# Coefficient Estimates for a New Subclass of Analytic and Bi-Univalent Functions Defined by Hadamard Product

Authors: Serap Bulut
Journal: Journal of Complex Analysis
 (2014)

---

## Abstract

We introduce and investigate a new general subclassℋ
Σ
λ
,
μ
(
φ
;
Θ
) of analytic and bi-univalent functions in the open unit disk U. For functions belonging to this class, we obtain estimates on the first two Taylor-Maclaurin coefficients |
a
2
| and |
a
3
|.


---

## Body

## 1. Introduction

LetA denote the class of all functions of the form
(1)
f
z
=
z
+
∑
n
=
2
∞
‍
a
n
z
n
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
+
⋯
.Denote byf
*
Θ the Hadamard product (or convolution) of the functions f and Θ; that is, if f is given by (1) and Θ is given by
(5)
Θ
z
=
z
+
∑
n
=
2
∞
‍
b
n
z
n
b
n
>
0
,
then
(6)
f
*
Θ
z
=
z
+
∑
n
=
2
∞
‍
a
n
b
n
z
n
.For two functionsf and Θ, analytic in U, we say that the function f is subordinate to Θ in U and write
(7)
f
z
≺
Θ
z
z
∈
U
,
if there exists a Schwarz function ω, which is analytic in U with
(8)
ω
0
=
0
,
ω
z
<
1
z
∈
U
such that
(9)
f
z
=
Θ
ω
z
z
∈
U
.
Indeed, it is known that
(10)
f
z
≺
Θ
z
z
∈
U
⟹
f
0
=
Θ
0
,
f
U
⊂
Θ
U
.
Furthermore, if the function Θ is univalent in U, then we have the following equivalence:
(11)
f
z
≺
Θ
z
z
∈
U
⟺
f
0
=
Θ
0
,
f
U
⊂
Θ
U
.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]). 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 Tang et al. [4], El-Ashwah [5], Frasin and Aouf [6], Aouf et al. [7], and others (see, e.g., [2, 8–15]).Throughout this paper, we assume thatφ is an analytic function with positive real part in the unit disk U, satisfying φ
0
=
1, φ
′
0
>
0, and φ
U is symmetric with respect to the real axis. Such a function has a series expansion of the form
(12)
φ
z
=
1
+
B
1
z
+
B
2
z
2
+
⋯
,
B
1
>
0
.With this assumption onφ, we now introduce the following subclass of bi-univalent functions.Definition 1.
Let the functionf, defined by (1), be in the analytic function class A and let Θ
∈
Σ. We say that
(13)
f
∈
H
Σ
λ
,
μ
φ
;
Θ
λ
≥
1
,
μ
≥
0
if the following conditions are satisfied:
(14)
f
∈
Σ
,
1
-
λ
f
*
Θ
z
z
μ
+
λ
f
*
Θ
′
z
f
*
Θ
z
z
μ
-
1
≺
φ
z
z
∈
U
,
1
-
λ
f
*
Θ
-
1
w
w
μ
+
λ
f
*
Θ
-
1
′
w
×
f
*
Θ
-
1
w
w
μ
-
1
≺
φ
w
w
∈
U
,
where the function (
f
*
Θ
)
-
1 is given by
(15)
f
*
Θ
-
1
w
=
w
-
a
2
b
2
w
2
+
2
a
2
2
b
2
2
-
a
3
b
3
w
3
-
5
a
2
3
b
2
3
-
5
a
2
b
2
a
3
b
3
+
a
4
b
4
w
4
+
⋯
.Remark 2.
If we let(16)
Θ
z
=
z
1
-
z
,
then the class H
Σ
λ
,
μ
(
φ
;
Θ
) reduces to the class denoted by H
Σ
μ
(
λ
,
φ
) which is the subclass of the functions f
∈
Σ satisfying
(17)
1
-
λ
f
z
z
μ
