Some New Subclasses Of Bi Univalent Functions Defined By Convolution Associated With Linear Differential Operator

103

Some New Subclasses Of Bi-Univalent Functions

Defined By Convolution Associated With Linear

Differential Operator

M.Thirucheran

1

, T. Stalin

2

Associate Professor and Head, Post Graduate and Research Department of Mathematics1, Assistant Professor, Department of Mathematics2,

LN Government College, University of Madras, Chennai, India1. Gojan School of Business and Technology, Anna University, Chennai, India2.

Email: [email protected], [email protected]

Abstract:The main object of this paper is investigating a new subclass of bi-univalent function in the open unit disk U which is defined by convolution of Al-Oboudi Differential Operator. And obtained the initial two Taylor -McLaurin co-efficient a₂ and a₃ for the subclass ,,,

,

b n m

r

S of Bi-Univalent function.

2010 Mathematics Subject classification:30C45,30C55 and 30C80.

Keywords:Analytic functions, univalent functions, bivalent functions, convolution and Al-Oboudi differential operator.

1. INTRODUCTION AND DEFINITIONS

Definition 1.1

Let A denote the class of functions f

normalized by







 

2

) (

j j jz

a z z

f (1.1)

Which are analytic in the open unit disc

} 1 : {z

U C z  .

Forf A, Al-Oboudi [1] introduces the

following operator.

) ( ) (

0

z f z f

D  ,

0 ), ( ) ( ) ( ) 1 ( )

(      

f z



f z



zf z Df z



D

,.... 3 , 2 , 1 )),

( ( )

(_z _{D D} 1_{f z n}_N

f

Dn n



1 ( 1)  ,  0. )

( 0

2

   

  

 _



N N n z a j z z f D

j

j j n

n  (1.2)

If



1_{, then we get Salagean [7]}

differential operator.

Koebe One -Quarter Theorem [5]

The range of every function of class A

contains the disk of radius

   



 _

4 1 : w

w .

It is well known that every function f Ahas an

inverse 1

f defined by f1(f(z))z_and w

w f

f( 1( ))

, .

4 1 ) ( );

( ₀

0 

  



 _{ }

f r f r w

For this inverse function 1

f , we have: ...

) 5

5 (

) 2 ( )

( : ) (

4 4 3 2 3 2

3 3 2 2 2 2 1

 

 

   

 

w a a a a

w a a w a w w f w g

(1.3).

Let j

n jz

r z z

r







 

2

)

( , (r_j 0)&





 

 



2

, , , ,

) (

j

b n m

r j jz S a z z

f 

Then the Hadamard product (f*r)defined by if

and only if





 

 



2

, , , ,

) ( ) * (

j

b n m

r j j ja z S r

z z r

f 

Definition 1.2

If both the function f and its inverse function 1

f are univalent in U , then the function

f is called bi-univalent.

For example, 

    

  



 z

z z

z z

1 1 l o g 2 1 , ) 1 l o g ( ,

1 ,

and so on.

However, the familiar Koebe function is not a bi-univalent function.

For example, ₂

2

1 ,

2 z

z z z



 , and so on.

Let the class of bi-univalent function first investigated by Levin[8]and found that

Afterward, Brannan and Clunie [2] conjectured

that a2  2.

Later, Brannan and Taha [3] introduced the new subclass of bi-univalent function of the class  like the familiar subclasses _S*()_{and C(}_{) of} starlike and convex functions of .(01)

,respectively. If a function f A is in the class )

(

* 



S of strongly bi-starlike function of order

) 1 0 (

. 





104

satisfied f(z)and

2 ) (

) (

arg _ 

     

z f

z f z

,

) (zU and

2 ) (

) (

arg _ 

    

w g

z wg

,(wU)

where the function g is the extension of 1

f to U

.Similarly, a function f A is in the class

) (





C of strongly bi-convex function of order

) 1 0 (

. 







if each of the following conditions

satisfied f(z)and

2 ) (

) ( 1

arg _ 

  

 

  

z f

z f z

,

) (zU and

2 ) (

) ( 1

arg _ 

  

 

  

w g

z g w

, (wU). Where the function g is the extension of 1

f to

U . For each function *()



S and C(



). They

found nonsharp estimates on the first two Taylor -McLaurin co-efficient a₂ and a₃ .

Recently, Qing-Hua Xu, Ying-Chun Gui and H.M. Srivastava [9], B.A. Frasin and M.K. Aouf [6], Seker.B [11], and R.M. El-Ashwah [10] investigated some subclasses of bi-univalent function and obtained non-sharp estimates on the first two co-efficient.

