Advances in Pure Mathematics, 2011, 1, 228-234
doi:10.4236/apm.2011.14040 Published Online July 2011 (http://www.SciRP.org/journal/apm)
On Certain Subclasses of Multivalent Functions Associated
with a Family of Linear Operators
Jae Ho Choi
Department of Mathematics Education, Daegu National University of Education, Daegu, South Korea
Received March 16, 2011; revised April 7, 2011; accepted April 20, 2011
Abstract
Making use of a linear operator , which is defined here by means of the Hadamard product (or
convolution), we introduce some new subclasses of multivalent functions and investigate various inclusion
properties of these subclasses. Some radius problems are also discussed.

,
pac



Keywords: Multivalent Functions, Hadamard Product (or Convolution), Linear Operators, Radius Problem
1. Introduction and Definitions
Let denote the class of functions

p

f
z of the
form

=1
=(:{1, 2,3,})
ppk
pk
k
fzza zp


 
 , (1)
which are analytic in the open unit disk

=: a <1zz ndz .
We define the Hadamard product (or convolution) of
two analytic functions
 
=0 =0
=and=
kk
kk
kk
f
zaz gzb


z

,
as

=0
:
k
kk
k
fgzabz z

 
.
For ,
a0 c  () H. Saitoh
[13] introduced a linear operator

0:,2,1,
 

0
 
,:
pac pp
defined by
 


,:,; (;
pp
acf zaczfzzfp  )
(2)
where


=0
,; :()
kp k
p
kk
a
aczz z
c



, (3)
and


k

is the Pochhammer symbol defined, in terms
of the Gamma function, by



  
1(
==
11(
=0)
)
k
k k
kk

 

 
 
.
The operator


,
pac is an extension of the Carlson-
Shaffer operator (see [2]). In [3], Cho et al. introduced
the following family of linear operators






,:
pac pp

 analogous to


,
pac (see
also [14]):






†
0
,:,;
,;>;;
pp
acf zaczf z
acpz fp





 

.
(4)
where


†,;
pacz
 is the function defined in terms of the
Hadamard product (or convolution) by the following
condition


†
,;,; =
1
p
pp
p
z
acz acz
z



, (5)
where
p

is given by (3). If

f
z is given by (1),
then from (3), (4) and (5), we deduce that





 
=1
,
=.
!
p
pk kk
kp
kk
ac fz
pc
zaz
ak








p
z
(6)
It is easily seen from (6) that






11,1 =
ppfzfz and



1,1=
p
zf z
pfzp



J. H. CHOI
229
.


  
1,
=, 1,
p
pp
zacfz
aacfzapacfz




 


(7)
and


 
1
,
=, ,
p
pp
zacfz
pacfz acfz








(8)
Clearly, from (7) and (8), we have





1,
Re> 0
1,
,
Re>( )
1,
p
p
p
p
zacfz
acfz
ac fzap ap
a
acfz























(9)
and




 
1
1,
Re> 0
,
,
Re>0 .
,
p
p
p
p
zacfz
ac fz
ac fz
p
ac fz
























(10)
When and
 
0
=:anpn 0==1c

,
the linear operator

1
1
,1 =
p
np
np  

, was introduced
and studied by Liu and Noor [5] (see also [9] and [10]).
Moreover, when , was first
introduced and studied by Noor [8] which is known as
Noor Integral operator.
=1p

1
11,1 =n
n
Let

k

 be the class of functions analytic
in the unit disk satisfying the properties

hz



0=1h
and

2
0
Re d
1
hz k 



, (11)
where =i zre

, and 2k0<1

. For =0

, the
class k was introduced in [11]. For

0=
k
 =0

,
, we have the well known class of functions
with and the class gives us the
class
=2k
=2

Reh z

>0k

 of functions with

zReh >

. Also we
can write, for

k
hz
 

 as
  
2
0
112
1
=d
21
it
it
ze
hz t
ze






, (12)
where is a function with bounded variation on
such that

t 
[0, 2]
 
