Applied Mathematics, 2011, 2, 1225-1235
doi:10.4236/am.2011.210171 Published Online October 2011 (http://www.SciRP.org/journal/am)
Copyright © 2011 SciRes. AM
Generalization of Certain Subclasses of Multivalent
Functions with Negative Coefficients Defined
by Cho-Kwon-Srivastava Operator
Elsayed A. Elrifai, Hanan E. Darwish, Abdusalam R. Ahmed
Department of Mathematics, Faculty of Science, University of
Mansoura, Mansoura, Egypt
E-mail: Rifai@mans.edu.eg, Darwish333@yahoo.com, Abdusalam5056@yahoo.com
Received November 1, 2010; revised July 6, 2011; accepted July 13, 2011
Abstract
Making use of the Cho-Kwon-Srivastava operator, we introduce and study a certain SCn (j, p, λ, α, δ) of
p-valently analytic functions with negative coefficients. In this paper, we obtain coefficient estimates, distor-
tion theorem, radii of close-to-convexity, starlikeness, convexity and modified Hadamard products of func-
tions belonging to the class SCn (j, p, λ, α, δ). Finally, several applications investigate an integral operator,
and certain fractional calculus operators also considered.
Keywords: Multivalent Functions, Cho-Kwon-Srivastava Operator, Modified-Hadamard Product,
Fractional Calculus
1. Introduction
Let denote the class of functions of the form:
,Tjp
 

0; ,1,2,3,,
pk
kk
kjp
fzzazapj N

 
(1.1)
which are analytic and p-valent in the open unit disc
:1
Uzz. A function

,
f
zTjp is said to be
p-valently starlike of order
if it satisfies the inequal-
ity :

 
Re;0; .
zf zzUpp N
fz





 (1.2)
We denote by
,
j
Tp
the class of all p-valently
starlike functions of order
. Also a function
f
z
is said to be p-valently convex of order
,Tjp
if
it satisfies the inequality:

 
Re1; 0;.
zf zzUpp N
fz







 (1.3)
We denote by
,
j
Cp
  

,,
0;.
jj
zf z
fzC pTp
p
pp N

 
(1.4)
The classes
,
j
Tp
and
,
j
Cp
are studied by
Owa [3].
In [4] Wang et al. defined Cho-kwon-Srivastava op-
erator which
  
,,:, ,
pj acfzTjpTjp
,
by
 

,,!
kp kp
p
k
pj k
kjp kp
pc
ac fzzaz
kpa



(1.5)
for
0
,0,1,2,3, ,,acRZz Up
 
and
 
1;
11;
k
k
kk
 

0
.N
Clearly,
,1 ,
pac
is the well-known Cho-kwon-
Srivastava operator (see [5]) where
the class of all p-valently
convex functions of order
. We note that (see for ex-
ample Duren [1] and Goodman [2])


11
,,
1,1,,1,
pj pj
zf z
pfzp
p
 
E. A. ELRIFAI ET AL.
1226
and

1
,1 ,,
p
paaDf zp



where is the well-known Ruscheweyh deriva-
tive of -th order.
1p
D

p
1
With the help of the Cho-Kwon-Srivastava

,,ac fz
,
pj we say that a function

f
z belonging
to is in the class

,Tjp
,,,,j p
n
SC

if and only
if




  


2
,,
,,
0
,,
Re 1, ,
,
pj pj
pj pj
zacfzz acfz
ac fzzac fz
pNjN













(1.6)
We note that:
1) when 0,
we have



,
,
,
Re ,
pj
pj
zacfz
ac fz





which is the class of starlike of order .
2) when 0
, , 1ap 1
, , we have the
class 1c


Re; 0
zf zp
fz







which is the class of starlike functions of order
stu-
died by Owa [3] and Y amakawa [6]
3) when 1,
we have




,
,
,
Re 1;
,
0
pj
pj
zacfz
ac fz
p








which is the class of convex operator of order .
4) when 1
, 1ap
, , 1c1
, we have

 
Re 1;0
zf zp
fz








which is the class of convex functions of order
stud-
ied by Owa [3] and Yam akawa [6].
In our present paper, we shall make use of the familiar
,cp
J
defined by (c.f. [7,8], see also [9])

 
1
,0d,
z
c
cp c
cp
J
fz tftt
z
 (1.7)
,; ,
f
zTjpcppN,
as well as the fractional calculus operator
z
D
for which
it is well known that (see, for details, [10,11]; see also
Section 5 below)




