International Research Journal of Engineering and Technology (IRJET)
e-ISSN: 2395-0056
Volume: 06 Issue: 10 | Oct 2019
p-ISSN: 2395-0072
www.irjet.net
ON CERTAIN SUBCLASSES OF UNIVALENT FUNCTIONS: AN
APPLICATION
Jayesh Jain1, Dr. Mahender Singh Poonia2
1Research
Scholar, Department of Mathematics, Shree JJT University, Jhunjhunu, Rajasthan
Professor, Department of Mathematics, Shree JJT University, Jhunjhunu, Rajasthan
----------------------------------------------------------------------***--------------------------------------------------------------------2Associate
Abstract - In the present paper, we have studied a class
wherever k(n,
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.
Proof:
 1
)
n
n
n2
, (an  0, n  N  1, 2,3,...)
+ If f
| |
U =*
g
specified in

b z
n2
n
n
(1)
f g(z) = z 
(
n
n n
n2
 ( g  f )( z )
R (f(z)) =
 1
)
(
(2)
(
 1
)
.
/
(
 k (n,  , )a z
n
n2
© 2019, IRJET
n
)
)
= z-

n2
 z  1   1 e x x dx 
[  0
 ]
 1
*
a z 
n
n2
n

0
(1   )  n e x x 1 n dx +
Impact Factor value: 7.34
n 1
1      na z n
 1
n

(3)
= z-
 k (n,  , )a z
n2
|
+dt



1
z
(1


)


1

an z n (1   )  n   n 


n2

 1 

f(zt)dt

= z-
n
=
R and defined as following
∫
n
 1
)

Lemma 1. The Rafid Operator of f
is denoted by
 a ( zt )
R (f(z)) =
, bn  0
a b z
f(zt)dt

n2
after that effective Hadamard product f g of f and g is clear
with
* zt 
/
Thus
is specified in (1) and

/
.
∫
 1 

   1 




 z t e  1   dt   an z n  t  1 n e  1   dt 
0
 0

n2
whichever analytic and univalent in the unit disk
g(z) = z -
.
∫
(
Let R stand in favor of mapping
a z
 1
)
=
1. INTRODUCTION

 1
R (f(z)) =
=
(
 n
)
(
Key Words:
Univalent function, Rafid operator,
Extreme point, Hadamard product, Weighted mean,
Arithmetic mean.
f(z) = z -
(
)
|
n
n
ISO 9001:2008 Certified Journal
|
Page 118
International Research Journal of Engineering and Technology (IRJET)
e-ISSN: 2395-0056
Volume: 06 Issue: 10 | Oct 2019
p-ISSN: 2395-0072
www.irjet.net
Definition1. A function f(z)  R , z  U is said to be
in the class WR( , , , , )
if and only if satisfies the inequality

'
2

''

 z ( R ((f  g)(z)))   z ( R ((f  g)(z))) 

Re 


'

 (1   ) R ((f  g)(z))   z ( R ((f  g)(z))) 

 z ( R ((f  g)(z)))   z ( R ((f  g)(z))) 
Re 


'
 (1   ) R ((f  g)(z))   z ( R ((f  g)(z))) 


|
'
2


z ( R ((f  g)(z)))'   z 2 ( R ((f  g)(z)))''




(1   ) R ((f  g)(z))   z ( R ((f  g)(z)))
somewhere
|
''
'
n2
n
n
, bn  0
Lemma 2. Let w = u+iv. Then Re w
|
(
|
)|
(
 z ( R ((f  g)(z)))'   z 2 ( R ((f  g)(z)))'' (1   ei ) 


(1   ) R ((f  g)(z))   z ( R ((f  g)(z))) '




Re 

  ei ((1   )( R ((f  g)(z))   z 2 ( R ((f  g)(z) ' )) 






'


(1


)
R
((f

g)(z))


z
(
R
((f

g)(z)))




iff
)|
Lemma 3. Let w = u+iv and
Re w

|
   ,or consistently,
-
b z
(6)
|+ if and only if
Re w(1   e )   e
i
i

Let F(z) =
[ z( R ((f  g)(z))),   z 2 ( R ((f  g)(z))),, ](1   ei )
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.
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

