Certain Subclass of Analytic Univalent Functions Using q - Differential Operator

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IJSRR, 8(2) April. – June., 2019 Page 1

Research article Available online www.ijsrr.org

ISSN: 2279–0543

International Journal of Scientific Research and Reviews

Certain Subclass of Analytic Univalent Functions Using q - Differential

Operator

G. Thirupathi

Department of Mathematics, Ayya Nadar Janaki Ammal College, Sivakasi - 626 124, Tamilnadu, India. Email: [email protected].

ABSTRACT

In this paper, we define a new subclass of analytic univalent function using q - differential

operator, which generalizes Ruschewayh differential operators. Coefficient inequalities,

Subordination, extreme points and integral means inequalities results are obtained.

KEYWORDS:

Univalent, subordinating factor sequence, integral means, Hadamard product, q-

derivative operator, subordination.

*Corresponding author:

G. Thirupathi

Department of Mathematics,

Ayya Nadar Janaki Ammal College,

Sivakasi - 626 124, Tamilnadu, India.

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INTRODUCTION, DEFINITIONS AND PRELIMINARIES

Let us denote U{z \∣ ∣z 1} and let A denote the class of functions of the form

2 3

( ) , _n 0,

f z  z a z a z  a  (1.1) which are analytic in the open disc U.

Mohammed and Darus1 studied approximation and geometric properties of the qoperators in some subclasses of analytic functions in compact disk. Recently, Purohit and Raina2,3 have used the fractional qcalculus operators in investigating certain classes of functions which are analytic in the open disk. Also Purohit4 studied these q operators, defined by using the convolution of normalized analytic functions and qhypergeometric functions.

The qderivative operator of a function f is defined by ( ) ( ) ( ) ( 0)

(1 ) q

f z f qz

D f z z

q z



 

 (1.2) and (D f_q )(0) f(0),provided that the function f is differentiable at 0.We note that

( ) ( ) q

D f z  f z as q_1

.

Also, from [1.2], we have D_

_{ }

q f z( )=1+

2

[ ] _n n n

n a z







_

, where

[ ] 1 . 1

n

q n

q

 

 (1.3)

The Hadamard product of two functions

2

( ) _n n

n

f z z a z





 

_

and

2

( ) _n n

n

g z z b z





 

_

is given by

2

( * )( ) _n _n n. n

f g z z a b z





 

_

(1.4) Recently, Kanas and Raducanu 5, defined and investigated Ruschewayh qdifferential operator as follows:

For fA , generalized Ruschewayh qdifferential operator is defined by





2

( )

( ) , .

( 1)! (1 )

q n

n q

n

R f z z a z z

n

 







 

  

  



U (1.5)

Here R f z_q0 ( ) f z R f z( ), _q1 ( )zD f z_q ( ) and

1

1 ( ( ))

( )

[ ]! m m q q

zD z f z R f z

m



 .

It can be seen that if we letq1, then R f z_q

 

reduces to the well-known Ruschewayh differential

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D_q



R f z_q ( )



= 1+

2

n 





[n] _q( , )n n 1

n

a z  (1.6)

where

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

q q

q

n n

n



 

   

   (1.7)

Using the generalized Ruschewayh qdifferential operator, we define the following class ( , , ).

q n  S

Definition 1.1: Let S_q( , , )n  be the class of functions fA satisfying



 



 





q q

D R f z Re

D R f z



 

 

 



 

 

 

(1.8)

for some 0

_{ }

1 1 1

01, _  _  0 , _ .

Definition 1.2: (Subordination) Given two functions f z

_{ }

andg z

_{ }

, which are analytic in U.Then

we say that the function ( )f z is subordinate to ( )g z in U, if there exists an analytic function ( )w z in Usuch that (0)w 0, | ( ) | 1(w z  zU such that ( )) f z g w z( ( )), denoted by ( )f z  g z( ).

Definition 1.3: (Subordinating Factor Sequence) A sequence

 

1

n _n

b 

 of complex numbers is said to

be a subordinating sequence if, whenever

1

( ) n

n n

f z a z





_

, a₁1 is regular, univalent and convex in U, we have

1

( ), ( ).

n n n n

b a z f z z









 U (1.9)

Motivated by the concept introduced by Serap Bulut7, Selvaraj8, in this paper, we obtain coefficient bounds, extreme points and integral means inequalities for the above said function class.