+
λ
f
′
z
f
z
z
μ
-
1
≺
φ
z
,
1
-
λ
g
w
w
μ
+
λ
g
′
w
g
w
w
μ
-
1
≺
φ
w
,
where the function g is defined by
(18)
g
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
+
⋯
,
which was introduced and studied recently by Tang et al. [4].Remark 3.
If we let(19)
φ
z
=
1
+
z
1
-
z
α
0
<
α
≤
1
,
then the class H
Σ
λ
,
μ
(
φ
;
Θ
) reduces to the new class denoted by H
Σ
λ
,
μ
(
α
;
Θ
) which is the subclass of the functions f
∈
Σ satisfying
(20)
arg
f
*
Θ
z
z
μ
-
1
1
-
λ
f
*
Θ
z
z
μ
+
λ
f
*
Θ
′
z
f
*
Θ
z
z
μ
-
1
<
α
π
2
,
arg
f
*
Θ
-
1
w
w
μ
-
1
1
-
λ
f
*
Θ
-
1
w
w
μ
+
λ
f
*
Θ
-
1
′
w
f
*
Θ
-
1
w
w
μ
-
1
<
α
π
2
,
where the function (
f
*
Θ
)
-
1 is defined by (15). Also we have the following classes:(i)
(21)
H
Σ
λ
,
1
α
;
Θ
=
B
Σ
Θ
,
α
,
λintroduced and studied recently by El-Ashwah [5];(ii)
(22)
H
Σ
λ
,
μ
α
;
z
1
-
z
=
N
Σ
μ
α
,
λintroduced and studied recently by Çağlar et al. [10];(iii)
(23)
H
Σ
λ
,
1
α
;
z
1
-
z
=
B
Σ
α
,
λintroduced and studied by Frasin and Aouf [6];(iv)
(24)
H
Σ
1,0
α
;
z
1
-
z
=
S
Σ
*
αof strongly bi-starlike functions of order α
0
<
α
≤
1 introduced and studied by Brannan and Taha [3];(v)
(25)
H
Σ
λ
,
1
α
;
z
+
∑
n
=
2
∞
‍
a
1
n
⋯
a
q
n
b
1
n
⋯
b
s
n
1
n
!
z
n
=
T
q
,
s
Σ
a
1
;
b
1
,
α
,
λintroduced and studied recently by Aouf et al. [7].Remark 4.
If we let(26)
φ
z
=
1
+
1
-
2
β
z
1
-
z
0
≤
β
<
1
,
then the class H
Σ
λ
,
μ
(
φ
;
Θ
) reduces to the new class denoted by H
Σ
λ
,
μ
(
β
;
Θ
) which is the subclass of the functions f
∈
Σ satisfying
(27)
R
f
*
Θ
z
z
μ
-
1
1
-
λ
f
*
Θ
z
z
μ
+
λ
f
*
Θ
′
z
f
*
Θ
z
z
μ
-
1
>
β
,
R
f
*
Θ
-
1
w
w
μ
-
1
1
-
λ
f
*
Θ
-
1
w
w
μ
+
λ
f
*
Θ
-
1
′
w
f
*
Θ
-
1
w
w
μ
-
1
>
β
,
where the function (
f
*
Θ
)
-
1 is defined by (15). Also we have the following classes:(i)
(28)
H
Σ
λ
,
1
β
;
Θ
=
B
Σ
Θ
,
β
,
λintroduced and studied recently by El-Ashwah [5];(ii)
(29)
H
Σ
λ
,
μ
β
;
z
1
-
z
=
N
Σ
μ
β
,
λintroduced and studied recently by Çağlar et al. [10];(iii)
(30)
H
Σ
λ
,
1
β
;
z
1
-
z
=
B
Σ
β
,
λintroduced and studied by Frasin and Aouf [6];(iv)
(31)
H
Σ
1,0
β
;
z
1
-
z
=
S
Σ
*
βof bi-starlike functions of order β
(
0
≤
β
<
1
) introduced and studied by Brannan and Taha [3];(v)
(32)
H
Σ
λ
,
1
α
;
z
+
∑
n
=
2
∞
‍
a
1
n
⋯
a
q
n
b
1
n
⋯
b
s
n
1
n
!
z
n
=
T
q
,
s
Σ
a
1
;
b
1
,
β
,
λintroduced and studied recently by Aouf et al. [7].For more results see also [2, 8, 13, 15–17].Firstly, in order to derive our main results, we need to go to following lemma.Lemma 5 (see [18]).
Ifp
∈
P, then |
c
n
|
≤
2 for each n, where P is the family of all functions p analytic in U for which
(33)
R
p
z
>
0
,
p
z
=
1
+
c
1
z
+
c
2
z
2
+
⋯
for z
∈
U.


