# Inclusion Properties of Certain Subclasses ofp-Valent Functions Associated with the Integral Operator

Authors: Tamer M. Seoudy
Journal: Journal of Mathematics
 (2014)

---

## Abstract

The purpose of the present paper is to introduce two subclasses ofp-valent functions by using the integral operator and to investigate various properties for these subclasses.


---

## Body

## 1. Introduction

LetA
(
p
) denote the class of functions of the following form:
(1)
f
(
z
)
=
z
p
+
∑
j
=
1
∞
‍
a
p
+
j
z
p
+
j
,
(
p
∈
N
=
{
1,2
,
3
,
.
.
.
}
)
,
which are analytic and p-valent in the open unit disc U
=
{
z
∈
C
:
|
z
|
<
1
}. Let P
k
(
p
,
γ
) be the class of functions g analytic in U satisfying g
(
0
)
=
p and
(2)
∫
0
2
π
‍
|
R
{
g
(
z
)
}
-
γ
p
-
γ
|
d
θ
≤
k
π
,
(
z
=
r
e
i
θ
;
k
≥
2
;
0
≤
γ
<
p
)
.
The class P
k
(
p
,
γ
) was introduced by Aouf [1] and we note the following:(i)
the classP
k
(
1
,
γ
)
=
P
k
(
γ
) was introduced by Padmanabhan and Parvatham [2];
(ii)
the classP
k
(
1,0
)
=
P
k was introduced by Pinchuk [3];
(iii)
P
2
(
p
,
γ
)
=
P
(
p
,
γ
) is the class of functions with positive real part greater than γ
(
0
≤
γ
<
p
);
(iv)
P
2
(
1
,
γ
)
=
P
(
γ
) is the class of functions with positive real part greater than γ
(
0
≤
γ
<
1
);
(v)
P
2
(
1,0
)
=
P is the class of functions with positive real part.From (1), we have g
∈
P
k
(
p
,
γ
) if and only if there exists g
1
,
g
2
∈
P
(
p
,
γ
) such that
(3)
g
(
z
)
=
(
k
4
+
1
2
)
g
1
(
z
)
-
(
k
4
-
1
2
)
g
2
(
z
)
,
(
z
∈
U
)
.
It is known that [4] the class P
k
(
γ
) is a convex set.Motivated essentially by Jung et al. [5], Liu and Owa [6] introduced the integral operator Q
β
,
p
α
:
A
(
p
)
→
A
(
p
)
(
α
≥
0
;
β
>
-
p
;
p
∈
N
) as follows:
(4)
Q
β
,
p
α
f
(
z
)
=

{
(
p

+

α
+
β

-

1
p

+

β

-

1
)
α
z
β
∫
0
z

(
1

-

t
z
)
α
-
1

t
β
-
1
f
(
t
)
d
t
(
α

>

0
)
,
f
(
z
)
(
α

=

0
)
.
For f
∈
A
(
p
) given by (1) and then from (4), we deduce that
(5)
Q
β
,
p
α
f
(
z
)
=
z
p

