# On a Subclass of Meromorphic Close-to-Convex Functions

Authors: Ming-Liang Li; Lei Shi; Zhi-Gang Wang
Journal: The Scientific World Journal
 (2014)

---

## Abstract

The main purpose of this paper is to introduce and investigate a certain subclass of meromorphic close-to-convex functions. Such results as coefficient inequalities, convolution property, inclusion relationship, distortion property, and radius of meromorphic convexity are derived.


---

## Body

## 1. Introduction and Preliminaries

LetΣ denote the class of functions f of the form
(1)
f
(
z
)
=
1
z
+
∑
n
=
1
∞
‍
a
n
z
n
,
which areanalytic in thepunctured open unit disk
(2)
U
*
∶
=
{
z
:
z
∈
C
0
<
|
z
|
<
1
}
=
:
U
∖
{
0
}
.For two functionsf and g, analytic in U, we say that the function f is subordinate to g in U and write
(3)
f
(
z
)
≺
g
(
z
)
(
z
∈
U
)
,
if there exists a Schwarz function ω, which is analytic in U with
(4)
ω
(
0
)
=
0
,
|
ω
(
z
)
|
<
1
(
z
∈
U
)
,
such that
(5)
f
(
z
)
=
g
(
ω
(
z
)
)
(
z
∈
U
)
.
Indeed, it is known that
(6)
f
(
z
)
≺
g
(
z
)
(
z
∈
U
)
⟹
f
(
0
)
=
g
(
0
)
,
11111111111111111111
f
(
U
)
⊂
g
(
U
)
.
Furthermore, if the function g is univalent in U, then we have the following equivalence:
(7)
f
(
z
)
≺
g
(
z
)
(
z
∈
U
)
⟺
f
(
0
)
=
g
(
0
)
,
11111111111111111111
f
(
U
)
⊂
g
(
U
)
.Letf
,
g
∈
Σ, where f is given by (1) and g is defined by
(8)
g
(
z
)
=
1
z
+
∑
n
=
1
∞
‍
b
n
z
n
.
Then the Hadamard product (or convolution) f
*
g of the functions f and g is defined by
(9)
(
f
*
g
)
(
z
)
∶
=
1
z
+
∑
n
=
1
∞
‍
a
n
b
n
z
n
=
:
(
g
*
f
)
(
z
)
.A functionf
∈
Σ is said to be in the class M
S
* ofmeromorphic starlike functions if it satisfies the inequality
(10)
R
(
z
f
′
(
z
)
f
(
z
)
)
<
0
(
z
∈
U
)
.
A function f
∈
Σ is said to be in the class M
K ofmeromorphic convex functions if it satisfies the inequality
(11)
R
(
1
+
z
f
′′
(
z
)
f
′
(
z
)
)
<
0
(
z
∈
U
)
.
Moreover, a function f
∈
Σ is said to be in the class M
C ofmeromorphic close-to-convex functions if it satisfies the condition
(12)
R
(
z
f
′
(
z
)
g
(
z
)
)
<
0
(
z
∈
U
;
g
∈
M
S
*
)
.Let(13)
f
(
z
)
=
z
+
a
2
z
2
+
a
3
z
3
+
⋯
be analytic in U. If there exists a function g
∈
K such that
(14)
|
(
z
f
′
(
z
)
/
g
(
z
)
)
-
1
(
z
f
′
(
z
)
/
g
(
z
)
)
+
(
1
-
2
α
)
|
<
β
(
z
∈
U
;
0
≦
α
<
1
;
0
<
β
≦
1
)
,
then we say that f
∈
C
(
α
,
β
), where K denotes the usual class of convex functions. The function class C
(
α
,
β
) was introduced and studied recently by Peng [1] (see also Peng and Han [2], Selvaraj [3], Gao and Zhou [4], Kowalczyk and Leś-Bomba [5], and Xu et al. [6]).Motivated essentially by the above mentioned function classC
