# Certain Admissible Classes of Multivalent Functions

Authors: M. K. Aouf; H. M. Srivastava; T. M. Seoudy
Journal: Journal of Complex Analysis
 (2014)

---

## Abstract

We investigate some applications of the differential subordination and the differential superordination of certain admissible classes of multivalent functions in the open unit diskU. Several differential sandwich-type results are also obtained.


---

## Body

## 1. Introduction

LetH
(
U
) be the class of functions analytic in the open unit disk
(1)
U
=
{
z
:
z
∈
C
,
|
z
|
<
1
}
.
Denote by H
[
a
,
n
] the subclass of H
(
U
) consisting of functions of the form
(2)
f
(
z
)
=
a
+
a
n
z
n
+
a
n
+
1
z
n
+
1
+
⋯
with
(3)
H
=
H
[
1,1
]
.
Also let A
(
p
) be the class of all analytic and p-valent functions of the form
(4)
f
(
z
)
=
z
p
+
∑
n
=
p
+
1
∞
‍
a
n
z
n
(
p
∈
N
=
{
1,2
,
3
,
…
}
;
z
∈
U
)
.
Let f and F be members of the function class H
(
U
). The function f
(
z
) is said to be subordinate to F
(
z
), or the function F
(
z
) is said to be superordinate to f
(
z
), if there exists a function ω
(
z
), analytic in U with
(5)
ω
(
0
)
=
0
,
|
ω
(
z
)
|
<
1
(
z
∈
U
)
,
such that
(6)
f
(
z
)
=
F
(
ω
(
z
)
)
.
In such a case we write f
(
z
)
≺
F
(
z
). If F is univalent in U, then f
(
z
)
≺
F
(
z
) if and only if f
(
0
)
=
F
(
0
) and f
(
U
)
⊂
F
(
U
) (see [1–3]; see also several recent works [4–8] dealing with various properties and applications of the principle of differential subordination and the principle of differential superordination).We denote byF the set of all functions q that are analytic and injective on U
¯
∖
E
(
q
), where
(7)
E
(
q
)
=
{
ζ
∈
∂
U
:
lim
⁡
z
→
ζ
q
(
z
)
=
∞
}
,
and are such that
(8)
q
′
(
ζ
)
≠
0
(
ζ
∈
∂
U
∖
E
(
q
)
)
.
We further let the subclass of F for which q
(
0
)
=
a be denoted by F
(
a
) and write
(9)
F
(
1
)
≡
F
1
.In order to prove our results, we will make use of the following classes of admissible functions.Definition 1 (see [2, p. 27, Definition 2.3a]).
LetΩ be a set in C,  q
∈
F, and n
∈
N. The class Ψ
n
[
Ω
,
q
] of admissible functions consists of those functions ψ
:
C
3
×
U
→
C that satisfy the following admissibility condition:
(10)
ψ
(
r
,
s
,
t
;
z
)
∉
Ω
whenever
(11)
r
=
q
(
ζ
)
,
s
=
k
ζ
q
′
(
ζ
)
,
R
(
t
s
+
1
)
≧
k
R
(
1
+
ζ
q
′
(
ζ
)
q
′
(
ζ
)
)
,
where z
∈
U
,
ζ
∈
∂
U
∖
E
(
q
), and k
≧
n. We write Ψ
1
[
Ω
,
q
] simply as Ψ
[
Ω
,
q
].
In particular, if(12)
q
(
z
)
=
(
M
z
+
a
M
+
a
¯
z
)
M
(
M
>
0
;
|
a
|
<
M
)
,
then
(13)
q
(
U
)
=
U
M
=
{
w
:
|
w
|
<
M
}
,
q
(
0
)
=
a
,
E
(
q
)
=
∅, and q
∈
F
(
a
). In this case, we set Ψ
n
[
Ω
,
M
,
a
]
=
Ψ
n
[
Ω
,
q
]. Moreover, in the special case, when we set Ω
=
U
M, the class is simply denoted by Ψ
n
[
M
,
a
].Definition 2 (see [3, p. 817, Definition 3]).