1,
1
1;
z
Dz z
R





 
(1.8)
in terms of Gamma functions.
2. Coefficient Estimates
Theorem 1. Let the function

f
z defined by (1.1).
Then
,,,,
n
fzSC jp

if and only if

 

11
!
11
kp kp
k
kjp kp
pc
kk
kpa
pp






 

a
(2.1)
0
;0,01, ,,.zUppjNnN

 
Proof. Assume that the inequality (2.1) holds true.
Then we have


 


  
11
!
11 1
!
kp kp
k
kjpkp
kp kp
k
kjpkp
pc
jpk a
kpa
pc
kkapp
kpa

 




 



1



that is, that

 
11
11
!
kp kp
k
kjp kp
pc pp
ka
kpa jp




.
 



Since
Copyright © 2011 SciRes. AM
E. A. ELRIFAI ET AL.
1227

 
 
 
 
  
11 11
!
11 11
!
11
11 11
!
kp kpkp
k
kjpkp
kp
kp kp
k
kjp kp
kp kp
k
kjp kp
pc
pkaz
kpa
pc
pkaz
kpa
pc jp
pka
kpa jp









  
 
 
 
 



 
 
 

0.
Then we find that




 



 
 
 


 
2
,,
,,
,,
1, ,
11
!
11 11
!
11
!
1
pj pj
pj pj
kp
kp kp
k
kjp kp
kp
kp kp
k
kjpkp
kp kp
k
kjpkp
zacfzzacfzp
acf zzacf z
pckpk az
kpa
pc
kk
kpa
pckpk a
kpa
p















 
 





 

az

.
111
!
kp kp
k
kjp kp
p
pc ka
kpa


 
 
(2.2)
This shows that the values of the function
 


  
2
,,
,,
,,
1, ,
pj pj
pj pj
zacfzz acfz
z
ac fzzac fz







(2.3)
lie in a circle which is centered at and whose
radius is wp
.p
Hence

f
z satisfies the condition
(1.6).
Conversely, assume that the function

f
z is in the
class
,,,,
n
SCj p

. Then we have




 

 
 
 
 
2
,,
,,
,,
Re 1, ,
11 11
!
Re
11 11
!
pj pj
pj pj
kp kpkp
k
kjp kp
kp kpkp
k
kjpkp
zacfzz acfz
ac fzzac fz
pc
pp kkaz
kpa
pc
pk
kpa




az













 
 

 




  
 




(2.4)
for some some , p,

0,p



01

 jN
,
0, and Choose values of z on the real axis
so that given by (2.3) is real. Upon clearing the
denominator in (2.4) and letting through real
values, we can see that
nN.zU

z
1z

 
 
 
11 11
!
11 11
!
kp kp
k
kjpkp
kp kp
k
kjpkp
pc
pp kka
kpa
pc
pk
kpa .
a

 




  
 


 
 

 


(2.5)
Copyright © 2011 SciRes. AM
E. A. ELRIFAI ET AL.
Copyright © 2011 SciRes. AM
1228
Thus we have the inequality (2.1).
Corollary 1. Let the function
f
z defined by (1.1)
be in the class

,,,,
n
SCj p

. Then



 

11
1
!
,, .
k
kp kp
kp
pp
apckk
kpa
kjppjN








 
1
(2.6)
The result is sharp for the function

f
z given by




 

0
11
11
!
,, ,.
p
k
kp kp
kp
pp
f
zz z
pckk
kpa
kjppjNnN








 
(2.7)
3. Distortion Theorem
Theorem 2. If a function

f
z defined by (1.1) is in
the class
,,,,
n
SCj p

then



 
 


 
11 !
!
!11 !
!
11 !
!
!11 !
!
jpm
jj
j
j
pm
m
jj
j
ppjp
pzz
pc
pm jpjp jpm
ja
ppjp
p
fzz z
pc
pm jpjp jpm
ja






 




 





 


 


 



(3.1)

0
;0;01,,,.zUppj NnN

 
The result is sharp for the function

f
z given by
 

 
0
11 ., ,.
11
!
pj
jj
j
pp
fzzzpj Nn N
pc jp jp
ja





 


p
(3.2)
Proof. In view of Theorem 1, we have

 



 

11 11
!!
!1
11 !11
jj kp kp
jkp
k k
kjpkjp
pc pc
jp jpkk
ja kpa
ka a
ppjp pp
 
 


 


 
 


 



which readily yields

 
11 !
!.
11
!
k
kjp jj
j
ppjp
ka pc jp jp
ja



 