(
α
,
β
), we now introduce and investigate the following class of meromorphic close-to-convex functions.Definition 1.
A functionf
∈
Σ is said to be in the class M
C
(
α
,
β
) if it satisfies the inequality
(15)
|
(
z
f
′
(
z
)
/
g
(
z
)
)
+
1
(
z
f
′
(
z
)
/
g
(
z
)
)
+
(
2
α
-
1
)
|
<
β
(
z
∈
U
;
g
∈
M
S
*
)
,
where (and throughout this paper unless otherwise mentioned) the parameters α and β are constrained as follows:
(16)
0
≦
α
<
1
,
0
<
β
≦
1
.
It is easy to verify thatf
∈
M
C
(
α
,
β
) if and only if
(17)
-
z
f
′
(
z
)
g
(
z
)
≺
1
+
(
1
-
2
α
)
β
z
1
-
β
z
(
z
∈
U
)
.
We observe that
(18)
R
(
1
+
(
1
-
2
α
)
β
z
1
-
β
z
)
>
0
(
z
∈
U
)
,
and, thus, the function class M
C
(
α
,
β
) is a subclass of meromorphic close-to-convex functions.
Clearly, the classM
C
(
α
,
1
)
=
:
M
C
(
α
) is the familiar class of meromorphic close-to-convex functions of order α.
For some recent investigations of meromorphic functions, see, for example, the works of [7–22] and the references cited therein.
To derive our main results, we need the following lemmas.Lemma 2 (see [23]).
Let(19)
f
1
(
z
)
=
1
+
∑
n
=
1
∞
c
n
z
n
be analytic in U and let
(20)
f
2
(
z
)
=
1
+
∑
n
=
1
∞
d
n
z
n
be analytic and convex in U. If f
1
≺
f
2, then
(21)
|
c
n
|
≦
|
d
1
|
(
n
∈
N
≔
{
1,2
,
…
}
)
.Lemma 3 (see [24]).
Suppose that(22)
h
(
z
)
=
1
z
+
∑
n
=
1
∞
c
n
z
n
∈
M
S
*
.
Then
(23)
|
c
n
|
≦
2
n
+
1
(
n
∈
N
)
.
Each of these inequalities is sharp, with the extremal function given by
(24)
h
(
z
)
=
z
-
1
(
1
+
z
n
+
1
)
2
/
(
n
+
1
)
.Lemma 4 (see [25]).
Let-
1
≦
D
<
C
≦
1 and -
1
≦
F
<
E
≦
1. Then,
(25)
1
+
C
z
1
+
D
z
≺
1
+
E
z
1
+
F
z
if and only if
(26)
|
E
D
-
F
C
|
≦
(
E
-
F
)
-
(
C
-
D
)
.Lemma 5 (see [26]).
Suppose that the functiong
∈
M
S
*. Then
(27)
(
1
-
r
)
2
r
≦
|
g
(
z
)
|
≦
(
1
+
r
)
2
r
(
|
z
|
=
r
;
0
<
r
<
1
)
.Lemma 6 (see [27]).
Suppose that(28)
p
(
z
)
≺
1
+
(
1
-
2
α
)
z
1
-
z
(
z
∈
U
;
0
≦
α
<
1
)
.
Then,(29)
R
(
z
p
′
(
z
)
p
(
z
)
)
≧
{
-
2
(
1
-
α
)
r
(
1
+
r
)
[
1
+
(
2
α
-
1
)
r
]
(
|
z
|
=
r
;
0
<
r
≦
r
0
)
,
(
4
α
(
1
-
r
2
)
[
1
+
(
1
-
2
α
)
r
2
]
-
[
1
+
(
1
-
2
α
)
r
2
]
4
α
(
1
-
r
2
)
[
1
+
(
1
-
2
α
)
r
2
]
)
×
(
(
1
-
α
)
(
1
-
r
2
)
)
-
1
-
α
1
-
α
(
|
z
|
=
r
;
r
0
<
r
<
1
)
,
where r
0 is the unique root of the equation
(30)
(
2
α
-
1
)
r
4
-
2
(
2
α
-
1
)
r
3
-
6
α
r
2
-
2
r
+
1
=
0
in the interval (
0,1
). The results are sharp.In the present paper, we aim at proving some coefficient inequalities, convolution property, inclusion relationship, distortion property, and radius of meromorphic convexity of the classM
C
(
α
,
β
).