LetΩ be a set in C
,
q
∈
H
[
a
,
n
] with q
′
(
z
)
≠
0. The class Ψ
n
′
[
Ω
,
q
] of admissible functions consists of those functions ψ
:
C
3
×
U
¯
→
C that satisfy the following admissibility condition:
(14)
ψ
(
r
,
s
,
t
;
ζ
)
∈
Ω
whenever
(15)
r
=
q
(
z
)
,
s
=
z
q
′
(
z
)
m
,
R
(
t
s
+
1
)
≦
1
m
R
(
1
+
z
q
′′
(
z
)
q
′
(
z
)
)
,
where z
∈
U, ζ
∈
∂
U, and m
≧
n
≧
1. In particular, we write Ψ
1
′
[
Ω
,
q
] simply as Ψ
′
[
Ω
,
q
].
In our investigation we need the following lemmas which are proved by Miller and Mocanu (see [2] and [3]).Lemma 3 (see [2, p. 28, Theorem 2.3b]).
Letψ
∈
Ψ
n
[
Ω
,
q
] with q
(
0
)
=
a. If the analytic function g
(
z
) given by
(16)
g
(
z
)
=
a
+
a
n
z
n
+
a
n
+
1
z
n
+
1
+
⋯
satisfies the inclusion relationship
(17)
ψ
(
g
(
z
)
,
z
g
′
(
z
)
,
z
2
g
′′
(
z
)
;
z
)
∈
Ω
,
then g
≺
q.Lemma 4 (see [3, p. 818, Theorem 1]).
Letψ
∈
Ψ
n
′
[
Ω
,
q
] with q
(
0
)
=
a. If g
∈
F
(
a
) and the function
(18)
ψ
(
g
(
z
)
,
z
g
′
(
z
)
,
z
2
g
′′
(
z
)
;
z
)
is univalent in U, then
(19)
Ω
⊂
{
ψ
(
g
(
z
)
,
z
g
′
(
z
)
,
z
2
g
′′
(
z
)
;
z
)
:
z
∈
U
}
implies that q
≺
g.In this paper, we determine the sufficient conditions for certain admissible classes of multivalent functions so that(20)
q
1
(
z
)
≺
(
f
(
z
)
z
p
)
μ
≺
q
2
(
z
)
,
where μ
>
0 and q
1 and q
2 are given univalent functions in U with
(21)
q
1
(
0
)
=
q
2
(
0
)
=
1
.
In addition, we derive several differential sandwich-type results. A similar problem for analytic functions involving certain operators was studied by Aghalary et al. [9], Ali et al. [10], Aouf et al. [11], Kim and Srivastava [12], and other authors (see [13–15]). In particular, unlike the earlier investigation by Aouf and Seoudy [16], we have not used any operators in our present investigation. Nevertheless, for the benefit of the targeted readers of our paper, in addition to oft-cited paper [11], we have included several further citations of recent works (see, e.g., [17–21]) in which various families of linear operators were applied in conjunction with the principle of differential subordination and the principle of differential superordination for the study of analytic or meromorphic multivalent functions.


## 2. A Set of Subordination Results

Unless otherwise mentioned, we assume throughout this paper thatp
∈
N, μ
>
0, z
∈
U, and all power functions are tacitly assumed to denote their principal values.Definition 5.
LetΩ be a set in C and q
∈
F
1
∩
H. The class Φ
[
Ω
,
q
,
p
,
μ
] of admissible functions consists of those functions ϕ
:
C
3
×
U
→
C that satisfy the following admissibility condition:
(22)
ϕ
(
u
,
v
,
w
;
z
)
∉
Ω
whenever
(23)
u
=
q
(
ζ
)
,
v
=
k
ζ
q
′
(
ζ
)
+
μ
p
q
(
ζ
)
μ
p
,
R
(
w
-
(
2
μ
p
-
1
)
v
+
μ
p
u
v
-
u
)
≧
k
R
(
1
+
ζ
q
′′
(
ζ
)
q
′
(
ζ
)
)
,
where z
∈
U, ζ
∈
∂
U
∖
E
(
q
), and k
≧
1. For simplicity, we write
(24)
Φ
[
Ω
,
q
,
p
,
1
]
=
Φ
[
Ω
,
q
,
p
]
.Theorem 6.
