# Some Properties for an Integral Operator Defined by Generalized Hypergeometric Function

Authors: Ibtisam Aldawish; Maslina Darus
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

We define a new general integral operator for meromorphic functions involving the generalized hypergeometric function. Furthermore, we study the characterization and other properties for this operator.


---

## Body

## 1. Introduction

LetΩ denote the class of functions of the form
(1)
f
(
z
)
=
1
z
+
∑
k
=
1
∞
‍
a
k
z
k
,
which are analytic in the punctured open unit disk U
*
=
{
z
:
z
∈
C
,
0
<
|
z
|
<
1
}
=
U
∖
{
0
}.We say that a functionf
∈
Ω is meromorphic starlike of order η
(
0
≤
η
<
1
) and belongs to the class Ω
*
(
η
), if it satisfies the following inequality:
(2)
-
R
e
(
z
f
′
(
z
)
f
(
z
)
)
>
η
.
A function f
∈
Ω is a meromorphic convex function of order η
(
0
≤
η
<
1
), if f satisfies the following inequality:
(3)
-
R
e
(
1
+
z
f
′′
(
z
)
f
′
(
z
)
)
>
η
,
and we denote this class by Ω
k
(
η
).Also, letΩ
N
(
ξ
)
(
ξ
>
1
,
z
∈
U
) be the subclass of Ω, consisting of the functions f, which satisfy the inequality
(4)
-
R
e
(
1
+
z
f
′′
(
z
)
f
′
(
z
)
)
<
ξ
(
ξ
>
1
,
z
∈
U
)
,
where the subclass Ω
N
(
ξ
) was introduced and studied by Wang et al. [1] and Nehari and Netanyahu [2].Aq-hypergeometric function is a power series in one complex variable z with power series coefficients which depend (apart from q) on r complex upper parameters α
i, β
j, (
i
=
1
,
…
,
r
,
j
=
1
,
…
,
s
,
α
i
∈
C
,
β
j
∈
C
∖
{
0
,
-
1
,
-
2
,
…
}
) as follows:
(5)
V
(
α
1
,
…
,
α
r
;
β
1
,
…
,
β
s
,
q
,
z
)
=
∑
k
=
0
∞
‍
(
α
1
,
q
)
k
⋯
(
α
r
,
q
)
k
(
q
,
q
)
k
(
β
1
,
q
)
k
⋯
(
β
s
,
q
)
k
[
(
-
1
)
k
q
(
k
2
)
]
1
+
s
-
r
z
k
,
with (
k
2
)
=
k
(
k
-
1
)
/
2, where q
≠
0 when r
>
s
+
1,  (
r
,
s
∈
N
0
=
N
∪
{
0
}
), and (
α
,
q
)
k is the Pochhammer symbol (
α
)
k defined by
(6)
(
α
,
q
)
k
=
{
1
,
k
=
0
;
(
1
-
α
)
(
1
-
α
q
)
(
1
-
α
q
2
)
⋯
(
1
-
α
q
k
-
1
)
,
k
∈
N
.
Tracing back the history of basic hypergeometric series defined by (5) has brought us to Heine dated in 1846. Therefore it is sometimes called Heine’s series. For brief survey on q-hypergeometric functions, one may refer to [3–5] (see also [6–8]).Forz
∈
U, |
q
|
<
1, and r
=
s
+
1, the basic hypergeometric function defined in (5) takes the form
(7)
r
Λ
s
(
α
1
,
…
,
α
r
;
β
1
,
…
,
β
s
,
q
,
z
)
=
∑
k
=
0
∞
‍
(
α
1
;
q
)
k
⋯
(
α
r
;
q
)
k
(
q
;
q
)
k
(
β
1
;
q
)
k
⋯
(
β
s
;
q
)
k
z
k
,
which converges absolutely in the open unit disk U.Corresponding to the functionr
Λ
s
(
α
1
,
…
,
α
r
;
β
1
,
…
,
β
s
,
q
,
z
), let
(8)
r
Ψ
s
(
α
1
,
…
,
α
r
;
β
1
,
…
,
β
s
,
q
,
z
)
=
1
z
r
Λ
s
(
α
1
,
…
,
α
r
;
β
1
,
…
,
β
s
,
q
,
z
)
=
1
z
+
∑
k
=
1
∞
‍
(
α
1
,
q
)
k
+
1
⋯
(
α
r
,
q
)
k
+
1
(
q
,
q
)
k
+
1
(
β
1
,
q
)
k
+
1
⋯
(
β
s
,
q
)
k
+
1
z
k
.
