# Certain Families of Multivalent Analytic Functions Associated with Iterations of the Owa-Srivastava Fractional Differintegral Operator

Authors: A. K. Mishra; S. N. Kund
Journal: Journal of Complex Analysis
 (2014)

---

## Abstract

By making use of a multivalent analogue of the
Owa-Srivastava fractional differintegral operator and its
iterations, certain new families of analytic functions are
introduced. Several interesting properties of these function
classes, such as convolution theorems, inclusion theorems, and
class-preserving transforms, are studied.


---

## Body

## 1. Introduction

LetA denote the class of analytic functions in the open unit disk
(1)
U
=
{
z
:
z
∈
C
,
|
z
|
<
1
}
and let A
p be the subclass of A consisting of functions represented by the followingTaylor-Maclaurin’s series:
(2)
f
(
z
)
=
z
p
+
∑
k
=
1
∞
‍
a
k
z
k
+
p
(
p
∈
N
∶
=
{
1,2
,
3
,
…
}
)
.In a recent paper Patel and Mishra [1] studied several interesting mapping properties of thefractional differintegral operator:
(3)
Ω
z
(
λ
,
p
)
:
A
p
⟶
A
p
(
p
∈
N
,
-
∞
<
λ
<
p
+
1
,
z
∈
U
)
,
defined by
(4)
Ω
z
(
λ
,
p
)
f
(
z
)
∶
=
z
p
+
∑
k
=
1
∞
‍
Γ
(
k
+
p
+
1
)
Γ
(
p
+
1
-
λ
)
Γ
(
p
+
1
)
Γ
(
k
+
p
+
1
-
λ
)
a
k
z
k
+
p
lllllllllllllllllllllllllllllllllllllllllllllllllll
(
z
∈
U
)
,
where f
∈
A
p is given by (2). In the particular case p
=
1 and -
∞
<
λ
<
1 thefractional-differintegral operator
(5)
Ω
z
(
λ
,
1
)
∶
=
Ω
z
λ
was earlier introduced by Owa and Srivastava [2] (see also [3]) and this is popularly known as theOwa-Srivastava operator [4–6]. Moreover, for 0
≤
λ
<
1 and p
∈
N, Ω
z
(
λ
,
p
) was investigated by Srivastava and Aouf [7] which was further extended to the range -
∞
<
λ
<
1,  p
∈
N by Srivastava and Mishra [8]. The following are some of the interesting particular cases of Ω
z
(
λ
,
p
):
(6)
Ω
z
(
-
1
,
p
)
f
(
z
)
=
p
+
1
z
∫
0
z
‍
f
(
ξ
)
d
ξ
,
Ω
z
(
0
,
p
)
f
(
z
)
=
f
(
z
)
,
Ω
z
(
1
,
p
)
f
(
z
)
=
z
f
′
(
z
)
p
,
Ω
z
(
n
,
p
)
f
(
z
)
=
(
p
-
n
)
!
z
n
f
n
(
z
)
p
!
,
(
n
∈
N
;
n
<
p
+
1
)
,
Ω
z
(
-
m
,
p
)
f
(
z
)
=
p
+
m
z
p
∫
0
z
‍
t
m
-
1
Ω
z
(
-
m
+
1
,
p
)
f
(
t
)
d
t
(
m
∈
N
)
.
Furthermore
(7)
z
(
Ω
z
(
λ
,
p
)
f
(
z
)
)
′
=
(
p
-
λ
)
Ω
z
(
λ
+
1
,
p
)
f
(
z
)
+
λ
Ω
z
(
λ
,
p
)
f
(
z
)
(
-
∞
<
λ
<
p
;
z
∈
U
)
.
The n-iterates of the operator Ω
z
(
λ
,
p
) are defined as follows:
(8)
Ω
z
,
0
(
λ
,
p
)
f
(
z
)
=
f
(
z
)
and, for n
∈
N,
(9)
Ω
z
,
n
(
λ
,
p
)
f
(
z
)
∶
=
Ω
z
(
λ
,
p
)
(
Ω
z
,
n
-
1
(
λ
,
p
)
f
(
z
)
)
=
∶
z
p
+
∑
k
=
1
∞
‍
[
Γ
(
k
+
p
+
1
)
Γ
(
p
+
1
-
λ
)
Γ
(
p
+
1
)
Γ
(
k
+
p
+
1
-
λ
)
]
n
a
k
z
k
+
p
.
Similarly, for f
∈
A
p, represented by (2), let theoperator
(10)
C
t
m
:
A
p
⟶
A
p
(
m
∈
N
,
t
∈
C
)
be defined by the following:
(11)
C
t
0
f
(
z
)
=
f
(
z
)
C
t
1
f
(
z
)
=
(
1
-
t
)
f
(
z
)
+
t
p
(
z
f
′
(
z
)
)
=
z
p
+
∑
k
=
1
∞
‍
{
1
+
t
k
p
}
a
k
z
k
+
pand, form
∈
N
∖
{
1
},
(12)
C
t
m
f
(
z
)
∶
=
C
t
1
(
C
t
m
-
1
f
(
z
)
)
=
∶
z
p
+
∑
k
=
1
∞
‍
{
1
+
t
k
p
}
m
a
k
z
k
+
p
.
Very recently Srivastava et al. [6] considered the composition of the operators C
t
m and Ω
z
,
n
(
λ
,
p
) and introduced the following operator:
(13)
D
t
(
λ
,
p
)
(
n
,
m
)
:
A
p
⟶
A
p
(
n
,
m
∈
N
∖
{
1
}
;
t
∈
C
)
.
That is, for f
∈
A
p, given by (2), we know that
(14)
D
t
(
λ
,
p
)
(
n
,
m
)
f
(
z
)
∶
=
Ω
z
,
n
(
λ
,
p
)
(
C
t
m
f
(
z
)
)
=
∶
C
t
m
(
Ω
z
,
n
(
λ
,
p
)
f
(
z
)
)
=
∶
z
p
+
∑
k
=
1
∞
‍
[
Γ
(
k
+
p
+
1
)
Γ
(
p
+
1
-
λ
)
Γ
(
p
+
1
)
Γ
(
k
+
p
+
1
-
λ
)
]
n
×
{
1
+
t
k
p
}
m
a
k
z
k
+
p
(
n
,
m
∈
N
0
∶
=
N
∪
{
0
}
,
-
∞
<
λ
<
p
+
1
,
t
≧
0
;
z
∈
U
)
.
The transformation D
t
(
λ
,
p
)
(
n
,
m
) includes, among many, the following two previously studied interesting operators as particular cases. (i)
Forn
=
m
∈
N, p
=
1, 0
≤
λ
<
1, the fractional derivative operator D
t
(
λ
,
1
)
(
n
,
n
)
:
=
D
t
n