 (1    n )[n(1   )  (   )]k (n,  , )a b
n n
n2
 ei [(1   )( R ((f  g)(z))   z( R ((f  g)(z))), ]
And
2. COEFFICIENT BOUNDS AND EXTREME POINTS:
(5)
 1
wherever
 1 |+
 z ( R ((f  g)(z)))'   z 2 ( R ((f  g)(z))) '' 


Re  (1   ) R ((f  g)(z))   z ( R ((f  g)(z))) '   


i
i
 X (1   e )   e

 1 |+ (4)
and g(z) are given by
g(z) = z -
(1   ) R ((f  g)(z))   z ( R ((f  g)(z)))'
subsequently through Lemma 3, we comprise
0

z ( R ((f  g)(z)))'   z 2 ( R ((f  g)(z))) ''
E(f) =
(1   ) R ((f  g)(z))   z( R ((f  g)(z)))'
next to Lemma 2. (6) is comparable to
| ( )
0
(
) ( )| | ( ) (
) ( )|
But | ( ) (
) ( )| for
=



  ei (1   )( z   k (n,  ,  )anbn z n ) 
n2





  ei  z    nk (n,  , )anbn z n 
n2





(1   )  z   (1    n )k (n,  ,  )anbn z n ) 
 n2

0
Proof; By clarification (1),we get
© 2019, IRJET
|
Impact Factor value: 7.34
|
ISO 9001:2008 Certified Journal
|
Page 119
International Research Journal of Engineering and Technology (IRJET)
e-ISSN: 2395-0056
Volume: 06 Issue: 10 | Oct 2019
p-ISSN: 2395-0072
= (2   ) z 
www.irjet.net

(n  n(n 1)  (1   )(1    n  k (n, , )anbn z n
n2

  ei   n  n (n  1)  (1    n )k (n,  , )anbn z n
n2

 (2   ) z   (n   n(n 1)  (1   )(1    n k (n, , )anbn z
n
   n   n(n  2)  1   k (n,  , )anbn z
3. HADAMARD PRODUCT

n2

 z ( R ((f  g)(z)))'   z 2 ( R ((f  g)(z)))'' (1   ei ) 




'
 (1   ) R ((f  g)(z))   z ( R ((f  g)(z)))

Re  i


2

'
  e ((1   )( R ((f  g)(z))   z ( R ((f  g)(z))) )) 


(1   ) R ((f  g)(z))   z ( R ((f  g)(z))) '


Theorem :
f(z) =z -
a z
n2
n
n2
n

n
and g(z) = z -
b z
n2
n
n
belong to WR(λ,β,α,μ,θ)
Also | ( )
(
) ( )|
afterward effective Hadamard product of f and g is given

az   (n   n(n  1)  (1   )(1    n ) k (n,  , )anbn z n
by f g(z) = z 

a b z
n2
n n
n
 ( g  f )( z )
n2
 e
i

 n  n (n  1)  (1    n )k (n,  , )a b z
Proof:
n
n n
n2

  z   (n   n(n  1)  (1   )(1    n k (n,  , )anbn z
Since f and g WR( ,  , ,  , )
n
We have
n2
 (1    n )[n(1   )  (   )]k(n,  , ) b n 
an  1
1

n2


    n   n(n  1)  (1    n )k (n,  , )anbn z
 
n
And
n2
 (1    n )[n(1   )  (   )]k(n,  , ) a n 
bn  1
1

n2
and by applying the Cauchy-Schwarz ineuqality, we have

 
Furthermore
| ( )
(
) ( )|
(
| ( ) (
)| |
) ( )|
(2n  2 n(n  1)  2 (1    n ) 
n

 (2n  2n (n  1)  2(1    n )) k (n,  , )anbn z  0
n2 


 (1    n )[n(1   )  (   )]k(n,  ,  ) anbn 
 anbn
1


n2 




1/2
   (1    n )[n(1   )  (   )]k(n,  , ) b n  
 
an 
1
 
 n2 
Or
 n(1   )  n  (n  1)(1   )  
 k (n,  , )anbn
n2 

 1

 (1    n )(   )
1/2
   (1    n )[n(1   )  (   )]k(n,  , ) a n  
  
bn 
1
 
 n2 
Consequently we attain
This is comparable to
 (1    n )[n(1   )  (   )]k (n,  , )a b
n n
n2
 (1    n )[n(1   )  (   )]k(n,  ,  ) anbn 

 anbn  1

1


n2 




 1
Now we want to prove
on the contrary, expect that (5) holds. afterward we
obliged to show
 (1    n )[n(1   )  (   )]k(n,  , ) 
anbn  1
1
n2