Letϕ
∈
Φ
[
Ω
,
q
,
p
,
μ
]. If f
∈
A
(
p
) satisfies the condition
(25)
{
ϕ
(
(
f
(
z
)
z
p
)
μ
,
(
f
(
z
)
z
p
)
μ
z
f
′
(
z
)
p
f
(
z
)
,
(
f
(
z
)
z
p
)
μ
z
2
f
′′
(
z
)
p
f
(
z
)
k
k
k
k
+
(
μ
-
1
)
p
(
f
(
z
)
z
p
)
μ
(
z
f
′
(
z
)
p
f
(
z
)
)
2
;
z
)
:
z
∈
U
}
⊂
Ω
,
then
(26)
(
f
(
z
)
z
p
)
μ
≺
q
(
z
)
.Proof.
We begin by defining the analytic functiong in U by
(27)
g
(
z
)
=
(
f
(
z
)
z
p
)
μ
(
z
∈
U
)
.
Then, in view of (27), we get
(28)
(
f
(
z
)
z
p
)
μ
z
f
′
(
z
)
p
f
(
z
)
=
z
g
′
(
z
)
+
μ
p
g
(
z
)
μ
p
.
Further computations show that
(29)
(
f
(
z
)
z
p
)
μ
z
2
f
′′
(
z
)
p
f
(
z
)
+
(
μ
-
1
)
p
(
f
(
z
)
z
p
)
μ
(
z
f
′
(
z
)
p
f
(
z
)
)
2
=
z
2
g
′′
(
z
)
+
2
μ
p
z
g
′
(
z
)
+
μ
p
(
μ
p
-
1
)
g
(
z
)
μ
p
.
We now define the transformations fromC
3 to C by
(30)
u
=
r
,
v
=
s
+
μ
p
r
μ
p
,
w
=
t
+
2
μ
p
s
+
μ
p
(
μ
p
-
1
)
r
μ
p
and suppose that
(31)
ψ
(
r
,
s
,
t
;
z
)
=
ϕ
(
u
,
v
,
w
;
z
)
=
ϕ
(
r
,
s
+
μ
p
r
μ
p
,
t
+
2
μ
p
s
+
μ
p
(
μ
p
-
1
)
r
μ
p
;
z
)
.
The proof will make use of Lemma 3. Indeed, by using (27) to (31), we obtain
(32)
ψ
(
g
(
z
)
,
z
g
′
(
z
)
,
z
2
g
′′
(
z
)
;
z
)
=
ϕ
(
(
f
(
z
)
z
p
)
μ
,
(
f
(
z
)
z
p
)
μ
z
f
′
(
z
)
p
f
(
z
)
,
k
k
k
k
k
(
f
(
z
)
z
p
)
μ
z
2
f
′′
(
z
)
p
f
(
z
)
k
k
k
k
k
+
(
μ
-
1
)
p
(
f
(
z
)
z
p
)
μ
(
z
f
′
(
z
)
p
f
(
z
)
)
2
;
z
)
.
Hence (25) becomes
(33)
ψ
(
g
(
z
)
,
z
g
′
(
z
)
,
z
2
g
′′
(
z
)
;
z
)
∈
Ω
.
