Advances in Pure Mathematics, 2011, 1, 193-200
doi:10.4236/apm.2011.14034 Published Online July 2011 (http://www.SciRP.org/journal/apm)
Copyright © 2011 SciRes. APM
Inclusion and Argument Properties for Certain
Subclasses of Analytic Functions Defined by
Using on Extended Multiplier Transformations
Oh Sang Kwon
Department of Mathematics, Kyungsung University, Busan, Korea
E-mail: oskwon@ks.ac.kr
Received March 28, 2011; revised April 27, 2011; accepted May 5, 2011
Abstract
Making use of a multiplier transformation, which is defined by means of the Hadamard product (or convolu-
tion), we introduce some new subclasses of analytic functions and investigate their inclusion relationships
and argument properties.
Keywords: Subordination, Starlike Functions, Convex Functions, Closed-to-Convex Functions, Multiplier
Transformation, Multivalent Functions, Argument Principle
1. Introduction
Let
p
A
denote the class of functions f normalized by

=1
=(:{1, 2,3,})
pkp
kp
k
fzza zp

 (1.1)
which are analytic and -valent in the open unit disk
p

=: and<1Uzz z
If
f
and
g
are analytic in U, we say that
f
is
subordinate to
g
, and write
 
or( )
f
gfzgzzU
if there exists a Schwarz function , analytic in
with and

z
U

0=0

<1z
in , such that zU
 
=
f
zg z
for . zU

We denote by *
p
S
and p

C
the subclasses of
p
A
consisting of all analytic functions which are,
respectively, -valent starlike of order
p
(0<p
)
in and -valent convex of order
U p
(0<p
)
in U.
Let
M
be the class of analytic functions
with
, which are convex and univalent in U and

0=
1
satisfy the following inequality:
Re> 0()zz
U
Making use of the aforementioned principle of
subordination between analytic functions, we define each
of the following subclasses of
p
A
:

 
*;
1
:Re :and
(0< ;;)
p
p
S
zf z
f
fA z
pfz
pz UM




 





(1.2)

 
;
1
:Re :and1
(0< ;;)
p
p
K
zf z
f
fA z
pfz
pz UM





 





(1.3)
For 0:{0,1, 2,}m
 , we define the multiplier
transformation
,,
m
J
pl
of functions
p
f
A
by
 
 
*1
,;, :Re:and;..
(0,<;; ,)
ppp
zf z
CffAgSst
pgz
pz UM
 
 



 







z
(1.4)
O. S. KWON
Copyright © 2011 SciRes. APM
194

=1
,, =
(>0; 0;)
m
mp
kp
k
lk kp
J
plfz zaz
l
lz




U
(1.5)
Put

,,
=1
=
(;>0;0;)
m
mp
pl
k
lk
zz z
l
ml zU




kp
(1.6)
The operators ,,
m
p
l
and ,1,
m
p
l
, are the multiplier
transformations introduced and studied earlier by Sarangi
and Uralegaddi [16] and Uralegaddi and Somanatha ([1]
and [2]), respectively. Correspending to the function

,,
m
plz
defined by (1.6), we introduce a function

,
,,
m
plz
given by the Hadamard product (or convolu-
tion):

,
,, ,,
*= (>
1
p
mm
pl plp
z
zz
z

 
)
p
Then, analogous to
,,
m
J
pl
, we have define a
new multiplier transformation

,, :
m
p
p
I
plA A
as follows:
 
,
,,
,, =*
mm
pl
I
plfz zfz


(1.7)
We note that
 
01
2
,1,1=and1,1, 2=
p
I
pfzfz Ifzzf
z
It is easily verifed from the above definition of the
operator
,,
m
I
pl
, that


 
1
,,
=,, ,,
m
mm
zIpl f z
pIpl fzIplf z



(1.8)
and


 
1
,,
=,, ,,
m
mm
zIplf z
lIpl f zplIplf z


 

(1.9)
The definition (1.6) of the multiplier transformation
,,
m
p
l
is motivated essentially by the Choi-Saigo-
Srivastava operator [3] for analytic functions, which
includes a simpler integral operator studied earlier by
Noor [7] and others (cf. [4-6]).
Next, by using the operator
,,
m
I
pl
defined by
(1.7), we introduce the following subclasses of analytic
functions:


*
=: and ,,;
(;,,>0;;0<1
m
pp
ff AI p lfzS
Ml m
,
,, ;
m
pl
S
)

 


(1.10)


,
,, ;
=:and ,,;
(;,,>0;;0<1
m
pl
m
pp
K
ff AI p lfzK
Ml m

)

 


(1.11)
and


,, ,;,
=:and ,,,;,
(,;,, >0;;0, <1)
pl
m
p
ff AI p lfzC
Ml m

,m
C

 


(1.12)
We also note that
 
,, ,,pl pl
zf zS

,m
,
;;
m
fz K

(1.13)
In particular, we set


,,
,, ,,
1
mm
Az


;=;,(1<<1)
1
pl pl
SS
ABBA
Bz





(1.14)
and

,,
,, ,,
1
;= ;,(1<<
1
mm
pl pl
Az
KKABB
Bz



 


1)A
(1.15)
In the present paper, we investigate some inc
re
lusion
lationships and argument properties associated with
such multivalent functions in the class
p
A
as those be-
longing to the subclasses

,
,, ;
m
pl
S
,
,
,;
,
m
Kpl
and
,
,, ,;,
m
pl
C

defined(1.12),
by (1.10), .11) and (1
respectively.
2. Inclusion Properties
emma 2.1: Let
L
be convex univalent in with U
01
and
Re> 0z
 
(,
). Ip is
n U
f
analytic iwith
0=1p, then

 
()
zp z
pzzz U
z
 

implies that
pz z
2.2: Let
(zU).
Theorem
M
with



Re>max,
min
zU
l
p
zpp










O. S. KWON195
then ;
 
,1 ,1,
,,,, ,,
;;
mmm
plpl pl
SSS
 

 


.
Proof. First of all, we show that
 
,1 ,
,, ,,
;;
mm
pl pl
SS



. Let

,1
,, ;
m
pl
fS
and set
 


,,
1m
zIpl f z
=,,
m
pz pIp fz
l



(2.1)
where the function
pz
obtain
is analytic in with
.
ng (2.1), we
U

0=1p
Applyi





1
,,
m
pp
pz
Iplf
,, =
m
Iplfz
z

 (2.2)
tiating both sides of (2.2)
and multiplying the reseulting equation by , we have
By logarithmically differen
z



 

,,
1
=()
m
zIplf z
zp z
pzz U
ppz





Since
, by app
ma 2.1 to (2.3), it follows that in , that
is, that
,,
m
pIplfz



(2.3)

Re> 0pz
 
 lying Lem-
 
pz z
U
 
,
,, ;
m
pl
fz S
.
prove the second To parm t of Theore 2.1, let

z S
 
,
,, ;
m
pl
f
and put
 


1
1
=,,
m
qz pIp
lfz
,,
1m
zIplf z




where the function is analytic in with
.
precisely the sanner, we can ult
that in , that is, that

qz
me ma
U
find the res

0=1q
In
 
qz z
U
 
1,
,, ;
m
pl
fz S
unhe hypothesis der t

Re> 0pzp
 




t

zM
with
l

Theorem 2.3: Le



>max ,
min
zU
l
p
Re zpp







then ;
 
,1 ,1,
,,,, ,,
;;
mmm
plpl pl
KKK
 




 
 
,1 ,1
,, ,,
,,
,, ,,
;;
;;
mm
pl pl
mm
pl pl
fzKzfzS
zfzSf zK








 
and
 
,,
,,,,
;;
mm
pl pl
fzKzfzS





1, 1,
,, ,,
;;
mm
pl pl
zfzSf zK




 
which evidently prove Theorem 2.3.
By setting

1
=(1<<1;
1
Az
zBA
Bz
)zU
in Theorems 2.2 and 2.3, we deduce the fo
corollary.
Corollary 2.4: Suppose that
llowing

.
Proof. Applying (1.11) and Theorem 2.2, we observe
that
1>max ,
1
l
p
A
Bpp









Then, for the function classes defined by (1.12) and
(1.13),

1,
,,,,,, ;,
m
plpl pl
SA

B
and
,1 ,
;, ;,
mm
SABSAB



,1 ,1,
,,,, ,,
;, ;,;,
mmm
plpl pl
K
AB KAB KAB
 

 