,
λ was recently introduced and investigated by Al-Oboudi and Al-Amoudi [9, 10], in the context of functions represented by conical domains.
(ii)
Forn
=
m
∈
N, p
=
1, λ
=
0, t
=
1, D
1
(
0,1
)
(
n
,
n
) is the Sălăgean operator [11], which is, in fact, the n-iterates of the popular Alexander’s differential transform f
(
z
)
→
z
f
′
(
z
) [12].We next recall the definition of subordination. Suppose thatf
∈
A and g in A isunivalent in U. We say that f
(
z
) issubordinate to g
(
z
) in U if f
(
0
)
=
g
(
0
) and f
(
U
)
⊆
g
(
U
). Considering the function w
(
z
)
=
g
-
1
(
f
(
z
)
), it is readily checked that w
(
z
) satisfies the conditions of theSchwarz lemma and
(15)
f
(
z
)
=
g
(
w
(
z
)
)
(
z
∈
U
)
.
In a broader sense the function f
∈
A is said to be subordinate to the function g
∈
A (g need not be univalent in U), written as
(16)
f
≺
g
in
U
or
f
(
z
)
≺
g
(
z
)
(
z
∈
U
)
,
if condition (15) holds for some Schwarz function w
(
z
) (see [12] for details). We also need the following definition ofHadamard product (or convolution). For the functions f and g in A
p, given by the followingTaylor-Maclaurin’s series
(17)
f
(
z
)
=
z
p
+
∑
k
=
1
∞
‍
a
k
z
k
+
p
,
g
(
z
)
=
z
p
+
∑
k
=
1
∞
‍
b
k
z
k
+
p
llllllllllllllllllllll
(
z
∈
U
)
,
theirHadamard product (orconvolution) f
*
g is defined by
(18)
(
f
*
g
)
(
z
)
=
z
p
+
∑
k
=
1
∞
‍
a
k
b
k
z
k
+
p
(
z
∈
U
)
.
It is easy to see that f
*
g
∈
A
p.The study of iterations of entire and meromorphic functions, as the number of iterations tends to infinity, is a popular topic in complex analysis. However, investigations have been initiated only recently regarding iterations of certain transforms defined on classes of analytic and meromorphic functions. For example, Al-Oboudi and Al-Amoudi [9, 10] investigated properties of certain classes of analytic functions associated with conical domains, by making use of the operator D
t
n
,
λ. Their work generalized several earlier results of Srivastava and Mishra [13]. This theme has been further pursued in our more recent papers [6, 14, 15]. In the sequel to these current investigations, in the present paper, we define the following subclass of A
p associated with the iterated operator D
t
(
λ
,
p
)
(
n
,
m
) and investigate its several interesting properties. Our work is also motivated by earlier works in [16–20], connecting subordination and Hadamard product.Definition 1.
The functionf
∈
A
p is said to be in the class H
p
n
,
m
(
λ
,
t
;
h
) if the following subordination condition is satisfied:
(19)
(
1
-
t
)
z
-
p
D
t
(
λ
,
p
)
(
n
,
m
)
f
(
z
)
+
t
p
z
-
p
+
1
(
D
t
(
λ
,
p
)
(
n
,
m
)
f
(
z
)
)
′
≺
h
(
z
)
(
z
∈
U
)
,
where t is a complex number and h is an analytic convex univalent function in U.The function classH
p
n
,
m
(
λ
,
t
;
h
) includes several previously studied subclasses of A as particular cases. For example,(i)
forn
=
1,  m
=
0 the class H
p
1,0
(
λ
,
t
;
h
)
∶
=
H
p
(
λ
,
t
;
h
) was recently studied by Liu [17];
(ii)
forn
=
1;  m
=
0
;
p
=
1;  λ
=
1;  t
=
1; and h
(
z
)
=
(
1
+
z
)
/
(
1
-
z
), the class H
1
1,0
(
1,1
;
(
1
+
z
)
/
(
1
-
z
)
) was earlier investigated by R. Singh and S. Singh [21];
(iii)
forn
=
1;  m
=
0;  p
=
1;  λ
=
1; and h
(
z
)
=
(
1
+
a
z
)
/
(
1
+
b
z
)  (
-
1
≤
b
<
a
≤
1
),  H
1
1,0
(
1
,
t
;
(
1
+
a
z
)
/
(
1
+
b
z
)
) reduces to H
(
t
,
a
,
b
) which was investigated by Yang [22];
(iv)
forn
=
1;  m
=
0;  p
=
1;  λ
=
1; and h
(
z
)
=
1
+
M
z
(
M
>
0
),  H
1
1,0
(
1
,
t
;
1
+
M
z
) reduces to S
(
t
,
M
), the class introduced and studied by Zhongzhu and Owa [23] and Jinlin [24]. In the present paper we primarily focus on a variety of convolution theorems for the class H
p
n
,
m
(
λ
,
t
;
h
). We also find inclusion theorems and study behavior of the Libera-Livingston integral operator.


