Aijrstem15 518

American International Journal of Research in Science, Technology, Engineering & Mathematics

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ISSN (Print): 2328-3491, ISSN (Online): 2328-3580, ISSN (CD-ROM): 23283629 AIJRSTEM is a refereed, indexed, peer-reviewed, multidisciplinary and open access journal published by International Association of Scientific Innovation and Research (IASIR), USA (An Association Unifying the Sciences, Engineering, and Applied Research)

New Classes Containing Ruscheweyh Derivative and a New Generalized Multiplier Differential Operator S R Swamy Department of Computer Science and Engineering, R V College of Engineering, Mysore Road, Bangalore-560 059, India Abstract: New subclasses of analytic functions, containing the linear operator obtained as a linear combination of Ruscheweyh derivative and a new generalized multiplier transformation have been considered. Sharp results concerning coefficients, distortion theorems of functions belonging to these classes are determined. Furthermore, Functions with negative coefficients belonging to these classes are also discussed.

2010 Mathematics Subject Classification: 30C45, 30A20, 34A40.

Key words and phrases: Analytic function, differential operator, multiplier transformation.

I.

Introduction.

U  {z  C : z  1}. Let H (U ) be the space of holomorphic functions in U . Let A denote the family of functions in H (U ) of the form Denote by

U

the open unit disc of the complex plane, 

f ( z)  z   ak z k .

(1.1)

k 2

The author has recently introduced the following new generalized multiplier differential operator in [12].

Definition 1.1 Let m  N 0  N  {0},   0,  a real number such that     0 . Then for f  A , a new generalized multiplier operator

I m,  was defined by

I 0 ,  f ( z )  f ( z ) , I 1 ,  f ( z ) 

f ( z )  zf ' ( z ) , …, I m,  f ( z )  I  ,  ( I m,1 f ( z )) .  

Remark 1.2 Observe that for f (z ) given by (1.1), we have 

I m,  f ( z )  z   Ak ( ,  , m)a k z k ,

(1.2)

k 2

where m

   k   .

Ak ( ,  , m)       m m We note that: i) I1 , , 0 f ( z )  D f ( z ),   0

(1.3) (See

F.

M.

Al-Oboudi

[1]),

ii)

I lm1 , ,0 f ( z )  I lm, f ( z ), l  1,   0 (See A. Catas [3] and he has considered for l  0 ) and iii) I m,1 f ( z )  I m f ( z ),   1 (See Cho and Srivastava [4]) and Cho and Kim [5]). D1m f ( z ) was introduced

by Salagean [8] and was considered for Definition

1.3

([7])

m  0 in [2].

For m  N 0 , f  A ,

the

operator

R m is

defined

by

Rm : A  A ,

R 0 f ( z )  f ( z ) , R 1 f ( z )  zf ' ( z ) ,…, (m  1) R m 1 f ( z )  z ( R m f ( z )) '  mR m f ( z ), z U .

Remark 1.4 If

f ( z )  z  k 2 a k z k  A , then R m f ( z )  z  k  2 Bk (m)a k z k , z U , where 

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

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Bk (m) 

(m  k  1)! . m! (k  1)!

(1.4)

The author has introduced the following operator in [13].

Definition 1.5 Let f  A, m  N 0  N  {0},   0,   [0,1),   0, 

    0.

Denote

by

RI

m  ,  ,,

,

the

operator

a real number such that

given

RI m,  , : A  A,

by

RI m,  , f ( z )  (1   ) R m f ( z )  I m,  f ( z ), z  U .

The operator was studied also in [10] and [11]. Clearly RI  ,  ,, 0  R m

Remark 1.6 If

m

and RI  ,  ,,1  I  ,  . m

m

f ( z )  z  k  2 a k z k , then from (1.2) and Remark 1.4, we have 

RI m,  , f ( z )  z  k  2 {(1   ) Bk (m)  Ak ( ,  , m)}a k z k , z U , 

where Ak ( ,  , m) and B k (m) are as defined in (1.3) and (1.4), respectively. m

We introduce new classes as below by making use of the generalized operator RI  ,  , .