 
Since
© 2019, IRJET
|
Impact Factor value: 7.34
|
ISO 9001:2008 Certified Journal
|
Page 120
International Research Journal of Engineering and Technology (IRJET)
e-ISSN: 2395-0056
Volume: 06 Issue: 10 | Oct 2019
p-ISSN: 2395-0072
www.irjet.net
 (1    n )[n(1   )  (   )]k(n,  , ) 
anbn
1
n2
Proof: By definition 2, we attain

 
hj(z)=
 (1    n )[n(1   )  (   )]k(n,  , ) anbn 
= 
 anbn
1
n2 



 cn z n ) (
)( z 
n2

d z
n2
n
n
)+

=
thus we search out the consequence.
4. WEIGHTED MEAN AND ARITHMETIC MEAN
1
(1  j )cn  (1  j )dn  z n
n2 2
z 
We necessity explain so as to hj(z) so by lemma 2 we get
Lemma 4.

If Re w   w  1  k , where 0  k
Then w 
Proof: Let
)( z 
*(

 (1    n )[n(1   )  (   )]k (n,  , )
1 ,   0.
n2
 k
 1
1

X  (1  j )cn  (1  j )d n  b n
2


1

 (1    n )[n(1   )  (   )]k (n, , )  2 (1  j)  c b
Re w   w  1  k , as w  Re ,
n n
n2

1
 (1    n )[n(1   )  (    )]k (n,  , )  (1 
2
n2
we acquire
w   w  1  k , or equivalent w (1   )    k ,
 (1  j )  (1  j ) (1   )  1  
 k
w
 1
subsequently
The proof is complete.
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) = ,(
) ( )
(

j)  d nbn

) ( )-
(7)
Thorem: Let fj(z) clear with

f j ( z )  z   an, j z n (a n, j  0, j  1, 2,......q)
(9)
n2
z (R ((f  g)(z))),   z 2 (R ((f  g)(z))),,
Definition 3. The arithmetic mean of fj




(1   ) R ((f  g)(z))   z (R ((f  g)(z)))
(j = 1,2,....,q) is
,

 
 1
Proof: Commencing (8) and (9) we container inscribe
1 q
clear within W(z) =  f j  z 
q j 1
(8)
W(z) =
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

1 q 

z

an, j z n 



q j 1 
n2

q
1 

z     an, j z n
j 1  q n  2

because fj(z)
WR( , , , , ) for every
(j=1,2,......q), so by using the theorem we get
WR( , , , , ) ,where

f ( z )  z   cn z n ,
n2
© 2019, IRJET
|

g( z )  z   d n z n
n2
Impact Factor value: 7.34
|
ISO 9001:2008 Certified Journal
|
Page 121
International Research Journal of Engineering and Technology (IRJET)
e-ISSN: 2395-0056
Volume: 06 Issue: 10 | Oct 2019
p-ISSN: 2395-0072
www.irjet.net
1 q

(1



n

)[
n
(1


)

(



)]
k
(
n
,

,

)

 q  an, j  bn
n2
 n2

q

1 

   (1    n )[n(1   )  (    )]k (n,  , )an, j bn 
q n2  n2



1 q
 (1   )  (1   )
q n2
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.
This is the absolute verification.
Theorem:
Let f(z) clear with (1) be in the class WR( , , , , )
.Then
z (R ((f  g)(z))),   z 2 (R ((f  g)(z))),,




(1   ) R ((f  g)(z))   z(R ((f  g)(z)))
Proof: As f(z)
,

 
 1
(10)
WR( , , , , ) after that by lemma 4
,we achieve
z (R ((f  g)(z))),   z 2 (R  ((f  g)(z))),,




(1   ) R ((f  g)(z))   z (R ((f  g)(z)))
,

 
 1
The verification is comprehensive.
5. CONCLUSION
Using Hadamard product with Rafid Operator, we
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
© 2019, IRJET
|
Impact Factor value: 7.34
|
ISO 9001:2008 Certified Journal
|
Page 122