## 2. Some More Definitions and Preliminary Lemmas

We need the following definitions and results for the presentation of our results. LetC
V
(
ρ
) and S
*
(
ρ
)
(
0
≤
ρ
<
1
) denote, respectively, the classes ofunivalent convex functions of order
ρ andstarlike functions of order
ρ (see [12] for details). The function f
∈
A
1 is said to be in the class P
S
*
(
ρ
) consisting ofprestarlike functions of order
ρ [25] if
(20)
z
(
1
-
z
)
2
(
1
-
ρ
)
*
f
(
z
)
∈
S
*
(
ρ
)
.
It is readily seen that
(21)
P
S
*
(
0
)
=
∶
C
V
(
0
)
∶
=
C
V
,
P
S
*
(
1
2
)
=
∶
S
*
(
1
2
)
.
Furthermore, it is well known [25] that
(22)
P
S
*
(
μ
)
⊂
P
S
*
(
λ
)
(
μ
≤
λ
)
.
We will also need the following lemmas in order to derive our main results.Lemma 2 (see [26]; also see [27]).
Letg be analytic in U and let h be analytic and convex univalent in U with h
(
0
)
=
g
(
0
). If
(23)
g
(
z
)
+
1
μ
z
g
′
(
z
)
≺
h
(
z
)
(
z
∈
U
)
,
where R
(
μ
)
≥
0 and μ
≠
0, then
(24)
g
(
z
)
≺
h
~
(
z
)
=
μ
z
-
μ
∫
0
z
‍
ξ
μ
-
1
h
(
ξ
)
d
ξ
≺
h
(
z
)
(
z
∈
U
)
and h
~ is the best dominant of (24).Lemma 3 (see [25]).
Letρ
<
1,  f
∈
S
*
(
ρ
), and g
∈
P
S
*
(
ρ
). Then, for any analytic function F in U,
(25)
g
*
(
f
F
)
g
*
f
(
U
)
⊂
C
O
¯
(
F
(
U
)
)
,
where C
O
¯
(
F
(
U
)
) denotes the closed convex hull of F
(
U
).Lemma 4 (see [28]).
LetH and G be univalent convex functions in U, and let h and g be functions in A. Suppose that h
≺
H and g
≺
G in U. Then h
*
g
≺
H
*
G in U.The following well known result is a consequence of theprinciple of subordination and can be found, for example, in [12, 29].Lemma 5.
Let the functiong
∈
A satisfy g
(
0
)
=
1 and R
(
g
(
z
)
)
>
β
0  (
0
≤
β
0
<
1
,
z
∈
U
). Then
(26)
R
(
g
(
z
)
)
>
β
0
+
(
1
-
β
0
)
1
-
|
z
|
1
+
|
z
|
(
z
∈
U
)
.