Definition

1.7

f  A, m  N 0  N  {0},   0,   [0,1),   0, 

Let

that     0 . Then f (z ) is in the class

that     0 . Then f (z ) is in the class

 m ,  , (  ) if and only if

  z ( RI m,  , f ( z )) ' Re    RI m,  , f ( z ) 

 z ( RI m,  , f ( z )) '' 1   ( RI m,  , f ( z )) ' 

1.9

that     0 . Then f (z ) is in the class

number

such



m  ,  ,

 2 z[ z ( RI m,  , f ( z )) ' ] ' Re  ( RI m f ( z )) '  ,  , 

(1.5) a

real

number

such

    , z U .  

f  A, m  N 0  N  {0},   0,   [0,1),   0, 

Let

real

m,  , (  ) if and only if

 [ z 2 ( RI m,  , f ( z )) ' ] '     , z U .

Re  2 RI m f ( z )   ,  ,   Definition 1.8 Let f  A, m  N 0  N  {0},   0,   [0,1),   0, 

Definition

a

(1.6)

a

real

number

such

(  ) if and only if

    , z U .  

(1.7)

In section 2 we study the characterization properties for the function f  A to belong to the classes  ,  , m

,

( )

 m ,  , (  ) and m,  , (  ) by obtaining the coefficient bounds. Distortion theorems of functions belonging

to these classes are obtained in section 3. Analytic functions with negative coefficients belonging to the above classes are considered in section 4.

II. Gene1ral Properties.

In this section we study the characterization properties following the paper of M. Darus and R. Ibrahim [6].

Throughout this paper, unless otherwise mentioned we shall assume that Ak ( ,  , m) and B k (m) are as defined in (1.3) and (1.4) respectively,.

Theorem 2.1 Let f  A, m  N 0  N  {0},   0,   [0,1),   0, 

a real number such that

    0 . i) If



 (k

2

k 2

then

 k  2  ){(1   ) B k (m)  Ak ( ,  , m)} a k  2(1   ),

(2.1)

f ( z )  m,  , (  ) .

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ii) If 

 {(2k

2

k 2

then

 (k  1)  )}{(1   ) B k (m)  Ak ( ,  , m)} a k  2(1   ),

(2.2)

f ( z )   m ,  , (  ) .

iii) If 

 {(k k 2

then

2

  )}{(1   ) B k (m)  Ak ( ,  , m)} a k  1   ,

(2.3)

f ( z )  m,  , (  ) . The results (2.1), (2.2) and (2.3) are sharp.

Proof. i) It suffices to show that the values of

[ z 2 ( RI m,  , f ( z )) ' ] ' / 2 RI m,  , f ( z ) lie in a circle centred at

1 with radius 1   . We have

[ z 2 ( RI m,  , f ( z )) ' ] ' 2 RI m,  , f ( z )

1 

[ z 2 ( RI m,  , f ( z )) ' ] '  2 RI m,  , f ( z ) 2 RI m,  , f ( z )

 

 k 2

(k 2  k  2){(1   ) B k (m)  Ak ( ,  , m)} a k

2[1  k  2 {(1   ) B k (m)  Ak ( ,  , m)} a k ] 

.

The last expression is bounded above by 1   if



 k 2

(k 2  k  2){(1   ) B k (m)  Ak ( ,  , m)} a k 





2(1   ) 1  k  2 {(1   ) Bk (m)  Ak ( ,  , m)} a k , which is equivalent to (2.1). Hence



[ z 2 ( RI m,  , f ( z )) ' ] ' 2 RI m,  , f ( z )

 1  1   , and the theorem is proved.

The proof of the second and third part of the theorem is similar and so omitted.

The assertions (2.1),(2.2) and (2.3) are sharp and extremal functions are given by

2(1   ) z k , z U , k  2 ( k  k  2  ){(1   ) Bk ( m)  Ak ( ,  , m)}  2(1   ) f ( z)  z   z k , z U , 2 k  2 {( 2k  ( k  1)  )}{(1   ) B k ( m)  Ak ( ,  , m)} 

f ( z)  z  

and

2

1  z k , z U ( k   ){( 1   ) B ( m )   A (  ,  , m )} k 2 k k 

f ( z)  z  

2

respectively.

Theorem 2.2 Let the hypothesis of the Theorem 2.1 i) ,ii) and iii) be satisfied. Then

i)

ii)

iii)

2(1   ) , k  2. (k  k  2  ){(1   ) B k (m)  Ak ( ,  , m)} 2(1   ) ak  , k  2.

2 {( 2k  (k  1)  }{(1   ) B k (m)  Ak ( ,  , m)} 1  ak  2 , k  2. (k   ){(1   ) Bk (m)  Ak ( ,  , m)} ak 

2

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The following inclusion results can be proved using Theorem 2.1.

