# Coefficient Estimates for New Subclasses of Meromorphic Bi-Univalent Functions

Authors: Serap Bulut
Journal: International Scholarly Research Notices
 (2014)

---

## Abstract

We introduce and investigate two new subclassesM
σ
α
,
λ and M
σ
β
,
λ of meromorphic bi-univalent functions defined on Δ
=
{
z
:
z
∈
C
,
1
<
|
z
|
<
∞
}. For functions belonging to these classes, estimates on the initial coefficients are obtained.


---

## Body

## 1. Introduction

LetA denote the class of all functions of the form
(1)
f
(
z
)
=
z
+
∑
k
=
2
∞
‍
a
k
z
k
,
which are analytic in the open unit disk
(2)
U
=
{
z
:
z
∈
C
,
|
z
|
<
1
}
.
We also denote by S the class of all functions in the normalized analytic function class A which are univalent in U.Since univalent functions are one-to-one, they are invertible and the inverse functions need not be defined on the entire unit diskU. In fact, the Koebe one-quarter theorem [1] ensures that the image of U under every univalent function f
∈
S contains a disk of radius 1
/
4. Thus, every function f
∈
A has an inverse f
-
1, which is defined by
(3)
f
-
1
(
f
(
z
)
)
=
z
(
z
∈
U
)
,
f
(
f
-
1
(
w
)
)
=
w
(
|
w
|
<
r
0
(
f
)
;
r
0
(
f
)
≥
1
4
)
.
In fact, the inverse function f
-
1 is given by
(4)
f
-
1
(
w
)
=
w
-
a
2
w
2
+
(
2
a
2
2
-
a
3
)
w
3
-
(
5
a
2
3
-
5
a
2
a
3
+
a
4
)
w
4
+
⋯
.A functionf
∈
A is said to be bi-univalent in U if both f and f
-
1 are univalent in U. Let Σ denote the class of bi-univalent functions in U given by (1). For a brief history and interesting examples of functions in the class Σ, see [2] (see also [3, 4]). In fact, the aforecited work of Srivastava et al. [2] essentially revived the investigation of various subclasses of the bi-univalent function class Σ in recent years; it was followed by such works as those by Murugusundaramoorthy et al. [5], Frasin and Aouf [6], Çağlar et al. [7], and others (see, e.g., [8–15]).In this paper, the concept of bi-univalency is extended to the class of meromorphic functions defined on(5)
Δ
=
{
z
:
z
∈
C
,
1
<
|
z
|
<
∞
}
.
For this purpose, let σ denote the class of all meromorphic univalent functions g of the form
(6)
g
(
z
)
=
z
+
∑
n
=
0
∞
‍
b
n
z
n
,
defined on the domain Δ. Since g
∈
σ is univalent, it has an inverse g
-
1 that satisfies
(7)
g
-
1
(
g
(
z
)
)
=
z
(
z
∈
Δ
)
,
g
(
g
-
1
(
w
)
)
=
w
(
M
<
|
w
|
<
∞
;
M
>
0
)
.
Furthermore, the inverse function g