## 3. Convolution Results

We state and prove the following convolution results.Theorem 6.
Letf
∈
H
p
n
,
m
(
λ
,
t
;
h
), and suppose that g in A
p satisfies the following:
(27)
R
{
z
-
p
g
(
z
)
}
>
1
2
(
z
∈
U
)
.
Then
(28)
f
*
g
∈
H
p
n
,
m
(
λ
,
t
;
h
)
.Proof.
For everyf and g in A
p, we have
(29)
(
1
-
t
)
z
-
p
D
t
(
λ
,
p
)
(
n
,
m
)
(
f
*
g
)
(
z
)
+
t
p
z
-
p
+
1
(
D
t
(
λ
,
p
)
(
n
,
m
)
(
f
*
g
)
(
z
)
)
′
=
(
1
-
t
)
(
z
-
p
g
(
z
)
)
*
(
z
-
p
D
t
(
λ
,
p
)
(
n
,
m
)
f
(
z
)
)
+
t
p
(
z
-
p
g
(
z
)
)
*
(
z
-
p
+
1
(
D
t
(
λ
,
p
)
(
n
,
m
)
f
(
z
)
)
′
)
=
(
z
-
p
g
(
z
)
)
*
Ψ
(
z
)
,
where
(30)
Ψ
(
z
)
=
(
1
-
t
)
z
-
p
D
t
(
λ
,
p
)
(
n
,
m
)
f
(
z
)
+
t
p
z
-
p
+
1
(
D
t
(
λ
,
p
)
(
n
,
m
)
f
(
z
)
)
′
.
Now, if f
∈
H
p
n
,
m
(
λ
,
t
;
h
), then
(31)
Ψ
(
z
)
≺
h
(
z
)
(
z
∈
U
)
.
Furthermore, condition (27) is equivalent to
(32)
z
-
p
g
(
z
)
≺
1
1
-
z
(
z
∈
U
)
.
Therefore, an application of Lemma 4 in (29) yields
(33)
(
1
-
t
)
z
-
p
D
t
(
λ
,
p
)
(
n
,
m
)
(
f
*
g
)
(
z
)
+
t
p
z
-
p
+
1
(
D
t
(
λ
,
p
)
(
n
,
m
)
(
f
*
g
)
(
z
)
)
′
≺
1
1
-
z
*
h
(
z
)
=
h
(
z
)
.
This shows thatf
*
g
∈
H
p
n
,
m
(
λ
,
t
;
h
). The proof of Theorem 6 is completed.Remark 7.
Takingn
=
1,  m
=
0, in Theorem 6 we get a recent result of Liu ([17], Theorem 3). The choices n
=
1,  m
=
0,  p
=
1,  λ
=
1, and h
(
z
)
=
(
1
+
a
z
)
/
(
1
+
b
z
)  (
-
1
≤
b
<
1
;
a
>
b
) yield a result of Yang ([22], Theorem 4).Corollary 8.
Let the functionf given by (2) be a member of H
p
n
,
m
(
λ
,
t
;
h
) and
(34)
s
r
(
z
)
=
z
p
+
∑
k
=
1
r
-
1
‍
a
k
z
k
+
p
(
r
∈
N
∖
{
1
}
;
z
∈
U
)
.
Then the function
(35)
σ
r
(
z
)
=
∫
0
1
‍
t
-
p
s
r
(
t
z
)
d
t
is in the class H
p
n
,
m
(
λ
,
t
;
h
).Proof.
Letf
∈
H
p
n
,
m
(
λ
,
t
;
h
). We note that
(36)
σ
r
(
z
)
=
z
p
+
∑
k
=
1
r
-
1
‍
a
k
k
+
1
z
k
+
p
=
(
f
*
g
r
)
(
z
)
(
r
∈
N
∖
{
1
}
)
,
where
(37)
f
(
z
)
=
z
p
+
∑
k
=
1
∞
‍
a
k
z
k
+
p
,
g
r
(
z
)
=
z
p
+
∑
k
=
1
r
-
1
‍
z
k
+
p
k
+
1
∈
A
p
.
Also, for r
∈
N
∖
{
1
}, it is well known [18] that
(38)
R
{
z
-
p
g
r
(
z
)
}
>
1
2
(
z
∈
U
)
.
In view of (36) and (38), an application of Theorem 6 gives
(39)
σ
r
∈
H
p
n
,
m
(
λ
,
t
;
h
)
.
The proof of Corollary 8 is completed.Theorem 9.
Let the functiong in A
p be such that z
-
p
+
1
g
(
z
) is a prestarlike function of order ρ
(
ρ
<
1
). If f
∈
H
p
n
,
m
(
λ
,
t
;
h
), then
(40)
f
*
g
∈
H
p
n
,
m
(
λ