The proof is completed if it can be shown that the admissibility condition for ϕ
∈
Φ
[
Ω
,
q
,
p
,
μ
] is equivalent to the admissibility condition for ψ as given in Definition 1. We note that
(34)
t
s
+
1
=
w
-
(
2
μ
p
-
1
)
v
+
μ
p
u
v
-
u
,
and hence ψ
∈
Ψ
1
[
Ω
,
q
]. By Lemma 3, we thus obtain
(35)
g
(
z
)
≺
q
(
z
)
or
(
f
(
z
)
z
p
)
μ
≺
q
(
z
)
.
which evidently proves Theorem 6.IfΩ
≠
C is a simply connected domain, then Ω
=
h
(
U
) for some conformal mapping h of U onto Ω. In this case, the class Φ
[
h
(
U
)
,
q
,
p
,
μ
] is written, for convenience, as Φ
[
h
,
q
,
p
,
μ
]. The following result is an immediate consequence of Theorem 6.Theorem 7.
Letϕ
∈
Φ
[
h
,
q
,
p
,
μ
]. If f
∈
A
(
p
) satisfies the condition,
(36)
ϕ
(
(
f
(
z
)
z
p
)
μ
,
(
f
(
z
)
z
p
)
μ
z
f
′
(
z
)
p
f
(
z
)
,
(
f
(
z
)
z
p
)
μ
z
2
f
′′
(
z
)
p
f
(
z
)
kkk
+
(
μ
-
1
)
p
(
f
(
z
)
z
p
)
μ
(
z
f
′
(
z
)
p
f
(
z
)
)
2
;
z
)
≺
h
(
z
)
,
then
(37)
(
f
(
z
)
z
p
)
μ
≺
q
(
z
)
.Puttingμ
=
1 in Theorem 7, we obtain the following corollary.Corollary 8.
Letϕ
∈
Φ
[
h
,
q
,
p
]. If f
∈
A
(
p
) satisfies the condition
(38)
ϕ
(
f
(
z
)
z
p
,
f
′
(
z
)
p
z
p
-
1
,
f
′′
(
z
)
p
z
p
-
2
)
≺
h
(
z
)
,
then
(39)
f
(
z
)
z
p
≺
q
(
z
)
.Our next result is an extension of Theorem6 to the case where the behavior of q on ∂
U is not known.Corollary 9.
LetΩ
⊂
C and suppose that the function q is univalent in U with q
(
0
)
=
1. Also let ϕ
∈
Φ
[
Ω
,
q
ρ
,
p
,
μ
] for some ρ
∈
(
0,1
), where q
ρ
(
z
)
=
q
(
ρ
z
). If f
∈
A
(
p
) and
(40)
ϕ
(
(
f
(
z
)
z
p
)
μ
,
(
f
(
z
)
z
p
)
μ
z
f
′
(
z
)
p
f
(
z
)
,
(
f
(
z
)
z
p
)
μ
z
2
f
′′
(
z
)
p
f
(
z
)
kkk
+
(
μ
-
1
)
p
(
f
(
z
)
z
p
)
μ
(
z
f
′
(
z
)
p
f
(
z
)
)
2
;
z
)
∈
Ω
,
then
(41)
(
f
(
z
)
z
p
)
μ
≺
q
(
z
)
.Proof.
Theorem6 readily yields
(42)
(
f
(
z
)
z
p
)
μ
≺
q
ρ
(
z
)
.
The asserted result is now deduced from the fact that q
ρ
(
z
)
≺
q
(
z
).Theorem 10.
Let the functionsh and q be univalent in U, with q
(
0
)
=
1, and set
(43)
q
ρ
(
z
)
=
q
(
ρ
z
)
,
h
ρ
(
z
)
=
h
(
ρ
z
)
.
Also letϕ
:
C
3
×
U
→
C satisfy one of the following conditions:(1)
ϕ
∈
Φ
[
h
,
q
ρ
,
p
,
μ
] for some ρ
∈
(
0,1
) or
(2)
there existsρ
0
∈
(
0,1
) such that ϕ
∈
Φ
[
h
ρ
,
q
ρ
,
p
,
μ
] for all ρ
∈
(
ρ
0
,
1
).
Iff
∈
A
(
p
) satisfies condition (36), then
(44)
(
f
(
z
)
z
p
)
μ
≺
q
(
z
)
.Proof.