Analogous to the differential operator defined in [9] which involves the q-hypergeometric functions on the normalized analytic functions, we define the following differential operator N
r
,
s
n
[
α
i
,
β
j
;
q
]
:
Ω
→
Ω on the space of meromorphic functions in the class Ω by
(9)
N
r
,
s
0
[
a
i
,
b
j
;
q
]
f
(
z
)
=
f
(
z
)
*
r
Ψ
s
(
a
i
,
b
j
;
q
;
z
)
N
r
,
s
1
[
a
i
,
b
j
;
q
]
f
(
z
)
=
(
1
-
λ
)
f
(
z
)
*
r
Ψ
s
(
a
i
,
b
j
;
q
;
z
)
hhh
+
λ
z
D
q
(
f
(
z
)
*
r
Ψ
s
(
a
i
,
b
j
;
q
;
z
)
)
⋮
N
r
,
s
n
[
a
i
,
b
j
;
q
]
f
(
z
)
=
N
r
,
s
1
(
N
r
,
s
n
-
1
(
f
(
z
)
)
)
.
If f
∈
Ω, then from (9) we may deduce that
(10)
N
r
,
s
n
[
a
i
,
b
j
;
q
]
f
(
z
)
=
1
z
+
∑
k
=
1
∞
‍
[
1
+
(
k
-
1
)
λ
]
n
Υ
k
a
k
z
k
,
where
(11)
Υ
k
=
(
α
1
,
q
)
k
+
1
⋯
(
α
r
,
q
)
k
+
1
(
q
,
q
)
k
+
1
(
β
1
,
q
)
k
+
1
⋯
(
β
s
,
q
)
k
+
1
,
λ
≥
0, and n
∈
N
0
=
N
∪
{
0
}.It should be remarked that the operator given by (10) is a generalization of many other operators considered earlier; for example, consider the following.(i)
Forλ
=
1, r
=
1, s
=
0, α
1
=
q, q
→
1 the operator N
1,0
[
q
,
q
]
f
(
z
)
=
I
n
f
(
z
) was defined by Frasin and Darus [10] and studied by El-Ashwah and Aouf [11].
(ii)
Forn
=
0, α
i
=
q
α
i, β
j
=
q
β
j, α
i
>
0, β
j
>
0,  (
i
=
1
,
…
,
r
;
j
=
1
,
…
,
s
,
r
=
s
+
1
), q
→
1 the operator N
r
,
s
[
q
α
i
,
q
β
j
,
q
]
f
(
z
)
=
H
r
,
s
[
α
i
,
β
j
]
f
(
z
) was imposed by Liu and Srivastava [12].
(iii)
Forn
=
0, r
=
2, s
=
1, α
2
=
q, q
→
1, the operator N
2,1
[
α
1
,
q
,
β
1
,
q
]
f
(
z
)
=
N
[
α
1
;
β
1
]
f
(
z
) was introduced and studied by Liu and Srivastava [13]. Further, we note in passing that this operator N
[
α
1
;
β
1
]
f
(
z
) is closely related to the Carlson-Shaffer operator N
[
α
1
;
β
1
]
f
(
z
) defined on the space of analytic univalent functions in U.
(iv)
Forn
=
0, r
=
1, s
=
0, α
1
=
λ
+
1, q
→
1, the operator N
1,0
[
λ
+
1
,
q
]
f
(
z
)
=
D
λ
f
(
z
)
=
(
1
/
z
(
1
-
z
)
λ
+
1
)
*
f
(
z
)
(
λ
>
-
1
), where D
λ is the differential operator, was introduced by Ganigi and Uralegaddi [14], and then it was generalized by Yang [15].Now, making use of the differential operator N
r
,
s
n
[
α
i
,
β
j
;
q
] given by (10), we introduce the following integral operator in the class Ω.Definition 1.
Letn
∈
N
,
i
∈
{
1,2
,
…
,
n
}, δ
i
>
0. We define the integral operator l
δ
1
,
…
,
δ
n
(
z
)
:
Ω
n
→
Ω by
(12)
l
δ
1
,
.
.
.
,
δ
n
(
z
)
=
1
z
2
∫
0
z
‍
(
-
u
2
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
1
)
′
(
u
)
)
δ
1
⋯
(
-
u
2
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
n
)
′
(
u
)
)
δ
n
d
u
.Remark 2.
Note that forn
=
0, r
=
1, s
=
0, α
1
=
q, we obtain the integral operator defined by Mohammed and Darus [16]; see also [17–19].With the aid of differential operatorN
r
,
s
n
[
α
i
,
β
j
;
q
]
f
(
z
) given by (10), we define a new subclass of functions in Ω as follows.Definition 3.