-
1 has a series expansion of the form
(8)
g
-
1
(
w
)
=
w
+
∑
n
=
0
∞
‍
B
n
w
n
,
where M
<
|
w
|
<
∞. Analogous to the bi-univalent analytic functions, a function g
∈
σ is said to be meromorphically bi-univalent if both g and g
-
1 are meromorphically univalent in Δ. We denote by σ
M the class of all meromorphic bi-univalent functions in Δ given by (6). A simple calculation shows that
(9)
g
-
1
(
w
)
=
w
-
b
0
-
b
1
w
-
b
2
+
b
0
b
1
w
2
-
⋯
.The coefficient problem was investigated for various interesting subclasses of the meromorphic univalent functions (see, e.g., [16–18]).In the present investigation, certain subclasses of meromorphic bi-univalent functions are introduced and estimates for the coefficientsb
0 and b
1 of functions in the newly introduced subclasses are obtained.Definition 1.
A functiong given by (6) is said to be in the class M
σ
(
α
,
λ
) if the following conditions are satisfied:
(10)
g
∈
σ
M
,
|
arg
(
z
g
′
(
z
)
(
1
-
λ
)
g
(
z
)
+
λ
z
g
′
(
z
)
)
|
<
α
π
2
m
m
m
m
m
m
m
m
m
m
l
(
0
<
α
≤
1
,
0
≤
λ
<
1
,
z
∈
Δ
)
,
|
arg
(
w
h
′
(
w
)
(
1
-
λ
)
h
(
w
)
+
λ
w
h
′
(
w
)
)
|
<
α
π
2
m
m
m
m
m
(
0
<
α
≤
1
,
0
≤
λ
<
1
,
w
∈
Δ
)
,
where the function h
=
g
-
1 is given by (9).Definition 2.
A functiong given by (6) is said to be in the class M
σ
(
β
,
λ
) if the following conditions are satisfied:
(11)
g
∈
σ
M
,
R
{
z
g
′
(
z
)
(
1
-
λ
)
g
(
z
)
+
λ
z
g
′
(
z
)
}
>
β
m
m
m
m
m
m
m
m
l
(
0
≤
β
<
1
,
0
≤
λ
<
1
,
z
∈
Δ
)
,
R
{
w
h
′
(
w
)
(
1
-
λ
)
h
(
w
)
+
λ
w
h
′
(
w
)
}
>
β
m
m
l
(
0
≤
β
<
1
,
0
≤
λ
<
1
,
w
∈
Δ
)
,
where the function h
=
g
-
1 is given by (9).Remark 3.
We note that, forλ
=
0, the classes M
σ
(
α
,
λ
) and M
σ
(
β
,
λ
) reduce to the classes
(12)
M
σ
(
α
,
0
)
=
Σ
~
B
*
(
α
)
,
M
σ
(
β
,
0
)
=
Σ
B
*
(
β
)
,
respectively, introduced and studied by Halim et al. [17].The object of the present paper is to extend the concept of bi-univalent to the class of meromorphic functions defined onΔ and find estimates on the coefficients |
b
0
| and |
b
1
| for functions in the above-defined classes M
σ
(
α
,
λ
) and M
σ
(
β
,
λ
) of the function class σ
M.Firstly, in order to derive our main results, we need the following lemma.Lemma 4 (see [19]).
Ifp
∈
P, then |
c
k
|
≤
2 for each k, where P is the family of all functions p analytic in U for which
(13)
R
(
p
(
z
)
)
>
0
,
p
(
z
)
=
1
+
c
1
z
+
c
2
z
2
+
⋯
,
for z
∈
U.