Theorem 2.3 Let 0  1   2  1 . Then i)

m,  , (  2 )  m,  , (  1 ) , ii)  m ,  , (  2 )   m ,  , (  1 ) and iii)  m,  , (  2 )  m,  , ( 1 ) .

III.

Distortion theorems

Theorem 3.1 Let f  A, m  N 0  N  {0},   0,   [0,1),   0, 

    0 .i) If



 [( k

2

k 2

z 

ii) If

 [( 2k

2

k 2



 (k

2

k 2

 k  2  )]{(1   ) B k ( m)  Ak ( ,  , m)} a k  2(1   ), then

2(1   ) z 5  2

2

 RI m,  , f ( z )  z 

2(1   ) 2 z , z U .

5  2

 (k  1)  )]{(1   ) B k (m)  Ak ( ,  , m)} a k  2(1   ), then z

iii) ) If

a real number such that

2(1   ) z 8  3

2

 RI m,  , f ( z )  z 

2(1   ) 2 z , z U .

8  3

  ){(1   ) B k ( m)  Ak ( ,  , m)} a k  1   , then

z

1  2 1  2 z  RI m,  , f ( z )  z  z , z U .

4 4 

Proof i) Note that (5  2  )



 (1   ) B k 2

k

(m)  Ak ( ,  , m) a k 



 (k

2

k 2



by Theorem 2.1. Thus

 (1   ) B k 2

k

 k  2  )(1   ) B k (m)  Ak ( ,  , m) a k  2(1   ) ,

(m)  Ak ( ,  , m) a k 

2(1   ) .Hence we obtain 5  2



RI m,  , f ( z )  z   (1   ) B k ( m)  Ak ( ,  , m) a k z

k

k 2

 zz



 (1   ) B

2

k 2

 z

k

( m)  Ak ( ,  , m) a k

2(1   ) 2 z .

5  2

Similarly 

RI m,  , f ( z )  z   (1   ) B k ( m)  Ak ( ,  , m) a k z

k

k 2

 zz

2



 (1   ) B k 2

 z

k

(m)  Ak ( ,  , m) a k

2(1   ) 2 z .

5  2

This completes the proof of (i). (ii) and iii) can be proved on similar lines.

Theorem

3.2

f  A, m  N 0  N  {0},   0,   [0,1),   0, 

Let

that     0 . i) If



 [( k k 2

2

a

real

number

such

 k  2  )]{(1   ) B k ( m)  Ak ( ,  , m)} a k  2(1   ), then

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f ( z)  z 

2(1   ) 2 z , z U (5  2  ){(m  1)(1   )  A2 ( ,  , m)}

f ( z)  z 

2(1   ) 2 z , z U .

(5  2  ){(m  1)(1   )  A2 ( ,  , m)}

and



ii) If

 [( 2k

2

k 2

 (k  1)  )]{(1   ) B k (m)  Ak ( ,  , m)} a k  2(1   ), then

f ( z)  z 

2(1   ) 2 z , z U (8  3 ){(m  1)(1   )  A2 ( ,  , m)}

f ( z)  z 

2(1   ) 2 z , z U .

(8  3 ){(m  1)(1   )  A2 ( ,  , m)}

and



iii) If

 [( k k 2

2

  )]{(1   ) B k ( m)  Ak ( ,  , m)} a k  1   , then f ( z)  z 

1  2 z , z U (4   ){(m  1)(1   )  A2 ( ,  , m)}

f ( z)  z 

1  2 z , z U .

(4   ){(m  1)(1   )  A2 ( ,  , m)}

and

Proof i) In virtue of Theorem 2.1, we have 

(5  2  ){(m  1)(1   )  A2 ( ,  , m)} a k  k 2



 (k

2

k 2



Thus

a k 2

k



 k  2  )(1   ) B k (m)  Ak ( ,  , m) a k  2(1   ) .

2(1   ) .So we get (5  2  ){(m  1)(1   )  A2 ( ,  ,  )}  2(1   ) 2 2 f ( z)  z  z  a k  z  z .

(5  2  ){(m  1)(1   )  A2 ( ,  , m)} k 2

On the other hand

f ( z)  z  z

2



a k 2

k

 z

2(1   ) 2 z .

(5  2  ){(m  1)(1   )  A2 ( ,  , m)}

This completes the proof. The proof of (ii) and iii) are similar.