Let a functionf
∈
Ω be analytic in U
*. Then f is in the class Ω
k
(
α
i
,
β
j
;
q
,
η
) if, and only if, f satisfies
(13)
-
R
e
{
1
+
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
)
′
(
z
)
)
>
η
,
where N
r
,
s
n
[
α
i
,
β
j
;
q
]
f is defined in (10) and 0
≤
η
<
1.Definition 4.
Let a functionf
∈
Ω be analytic in U
*. Then f is in the class Ω
r
,
s
K
(
α
i
,
β
j
,
q
,
η
,
b
) if, and only if, f satisfies
(14)
R
e
{
1
-
1
b
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
)
′
(
z
)
+
2
)
}
>
η
,
where N
r
,
s
n
[
α
i
,
β
j
;
q
]
f is defined in (10) and b
∈
C
∖
{
0
}, 0
≤
η
<
1.Definition 5.
Let a functionf
∈
Ω be analytic in U
*. Then f is in the class Ω
r
,
s
K
U
(
α
i
,
β
j
;
q
,
α
,
η
,
b
) if, and only if, f satisfies
(15)
R
e
{
1
-
1
b
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
)
′
(
z
)
+
2
)
}
>
α
|
1
b
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
)
′
(
z
)
+
2
)
|
+
η
,
where N
r
,
s
n
[
α
i
,
β
j
;
q
]
f is defined in (10) and α
≥
0
,
-
1
≤
η
<
1
,
α
+
η
≥
0
,
b
∈
C
∖
{
0
}.Definition 6.
Let a functionf
∈
Ω be analytic in U
*. Then f is in the class Ω
r
,
s
K
U
H
(
α
i
,
β
j
,
q
,
α
,
b
) if, and only if, f satisfies
(16)
|
1
-
1
b
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
)
′
(
z
)
+
2
)
-
2
α
(
2
-
1
)
|
<
R
e
{
2
(
1
-
1
b
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
)
′
(
z
)
+
2
)
)
}
+
2
α
(
2
-
1
)
,
where N
r
,
s
n
[
α
i
,
β
j
;
q
]
f is defined in (10) and α
>
0
,
b
∈
C
∖
{
0
}.Remark 7.
Puttingn
=
0, r
=
1, s
=
0 and α
1
=
q in Definition 3, we obtain the result corresponding to the class mentioned in (3).Forn
=
0, r
=
1, s
=
0 and α
1
=
q in Definitions 4, 5, and 6, we obtain Ω
K
b
(
η
)
,
Ω
K
U
(
α
,
η
,
b
) and Ω
K
U
H
(
α
,
b
) the classes of meromorphic functions, introduced and studied by Mohammed and Darus [20].


## 2. Preliminary Definitions

We begin by recalling each of the following definitions of subclasses of meromorphic functionsΩ
G
1
(
η
,
b
), Ω
G
2
(
α
,
η
,
b
) and Ω
G
3
(
α
,
b
) which will be required in our investigation.Definition 8 (see [20]).
Let a functionf
∈
Ω be analytic in U
*. Then f is in the class Ω
G
1
(
η
,
b
) if, and only if, f satisfies
(17)
R
e
{
1
-
1
b
(
z
(
z
f
′′
(
z
)
+
3
f
′
(
z
)
)
z
f
′
(
z
)
+
2
f
(
z
)
+
1
)
}
>
η
,
where b
∈
C
∖
{
0
}, 0
≤
η
<
1.Definition 9 (see [20]).
Let a functionf
∈
Ω be analytic in U
*. Then f is in the class Ω
G
2
(
α
,
η
,
b
) if, and only if, f satisfies
(18)
R
e
{
1
-
1
b
(
z
(
z
f
′′
(
z
)
+
3
f
′
(
z
)
)
z
f
′
(
z
)
+
2
f
(
z
)
+
1
)
}
>
α
|
1
b
(
z
(
z
f
′′
(
z
)
+
3
f
′
(
z
)
)
z
f
′
(
z
)
+
2
f
(
z
)
+
1
)
|
+
η
,
where α
≥
0
,
-
1
≤
η
<
1
,
α
+
η
≥
0
,
b
∈
C
∖
{
0
}.Definition 10 (see [20]).