22
00
d=2anddtt




From (11) and (12) it can be seen that ()
k
h


 if
and only if there exist 12 ,(hh )

 such that
  
12
11
=42 42
kk
hzh zhz
 

 
 . (14)
It is known [7] that the class


k

 is a convex set.
We also note that


k
hz


 if and only if there
exists k q

 such that




=1hz qz

 

. (15)
By using the linear operator , we now define
some subclasses of

,
pac




p
ap
as follows:
Definition 1: Let , 0 c  , >p


, >0

,
0

, and 2k0<1


. A function




f
zp

,, ,
is said to be in the class ,,,
pkac


 if and
only if it satisfies





 
1
1,
(1 )1,
,1, ,
,1,
p
p
pp
k
pp
acfz
acgz
acfza cfz
acgza cgz


















 


 

 

 



 

(16)
where




g
zp satisfies the condition






,(0< 1;)
1,
p
p
ac g zz
acgz













. (17)
We note that g is starlike univalent in when 
=== =1ac p

in (16).
Definition 2: Let 0
,ac   , 0

, >0

, 0

,
and 2k0<1


. A function



f
zp is said
to be in the class


,,,,,,kac
p


 if and only if it
satisfies





 
1
1
1
,
(1 ),
,, ,
,,
p
p
pp
k
pp
ac fz
ac g z
ac fzac fz
ac gzac gz





























 

(18)
where




g
zp satisfies the condition




  
1,0<1;
,
p
p
ac fzz
ac gz.











 
(19)
k  . (13)
In this manuscript, we investigate several inclusion
and other properties of functions in the classes


,,, , , ,
pkac


 and

,,, , , ,
pkac


 which
are introduced above. Furthermore, some radius pro-
blems are also considered.

230 J. H. CHOI
2. Main Results
In order to establish our results, we require the following
lemmas.
Lemma 1: [6] Let 12 =uu iu

and 12 =vviv

and
let be a complex-valued function satisfying the
conditions:
(,)uv
1) is continuous in a domain (,)uv

1,0 
2,
2) and .

1,0>0
0 3) whenever
21
)vRe (,iu


21
,iu v  and

2
12
12vu .
If is analytic in , with , such that

pz

,zp



0=1p


pzz  and



> 0p z

Re for ,pz z
z


,
then .

Re>0pz
Lemma 2: [12] If is analytic in with
, and if

hz 

0=1p

is a complex number satisfying
Re0 (0)

, then
 

Re>(0<1)hzzh z 

 implies
 
1
Re>121hz 

, (20)
where 1

is given by
1
1R
0
d
=1e
t
t




which is an increasing function of Re

and
1
12 <1

. The estimate (20) cannot be improved in
general.
Lemma 3: [4] Let be analytic in with
and . Then, for

qz
R>0


0=1q

eq z(z)
=<zr1,
 
11
Re
11
rr
qzqz
r


r 
and
 
2Re
1z
qz
qz r
.
We begin by proving the following.
Theorem 1: Let 0



,,
. If
,,,,
p
fkac


, then

 
1,
1,
p
k
p
acfz
acgz











,
where
2
=2
a
a






, (21)
and

g
p satisfies the condition (16) and


  

0
0
2
0
, Re
=,=
1,
p
p
ac g z hz hz acgz
hz





. (22)
Proof. Let


,,,,,,
p
fkac


  and set



1, =1
1,
p
p
acfz hz
acgz













, (23)
where


hz is analytic in with 


0=1h and we
write
  
1
11
=42 42
kk
hzh zhz
 

 
 
2
. (24)
A simple computation using (23) and (24) gives








 

1
1
1
0
2
2
0
1,
(1 )1,
,1,
,1,
1
1
=1
42
1
11.
42
p
p
pp
pp
acfz
acgz
acfza cfz
acgza cgz
zh z
khz ah z
zh z
khz ah z

































 
 














 











