The document is a research paper that presents results on the Hadamard product of p-valent functions. It begins with introducing definitions of p-valent starlike and convex functions. It then states that the class of p-valent functions is closed under Hadamard products. The main results proved in the paper are two lemmas that provide coefficient conditions for a p-valent function to belong to the classes M*(A,B) and C(A,B) under Hadamard products.

1. IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 6, Issue 1 (Mar. - Apr. 2013), PP 80-88
www.iosrjournals.org
www.iosrjournals.org 80 | Page
On Hadamard Product of P-Valent Functions with Alternating
Type
1
S.S.Pokley, 2
K.D.Patil, 3
M.G.Shrigan
1
Department of Mathematics, Kavikulguru Institute of Technology and Science, Ramtek-441106, (M.S.), India
2
Department of Mathematics, Bapurao Deshmukh College of Engineering, Sevagram Wardha-442102(M.S.),
India
3
Department of Mathematics, G.H.Raisoni Institute of Engineering & Technology, Pune - 412207, (M.S.), India
Abstract: Padmanabhan and Ganeshan have obtained some results on Hadamard product of univalent
functions with negative coefficients of the type .0a,zaz)z(f n2
1n
n2
n2 


In this paper we have
obtained coefficient bounds and convolution results of p-valent function.
Keywords: Hadamard (convolution) product, p-Valent function, Cauchy-Schwarz inequality.
Mathematics Subject Classification (2000): 30C45, 30C50
I. Introduction
Let A (p) denote the class of f normalized univalent functions of the form
(1.1)01na,1nz1na1n1)(
pn
p
z(z)f 






holomorphic & p-valent in the unit disc E = {z: z Є C;|z| < 1}.
A function f Є A (p) is said to belong to the class S (p, α) of p-valently starlike function of order α (0 ≤
α ≤ p) if it satisfies, for z ЄE, the condition
(1.2)
'
z f (z)
Re
f(z)

 
 
 
A function f Є A (p) is said to belong to the class S (p, α) of p-valently convex functions of order α (0 ≤ α ≤ p) if
it satisfies, for z ЄE, the condition
1 (1.3)
''
'
z f (z)
Re
f (z)

 
  
 
The class S (p, α) was introduced by Goodman [1].
Silverman [2] studied the properties of these functions. Let K = {w/w is analytic in E; w (0) = 0, |w (z)|
< 1 in E}. Let G (A, B) denote a subclass of analytic functions in E, which are of the form
Bw(z)+1
Aw(z)+1
, −1 ≤
A < B ≤ 1 where w (z) ∈ K.
Consider the following definitions,
S* (A, B) =






 ),(andS BAG
f
z f
f/f
,
H (A, B) =














 ),(andS
'
'
BAG
f
z f
f/f
M* (A, B) =






 ),(and BAG
f
z f
Mf/f
,

2. On Hadamard Product Of P-Valent Functions With Alternating Type
www.iosrjournals.org 81 | Page














 )B,A(G
f
z f
andSf/f)B,A(C
'
'
'
If 01na,1nz1na1n1)(
pn
p
zf(z) 






and
01nb,1nz1nb1n1)(
pn
p
zg(z) 






The class A (p) is closed under the Hardmard product (or Convolution) of f & g as,
(1.4)01nb,1na,1nz1nb1na1n1)(
pn
p
zg(z))z(f(z)h 





 .
K..S. Padmanabhan and M.S. Ganeshan [3] also S.M.Khainar and Meena More [4] studied some convolution
properties of functions with negative coefficients. In this paper we extend the results on convolution property for
p-valent function in the class M*
(A, B) and C (A, B).
II. Preliminary and Main Results
We start with lemma which will be required for further investigation convolution results.
)z(wB1
)z(wA1
1nz
1n
a1n1)(
pn
p
z
1nz1)(n
1n
a1n1)(
pn
p
zp
iff















 
2. :
p n 1 n 1f(z) ( 1) a z , a 0
n 1 n 1
n p
,
( 1) ( 1) ( )
1
1.
( )
:We have
p n 1 n 1f(z) ( 1) a z , a 0
n 1 n 1
n p
then
A function
z
is in M (A,B) iff
p n B A B a
nn p
A pB
z
       


      
 
  
 
       

Leema 1
Proof
p n 1 n 1p ( 1) a (n 1) zn 1'z ( ) n p
( , )
( ) p n 1 n 1( 1) a zn 1
n p
z
f z
G A B
f z
z
     

     


3. On Hadamard Product Of P-Valent Functions With Alternating Type
www.iosrjournals.org 82 | Page
  1nz1na1n1)(
pn
n
p
z)1p(
pn
1nz
1n
a1n1)(1)]B(n-[ApzpBA)z(w
getwe,tionSimplificaOn




