+

Γ
(
α

+

β

+

p
)
Γ
(
β

+

p
)
∑
j
=
1
∞
Γ
(
β

+

p

+

j
)
Γ
(
α

+

β

+

p

+

j
)
a
p
+
j
z
p
+
j
0000000000000000000000000000000000000
(
α
≥
0
;
β
>
-
p
)
.
It is easily verified from (5) that (see [6])
(6)
z
(
Q
β
,
p
α
+
1
f
(
z
)
)
′
=
(
α
+
β
+
p
)
Q
β
,
p
α
f
(
z
)
-
(
α
+
β
)
Q
β
,
p
α
f
(
z
)
.
We note that (i) the one-parameter family of integral operator Q
β
,
1
α
=
Q
β
α was defined by Jung et al. [5] and studied by Aouf [7] and Gao et al. [8].(ii) Consider(7)
Q
c
,
p
1
f
(
z
)
=
F
c
,
p
(
f
)
(
z
)
=
c
+
p
z
c
∫
‍
t
c
-
1
f
(
z
)
d
t
,
(
c
>
-
p
)
,
where the operator F
c
,
p is the generalized Bernardi-Libera-Livingston integral operator (see [9]).We have the following known subclassesS
k
(
p
,
γ
) and C
k
(
p
,
γ
) of the class A
(
p
) for 0
≤
γ, η
<
p, and k
≥
2 which are defined by
(8)
S
k
(
p
,
γ
)
=
{
f
∈
A
(
p
)
:
z
f
′
(
z
)
f
(
z
)
∈
P
k
(
p
,
γ
)
,
z
∈
U
}
,
C
k
(
p
,
γ
)
=
{
f
∈
A
(
p
)
:
(
z
f
′
(
z
)
)
′
f
′
(
z
)
∈
P
k
(
p
,
γ
)
,
z
∈
U
}
.Next, by using the integral operatorQ
β
,
p
α, we introduce the following classes of analytic functions for 0
≤
γ
<
p and k
≥
2:
(9)
S
k
(
p
,
α
;
γ
)
=
{
f
∈
A
(
p
)
:
Q
β
,
p
α
f
(
z
)
∈
S
k
(
p
,
γ
)
}
,
C
k
(
p
,
α
;
γ
)
=
{
f
∈
A
(
p
)
:
Q
β
,
p
α
f
(
z
)
∈
C
k
(
p
,
γ
)
}
.
We also note that
(10)
f
∈
C
k
(
p
,
α
;
γ
)
⟺
z
f
′
p
∈
S
k
(
p
,
α
;
γ
)
.
In particular, we set S
k
(
1
,
α
;
γ
)
=
S
k
(
α
;
γ
) and C
k
(
1
,
α
;
γ
)
=
C
k
(
α
;
γ
).The following lemma will be required in our investigation.Lemma 1 (see [10]).
Letu
=
u
1
+
i
u
2 and v
=
v
1
+
i
v
2 and let Ψ
(
u
,
v
) be a complex-valued function satisfying the following conditions:(i)
Ψ
(
u
,
v
) is continuous in a domain D
∈
C
2;
(ii)
(
0,1
)
∈
D and Ψ
(
1,0
)
>
0;
(iii)
R
{
Ψ
(
i
u
2
,
v
1
)
}
>
0 whenever (
i
u
2
,
v
1
)
∈
D and v
1
≤
-
(
1
/
2
)
(
1
+
u
2
2
).
Ifh
(
z
)
=
1
+
c
1
z
+
c
2
z
2
+
⋯ is analytic in U such that (
h
(
z
)
,
z
h
′
(
z
)
)
∈
D and R
{
Ψ
(
h
(
z
)
,
z
h
′
(
z
)
)
}
>
0 for z
∈
U, then R
{
Ψ
(
h
(
z
)
,
z
h
′
(
z
)
)
}
>
0 in U.Lemma 2 (see [11]).
Letp
(
z
) be analytic in U with p
(
0
)
=
a and R
{
p
(
z
)
}
>
0,  z
∈
U. Then, for s
>
0 and μ
∈
C
∖
{
-
1
},
(11)
R
{
p
(
z
)
+
s
z
p
′
(
z
)
p
(
z
)
+
μ
}
>
0
,
(
|
z
|
<
r
0
)
,
where r
0 is given by
(12)
r
0
=
|
μ
+
1
|
A
+
(
A
2
-
|
μ
2
-
1
|
)
1
/
2
,
A
=
2
(
s
+
1
)
2
+
|
μ
|
2
-
1
,
and this radius is the best possible.Lemma 3 (see [12]).
Letψ be convex and let g be starlike in U. Then, for F analytic in U with F
(
0
)
=
1, (
(
ψ
*
F
g
)
/
(
ψ
*
g
)
) is contained in the convex hull of F
(
U
).In this paper, we obtain several inclusion properties of the classesS
k
(
p
,
α
;
γ
) and C
k
(
p
,
α
;
γ
) associated with the operator Q
β
,
p
α.