,
t
;
h
)
.Proof.
Letf
∈
H
p
n
,
m
(
λ
,
t
;
h
) and g
∈
A
p. Then (29) gives
(41)
(
1
-
t
)
z
-
p
D
t
(
λ
,
p
)
(
n
,
m
)
(
f
*
g
)
(
z
)
+
t
p
z
-
p
+
1
(
D
t
(
λ
,
p
)
(
n
,
m
)
(
f
*
g
)
(
z
)
)
′
=
(
z
-
p
+
1
g
(
z
)
)
*
(
z
Ψ
(
z
)
)
(
z
-
p
+
1
g
(
z
)
)
*
z
(
z
∈
U
)
,
where Ψ
(
z
) is defined as in (30). We noted in the proof of Theorem 6 that Ψ
(
z
)
≺
h
(
z
). Since z
-
p
+
1
g
(
z
)
∈
P
S
*
(
ρ
),  z
∈
S
*
(
ρ
), and h
(
z
) is convex univalent in U, an application of Lemma 3 in (41) yields the following:
(42)
(
1
-
t
)
z
-
p
D
t
(
λ
,
p
)
(
n
,
m
)
(
f
*
g
)
(
z
)
+
t
p
z
-
p
+
1
(
D
t
(
λ
,
p
)
(
n
,
m
)
(
f
*
g
)
(
z
)
)
′
≺
h
(
z
)
(
z
∈
U
)
.
Therefore, f
*
g
∈
H
p
n
,
m
(
λ
,
t
;
h
). The proof Theorem 9 is completed.Takingρ
=
1
/
2 in Theorem 9 we get the following.Corollary 10.
Letf
∈
H
p
n
,
m
(
λ
,
t
;
h
) and suppose that g in A
p is such that z
-
p
+
1
g
∈
S
*
(
1
/
2
). Then
(43)
f
*
g
∈
H
p
n
,
m
(
λ
,
t
;
h
)
.
In particular, if z
-
p
+
1
g is univalent convex then f
*
g
∈
H
p
n
,
m
(
λ
,
t
;
h
).In the following theorem we discuss convolution properties of the function classH
(
λ
,
t
;
h
) when h is a right half plane mapping.Theorem 11.
Lett
≥
0 and suppose that each of the functions f
j
(
j
=
1,2
) is a member of the class H
p
n
,
m
(
λ
,
t
;
h
j
), where
(44)
h
j
(
z
)
=
1
+
(
1
-
2
β
j
)
z
1
-
z
(
0
≤
β
j
<
1
)
.
If f
∈
A
p is defined by the following
(45)
f
(
z
)
=
D
t
(
λ
,
p
)
(
n
,
m
)
(
f
1
*
f
2
)
(
z
)
(
z
∈
U
)
,
then f
∈
H
p
n
,
m
(
λ
,
t
;
h
), where
(46)
h
(
z
)
=
1
+
(
1
-
2
β
)
z
1
-
z
and β is given by
(47)
β
=
{
1
-
4
(
1
-
β
1
)
(
1
-
β
2
)
×
(
1
-
p
t
∫
0
1
‍
u
(
p
/
t
)
-
1
1
+
u
d
u
)
(
λ
>
0
)
1
-
2
(
1
-
β
1
)
(
1
-
β
2
)
(
λ
=
0
)
.
The bound on β is the best possible.Proof.
We consider the caset
>
0. Suppose that f
j
∈
H
p
n
,
m
(
λ
,
t
;
h
j
),  (
j
=
1,2
), where h
j
(
z
) is given by (44). By setting
(48)
Φ
j
(
z
)
=
(
1
-
t
)
z
-
p
D
t
(
λ
,
p
)
(
n
,
m
)
f
j
(
z
)
+
t
p
z
-
p
+
1
(
D
t
(
λ
,
p
)
(
n
,
m
)
f
j
(
z
)
)
′
we see, in the light of Definition 1, that
(49)
Φ
j
(
z
)
≺
h
j
(
z
)
(
j
=
1,2
)
.
A routine calculation yields the following:
(50)
D
t
(
λ
,
p
)
(
n
,
m
)
f
j
(
z
)
=
p
t
z
-
p
(
1
-
t
)
/
t
∫
0
z
‍
ξ
(
p
/
t
)
-
1
Φ
j
(
ξ
)
d
ξ
(
j
=
1,2
)
.
Now, if f
(
z
) is defined by (45), then using (50) we get that
(51)
D
t
(
λ
,
p
)
(
n
,
m
)
f
(
z
)
=
D
t
(
λ