By motivated this study we introduce and obtained the initial two co-efficient a2 and a3 for

the subclass ,,, ,

b n m

r S . Definition: 1.3

A function f(z) given by (1.1) is said to

be in the class f  ,,, ( )

,





b n m

r S

) 1 0 , , , (

, mnN0 mn 



 . if the following

conditions are satisfied: f(z),r(z)Aand 2

1 ) )( * (

) )( * ( 1 1

arg _ 

  

  

   

 

 

z r f D

z r f D

b n

m

, (zU)

(1.4)

and

2 1 ) )( * (

) )( * ( 1 1

arg _ 

  

  

   

 

 

w r g D

w r g D

b n

m

,

)

(wU (1.5)

Where the function g is given by (1.3).

Definition: 1.4

A function f(z) given by (1.1) is said to

be in the class f ,,, ( )

,





b n m

r S

) 1 0 , , , (

, mnN0 mn 



 , if the following

conditions are satisfied: f(z),r(z)Aand



     

  

   

 

 

 1

) )( * (

) )( * ( 1 1

z r f D

z r f D

b n

m

,(zU) (1.6)

& _

  

  

   

 

 

 1

) * (

) )( * ( 1 1

r g D

w r g D

b n

m

,(wU)(1.7) Where the function g is given by (1.3).

Definition: 1.5

A function h(z),p(z):UC satisfy the conditions min







h(z)

 

,p(z)





0 , zU and h(0)p(0)1.

For a function

f



,,, ( ( ), ( ))

, h z p z

Smnb r





defined by (1.1),(m,nN0,mn,0



1). if

the following conditions are satisfied: f,rAand )

( 1 ) )( * (

) )( * ( 1

1 h z

z r f D

z r f D

b n

m

     

  

   

 



 ,(zU) (1.8)

& 1 ( )

) )( * (

) )( * ( 1

1 p z

w r g D

w r g D

b n

m

     

  

   

 



 ,(wU)

(1.9)

Where the function g is given by (1.3).

Remarks

(i). ,,1,1( )

1 ,



n m

S =Hm,n(),( Seker.B [11])

(ii). 1,0,1,1( )

1 ,





S = *()



S , (Brannan and Taha [3])

(iii). 2,1,1,1( )

1 ,





S =C(), (Brannan and Taha [3])

2. MAIN RESULT

To derive our main results, we should recall the following lemma [4].

Lemma 2.1

Let hPthe family of all functions h

analytic in U for which Re



h(z)0



and have the

form ( ) 1 3 ...

3 2 2

1   



 pz p z p z

z

h for zU

.Then pn 2, for each

n

.

Theorem 2.2

Let the function f(z) given by (1.1) be in the bi-univalent function class _Sm,n,b,(_h,_p)



.Then











  

 

   

   

n n

m

n m

r

b a

2 2

2

) 1 ( ) 1 (

) 2 1 ( ) 2 1 ( 2

2

 

 

(2.1)

and











   

 

    

 

   

   

n m

n m

b b

r a

) 2 1 ( ) 2 1 (

2 ) 1 ( ) 1 (

4

1 2

2

3 3

 



 (2.2)

Proof

Let consider the function hand p

Satisfying the conditions of the definition (1.5)

with the form

U z z

h z h z h z

h( )1   3... , 

3 2 2

1 &

U w w

p w p w p w

p( )1   3... , 

3 2 2 1

105

Since

,

,,,

(

,

)

,

h

p

S

r

f

mnb

r  



,then ) ( 1 ) )( * ( ) )( * ( 1

1 h z

z r f D z r f D b n m                

 , (zU) (2.3)

and ) ( 1 ) )( * ( ) )( * ( 1

1 p z

w r g D w r g D b n m                

 ,(wU) (2.4)

respectively.

By equating the coefficient of (2.3 and (2.4), we get



(1 ) (1 )



a2r2 bh1

n

m  







(2.5)









2

2 2 2 2 2 3 3 ) 1 ( ) 1 ( ) 2 1 ( ) 2 1 ( bh r a r a n n m n m          _    (2.6)



(1 ) (1 )



a2r2 bp1

n

m  









(2.7)

and













2

2 2 2 2 2 3 3 ) 1 ( ) 1 ( ) 2 1 ( ) 2 1 ( 2 ) 2 1 ( ) 2 1 ( bp r a r a n n m n m n m                     _      (2.8) From (2.5)&(2.7), 1 1 p

h  (2.9)

And



(1 ) (1 )



( )

2 2 1 2 1 2 2 2 2 2 2 p h b r a n

m   

  (2.10)





2

2 2 2 1 2 1 2 2 2 ) 1 ( ) 1 ( 2 ) ( n m r p h b a       

 (2.11)