Theorem 2.5: Let with ,
M



>max ,
min
zU
l
p
Re zpp










then

,1 ,
,, ,,
,,
,;, ,;,
;,
mm
pl pl
pl
CC
C



1, ,
m
 

Proof. We begin by proving that
,1 ,
,, ,,
,;, ,;,
mm
pl pl
CC


 
, which is tfirst
inclusion relationship asserted by Theorem 2.5.
he
Let
,1
,, ,;,
m
pl
fz C

. Then there exists a
function
;
p
kz S
* such that


 
1,,
1()
m
zIpl f zzzU
pkz


Choose the function




g
z such that

1,, =
m
p
Iplgzk
 
*;zS

Copyright © 2011 SciRes. APM
O. S. KWON
Copyright © 2011 SciRes. APM
196
Then
 
,
,,
;;
pl
S
 
,1
,,
mm
pl
gz S



, and



 
1
1
,,
1()
,,
m
m
zIpl f zzzU
pIplgz







(2.4)
Now let


,,
=,,
m
m
zIplf z
Iplgz
pz





(2.5)
where the function
pz
is analytic in with U
0=1p.
nd th
Using (1.9), we fiat









 






1
1
2
,, ,,
,,
11
=
,, ,, ,,
1,, ,,
1
=
,, ,,
1
=
mm
m
mmm
mm
mm
zzI p lfzI p lfz
zIpl f z
pp
IplgzzI p lgzI p lgz
zIplfzzIplfz
pzI p lgzI p lgz
p







 










 
















 








2
,, ,,
1,, ,,
,,
,,
mm
mm
m
m
zIplfz zIplfz
Iplgz Iplgz
zI p lgz
Iplgz













Since
 
,
,, ;
m
pl
gz S
, then we set
 


,,
1
=,,
m
m
zIplf z
qz pIplgz








 

,,
,,
=1
m
m
zI p lfz
Iplfz
pzpz
pqz ppz




(2.8)
Hence
(2.6)
w in Uumption that here q with the ass
 
z z
M
. By (2.5),







2,,
,,
=1
m
m
zI plfz
Iplgz
pzpzpqz
ppz







,, =
,,
m
m
zIpl f zppz
Iplgz

and mu, we obtain
(2.7)
 
 
Differentiating both side of (2.7) with respect to z
ltiplying by zComputing the above equations, we can obtain



 

 

 



 

1
1
,,
1
,,
=
m
m
zIpl f z
pIplgz
pz
ppqz
zp z
pz pqz
1
1
=pp
z pz

 1
pqz ppz








 

 
 
O. S. KWON197
Since , applying Lemma
2.1 wit


Re> 0pz
 

h
 
1
=wz pz
 

, we can show
that
p
zz
in U, so that
,
,, ,;,
m
pl
fz C

.
3. Argument Properties
emma 3.1: Let L
be convex univalent in and U
betic in with analy U

Re z
0. If
pz is
analytic in U and

0= 0
, thepn
)pzz U
 
(zzp zz


impli
 
z
(z
es that
pz U
).
Lemma 3.2: Let be analytic in with p U

0=1p and

=0pz
for
o points ,zz U such that
all . If there exist
zU
tw 12




11 22
=arg <arg
2
p pz

<arg=z pz2
(3.1)
for some 1
and 2
(1
,2>0
) and fr all oz
12
(<= )zz z
.



112 2
12 12
12
=and=
22
zp zzpz
im i
pz pz
 
m




(3.2)
where 1
1
b
mb
and 21
12
=tan
4
bi



.
Theorem 3.3: Let
p
f
A
. 12
0< ,1
. 0<< p
.
If



1
12
1
,,
<arg <
22
,,
m
m
zIplf z
Iplgz







,1
,, ,;,
m
pl
g
SpA
B, then for some



1
<a
rg <
zI
2
,,
22
,,
m
m
pl f z
Iplgz




where


, 2
1
are the solutions for the following
equatio ns:





1
1
1
(
2
=tan
2
)1cb
1
1
12 1
os
2
1
211sin
12
t
pA bbt
B

 

 