## 2. Coefficient Estimates

We begin this section by finding the estimates on the coefficients|
b
0
| and |
b
1
| for functions in the class M
σ
(
α
,
λ
).Theorem 5.
Let the functiong
(
z
) given by the series expansion (6) be in the function class
(14)
M
σ
(
α
,
λ
)
(
0
<
α
≤
1
,
0
≤
λ
<
1
)
.
Then
(15)
|
b
0
|
≤
2
α
1
-
λ
,
(16)
|
b
1
|
≤
{
2
α
2
1
-
λ
,
0
<
α
≤
1
2
,
α
1
-
λ
,
1
2
≤
α
≤
1
.Proof.
It follows from (10) that
(17)
z
g
′
(
z
)
(
1
-
λ
)
g
(
z
)
+
λ
z
g
′
(
z
)
=
[
p
(
z
)
]
α
(
z
∈
Δ
)
,
w
h
′
(
w
)
(
1
-
λ
)
h
(
w
)
+
λ
w
h
′
(
w
)
=
[
q
(
w
)
]
α
(
w
∈
Δ
)
,
respectively, where p
(
z
) and q
(
w
) are functions with positive real part in Δ and have the forms
(18)
p
(
z
)
=
1
+
p
1
z
+
p
2
z
2
+
⋯
,
(19)
q
(
w
)
=
1
+
q
1
w
+
q
2
w
2
+
⋯
,
respectively. Now, upon equating the coefficients in (17), we get
(20)
-
(
1
-
λ
)
b
0
=
α
p
1
,
(21)
(
1
-
λ
)
[
(
1
-
λ
)
b
0
2
-
2
b
1
]
=
α
p
2
+
α
(
α
-
1
)
2
p
1
2
,
(22)
(
1
-
λ
)
b
0
=
α
q
1
,
(23)
(
1
-
λ
)
[
(
1
-
λ
)
b
0
2
+
2
b
1
]
=
α
q
2
+
α
(
α
-
1
)
2
q
1
2
.
From (20) and (22), we find that
(24)
p
1
=
-
q
1
,
(25)
2
(
1
-
λ
)
2
b
0
2
=
α
2
(
p
1
2
+
q
1
2
)
.
Also, from (21) and (23) we obtain
(26)
2
(
1
-
λ
)
2
b
0
2
=
α
(
p
2
+
q
2
)
+
α
(
α
-
1
)
2
(
p
1
2
+
q
1
2
)
.
Since R
(
p
(
z
)
)
>
0 and R
(
q
(
z
)
)
>
0 in Δ, the functions p
(
1
/
z
)
,
q
(
1
/
z
)
∈
P and hence the coefficients p
k and q
k for each k satisfy the inequality in Lemma 4. Applications of triangle inequality followed by Lemma 4 in (25) and (26) give us the required estimates on |
b
0
| as asserted in (15).
Next, in order to find the bound on the coefficient|
b
1
|, we subtract (23) from (21). We thus get
(27)
-
4
(
1
-
λ
)
b
1
=
α
(
p
2
-
q
2
)
.
Hence
(28)
|
b
1
|
≤
α
1
-
λ
.
On the other hand, using (21) and (23) yields
(29)
2
(
1
-
λ
)
2
[
(
1
-
λ
)
2
b
0
4
+
4
b
1
2
]
=
α
2
(
α
-
1
)
2
4
(
p
1
4
+
q
1
4
)
+
α
2
(
p
2
2
+
q
2
2
)
+
α
2
(
α
-
1
)
(
p
1
2
p
2
+
q
1
2
q
2
)
.
By using (25) we have from the above equality
(30)
b
1
2
=
α
2
(
α
-
1
)
2
32
(
1
-
λ
)
2
(
p
1
4
+
q
1
4
)
+
α
2
8
(
1
-
λ
)
2
(
p
2
2
+
q
2
2
)
+
α
2
(
α
-
1
)
8
(
1
-
λ
)
2
(
p
1
2
p
2
+
q
1
2
q
2
)
-
α
4
16
(
1
-
λ
)
2
(
p
1
4
+
q
1
4
)
-
α
4
8
(
1
-
λ
)
2
p
1
2
q
1
2
.
From Lemma 4, we obtain
(31)
|
b
1
|
≤
5
α
2
1
-
λ
.
Also, by using (26) we have, from equality (29),
(32)
|
b
1
|
≤
2
α
2
1
-
λ
.
Comparing (28), (31), and (32) we get the desired estimate on the coefficient |
b
1
| as asserted in (16).Forλ
=
0, we have the following corollary of Theorem 5.Corollary 6.
Let the functiong
(
z
) given by the series expansion (6) be in the function class
(33)
Σ
~
B
*
(
α
)
(
0
<
α
≤
1
)
.
Then
(34)
|
b
0
|
≤
2
α
,
|
b
1
|
≤
{
2
α
2
,
0
<
α
≤
1
2
,
α
,
1
2
≤
α
≤
1
.Remark 7.
Corollary6 is an improvement of the following estimates which were given by Halim et al. [17].Corollary 8 (see [17]).
Let the functiong
(
z
) given by the series expansion (6) be in the function class
(35)
Σ
~
B
*
(
α
)
(
0
<
α
≤
1
)
.
Then
(36)
|
b