From (2.6)&(2.8),









( ) ) 1 ( ) 1 ( ) 2 1 ( ) 2 1 (

2 2 2

2 2 2 2

2n a r bh p n m n m                









(2.12)









               

n n

m n m r p h b a 2 2 2 2 2 2 2 ) 1 ( ) 1 ( ) 2 1 ( ) 2 1 ( 2 ) (   

 (2.13)







(1 2 ) (1 2 )



( )

2 ) 2 1 ( ) 2 1 ( 2 2 2 2 2 2 2 3 3 p h b r a r a n m n m              (2.14)









                        n m n m p h b p h b r a ) 2 1 ( ) 2 1 ( 2 ) ( ) 1 ( ) 1 ( 2 ) ( 1 2 2 2 2 1 2 1 2 3 3









(2.15)

From Lemma (2.1),(2.11),(2.13) and (2.15), we get









             

n n

m n m r b a 2 2 2 ) 1 ( ) 1 ( ) 2 1 ( ) 2 1 ( 2 2    

and









                      n m n m b b r a ) 2 1 ( ) 2 1 ( 2 ) 1 ( ) 1 ( 4 1 2 2 3 3    

This completes the theorem (2.2).

Theorem 2.3

Let the function f,r given by (1.1) be in the bi-univalent function class ,,, ( )

,



 b n m r

S .Then









             

n n

m n m r b a 2 2 2 ) 1 ( ) 1 ( ) 2 1 ( ) 2 1 ( 2 2       (2.16)

and









                      n m n m b b r a ) 2 1 ( ) 2 1 ( 2 ) 1 ( ) 1 ( 4 1 2 2 2 3 3       (2.17) Proof

Let consider the function

h

and

p

Satisfying the conditions of the definition (1.5) with the form

U z z h z h z h z

h( )1   3... , 

3 2 2 1 and U w w p w p w p w

p( )1   3... , 

3 2 2 1

respectively.

Since

,

,,,

(

)

,



 b n m r

S

r

f



 ,then

 

 ) ( 1 ) )( * ( ) )( * ( 1

1 h z

z r f D z r f D b n m                

 ,(zU)

(2.18)

and 1



( )





) )( * ( ) )( * ( 1

1 p z

w r g D w r g D b n m                

 ,(wU)

(2.19) respectively.

From Lemma (2.1),(2.11),(2.13) and (2.15), we get









             

n n

m n m r b a 2 2 2 ) 1 ( ) 1 ( ) 2 1 ( ) 2 1 ( 2 2       and









                      n m n m b b r a ) 2 1 ( ) 2 1 ( 2 ) 1 ( ) 1 ( 4 1 2 2 2 3 3      

This completes the theorem (2.3).

Theorem 2.4

Let the function f,r given by (1.1) be in the

bi-univalent function class ,,, ( )

,



 b n m r

106











  

 

   

   

 

n n

m

n m

r

b a

2 2

2

) 1 ( ) 1 (

) 2 1 ( ) 2 1 ( ) 1 ( 2

) 1 ( 2

 

 





(2.20)

and











   

 

    

 

  

 

  





n m

n m

b b r

a

) 2 1 ( ) 2 1 (

) 1 ( 2

) 1 ( ) 1 (

) 1 ( 4

1 2

2 2

3 3

 

  



(2.21)

Proof

Let consider the function

h

and

p

Satisfying the conditions of the definition (1.5) with the form

U z z

h z h z h z

h( )1   3... , 

3 2 2

1 and

U w w

p w p w p w

p( )1   3... , 

3 2 2 1

respectively.

Since

,

,,,

(

)

,



 b n m

r

S

r

f



 ,then

) ( ) 1 ( 1

) )( * (

) )( * ( 1

1 h z

z r f D

z r f D

b n

m

         

  

   

 



 ,

)

(zU (2.22)

and 1 (1 ) ( )

) )( * (

) )( * ( 1

1 p z

w r g D

w r g D

b n

m

        

  

   

 





, (wU) (2.23)

respectively.

From Lemma (2.1),(2.11),(2.13) and (2.15), we get











  

 

   

   

 

n n

m

n m

r

b a

2 2

2

) 1 ( ) 1 (

) 2 1 ( ) 2 1 ( ) 1 ( 2

) 1 ( 2

 

 





and











   

 

    

 

  

 

  





n m

n m

b b r

a

) 2 1 ( ) 2 1 (

) 1 ( 2

) 1 ( ) 1 (

) 1 ( 4

1 2

2 2

3 3

 

  



Letting



1in theorem (2.2), (2.3) and (2.4), we obtain the following corollaries.

