Coefficient bounds for a subclass of bi univalent functions using new differential operator

(1)

COEFFICIENT BOUNDS FOR A SUBCLASS OF BI

Presidency College, Chennai

ARTICLE INFO ABSTRACT

In this present paper, we introduce new subclasses

defined in the open disk

third coefficients for functions in these new subclasses using differential operator.

use, distribution, and reproduction in any medium, provided the original work is properly cited.

INTRODUCTION

Let A denote the class of functions f z

 

of the form

 

2

, n n n

f z z a z





 

_

which are analytic in the open unit disk U



 

Further, by

S

we shall denote the class of functions which are univalent in

U

. Since univalent functions are one to-one, they are invertible and the inverse functions need not be defined on the entire unit disk

U

. However, the famous Koebe one-quarter theorem ensures that the image of the unit disk

under every function f A contains a disk of radius Thus every univalent function f has an inverse





1

( )

f f z z

, (zU) and



1



0 0

1 ( ) , ( ), ( )

4

f f w ww r f r f 

 _   _

 

*Corresponding author: Elumalai, M.

Presidency College, Chennai-600 005, Tamilnadu, India.

ISSN: 0975-833X

Article History:

Received 08th_{November, 2016} Received in revised form 28th_{December, 2016} Accepted 18th_{January, 2017} Published online 28th_February,₂₀₁₇

Key words:

Bi-univalent functions, Coefficient estimates, Starlike functions, Convex function, Differential operator.

Citation: Elumalai, M. and Selvaraj, C.2017. “Coefficient bounds for a subclass of bi study of tunisie telecom operators””, International Journal of Current Research

RESEARCH ARTICLE

COEFFICIENT BOUNDS FOR A SUBCLASS OF BI-UNIVALENT FUNCTIONS USING NEW

DIFFERENTIAL OPERATOR

*Elumalai, M. and Selvaraj, C.

Presidency College, Chennai-600 005, Tamilnadu, India

ABSTRACT

this present paper, we introduce new subclasses S( , )  and

defined in the open diskU

_

z :z 1 .

_

Furthermore, we find upper bounds for the second and third coefficients for functions in these new subclasses using differential operator.

is an open access article distributed under the Creative Commons Attribution License, which use, distribution, and reproduction in any medium, provided the original work is properly cited.

f z

of the form

(1.1)



: 1 .

z z

  

we shall denote the class of functions fA Since univalent functions are one-one, they are invertible and the inverse functions need not be

. However, the famous Koebe quarter theorem ensures that the image of the unit disk

U

contains a disk of radius

1 / 4

.

has an inverse

1

f

satisfying

600 005, Tamilnadu, India.

where

1 2 2 3

2 2 3

3 4

2 2 3 4

( ) (2 )

(5 5 ) .

f w w a w a a w

a a a a w



  

 

  

A function f A is said to be bi

 

f z

and

1 ( )

f z

are univalent in

We let



to denote the class of bi

given by (1.1). If f z

 

is

bi-the boundary of bi-the domain and such that it can be continued

across the boundary of the domain so that

and analytic throughout w 1.

class



are





, 1

1

z

log z

z  

 _{and so on.}

The coefficient estimate problem for the class Bieberbach conjecture, is settled by de

proved that for a function



f z z a z

n

a n

, for n2,3,, with equality only for the rotations of the Koebe function

International Journal of Current Research

Vol. 9, Issue, 02, pp.46091-46095, February, 2017

INTERNATIONAL

Coefficient bounds for a subclass of bi-univalent functions using new differential operator

International Journal of Current Research, 09, (02), 46091-46095.

UNIVALENT FUNCTIONS USING NEW

and C( , )  of bi-univalent functions

Furthermore, we find upper bounds for the second and

third coefficients for functions in these new subclasses using differential operator.

ribution License, which permits unrestricted

1 2 2 3

2 2 3

3 4

2 2 3 4

( ) (2 )

(5 5 ) .

f w w a w a a w

a a a a w



  

(1.2)

is said to be bi-univalent in

U

if both are univalent in

U

.

to denote the class of bi-univalent functions in

U

-univalent, it must be analytic in the boundary of the domain and such that it can be continued

across the boundary of the domain so that

1 ( )

f z _{is defined}

1. 