## 2. Main Results

We begin by stating the following coefficient inequality of the classM
C
(
α
,
β
).Theorem 7.
Suppose that(31)
f
(
z
)
=
1
z
+
∑
n
=
1
∞
a
n
z
n
∈
M
C
(
α
,
β
)
.
Then
(32)
|
a
1
|
≦
1
,
(33)
|
a
n
|
≦
2
(
1
-
α
)
β
n
(
1
+
∑
k
=
1
n
-
1
2
k
+
1
)
+
2
n
(
n
+
1
)
111111111111111111111111111
(
n
∈
N
∖
{
1
}
)
.Proof.
Letf
∈
M
C
(
α
,
β
) and suppose that
(34)
p
(
z
)
∶
=
-
z
f
′
(
z
)
g
(
z
)
(
z
∈
U
)
,
where
(35)
g
(
z
)
=
1
z
+
∑
n
=
1
∞
b
n
z
n
∈
M
S
*
.
It follows that
(36)
p
(
z
)
=
1
+
∑
n
=
1
∞
p
n
z
n
≺
1
+
(
1
-
2
α
)
β
z
1
-
β
z
(
z
∈
U
)
.
In view of Lemma 2, we know that
(37)
|
p
n
|
≦
2
(
1
-
α
)
β
(
n
∈
N
)
.
By substituting the series expressions of functions f, g, and p into (34), we get
(38)
(
1
+
p
1
z
+
p
2
z
2
+
⋯
+
p
n
z
n
+
p
n
+
1
z
n
+
1
+
⋯
)
×
(
1
z
+
b
1
z
+
b
2
z
2
+
⋯
+
b
n
z
n
+
⋯
)
=
1
z
-
a
1
z
-
2
a
2
z
2
-
⋯
-
n
a
n
z
n
-
⋯
.
Since f is univalent in U
*, it is well known that |
a
1
|
≦
1.
On the other hand, we find from (38) that
(39)
-
n
a
n
=
b
n
+
p
1
b
n
-
1
+
p
2
b
n
-
2
+
⋯
+
p
n
-
1
b
1
+
p
n
+
1
111111111111111111111111111111
(
n
∈
N
∖
{
1
}
)
.
By noting that g
∈
M
S
*, it follows from Lemma 3 that
(40)
|
b
n
|
≦
2
n
+
1
(
n
∈
N
)
.
Combining (37), (39), and (40), we have
(41)
n
|
a
n
|
≦
2
(
1
-
α
)
β
(
2
n
+
2
n
-
1
+
⋯
+
2
3
+
1
+
1
)
+
2
n
+
1
11111111111111111111111111111111111
(
n
∈
N
∖
{
1
}
)
.
Thus, the assertion (33) of Theorem 7 follows directly from (41).Theorem 8.
Let(42)
g
(
z
)
=
1
z
+
∑
n
=
1
∞
b
n
z
n
∈
M
S
*
.
Iff
∈
Σ satisfies the condition
(43)
(
1
+
β
)
∑
n
=
1
∞
n
|
a
n
|
+
(
1
+
|
2
α
-
1
|
β
)
∑
n
=
1
∞
|
b
n
|
≦
2
(
1
-
α
)
β
,
then f
∈
M
C
(
α
,
β
).Proof.
To provef
∈
M
C
(
α
,
β
), it suffices to show that (15) holds. From (43), we know that
(44)
β
(
2
(
1
-
α
)
-
∑
n
=
1
∞
n
|
a
n
|
-
|
2
α
-
1
|
∑
n
=
1
∞
|
b
n
|
)
≧
∑
n
=
1
∞
n
|
a
n
|
+
∑
n
=
1
∞
|
b
n
|
>
0
.
Now, by the maximum modulus principle, we deduce from (1) and (44) that