,
p
)
(
n
,
m
)
f
1
(
z
)
*
D
t
(
λ
,
p
)
(
n
,
m
)
f
2
(
z
)
=
(
p
t
z
p
∫
0
1
‍
u
(
p
/
t
)
-
1
Φ
1
(
u
z
)
d
u
)
*
(
p
t
z
p
∫
0
1
‍
u
(
p
/
t
)
-
1
Φ
2
(
u
z
)
d
u
)
=
p
t
z
p
∫
0
1
‍
u
(
p
/
t
)
-
1
Φ
0
(
u
z
)
d
u
,
where
(52)
Φ
0
(
z
)
=
p
t
z
p
∫
0
1
‍
u
(
p
/
t
)
-
1
(
Φ
1
*
Φ
2
)
(
u
z
)
d
u
.
Since
(53)
Φ
1
(
z
)
-
β
1
1
-
β
1
≺
1
+
z
1
-
z
,
1
2
+
Φ
2
(
z
)
-
β
2
2
(
1
-
β
2
)
≺
1
1
-
z
,
by using Lemma 4, we get
(54)
(
Φ
1
(
z
)
-
β
1
1
-
β
1
)
*
(
1
2
+
Φ
2
(
z
)
-
β
2
2
(
1
-
β
2
)
)
≺
1
+
z
1
-
z
.
A simple calculation gives that
(55)
Φ
1
(
z
)
*
Φ
2
(
z
)
≺
1
±
4
(
1
-
β
1
)
(
1
-
β
2
)
z
1
-
z
.
Therefore, by the Lindelöff principle of subordination we have
(56)
R
{
Φ
1
(
z
)
*
Φ
2
(
z
)
}
>
β
0
∶
=
1
-
2
(
1
-
β
1
)
(
1
-
β
2
)
(
z
∈
U
)
.
Since
(57)
(
1
-
t
)
z
-
p
D
t
(
λ
,
p
)
(
n
,
m
)
f
(
z
)
+
t
p
z
-
p
+
1
(
D
t
(
λ
,
p
)
(
n
,
m
)
f
(
z
)
)
′
=
Φ
0
(
z
)
,
by using (52) in conjunction with (56) and Lemma 4 we get the following:
(58)
R
{
t
p
(
1
-
t
)
z
-
p
D
t
(
λ
,
p
)
(
n
,
m
)
f
(
z
)
+
t
p
z
-
p
+
1
(
D
t
(
λ
,
p
)
(
n
,
m
)
f
(
z
)
)
′
}
=
R
{
Φ
0
(
z
)
}
=
p
t
∫
0
1
‍
u
(
p
/
t
)
-
1
R
{
(
Φ
1
*
Φ
2
)
(
u
z
)
}
d
u
≥
p
t
∫
0
1
‍
u
(
p
/
t
)
-
1
(
β
0
+
(
1
-
β
0
)
1
-
u
|
z
|
1
+
u
|
z
|
)
d
u
>
β
0
+
p
(
1
-
β
0
)
t
∫
0
1
‍
u
(
p
/
t
)
-
1
1
-
u
1
+
u
d
u
=
1
-
4
(
1
-
β
1
)
(
1
-
β
2
)
(
1
-
p
t
∫
0
1
‍
u
(
p
/
t
)
-
1
1
+
u
d
u
)
=
β
(
z
∈
U
)
.
This proves that f
(
z
)
∈
H
p
n
,
m
(
λ
,
t
;
h
), where the function h
(
z
) is given by (46).
In order to show that the value ofβ is the least possible, we take the functions f
j
(
z
)
∈
A
p  (
j
=
1,2
) defined by
(59)
D
t
(
λ
,
p
)
(
n
,
m
)
f
j
(
z
)
=
p
t
z
-
p
(
1
-
t
)
/
t
∫
0
z
‍
ξ
(
p
/
t
)
-
1
×
(
β
j
+
(
1
-
β
j
)
1
+
ξ
1
-
ξ
)
d
ξ
,
for which we have
(60)
Φ
j
(
z
)
=
(
1
-
t
)
z
-
p
D
t
(
λ
,
p
)
(
n
,
m
)
f
j
(
z
)
+
p
t
z
-
p
+
1
(
D
t
(
λ
,
p
)
(
n
,
m
)
f
j
(
z
)
)
′
=
β
j
+
(
1
-
β
j
)
1
+
z
1
-
z
(
j
=
1,2
)
,
(61)
(
Φ
1
*
Φ
2
)
(
z
)
=
1
+
4
(
1
-
β
1
)
(
1
-
β
2
)
z
1
-
z
.
Hence, for f
∈
A
p, given by (45), we obtain
(62)
(
1
-
t
)
z
-
p
D
t
(
λ
,
p
)
(
n
,
m
)
f
(
z
)
+
p
t
z
-
p
+
1
(
D
t
(
λ
,
p
)
(
n
,
m
)
f
(
z
)
)
′
=
p
t
∫
0
1
‍
u
(
p
/
t
)
-
1
(
1
+
4
(
1
-
β
1
)
(
1
-
β
2
)
u
z
1
-
u
z
)
d
u
⟶
β
(
as
z
⟶
-
1
)
.
Finally, for the case t
=
0, the proof of Theorem 11 is simple, so we choose to omit the details involved.