Examples of functions in the

log z

and so on.

The coefficient estimate problem for the class

S

known as the Bieberbach conjecture, is settled by de-Branges [4], who



2

, n n n

f z z a z





 

_

in the class S, , with equality only for the rotations of

INTERNATIONAL JOURNAL OF CURRENT RESEARCH

(2)

0( ) 2.

(1 )

z K z

z

 

In 1967, Lewin [8] introduced the class



of bi-univalent

functions and showed that a2 1.51_{for the functions}

belonging to



. It was earlier believed that for f  , the

bound was an 1_{for every}

n

_{and the extremal function in}

the class was 1

z z

 _{. E.Netanyahu [10] in 1969, ruined this}

conjecture by proving that in the set



, 2

4

ax / 3

m

f a 

. In

1969, Suffridge [13] gave an example of f   for which

2 4 / 3

a 

and conjectured thata2 4 / 3._{In 1981, Styer and}

Wright [12] disproved the conjecture that a2 4 / 3_{. Brannan}

and Clunie [2] conjectured that a2  2_{. Kedzierawski [6] in}

1985 proved this conjecture for a special case when the

function f and

1

f

are starlike functions. Brannan and Clunie

[2] conjectured that a2  2_{.Tan [14] in proved that} 2 1.485

a 

which is the best known estimate for functions in the class of bi-univalent functions.

Brannan and Taha [3] introduced certain subclasses of the bi-univalent function class



similar to the familiar subclasses

 

S 

and C

 

 of the univalent function class



. Recently, Ali et al. [1] extended the results of Brannan and Taha [3] by generalising their classes using subordination.

An analytic function f is subordinate to an analytic function

g

,written f z( )g z( ), provided there is a Schwarz function

w

_{defined on}

U

_with w(0)0_and w z( ) 1_satisfying

 



( )



f z g w z

. Ma and Minda [9], unified various subclasses of starlike and convex functions for which either of

the quantity

( ) ( )

zf z f z



or

( ) 1

( )

zf z f z

 



is subordinate to a more general superordinate function. For this purpose, they considered an analytic function  with positive real part in the

unit disk

U

,(0)1, (0)0 and  maps

U

onto a region starlike with respect to 1 and symmetric with respect to the real axis. Such a function has a series expansion of the form

2 3

1 2 3 1

( )z 1 B z B z B z , (B 0).



     

(1.3)

In this paper, for f z( )A. Let a new differential operator be defined [5] on a class of analytic functions of the form (1.1) as follows:

 

0 _,

F f z  f z

 

1 _:

F f z zf z  Ff z

and in general

 



1

 





 



0 0 .

n n

F f z F F  f z n  

We easily find that

 



0



2

.

k n

nk n n

F f z z C a z n





 

_



(1.4)

where

! | ( ) | !

n k

n C

n k

 

Definition: 1.1. Let  be a non-zero complex number. A

function f z

 

given by (1.1) is said to be in the class S



 ,



if the following conditions are satisfied:

f _and

 





 

' 1

1 1 ,

( )

j

z F f z

z z

F f z 



 

 

  

 

 

 U

(1.5) and

 





 

1

1 1 , ,

( )

j

w F g w

w w

F g w 





 

 

  

 

 

 U

(1.6)

where the function g is given by (1.2).

Definition: 1.2. Let  be a non-zero complex number. A

function f z

 

given by (1.1) is said to be in the class C



 ,



if the following conditions are satisfied:

f _and

 









 

1

1 ,

( )

j

z F f z

z z F f z

 





 

 

 

 

 

 U

(1.7) and

 









 

1

1 , ,

( )

j

w F g w

w w F g w

 





 

 

 

 

 

 U

(1.8)

where the function g is given by (1.2).