(45)
|
(
z
f
′
(
z
)
/
g
(
z
)
)
+
1
(
z
f
′
(
z
)
/
g
(
z
)
)
+
(
2
α
-
1
)
|
=
|
∑
n
=
1
∞
n
a
n
z
n
+
1
+
∑
n
=
1
∞
b
n
z
n
+
1
∑
n
=
1
∞
n
a
n
z
n
+
1
+
(
2
α
-
1
)
∑
n
=
1
∞
b
n
z
n
+
1
+
2
(
α
-
1
)
|
<
∑
n
=
1
∞
n
|
a
n
|
+
∑
n
=
1
∞
|
b
n
|
2
(
1
-
α
)
-
∑
n
=
1
∞
n
|
a
n
|
-
|
2
α
-
1
|
∑
n
=
1
∞
|
b
n
|
≦
β
.
This evidently completes the proof of Theorem 8.Example 9.
By applying Theorem8, it is obvious to see that the function
(46)
f
(
z
)
=
1
z
+
z
∈
M
C
(
0,1
)
.Theorem 10.
Let|
ξ
|
=
1 and g
∈
M
S
*. A function f
∈
M
C
(
α
,
β
) if and only if
(47)
f
(
z
)
*
[
(
1
-
β
ξ
)
1
-
2
z
z
(
1
-
z
)
2
]
-
g
(
z
)
*
{
[
1
+
(
1
-
2
α
)
β
ξ
]
z
2
-
z
+
1
z
(
1
-
z
)
}
≠
0
11111111111111111111
1111
(
z
∈
U
*
)
.Proof.
A functionf
∈
M
C
(
α
,
β
) if and only if
(48)
-
z
f
′
(
z
)
g
(
z
)
≠
1
+
(
1
-
2
α
)
β
ξ
1
-
β
ξ
(
z
∈
U
;
|
ξ
|
=
1
)
.
It is easy to see that condition (48) can be written as
(49)
z
f
′
(
z
)
(
1
-
β
ξ
)
+
g
(
z
)
[
1
+
(
1
-
2
α
)
β
ξ
]
≠
0
11111111111111111111111
(
z
∈
U
*
;
|
ξ
|
=
1
)
.
We observe that
(50)
-
z
f
′
(
z
)
=
f
(
z
)
*
(
1
z
-
z
(
1
-
z
)
2
)
=
f
(
z
)
*
1
-
2
z
z
(
1
-
z
)
2
,
g
(
z
)
=
g
(
z
)
*
(
1
z
+
z
1
-
z
)
=
g
(
z
)
*
z
2
-
z
+
1
z
(
1
-
z
)
.
By substituting (50) into (49), we get the desired assertion (47) of Theorem 10.Theorem 11.
Let(51)
0
≦
α
2
≦
α
1
≦
1
2
,
0
<
β
1
≦
β
2
<
1
.
Then,
(52)
M
C
(
α
1
,
β
1
)
⊂
M
C
(
α
2
,
β
2
)
.Proof.
Suppose thatf
∈
M
C
(
α
1
,
β
1
). We easily know that
(53)
-
z
f
′
(
z
)
g
(
z
)
≺
1
+
(
1
-
2
α
1
)
β
1
z
1
-
β
1
z
(
z
∈
U
)
.
By setting C
=
1
+
(
1
-
2
α
1
)
β
1, D
=
-
β
1, E
=
1
+
(
1
-
2
α
2
)
β
2, and F
=
-
β
2, it follows from (51) that
(54)
|
E
D
-
F
C
|
=
|
-
(
1
-
2
α
2
)
β
1
β
2
+
(
1
-
2
α
1
)
β
1
β
2
|
≦
|
-
(
1
-
2
α
2
)
β
1
β
2
+
(
1
-
2
α
1
)
β
1
2
|
111
+
|
-
(
1
-
2
α
1
)
β
1
2
+
(
1
-
2
α
1
)
β
1
β
2
|
≦
[
(
1
-
2
α
2
)
β
2
-
(
1
-
2
α
1
)
β
1
]
+
(
