Journal of Quality Measurement and Analysis JQMA 6(1) 2010, 23-32 Jurnal Pengukuran Kualiti dan Analisis

INCLUSION PROPERTIES FOR CERTAIN SUBCLASSES OF ANALYTIC FUNCTIONS ASSOCIATED WITH A MULTIPLIER TRANSFORMATION (Sifat Rangkuman bagi beberapa Subkelas Fungsi Analisis Bersekutu dengan Penjelmaan Berganda)

AABED MOHAMMED & MASLINA DARUS

ABSTRACT

Let f be the normalised analytic function in the open unit disk .U In the present paper, we define a new operator related to generalised Hurwitz–Lerch zeta function which involved the convolution (Hadamard product). Via this operator, we introduce new classes of functions and derive some interesting properties for these classes.

Keywords: Hurwitz–Lerch zeta function; Hadamard product; multiplier transformation; strongly convex and strongly starlike

ABSTRAK

Andaikan f fungsi analisis ternormal dalam cakera unit .U Dalam makalah ini, pengoperasi baru berkenaan dengan fungsi Hurwitz–Lerch zeta teritlak yang melibatkan konvolusi (hasil darab Hadamard) diperkenalkan. Menerusi pengoperasi ini, kelas fungsi baru diperkenalkan dan beberapa sifat menarik diperoleh bagi kelas tersebut.

Kata kunci: fungsi Hurwitz–Lerch zeta; hasil darab Hadamard; penjelmaan multiplier; cembung kuat dan bakbintang kuat

1. Introduction

Let : , 1U z z C z be the open unit disk and A denotes the class of functions f with the form

2

( ) ,kk

k

f z z a z

which are analytic in the open unit disk U and satisfy the condition (0) '(0) 1 0.f f We denote by ( )S and ( )K for 0 1 the familiar subclasses of A consisting of functions which respectively starlike and convex functions are of order . Thus by definition, we have

'( )( ) : , 0 1;( )

f zS f f A and z z Uf z

and

Aabed Mohammed & Maslina Darus

24

"( )( ) : 1 , 0 1; .'( )

zf zK f f Aand z Uf z

If f A satisfies

'( )arg (0 1; 0 1; ),( ) 2

zf z z Uf z

then f is said to be strongly starlike of order and type in .U If f A satisfies

''(arg 1 (0 1; 0 1; ),'( ) 2

zf z z Uf z

then f is said to be strongly convex of order and type in .U

We denote by ( , )S and ( , ),C respectively, the subclasses of A consisting of all strongly starlike and strongly convex of order and type in .U It is obvious that f A belongs to ( , ),C if and only if '( ) ( , ).zf z S We also note that (1, ) ( )S S and

(1, ) ( ).C C In particular, the classes ( ,0)S and ( ,0)C have been extensively studied by Mocanu (1989) and Nunokawa (1993).

Now let us consider the generalised Hurwitz–Lerch Zeta function

0

( )( , , ) ,! ( )

kk

mk

zz m ak k a

for , 1, /{0, 1, 2,...}, , ,z z a m

introduced by Goyal and Laddha (1997). Here ( )kx is Pochhammer symbol (or the shifted factorial, since (1) !k k ) and ( )k given in terms of the Gamma functions can be written as

1, 0 and \ {0};( )( )

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

if k x Cx kxx x x k if k N x Cx

Then

1

1

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

kk

mk

zz z m ak k a

where , 1, /{ 1, 2,...}, , .z z a m Now we define the function ( , , )G z m a by

Inclusion properties for certain subclasses of analytic functions associated with a multiplier transformation

25

1

1

1 ( )( , , ) (1 ) [ ( , , )]( 1)!

mm kk

k

aG z m a a z z m a zk a k

and obtain the following operator , ( ) ( , , ) ( )m

aD f z G z m a f z then

1

,2

1 ( )( ) ,( 1)!

( , \ , , ).

mm kk

a kk

aD f z z a zk a k

z U a m

(1)

It is clear that ,

maD are multiplier transformations. For 1m and 1 the operator ,

maD is

the integral operator studied by Owa and Srivastava (1986), for any nonnegative real number m and 1, 1a the operator ,

maD is the integral operator studied by Jung et al. (1993), for

any negative integer number m and 1, 1a the operator ,m

aD is the differential operator

defined by Salagean (1983), for m and 2, 0a the operator 2,m

aD is the multiplier transformation defined by Cho and Kim (2003). In Particular, we note that 0

1, ( )aD f z and 1 0

1,0 2,0 '( ).D D zf z In view of (1), we obtain

1 1, , ,( ( )) ' ( 1) ( ) ( )m m ma a az D f z a D f z aD f z (2)

, 1, ,( ( )) ' ( ) ( 1) ( ).m m m

a a az D f z D f z D f z (3) The relations (2) and (3) play important and significant roles in obtaining our results. Using the operator ,

maD , we now introduce the following classes:

,

, ,,

( ) '( , ) ( ) : ( ) ( , ) , ,

( )

mam m

a a ma

z D f zST f z A D f z S z U

D f z

,, ,

,

( ) ' '( , ) ( ) : ( ) ( , ), , .