Corollary2.5

Let the function f(z) given by (1.1) be in the bi-univalent function class ,,,1( , )

1 , h p

Smnb



.Then a2





_{m n}

 

b _{m n 2n}





) 2 ( ) 2 ( ) 3 ( ) 3 (

2

 



 _

(2.24)

and



m n

 

m n



b b

a

) 3 ( ) 3 (

2

) 2 ( ) 2 (

4

2 2 3

  



(2.25)

Corollary2.6

Let the function f(z) given by (1.1) be in the bi-univalent function class ,,,1

(

)

1

,



b n m

S

 .Then

(using definition (1.3))



 





m n m n n



b

a_{2 2}

) 2 ( ) 2 ( ) 3 ( ) 3 (

2

 



  _ (2.26)

and



m n

 

m n



b b

a

) 3 ( ) 3 (

2

) 2 ( ) 2 (

4

2 2 2 3

  

  

(2.27)

Corollary2.7

Let the function f(z) given by (1.1) be in the bi-univalent function class ,,,1

(

)

1

,



b n m

S

 .Then

(using definition (1.4))



 





m n m n n



b

a2 ₂

) 2 ( ) 2 ( ) 3 ( ) 3 (

) 1 ( 2

 





  _

(2.28)

and



m n

 

m n



b b

a

) 3 ( ) 3 (

) 1 ( 2

) 2 ( ) 2 (

) 1 ( 4

2 2 2 3

  

 

  

(2.29)

Letting



1&b1in theorem (2.2), (2.3) and (2.4), we obtain the following corollaries Seker [6].

Corollary2.8

Let the function f(z) given by (1.1) be in the bi-univalent function class ,,1,1( , )

1 , h p

Smn



.Then a2





_{m n}

 

_{m n 2n}





) 2 ( ) 2 ( ) 3 ( ) 3 (

2

 



 _

(2.30)

and



m n

 

m n



a

) 3 ( ) 3 (

2

) 2 ( ) 2 (

4

2 3

  

 (2.31)

Corollary2.9

Let the function f(z) given by (1.1) be in the bi-univalent function class ,,1,1

(

)

1

,



n m

S

 .Then

(using definition (1.3))



 





m n m n n



a2 ₂

) 2 ( ) 2 ( ) 3 ( ) 3 (

2

 







_ (2.32)

and



m n

 

m n



a

) 3 ( ) 3 (

2

) 2 ( ) 2 (

4

2 2 3

  

   (2.33)

Corollary2.10

Let the function f(z) given by (1.1) be in the bi-univalent function class ,,1,1

(

)

1

,



n m

S

 .Then

(using definition (1.4))



 





m n m n n



a2 ₂

) 2 ( ) 2 ( ) 3 ( ) 3 (

) 1 ( 2

 









_ (2.34)

and



m n

 

m n



a

) 3 ( ) 3 (

) 1 ( 2

) 2 ( ) 2 (

) 1 ( 4

2 2 3

   



107

ACKNOWLEDGMENT

The authors thank the referees for their valuable suggestions which lead to the improvement of this paper.

REFERENCES

[1] Al-Oboudi.(2004), On univalent functions defined by a generalized Salagean operator, Int. Journal of Mathematics and Mathematical Science, Vol.27, 1429--1436.

[2] Brannan D.A, Clunie (1980), Aspects of contemporary complex analysis, Academic press New York, London.

[3] Brannan D.A, Taha T.S (1986), On some classes of bi-valent functions, Studia Univ Babes-Bolyoni Math, Vol.31, 70-77.

[4] Ch. Pommerenke (1975), Univalent functions, Vandenhoeck and Rupercht, Gottingen. [5] Duren P.L (1983), Univalent functions,

Springer-Verlag, New York.

[6] Frasin B.A. and M.K. Aouf (2011), New Subclasses of bi-univalent functions, Applied Mathematics letters, Vol.24, 1569-1573. [7] G. St. Salagean (1983), Subclasses of

univalent functions, Lecture Notes in Math. Vol.1013, Springer-Verlag, Berlin, 362-372. [8] Lewin M (1967). On a coefficient problem for bi-valent functions, Proc Amer Math Soc, Vol.18, 63-68.

[9] Qing-Hua Xu, Ying-Chun Gui and H.M.

Srivastava (2012), Coefficient estimates for a certain subclass of analytic and bi-univalent functions, Applied Mathematics letters, Vol.25, 990-994.

[10]R. M.El-Ashwah, subclass of bi-univalent function defined by convolution, J. Egypt. Math. Soc., 2014, 22, 348-351.