IRJET- On Certain Subclasses of Univalent Functions: An Application

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056
Volume: 06 Issue: 10 | Oct 2019 www.irjet.net p-ISSN: 2395-0072
© 2019, IRJET | Impact Factor value: 7.34 | ISO 9001:2008 Certified Journal | Page 118
ON CERTAIN SUBCLASSES OF UNIVALENT FUNCTIONS: AN
APPLICATION
Jayesh Jain1, Dr. Mahender Singh Poonia2
1Research Scholar, Department of Mathematics, Shree JJT University, Jhunjhunu, Rajasthan
2Associate Professor, Department of Mathematics, Shree JJT University, Jhunjhunu, Rajasthan
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Abstract - In the present paper, we have studied a class
WR(λ,β,α,μ,θ) which consist of analytic and univalent
functions with negative coefficients in the open disk U =
* | | + defined by Hadamard product with Rafid
Operator,we obtain coefficient bounds , extreme points for
this class ,Also weighted mean , arithmetic mean and some
results.
Key Words: Univalent function, Rafid operator,
Extreme point, Hadamard product, Weighted mean,
Arithmetic mean.
1. INTRODUCTION
Let R stand in favor of mapping
f(z) = z -  
2
,( 0, 1,2,3,... )n
n n
n
a z a n N


   (1)
whichever analytic and univalent in the unit disk
U =* | | + If f is specified in (1) and
g specified in
g(z) = z -
2
,b 0n
n n
n
b z



after that effective Hadamard product f g of f and g is clear
with f g(z) =
2
( )( )n
n n
n
z a b z g f z


   (2)
Lemma 1. The Rafid Operator of f
is denoted by R
 and defined as following
R
 (f(z)) =
( ) 1 
∫
. /
f(zt)dt
= z-
2
( , , ) n
n
n
k n a z 


 (3)
wherever k(n, )
( ) n 
1 
Proof: R
 (f(z)) =
( ) 1 
∫
. /
f(zt)dt
=
( ) 1 
∫
. /
*
2
( )n
n
n
zt a zt


  +dt
=
( ) 1 
1
1
0
2
1
0
1
1
n n
n
n
tz dt a z t e dte 
   

   
    

 
 
  
 
Thus
R
 (f(z)) =
( ) 1 
[
 
1
0
1 x
z e x dx
 

   
   ]
( ) 1 
*
1
0
2
(1 )n n x n
n
n
a z e x dx 

 
   

  +
=
( ) 1 
1
2
(1 ) 1 (1 )n n
n
n
z a z n 
   

 

 
     
 

= z-
 
1
2
1
1
n
n
n
n
n
a z
 




 


= z-
2
( , , ) n
n
n
k n a z 




Definition1. A function f(z) R , z U is said to be
in the class WR( , , , , )    
 
if and only if satisfies the inequality
' 2 ''
'
( ((f g)(z))) ( ((f g)(z)))
Re
(1 ) ((f g)(z)) ( ((f g)(z)))
z R z R
R z R
 
 
 
 

 
    
 
     
|
' 2 ''
'
( ((f g)(z))) ( ((f g)(z)))
1
(1 ) ((f g)(z)) ( ((f g)(z)))
z R z R
R z R
 
 
 
 

 
  

   
|+ (4)
somewhere 0
and g(z) are given by
g(z) = z -
2
,b 0n
n n
n
b z



Lemma 2. Let w = u+iv. Then Re w iff
| ( )| | ( )|
Lemma 3. Let w = u+iv and
Re w | |+ if and only if
Re  (1 )i i
w e e 
  
We endeavor to study the coefficient bounds, extreme
points, Hadamard product of the class WR(λ,β,α,μ,θ) ,
wighted mean, arithmetic can and some results.
2. COEFFICIENT BOUNDS AND EXTREME POINTS:
We acquire the essential and satisfactory circumstance
and extreme points for the functions f(z) in the class
WR(λ,β,α,μ,θ).
Therom2.1 The mapping f(z) clear with (1) is in the class
WR(λ,β,α,μ,θ) iff
2
(1 )[ (1 ) ( )] ( , , )
1
n n
n
n n k n a b      



    
 
 (5)
wherever 0
Proof; By clarification (1),we get
' 2 ''
'
( ((f g)(z))) ( ((f g)(z)))
Re
(1 ) ((f g)(z)) ( ((f g)(z)))
z R z R
R z R
 
 
 
 

 
    
 
     
|
' 2 ''
'
( ((f g)(z))) ( ((f g)(z)))
1
(1 ) ((f g)(z)) ( ((f g)(z)))
z R z R
R z R
 
 
 
 

 
  

   
|+
subsequently through Lemma 3, we comprise
' 2 ''
'
( ((f g)(z))) ( ((f g)(z)))
(1 ) ((f g)(z)) ( ((f g)(z)))Re
(1 )i i
z R z R
R z R
X e e
 
 
 
 
 

  
 
   
 
     
 