 





















pn
1nz
1n
a1n1)(1)]B(n-[ApzpBA
1nz1na1n1)(
pn
n
p
z)1p(
)z(w
Therefore
 
1
pn
1nz
1n
a1n1)(1)]B(n-[ApzpBA
1nz1na1n1)(
pn
n
p
z)1p(
havewe1,w(z)thatNotice






















 
Allowing z r 1,we get
( 1)
1
1
( ) [A-B(n 1)] a
n 1n p
Therefore
( 1) [n(B 1)-(A-B)] a
n 1n p
( 1) ( 1) ( )
1
1
( )
p n a
nn p
A pB
p A pB
p n B A B a
nn p
A pB
 

   

   

     
      
 
  
 
 
2. :
1
:We have
p n 1 n 1f (z) ( 1) a z , a 0
n 1 n 1
n p
A function
p n 1 n 1f(z) z ( 1) a z , a 0
n 1 n 1
n p
is in c(A,B) iff,
2(p p) (n 1) n(B 1) (A B) a
nn p
1.
p(A pB)
z
       

       
 
  
 
       

Leema 2
Proof

4. On Hadamard Product Of P-Valent Functions With Alternating Type
www.iosrjournals.org 83 | Page
 
 
 
1
1na
pn
21)B(n1)(nAPB)-p(A
pn
nz1n1)]a[n(n1)p(p
getwe1rzAllowing
1
nz1na
pn
21)B(n1)(nA1n1)(
1p
B)z2P-(PA
nz1n1)]a[n(n1n1)(
pn
1-p
zp)2(p
getwe1,w(z)sinceAgain
nz1n1)]a[n(n1n1)(
pn
1-p
zp)2(p
nz
1n
a
pn
21)B(n1)(nA1n1)(1pB)z2P-(PAw(z)
w(z)B1
w(z)A1
nz1)(n
1n
a1n1)(
pn
1-p
zp
nz21)(n
1n
a1n1)(
pn
1-p
z2p
iff
)B,A(G
nz1)(n
1n
a1n1)(
pn
1-p
zp
nz21)(n
1n
a1n1)(
pn
1-p
z2p
)z('f
'
)z('fz
then







 





























































2
np(p 1) [n(n 1)]a z
n 1n p
1
2p(A-PB) A (n 1) B(n 1) a
n 1n p
2( ) ( 1)[( 1) ( )] ( )
1
( ) ( 1)[( 1) ( )]
1
1
( )
p p n B n A B a p A PB
nn p
p p n B n A B a
nn p
p A PB

   


        

        

      


).B,A(Cand)B,A(Mofmembersg(z)and(z)ffor
01nb,1na,1nz1nb1nan1)(
pn
p
zg(z))z(fh(z)
defineWe








5. On Hadamard Product Of P-Valent Functions With Alternating Type
www.iosrjournals.org 84 | Page
 
: 1,We have
( 1) ( 1) ( )
1
1
( )
M (A,B). (2.1)
Bylemma
p n B A B a
nn p
A pB
Since f
      
 
  
 

Proof
 ( 1) ( 1) ( )
1
1
( )
M (A,B). (2.2)
We want to find & 1 1, ( ) ( ) ( ) M (A,B).
1 1 1 1
Now ( ) M (A,B) if
p n B A B b
nn p
A pB
Since g
A B such that A B for h z f z g z
h z
      
 
  
 

      

2.
,
: A function
p n 1 n 1f(z) z ( 1) a z , a 0
n 1 n 1
n p
p n 1 n 1g(z) z ( 1) b z , b 0
n 1 n 1
n p
are elements of class M (A,B) then
p n n 1h(z) f(z) g(z) z ( 1) a b z a ,b 0
n 1 n 1 n 1 n 1
n p
is
       

       


          

Theroem 1
A 2K1element of M (A,B) with 1 A B 1, where A 1,B
1 1 1 1 1 2K
these bounds for A and B are sharp.
1 1
       


6. On Hadamard Product Of P-Valent Functions With Alternating Type
www.iosrjournals.org 85 | Page
( 1) ( 1) ( )
1 1 1 1 1
1 ; 2.1 (2.3)
( )
1 1
( 1) ( )( 1) 1 1 1 1 1 1
( ) ( )
1 1 1 1
( 1)
1
1 1 1( )
1 1
( 1) ( )
1 1 1
1 (
p n B A B a b
n nn p
by lemma
A pB
n B A B a bp n n
A pB A pBn p
p
u a b
n nA pB n p
n B A B
where u
A
          
 
  
 
           
  
 


    
   
 
 
)
1 1
,
1 1
2 2
1 (2.4)
1 1 1 1
( 1) ( )
(2.5)
( )
(1.3)
( 1) ( 1)
1 1 1 1 1( ) ( )
1 1
1 1
pB
By Cauchy Schwarz inequlity
ua b ua ub
n n n nn p n p n p
n B A B
where u
A pB
is true if
p p
u a b u a b
n n n nA pB A pB
If A A and B B then we get

     
              
  


 
  
    
  ,
(2.6)
1 1 1 1 1
. .
1 1 1
1 2. . (2.7)
1
1
u a b u a b
n n n n
i e u a b u
n n
Therefore it is enough to find u such that
u
i e u u
u u

   

 
 