  
 




and





1 1
1
22
12 1
22
=tan1
211sin
12
pA bbt
B
 







 
 




is given by (3.2), and
2
1cosbt



()

b
 
 

1
11
=
tt 2
2
=
cos 11
pAB
pAB B







(3.3)
Proof. Let
 


,,
1
=,,
m
m
zIpl f z
pz pIplgz





.
in with . By
using (1we obtain
(3.4)
Differentiating both sides of the above equation and
multiplying the resulting equation by , we find that
Then

pz is analytic U

0=1p
.9),




 
1
,,
=,, ,,
m
mm
ppz Iplgz
pIpl f zIpl fz





z
 








1
,,
(),,
=,, ,,
m
m
mm
ppzIplgz
ppz Iplgz
pIplfzI plfz


 

 


Since
,1
,, ,;,
m
pl
g
zS pAB
hat
, by Corollary 2.4, it
follows t
,
,, ,;,
m
pl
g
zS pA
B.
Next we let
 


,,
1
=
m
m
zI p lgz
qz gz
,,
pIpl





.
Copyright © 2011 SciRes. APM
O. S. KWON
198
Then, using (1.9), we have




,, =
,,
m
m
Iplgz
ppq
Iplgz
z


 (3.5)
From (3.4) and (3.5), we obtain


 

1
1,,
=
m
pIp
lgz
zp z
pz pqz
,,
1m
zIplf z





Furthermore, by using a known result, we have



22
1<
11
A
BAB
qz BB

(3.6)
Thus, from (3.6), we obtain

=exp2
i
pqz
 

 

where, in terms of 1
t given by (3.3).

11
<<tt
11
<<
11
pA p
BB

A

 



We note that is analytic in with
p U
0=1p.
Let
=hz
lar domain
be function whicaps
angu
theh mU onto the
 
12
:<< with0
22
arg h

=1


Applying Lemma 3.1 for this function with


h
 
1
=
zpqz


we see that
e> 0pz (), and hence RzU
=0pz
(zU
). By using
12
zz U
Lem if there exist ma 3.2,
two points ,
such th
we obtain (3.2
at the
h
condition (3) is
) under te constraint (3.2).
A
.1
satisfied, then
nd we obtain
 

 
 



11
1
1
11
12
1
12
12 1
1
1
arg
=arg1 exp
222
2
1
1
cos 1
2
tan
2
2c
os1
2
1cos
2
tan
21
21
zp z
pz pqz
i
im
m
bt
pA
B



 











 












 

 
m













1
12 1
=2
11cos
2
bbt
 







 
 




and
 







12 1
22 1
22 2
2
12 1
1cos
π2
arg =
tan
22
1
211cos
12
bt
zp z
pz pqz pA bbt
B

  



 












 
 




which would obviously contradict the assertion of
Theorem 3.3. We thus complete the proof of Theorem
3.3.
If we let 12
=
onseque
in Theorem 3.5, we easily obtain
the following cnce.
rollary 3.4: Let Co
p
f
A. 0< 1
. 0<< p
. If



,,
arg< 2
,,
m
m
zIpl f z
Iplgz





,1
,, ,;,
m
pl
g
SpA
B, then for some



,,
arg ,,
<2
m
m
zIpl f z
Iplgz





where
is the solutions for the following equation:
Copyright © 2011 SciRes. APM
O. S. KWON199



1
1
1
1cos
22
=tan 111cos
12
bt
pA bbt
B
  










 





b is given by (3.2), and



1
,,
<arg<
2
m
m
zIplf z
Ip
2
2
,,
lgz



(3.7)


Theorem 3.5: Let


p
f
A. 12
0< ,1
. 0<<p
. If



12
,,
<arg<
22
,,
m
m
zIplf z
Iplgz




for some
,
,, ,;,
m
pl
g
SpA
B, then



1
11
2
,,
<arg
2,,
<2
m
m
zIpl f z
Iplgz





where 1
, 2
are the solutions for the following
equations:






12 1
1
11
12 1
1cos
22
=tan 1
211
12
bt
pA l
pbb
B

 



 





 
 




cost
and






12 1
1
22
12 1
1cos
22
=tan 1
211cos
bt
pA lbbt
B




 