Let a functionf
∈
Ω be analytic in U
*. Then f is in the class Ω
G
3
(
α
,
b
) if, and only if, f satisfies
(19)
|
1
-
1
b
(
z
(
z
f
′′
(
z
)
+
3
f
′
(
z
)
)
z
f
′
(
z
)
+
2
f
(
z
)
+
1
)
-
2
α
(
2
-
1
)
|
<
R
e
{
2
(
1
-
1
b
(
z
(
z
f
′′
(
z
)
+
3
f
′
(
z
)
)
z
f
′
(
z
)
+
2
f
(
z
)
+
1
)
)
}
+
2
α
(
2
-
1
)
,
where α
>
0
,
b
∈
C
∖
{
0
}.


## 3. Main Results

In this section, we study some properties for the integral operatorl
δ
1
,
…
,
δ
n
(
z
) defined by (12) of the subclasses given by Definitions 3, 4, 5, and 6.Theorem 11.
Letδ
i
>
0,  for i
∈
{
1,2
,
…
,
n
} and f
i
∈
Ω. If
(20)
∑
i
=
1
n
‍
δ
i
(
1
-
η
i
)
<
2
,
(
0
≤
η
i
<
1
)
and f
i
∈
Ω
k
(
α
i
,
β
j
,
q
,
η
i
), then l
δ
1
,
…
,
δ
n
(
z
) is in the class Ω
N
(
ξ
), ξ
>
1.Proof.
By differentiating the equality given by (12), we get
(21)
z
2
l
δ
1
,
…
,
δ
n
′
(
z
)
+
2
z
l
δ
1
,
…
,
δ
n
(
z
)
=
(
-
z
2
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
1
)
′
(
z
)
)
δ
1
⋯
(
-
z
2
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
n
)
′
(
z
)
)
δ
n
.
Again, differentiating (21), we deduce that
(22)
z
2
l
δ
1
,
…
,
δ
n
′′
(
z
)
+
4
z
l
δ
1
,
…
,
δ
n
′
(
z
)
+
2
l
δ
1
,
…
,
δ
n
(
z
)
=
∑
i
=
1
n
‍
δ
i
(
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
2
z
)
,
(23)
[
(
-
z
2
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
1
)
′
(
z
)
)
δ
1
⋯
(
-
z
2
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
n
)
′
(
z
)
)
δ
n
]
.
By using (21) and (23), we have
(24)
z
2
l
δ
1
,
…
,
δ
n
′′
(
z
)
+
4
z
l
δ
1
,
…
,
δ
n
′
(
z
)
+
2
l
δ
1
,
…
,
δ
n
(
z
)
z
2
l
δ
1
,
…
,
δ
n
′
(
z
)
+
2
z
l
δ
1
,
…
,
δ
n
(
z
)
=
∑
i
=
1
n
‍
δ
i
(
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
2
z
)
.
Multiplying (24) by z yields that
(25)
z
2
l
δ
1
,
…
,
δ
n
′′
(
z
)
+
4
z
l
δ
1
,
…
,
δ
n
′
(
z
)
+
2
l
δ
1
,
…
,
δ
n
(
z
)
z
l
δ
1
,
…
,
δ
n
′
(
z
)
+
2
l
δ
1
,
…
,
δ
n
(
z
)
=
∑
i
=
1
n
‍
δ
i
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
2
)
,
or, equivalently, that
(26)
z
(
z
l
δ
1
,
…
,
δ
n
′′
(
z
)
+
3
l
δ
1
,
…
,
δ
n
′
(
z
)
)
z
l
δ
1
,
…
,
δ
n
′
(
z
)
+
2
l
δ
1
,
…
,
δ
n
(
z
)
+
1
=
∑
i
=
1
n
‍
δ
i
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
2
)
.
Thus, we get
(27)
-
(
z
l
δ
1
,
…
,
δ
n
′′
(
z
)
l
δ
1
,
…
,
δ
n
′
(
z
)
+
1
)
-
2
=
(
-
∑
i
=
1
n
‍
δ
i
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
2
)
+
1
)
×
(
1
+
2
l
δ
1
,
…
,
δ
n
(
z
)
z
l
δ
1
,
…
,
δ
n
′
(
z
)
)
.
That is,
(28)
-
(
z
l
δ
1
,
…
,
δ
n
′′
(
z
)
l
δ
1
,
…
,
δ
n
′
(
z
)
+
1
)
=
(
2
l
δ
1
,
…
,
δ
n
(
z
)
z
l
δ
1
,
…
,
δ
n
′
(
z
)
)
×
(
-
∑
i
=
1
n
‍
δ
i
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
2
)
+
1
)
+
∑
i
=
1
n
‍
δ
i
(
-
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
1
)
)
+
3
-
∑
i
=
1
n
‍
δ
i
.