The proof of Theorem10 is similar to the proof of a known result [2, p. 30, Theorem 2.3d] and is, therefore, omitted.The next theorem yields the best dominant of differential subordination (36).Theorem 11.
Let the functionh be univalent in U. Also let ϕ
:
C
3
×
U
→
C. Suppose that the differential equation
(45)
ϕ
(
q
(
z
)
,
z
q
′
(
z
)
,
z
2
q
′′
(
z
)
;
z
)
=
h
(
z
)
has a solution q with q
(
0
)
=
1 and satisfies one of the following conditions:(1)
q
∈
F
1 and ϕ
∈
Φ
[
h
,
q
,
p
,
μ
];
(2)
the functionq is univalent in U and ϕ
∈
Φ
[
h
,
q
ρ
,
p
,
μ
] for some ρ
∈
(
0,1
); or
(3)
the functionq is univalent in U and there exists ρ
0
∈
(
0,1
) such that ϕ
∈
Φ
[
h
ρ
,
q
ρ
,
p
,
μ
] for all ρ
∈
(
ρ
0
,
1
). If f
∈
A
(
p
) satisfies (36), then
(46)
(
f
(
z
)
z
p
)
μ
≺
q
(
z
) and q is the best dominant.Proof.
Following the same arguments in [2, p. 31, Theorem 2.3e], we deduce that q is a dominant from Theorems 7 and 10. Since q satisfies (45), it is also a solution of (36) and, therefore, q will be dominated by all dominants. Hence q is the best dominant.In the particular case whenq
(
z
)
=
1
+
M
z
(
M
>
0
), in view of Definition 5, the class Φ
[
Ω
,
q
,
p
,
μ
] of admissible functions, denoted by Φ
[
Ω
,
M
,
p
,
μ
], is described below.Definition 12.
LetΩ be a set in C and M
>
0. The class Φ
[
Ω
,
M
,
p
,
μ
] of admissible functions consists of those functions ϕ
:
C
3
×
U
→
C such that
(47)
ϕ
(
L
+
μ
p
[
(
2
k
+
μ
p
-
1
)
M
e
i
θ
+
μ
p
-
1
]
μ
p
;
z
1
+
M
e
i
θ
,
1
+
k
+
μ
p
μ
p
M
e
i
θ
,
L
+
μ
p
[
(
2
k
+
μ
p
-
1
)
M
e
i
θ
+
μ
p
-
1
]
μ
p
;
z
)
∉
Ω
whenever z
∈
U, θ
∈
R, and
(48)
R
(
L
e
-
i
θ
)
≧
(
k
-
1
)
k
M
for all real θ and k
≧
μ
p.Corollary 13.
Letϕ
∈
Φ
[
Ω
,
M
,
p
,
μ
]. If f
∈
A
(
p
) satisfies the condition
(49)
ϕ
(
(
f
(
z
)
z
p
)
μ
,
(
f
(
z
)
z
p
)
μ
z
f
′
(
z
)
p
f
(
z
)
,
(
f
(
z
)
z
p
)
μ
z
2
f
′′
(
z
)
p
f
(
z
)
kkkk
+
(
μ
-
1
)
p
(
f
(
z
)
z
p
)
μ
(
z
f
′
(
z
)
p
f
(
z
)
)
2
;
z
)
∈
Ω
,
then
(50)
|
(
f
(
z
)
z
p
)
μ
-
1
|
<
M
(
z
∈
U
)
.In the special case when(51)
Ω
=
q
(
U
)
=
{
ω
:
|
ω
-
1
|
<
M
}
,
the class Φ
[
Ω
,
M
,
p
,
μ
] is simply denoted by Φ
[
M
,
p
,
μ
].Corollary 14.