( ) '

mam m

a a ma

z D f zCV f z A D f z C z U

D f z

It is obvious that , ( , )m

af CV if and only if ,' ( , ).mazf ST

Aabed Mohammed & Maslina Darus

26

In the present paper, we investigate some properties of the classes , ( , )maST and , ( , ).m

aCV

The integral preserving properties in connection with the operator ,m

aD defined by (1) are also considered.

The basic tools of our investigation are the following lemmas.

Lemma 1 (Nunokawa 1993). Let p is analytic in U with (0) 1p and ( ) 0 in .p z U Suppose that there exists a point 0z U such that

0arg ( )2

p z for z z and 0arg ( ) (0 1).

2p z for

(4)

Then we have

0 0

0

'( ) ,( )

z p z ikp z

(5)

where

01 1 arg ( ) ,2 2

k b when p zb

(6)

01 1 arg ( ) .2 2

k b when p zb

(7)

and

1

0( ) ( 0).p z ib b (8)

Lemma 2 (Eenigenburg et al. 1981; 1983). Let , be complex numbers. Let ( )h z be convex univalent in U with (0) 1h and [ ( ) ] 0, .h z z U If 2

1 2( ) 1 ...p z p z p z is analytic in U with (0) 1p then,

'( )( ) ( ) ( ) ( ).( )

zp zp z h z p z h zp z

Our first inclusion theorem is stated as

Inclusion properties for certain subclasses of analytic functions associated with a multiplier transformation

27

Theorem 1. For , and \ ,m a 1, ,( , ) ( , )m ma aST ST

Proof. Let ,( ) ( , )m

af z ST Define the function ( )p z by

1

,1

,

( ) '(1 ) ( ),

( )

ma

ma

z D f zp z

D f z

(9)

where 2

1 2( ) 1 ...p z p z p z is analytic in U and ( ) 0p z for all .z U

Using the identity

1 1, , ,( ( )) ' ( 1) ( ) ( )m m ma a az D f z a D f z aD f z

we get

,1

,

( )(1 ) (1 ) ( )

( )

ma

ma

D f za a p z

D f z

(10)

Differentiating both sides of (10) logarithmically and multiplying by ,z we obtain

,

,

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

ma

ma

z D f z zp zp zp z aD f z

Suppose now that there exists a point 0z U such that

0arg ( )2

p z for z z and 0arg ( ) .

2p z

Then, by applying Lemma 1, we can write that 0 0 0'( ) ( )z p z p z ik and 1

0( ) ( 0).p z ib b

Therefore, if 0arg ( ) ( 2) ,p z then

0 , 0 0 0 00

, 0 0

22

( ) ' '( ) ( )(1 ) ( ) 1( ) (1 ) ( )

(1 ) 1 .(1 )

ma

ma

ii

z D f z z p z p zp zD f z p z a

ikb eb e a

Thus we have

Aabed Mohammed & Maslina Darus

28

0 , 02

, 0

12 2 2

( ) 'arg arg 1

( ) 2 (1 )

[ (1 ) cos( 2)]tan2 ( ) 2( )(1 ) cos( 2) (1 ) (1 ) sin( 2)

ma

m ia

z D f z ikD f z b e a

k a ba a b b k b

1 1where 1 ,2 2

k bb

and this contradicts the condition , ( , )m

af ST . Similarly, if 0arg ( ) ( 2) ,p z then we have

0 , 0

, 0

( ) 'arg

( ) 2

ma

ma

z D f zD f z

which contradicts the condition ,( ) ( , )maf z ST .

Thus the function ( )p z has to satisfy arg ( ) ( 2) ( ),p z z U which leads us to the following

,

,

( ( )) 'arg .