Taking the real parts of both terms of the last expression and using the fact that Re
(
z
)
≤
|
z
| for all z, we impose
(29)
-
R
e
(
z
l
δ
1
,
…
,
δ
n
′′
(
z
)
l
δ
1
,
…
,
δ
n
′
(
z
)
+
1
)
=
R
e
{
(
-
∑
i
=
1
n
‍
δ
i
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
2
)
+
1
)
(
2
l
δ
1
,
…
,
δ
n
(
z
)
z
l
δ
1
,
…
,
δ
n
′
(
z
)
)
h
h
h
h
i
×
(
-
∑
i
=
1
n
‍
δ
i
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
2
)
+
1
)
}
h
h
h
+
∑
i
=
1
n
‍
δ
i
R
e
{
-
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
1
)
}
h
h
h
+
3
-
∑
i
=
1
n
‍
δ
i
≤
|
(
-
∑
i
=
1
n
‍
δ
i
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
,
q
]
f
i
)
′
(
z
)
+
2
)
+
1
)
2
l
δ
1
,
…
,
δ
n
(
z
)
z
l
δ
1
,
…
,
δ
n
′
(
z
)
h
h
h
h
i
×
(
-
∑
i
=
1
n
‍
δ
i
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
,
q
]
f
i
)
′
(
z
)
+
2
)
+
1
)
|
h
h
h
+
∑
i
=
1
n
‍
δ
i
R
e
{
-
(
z
(
N
r
,
s
n
[
α
i
,
β
j
,
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
1
)
}
h
h
h
+
3
-
∑
i
=
1
n
‍
δ
i
.
Setting
(30)
ξ
=
|
(
-
∑
i
=
1
n
‍
δ
i
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
2
)
+
1
)
2
l
δ
1
,
…
,
δ
n
(
z
)
z
l
δ
1
,
…
,
δ
n
′
(
z
)
×
(
-
∑
i
=
1
n
‍
δ
i
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
2
)
+
1
)
|
+
∑
i
=
1
n
‍
δ
i
R
e
(
-
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
1
)
)
+
3
-
∑
i
=
1
n
‍
δ
i
,
and since
(31)
|
(
-
∑
i
=
1
n
‍
δ
i
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
2
)
+
1
)
2
l
δ
1
,
…
,
δ
n
(
z
)
z
l
δ
1
,
…
,
δ
n
′
(
z
)
×
(
-
∑
i
=
1
n
‍
δ
i
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
2
)
+
1
)
|
>
0
,
f
i
∈
Ω
k
(
α
i
,
β
j
,
q
,
η
i
)
,
we conclude that
(32)
ξ
>
∑
i
=
1
n
‍
δ
i
(
η
i
-
1
)
+
3
.
Hence, by (20), we have ξ
>
1, so that l
δ
1
,
.
.
.
,
δ
n
(
z
)
∈
Ω
N
(
ξ
), which evidently completes the proof of Theorem 11.Theorem 12.
Letδ
i
>
0, for i
∈
{
1,2
,
…
,
n
}, f
i
∈
Ω
r
,
s
K
(
α
i
,
β
j
,
q
,
η
i
,
b
)
(
0
≤
η
i
<
1
) and b
∈
C
∖
{
0
}. If
(33)
0
<
∑
i
=
1
n
‍
δ
i
(
1
-
η
i
)
≤
1
,
then l
δ
1
,
.
.
.
,
δ
n
(
z
) is in the class Σ
G
1
(
γ
,
b
), where γ
=
1
-
∑
i
=
1
n
‍
δ
i
(
1
-
η
i
).Proof.
We know from the proof of Theorem11 that
(34)
z
(
z
l
δ
1
,
…
,
δ
n
′′
(
z
)
+
3
l
δ
1
,
…
,
δ
n
′
(
z
)
)
z
l
δ
1
,
…
,
δ
n
′
(
z
)
+
2
l
δ
1
,
…
,
δ
n
(
z
)
+
1
=
∑
i
=
1
n
‍
δ
i
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
2
)
,
which is equivalent to
(35)
1
-
1
b
{
z
(
z
l
δ
1
,
…
,
δ
n
′′
(
z
)
+
3
l
δ
1
,
…
,
δ
n
′
(
z
)
)
z
l
δ
1
,
…
,
δ
n
′
(
z
)
+
2
l
(
z
)
+
1
}
=
1
-
∑
i
=
1
n
‍
δ
i
+
∑
i
=
1
n
‍
δ
i
{
1
-
1
b
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
2
)
}
.