Now we form the functional by choosing

,uv



1
==
i
uhzuiu
2

and . Thus

1
==zv2i vzhiv

 

0
1
,=1 v uv uah z




. (25)
The conditions 1) and 2) of Lemma 1 are clearly
satisfied. Therefore, we show that the condition 3) of
Lemma 1 is satisfied.
By virtue of (25), we have
  


0
21 1
2
0
1
1Re
Re, =
1
=,
hz
iu vv
ahz
v
a

 

 




where

is given by (22). Thus, for


2
12
12vu,
we obtain




22
22
21
11
Re ,=,
22
u
A
Bu
iu vaC
 
 
 

where

J. H. CHOI
231

=2 1Aa

   ,

=1 and=BCa

  

.
Since , and by (21), we get
. Hence, by applying Lemma 1, it
follows that which implies that
. The proof of Theorem 1 is thus completed.
0B

21
,v
i
>0C

=1hi 
0A
z 
Re 0iu
h

,2;
k
Remark: If we put and =anp==1c

in
Theorem 1, we have the result due to Noor and Arif [9,
Theorem 3.1].
Theorem 2: Let 0

. If

,,,,,,
p
fkac



 ,
then



 
,
,
p
k
p
ac f
ac g


z
z








, where


2
=2
p
p

 




 
,
and

g
p satisfies the condition (18) and



0
2
0
Re
=||
hz
hz
 ,




1
0
,
=,
p
p
ac fz
hz ac gz




.
Proof. Let

,,,,,,
p
fkac


  and set

 
,=1
,
p
p
ac fzhz
ac g z











,
where is analytic in with . Then, by
using same techniques as in the proof of Theorem 1, we
obtain the desired result.

hz 

0=1h
We note that =

 when =0

in Theorem 1.
Corollary 1: Let 1

. If

,,, ,0,1,
p
fkac


 ,
then


,()
,
p
k
p
ac fzz
ac g z


 

.
Proof. It is clear that, for 1

,








,
,
1, ,
=(1 )1,,
1,
(1).
1,
p
p
pp
pp
p
p
ac fz
ac g z
a cfzacfz
a cgzacgz
acfz
acgz

















 






This implies that












12
,
,
1, ,
1
=1 1, ,
1,
11
1=
1,
p
p
pp
pp
p
p
ac fz
ac g z
a cfzacfz
acgz acgz
acfz PP
acgz









1
1.













 
 
 

 






Since


k

 is a convex set (see [7]), by using
Theorem 1 and Definition 1, we observe that


k12
,PP

 and





,
,
p
k
p
ac fz
ac g z





,
which completes the proof of Corollary 1.
Making use of Theorem 2 and Definition 2, we can
prove the following result.
Corollary 2: Let 1

. If

,,,,0,1,
p
fkac


 ,
then




 
1
1
,()
,
p
k
p
ac fzz
ac g z





.
Next, by using Lemma 2, we prove the following.
Theorem 3: Let

be a complex number satisfying
R>0e

and let , >0a0 c , 0

 and >0

.
If


f
p satisfies the condition

 
1
1,
1
,1, ,
p
p
pp
k
pp
acfz
z
acfzacfz
zz

















 






 
then

1, () ()
p
k
p
acfz z
z









,
where

 
1
Re
1
11
0
=121 with=1d
a
tt


 


  


.
(26)
The value of

is best possible and cannot be
improved.
Proof. If we set

232 J. H. CHOI

 
12
1,
11
==
42 42
p
p
acfz
z
kk
hzh zhz







 

 
 

,
then and is analytic in . By applying
(7), we have

0=1hh

 
 
