 ,b

n,m Fc[ f (z)]( )⎛

⎝⎜⎜

⎠⎟⎟

z Dλ1,λ2 ,bn,m Fc[ f (z)]⎡

⎣⎢⎤⎦⎥′⎛

⎝⎜

⎠⎟ Dλ1,λ2 ,b

n,m Fc[ f (z)]( )+ c (3.3)

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Entisar El-Yagubi & Maslina Darus

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Setting

( )1 2 1 2

, ,, , , ,( ) [ ( )] [ ( )] ,n m n m

b c b cp z z D F f z D F f zλ λ λ λ⎛ ⎞′⎡ ⎤= ⎜ ⎟⎣ ⎦⎝ ⎠

 

we have

1 2

1 2

,, ,,, ,

[ ( )] ( )( ) .( ) ( )

n mb

n mb

z D f z zp zp zD f z p z c

λ λ

λ λ

′ ′= +

+ (3.4)

Lemma 2.1 implies z Dλ1,λ2 ,bn,m Fc[ f (z)]⎡

⎣⎤⎦′⎛

⎝⎜⎞

⎠⎟Dλ1,λ2 ,bn,m Fc[ f (z)]( )φ(z). Hence

1 2

,, , ( ).n m

c bF f S λ λ φ∈

Theorem 3.2. Let 1 2

,, , ( ),n m

bf C λ λ φ∈  then 1 2

,, , ( ).n m

c bF f C λ λ φ∈Proof. By using (1.12) and Theorem 3.1, it follows that

1 2

,, , ( )n m

bf C λ λ φ∈  ⟺1 2

,, ,( ) ( )n m

bzf z S λ λ φ′ ∈  

⟹1 2

,, ,[ ( )] ( )n m

c bF zf z S λ λ φ′ ∈  

⟺1 2

,, ,[ [ ( )]] ( )n m

c bz F f z S λ λ φ′∈ ⟺1 2

,, ,[ ( )] ( ).n m

c bF f z C λ λ φ∈ (3.5)

Theorem 3.3. Let , Nφ ψ ∈  and 1 2

,, , ( , ),n m

bf K λ λ φ ψ∈ then 1 2

,, , ( , ).n m

c bF f K λ λ φ ψ∈

Proof. Let 1 2

,, , ( , ),n m

bf K λ λ φ ψ∈ then there exists a function 1 2

,, , ( ),n m

bg S λ λ φ∈  such that

z[Dλ1,λ2 ,bn,m f (z) ′]( ) Dλ1,λ2 ,b

n,m g(z)( )ψ (z). Since 1 2

,, , ( ),n m

bg S λ λ φ∈  therefore from Theorem 3.1,

1 2

,, ,[ ( )] ( ).n m

c bF g z S λ λ φ∈ Then let

q(z) =z Dλ1,λ2 ,b

n,m Fc[g(z)]⎡⎣

⎤⎦′

Dλ1,λ2 ,bn,m Fc[g(z)]

φ(z). (3.6)

Set

1 2

1 2

,, ,

,, ,

[ ( )]( ) .

[ ( )]

n mb c

n mb c

z D F f zp z

D F g zλ λ

λ λ

′⎡ ⎤⎣ ⎦= (3.7)

By rearranging and differentiating (3.7), we obtain that

1 2 1 2 1 2

1 2 1 2 1 2

, , ,, , , , , ,, , ,, , , , , ,

[ [ ( )]] ( )[ [ ( )]] ( )[ [ ( )]].

[ ( )] [ ( )] [ ( )]

n m n m n mb c b c b c

n m n m n mb c b c b c

D F zf z p z D F g z p z D F g zD F g z D F g z D F g zλ λ λ λ λ λ

λ λ λ λ λ λ

′ ′ ′ ′= +   (3.8)

Making use of (3.2), (3.7), and (3.6), it can be derived that

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Subclasses of analytic functions defined by new generalised derivative operator

54 55

1 2

1 2

,, ,,, ,

[ ( )] ( )( ) .( ) ( )

n mb

n mb

z D f z zp zp zD g z q z c

λ λ

λ λ

′ ′= +

+   (3.9)

Hence, applying Lemma 2.2, we conclude that p(z) ψ (z), and it follows that

1 2

,, ,[ ( )] ( , )n m

c bF f z Kλ λ φ ψ∈ . For analytic functions in the classes defined by generalised differential operator, the generalised Bernardi-Libera-Livington integral operator has been shown to be preserved in these classes.