( ) 2

ma

ma

z D f zD f z

This evidently completes the proof of Theorem 1. Theorem 2. For , and \ ,m a 1, ,( , ) ( , )m m

a aST ST Proof. Let

,

,

( ) '(1 ) ( ),

( )

ma

ma

z D f zp z

D f z

(11)

where 21 2( ) 1 ...p z p z p z is analytic in U and ( ) 0p z for all .z U

Using the identity

, 1, ,( ( )) ' ( ) ( 1) ( )m m ma a az D f z D f z D f z

Then

Inclusion properties for certain subclasses of analytic functions associated with a multiplier transformation

29

1,

,

( )(1 ) ( ) ( 1).

( )

ma

ma

D f zp z

D f z

(12)

Differentiating both sides of (12) logarithmically and multiplying by ,z we obtain

1,

1,

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

ma

ma

z D f z zp zp zp zD f z

The remaining part of the proof in Theorem 2 is similar to that of Theorem 1 and so we omit the details of the proofs. We next state Theorem 3. For , and \ ,m C a 1

, ,( , ) ( , ).m ma aCV CV

Proof.

, , ,( ) ( , ) ( ) ( , ) ( ) ' ( , )m m ma a af z CV D f z C z D f z S

, ,'( ) ( , ) '( ) ( , )m m

a aD zf z S zf z ST

1 1, ,'( ) ( , ) '( ) ( , )m ma azf z ST D zf z S

1 1 1, , ,( ) ' ( , ) ( ) ( , ) ( ) ( , ).m m ma a az D f z S D f z C f z CV

Theorem 4. 1, ,( , ) ( , ).m m

a aCV CV Proof. Similar proving as in Theorem 3. Now, with the help of Lemma 2, we obtain the following: Theorem 5. Let ( )h z be convex univalent in U with (0) 1h and ( ) 0.h z If f A satisfies

the condition

1,

1,

( ( )) '1 ( ) (0 1 ; )1 ( )

ma

ma

z D f zh z z U

D f z

Then

,

,

( ( )) '1 ( ) (0 1 ; )1 ( )

ma

ma

z D f zh z z U

D f z

Aabed Mohammed & Maslina Darus

30

Proof. Let

,

,

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

ma

ma

z D f zp z

D f z

where ( )p z is analytic function with (0) 1p . By using the identity , 1, ,( ( )) ' ( ) ( 1) ( )m m m

a a cz D f z D f z D f z we get

1,

,

( )(1 ) ( ) 1 .

( )

ma

ma

D f zp z

D f z

(13)

Taking logarithmic derivatives in both sides of (13) and multiplying by z , we have

1,

1,

( ( )) ''( ) 1( )(1 ) ( ) 1 1 ( )

ma

ma

z D f zzp zp zp z D f z

Applying Lemma 2, it follows that ,p h that is

,

,

( ( )) '1 ( ), ( ).1 ( )

ma

ma

z D f zh z z U

D f z

Theorem 6. Let ( )h z be convex univalent in U with (0) 1h and ( ) 0.h z If a function

f A satisfies the condition

,

,

( ( )) '1 ( ) (0 1; ),1 ( )

ma

ma

z D f zh z z U

D f z

Then

,

,

( ( )) '1 ( ) (0 1; ),1 ( )

ma

ma

z D zh z z U

D z

where be the integral operator introduced by Bernardi(1969) and defined by

1

0

1( ) ( ) ( 1).z

cc

cz t f t dt cz

(14)

Inclusion properties for certain subclasses of analytic functions associated with a multiplier transformation

31

Proof. Let

,

,

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

ma

ma

z D zp z

D z

where ( ) .p z p From (14), we have , , ,( ( )) ' ( 1) ( ) ( ).m m m

a a az D z c D f z cD z (15) Then by using (15), we get

,

,

( 1) ( )(1 ) ( ) .

( )

ma

ma

c D f zp z c

D z

(16)

Taking logarithmic derivatives in both sides of (16), we obtain

,

,

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

ma

ma

z D f zzp zp zc p z D f z

Finally, by using Lemma 2, we obtain that

,

,

( ( )) '1 ( ) (0 1; ).1 ( )

ma

ma

z D zh z z U

D z

Acknowledgement The work here is partially supported by UKM-ST-06-FRGS0107-2009.

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Mocanu P.T. 1989. Alpha-convex integral operators and strongly starlike functions. Studia Univ. Babes-Bolyai Mathematica 34: 18–24.

Nunokawa M. 1993. On the order of strongly starlikeness of strongly convex functions. Proc. Japan Acad. Ser. A Math. Sci. 69: 234–237.

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Salagean G.S. 1983. Subclasses of univalent functions. Lecture notes in Math. (Springer-Verlag) 1013: 362-372. School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia 43600 UKM Bangi Selangor D.E. MALAYSIA E-mail: aabedukm@yahoo.com, maslina@ukm.my*

* Corresponding author