 
 




is given by (3.2), and
12
p


b




11
1
2
=
()
2
=co s
11
tt
pAB
l
pABp



 


B
(3.8)
we letIf 12
=
in Theorem 3.5, we easily obtain
onsequence.
Corollary 3.6: Let
the following c
p
f
A



,,
arg
m
<2
,,
m
zIpl f z
Iplgz





for some
,
,, ,;,
m
pl
g
SpA
B, then



1,,
arg< 2
,,
m
m
zIplf z
Iplgz





. 0< 1
. 0<< p
. If
w here is the solutions for the following equation:



1
1
1
1cos
22
=tan 111cos
12
bt
pA l
pbb
B
 










 






t
Copyright © 2011 SciRes. APM
O. S. KWON
200
is given by (1.17), and
b




11
1
2
=
2
=cos
1
tt
pAB
l
pABp B

1






 

(3.9)
4. Acknow
The research was supported by Kyungsung University
Research Grants in 2011.
[1] B. A. Uralegaddi and C. Somanatha, “Certain Differential
Operators for Meromorphic Functions,” Houston Journal
of Mathematics, Vol. 17, 1991, pp. 279-284.
[2] B. A. Uralegaddi and C. Somanatha, “New Critetia for
Meromorphic Starlike Functions,” Bulletin of the Austra-
lian Mathematical Society, Vol. 43, No. 1, 1991, pp.
137-140. doi:10.1017/S0004972700028859

ledgements
5. References
[3 H. M. Srivastava, “Some Inclu-
sion Properties of a Certain Family of Integral Ope
tors,” Journal of Mathematical Analysis and Applications
Vol. 276, No. 1, 2002, pp. 432-445.
doi:10.1016/S0022-247X(02)00500-0
] J. H. Choi, M. Saigo and
ra-
,
[4] J.-L. Liu and K. I. Noor, “Some Properties of Noor Inte-
gral Operator,” Journal of Natural Geometry, Vol.
2002, pp. 81-90.
[5] J.-L. Liu, “The Noor Integral and Strongly Starlike
Journal of Mathemalysis and Applica-
, Vol. 261, No. 2, 2001, pp.
doi:10.1006/jmaa.2001.7489
21,
Func-
tions, tical Ana
tions 441-447.
[6] K. I. Noor and M. A. Noor, “On Integral O erators,”
Journal of Mathematical Analysis and Applicans, Vol.
238, No. 2, 1999, pp. 341-352.
doi:10.1006/jmaa.1999.6501
p
tio
I”
Journal of Natural Geometry, Vol. 16, 1999, pp. 71-80.
[8] K. S. Padmanabhan and R. Parvatham, “On Analytic
Functions and Differential Subordination,” Bulletin
Mathématique de la Société des Sciences, Mathématiques
de Roumanie, Vol. 31, 1987, pp. 237-248.
[9] M. Nunokawa, S. Owa, H. Saitoh, N. E. Cho and N.
Ta-kahashi, “Some Properties of Analytic Functions at
Extremal Points for Arguments,” preprint, 2003.
[10] P. Eenigenburg, S. S. Miller, P. T. Mocanu and M. O.
Reade, “On a Briot-Bouquet Differential Subordination,”
General Inequalities, Vol. 3, 1983, pp. 339-348.
[11] R. J. Libera and M. S. Robertso “Meromorphic Close-
to-Convex Functions,” Michigan ,
Vol. 8, No. 2, 1961, pp. 167-176.
doi:10.1307/mmj/1028998568
[7] K.. Noor, “On New Classes of Integral Operators,
n,
Mathematical Journal
[12] S. K. Bajpai, “A Note on a Class of Meromorphic Univa-
lent Functions,” Revue Roumaine de Mathématiques
u, “Differential Subordina-
Pures et Appliquées, Vol. 22, 1997, pp. 295-297.
S. M. Sarangi and[13] S. B. Uralegaddi, “Certain Differential
Operators for Meromorphic Functions,” Bulletin of the
Calcutta Mathematical Society, Vol. 88, 1996, pp.
333-336
[14] S. S. Miller and P. T. Mocan
tions and Univalent Functions,” Michigan Mathematical
Journal, Vol. 28, No. 2, 1981, pp. 157-171.
Copyright © 2011 SciRes. APM