Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 790730, 11 pages
doi:10.1155/2010/790730
Research Article
Some Applications of Srivastava-Attiya Operator to
p-Valent Starlike Functions
E. A. Elrifai, H. E. Darwish, and A. R. Ahmed
Faculty of Science, Mansoura University, Mansoura 35516, Egypt
Correspondence should be addressed to H. E. Darwish, [email protected]
Received 25 March 2010; Accepted 14 July 2010
Academic Editor: Ram N. Mohapatra
Copyright q 2010 E. A. Elrifai et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We introduce and study some new subclasses of p-valent starlike, convex, close-to-convex, and
quasi-convex functions defined by certain Srivastava-Attiya operator. Inclusion relations are
established, and integral operator of functions in these subclasses is discussed.
1. Introduction
Let Ap denote the class of functions of the form
f
z
z
p
∞
n1
a
1.2
be given by f
1
∗ f
2
zz
p
∞
n1
a
np,1
a
np,2
z
np
f
2
∗ f
1
z.
A function fz ∈ Ap is said to be in the class S
∗
p
α of p-valent functions of order α
if it satisfies
Re
zf
z
f
z
>α
0 ≤ α<p,z∈ U
. 1.4
The class of p-valent convex functions in U is denoted by C
p
C
p
0.
It follows from 1.3 and 1.4 that
f
z
∈ C
p
α
zf
z
g
z
>β
0 ≤ β, γ < p, z ∈ U
. 1.6
We denote this class by K
p
β, γ The class K
p
β, γ was studied by Aouf 2.Wenote
that K
1
β, γKβ, γ was studied by Libera 3.
A function f ∈ Ap is called quasi-convex of order β type γ, if there exists a function
gz ∈ C
p
γ such that
Re
s,b
f
z
G
s,b
z
∗ f
z
z ∈ U : b ∈ C \
Z
0
{
0, −1, −2, −3,
}
,s∈ C,p∈ N
1.8
where
G
s,b
z
z
1p
2 b
s
··· .
1.9
It is not difficult to see from 1.8 and 1.9 that
J
s,b
f
z
z
p
∞
n1
1 b
n 1 b
s
a
np
z
∈ S
∗
p
γ
,
C
p,s,b
γ
f
z
∈ A
p
: J
s,b
f
z
β, γ
,
K
∗
p,s,b
β, γ
f
z
∈ A
p
: J
s,b
f
z
∈ K
∗
p
β, γ
Lemma 2.2 see 12. Let u u
1
iu
2
,v v
1
iv
2
, and let ψu, v be a complex function,
ψ : D → C,D⊂ C × C. Suppose that ψ satisfies the following conditions:
i ψu, v is continuous in D,
ii1, 0 ∈ D and Re{ψ1, 0} > 0,
iii Re{ψiu
2
,v
1
}≤0 for aliu
2
,v
1
∈ D such that v
1
≤−1 u
2
2
/2.
Let hz1 c
1
z c
1
J
s1,b
f
z
− γ
p − γ
h
z
, 2.1
4 Journal of Inequalities and Applications
where hz1 c
1
z c
2
z
2
··· . Using the identity
z
J
s1,b
f
z
f
z
J
s1,b
f
z
1
1 b
z
J
s1,b
f
z
J
s1,b
f
z
2.3
Differentiating 2.3, logarithmically with respect to z,weobtain
z
J
s,b
f
z
J
s,b
f
z
− γ
p − γ
h
z
p − γ
p − γ
u γ − p b 1
, 2.5
it is easy to see that the function ψu, v satisfies condition i and ii of Lemma 2.2,inD
C −{γ − p b 1/γ − p} × C. To verify condition iii, we calculate as follows:
Re
ψ
iu
2
,v
1
Re
p − γ
v
1
p − γ
iu
2
γ − p b 1
Re
p − γ
γ − p b 1
v
1
− i
p − γ
2
v
1
u
2
p − γ
2
u
2
2
1 − p b γ
2
2
p − γ
2
u
2
2
1 − p b γ
2
< 0,
2.6
where v
1
≤−1 u
2
2
/2andiu
2
,v
1
∈ D. Therefore, the function ψu, v satisfies the
conditions of Lemma 2.2.
. 2.8
This completes the proof of Theorem 2.3.
Theorem 2.4. C
p,s,b
γ ⊂ C
p,s1,b
γ, for any complex number s.
Proof. Consider the following:
f
z
∈ C
p,s,b
γ
⇐⇒ J
s,b
f
z
∈ C
p
γ
⇐⇒
γ
⇐⇒
zf
z
p
∈ S
∗
p,s,b
γ
⇒
zf
z
p
∈ S
∗
p,s1,b
γ
⇐⇒ J
γ
⇐⇒ J
s1,b
f
z
∈ C
p
γ
⇐⇒ f
z
∈ C
p,s1,b
γ
,
2.9
which evidently proves Theorem 2.4.
Theorem 2.5. K
p,s,b
β, γ ⊂ K
p,s1,b
s,b
kzgz, we have kz ∈ S
p
γ and
Re{zJ
s,b
fz
/J
s,b
kz} >βz ∈ U.
Now, put zJ
s1,b
fz
/J
s1,b
kz − β p − βhz, where hz1 c
1
z c
2
z
2
··· .