  
-   ,or consistently,
' 2 ''
'
2 '
'
( ((f g)(z))) ( ((f g)(z))) (1 )
(1 ) ((f g)(z)) ( ((f g)(z)))
Re
e ((1 )( ((f g)(z)) ( ((f g)(z) ))
(1 ) ((f g)(z)) ( ((f g)(z)))
i
i
z R z R e
R z R
R z R
R z R
  
 
 
 
  
 
 
 
 
 

  
 
    
 
    
 
  
     
     
(6)
Let F(z) =
, 2 ,,
[ ( ((f g)(z))) ( ((f g)(z))) ](1 )i
z R z R e  
     
,
[(1 )( ((f g)(z)) ( ((f g)(z))) ]i
e R z R  
       
And
E(f) =
'
(1 ) ((f g)(z)) ( ((f g)(z)))R z R 
     
next to Lemma 2. (6) is comparable to
| ( ) ( ) ( )| | ( ) ( ) ( )| for
0 But | ( ) ( ) ( )| =
2
2
2
(1 )( ( , , ) )
( , , )
(1 ) (1 ) ( , , ) )
i n
n n
n
i n
n n
n
n
n n
n
e z k n a b z
e z nk n a b z
z n k n a b z


   
    
    






 
   
 
 
  
 
 
     
 




=  
2
(2 ) ( ( 1) (1 )(1 ( , , ) n
n n
n
z n n n n k n a b z      


       
 
2
( 1) (1 ) ( , , )i n
n n
n
e n n n n k n a b z
     


     
 
2
(2 ) ( ( 1) (1 )(1 ( , , )
n
n n
n
z n n n n k n a b z      


        
 
2
( 2) 1 ( , , )
n
n n
n
n n n k n a b z    


    
Also | ( ) ( ) ( )|
 
2
( ( 1) (1 )(1 ) ( , , ) n
n n
n
az n n n n k n a b z     


       
 
2
( 1) (1 ) ( , , )i n
n n
n
e n n n n k n a b z
     


     
 
2
( ( 1) (1 )(1 ( , , )
n
n n
n
z n n n n k n a b z      


       
 
2
( 1) (1 ) ( , , )
n
n n
n
n n n n k n a b z     


     
Furthermore
| ( ) ( ) ( )| | ( ) ( ) ( )|
( )| |
2
(2 2 ( 1) 2 (1 )
( , , ) 0
(2 2 ( 1) 2(1 ))
n
n n
n
n n n n
k n a b z
n n n n
   
 
   


     
       

Or
2
(1 ) n (n 1)(1 )
( , , )
(1 )( )
1
n n
n
n
k n a b
n
  
 
   



     
    
 

This is comparable to
2
(1 )[ (1 ) ( )] ( , , ) 1n n
n
n n k n a b       


      
on the contrary, expect that (5) holds. afterward we
obliged to show
' 2 ''
'
2 '
'
( ((f g)(z))) ( ((f g)(z))) (1 )
(1 ) ((f g)(z)) ( ((f g)(z)))
Re
e ((1 )( ((f g)(z)) ( ((f g)(z))) ))
(1 ) ((f g)(z)) ( ((f g)(z)))
i
i
z R z R e
R z R
R z R
R z R
  
 
 
 
  
 
 
 
 
 

  
 
    
 
    
 
    
    
3. HADAMARD PRODUCT
Theorem : f(z) =z -
2
n
n
n
a z


 and g(z) = z -
2
n
n
n
b z



belong to WR(λ,β,α,μ,θ)
afterward effective Hadamard product of f and g is given
by f g(z) =
2
( )( )n
n n
n
z a b z g f z


  
Proof:
Since f and g ( , , , , )WR     
We have
2
(1 )[ (1 ) ( )]k(n, , )b
1
1
n
n
n
n n
a
      



     
  

And
2
(1 )[ (1 ) ( )]k(n, , )a
1
1
n
n
n
n n
b
      



     
  

and by applying the Cauchy-Schwarz ineuqality, we have
2
(1 )[ (1 ) ( )]k(n, , )
1
n n
n n
n
n n a b
a b
      



     
 
  

1/2
2
(1 )[ (1 ) ( )]k(n, , )b
1
n
n
n
n n
a
      



       
     

1/2
2
(1 )[ (1 ) ( )]k(n, , )a
1
n
n
n
n n
b
      



       
    

Consequently we attain
2
(1 )[ (1 ) ( )]k(n, , )
1
1
n n
n n
n
n n a b
a b
      



     
 
  

Now we want to prove
2
(1 )[ (1 ) ( )]k(n, , )
1
1
n n
n
n n
a b
      



     
  

Since
 

2
(1 )[ (1 ) ( )]k(n, , )
1
n n
n
n n
a b
      



     
  

=
2
(1 )[ (1 ) ( )]k(n, , )
1
n n
n n
n
n n a b
a b
      



     
 
  

thus we search out the consequence.
4. WEIGHTED MEAN AND ARITHMETIC MEAN
Lemma 4.
If Re w 1 , where 0 1 , 0.
Then
1
w k k
k
w
 


    