On Simplification, we get

7. On Hadamard Product Of P-Valent Functions With Alternating Type
www.iosrjournals.org 86 | Page
   
 
2
1 1 1 2
1 1
1 1 1 2
1 1
2
1 1 1 1 1
2
1 1
1 2
1
( 1) ( 1) ( ) ( 1) ( 1) ( )
( ) ( )
( 1) ( 1) ( )
( )
. . ( 1) ( 1) ( ) ( )
( 1) ( 1) ( 1)
(2.8)
(1 )
1 (1.8) (1.5)
p n B A B p n B A B
u
A pB A pB
p n B A B
u
A pB
i e p n B A B A pB u
p B n B pu
A
u
Taking B and n p in and
          
  
  
    


      
    


 
2
1 2
2
1 2
2
2 2
1
2
2 2
'
( 1) 2 ( 1)
(1 )
(3 )
( 1)
( )
1 2
( ) ( 1) ( 1) ( )
(1 2 )
( )
(2.9)
( ) ( 1) ( 1) ( )
: ( ) ( , ) ( ) (
abovewe get
p p pu
A
u
p u
A
u
A pB
p
A pB p p B A B
A p k
A pB
where k
A pB p p B A B
If f z M A B and g z M A 
   





 
  
          
 


        
 Theroem 2.2
 
'
1 1
1
1 1
' '
' ' ' '
1 1 1 1
1 1
, ) ( ) ( ) ( , )
2
1,
1 2
( )
(2.10)
(3 1)(3 1) [( ) ( 1)]( )
Pr arg 1,
( 1) ( 1) (
B then f z g z M A B
A k
where A B with
k
A pB A pB
k
B p A B p A A B P A PB
oceeding with the ument developed in Theroem werequire
p n B A B

 

 

 

          
    
Proof :
    ' ' '
1 1 1 1
' '
1 1 1 1
1 1 1
1 1 1 1
1 1 1 1
1 1 1 1 1
1 1
1 1 1 1
( 1) ( 1) ( )) ( 1) ( 1) ( )
( ) ( ) ( )
( 1) ( 1)
( )
( )
1 ( ) ( ) ( 1)
( )
( ) ( ) ( 1
p n B A Bp n B A B
A pB A pB A pB
That is
p n B A B
A pB A pB
A pB n A pB
B A pB A B p
n A pB
Notice that
A pB A B p




            
  
   
 
 
 

     

    
. ,
)
decreases as n increases Simplifying we get

8. On Hadamard Product Of P-Valent Functions With Alternating Type
www.iosrjournals.org 87 | Page
 
 
 
' '
1 1
' ' ' '
1 1 1 1 1
' '
1 1
' ' ' '
1 1 1 1 1
1 1
1
( )
(2.11)
1 (3 1)(3 1) [( ) ( 1)]( )
2,
2 ( )
1 (3 1)(3 1) [( ) ( 1)]( )
. . 2 (2.12)
1
n A pB A pBA pB
B B p A B p A A B P A PB
Taking n we get
A pB A pBA pB
B B p A B p A A B P A PB
A pB
i e k
B
n A pB
where k
 

           

 

           





' '
' ' ' '
1 1 1 1
( )
(3 1)(3 1) [( ) ( 1)]( )
A pB
B p A B p A A B P A PB

          
 
:
(2.13)
(
' 'If f(z) C(A,B) and g(z) C(A ,B ) then f(z) g(z) C(A ,B ) where
1 1
A 2k
1A 1,B with
1 1 1 2k
' 'A pB (A pB )
k
' ' ' '9(3B p A 1)(3B p A 1) [(A B ) (P 1)](A PB )
1 1 1 1
Letus verify with the following example
f z
   

 

 

          
Theroem2.3
Proof :
( ) 2) ( , )
3(3 2)
' '( ) 2 ' '( ) ( , )
' '3(3 2)
' '( )( ) 1
( ) ( ) ( , ) (2.14)
1 1' '9(3 2)(3 2)
' '( 1) ( 1) ( ) 9(3 2)(3 21 1 1
1 1
A pBp Pz z C A B
B p A
A pBp p
g z z z C A B
B p A
Then
A pB A pBp p
f z g z z z C A B
B p A B p A
if
p n B A B B p A B p A
A pB
   
  
 
  
  
  
   
     
          


 
)
' '( )( )
,
(1 2 )
1
' 'A pB (A pB )
' ' ' '9(3B p A 1)(3B p A 1) [(A B ) (P 1)](A PB )
1 1 1 1
A pB A pB
Simplifying we get
A p K
where K
 
 
 

          

9. On Hadamard Product Of P-Valent Functions With Alternating Type
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References
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[2]. H.Silverman, On a close-to-convex function, Proc.Amer.Math.Soc.36, No.2 (1972), 477-484.
[3]. K.S.Padmanabhan and Ganeshan, Convolution of certain class of univalent function with negative coefficient, Indian J Pure
Appl.Math.Sept.19 (9) (1998), 880-889.
[4]. S.M.Khairnar and Meena More, Convolution properties of univalent functions with Missing Second Coefficient of Alternating
Series,Int.J.Of Math Analysis,Vol.2,2008, No.12,569-580.
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