## 4. Properties of the Libera-Livingston Transform

For the functionf
∈
A
p, the function F defined by
(63)
F
(
z
)
=
μ
+
p
z
μ
∫
0
z
‍
ξ
μ
-
1
f
(
ξ
)
d
ξ
(
R
(
μ
)
>
-
p
;
z
∈
U
)
is popularly known as theLibera-Livingston transform of f. We state and prove the following.Theorem 12.
Letf
∈
H
p
n
,
m
(
λ
,
t
;
h
). Then the function F defined by (63) is in the class H
p
n
,
m
(
λ
,
t
;
h
~
), where
(64)
h
~
(
z
)
=
(
μ
+
p
)
z
-
(
μ
+
p
)
∫
0
z
‍
ξ
μ
+
p
-
1
h
(
ξ
)
d
ξ
≺
h
(
z
)
(
z
∈
U
)
.
Consequently, F
∈
H
p
n
,
m
(
λ
,
t
;
h
). The function h
~
(
z
) is the best dominant in (64).Proof.
We define the functionK on U by
(65)
z
p
K
(
z
)
=
(
1
-
t
)
D
t
(
λ
,
p
)
(
n
,
m
)
F
(
z
)
+
t
p
z
(
D
t
(
λ
,
p
)
(
n
,
m
)
F
(
z
)
)
′
.
Differentiating both the sides of (65) with respect to z, we get
(66)
p
K
(
z
)
+
z
K
′
(
z
)
=
p
(
1
-
t
)
z
-
p
D
t
(
λ
,
p
)
(
n
,
m
)
(
z
F
′
(
z
)
p
)
+
t
z
-
p
+
1
(
D
t
(
λ
,
p
)
(
n
,
m
)
{
z
F
'
(
z
)
p
}
)
′
.
Also, the defining relation (63) yields
(67)
(
μ
+
p
)
f
(
z
)
=
μ
F
(
z
)
+
z
F
′
(
z
)
.
Now, a routine calculation using (65), (66), and (67) gives
(68)
(
1
-
t
)
z
-
p
D
t
(
λ
,
p
)
(
n
,
m
)
f
(
z
)
+
t
p
z
-
p
+
1
(
D
t
(
λ
,
p
)
(
n
,
m
)
f
(
z
)
)
′
=
K
(
z
)
+
z
K
′
(
z
)
μ
+
p
(
z
∈
U
)
.
Since f
∈
H
p
n
,
m
(
λ
,
t
;
h
), we get the following from the preceding equation (68):
(69)
K
(
z
)
+
z
K
′
(
z
)
μ
+
p
≺
h
(
z
)
.
Therefore, by applying Lemma 2 we have
(70)
K
(
z
)
≺
h
~
(
z
)
=
(
μ
+
p
)
z
-
(
μ
+
p
)
∫
0
z
‍
t
μ
+
p
-
1
h
(
t
)
d
t
≺
h
(
z
)
.
This last subordination (70) is equivalent to
(71)
F
∈
H
p
n
,
m
(
λ
,
t
;
h
~
)
⊂
H
p
n
,
m
(
λ
,
t
;
h
)
.
The proof of Theorem 12 is completed.Theorem 13.
Letf
∈
A
p and suppose that the function F is defined as in (63). If
(72)
(
1
-
γ
)
z
-
p
D
t
(
λ
,
p
)
(
n
,
m
)
F
(
z
)
+
γ
z
-
p
D
t
(
λ
,
p
)
(
n
,
m
)
f
(
z
)
≺
h
(
z
)
(
γ
>
0
)
,
then F
∈
H
p
n
,
m
(
λ
,
0
;
h
~
), where
(73)
h
~
(
z
)
=
μ
+
p
γ
z
-
(
μ
+
p
)
/
γ
×
∫
0
z
‍
ξ
(
(
μ
+
p
)
/
γ
)
-
1
h
(
ξ
)
d
ξ
≺
h
(
z
)
(
R
(
μ
)
>
-
p
)
.
The function h
~
(
z
) is the best dominant in (73).Proof.
We define the functionH on U by
(74)
H
(
z
)
=
z
-
p
D
t
(
λ
,
p
)
(
n
,
m
)
F
(
z
)
.
Differentiation of both the sides of (74) combined with the identity
(75)
z
(
D
t
(
λ
,
p
)
(
n
,
m
)
F
(
z
)
)
′
p
=
D
t
(
λ
,
p
)
(
n
,
m
)
(
z
F
′
(
z
)
p
)
gives
(76)
z
-
p
D
t
(
λ
,
p
)
(
n
,
m
)
(
z
F
′
(
z
)
p
)
=
H
(
z
)
+
z
H
′
(
z
)
p
.
By making use of (67) we simplify the subordinate of (72) as follows:
(77)
(
1
-
γ
)
z
-
p
D
t
(
λ
,
p
)
(
n
,
m
)
F
(
z
)
+
γ
z
-
p
D
t
(
λ
,
p
)
(
n
,
m
)
f
(
z
)
=
(
1
-
γ
)
H
(
z
)
+
γ
z
-
p
D
t
(
λ
,
p
)
(
n
,
m
)
×
(
μ
μ
+
p
F
(
z
)
+
p
μ
+
p
z
F
′
(
z
)
p
)
=
(
1
-
γ
)
H
(
z
)
μ
γ
μ
+
p
z
-
p
D
t
(
λ
,
p
)
(
n
,
m
)
F
(
z
)
+
p
γ
μ
+
p
z
-
p
D
t
(
λ
,
p
)
(
z
F
'
(
z
)
p
)
.
Next, by using (74) and (76), the above identity further simplifies to the following:
(78)
(
1
-
γ
)
z
-
p
D
t
(
λ
,
p
)
(
n
,
m
)
F
(
z
)
+
γ
z
-
p
D
t
(
λ
,
p
)
(
n
,
m
)
f
(
z
)
=
(
1
-
p
γ
μ
+
p
)
H
(
z
)
+
p
γ
μ
+
p
(
H
(
z
)
+
z
H
′
(
z
)
p
)
=
H
(
z
)
+
γ
μ
+
p
z
H
′
(
z
)
.
The subordination (72) is thus equivalent to
(79)
H
(
z
)
+
γ
μ
+
p
z
H
′
(
z
)
≺
h
(
z
)
,
(
R
(
μ
)
>
-
p
,
γ
>
0
;
z
∈
U
)
.
Therefore, an application of Lemma2 yields the assertion of Theorem 13. The proof of Theorem 13 is completed.