2. Coefficient estimates

Lemma: 2.1. [11] If pP , then ck 2_{for each}k_{, where} P is the family of functions p analytic in

U

for which

 

Rep z 0 ,

 

2

1 2

1

p z  c zc z 

for z_{U .}

Theorem: 2.2. Let the function f z

 

A be given by (1.1). If



,



f S   _{, then}









1 1 2

2 2 2

3 2 1 1 2 2

2 j j j

B B a

C C B B B C





  

and



1 2 1



3 2

.

2 j j

B B B

a

C C



 





(3)

Proof: Sincef S



 ,



_{, there exists two analytic functions}

, :

r s UU_{, with}r(0)0s(0),_{such that}

 









1

1 1 ( )

( )

j

z F f z

r z

F f z 

             and

 









1

1 1 ( ) .

( )

j

w F g w

s z

F g w 

          

  _(2.2)

Define the functions p and q by

 

2

1 2

1 ( )

1

1 ( )

r z

p z p z p z

r z



    

 

and

 

2

1 2

1 ( )

1 .

1 ( )

s z

q z q z q z

s z         (2.3) Or equivalently,

 

2 2 1 1 2 2 3

1 1 1 2

3 2

( ) 1 ( ) 1

2 1

2

2 2 2

p z r z

p z

p

p z p z

p p p p

p p z

                   _ _     _ _ _ _ _ _       _ _       (2.4) and

 

2 2 1 1 2 2 3

1 1 1 2

3 2

( ) 1 ( ) 1

2 1

. 2

2 2 2

q z s z

q z

q

q z q z

q q q q

q q z

                  _ _     __ _ _ _ _ _       _ _       (2.5)

It is clear that p and q are analytic in U andp(0) 1 q(0). Also p and q have positive real part in U and hence pi 2

and qi 2.

In the view of (2.3), (2.4) and (2.5), clearly,

 





1 ( ) 1

1 1

( ) 1 ( )

j

z F f z _{p z}

p z

F f z 

            _ _   _  _   and

 





1 ( ) 1

1 1 .

( ) 1 ( )

j

w F g w _{q w}

q w

F g w 

            _ _   _  _

  _(2.6)

Using (2.5) and (2.6) together with (1.3), one can easily verify that

2

2 2

1 1 1 1

2 2 1

( ) 1 1

1

( ) 1 2 2 2 4

B p B p

p z

z p B p z

p z     _    _          (2.7) and 2 2 2

1 1 1 1 2 1

2

( ) 1

1 .

( ) 1 2 2 2 4

B q B q B q

q w

w q w

q w     _    _          (2.8)

Since f  has the Maclaurin series given by (1.1),

computation shows that its inverse

1

g f _{has the expansion}

given by (1.2). It follows from (2.6), (2.7) and (2.8) that

2 2 1 1

1 , 2 j

C a  B p

(2.9)

2 2 2 2

3 3 2 2 1 2 1 2 1

1 1 1

2

2 2 4

j j

C a C a  B _p  p _ B p

  _(2.10)

and

2 2 1 1

1 , 2 j

C a B q

 

(2.11)



2



2 2 2

3 2 2 3 3 1 2 1 2 1

1 1 1

4 2 .

2 2 4

j j j

C C a  C a  B q_  q _ B q

  _(2.12)

From (2.9) and (2.11), it follows that

1 1.

p  q

(2.13)

Now (2.10), (2.12) and (2.13) gives

















3

1 2 2

2

2 ₂ ₂ ₂

3 2 1 2 1 2

.

4 2 j j j

B p q a

C C B C B B

       (2.14)

Using the fact that p2 2_and q2 2_{gives the desired}

estimate on a2 _,









1 1 2

2 2 2

3 2 1 1 2 2

.

2 _j _j _j

B B a

C C B B B C





  

From (2.10)-(2.12), gives

















2 2 2

1

3 2 2 2 2 3 1 2 1

3 2 2

3 3 2

4

2 _.

4 2

j j j j

j j j

B

C C p C q C p B B

a

C C C



   





Using the inequalities p1 2_, p2 2_and q2 2_for

functions with positive real part yields the desired estimation of

3

a

.