β
2
-
β
1
)
=
[
(
1
-
2
α
2
)
β
2
-
(
-
β
2
)
]
111
-
[
(
1
-
2
α
1
)
β
1
-
(
-
β
1
)
]
=
(
E
-
F
)
-
(
C
-
D
)
.
In view of Lemma 4, we deduce that
(55)
-
z
f
′
(
z
)
g
(
z
)
≺
1
+
(
1
-
2
α
1
)
β
1
z
1
-
β
1
z
≺
1
+
(
1
-
2
α
2
)
β
2
z
1
-
β
2
z
11111111111111111111111111111111111
(
z
∈
U
)
,
which implies that f
∈
M
C
(
α
2
,
β
2
). Thus, the assertion (52) of Theorem 11 holds.Theorem 12.
Letf
∈
M
C
(
α
,
β
). Then,
(56)
(
1
-
r
)
2
[
1
-
(
1
-
2
α
)
β
r
]
r
2
(
1
+
β
r
)
≦
|
f
′
(
z
)
|
≦
(
1
+
r
)
2
[
1
+
(
1
-
2
α
)
β
r
]
r
2
(
1
-
β
r
)
1111111111111
(
|
z
|
=
r
;
0
<
r
<
1
)
.Proof.
Letf
∈
M
C
(
α
,
β
). By definition, we know that
(57)
-
z
f
′
(
z
)
g
(
z
)
≺
1
+
(
1
-
2
α
)
β
z
1
-
β
z
(
z
∈
U
)
.
Suppose that the function p is defined by (36). Then, we have
(58)
1
-
(
1
-
2
α
)
β
r
1
+
β
r
≦
|
p
(
z
)
|
≦
1
+
(
1
-
2
α
)
β
r
1
-
β
r
,
11111111111111111
(
|
z
|
=
r
;
0
<
r
<
1
)
.
Since g
∈
M
S
*, by Lemma 5, we know that
(59)
(
1
-
r
)
2
r
≦
|
g
(
z
)
|
≦
(
1
+
r
)
2
r
(
|
z
|
=
r
;
0
<
r
<
1
)
.
Thus, by virtue of (36), (58), and (59), we readily get the assertion (56) of Theorem 12.Finally, we derive the radius of meromorphic convexity for the classM
C
(
α
).Theorem 13.
Letf
∈
M
C
(
α
) with 0
<
α
<
1. Then,(1)
forr
1
≦
r
0, f is meromorphic convex in 0
<
|
z
|
<
r
1;
(2)
forr
1
>
r
0, f is meromorphic convex in 0
<
|
z
|
<
r
2,where r
0 is the unique root of the equation
(60)
(
2
α
-
1
)
r
4
-
2
(
2
α
-
1
)
r
3
-
6
α
r
2
-
2
r
+
1
=
0
in the interval (
0,1
) and r
1 and r
2 are the smallest root of the equations
(61)
(
2
α
-
1
)
r
3
+
3
(
2
α
-
1
)
r
2
+
3
r
+
1
=
0
,
(
1
-
2
α
)
r
4
+
2
(
1
-
2
α
)
r
3
+
3
(
1
-
α
)
r
2
+
2
r
+
1
=
0
in the interval (
0,1
), respectively.Proof.
Letf
∈
M
C
(
α
) and suppose that
(62)
q
(
z
)
∶
=
-
z
f
′
(
z
)
g
(
z
)
(
z
∈
U
)
.
Then,
(63)
q
(
z
)
≺
1
+
(
1
-
2
α
)
z
1
-
z
(
z
∈
U
)
.
It follows from (62) that
(64)
-
z
f
′
(
z
)
=
q
(
z
)
g
(