(3.3)
Now, by differentiating both sides of (1.1) m times,
we obtain

  

!!
!!
,, .
m
p
m
k
kjp
pk
km
f
zz a
pm km
kjpjpN



 
z
(3.4)
Theorem 2, follows from (3.3) and (3.4).
Finally, it is easy to see that the bou nds in (3.1) are at-
tained for the function
f
z given by (3.2).
4. Radii of Close-to-Convexity, Starlikeness
and Convexity
Theorem 3. Let the function

f
zdefined by (1.1) be
in the class
,,,,
n
SCj p

then
1)
f
z is p-valently close-to-convex of order
E. A. ELRIFAI ET AL.
1229

0p

 in 1,zr where


 


1
1 0
11
!
inf,, ,,,
11
kp
kp kp
kp
k
pckk
kpa p
rk
k
pp


















jppjNnN
(4.1)
2)

f
z is p-valently starlike of order in

0p

 2,zr
where


 

1
2
11
!
inf,,
11
kp
kp kp
kp
k
pckk
kpa p
rk
k
pp



















jppjN
(4.2)
3)

f
z is p-valently convex of order in

0p

 3,zr
where


 



1
3
11
!
inf,,,.
11
kp
kp kp
kp
k
pckk
kpa pp
rk
kk
pp
















jpjpN
(4.3)
Each of these results is sharp for the function
f
z
given by (2. 7) .
Proof. It is sufficient to show that

1
1;0, ,
p
fz pzr ppN
z
 
 
(4.4)



2;0, ,
zf zppzrppN
fz

 (4.5)
and



3
1;0
zf zppzrppN
fz



,,
(4.6)
for a function

,,,,
n
fzSC jp

where 12
and 3 are defined by (4.1) - (4.3) respectively. The
details involved are fairly straightforward and may omi-
,rr
r
tited.
5. Modified Hadamard Products
For the func tions
1, 2
i
fzi
defined by


,,
0;1,2 ,
pk
ikiki
kjp
fzzaz ai

 
(5.1)
we denote by
12
f
fz the modified Hadamard
product (or convolution) of the functions
1
f
z and
2,
f
z defined by

12 ,1,2
.
p
k
kk
kjp
f
fzz aaz


(5.2)
Theorem 4. Let the functions defined
by (5.1) be in the class
 
1, 2
i
fzi
,,,,
n
SCj p


then
12 ,,,,
n
ffz SCjp,

 where
 

 
2
22
11 .
11 1
!
jj
j
jp p
ppc jpjp pp
ja

 



 
1


(5.3)
The result is sharp for the functions given by
 
1, 2
i
fzi
 

 
11 ,(,,1,2).
11
!
pj
i
jj
j
pp
fzzzpjNi
pc jp jp
ja





 



p
(5.4)
Copyright © 2011 SciRes. AM
E. A. ELRIFAI ET AL.
Copyright © 2011 SciRes. AM
1230
Proof. Employing the technique used earlier by Schild
and Silverman [12], we need to find the largest γ such
that


 
 
,1 ,2
11
!1.
11
kp kp
kp
kk
kjp
pckk
kpa
aa
pp









(5.5)
Since we readily
see that
 
,,,, 1,2,
in
fzSC jpi




 


,
11
!1
11
1, 2.
kp kp
kp
ki
kjp
pckk
kpa
a
pp
i








(5.6)
Therefore, by the Cauchy-Schwarz inequality, we ob-
tain

 

,1 ,2
11
!1.
11
kpkp
kp
kk
kjp
pc kk
kpa
aa
pp









(5.7)
Thus we onl y need to show that




,1 ,2,1 ,2
,, ,
kk kk
kk
aa aa
pp
kjppjN



 

 
(5.8)
or, equivalently, that
 


,1 ,2
,, ,
kk
pk
aa pk
kjppjN



 
(5.9)
Hence, in light of the inequality (5.7), it is sufficient to
prove that


 


11 ,,,
11
!
kp kp
kp
pp pk kjppjN
pc pk
kk
kpa
 



 






,
(5.10)
It follows fro m (5.10) that
 


  

2
22
11 ,,,
11 11
!
kp kp
kp
kppp
pk
pckkpp
kpa

 

 





.jpjpN
(5.11)
Now, defining the function
Gk by
 


 

2
22
11 ,,,
11 11
!
kp kp
kp
kppp
Gkpkjp jpN
pckkpp
kpa

 

 






.
(5.12)
We see that is an increasing function of k. Ther efore, we con clude that

Gk
  

 
2
22
11 ,
11
!
jj
j
jp p
Gj pppc jpjp pp
ja

 




  1


(5.13)
which eviden tly completes the proof of Theorem 4.
Remark: Putting 1) , 1ap 1
, , 1c0
and 2) ,
1ap 1
, 1c
, 1
in Theorem 4,
we obtain
Corolla ry 2. Let the functions defined
by (5.1) be in the class
 
1, 2
i
fzi
.
*,
j
Tp
Then

12
f
fz

*,,
j
Tp
where


2
22
.
jp
p
jp p

   (5.14)
The result is sharp.
Corollary 3. Let the functions defined
by (5.1) be in the class.
 