4. Conclusion

Results involving functions defined using the generalised differential operator, namely, inclusion properties and the Bernardi-Libera-Livington integral operator were obtained using subordination principles. In Omar and Halim (2012), similar results were discussed for functions defined using the Dziok-Srivastava operator.

Acknowledgment

The work presented here was partially supported by GUP-2013-023.

References

Al-Oboudi F.M. 2004. On univalent functions defined by derivative operator. International Journal of Mathematics and Mathematical Sciences 27: 1429–1436.

Al-Shaqsi K. & Darus M. 2009. On univalent functions with respect to k-symmetric points defined by a generalized Ruscheweyh derivative operators. Journal of Analysis and Applications 7(1):53–61.

Bernardi S.D. 1969. Convex and starlike univalent functions. Transactions of the American Mathematical Society 135: 429–446.

Cǎtas A. 2008. On certain classes of p-valent functions defined by new multiplier transformations, TC Istanbul Kultur University Publications, Proceedings of the International Symposium on Geometric Function Theory and Applications (GFTA ‘07), Istanbul, Turkey, August 2007, vol. 91, pp. 241–250.

Cho N.E. & Srivastava H. M. 2003. Argument estimates of certain analytic functions defined by a class of multiplier transformations. Mathematical and Computer Modelling 37(1-2): 39–49.

Darus M. & Al-Shaqsi K. 2008. Differential sandwich theorems with generalised derivative operator. International Journal of Computing and Mathematical Sciences 22:75–78.

Eenigenburg P., Miller S.S., Mocanu P.T. & Reade M. O. 1983. On a Briot-Bouquet differential subordination. General Inequalities 3: 339–348.

Eljamal E.A. & Darus M. 2011. Subordination results defined by a new differential operator. Acta Universitatis Apulensis 27:121–126.

Janowski W. 1973. Some extremal problems for certain families of analytic functions I. Annales Polonici Mathematici 28: 297–326.

Jung I.B., Kim Y. C. & Srivastava H. M. 1993. The Hardy space of analytic functions associated with certain one-parameter families of integral operators. Journal of Mathematical Analysis and Applications 176(1):138–147.

Libera R.J. 1965. Some classes of regular univalent functions. Proceedings of the American Mathematical Society 16: 755–758.

Livington A.E. 1966. On the radius of univalence of certain analytic functions. Proceedings of the American Mathematical Society 17: 352–357.

Ma W. & Minda D. 1992. A unified treatment of some special classes of univalent functions. In Z. Li, F. Ren, L. Yang & S. Zhang (Eds.). Proceedings of the Conference on Complex Analysis, pp. 157–169, International Press, Cambridge, Mass, USA.

Page 10: SUBCLASSES OF ANALYTIC FUNCTIONS DEFINED BY ...Subclasses of analytic functions defined by new generalised derivative operator 48 49 Srivastave derivative operator in the case 0,1,0,

Entisar El-Yagubi & Maslina Darus

56 57

Miller S.S. & Mocanu P.T. 1981. Differential subordination and univalent functions. The Michigan Mathematical Journal 28: 157–171.

Omar R. & Halim S.A. 2012. Classes of functions defined by Dziok-Srivastava operator. Far East Journal of Mathematical Sciences 66(1): 75–86.

Ruscheweyh S. 1975. New criteria for univalant function. Proceedings of the American Mathematical Society 49:109–115.

Salagean G.S. 1983. Subclasses of univalent functions. In Proceedings of the Complex Analysis 5th Romanian-Finnish Seminar Part 1, 1013: 362–372.

Uralegaddi B.A. & Somanatha C. 1992. Certain classes of univalent function. In Current Topics in Analytic Function Theory. River Edge, NJ: World Scientific, pp. 371–374.

School of Mathematical SciencesFaculty of Science and TechnologyUniversity Kebangsaan Malaysia43600 UKM BangiSelangor DE, MALAYSIAE-mail: [email protected], [email protected]*

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