Using the identity 2.2 we have
z
J
s,b
J
s1,b
zf
z
−
p −
1 b
J
s1,b
zf
z
z
z
/J
s1,b
k
z
−
p −
1 b
J
s1,b
zf
z
/J
s1,b
k
γ and S
∗
p,s,b
γ ⊂ S
∗
p,s1,b
γ,weletzJ
s1,b
kz
/J
s1,b
kzp − γHz
γ, where Re Hz > 0 z ∈ U thus 2.11 can be written as
z
J
s,b
f
z
J
s,b
k
z
h
z
p − γ
H
z
γ −
p −
1 b
.
2.12
Consider that
z
J
s1,b
f
z
z
J
s1,b
k
z
p − β
zh
z
β
p − β
h
z
p − β
h
z
p − β
zh
z
p − γ
H
z
γ −
p −
1 b
. 2.15
. 2.16
It is not difficult to see that ψu, v satisfies the conditions i and ii of Lemma 2.2 in D
C × C. To verify condition iii, we proceed as follows:
Re ψ
iu
2
,v
1
p − β
v
1
p − γ
h
1
x, y
γ −
p −
1 b
x, yih
2
x, y,h
1
x, y and h
2
x, y being the functions of x and y and
Re Hzh
1
x, y > 0.
By putting v
1
≤−1/21 u
2
2
, we have
Re ψ
iu
2
,v
1
≤−
p − β
1 u
2
2
2
p − γ
h
2
x, y
2
< 0. 2.18
Hence, Re hz > 0 z ∈ U and fz ∈ K
p,s1,b
β, γ. The proof of Theorem 2.5 is complete.
Theorem 2.6. K
∗
p,s,b
β, γ ⊂ K
∗
p,s1,b
β, γ for any complex number s.
Journal of Inequalities and Applications 7
Proof. Consider the following:
f
z
∈ K
p
β, γ
⇐⇒ J
s,b
zf
z
p
∈ K
p
β, γ
⇒
zf
z
p
∈ K
p,s,b
z
p
J
s1,b
f
z
∈ K
p
β, γ
⇐⇒ J
s1,b
f
z
∈ K
∗
p
β, γ
⇒ f
f
t
dt
z
p
∞
n1
c p
c p n
a
np
z
np
.
3.1
The operator L
c
fz when c ∈ N {1, 2, 3, } was studied by Bernardi 13,for
c 1,L
1
fz was investigated earlier by Libera 14. Now, we have
J
s,b
s,b
L
c
f
z
c p
J
s,b
f
z
− c
L
c
f
z
. 3.3
L
c
f
z
c p
J
s,b
f
z
J
s,b
L
c
f
z
− c
1
1 − 2γ
z
1 − c − 2γ
ω
z
c p
1 − ω
z
. 3.5
Differentiating 3.5,weobtain
z
J
s,b
f
z
1 − c − 2γ
zw
z
p c
1 − c − 2γ
w
z
. 3.6
Now we assume that |wz| < 1 z ∈ U. Otherwise, there exists a point z
0
∈ U such that
max |wz| |wz
0
| 1. Then by Lemma 2.1, we have z
0
w
z
0
kwz
0
Re
2
1 − γ
ke
iθ
1 − e
iθ
p c
1 − c − 2γ
e
iθ
−2k
1 − γ
c γ
1 c
c
fz ∈ C
p,s,b
γ.
Proof. Consider the following:
f
z
∈ C
p,s,b
γ
⇐⇒
zf
z
p
∈ S
∗
p,s,b
γ
⇒ L
c
f
z
∈ C
p,s,b
γ
.
3.8
This completes the proof of Theorem 3.2.
Theorem 3.3. Let c>−γ,0 ≤ γ<p.If fz ∈ K
p,s,b
β, γ then L
c
fz ∈ K
p,s,b
β, γ.
Journal of Inequalities and Applications 9
Proof. Let fz ∈ K
p,s,b
β, γ. Then, by definition, there exists a function gz ∈ S
∗
p,s,b
γ such
that
Re
z
z
J
s,b
L
c
g
z
− β
p − β
h
z
3.10
where hzc
1
z c
2
z
2
··· . From 3.3 and 3.10, we have
z
z
J
s,b
L
c
zf
z
cJ
s,b
L
c
zf
z
z
J
s,b
/J
s,b
L
c
g
z
cJ
s,b
L
c
zf
z
/J
s,b
L
c
g
z
γ, then from Theorem 3.1, we have L
c
g ∈ S
∗
p,s,b
γ.
Let
z
J
s,b
L
c
g
z
J
s,b
L
c
g
z
s,b
L
c
zf
z
/J
s,b
L
c
g
c
p − β
h
z
β
p − γ
p − β
h
z
β
. 3.14
Differentiating both sides, we have
z
z
J
s,b
L
c
f
z
z
J
c
g
z
,
3.15
or
z
z
J
s,b
L
c
f
z
J
s,b
L
c
g
p − β
zh
z
p − β
h
z
β
1 − γ
H
z
γ
.
3.16
10 Journal of Inequalities and Applications
Now, from 3.13 we have
z
p − γ
H
z
γ c
.
3.17
We form the function ψu, v by taking u hz,v zh
z in 3.17 as follows
ψ
u, v
p − β
u
p − β
v
γ c
p − γ
h
1
x, y
γ c
2
p − γ
h
2
x, y
2
, 3.19
where Hzh
1
x, yih
2
x, y,h
h
1
x, y
γ c
2
p − γ
h
1
x, y
γ c
2
p − γ
h
2
x, y
2
β, γ
⇐⇒ zf
z
∈ K
p,s,,b
β, γ
⇒ L
c
zf
z
∈ K
p,s,b
β, γ
⇐⇒ z
L
c
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10 G. S¸. S
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