# Initial Coefficient Bounds for a General Class of Biunivalent Functions

Authors: Şahsene Altınkaya; Sibel Yalçın
Journal: International Journal of Analysis
 (2014)

---

## Abstract

We find estimates on the coefficientsa
2 and a
3 for
functions in the function class B
λ
,
μ
,
ϕ. The results presented in this paper improve or generalize the recent work of Srutha Keerthi and  Raja (2013).


---

## Body

## 1. Introduction and Definitions

LetA denote the class of analytic functions in the unit disk
(1)
U
=
{
z
∈
ℂ
:
|
z
|
<
1
}
that have the form
(2)
f
(
z
)
=
z
+
∑
n
=
2
∞
a
n
z
n
and let S be the class of all functions from A which are univalent in U.The Koebe one-quarter theorem [1] states that the image of U under every function f from S contains a disk of radius 1
/
4. Thus every such univalent function has an inverse f
-
1 which satisfies
(3)
f
-
1
(
f
(
z
)
)
=
z
(
z
∈
U
)
,
f
(
f
-
1
(
w
)
)
=
w
,
(
|
w
|
<
r
0
(
f
)
,
r
0
(
f
)
≥
1
4
)
,
where
(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
(
z
)
∈
A is said to be biunivalent in U if both f
(
z
) and f
-
1
(
z
) are univalent in U.If the functionsf and g are analytic in U, then f is said to be subordinate to g, written as f
(
z
)
≺
g
(
z
), if there exists a Schwarz function w such that f
(
z
)
=
g
(
w
(
z
)
).LetΣ denote the class of biunivalent functions defined in the unit disk U. For a brief history and interesting examples in the class Σ, (see [2]).Lewin [3] studied the class of biunivalent functions, obtaining the bound 1.51 for modulus of the second coefficient |
a
2
|. Subsequently, Brannan and Clunie [4] conjectured that |
a
2
|
≤
2 for f
∈
Σ. Netanyahu [5] showed that max
⁡
|
a
2
|
=
4
/
3 if f
(
z
)
∈
Σ.Brannan and Taha [6] introduced certain subclasses of the biunivalent function class Σ similar to the familiar subclasses,  δ
⋆
(
α
) and K
(
α
) of starlike and convex function of order α
(
0
<
α
≤
1
), respectively (see [5]). Thus, following Brannan and Taha [6], a function f
∈
A is the class K
Σ
(
α
) of strongly biconvex functions of order α
(
0
<
α
≤
1
) if each of the following conditions is satisfied:
(5)
f
∈
Σ
,
|
arg
(
z
2
f
′′
(
z
)
+
z
f
′
(
z
)
z
f
′
(
z
)
)
|
<
α
π
2
5525555555555555555
(
0
<
α
≤
1
,
z
∈
U
)
,
|
arg
(
w
2
g
′′
(
w
)
+
w
g
′
(
w
)
w
g
′
(
w
)
)
|
<
α
π
2
(
0
<
α
≤
1
,
w
∈
U
)
,
where g is the extension of f
-
1 to U. The classes δ
Σ
⋆
(
α
) and K
Σ
(
α
) of bistarlike functions of order α and biconvex functions of order α, corresponding to the function classes δ
⋆
(
α
) and K
(
α
), were also introduced analogously. For each of the function classes δ
Σ
⋆
(
α
) and K
Σ
(
α
), they found nonsharp estimates on the initial coefficients. Recently, many authors investigated bounds for various subclasses of biunivalent functions ([2, 7, 8]). The coefficient estimate problem for each of the following Taylor-Maclaurin coefficients |
a
n
| for n
∈
ℕ
∖
{
1,2
}; ℕ
=
{
1,2
,
3
,
…
} is presumably still an open problem.In this paper, by using the method [9] different from that used by other authors, we obtain bounds for the coefficients |
a
2
| and |
a
3
| for the subclasses of biunivalent functions considered by Srutha Keerthi and Raja and get more accurate estimates than that given in [10].