1, 2
i
fzi

,
j
Cp
Then

12
f
fz
,,
j
Cp
where

 
2
22
.
1
jp p
p
jpjppp

  
(5.15)
The result is sharp.
Using arguments similar to those in the proof of
E. A. ELRIFAI ET AL.
1231
Theorem 4, we obtain the following result.
Theorem 5. Let the function

1
f
z
defined by (5.1)
be in the class
,,,,
n
SCj p

Suppose also that the
function
2
f
z defined by (5.1) be in the class
,,,,SCj p
n

, Then
 
12 ,,,,
n
ff z SCjp,


where



11 ,
11
!
kp kp
kp
jp pp
ppcjpjpjp
kpa










(5.16)
and

11pp p

 .
(5.17)
The result is the best possible for the functions
 

 
1
11 ,,
11
!
pj
jj
j
pp
fzzzpj N
pc jp jp
ja





 


p
(5.18)
and
 

 
2
11 ,,
11
!
pj
jj
j
pp
fz zzpjN
pc jpjp
ja






 


.
p
(5.19)
Theorem 6. Let the functions
 
1, 2,,
i
f
zi m

,,,,
nj p
defined by (5.1) be in the class SC

. Then
the function

2,
1,
p
ki
kjpi
hz za z

 

 

 k
(5.20)
belongs to the class
,,,,
n
SCj p

where
 

 
2
22
11 .
11 1
!
jj
j
jm pp
ppc jpjp mpp
ja





  
1


(5.21)
The result is sharp for the functions
 
1, 2,,
i
f
zi m given by (5.4).
Proof. Noting that


 



 

 

2 2
2, ,
11 11
!!
1,
11 11
,,,,1,2, ,
kp kpkp kp
kp kp
ki ki
kjp kjp
in
pc pc
kk kk
kpa kpa
aa
pp pp
fzSC jpim

 
 

 
 
 
 


 
 

 



 
 

 




(5.22)
we have


 

2
2,
1
11
!
11.
11
kp kp
m
kp
ki
kjp i
pckk
kpa
a
mpp


















(5.23)
Therefore, we have find largest
such that
Copyright © 2011 SciRes. AM
1232 E. A. ELRIFAI ET AL.




 
 
2
2
11
!,, ,
11
kp kp
kp
pckk
kpa
kkjppjN
pmpp








(5.24)
that is, that
 


  

2
22
11 ,,,
11 11
!
kp kp
kp
mkp pp
pk
pckkmpp
kpa

 

 




 
 
.jppjN
(5.25)
Now, defining the function by

k
  


  

2
22
11 ,,,
11 11
!
kp kp
kp
mkp pp
kp kjppjN
pckkmpp
kpa

 

 




 
 
(5.26)
we observe that is an increasing function of k. We thus conclude that

k
  

 
2
22
11 ,
11 1
!
jj
j
mj pp
jp ppc jpjp mpp
ja

 




  1


(5.27)
which completes the proo f of Theorem 6.
6. Applications of Fractional Calculus
Various operators of fractional calculus (that is, frac-
tional integral and fractional derivatives) have been
studied in (cf., e.g., [9,10,13-15]; see also the various
references cited therein).
For our present investigation, we recall the following
definitions.
Definition 1. The fractional integral of order
is
defined, for a function

f
z, by
 

1
0
1d
z
z
f
Dfz
z

0
,
(6.1)
where the function
f
z is analytic in a simply-con-
nected domain of the complex z-plane containing the
origin and the multiplicity of

1
z
is removed by
requiring
log z
to be real when

0.
z
Definition 2. The fractional derivative of order
is
defined, for a function
f
z, by
 


0
1d01
1
,
z
f
Dfz
z



z (6.2)
where the function
f
z
is constrained, and the multi-
plicity of
z
is r emoved, as in D efinition 1.
Definition 3. Under the hypotheses of Definition 2,
the fractional derivative of order n
is defined, for a
function
f
z, by
 