## 2. Main Results

Unless otherwise mentioned, we assume throughout this paper thatk
≥
2, α
≥
0, β
>
0, 0
≤
γ
<
p, and p
∈
N.Theorem 4.
One has(13)
S
k
(
p
,
α
+
1
;
γ
)
⊂
S
k
(
p
,
α
;
γ
)
.Proof.
We begin by setting(14)
z
(
Q
β
,
p
α
+
1
f
(
z
)
)
′
Q
β
,
p
α
+
1
f
(
z
)
=
(
p
-
γ
)
h
(
z
)
+
γ
=
(
k
4
+
1
2
)
{
(
p
-
γ
)
h
1
(
z
)
+
γ
}
-
(
k
4
-
1
2
)
{
(
p
-
γ
)
h
2
(
z
)
+
γ
}
,
where h
i is analytic in U with h
i
(
0
)
=
1, i
=
1,2. Using the identity (6) in (14) and differentiating the resulting equation with respect to z, we obtain
(15)
z
(
Q
β
,
p
α
f
(
z
)
)
′
Q
β
,
p
α
f
(
z
)
=
{
(
p
-
γ
)
z
h
′
(
z
)
(
p
-
γ
)
h
(
z
)
+
γ
+
α
+
β
γ
+
(
p
-
γ
)
h
(
z
)
00
z
(
Q
β
,
p
α
f
(
z
)
)
′
Q
β
,
p
α
f
(
z
)
=
+
(
p
-
γ
)
z
h
′
(
z
)
(
p
-
γ
)
h
(
z
)
+
γ
+
α
+
β
}
∈
P
k
(
p
,
γ
)
.
This implies that
(16)
h
i
(
z
)
+
z
h
i
′
(
z
)
(
p
-
γ
)
h
i
(
z
)
+
γ
+
α
+
β
∈
P
,
(
z
∈
U
;
i
=
1,2
)
.
We form the functional Ψ
(
u
,
v
) by choosing u
=
h
i
(
z
) and v
=
z
h
i
′
(
z
):
(17)
Ψ
(
u
,
v
)
=
u
+
v
(
p
-
γ
)
u
+
γ
+
α
+
β
.
Clearly, the first two conditions of Lemma1 are satisfied. Now, we verify condition (iii) as follows:
(18)
R
{
Ψ
(
i
u
2
,
v
1
)
}
=
R
{
v
1
(
p
-
γ
)
i
u
2
+
γ
+
α
+
β
}
≤
-
(
γ
+
α
+
β
)
(
1
+
u
2
2
)
2
[
(
p
-
γ
)
2
u
2
2
+
(
γ
+
α
+
β
)
2
]
<
0
.
Therefore applying Lemma 1, h
i
∈
P
(
i
=
1,2
) and consequently h
∈
P
kfor z
∈
U. This completes the proof of Theorem 4.Theorem 5.
One has(19)
C
k
(
p
,
α
+
1
;
γ
)
⊂
C
k
(
p
,
α
;
γ
)
.Proof.
Applying (10) and Theorem 4, we observe that
(20)
f
∈
C
k
(
p
,
α
+
1
;
γ
)
⟺
z
f
′
p
∈
S
k
(
p
,
α
+
1
;
γ
)
⟹
z
f
′
p
∈
S
k
(
p
,
α
;
γ
)
⟺
f
∈
C
k
(
p
,
α
;
γ
)
,
which evidently proves Theorem 5.Theorem 6.
Iff
∈
S
k
(
p
,
α
;
γ
), then F
c
,
p
(
f
)
∈
S
k
(
p
,
α
;
γ
)
(
c
≥
0
), where the generalized Libera integral operator F
c
,
p is defined by (7).Proof.
Letf
∈
S
k
(
p
,
α
;
γ
) and set
(21)
z
(
Q
β
,
p
α
F
c
,
p
(
f
)
(
z
)
)
′
Q
β
,
p
α
F
c
,
p
(
f
)
(
z
)
=
(
p
-
γ
)
h
(
z
)
+
γ
=
(
k
4
+
1
2
)
{
(
p
-
γ
)
h
1
(
z
)
+
γ
}
-
(
k
4
-
1
2
)
{
(
p
-
γ
)
h
2
(
z
)
+
γ