## 2. Coefficient Estimates

In the following, letϕ be an analytic function with positive real part in U, with ϕ
(
0
)
=
1 and ϕ
′
(
0
)
>
0. Also, let ϕ
(
U
) be starlike with respect to 1 and symmetric with respect to the real axis. Thus, ϕ has the Taylor series expansion
(6)
ϕ
(
z
)
=
1
+
B
1
z
+
B
2
z
2
+
B
3
z
3
+
⋯
(
B
1
>
0
)
.
Suppose that u
(
z
), and v
(
z
) are analytic in the unit disk U with u
(
0
)
=
v
(
0
)
=
0, |
u
(
z
)
|
<
1, |
v
(
z
)
|
<
1, and suppose that
(7)
u
(
z
)
=
b
1
z
+
∑
n
=
2
∞
‍
b
n
z
n
,
v
(
z
)
=
c
1
z
+
∑
n
=
2
∞
‍
c
n
z
n
(
|
z
|
<
1
)
.
It is well known that
(8)
|
b
1
|
≤
1
,
|
b
2
|
≤
1
-
|
b
1
|
2
,
|
c
1
|
≤
1
,
|
c
2
|
≤
1
-
|
c
1
|
2
.
Next, (6) and (7) lead to
(9)
ϕ
(
u
(
z
)
)
=
1
+
B
1
b
1
z
+
(
B
1
b
2
+
B
2
b
1
2
)
z
2
+
⋯
,
|
z
|
<
1
,
ϕ
(
v
(
w
)
)
=
1
+
B
1
c
1
w
+
(
B
1
c
2
+
B
2
c
1
2
)
w
2
+
⋯
,
|
w
|
<
1
.Definition 1.
A functionf
∈
Σ is said to be in the class B
(
λ
,
μ
,
ϕ
)
,
λ
≥
0, 0
≤
μ
≤
λ
≤
1, if the following subordinations hold:
(10)
λ
μ
z
3
f
′′′
(
z
)
+
(
2
λ
μ
+
λ
-
μ
)
z
2
f
′′
(
z
)
+
z
f
′
(
z
)
λ
μ
z
2
f
′′
(
z
)
+
(
λ
-
μ
)
z
f
′
(
z
)
+
(
1
-
λ
+
μ
)
f
(
z
)
≺
ϕ
(
z
)
,
λ
μ
w
3
g
′′′
(
w
)
+
(
2
λ
μ
+
λ
-
μ
)
w
2
g
′′
(
w
)
+
w
g
′
(
w
)
λ
μ
w
2
g
′′
(
w
)
+
(
λ
-
μ
)
w
g
′
(
w
)
+
(
1
-
λ
+
μ
)
g
(
w
)
≺
ϕ
(
w
)
,
where g
(
w
)
=
f
-
1
(
w
).Theorem 2.
Letf given by (2) be in the class B
(
λ
,
μ
,
ϕ
). Then
(11)
|
a
2
|
≤
B
1
B
1
×
(
|
2
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
B
1
2
-
(
1
+
λ
-
μ
+
2
λ
μ
)
2
(
B
1
2
+
B
2
)
|
+
(
1
+
λ
-
μ
+
2
λ
μ
)
2
B
1
)
-
1
/
2
,
(12)
|
a
3
|
≤
{
B
1
2
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
,
i
f
B
1
≤
(
1
+
λ
-
μ
+
2
λ
μ
)
2
2
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
,
(
|
2
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
B
1
2
-
(
1
+
λ
-
μ
+
2
λ
μ
)
2
(
B
1
2
+
B
2
)
|
B
1
+
2
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
B
1
3
)
×
(
2
[
|
2
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
B
1
2
-
(
1
+
λ
-
μ
+
2
λ
μ
)
2
(
B
1
2
+
B
2
)
|
+
(
1
+
λ
-
μ
+
2
λ
μ
)
2
B
1
]
×
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
(
2
[
|
2
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
B
1
2
)
-
1
,
i
f
B
1
>
(
1
+
λ
-
μ
+
2
λ
μ
)
2
2
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
.Proof.
Letf
∈
B
(
λ
,
μ
,
ϕ
), λ
≥
0, and 0
≤
μ
≤
λ
≤
1. Then there are analytic functions u
,
v
:
U
→
U given by (7) such that
(13)
λ
μ
z
3
f
′′′
(
z
)
+
(
2
λ
μ
+
λ
-
μ
)
z
2
f
′′
(
z
)
+
z
f
′
(
z
)
λ
μ
z
2
f
′′
(
z
)
+
(
λ
-
μ
)
z
f
′
(
z
)
+
(
1
-
λ
+
μ
)
f
(
z
)
=
ϕ
(
u
(
z
)
)
,
λ
μ
w
3
g
′′′
(
w
)
+
(
2
λ
μ
+
λ
-
μ
)
w
2
g
′′
(
w
)
+
w
g
′
(
w
)
λ
μ
w
2
g
′′
(
w
)
+
(
λ
-
μ
)
w
g
′
(
w
)
+
(
1
-
λ
+
μ
)
g
(
w
)
=
ϕ
(
v
(
w
)
)
,
where g
(
w
)
=
f
-
1
(
w
). Since
(14)
λ
μ
z
3
f
′′′
(
z
)
+
(
2
λ
μ
+
λ
-
μ
)
z
2
f
′′
(
z
)
+
z
f
′
(
z
)
λ
μ
z
2
f
′′
(
z
)
+
(