0
01,
n
n
zz
n
d
Dfz DfznN
dz

. (6.3)
In this section, we shall investigate the growth and
distortion properties of functions in the class
,,,
n
SCj p
,
involving the operators ,cp
J
and
.
z
D
In order to derive our results, we need the following
Lemma given by Chen et al. [14].
Lemma 1 (see [14]). Let the function

f
z defined
by (1.1). Then




 
,
11
,;,,
11
pk
zcp k
kjp
pcpk
DJfzzazRc ppjN
pckk
 



 
 
 
(6.4)
and






,
11
,;,,
11
pk
cp zk
kjp
cppcp k
J
DfzzazRc ppjN
cppckk

 


 
 


(6.5)
provided that no zeros appear in the denominators in (6.4) and (6.5).
Copyright © 2011 SciRes. AM
E. A. ELRIFAI ET AL.
1233
Theorem 7. Let the function

f
z defined by (1.1) be in the class
,,,,
n
SCj p

. Then






 

 

,
111
1,
1111
!
;0;01;0;,,
jp
zcp
jj
j
cp jppp
p
DJfzzz
pc
pcjp jpjpjp
ja
zUpcppjN



 
 


 


 

 
  

 
 
 
(6.6)
and






 

 

,
111
1,
1111
!
;0;01;0;,,.
jp
zcp
jj
j
cp jppp
p
DJfzzz
pc
pcjp jpjpjp
ja
zUpcppj N



 
 


 


 

 
 

 
 
 
(6.7)
Each of the assertion (6.6) and (6.7) is sharp.
Proof. In view of Theorem 1, we have

 




11 11
!!
1,
11 11
jj kp kp
jkp
k k
kjpkjp
pcp c
jpjpjp k
jakpa
aa
pp pp

 
 


 

   
 
 
 
 
 (6.8)
which readily yields

 
11 .
11
!
k
kjpjj
j
pp
apc jp jp
ja






 
(6.9)
Consider the function
f
z defined in U by
 




 
 
,
11
11
,
1
1
p
k
zcp k
kjp
pk
k
kjp
pcpkp
F
zzDJfzz
pckk
zkazzU





 

 
 
az
p
where
 
 
11 ,,, 0.
11
cp kp
kkjp
ck kp



 
pjN
(6.10)
Since is a decreasing function of k when

k
0
, we get
 
 

11
0,
11
cp jpp
kjp cppjN
cjp jpp
 


 ,,0.
(6.11)
Thus, by using (6. 9) an d (6 .1 1), we deduce that
 

 

 
111 1,
1111
!
pjp
k
kjp
p
jp
jj
j
Fzzj pza
cp jpppp
z z
pc
cjp jppjpjp
ja







 

Copyright © 2011 SciRes. AM
E. A. ELRIFAI ET AL.
Copyright © 2011 SciRes. AM
1234
and
 
 


 
111 1,
1111
!
pjp
k
kjp
p
jp
jj
j
Fzzj pza
cp jpppp
z z
pc
cjp jppjpjp
ja







  

which yield the inequ alities (6.6 ) and (6 .7 ) of Theorem 7.
The equalities in (6.6) and (6.7) are attained for the func-tion
f
z given by





 
 

 
,
11 1
1
111
!
jp
zcp
jj
j
cppjpp
p
DJfzzz
pc
pcjpjpjpjp
ja 1


 
 
 



 


 

 
  

 
 
(6.12)
or, equivalently, by


 

 
,
11 .
11
!1
p
jp
cp
jj
j
cppp
Jfz zz
pc
cjp jpjpjp
ja






   


(6.13)
Theorem 8. Let the function

f
z defined by (1.1)
be in the class
.,,,,
n
SCj p

Then
Thus we complete the proof of Theorem 7.
Using arguments similar to those in the proof of
Theorem 7, we obtain the following result.






 

 

,
111
1,
1111
!
;0,01,01,, ,,
jp
zcp
jj
j
cp jppp
p
DJfzzz
pc
pcjpjpjpjp
ja
zUpcpjp N



 
 


 


 

 
 

 
 

(6.14)
and






 

 

,
111
1,
1111
!
;0,01,01, ,.
jp
zcp
jj
j
cp jppp
p
DJfzzz
pc
pcjpjpjpjp
ja
zUpjp N



 
 


 


 

 
 

 
 
 
(6.15)
Each of the assertions (6.14) and (6.15) is sharp.
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