Letϕ
∈
Φ
[
M
]. If f
∈
A
(
p
) satisfies the condition
(52)
|
ϕ
(
(
f
(
z
)
z
p
)
μ
,
(
f
(
z
)
z
p
)
μ
z
f
′
(
z
)
p
f
(
z
)
,
(
f
(
z
)
z
p
)
μ
z
2
f
′′
(
z
)
p
f
(
z
)
kk
k
+
(
μ
-
1
)
p
(
f
(
z
)
z
p
)
μ
(
z
f
′
(
z
)
p
f
(
z
)
)
2
;
z
)
-
1
|
<
M
,
then
(53)
|
(
f
(
z
)
z
p
)
μ
-
1
|
<
M
.Corollary 15.
Ifk
≧
1 and f
∈
A
(
p
) satisfies the condition
(54)
|
(
f
(
z
)
z
p
)
μ
z
f
′
(
z
)
p
f
(
z
)
-
1
|
<
M
,
then
(55)
|
(
f
(
z
)
z
p
)
μ
-
1
|
<
M
.Proof.
Corollary15 follows from Corollary 14 upon setting
(56)
ϕ
(
u
,
v
,
w
;
z
)
=
v
=
1
+
(
k
+
μ
p
μ
p
)
M
e
i
θ
.


## 3. Superordination and Sandwich-Type Results

In this section we investigate the dual problem of differential subordination, that is, differential superordination of multivalent functions. For this purpose, the class of admissible functions is given in the following definition.Definition 16.
LetΩ be a set in C and q
∈
H with z
q
′
(
z
)
≠
0. The class Φ
′
[
Ω
,
q
,
p
,
μ
] of admissible functions consists of those functions ϕ
:
C
3
×
U
¯
→
C that satisfy the following admissibility condition:
(57)
ϕ
(
u
,
v
,
w
;
ζ
)
∈
Ω
whenever
(58)
u
=
q
(
z
)
,
v
=
z
q
′
(
z
)
+
m
μ
p
q
(
z
)
m
μ
p
,
R
(
w
-
(
2
μ
p
-
1
)
v
+
μ
p
u
v
-
u
)
≧
1
m
R
(
1
+
ζ
q
′′
(
ζ
)
q
′
(
ζ
)
)
,
where z
∈
U, ζ
∈
∂
U, and m
≧
1. For convenience, we write
(59)
Φ
′
[
Ω
,
q
,
p
,
1
]
=
Φ
′
[
Ω
,
q
,
p
]
.Theorem 17.
Letϕ
∈
Φ
′
[
Ω
,
q
,
p
,
μ
]. If f
∈
A
(
p
),
(60)
(
f
(
z
)
z
p
)
μ
∈
F
1
ϕ
(
(
f
(
z
)
z
p
)
μ
,
(
f
(
z
)
z
p
)
μ
z
f
′
(
z
)
p
f
(
z
)
,
(
f
(
z
)
z
p
)
μ
z
2
f
′′
(
z
)
p
f
(
z
)
kkk
+
(
μ
-
1
)
p
(
f
(
z
)
z
p
)
μ
(
z
f
′
(
z
)
p
f
(
z
)
)
2
;
z
)
is univalent in U, then
(61)
Ω
⊂
{
+
(
μ
-
1
)
p
(
f
(
z
)
z
p
)
μ
(
z
f
′
(
z
)
p
f
(
z
)
)
2
;
z
)
ϕ
(
(
f
(
z
)
z
p
)
μ
,
(
f
(
z
)
z
p
)
μ
z
f
′
(
z
)
p
f
(
z
)
,
(
f
(
z
)
z
p
)
μ
z
2
f
′′
(
z
)
p
f
(
z
)
kkkk
k
+
(
μ
-
1
)
p
(
f
(
z
)
z
p
)
μ
(
z
f
′
(
z
)
p
f
(
z
)
)
2
;
z
)
:
z
∈
U
}
implies that
(62)
q
(
z
)
≺
(
f
(
z
)
z
p
)
μ
.Proof.
From (32) and (61), we find that
(63)
Ω
⊂
{
ψ
(
g
(
z
)
,
z
g
′
(
z
)
,
z
2
g
′′
(
z
)
;
z
)
:
z
∈
U
}
.