λ
-
μ
)
z
f
′
(
z
)
+
(
1
-
λ
+
μ
)
f
(
z
)
=
1
+
(
1
+
λ
-
μ
+
2
λ
μ
)
a
2
z
+
[
-
(
1
+
λ
-
μ
+
2
λ
μ
)
2
a
2
2
2
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
a
3
-
(
1
+
λ
-
μ
+
2
λ
μ
)
2
a
2
2
]
z
2
+
⋯
,
λ
μ
w
3
g
′′′
(
w
)
+
(
2
λ
μ
+
λ
-
μ
)
w
2
g
′′
(
w
)
+
w
g
′
(
w
)
λ
μ
w
2
g
′′
(
w
)
+
(
λ
-
μ
)
w
g
′
(
w
)
+
(
1
-
λ
+
μ
)
g
(
w
)
=
1
-
(
1
+
λ
-
μ
+
2
λ
μ
)
a
2
w
+
[
2
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
(
2
a
2
2
-
a
3
)
-
(
1
+
λ
-
μ
+
2
λ
μ
)
2
a
2
2
]
w
2
+
⋯
,
it follows from (9) and (13) that
(15)
(
1
+
λ
-
μ
+
2
λ
μ
)
a
2
=
B
1
b
1
,
(16)
2
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
a
3
-
(
1
+
λ
-
μ
+
2
λ
μ
)
2
a
2
2
=
B
1
b
2
+
B
2
b
1
2
,
(17)
-
(
1
+
λ
-
μ
+
2
λ
μ
)
a
2
=
B
1
c
1
,
(18)
2
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
(
2
a
2
2
-
a
3
)
-
(
1
+
λ
-
μ
+
2
λ
μ
)
2
a
2
2
=
B
1
c
2
+
B
2
c
1
2
.
From (15) and (17) we obtain
(19)
c
1
=
-
b
1
.
By adding (18) to (16), further computations using (15) to (19) lead to
(20)
[
4
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
B
1
2
-
2
(
1
+
λ
-
μ
+
2
λ
μ
)
2
(
B
1
2
+
B
2
)
]
a
2
2
=
B
1
3
(
b
2
+
c
2
)
.
Equations (19) and (20), together with (8), give
(21)
|
2
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
B
1
2
-
(
1
+
λ
-
μ
+
2
λ
μ
)
2
(
B
1
2
+
B
2
)
|
×
|
a
2
|
2
≤
B
1
3
(
1
-
|
b
1
|
2
)
.
From (15) and (21) we get
(22)
|
a
2
|
≤
B
1
B
1
×
(
|
2
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
B
1
2
-
(
1
+
λ
-
μ
+
2
λ
μ
)
2
(
B
1
2
+
B
2
)
-
(
1
+
λ
-
μ
+
2
λ
μ
)
2
(
B
1
2
+
B
2
)
|
+
(
1
+
λ
-
μ
+
2
λ
μ
)
2
B
1
)
-
1
/
2
.
Next, in order to find the bound on |
a
3
|, by subtracting (18) from (16), we obtain
(23)
4
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
a
3
-
4
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
a
2
2
=
B
1
(
b
2
-
c
2
)
.
Then, in view of (8) and (19), we have
(24)
2
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
B
1
|
a
3
|
≤
[
-
(
1
+
λ
-
μ
+
2
λ
μ
)
2
2
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
B
1
-
(
1
+
λ
-
μ
+
2
λ
μ
)
2
]
|
a
2
|
2
+
B
1
2
.
Notice that (11), we get
(25)
|
a
3
|
≤
{
B
1
2
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
,
if
B
1
≤
(
1
+
λ
-
μ
+
2
λ
μ
)
2
2
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
,
(
|
2
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
B
1
2
+
2
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
B
1
3
-
(
1
+
λ
-
μ
+
2
λ
μ
)
2
(
B
1
2
+
B
2
)
|
B
1
+
2
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
B
1
3
)
×
(
2
[
|
2
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
B
1
2
-
(
1
+
λ
-
μ
+
2
λ
μ
)
2
(
B
1
2
+
B
2
)
|
+
(
1
+
λ
-
μ
+
2
λ
μ
)
2
B
1
]
×
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
(
2
[
|
2
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
B
1
2
)
-
1
,
if
B
1
>
(
1
+
λ
-
μ
+
2
λ
μ
)
2
2
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
