• No results found

 SomeNewSubclassesof Bi-Univalent Functions Definedby Al-Oboudi Differential Operator

N/A
N/A
Protected

Academic year: 2022

Share " SomeNewSubclassesof Bi-Univalent Functions Definedby Al-Oboudi Differential Operator"

Copied!
10
0
0

Loading.... (view fulltext now)

Full text

(1)

ISSN 2319-8133 (Online)

Some New Subclasses of Bi-Univalent Functions Defined by Al-Oboudi Differential Operator

T. Stalin

*1

, M. Thirucheran

2

and M. Vinoth Kumar

3

1

Assistant Professor, Department of Mathematics, Gojan School of Business and Technology, Redhills-600052.

Anna University, Chennai, INDIA.

2

Associate Professor and Head, Post Graduate and Research Department of Mathematics, LN Government College, Ponneri-601204, University of Madras, Chennai, INDIA.

3

Research Scholar, Post Graduate and Research Department of Mathematics, LN Government College, Ponneri-601204, University of Madras, Chennai, INDIA.

(Received on: April 3, 2019) 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 Al-Oboudi Differential Operator.

And obtained the initial two Taylor -McLaurin co-efficient

a

2 and

a

3 for the subclass

S

m,n,b,

( )

of Bi-Univalent function.

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

Keywords: Analytic functions, univalent functions, bivalent functions, 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 j

z a z z

f (1.1)

Which are analytic in the open unit disc U  {z  C : z  1 } .

(2)

For fA , Al-Oboudi

1

introduces the following operator.

) ( )

0

(

z f z f

D  ,

0 ),

( )

( )

( ) 1 ( )

(      

f z f z z f z D

f z D

,....

3 , 2 , 1 )),

( (

)

( zD D

1

f z nNf

D

n 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 : w 1

w .

It is well known that every function fA has an inverse f

1

defined by f

1

( f ( z ))  z

and f ( f

1

( w ))  w , .

4 ) 1 ( );

(

0

0

 

wr f r f

For this inverse function f

1

, we have:

...

) 5

5 ( ) 2

( )

( : )

( wf

1

wwa

2

w

2

a

22

a

3

w

3

a

23

a

2

a

3

a

4

w

4

g (1.3)

Definition 1.2

If both the function f and its inverse function f

1

are univalent in U , then the function f is called bi-univalent.

For example, 

 

 

  z

z z z

z

1 log 1 2 , 1 ) 1 log(

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 51

.

2

 1

a .

Afterward, Brannan and Clunie

2

conjectured that a

2

 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

) 1 0

(

. 

,respectively. If a function fA is in the class S

*

( ) of strongly bi-starlike

function of order . ( 0   1 ) if each of the following conditions satisfied

(3)

 ) (z

f and

2 )

( )

arg (   

 

  z f

z f

z , ( zU )

And

2 )

( )

arg (    

 

w g

z

wg , ( wU )

Where the function g is the extension of f

1

to U .

Similarly, a function fA is in the class C

( ) of strongly bi-convex function of order . ( 0   1 ) if each of the following conditions satisfied

 ) (z

f and

2 )

( ) 1 (

arg    

 

 

z f

z f

z , ( zU )

And

2 )

( ) 1 (

arg   

 

 

w g

z g

w , ( wU )

Where the function g is the extension of f

1

to U . For each function S

*

( ) and C

( ) . They found non-sharp estimates on the first two Taylor -McLaurin co-efficient a

2

and a

3

. Recently, Qing-Hua Xu, Ying-Chun Gui and H.M. Srivastava

9

, B.A. Frasin and M.K.

Aouf

6

, Seker.B

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 a

2

and a

3

for the subclass S

m,n,b,

( ) .

Definition: 1.3

A function f (z ) given by (1.1) is said to be in the class

f S

m,n,b,

( ) , ( mN , nN

0

, mn , 0   1 , bC    0 ,  0 if the following conditions are satisfied:

A z

f ( )  and

1 2 ) (

) ( 1 1

arg    

 

 

 

 

D f z z f D b

n

m

, ( zU ) (1.4)

And

1 2 ) (

) ( 1 1

arg    

 

 

 

 

D g w w g D

b

n

m

, ( wU ) (1.5)

(4)

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 S

m,n,b,

( ) , ( mN , nN

0

, mn , 0   1 , bC    0 ,  0 if the following conditions are satisfied:

A z

f ( )  and  

 

  

 

 

 1

) (

) ( 1 1

z f D

z f D b

n

m

, ( zU ) (1.6)

And

 

 

 

 

 

 1

) (

) ( 1 1

w g D

w g D b

n

m

, ( wU ) (1.7)

Where the function g is given by (1.3).