## 5. Inclusion Theorems

Theorem 14.
Lett
>
0,  γ
>
0, and f
∈
H
p
n
,
m
(
λ
,
t
;
γ
h
+
1
-
γ
). If γ
≤
γ
0, then f
(
z
)
∈
H
p
n
,
m
(
λ
,
0
;
h
), where
(80)
γ
0
=
1
2
(
1
-
p
t
∫
0
1
‍
u
(
p
/
t
)
-
1
1
+
u
d
u
)
-
1
.
The bound γ
0 is sharp when h
(
z
)
=
1
/
(
1
-
z
).Proof.
Letf
∈
H
p
n
,
m
(
λ
,
t
;
γ
h
+
1
-
γ
). By setting
(81)
g
(
z
)
=
z
-
p
D
t
(
λ
,
p
)
(
n
,
m
)
f
(
z
)
(
z
∈
U
)
,
we have
(82)
(
1
-
t
)
z
-
p
D
t
(
λ
,
p
)
(
n
,
m
)
f
(
z
)
+
t
p
z
-
p
+
1
(
D
t
(
λ
,
p
)
(
n
,
m
)
f
(
z
)
)
′
=
g
(
z
)
+
t
p
z
g
′
(
z
)
.
Therefore, by using Definition 1, we get
(83)
g
(
z
)
+
t
p
z
g
′
(
z
)
≺
γ
h
(
z
)
+
1
-
γ
and an application of Lemma 2 yields
(84)
g
(
z
)
≺
γ
p
t
z
-
p
/
t
∫
0
z
‍
ξ
(
p
/
t
)
-
1
h
(
ξ
)
d
ξ
+
1
-
γ
=
(
h
*
ψ
)
(
z
)
,
where
(85)
ψ
(
z
)
=
γ
p
t
z
-
p
/
t
∫
0
z
‍
ξ
(
p
/
t
)
-
1
1
-
ξ
d
ξ
+
1
-
γ
.
It follows from (85) that if 0
<
γ
≤
γ
0, where γ
0
>
1 is given by (80), then
(86)
R
(
ψ
(
z
)
)
=
γ
p
t
∫
0
1
‍
u
(
p
/
t
)
-
1
R
(
1
1
-
u
z
)
d
u
+
1
-
γ
>
γ
p
t
∫
0
1
‍
u
(
p
/
t
)
-
1
1
+
u
d
u
+
1
-
γ
≥
1
2
(
z
∈
U
)
.
Equivalently,
(87)
ψ
(
z
)
≺
1
1
-
z
,
(
z
∈
U
)
.
Since h
(
z
) and ψ
(
z
) are both convex univalent functions in U, using Lemma 4, we obtain from (84) that
(88)
g
(
z
)
≺
(
h
*
ψ
)
(
z
)
≺
h
(
z
)
.
Therefore in view of (81) we have
(89)
z
-
p
D
t
(
λ
,
p
)
(
n
,
m
)
f
(
z
)
≺
h
(
z
)
.
Or equivalently, f
(
z
)
∈
H
p
n
,
m
(
λ
,
0
;
h
).
In order to prove that the bound onγ
0 is the best possible, set h
(
z
)
=
1
/
(
1
-
z
) and let the function f be defined on U by
(90)
z
-
p
D
t
(
λ
,
p
)
(
n
,
m
)
f
(
z
)
=
γ
p
t
z
-
p
/
t
∫
0
z
‍
ξ
(
p
/
t
)
-
1
1
-
ξ
d
ξ
+
1
-
γ
.
It can be readily verified that f
∈
A
p and
(91)
(
1
-
t
)
z
-
p
D
t
(
λ
,
p
)
(
n
,
m
)
f
(
z
)
+
t
p
z
-
p
+
1
(
D
t
(
λ
,
p
)
(
n
,
m
)
f
(
z
)
)
′
=
γ
h
(
z
)
+
1
-
γ
.
Thus f
(
z
)
∈
H
p
n
,
m
(
λ
,
t
;
γ
h
+
1
-
γ
). Also, for γ
>
γ
0, and z
→
-
1, we have
(92)
R
{
z
-
p
D
t
(
λ
,
p
)
(
n
,
m
)
f
(
z
)
}
⟶
γ
p
t
∫
0
1
‍
u
(
p
/
t
)
-
1
1
+
u
d
u
+
1
-
γ
<
1
2
which shows that
(93)
f
(
z
)
∉
H
p
n
,
m
(
λ
,
0
;
h
)
.
Hence the bound γ
0 cannot be increased when h
(
z
)
=
1
/
(
1
-
z
). The proof of Theorem 14 is completed.Takingm
=
0, we use the notation
(94)
H
p
n
,
0
(
λ
,
t
;
h
)
∶
=
H
p
n
(
λ
,
t
;
h
)
in the following theorem.Theorem 15.
Let0
≤
t
1
<
t
2. Then
(95)
H
p
n
(
λ
,
t
2
;
h
)
⊂
H
p
n
(
λ
,
t
1
;
h
)
.Proof.
Let0
≤
t
1
<
t
2 and suppose that f
∈
H
p
n
(
λ
,
t
2
;
h
). We define
(96)
g
(
z
)
=
z
-
p
D
t
(
λ
,
p
)
(
n
,
0
)
f
(
z
)
(
z
∈
U
)
.
The functiong is analytic in U with g
(
0
)
=
1. Differentiating the expressions on both the sides of (96) with respect to z and using Definition 1, we have
(97)
(
1
-
t
2
)
z
-
p
D
t
(
λ
,
p
)
(
n
,
0
)
f
(
z
)
+
p
t
2
z
-
p
+
1
(
D
t
(
λ
,
p
)
(
n
,
0
)
f
(
z
)
)
′
=
g
(
z
)
+
t
2
p
z
g
′
(
z
)
≺
h
(
z
)
(
z
∈
U
)
.
Hence an application of Lemma 2 yields
(98)
g
(
z
)
≺
h
(
z
)
(
z
∈
U
)
.
Noting that 0
≤
t
1
/
t
2
<
1 and h
(
z
) is convex univalent in U, it follows from (96), (97), and (98) that
(99)
(
1
-
t
1
)
z
-
p
D
t
(
λ
,
p
)
(
n
,
0
)
f
(
z
)
+
p
t
1
z
-
p
+
1
(
D
t
(
λ
,
p
)
(
n
,
0
)
f
(
z
)
)
′
=
t
1
t
2
{
p
t
2
(
1
-
t
2
)
z
-
p
D
t
(
λ
,
p
)
(
n
,
0
)
f
(
z
)
+
p
t
2
z
-
p
+
1
(
D
t
(
λ
,
p
)
(
n
,
0
)
f
(
z
)
)
′
}
+
(
1
-
t
1
t
2
)
g
(
z
)
.
≺
h
(
z
)
(
z
∈
U
)
.
Thus f
∈
H
p
n
(
λ
,
t
1
;
h
). Consequently,
(100)
H
p
n
(
λ
,
t
2
;
h
)
⊂
H
p
n
(
λ
,
t
1
;
h
)
.
The proof of Theorem 15 is completed.Remark 16.
The particular casen
=
1 of Theorem 15 gives a result of Liu ([17], Theorem 1).



---

*Source: 915385-2014-10-14.xml*