0
|
≤
2
α
,
|
b
1
|
≤
5
α
2
.Remark 9.
Corollary6 is also an improvement of the estimates which were given by Panigrahi [20, Corollary 2.3].Next we estimate the coefficients|
b
0
| and |
b
1
| for functions in the class M
σ
(
β
,
λ
).Theorem 10.
Let the functiong
(
z
) given by the series expansion (6) be in the function class
(37)
M
σ
(
β
,
λ
)
(
0
≤
β
<
1
,
0
≤
λ
<
1
)
.
Then
(38)
|
b
0
|
≤
2
(
1
-
β
)
1
-
λ
,
(39)
|
b
1
|
≤
1
-
β
1
-
λ
.Proof.
It follows from (11) that
(40)
z
g
′
(
z
)
(
1
-
λ
)
g
(
z
)
+
λ
z
g
′
(
z
)
=
β
+
(
1
-
β
)
p
(
z
)
(
z
∈
Δ
)
,
w
h
′
(
w
)
(
1
-
λ
)
h
(
w
)
+
λ
w
h
′
(
w
)
=
β
+
(
1
-
β
)
q
(
w
)
(
w
∈
Δ
)
,
respectively, where p
(
z
) and q
(
w
) are functions with positive real part in Δ and have the forms (18) and (19), respectively. Now, upon equating the coefficients in (40), we get
(41)
-
(
1
-
λ
)
b
0
=
(
1
-
β
)
p
1
,
(42)
(
1
-
λ
)
[
(
1
-
λ
)
b
0
2
-
2
b
1
]
=
(
1
-
β
)
p
2
,
(43)
(
1
-
λ
)
b
0
=
(
1
-
β
)
q
1
,
(44)
(
1
-
λ
)
[
(
1
-
λ
)
b
0
2
+
2
b
1
]
=
(
1
-
β
)
q
2
.
From (41) and (43), we obtain
(45)
p
1
=
-
q
1
,
(46)
2
(
1
-
λ
)
2
b
0
2
=
(
1
-
β
)
2
(
p
1
2
+
q
1
2
)
.
Also, from (42) and (44), we obtain
(47)
2
(
1
-
λ
)
2
b
0
2
=
(
1
-
β
)
(
p
2
+
q
2
)
.
SinceR
(
p
(
z
)
)
>
0 and R
(
q
(
z
)
)
>
0 in Δ, the functions p
(
1
/
z
)
,
q
(
1
/
z
)
∈
P and hence the coefficients p
k and q
k for each k satisfy the inequality in Lemma 4. Therefore, we find from (46) and (47) that
(48)
|
b
0
|
≤
2
(
1
-
β
)
1
-
λ
,
|
b
0
|
≤
2
(
1
-
β
)
1
-
λ
,
respectively. So we get the desired estimate on the coefficient |
b
0
| as asserted in (38).
Next, in order to find the bound on the coefficient|
b
1
|, we subtract (44) from (42). We thus get
(49)
-
4
(
1
-
λ
)
b
1
=
(
1
-
β
)
(
p
2
-
q
2
)
.
Hence
(50)
|
b
1
|
≤
1
-
β
1
-
λ
.
On the other hand, using (42) and (44) yields
(51)
(
1
-
λ
)
2
[
(
1
-
λ
)
2
b
0
4
-
4
b
1
2
]
=
(
1
-
β
)
2
p
2
q
2
or equivalently
(52)
4
b
1
2
=
(
1
-
λ
)
2
b
0
4
-
(
1
-
β
)
2
(
1
-
λ
)
2
p
2
q
2
.
Upon substituting the value of b
0
2 from (46) and (47) into (52), respectively, it follows that
(53)
|
b
1
|
≤
1
-
β
1
-
λ
4
β
2
-
8
β
+
5
,
|
b
1
|
≤
2
(
1
-
β
)
1
-
λ
.
Comparing (50) and (53), we get the desired estimate on the coefficient |
b
1
| as asserted in (39).Forλ
=
0, we have the following corollary of Theorem 10.Corollary 11.
Let the functiong
(
z
) given by the series expansion (6) be in the function class
(54)
Σ
B
*
(
β
)
(
0
≤
β
<
1
)
.
Then
(55)
|
b
0
|
≤
2
(
1
-
β
)
,
|
b
1
|
≤
1
-
β
.Remark 12.
Corollary11 is an improvement of the following estimates which were given by Halim et al. [17].Corollary 13 (see [17]).
Let the functiong
(
z
) given by the series expansion (6) be in the function class
(56)
Σ
B
*
(
β
)
(
0
≤
β
<
1
)
.
Then
(57)
|
b
0
|
≤
2
(
1
-
β
)
,
|
b
1
|
≤
(
1
-
β
)
4
β
2
-
8
β
+
5
.Remark 14.
Corollary11 is also an improvement of the estimates which were given by Panigrahi [20, Corollary 3.3].



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