.
Remark
3. If we let
(26)
ϕ
(
z
)
=
(
1
+
z
1
-
z
)
α
=
1
+
2
α
z
+
2
α
2
z
2
+
⋯
(
0
<
α
≤
1
)
,
then inequalities (11) and (12) become
(27)
|
a
2
|
≤
2
α
×
(
|
-
3
(
1
+
λ
-
μ
+
2
λ
μ
)
2
4
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
+
(
1
+
λ
-
μ
+
2
λ
μ
)
2
-
3
(
1
+
λ
-
μ
+
2
λ
μ
)
2
-
3
(
1
+
λ
-
μ
+
2
λ
μ
)
2
|
α
+
(
1
+
λ
-
μ
+
2
λ
μ
)
2
)
-
1
/
2
,
|
a
3
|
≤
{
α
1
+
2
λ
-
2
μ
+
6
λ
μ
,
if
0
<
α
≤
(
1
+
λ
-
μ
+
2
λ
μ
)
2
4
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
,
(
[
|
-
3
(
1
+
λ
-
μ
+
2
λ
μ
)
2
4
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
-
3
(
1
+
λ
-
μ
+
2
λ
μ
)
2
|
+
4
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
|
-
3
(
1
+
λ
-
μ
+
2
λ
μ
)
2
4
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
]
α
2
)
×
(
[
+
(
1
+
λ
-
μ
+
2
λ
μ
)
2
|
-
3
(
1
+
λ
-
μ
+
2
λ
μ
)
2
4
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
-
3
(
1
+
λ
-
μ
+
2
λ
μ
)
2
|
α
+
(
1
+
λ
-
μ
+
2
λ
μ
)
2
]
×
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
(
[
+
(
1
+
λ
-
μ
+
2
λ
μ
)
2
|
-
3
(
1
+
λ
-
μ
+
2
λ
μ
)
2
4
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
)
-
1
,
if
(
1
+
λ
-
μ
+
2
λ
μ
)
2
4
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
<
α
≤
1
.
The bounds on |
a
2
| and |
a
3
| given by (27) are more accurate than those given in Theorem  2.2 in [10].
Remark  4. If we let
(28)
ϕ
(
z
)
=
1
+
(
1
-
2
α
)
z
1
-
z
=
1
+
2
(
1
-
α
)
z
+
2
(
1
-
α
)
z
2
+
⋯
5555555555555555555552155555555555555
(
0
<
α
≤
1
)
,
then inequalities (11) and (12) become
(29)
|
a
2
|
≤
2
(
1
-
α
)
×
(
+
(
1
+
λ
-
μ
+
2
λ
μ
)
2
|
-
(
3
-
2
α
)
(
1
+
λ
-
μ
+
2
λ
μ
)
2
4
(
1
-
α
)
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
-
(
3
-
2
α
)
(
1
+
λ
-
μ
+
2
λ
μ
)
2
|
+
(
1
+
λ
-
μ
+
2
λ
μ
)
2
)
-
1
/
2
,
|
a
3
|
≤
{
1
-
α
1
+
2
λ
-
2
μ
+
6
λ
μ
,
if
4
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
-
(
1
+
λ
-
μ
+
2
λ
μ
)
2
4
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
≤
α
<
1
,
(
[
|
4
(
1
-
α
)
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
-
(
3
-
2
α
)
(
1
+
λ
-
μ
+
2
λ
μ
)
2
-
(
3
-
2
α
)
(
1
+
λ
-
μ
+
2
λ
μ
)
2
-
(
3
-
2
α
)
(
1
+
λ
-
μ
+
2
λ
μ
)
2
|
]
×
(
1
-
α
)
+
4
(
1
-
α
)
2
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
)
×
(
[
+
(
1
+
λ
-
μ
+
2
λ
μ
)
2
|
-
(
3
-
2
α
)
(
1
+
λ
-
μ
+
2
λ
μ
)
2
4
(
1
-
α
)
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
-
(
3
-
2
α
)
(
1
+
λ
-
μ
+
2
λ
μ
)
2
|
+
(
1
+
λ
-
μ
+
2
λ
μ
)
2
]
×
(
[
+
(
1
+
λ
-
μ
+
2
λ
μ
)
2
|
-
(
3
-
2
α
)
(
1
+
λ
-
μ
+
2
λ
μ
)
2
4
(
1
-
α
)
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
)
-
1
,
if
0
≤
α
<
4
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
-
(
1
+
λ
-
μ
+
2
λ
μ
)
2
4
(
1
+
2
λ
-
2
μ
+
6
λ
μ
)
.
The bounds on |
a
2
| and |
a
3
| given by (29) are more accurate than those given in Theorem  3.3 in [10].



---

*Source: 867871-2014-04-16.xml*