1
1,
1
,1,
=.
p
p
pp
pp
k
acfz
z
ac fzac fz
zz
hzzhz
a
























Therefore, by virtue of Lemma 2, we see that
, where

(= 1, 2)
i
hi  

is given by (26). Hence
we conclude that

k
h

, which evidently proves
Theorem 3.
By using (8) instead of (7) in Theorem 3, we have the
following.
Theorem 4: Let

be a complex number satisfying
Re> 0

and let 0 ,ac

 , 0

 and >0

. If

f
p satisfies the condition

 
1
1
1,
1
,, ,
p
p
pp
k
pp
acfz
z
ac fzacfz
zz


























 
then
 
,()
p
k
p
ac fzz
z









,
where

is given by (26) with
1
R
1()
10
=1 d
e
p
t






 


t The value of

is best possi-
ble and cannot be improved.
Theorem 5. Let 2 0<
1

 

. Then

1
,,,,0,, ,,,,0,,
pp
kac kac

2

  .
Proof. If 2=0

, then the proof is immediate from
Theorem 1. Let 2>0

and

1
,,, ,0, ,
p
fkac


 
1
.
Then there exist two functions
H
,


2k
H

 such
that






1
1
11
1,
11,
,1,
=
,1,
p
p
pp
pp
acfz
acgz
acfza cfz
H
z
acgza cgz















 

 
 

 




and

 
2
1, =
1,
p
p
acfz
H
z
acgz










.
Then







 
2
1
2
22
12
11
1,
11,
,1,
,1,
=1.
p
p
pp
pp
acfz
acgz
ac fzac fz
ac g zac gz
HzH z



































(27)
Since


k

 is convex set (see [7]), it follows that
the right hand side of (2.8) belongs to


k

, which
proves Theorem 5.
Next, we consider the generalized Bernardi-Libera-
Livingston integral operator defined by
(cf. [1,8], and [15])

>p

 

  
1
0
=d(;
z
p>)
f
ztfttfp
z






p.
(28)
Theorem 6: Let

be a complex number satisfying
R>0e

and let


f
zp and

f

 be given
by (2.9). If







,,
1pp
k
pp
acfzacfz
zz










 
then



 
,
p
k
p
acfz
z




 ,
where

is given by (26) with
1
Re
10
=1 d
zp
tt






 


.
Proof. From (28), we obtain


 
,
=, ,
p
pp
zac fz
pacfzacf









.z
(29)

J. H. CHOI
233
Let
 
,=
p
p
acfzhz
z


 . Then, by virtue of
(29), we have



  
,,
(1 )
=.
pp
pp
k
acfzac fz
zz
zh z
hz p









 

Hence, by using Lemma 2, we obtain the desired
result.
Finally, we consider the converse case of Theorem 1
as follows.
Theorem 7: Let


,,,,,,0
p
fkac

 

,,
. Then
p

,,,,fkac


   for <zR, where is
given by
R
 
22
=
12
a
R
aaa

1

   
. (30)
Proof. Let
 

1,
=1,
p
p
acfz
Hz acgz











and




0
,
=1,
p
p
ac g z
Hz acgz





and since , it follows that

,,,,0, ,0
p
fkac

 



k
H

 and

0k
H



hz
. Proceeding as in
Theorem 1, for
 
=1Hz

 

and
 


=1 hz
0
Hz

 
with , , we obtain
k0
hh






 

 

 

1
2
0
1
1
0
2
2
0
1,
11
11,
,1,
,1,
=1
1
=42 1
1
421
p
p
pp
d
pp
acfz
acgz
acfza cfzV
ac g zac gz
zh z
hz ah
zh z
khz ah
zh z
khz ah
























 




 

 


 








 







 










By applying Lemma 3, for and
(=0,1, 2)
i
hi
=<1zr , we have
 

 
  

 

 

0
2
2
Re 1
21
Re 1112
1
2
Re11112
12 21
Re .
1112
i
i
i
i
i
zh z
hz ah
rr
hz ar
r
r
hz ar r
arar a
hz ar r



















 

 





















(31)
Hence, the hand right side of (31) is positive for
=<zrR , where is given by (30). This completes
the proof of Theorem 7.
R
Theorem 8. Let . Then

,,,,,,0
p
fkac

 


,,,,,,fkac

p

   for <zR , where is
given by
R

 
1
22
=1
21
Rp p
pp
 
  .