Let T denote the subclass of f A consisting of functions of the form

2

( ) n

n n

f z z a z





 

_

(1.10)

COEFFICIENT INEQUALITIES

Theorem 2.1: Let f z( )A of the form [1.1]. If the inequality





2

( , , ) 2(1 ), ,

n n

n

a z

   





  



B U (2.1)

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 

( , , ) ( , ) (1 ) ( , ) ( , ) (1 ) ( , ) n     n _q n   q n q n   q n  _

B (2.2)

and _q( , )n is given by (1.7) then fS_q( , , )n  .

Proof: Suppose that the inequality (2.1) holds. Then forzU, we define the function F by



 



 





( ) q q .

q q

D R f z F z

D R f z



 

  (2.3)

It is sufficient to show that

( ) 1 1,

_

_

. ( ) 1

F z

z F z



 

 U (2.4)

Now,

 





_

 

_

 





_

 

_

(1 ) ( ) 1

( ) 1 (1 )

q q q q

D R f z D R f z F z

F z D R f z D R f z

 

    



  

 

1 2

2

( , ) (1 ) ( , )

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

( , ) (1 ) ( , )

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

( , ) (1 ) ( , )

(2

n

q q n

n

q q n

n

q q n

n

q q n

n

q q n

n

n n n a z

n n n a z

n n n a

     





 



 



 



 



 

 _   _



 

  _   _

 

  

 



 

  _   _

 

 _   _

 



 

2

1 ) _q( , ) (1 ) _q( , ) _n

n

n n n a

     







 



_

_   _

Therefore, fS_q( , , )n  .

INTEGRAL MEANS INEQUALITIES

Lemma 1: [Selvaraj et al.9] If the functions f and gare analytic in U with ( )f z g z( ), then for  0 and zrei(0 r 1),

2 2

0 f z( ) d 0 g z( ) d .

 _  _

  

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IJSRR, 8(2) April. – June., 2019 Page 5 Silverman10 found that the function

2 2( )

2

z

f z  z is often extremal over the family T and applied this function to resolve his integral means inequality, conjectured and settled in11 , that 2 2 ₂

0 f z( ) d 0 f z( ) d .

   

  



(3.2)

for all fT . In12, Silverman also proved his conjecture for the subclasses T*( ) and K( ) of T .

Theorem 3.1: Suppose fS_q( , , ),n   0, 0 1 and f z₂( )is defined by ₂ 2

2

2(1 ) ( )

( , , )

f z z  z

     

B , where

2( , , )   _q(2, ) (1   ) q(2, ) q(2, ) (1 ) q(2, ) _

B (3.3)

and (2, ) [2] (2 ) . (2 1)! (1 )

q q

q



 

   

   (3.4)

Then for zrei(0 r 1),we have

2 2 ₂

0 f z( ) d 0 f z( ) d .

 _  _

 



(3.5)

Proof: Using (1.10) and (3.5), it is enough to prove that

2 1 2

0 0

2 2

2(1 )

1 1 .

( , , ) n

n n

a z d z d

 

  _

 

  

 



 

_

 



_B (3.6)

By Lemma 1, it suffices to show that

1

2 2

2(1 )

1 1 .

( , , ) n

n n

a z  z

  

 





_

 

B

Setting 1

2 2

2(1 )

1 1 ( ),

( , , ) n

n n

a z  w z

  

 





_

 

B

and using (2.1), we obtain w z( ) is analytic inU, (0)w 0, and 2 1

2 2

( , , ) ( , , )

( ) ,

2(1 ) 2(1 )

n n

w z    a z z    a z

 

 



 

  

 



B



B

where B_n( , , )   is given by (2.2). This completes the proof of the theorem.