}
,
where h is analytic in U with h
(
0
)
=
1. From (21), we have
(22)
z
(
Q
β
,
p
α
F
c
,
p
(
f
)
(
z
)
)
′
=
(
c
+
p
)
Q
β
,
p
α
f
(
z
)
-
c
Q
β
,
p
α
F
c
,
p
(
f
)
(
z
)
.
Then, by using (21) and (22), we obtain
(23)
(
c
+
p
)
Q
β
,
p
α
f
(
z
)
Q
β
,
p
α
F
c
,
p
(
f
)
(
z
)
=
(
p
-
γ
)
h
(
z
)
+
γ
+
c
.
Taking the logarithmic differentiation on both sides of (23) with respect to z and multiplying by z, we have
(24)
1
p
-
γ
(
z
(
Q
β
,
p
α
f
(
z
)
)
′
Q
β
,
p
α
f
(
z
)
-
γ
)
=
h
(
z
)
+
z
h
′
(
z
)
(
p
-
γ
)
h
(
z
)
+
γ
+
c
∈
P
k
.
This implies that
(25)
{
h
i
(
z
)
+
z
h
i
′
(
z
)
(
p
-
γ
)
h
i
(
z
)
+
γ
+
c
}
∈
P
,
(
z
∈
U
;
i
=
1,2
)
.
We form the functional Ψ
(
u
,
v
) by choosing u
=
h
i
(
z
) and v
=
z
h
i
′
(
z
):
(26)
Ψ
(
u
,
v
)
=
u
+
v
(
p
-
γ
)
u
+
γ
+
c
.
Then clearly Ψ
(
u
,
v
) satisfies all the properties of Lemma 1. Hence, h
i
∈
P
(
i
=
1,2
) and consequently h
∈
P
kfor z
∈
U, which implies that F
c
,
p
(
f
)
∈
S
k
(
p
,
α
;
γ
).Next, we derive an inclusion property for the subclassC
k
(
α
;
γ
) involving F
c
,
p
(
f
), which is given by the following theorem.Theorem 7.
Iff
∈
C
k
(
p
,
α
;
γ
), then F
c
,
p
(
f
)
∈
C
k
(
p
,
α
;
γ
)
(
c
≥
0
), where F
c
,
p is defined by (7).Proof.
By applying Theorem6, it follows that
(27)
f
∈
C
k
(
p
,
α
;
γ
)
⟺
z
f
′
p
∈
S
k
(
p
,
α
;
γ
)
⟹
F
c
,
p
(
z
f
′
p
)
∈
S
k
(
p
,
α
;
γ
)
(
by
Theorem
5
)
⟺
z
(
F
c
,
p
(
f
)
)
′
p
∈
S
k
(
p
,
α
;
γ
)
⟺
F
c
,
p
(
f
)
∈
C
k
(
p
,
α
;
γ
)
,
which proves Theorem7.Theorem 8.
Iff
∈
C
k
(
p
,
α
+
1
;
γ
), for z
∈
U, then f
∈
C
k
(
p
,
α
;
γ
) for
(28)
|
z
|
<
r
0
=
|
μ
+
1
|
A
+
(
A
2
-
|
μ
2
-
1
|
)
1
/
2
,
where A
=
2
(
s
+
1
)
2
+
|
μ
|
2
-
1, with μ
=
(
(
γ
+
α
+
β
)
/
(
p
-
γ
)
)
≠
-
1 and s
=
(
1
/
(
p
-
γ
)
). This radius is the best possible.Proof.
Letf
∈
C
k
(
p
,
α
+
1
;
γ
) for z
∈
U and let
(29)
z
(
Q
β
,
p
α
+
1
f
(
z
)
)
′
Q
β
,
p
α
+
1
f
(
z
)
=
(
p
-
γ
)
h
(
z
)
+
γ
=
(
k
4
+
1
2
)
{
(
p
-
γ
)
h
1
(
z
)
+
γ
}
-
(
k
4
-
1
2
)
{
(
p
-
γ
)
h
2
(
z
)
+
γ
}
,
where h
i is analytic in U with h
i
(
0
)
=
1 and R
{
h
i
(
z
)
}
>
0 for i