Definition: 1.5

A function h ( z ), p ( z ) : UC satisfy the conditions

   

h ( z ) , p ( z ) 0 , z U

min and h ( 0 )  p ( 0 )  1

For a function fA defined by (1.1) ,

we say that fS

m,n,b,

( ) , ( mN , nN

0

, mn , 01 , bC    0 , 0 if the following conditions are satisfied:

A z

f ( )  and 1 ( )

) (

) (

1 1 h z

z f D

z f D b

n

m

 

 

  

 

 

 , ( zU ) (1.8)

And

) ( ) 1

( ) (

1 1 p z

w g D

w g D b

n

m

 

 

  

 

 

 , ( wU ) (1.9)

Where the function g is given by (1.3).

Remarks

(i). S

m,n,1,1

( ) = H

m,n

( ) ,(Seker.B

10

)

(ii). S

1,0,1,1

( ) = S

*

( ) , (Brannan and Taha

3

)

(iii). S

2,1,1,1

( ) = C

( ) , (Brannan and Taha

3

)

(5)

2. MAIN RESULT

To derive our main results, we should recall the following lemma

4

. Lemma 2.1

Let hP the family of all functions h analytic in U for which Reh ( z ) 0  and have the form h ( z )  1  p

1

zp

2

z

2

p

3

z

3

 ... for zU .Then p

n

 2 , for each n . Theorem 2.2

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

,

,

(

,

,

h p

S

mnb

.Then

        



 

 

m

b

n m n

b

mn n

a

2 2

) 1 ( ) 1 ( ) 2 1 ( ) 2 1 ( , 2 ) 1 ( ) 1 ( min 2

(2.1)

and

   

   

   

 

 

 

 

 

 

n n m n

m n

m

n m

n m

b b

b b

a

) 2 1 ( ) 2 1 (

2 )

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

2

, ) 2 1 ( ) 2 1 (

2 )

1 ( ) 1 (

4 min

2 2

2

3

(2.2)

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 

1

2 2

3 3

 ... ,  and

U w w

p w p w p w

p ( )  1 

1

2 2

3 3

 ... ,  respectively.

Since fS

m,n,b,

( h , p ) ,then

) ( ) 1

( ) (

1 1 h z

z f D

z f D b

n

m

 

 

  

 

 

 , ( zU ) (2.3)

and

) ( ) 1

( ) (

1 1 p z

w g D

w g D b

n

m

  

 

 

 

 

 , ( wU ) (2.4)

respectively.

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

 ( 1  )

m

 ( 1  )

n

a

2

bh

1

(2.5)

(6)

   

2 2 2 2

3

( 1 ) ( 1 )

) 2 1 ( ) 2 1

( 

m

 

n

a  

mn

 

n

abh (2.6)

 ( 1  )

m

 ( 1  )

n

a

2

bp

1

(2.7)

and        

2

2 2 2

3

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

) 2 1 ( ) 2 1

( 

m

 

n

a  

m

 

n

 

m n

 

n

abp

(2.8)

From (2.5)&(2.7),

1

1

p

h   (2.9)

and 2( 1 ) ( 1 )(

12

)

2 1 2 2 2

2

a b h p

n

m

   

(2.10)

 

2

2 1 2 1 2 2

2

2 ( 1 ) ( 1 )

) (

n m

p h a b

 

 

 (2.11)

From (2.6)&(2.8),

   

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

2 

m

 

n

 

mn

 

2n

a

22

b h

2

p

2

(2.12)

   

m

b h

n

p

m n n

a

22 2 2 2

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

) (

     

 

(2.13)

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

2 

m

 

n

a

3

 

m

 

n

a

22

b h

2

p

2

(2.14)

b h

m

p

n

  b h

m

p

n

a 2 ( 1 2 ) ( 1 2 )

) (

) 1 ( ) 1 ( 2

)

(

2 2

2 2 1 2 1 2

3

 

 

 

 (2.15)

and

   

m

b h

n

p

m n n

  b h

m

p

n

a 2 ( 1 2 ) ( 1 2 )

) (

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

)

(

2 2

2 2

2

3

 

 

 

(2.16)

From Lemma (2.1),(2.11),(2.13),(2.15) and (2.16), we get

m

b

n

a ( 1 ) ( 1 ) 2

2

    ,

   

m n

b

m n n

a

2 2

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

2

     

 

,

m

b

n

 

m

b

n

a ( 1 2 ) ( 1 2 )

2 )

1 ( ) 1 (

4

2 2

3

   

  and

   

m n

b

m n n

 

m

b

n

a 2 ( 1 2 ) ( 1 2 )

2 )

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

2

3 2

   

 

(7)

This completes the theorem (2.2).