We also see from (30) that the admissibility condition for the function class ϕ
∈
Φ
′
[
Ω
,
q
,
p
,
μ
] is equivalent to the admissibility condition for ψ as given in Definition 2. Hence ψ
∈
Ψ
1
′
[
Ω
,
q
]. Thus, by Lemma 4, we have
(64)
q
(
z
)
≺
g
(
z
)
or
q
(
z
)
≺
(
f
(
z
)
z
p
)
μ
,
which evidently completes the proof of Theorem 17.IfΩ
≠
C is a simply connected domain, then Ω
=
h
(
U
) for some conformal mapping h of U onto Ω. In this case, the class Φ
′
[
h
(
U
)
,
q
,
p
,
μ
] is written simply as Φ
′
[
h
,
q
,
p
,
μ
].Proceeding similarly as in Section2, the following result can be derived as an immediate consequence of Theorem 17.Theorem 18.
Let the functionh be analytic in U and ϕ
∈
Φ
′
[
h
,
q
,
p
,
μ
]. If f
∈
A
(
p
),
(65)
(
f
(
z
)
z
p
)
μ
∈
F
1
,
ϕ
(
(
f
(
z
)
z
p
)
μ
,
(
f
(
z
)
z
p
)
μ
z
f
′
(
z
)
p
f
(
z
)
,
(
f
(
z
)
z
p
)
μ
z
2
f
′′
(
z
)
p
f
(
z
)
kkk
+
(
μ
-
1
)
p
(
f
(
z
)
z
p
)
μ
(
z
f
′
(
z
)
p
f
(
z
)
)
2
;
z
)
is univalent in U, then
(66)
h
(
z
)
≺
ϕ
(
(
f
(
z
)
z
p
)
μ
,
(
f
(
z
)
z
p
)
μ
z
f
′
(
z
)
p
f
(
z
)
,
(
f
(
z
)
z
p
)
μ
z
2
f
′′
(
z
)
p
f
(
z
)
kkkkk
+
(
μ
-
1
)
p
(
f
(
z
)
z
p
)
μ
(
z
f
′
(
z
)
p
f
(
z
)
)
2
;
z
)
implies that
(67)
q
(
z
)
≺
(
f
(
z
)
z
p
)
μ
.Puttingμ
=
1 in Theorem 18, we obtain the following corollary.Corollary 19.
Let the functionh be analytic in U and ϕ
∈
Φ
′
[
h
,
q
,
p
]. If f
∈
A
(
p
),
(68)
f
(
z
)
z
p
∈
F
1
,
ϕ
(
f
(
z
)
z
p
,
f
′
(
z
)
p
z
p
-
1
,
f
′′
(
z
)
p
z
p
-
2
;
z
)
is univalent in U, then
(69)
h
(
z
)
≺
ϕ
(
f
(
z
)
z
p
,
f
′
(
z
)
p
z
p
-
1
,
f
′′
(
z
)
p
z
p
-
2
;
z
)
implies that
(70)
q
(
z
)
≺
f
(
z
)
z
p
.Theorems17 and 18 can only be used to obtain subordinants  of the differential superordination of the form (61) or (66). The following theorem proves the existence of the best subordinant of (66) for a specified ϕ.Theorem 20.