Proof: Let Re w 1w k   , as w  Re ,
we acquire
1w w k   , or equivalent (1 )w k    ,
subsequently
1
k
w





Defination 2. Allow f(z) and g(z) belong to R.
subsequently the weighed mean hj(z) of f(z) and g(z) is
given by
hj(z) = ,( ) ( ) ( ) ( )- (7)
Definition 3. The arithmetic mean of fj
(j = 1,2,....,q) is
clear within W(z) =  
1
1 q
j
j
f z
q 
 (8)
In the next theorem we will show the weighted mean
and arithmetic mean in the class
Theorem.
If f(z) and g(z) are in the class WR( , , , , )    
Afterward the weighted mean defined by Definition 2 is in
the class
WR( , , , , )     ,where
2 2
( ) , g( )n n
n n
n n
f z z c z z z d z
 
 
    
Proof: By definition 2, we attain
hj(z)=
*( ) (
2
n
n
n
z c z


  ) ( ) (
2
n
n
n
z d z


  )+
=  
2
1
(1 ) (1 )
2
n
n n
n
z j c j d z


   
We necessity explain so as to hj(z) so by lemma 2 we get
2
(1 )[ (1 ) ( )] ( , , )
1
(1 ) (1 )
2
n
n
n n
n n k n
X j c j d b
      


    
 
    

2
1
(1 )[ (1 ) ( )] ( , , ) (1 )
2
n n
n
n n k n j c b      


 
       

2
1
(1 )[ (1 ) ( )] ( , , ) (1 )
2
n n
n
n n k n j d b      


 
        

 (1 ) (1 ) (1 ) 1j j        
The proof is complete.
Thorem: Let fj(z) clear with
, ,
2
( ) (a 0, 1,2,......q) (9)n
j n j n j
n
f z z a z j


   
, 2 ,,
,
(R ((f g)(z))) (R ((f g)(z)))
(1 )R ((f g)(z)) (R ((f g)(z))) 1
z z
z
 
 
 
 
  
  
   

    
Proof: Commencing (8) and (9) we container inscribe
W(z) = ,
1 2
1 q
n
n j
j n
z a z
q

 
 
 
 
 
= ,
1 2
1q
n
n j
j n
z a z
q

 
 
  
 
 
because fj(z) WR( , , , , )     for every
(j=1,2,......q), so by using the theorem we get

,
2 2
1
(1 )[ (1 ) ( )] ( , , )
q
n j n
n n
n n k n a b
q
      

 
 
      
 
 
,
2 2
1
(1 )[ (1 ) ( )] ( , , )
q
n j n
n n
n n k n a b
q
      

 
 
      
 
 
2
1
(1 ) (1 )
q
nq
 

   
This is the absolute verification.
Theorem:
Let f(z) clear with (1) be in the class WR( , , , , )    
.Then
, 2 ,,
,
(R ((f g)(z))) (R ((f g)(z)))
(10)
(1 )R ((f g)(z)) (R ((f g)(z))) 1
z z
z
 
 
 
 
  
  
   

    
Proof: As f(z) WR( , , , , )     after that by lemma 4
,we achieve
, 2 ,,
,
(R ((f g)(z))) (R ((f g)(z)))
(1 )R ((f g)(z)) (R ((f g)(z))) 1
z z
z
 
 
 
 
  
  
   

    
The verification is comprehensive.
obtained coefficient bounds, extreme points of the class
WR(λ,β,α,μ,θ), Also described weighted mean, arithmetic
mean and some results.
REFERENCES
1) E. S. Aqlan, Some Problems Connected with
Geometric Function Theory, Ph.D. Thesis, Pune
University, Pune (unpublished), (2004).
2) S. Owa, On the distortion theorems, Kyungpook
Math. J., 18: 53-59, 1978.
3) H. M. Srivastava and R. G. Buschman, Convolution
integral equation with special function kernels,
John Wiley and Sons, New York, London, Sydney
and Toronto, 1977.
4) H. M. Srivastava and S. Owa, An application of the
fractional derivative, Math. Japon,29:384-389,
1984.
5) H. M. Srivastava and S. Owa, (Editors), Univalent
Functions, Fractional Calculus and Their
Applications, Halsted press (Ellis Harwood
Limited, Chichester), John Wiley and Sons, New
York, Chichester, Brisbane and Toronto, 1989.
6) W.G. Atshan and R.H. Buti , Fractional Calculus of a
Class of UnivalentFunctions With Negative
Coefficients Defined By Hadamard Product
WithRafid –Operator , European J. Of Pure And
Applied Math. 4(2), 2011,162-173
7) M.K. Aouf, A. Shamandy and M.F. Yassen, Some
applications of fractional calculus operators to
certain subclass of univalent functions, Soochow
Journal Of Mathematics, 21(1)(1995), 139-144.
8) G.L. Reddy and K.S. Padmanabhan , Some
properties of fractional integrals and derivatives
of univalent functions, Indian J. pure appl.
Math.16(3) (1985) , 291-302.
5. CONCLUSION
Using Hadamard product with Rafid Operator, we

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