=
1,2. Using the identity (6) in (29) and differentiating the resulting equation with respect to z, we obtain
(30)
1
p
-
γ
{
z
(
Q
β
,
p
α
f
(
z
)
)
′
Q
β
,
p
α
f
(
z
)
-
γ
}
=
h
(
z
)
+
(
1
/
(
p
-
γ
)
)
z
h
′
(
z
)
h
(
z
)
+
(
(
γ
+
α
+
β
)
/
(
p
-
γ
)
)
=
(
k
4
+
1
2
)
{
h
1
(
z
)
+
(
1
/
(
p
-
γ
)
)
z
h
1
′
(
z
)
h
1
(
z
)
+
(
(
γ
+
α
+
β
)
/
(
p
-
γ
)
)
}
-
(
k
4
-
1
2
)
{
(
1
/
(
p
-
γ
)
)
z
h
2
′
(
z
)
h
2
(
z
)
+
(
(
γ
+
α
+
β
)
/
(
p
-
γ
)
)
}
h
2
(
z
)
00
-
(
k
4
-
1
2
)
+
(
1
/
(
p
-
γ
)
)
z
h
2
′
(
z
)
h
2
(
z
)
+
(
(
γ
+
α
+
β
)
/
(
p
-
γ
)
)
}
,
where R
{
h
i
(
z
)
}
>
0 for i
=
1,2. Applying Lemma 2 with s
=
(
(
γ
+
α
+
β
)
/
(
p
-
γ
)
) and μ
=
(
(
γ
+
α
+
β
)
/
(
p
-
γ
)
)
≠
-
1, we get
(31)
R
{
h
i
(
z
)
+
(
1
/
(
p
-
γ
)
)
z
h
i
′
(
z
)
h
i
(
z
)
+
(
(
γ
+
α
+
β
)
/
(
p
-
γ
)
)
}
>
0
000000000
000
0
00000000000000
for
|
z
|
<
r
0
,
where r
0 is given by (28). This completes the proof of Theorem 8.Theorem 9.
Letϕ be a convex function and f
∈
S
2
(
α
;
γ
). Then G
∈
S
2
(
α
;
γ
), where G
=
ϕ
*
f.Proof.
Let=
ϕ
*
f. Then
(32)
Q
β
,
p
α
G
(
z
)
=
Q
β
,
p
α
(
ϕ
*
f
)
(
z
)
=
ϕ
(
z
)
*
Q
β
,
p
α
f
(
z
)
.
Also, f
∈
S
2
(
α
;
γ
). Therefore, Q
β
,
p
α
f
∈
S
2
(
γ
). By logarithmic differentiation of (32) and after some simplification, we obtain
(33)
z
(
Q
β
,
p
α
G
(
z
)
)
′
p
Q
β
,
p
α
G
(
z
)
=
ϕ
(
z
)
*
F
(
z
)
Q
β
,
p
α
f
(
z
)
ϕ
(
z
)
*
Q
β
,
p
α
f
(
z
)
,
where F
=
z
(
Q
β
,
p
α
f
(
z
)
)
′
/
p
Q
β
,
p
α
f
(
z
) is analytic in U and F
(
0
)
=
1. From Lemma 3, we can see that z
(
Q
β
,
p
α
G
(
z
)
)
′
/
p
Q
β
,
p
α
G
(
z
) is contained in the convex hull of F
(
U
). Since z
(
Q
β
,
p
α
G
(
z
)
)
′
/
p
Q
β
,
p
α
G
(
z
) is analytic in U and
(34)
F
(
U
)
=
Ω
=
{
w
:
z
(
Q
β
,
p
α
w
(
z
)
)
′
p
Q
β
,
p
α
w
(
z
)
∈
P
(
γ
)
}
,
then z
(
Q
β
,
p
α
G
(
z
)
)
′
/
p
Q
β
,
p
α
G
(
z
) lies in Ω; this implies that G
=
ϕ
*
f
∈
S
2
(
α
;
γ
).Remark 10.
Puttingp
=
1 in the above results, we obtain corresponding results for the operator Q
β
α.



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*Source: 749251-2014-05-29.xml*