z
)
.
Differentiating both sides of (64) logarithmically, we get
(65)
-
(
1
+
z
f
′′
(
z
)
f
′
(
z
)
)
=
z
q
′
(
z
)
q
(
z
)
+
z
g
′
(
z
)
g
(
z
)
.
Since g
∈
M
S
*, we know that
(66)
R
(
z
g
′
(
z
)
g
(
z
)
)
≧
-
1
+
r
1
-
r
(
|
z
|
=
r
)
.
Combining (63), (65), (66), and Lemma 6, we obtain
(67)
-
R
(
1
+
z
f
′′
(
z
)
f
′
(
z
)
)
≧
{
-
1
+
r
1
-
r
-
2
(
1
-
α
)
r
(
1
+
r
)
[
1
+
(
2
α
-
1
)
r
]
=
:
H
~
α
(
r
)
11111111111111111
(
|
z
|
=
r
;
0
<
r
≦
r
0
)
,
-
1
+
r
1
-
r
+
(
4
α
(
1
-
r
2
)
[
1
+
(
1
-
2
α
)
r
2
]
-
[
1
+
(
1
-
2
α
)
r
2
]
4
α
(
1
-
r
2
)
[
1
+
(
1
-
2
α
)
r
2
]
)
×
(
(
1
-
α
)
(
1
-
r
2
)
)
-
1
-
α
1
-
α
=
:
F
~
α
(
r
)
111111111
1
(
|
z
|
=
r
;
r
0
<
r
<
1
)
,
where r
0 is the unique root of (30) in the interval (
0,1
). It follows from (67) that the bound of meromorphic convexity for the class M
C
(
α
) is determined either by the equation
(68)
-
1
+
r
1
-
r
-
2
(
1
-
α
)
r
(
1
+
r
)
[
1
+
(
2
α
-
1
)
r
]
=
0
,
or by the equation
(69)
-
1
+
r
1
-
r
+
4
α
(
1
-
r
2
)
[
1
+
(
1
-
2
α
)
r
2
]
-
[
1
+
(
1
-
2
α
)
r
2
]
(
1
-
α
)
(
1
-
r
2
)
-
α
1
-
α
=
0
.
We note that (68) and (69) can be rewritten as follows:(70)
H
α
(
r
)
∶
=
(
2
α
-
1
)
r
3
+
3
(
2
α
-
1
)
r
2
+
3
r
+
1
=
0
,
F
α
(
r
)
∶
=
(
1
-
2
α
)
r
4
+
2
(
1
-
2
α
)
r
3
+
3
(
1
-
α
)
r
2
+
2
r
+
1
=
0
.
Letr
1 and r
2 be the smallest root of the equations H
α
(
r
)
=
0 and F
α
(
r
)
=
0 in the interval (
0,1
), respectively. By observing that H
α
(
0
)
=
1
>
0, we deduce that H
α
(
r
)
>
0 for r
<
r
1.
Similarly, we know thatF
α
(
r
)
>
0 for r
<
r
2, since F
α
(
0
)
=
1
>
0.
We observe that(71)
H
~
α
(
r
)
≧
0
⟺
H
α
(
r
)
≧
0
,
F
~
α
(
r
)
≧
0
⟺
F
α
(
r
)
≧
0
.
Thus, when r
1
≦
r
0, f is meromorphic convex in 0
<
|
z
|
<
r
1; when r
1
>
r
0, f is meromorphic convex in 0
<
|
z
|
<
r
2.
The proof of Theorem13 is thus completed.



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*Source: 806168-2014-08-28.xml*