SUBORDINATION RESULTS

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IJSRR, 8(2) April. – June., 2019 Page 6 Lemma 2: The sequence

_{ }

1

n _n

b 

 is a subordinating factor sequence if and only if

1

1 2 _n n 0, ( ).

n

Re b z z

         





U (4.1)

Theorem 4.1: Let fS_q( , , )n  and ( )g z be any function in the usual class of convex functions K, then





2 2 ( , , ) ( * )( ) ( ) ( )

2 2(1 ) ( , , ) f g z g z z

  

    

   U

B

B (4.2)

where 01with _q( , )n given by (3.4) and







2



2

2(1 ) ( , , )

( ) ( ).

( , , )

Re f z     z

    

  B U

B (4.3)

Proof: Let fS_q( , , )n  and suppose that

2

( ) _n n .

n

g z z b z





 

_

K Then, for fA given by (1.1), we have









2 2 2 2 2 ( , , ) ( , , ) ( * )( ) .

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

n n n n

f g z z a b z

                   _  _     _



_ B B

B B (4.4)

Thus, by Definition 3, the subordination result holds true if





2

2 ₁

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

n               _ _    B B

is a subordinating factor sequence, witha₁1. In view of Lemma 2, this is equivalent to the following inequality





2 1 2 ( , , )

1 0, ( ).

2(1 ) ( , , ) n n n

Re    a z z

              _ _  



 U B

B (4.5)

Now, for z  r 1, we have













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

2(1 ) ( , , )

( , , ) ( , , )

1

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

n n n n n n

Re a z

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



















2 1 2

2 2

2

2 2

( , , ) ( , , )

1

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

( , , ) 2(1 )

1

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

1 0, 1

n n n

a r r

r r

r z r

     

       

   

       





  

   



  

   

    



B

B B

B

B B

where we have also made use of the assertion (2.1) of Theorem 2.1. This evidently proves the

inequality (4.5) and hence also the subordination result (4.4) asserted by Theorem 4.1. The

inequality (4.3) asserted by Theorem 4.1 would follow from (4.2) upon setting

1

( ) .

1

j

z

g z z

z





  





K

Finally, we consider the function ( )q z is given by

2 2

2(1 ) ( )

( , , )

q z z  z

     

B (4.7)

and _q(2, ) is given by (3.4). ClearlyqS_q( , , )n  . For this function (4.2) becomes





2 2

( , , )

( ) .

2 2(1 ) ( , , ) 1

z q z

z

  

   

   

B B

Moreover, it can easily be verified for the function ( )q z given by (4.7) that





2 2

( , , ) 1

( ) ( ),

2 2(1 ) ( , , ) 2

min Re    q z z

   

  

 

  

 

 _ _

 

  

 

U B

B

which evidently completes the proof of Theorem 4.1.

EXTREME POINTS

Theorem 5.1: Let f z₁( )z and

( ) 2(1 ) ( 2,3, ) ( , , )

k k

k

f z z  z k

   

   

B (5.1)

where B_k( , , )   given by (2.2). Then f S_q( , , )n  if and only if it can be expressed in the form

1

( ) _k _k( ), k

f z  f z





_

(5.2)

where _k 0 and

1

1 k k









.

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1 1

2

1 2

1 2 2

( ) ( ) ( ),

2(1 ) ( , , )

2(1 ) 2(1 )

.

( , , ) ( , , )

k k k

k k

k k k

k k k k k

f z f z f z

z z z

z z z z

 



 

  

 

  

     



 



  

  

 

  

  _  _

 

     

 

_ _  _ _   _ _

     



B

B B

(5.3)

Thus

₁

2 2

2(1 )

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

k k k



        

  

 

 



      



B



B (5.4)

Therefore, we have fS_q( , , )n  .

Conversely, suppose that f S_q( , , )n  . Since 2(1 ) , ( 2,3, ), ( , , )

k k

a  k

   

  

B

we can set ( , , ) , ( 2,3, ) 2(1 )

k

k ak k

   



  



B

and ₁

2

1 _k.

k

 





 

_

Now

2 1 2

1 2

1 1

2 1

2(1 ) ( )

( , , )

2(1 ) ( , , )

( ) ( ) ( ).

k k

k k k

k k k k

k k

k k k k

k k

f z z a z z z

z z z

f z f z f z



 

   

 

  

  

  

  





 

 

    

  _ _  _ _

   

  

  _  _

 

  



B

This completes the proof of the Theorem 5.1.

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