Taking the real part of both terms of (35), we get
(36)
R
e
{
1
-
1
b
(
z
(
z
l
δ
1
,
…
,
δ
n
′′
(
z
)
+
3
l
δ
1
,
…
,
δ
n
′
(
z
)
)
z
l
δ
1
,
…
,
δ
n
′
(
z
)
+
2
l
δ
1
,
…
,
δ
n
(
z
)
+
1
)
}
=
∑
i
=
1
n
‍
δ
i
R
e
{
1
-
1
b
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
2
)
}
+
1
-
∑
i
=
1
n
‍
δ
i
.
Now, since f
i
∈
Ω
r
,
s
K
(
α
i
,
β
j
,
q
,
η
i
,
b
), we have
(37)
R
e
{
1
-
1
b
(
z
(
z
l
δ
1
,
…
,
δ
n
′′
(
z
)
+
3
l
δ
1
,
…
,
δ
n
′
(
z
)
)
z
l
δ
1
,
…
,
δ
n
′
(
z
)
+
2
l
δ
1
,
…
,
δ
n
(
z
)
+
1
)
}
>
∑
i
=
1
n
‍
δ
i
η
i
+
1
-
∑
i
=
1
n
‍
δ
i
=
1
-
∑
i
=
1
n
‍
δ
i
(
1
-
η
i
)
.
Letting
(38)
γ
=
1
-
∑
i
=
1
n
‍
δ
i
(
1
-
η
i
)
,
and in view of (33), it follows that 0
≤
γ
<
1, so that l
δ
1
,
…
,
δ
n
(
z
)
∈
Ω
G
1
(
γ
,
b
). This completes the proof of Theorem 12.Theorem 13.
Letδ
i
>
0, for i
∈
{
1,2
,
.
.
.
,
n
}, f
i
∈
Ω
r
,
s
K
U
(
α
i
,
β
j
,
q
,
α
,
η
,
b
)
(
α
≥
0
,
-
1
≤
η
<
1
,
α
+
η
≥
0
) and b
∈
C
∖
{
0
}. If
(39)
∑
i
=
1
n
‍
δ
i
-
1
≤
0
,
then l
δ
1
,
.
.
.
,
δ
n
(
z
) is in the class Ω
G
2
(
α
,
η
,
b
).Proof.
We want to show that(40)
R
e
{
1
-
1
b
(
z
(
z
l
δ
1
,
…
,
δ
n
′′
(
z
)
+
3
l
δ
1
,
…
,
δ
n
′
(
z
)
)
z
l
δ
1
,
…
,
δ
n
′
(
z
)
+
2
l
δ
1
,
…
,
δ
n
(
z
)
+
1
)
}
-
α
|
1
b
(
z
(
z
l
δ
1
,
…
,
δ
n
′′
(
z
)
+
3
l
δ
1
,
…
,
δ
n
′
(
z
)
)
z
l
δ
1
,
…
,
δ
n
′
(
z
)
+
2
l
δ
1
,
…
,
δ
n
(
z
)
+
1
)
|
-
η
>
0
.
We find from (26) and (36) that
(41)
R
e
{
1
-
1
b
(
z
(
z
l
δ
1
,
…
,
δ
n
′′
(
z
)
+
3
l
δ
1
,
…
,
δ
n
′
(
z
)
)
z
l
δ
1
,
…
,
δ
n
′
(
z
)
+
2
l
δ
1
,
…
,
δ
n
(
z
)
+
1
)
}
-
α
|
1
b
(
z
(
z
l
δ
1
,
…
,
δ
n
′′
(
z
)
+
3
l
′
(
z
)
)
z
l
δ
1
,
…
,
δ
n
′
(
z
)
+
2
l
δ
1
,
…
,
δ
n
(
z
)
+
1
)
|
-
η
=
1
-
∑
i
=
1
n
‍
δ
i
+
∑
i
=
1
n
‍
δ
i
R
e
{
1
-
1
b
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
2
)
}
-
α
|
∑
i
=
1
n
‍
δ
i
1
b
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
2
)
|
-
η
.