For a choice of

 

1 1 Az z Bz   

 _,  1 BA1_{, we have the} following

Corollary: 2.3. Let 1 BA1_{. If}

1 , 1 Az f Bz       _ _    S , then

















2 2 2

3 2 2

2 _j _j 1 _j

A B

a

C C A B B C



 

(4)

and









3 ₂

3 2

1 1

.

2 j j

A B B

a

C C



  





If we let

 

1 2 2

1

z

z z z

z





_  _  











  _{, 0}1_{, in} the above theorem, we get the following:

Corollary: 2.4. Let 01. If f S



 ,



_{, then}









2

2 2

3 2 2

2

2 2 _j _j 1 _j

a

C C C

 

  



  

and





3 2

1 1 2

.

2 j j

a

C C



 

 





Theorem: 2.5. Let the function f z

 

A be given by (1.1). If



,



fC_  

, then









1 1 2

2 2 2

3 2 1 1 2 2

2 3 j 2 j 2 j

B B a

C C B B B C





  

and



1 2 1



3 2

.

2(3 j 2 j)

B B B

a

C C



 





(2.15)

Proof: Since fC



 ,



_{, there exists two analytic functions}

, :

r s UU_{, with}r(0)0s(0)

, such that

 













1

1 ( )

( )

j

z F f z

r z F f z

 





 

 

 

 

 

and

 













1

1 ( ) .

( )

j

w F g w

s z F g w

 





 

 

 

 

  _(2.16)

Using (2.3), (2.4), (2.7) and (2.8), one can easily verified that

2 2 1 1

1

2 ,

2 j

C a  B p

(2.17)

2 2 2 2

3 3 2 2 1 2 1 2 1

1 1 1

6 4

2 2 4

j j

C a  C a  B _p  p _ B p

  _(2.18)

and

2 2 1 1

1

2 ,

2 j

C a B q

 

(2.19)



2



2 2 2

3 2 2 3 3 1 2 1 2 1

1 1 1

12 4 6 .

2 2 4

j j j

C  C a  C a  B q_  q _ B q

  _(2.20)

From (2.17) and (2.19), it follows that

1 1.

p  q _(2.21)

Now (2.18), (2.20) and (2.21) gives

















3

1 2 2

2

2 ₂ ₂ ₂

3 2 1 1 2 2

.

8 3 _j 2 _j 2 _j

B p q a

C C B B B C



  

  

(2.22)

Using the fact that p2 2_and q2 2_{gives the desired}

estimate on a2 _,









1 1 2

2 2 2

3 2 1 1 2 2

.

2 3 j 2 j 2 j

B B a

C C B B B C





  

From (2.18)-(2.20), gives

















2 2 2 2

1

3 2 2 2 2 2 1 1 3

3 ₂

3 3 2

12 4 4 3

2 _.

24 3 2

j j j j

j j j

B

C C p C q B B p C

a

C C C



   





Using the inequalities p1 2_, p2 2_and q2 2_for

functions with positive real part yields









1 2 1

3 2

.

2 3 j 2 j

B B B

a

C C



 





For a choice of

 

1 1

Az z

Bz

  

 _,  1 BA1_{, we have the} following corollary.

Corollary: 2.6. Let 1 BA1. If

1 ,

1

Az f

Bz

 



 

 _ _ 

 

S

, then

















2

2 2

3 2 2

2 3 _j 2 _j 2 1 _j

A B a

C C A B B C



 

   

and









3 ₂

3 2

1 1

.

2 3 j 2 j

A B B

a

C C



  





If we let

 

1 2 2

1

z

z z z

z





_  _  











  _{, 0}1_{, in} the above theorem, we get the following:

Corollary: 2.7. Let 01. If fS



 ,



_{, then}









2

2 2

3 2 2

3 _j 2 _j 1 _j

a

C C C

 

 



(5)

and









3 ₂

3 2

1 1

.

3 j 2 j

a

C C

  

 





Remark: 2.1. If we let  1,j0, Theorem: 2.2 and Theorem: 2.5 reduce to the result of R.M.Ali et.al [1], corollary 2.1 and corollary 2.2.

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