 


(32)
Proof. Let
 

,
=,
p
p
ac fz
Hz acgz









and
 

0
,
=1,
p
p
ac g z
Hz acgz





and since


,,,,0, ,0
p
fkac

 , it follows that


k
H

 and


0
H

. Then, by using same
methods as in the proof of Theorem 7, we obtain the
required result.
3.Acknowledgements
This work was supported by Daegu National University
of Education Research Grant in 2010.
4. References
[1] S. D. Bernardi, “Convex and Starlike Univalent Func-
tions,” Transactions of the American Mathematical Soci-
ety, Vol. 135, 1969, pp. 429-446.
doi:10.1090/S0002-9947-1969-0232920-2

J. H. CHOI
234
[2] B. C. Carlson and D. B. Shaffer, “Starlike and Prestarlike
Hypergeometric Functions,” SIAM Journal on Mathe-
matical Analysis, Vol. 15, No. 4, 1984, pp. 737-745.
doi:10.1137/0515057
[3] N. K. Cho, O. S. Kwon and H. M. Srivastava, “Inclusion
Relationships and Argument Properties for Certain Sub-
classes of Multivalent Functions Associated with a Fam-
ily of Linear Operator,” Journal of Mathematical Analy-
sis and Applications, Vol. 292, No. 2, 2004, pp. 470-483.
doi:10.1016/j.jmaa.2003.12.026
[4] A. W. Goodman, “Univalent Functions, Vol. I, II,” Po-
lygonal Publishing House, Washington, 1983.
[5] J.-L. Liu and K. I. Noor, “Some Properties of Noor Inte-
gral Operator,” Journal of Natural Geometry, Vol. 21,
2002, pp. 81-90.
[6] S. S. Miller, “Differential Inequalities and Caratheodory
Functions,” Bulletin of the American Mathematical Soci-
ety, Vol. 81, 1975, pp. 78-81.
doi:10.1090/S0002-9904-1975-13643-3
[7] K. I. Noor, “On Subclasses of Close-to-Convex Functions
of Higher Order,” International Journal of Mathematics
and Mathematical Sciences, Vol. 15, No. 2, 1992, pp.
279-289. doi:10.1155/S016117129200036X
[8] K. I. Noor, “On New Classes of Integral Operators,”
Journal of Natural Geometry, Vol. 16, 1999, pp. 71-80.
[9] K. I. Noor and M. Arif, “Generalized Integral Operators
Related with p-Valent Analytic Functions,” Mathematical
Inequalities Applications, Vol. 12, No. 1, 2009, pp.
91-98.
[10] J. Patal and N. E. Cho, “Some Classes of Analytic Func-
tions Involving Noor Integral Operator,” Journal of
Mathematical Analysis and Applications, Vol. 312, No. 2,
2005, pp. 564-575. doi:10.1016/j.jmaa.2005.03.047
[11] B. Pinchuk, “Functions with Bounded Boundary Rota-
tion,” Israel Journal of Mathematics, Vol. 10, No. 1,
1971, pp. 7-16. doi:10.1007/BF02771515
[12] S. Ponnusamy, “Differential Subordination and Bazilevic
Functions,” Proceedings Mathematical Sciences, Vol.
105, No. 2, 1995, pp. 169-186. doi:10.1007/BF02880363
[13] H. Saitoh, “A Linear Operator and Its Applications of
First Order Differential Subordinations,” Mathematica
Japonica, Vol. 44, 1996, pp. 31-38.
[14] J. Sokół and L. Trojnar-Spelina, “Convolution Properties
for Certain Classes of Multivalent Functions,” Journal of
Mathematical Analysis and Applications, Vol. 337, No. 2,
2008, pp. 1190-1197. doi:10.1016/j.jmaa.2007.04.055
[15] H. M. Srivastava and S. Owa (Eds.), “Current Topics in
analytic Function Theory,” World Scientific, Singapore,
1992.