Let the functionh be analytic in U and ϕ
:
C
3
×
U
¯
→
C. Suppose that the differential equation
(71)
ϕ
(
q
(
z
)
,
z
q
′
(
z
)
,
z
2
q
′′
(
z
)
;
z
)
=
h
(
z
)
has a solution q
∈
F
1. If ϕ
∈
Φ
′
[
h
,
q
,
p
,
μ
],f
∈
A
(
p
),
(72)
(
f
(
z
)
z
p
)
μ
∈
F
1
ϕ
(
(
f
(
z
)
z
p
)
μ
,
(
f
(
z
)
z
p
)
μ
z
f
′
(
z
)
p
f
(
z
)
,
(
f
(
z
)
z
p
)
μ
z
2
f
′′
(
z
)
p
f
(
z
)
kkk
+
(
μ
-
1
)
p
(
f
(
z
)
z
p
)
μ
(
z
f
′
(
z
)
p
f
(
z
)
)
2
;
z
)
is univalent in U, then
(73)
h
(
z
)
≺
ϕ
(
(
f
(
z
)
z
p
)
μ
,
(
f
(
z
)
z
p
)
μ
z
f
′
(
z
)
p
f
(
z
)
,
(
f
(
z
)
z
p
)
μ
z
2
f
′′
(
z
)
p
f
(
z
)
kkk
kk
+
(
μ
-
1
)
p
(
f
(
z
)
z
p
)
μ
(
z
f
′
(
z
)
p
f
(
z
)
)
2
;
z
)
implies that
(74)
q
(
z
)
≺
(
f
(
z
)
z
p
)
μ
and q is the best subordinant.Proof.
The proof is similar to the proof of Theorem11. We, therefore, omit the details involved.Combining Theorems7 and 18, we obtain the following sandwich-type theorem.Theorem 21.
Let the functionsh
1 and q
1 be analytic in U, the function h
2 univalent in U
,
q
2
∈
F
1 with
(75)
q
1
(
0
)
=
q
2
(
0
)
=
1
,
ϕ
∈
Φ
[
h
2
,
q
2
,
p
,
μ
]
∩
Φ
′
[
h
1
,
q
1
,
p
,
μ
]
.
If f
∈
A
(
p
),
(76)
(
f
(
z
)
z
p
)
μ
∈
H
∩
F
1
,
ϕ
(
(
f
(
z
)
z
p
)
μ
,
(
f
(
z
)
z
p
)
μ
z
f
′
(
z
)
p
f
(
z
)
,
(
f
(
z
)
z
p
)
μ
z
2
f
′′
(
z
)
p
f
(
z
)
k
k
k
+
(
μ
-
1
)
p
(
f
(
z
)
z
p
)
μ
(
z
f
′
(
z
)
p
f
(
z
)
)
2
;
z
)
is univalent in U, then
(77)
h
1
(
z
)
≺
ϕ
(
(
f
(
z
)
z
p
)
μ
,
(
f
(
z
)
z
p
)
μ
z
f
′
(
z
)
p
f
(
z
)
,
(
f
(
z
)
z
p
)
μ
z
2
f
′′
(
z
)
p
f
(
z
)
kkk
k
k
+
(
μ
-
1
)
p
(
f
(
z
)
z
p
)
μ
(
z
f
′
(
z
)
p
f
(
z
)
)
2
;
z
)
≺
h
2
(
z
)
implies that
(78)
q
1
(
z
)
≺
(
f
(
z
)
z
p
)
μ
≺
q
2
(
z
)
.Upon settingμ
=
1 in Theorem 21, we get the following result.Corollary 22.
Let the functionsh
1 and q
1 be analytic in U, the function h
2 univalent in U
,
q
2
∈
F
1 with
(79)
q
1
(
0
)
=
q
2
(
0
)
=
1
,
and ϕ
∈
Φ
[
h
2
,
q
2
,
p
]
∩
Φ
′
[
h
1
,
q
1
,
p
]. If f
∈
A
(
p
),
(80)
f
(
z
)
z
p
∈
H
∩
F
1
,
ϕ
(
f
(
z
)
z
p
,
f
′
(
z
)
p
z
p
-
1
,
f
′′
(
z
)
p
z
p
-
2
)
is univalent in U, then
(81)
h
1
(
z
)
≺
ϕ
(
f
(
z
)
z
p
,
f
′
(
z
)
p
z
p
-
1
,
f
′′
(
z
)
p
z
p
-
2
)
≺
h
2
(
z
)
implies that
(82)
q
1
(
z
)
≺
f
(
z
)
z
p
≺
q
2
(
z
)
.



---

*Source: 936748-2014-09-16.xml*