IV.

Functions with negative coefficients 

k Let T denote the subclass of A consisting of functions of the form f ( z )  z   a k z , a k  0 . k 2

m  ,  ,

We denote by T

(  ) , T  ,  , (  ) and T m

m  ,  ,,

(  ) , the classes of functions f ( z )  T satisfying

(1.5),(1.6) and (1.7) respectively. We study the coefficient estimates, distortion theorems and other properties of these classes, following the paper of H. Silverman [9].

For functions in T , the converses of Theorem 2.1 is also true.

Theorem 4.1 i) A function f ( z )  T is in

Tm,  , (  ) if and only if

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 (k

2

k 2

 k  2  )(1   ) B k (m)  Ak ( ,  , m)a k  2(1   ).

ii) A function f ( z )  T is in 

 ( 2k k 2

2

 (k k 2

2

T m ,  , (  ) if and only if

 (k  1)  )(1   ) B k (m)  Ak ( ,  , m)a k  2(1   ). .

iii) A function f ( z )  T is in 

(4.1)

(4.2)

Tm,  , (  ) if and only if

  )(1   ) B k ( m)  Ak ( ,  , m)a k  1   .

(4.3)

The results are sharp.

Proof i) In view of Theorem 2.1, it suffices to prove the only if part. Assume that

 2 z   {(1   ) B k (m)  Ak ( ,  , m)}k (k  1)a k z k  [ z 2 ( RI m,  , f ( z )) ' ] '  k 2   Re Re   k  2 RI m f ( z )   ,  ,    2 z  k  2 {(1   ) B k (m)  Ak ( ,  , m)}2a k z

  .  

(4.4)



letting z  1 through real values in (4.4), we obtain     2   {(1  ) B k (m)  Ak ( ,  , m)}k (k  1)a k  2  1   {(1   ) B k (m)  Ak ( ,  , m)}a k . k 2  k 2 

Hence we obtain (4.1), and the proof is complete. The proof of ii) and iii) are similar.

Finally, we note that assertions (4.1), (4.2) and (4.3) of Theorem 4.1 are sharp, extremal functions being

1  z k , z U , k  2 ( k  k  2  ){(1   ) B k ( m)  Ak ( ,  , m)} 

f ( z)  z  

2

1  z k , z U , k  2 {( 2k  ( k  1)  )}{(1   ) B k ( m)  Ak ( ,  , m)} 

f ( z)  z   and

2

1  z k , z U , 2 k  2 ( k   ){(1   ) B k ( m)  Ak ( ,  , m)} 

f ( z)  z   respectively.

Our coefficient bounds enable us to prove the following. f  A, m  N 0  N  {0},   0,   [0,1),   0,  Theorem 4.2 Let that     0 .i) If A function f ( z )  T is in

m  ,  ,

T

a

real

2(1   ) 2 z , z U (5  2  ){(m  1)(1   )  A2 ( ,  , m)}

f ( z)  z 

2(1   ) 2 z , z U .

(5  2  ){(m  1)(1   )  A2 ( ,  , m)}

T m ,  , (  ) then

ii) If A function f ( z )  T is in

f ( z)  z 

2(1   ) 2 z , z U (8  3 ){(m  1)(1   )  A2 ( ,  , m)}

f ( z)  z 

2(1   ) 2 z , z U .

(8  3 ){(m  1)(1   )  A2 ( ,  , m)}

and

such

(  ) then

f ( z)  z  and

number

iii) If A function f ( z )  T is in

Tm,  , (  ) then

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S R Swamy et al., American International Journal of Research in Science, Technology, Engineering & Mathematics, 11(1), June-August, 2015, pp. 65-71

f ( z)  z 

1  2 z , z U (4   ){(m  1)(1   )  A2 ( ,  , m)}

f ( z)  z 

1  2 z , z U .

(4   ){(m  1)(1   )  A2 ( ,  , m)}

and

Remark 4.3 The bounds of i) and ii) of Theorem 4.2 are sharp since the equalities are attained for the functions

2(1   ) z 2 ( z  r ) , (5  2  ){(m  1)(1   )  A2 ( ,  , m)} 1  f ( z)  z  z 2 ( z   r ) and (8  3 ){(m  1)(1   )  A2 ( ,  , m)} 1  f ( z)  z  z 2 ( z   r ) ,respectively.

(4   ){(m  1)(1   )  A2 ( ,  , m)}

f ( z)  z 

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