Since f
i
∈
Ω
r
,
s
K
U
(
α
i
,
β
j
,
q
,
α
,
δ
,
b
), it follows from Definition 5 that
(42)
R
e
{
1
-
1
b
(
z
(
z
l
δ
1
,
…
,
δ
n
′′
(
z
)
+
3
l
δ
1
,
…
,
δ
n
′
(
z
)
)
z
l
δ
1
,
…
,
δ
n
′
(
z
)
+
2
l
δ
1
,
…
,
δ
n
(
z
)
+
1
)
}
-
α
|
1
b
(
z
(
z
l
δ
1
,
…
,
δ
n
′′
(
z
)
+
3
l
δ
1
,
…
,
δ
n
′
(
z
)
)
z
l
δ
1
,
…
,
δ
n
′
(
z
)
+
2
l
δ
1
,
…
,
δ
n
(
z
)
+
1
)
|
-
η
>
1
-
∑
i
=
1
n
‍
δ
i
+
∑
i
=
1
n
‍
δ
i
{
α
|
1
b
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
2
)
|
+
η
}
-
α
∑
i
=
1
n
‍
δ
i
|
1
b
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
2
)
|
-
η
=
1
-
∑
i
=
1
n
‍
δ
i
+
∑
i
=
1
n
‍
δ
i
η
-
η
.
From hypothesis (39), we note that
(43)
(
1
-
η
)
(
1
-
∑
i
=
1
n
‍
δ
i
)
≥
0
.
Hence l
δ
1
,
…
,
δ
n
∈
Ω
G
2
(
α
,
η
,
b
). Therefore, we complete the proof of Theorem 13.Theorem 14.
Letδ
i
>
0 for i
∈
{
1,2
,
…
,
n
}, f
i
∈
Ω
r
,
s
K
U
H
(
α
i
,
β
j
,
q
;
α
,
b
)
(
α
>
0 and b
∈
C
∖
{
0
}
). If
(44)
∑
i
=
1
n
‍
δ
i
-
1
≤
0
,
then l
δ
1
,
…
,
δ
n
(
z
) is in the class Ω
G
3
(
α
,
b
).Proof.
Our aim is to prove that(45)
R
{
2
(
1
-
1
b
(
z
(
z
l
δ
1
,
…
,
δ
n
′′
(
z
)
+
3
l
δ
1
,
…
,
δ
n
′
(
z
)
)
z
l
δ
1
,
…
,
δ
n
′
(
z
)
+
2
l
δ
1
,
…
,
δ
n
(
z
)
+
1
)
)
}
+
2
α
(
2
-
1
)
-
|
1
-
1
b
(
z
(
z
l
δ
1
,
…
,
δ
n
′′
(
z
)
+
3
l
δ
1
,
…
,
δ
n
′
(
z
)
)
z
l
δ
1
,
…
,
δ
n
′
(
z
)
+
2
l
δ
1
,
…
,
δ
n
(
z
)
+
1
)
-
2
α
(
2
-
1
)
1
-
1
b
(
z
(
z
l
δ
1
,
…
,
δ
n
′′
(
z
)
+
3
l
δ
1
,
…
,
δ
n
′
(
z
)
)
z
l
δ
1
,
…
,
δ
n
′
(
z
)
+
2
l
δ
1
,
…
,
δ
n
(
z
)
+
1
)
|
>
0
.
With the aid of (26), (36) can be written as follows:
(46)
R
{
2
(
1
-
1
b
(
z
(
z
l
δ
1
,
…
,
δ
n
′′
(
z
)
+
3
l
δ
1
,
…
,
δ
n
′
(
z
)
)
z
l
δ
1
,
…
,
δ
n
′
(
z
)
+
2
l
δ
1
,
…
,
δ
n
(
z
)
+
1
)
)
}
+
2
α
(
2
-
1
)
-
|
1
-
1
b
(
z
(
z
l
δ
1
,
…
,
δ
n
′′
(
z
)
+
3
l
δ
1
,
…
,
δ
n
′
(
z
)
)
z
l
δ
1
,
…
,
δ
n
′
(
z
)
+
2
l
δ
1
,
…
,
δ
n
(
z
)
+
1
)
-
2
α
(
2
-
1
)
1
-
1
b
(
z
(
z
l
δ
1
,
…
,
δ
n
′′
(
z
)
+
3
l
δ
1
,
…
,
δ
n
′
(
z
)
)
z
l
δ
1
,
…
,
δ
n
′
(
z
)
+
2
l
δ
1
,
…
,
δ
n
(
z
)
+
1
)
|
=
R
{
2
[
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
2
)
1
-
∑
i
=
1
n
‍
δ
i
1
b
×
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
2
)
]
}
+
2
α
(
2
-
1
)
-
|
1
-
∑
i
=
1
n
‍
δ
i
1
b
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
2
)
-
2
α
(
2
-
1
)
∑
i
=
1
n
‍
δ
i
1
b
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
2
)
|
=
2
-
2
∑
i
=
1
n
‍
δ
i
R
1
b
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
2
)
+
2
α
(
2
-
1
)
-
|
1
-
∑
i
=
1
n
‍
δ
i
1
b
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
2
)
-
2
α
(
2
-
1
)
∑
i
=
1
n
‍
δ
i
1
b
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