Theorem 2.3

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

m,n,b,

( ) . Then (using definition (1.3))

        



 

 

m

b

n m n

b

mn n

a

2 2

) 1 ( ) 1 ( ) 2 1 ( ) 2 1 ( , 2 ) 1 ( ) 1 ( min 2

(2.17)

and

   

   

   

 

 

 

 

 

 

n n m n

m n

m

n m

n m

b b

b b

a

) 2 1 ( ) 2 1 (

2 )

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

2

, ) 2 1 ( ) 2 1 (

2 )

1 ( ) 1 (

4 min

2 2

2 2

3

(2.18)

Proof

m

b

n

a ( 1 ) ( 1 ) 2

2

  ,

   

m n

b

m n n

a

2 2

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

2

 

,

m

b

n

 

m

b

n

a ( 1 2 ) ( 1 2 )

2 )

1 ( ) 1 (

4

2 2 2

3

 

  and

   

m n

b

m n n

 

m

b

n

a 2 ( 1 2 ) ( 1 2 )

2 )

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

2

3 2

 

 

This completes the theorem (2.3).

Theorem 2.4

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

m,n,b,

( ) . Then (using definition (1.4))

        



 

b

m

n m

b

n mn n

a

2 2

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

) 1 ( , 2

) 1 ( ) 1 (

) 1 ( min 2

(2.19)

and

(8)

   

   

   

 

 

 

 

 

n n m n

m n

m

n m

n m

b b

b b

a

) 2 1 ( ) 2 1 (

) 1 ( 2 )

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

) 1 ( 2

, ) 2 1 ( ) 2 1 (

) 1 ( 2 )

1 ( ) 1 (

) 1 ( 4 min

2 2

2 2

3

(2.20)

Proof

b

m n

a ( 1 ) ( 1 ) ) 1 ( 2

2

  ,

   

m

b

n m n n

a

2 2

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

) 1 ( 2

 

,

b

m n

  b

m n

a ( 1 2 ) ( 1 2 )

) 1 ( 2 )

1 ( ) 1 (

) 1 ( 4

2 2 2

3

 

 

and

   

m

b

n m n n

  b

m n

a 2 ( 1 2 ) ( 1 2 )

) 1 ( 2 )

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

) 1 ( 2

3 2

 

 

This completes the theorem (2.4).

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

Corollary 2.5

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

Sm,n,b,1(h,p)

. Then  

m

b

n

  

m

n

   b

mn

n

  

a

2 2

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

min 2 (2.21)

and

   

   

   

 

 

 

 

 

 

n n m n

m n

m

n m n

m

b b

b b

a

) 3 ( ) 3 (

2 )

2 ( ) 2 ( ) 3 ( ) 3 (

2

, ) 3 ( ) 3 (

2 )

2 ( ) 2 (

4 min

2 2

2

3

(2.22)

Corollary 2.6

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

m,n,b,1

( ) . Then (using definition (1.3))

        



 

m

b

n m n

b

mn n

a

2 2

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

min 2

(2.23)

(9)

and

   

   

   







 

 

 

n n m n

m n

m

n m n

m

b b

b b

a

) 3 ( ) 3 (

2 )

2 ( ) 2 ( ) 3 ( ) 3 (

2

, ) 3 ( ) 3 (

2 )

2 ( ) 2 (

4 min

2 2

2 2

3

(2.24)

Corollary 2.7

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

m,n,b,1

( ) . Then (using definition (1.4))

       







bmn m nb mn n

a2 2

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

) 1 ( , 2

) 2 ( ) 2 (

) 1 (

min 2

(2.25)

and

   

   

   







 

 

n n m n

m n m

n m n

m

b b

b b

a

) 3 ( ) 3 (

) 1 ( 2 )

2 ( ) 2 ( ) 3 ( ) 3 (

) 1 ( 2

, ) 3 ( ) 3 (

) 1 ( 2 )

2 ( ) 2 (

) 1 ( 4 min

2 2

2 2

3

(2.26)

Letting  1 & b  1 in theorem (2.2), (2.3) and (2.4), we obtain the following corollaries Seker

6

.

Corollary 2.8

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

Sm,n,1,1(h,p)

. Then a

2

min ( 2 )

m

2 ( 2 )

n

,   ( 3 )

m

( 3 )

n

  2 ( 2 )

mn

( 2 )

2n

  (2.27)

and

   

   

   

 

 

 

 

 

 

n n m n

m n

m

n m n

m

a

) 3 ( ) 3 (

2 )

2 ( ) 2 ( ) 3 ( ) 3 (

2

, ) 3 ( ) 3 (

2 )

2 ( ) 2 (

4 min

2 2

3

(2.28)

Corollary 2.9

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

m,n,1,1

( ) . Then (using definition (1.3))

       

 

m

n m n mn n

a

2 2

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

min 2

(2.29)

and

(10)

   

   

   







 

 

 

n n m n

m n m

n m n

m

a

) 3 ( ) 3 (

2 )

2 ( ) 2 ( ) 3 ( ) 3 (

2

, ) 3 ( ) 3 (

2 )

2 ( ) 2 (

4 min

2 2

2

3

(2.30)

Corollary 2.10

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

m,n,1,1

( ) . Then (using definition (1.4))

       



mn m n mn n

a2 2

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

) 1 ( , 2

) 2 ( ) 2 (

) 1 (

min 2

(2.31)

and

   

   

   







 

 

n n m n

m n m

n m n

m

a

) 3 ( ) 3 (

) 1 ( 2 )

2 ( ) 2 ( ) 3 ( ) 3 (

) 1 ( 2

, ) 3 ( ) 3 (

) 1 ( 2 )

2 ( ) 2 (

) 1 ( 4 min

2 2

2

3

(2.32)

ACKNOWLEDGMENT

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

REFERENCES

1. Al-Oboudi., On univalent functions defined by a generalized Salagean operator, Int.

Journal of Mathematics and Mathematical Science, Vol.27, 1429-1436 (2004).

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

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

4. Ch. Pommerenke, Univalent functions, Vandenhoeck and Rupercht, Gottingen (1975).

5. Duren P.L, Univalent functions, Springer-Verlag, New York (1983).

6. Frasin B.A. and M.K. Aouf, New Subclasses of bi-univalent functions, Applied Mathematics Letters, Vol.24, 1569-1573 (2011).

7. G. St. Salagean, Subclasses of univalent functions, Lecture Notes in Math. Vol.1013, Springer-Verlag, Berlin, 362-372 (1983).

8. Lewin M. On a coefficient problem for bi-valent functions, Proc Amer Math Soc, Vol.18, 63-68 (1967).

9. Qing-Hua Xu, Ying-Chun Gui and H.M. Srivastava, Coefficient estimates for a certain subclass of analytic and bi-univalent functions, Applied Mathematics Letters, Vol.25, 990-994 (2012).

10. Seker.B, On a new Subclasses of bi-univalent functions defined by using Salagean

operator, Turk J Math, Vol.42, 2891-2896 (2018).

References

Related documents

She emphasized the necessity to introduce an economic assessment, including efficiencies, into the approach of the European Commission’s to vertical agreements and the resulting

* A spectacular tour of the North Island of New Zealand, visiting Ayrlies Garden, Waiheke Island Gardens, Hamilton Gardens, Rotorua Government Gardens, Taranaki Garden.. Festival,

www.ijper.org Antitumor Activity of Methanolic Fractions Extracted From the Aerial Part of Algerian Hyoscyamus albus and apoptotic cell aspect screening.. Massinissa Yahia 1,2

1, 25(OH)2D: 1, 25 Dihydroxyvitamin D; CKD: Chronic kidney disease; CKD- MBD: Chronic kidney disease-mineral and bone disorder; CRIC: Chronic Renal Insufficiency Cohort;

Performance and Durability of Electrodes with Platinum Catalysts in Polymer Electrolyte Cells Prepared by Ultrasonic Spray Deposition..

Submissions to Research Notes will be reviewed for readability, rationale, clarity, logic, organization, length, and adherence to all Alberta Journal of Educational Research (

The lack of substantial differences in long term effects between FP and IP methods is in keeping with recent findings from the Radiation Therapy Oncology Group (RTOG) trial