2
)
|
=
2
+
2
∑
i
=
1
n
‍
δ
i
R
×
{
1
-
1
b
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
2
)
}
-
2
∑
i
=
1
n
‍
δ
i
+
2
α
(
2
-
1
)
-
|
1
+
∑
i
=
1
n
‍
δ
i
[
1
-
1
b
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
2
)
-
2
α
(
2
-
1
)
1
b
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
2
)
]
-
∑
i
=
1
n
‍
δ
i
+
2
α
(
2
-
1
)
∑
i
=
1
n
‍
δ
i
-
2
α
(
2
-
1
)
[
1
-
1
b
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
2
)
|
=
2
(
1
-
∑
i
=
1
n
‍
δ
i
)
+
2
α
(
2
-
1
)
+
2
∑
i
=
1
n
‍
δ
i
R
×
{
1
-
1
b
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
2
)
}
-
|
1
b
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
2
)
[
1
-
2
α
(
2
-
1
)
]
(
1
-
∑
i
=
1
n
‍
δ
i
)
+
∑
i
=
1
n
‍
δ
i
[
1
-
1
b
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
2
)
-
2
α
(
2
-
1
)
1
b
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
2
)
]
|
≥
2
(
1
-
∑
i
=
1
n
‍
δ
i
)
+
2
α
(
2
-
1
)
+
2
∑
i
=
1
n
‍
δ
i
R
×
{
1
-
1
b
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
2
)
}
-
∑
i
=
1
n
‍
δ
i
|
1
-
1
b
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
2
)
-
2
α
(
2
-
1
)
1
-
1
b
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
2
)
|
-
|
1
-
2
α
(
2
-
1
)
|
(
1
-
∑
i
=
1
n
‍
δ
i
)
=
∑
i
=
1
n
‍
δ
i
{
[
1
-
1
b
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
2
)
]
R
e
2
×
[
1
-
1
b
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
2
)
]
+
2
α
(
2
-
1
)
-
|
1
-
1
b
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
2
)
-
2
α
(
2
-
1
)
R
e
2
[
1
-
1
b
(
z
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′′
(
z
)
(
N
r
,
s
n
[
α
i
,
β
j
;
q
]
f
i
)
′
(
z
)
+
2
)
]
|
}
+
2
(
1
-
∑
i
=
1
n
‍
δ
i
)
+
2
α
(
2
-
1
)
-
2
α
(
2
-
1
)
∑
i
=
1
n
‍
δ
i
-
|
1
-
2
α
(
2
-
1
)
|
(
1
-
∑
i
=
1
n
‍
δ
i
)
.
Since f
i
∈
Ω
r
,
s
K
U
H
(
α
i
,
β
j
,
q
,
α
,
b
), it follows from Definition 6 and by a simple calculation that
(47)
R
{
2
(
1
-
1
b
(
z
(
z
l
δ
1
,
…
,
δ
n
′′
(
z
)
+
3
l
δ
1
,
…
,
δ
n
′
(
z
)
)
z
l
δ
1
,
…
,
δ
n
′
(
z
)
+
2
l
δ
1
,
…
,
δ
n
(
z
)
+
1
)
)
}
+
2
α
(
2
-
1
)
-
|
1
-
1
b
(
z
(
z
l
δ
1
,
…
,
δ
n
′′
(
z
)
+
3
l
δ
1
,
…
,
δ
n
′
(
z
)
)
z
l
δ
1
,
…
,
δ
n
′
(
z
)
+
2
l
δ
1
,
…
,
δ
n
(
z
)
+
1
)
-
2
α
(
2
-
1
)
1
b
(
z
(
z
l
δ
1
,
…
,
δ
n
′′
(
z
)
+
3
l
δ
1
,
…
,
δ
n
′
(
z
)
)
z
l
δ
1
,
…
,
δ
n
′
(
z
)
+
2
l
δ
1
,
…
,
δ
n
(
z
)
+
1
)
|
>
[
2
+
2
α
(
2
-
1
)
-
|
1
-
2
α
(
2
-
1
)
|
]
(
1
-
∑
i
=
1
n
‍
δ
i
)
>
(
1
-
∑
i
=
1
n
‍
δ
i
)
min
⁡
{
(
2
-
1
)
(
1
+
4
α
)
,
2
+
1
}
≥
0
,
which means that l
δ
1
,
…
,
δ
n
∈
Ω
G
3
(
α
,
b
), and we obtain the proof of the theorem.



---

*Source: 134910-2014-08-04.xml*