## 2. A Set of General Coefficient Estimates

In this section, we state and prove our general results involving the bi-univalent function classH
Σ
λ
,
μ
(
φ
;
Θ
) given by Definition 1.Theorem 6.
Let the functionf
z given by the Taylor-Maclaurin series expansion (1) be in the function class
(34)
H
Σ
λ
,
μ
φ
;
Θ
λ
≥
1
,
μ
≥
0
with
(35)
Θ
∈
Σ
,
Θ
z
=
z
+
∑
n
=
2
∞
‍
b
n
z
n
b
n
>
0
.
Then
(36)
a
2
≤
1
b
2
min
⁡
⁡
B
1
2
B
1
B
1
2
λ
+
μ
2
B
1
λ
+
μ
,
2
B
1
+
B
2
-
B
1
μ
+
1
2
λ
+
μ
,
1
b
2
min
⁡
B
1
2
B
1
B
1
2
μ
+
1
2
λ
+
μ
-
2
B
2
-
B
1
λ
+
μ
2
,
(37)
a
3
≤
1
b
3
min
⁡
B
1
2
λ
+
μ
2
+
B
1
2
λ
+
μ
,
≤
1
b
3
min
⁡
B
1
μ
+
3
+
1
-
μ
2
μ
+
1
2
λ
+
μ
+
2
B
2
-
B
1
μ
+
1
2
λ
+
μ
,
≤
1
b
3
min
⁡
2
B
1
3
B
1
2
μ
+
1
2
λ
+
μ
-
2
B
2
-
B
1
λ
+
μ
2
≤
1
b
3
min
⁡
+
B
1
2
λ
+
μ
B
1
2
λ
+
μ
2
.Proof.
Letf
∈
H
Σ
λ
,
μ
(
φ
;
Θ
). Then there are analytic functions u
,
v
:
U
→
U, with u
0
=
v
0
=
0, satisfying
(38)
1
-
λ
f
*
Θ
z
z
μ
+
λ
f
*
Θ
′
z
f
*
Θ
z
z
μ
-
1
=
φ
u
z
,
1
-
λ
f
*
Θ
-
1
w
w
μ
+
λ
f
*
Θ
-
1
′
w
f
*
Θ
-
1
w
w
μ
-
1
=
φ
v
w
.
Define the functions p and q by
(39)
p
z
=
1
+
u
z
1
-
u
z
=
1
+
p
1
z
+
p
2
z
2
+
⋯
,
q
(
z
)
=
1
+
v
z
1
-
v
z
=
1
+
q
1
z
+
q
2
z
2
+
⋯
,
or, equivalently,
(40)
u
z
=
p
z
-
1
p
z
+
1
=
1
2
p
1
z
+
p
2
-
p
1
2
2
z
2
+
⋯
,
(41)
v
z
=
q
z
-
1
q
z
+
1
=
1
2
q
1
z
+
q
2
-
q
1
2
2
z
2
+
⋯
.
Then p and q are analytic in U with p
0
=
1
=
q
0. Since u
,
v
:
U
→
U, the functions p and q have a positive real part in U, and by Lemma 5  |
p
n
|
≤
2 and |
q
n
|
≤
2 for each n. Also it is clear that
(42)
1
-
λ
f
*
Θ
z
z
μ
+
λ
f
*
Θ
′
z
f
*
Θ
z
z
μ
-
1
=
φ
p
z
-
1
p
z
+
1
,
1
-
λ
f
*
Θ
-
1
w
w
μ
+
λ
f
*
Θ
-
1
′
w
×
f
*
Θ
-
1
w
w
μ
-
1
=
φ
q
w
-
1
q
w
+
1
.
Using (40) and (41) together with (12), we get
(43)
φ
p
z
-
1
p
z
+
1
=
1
+
1
2
B
1
p
1
z
+
1
2
B
1
p
2
-
p
1
2
2
+
1
4
B
2
p
1
2
z
2
+
⋯
,
φ
q
w
-
1
q
w
+
1
=
1
+
1
2
B
1
q
1
w
+
1
2
B
1
q
2
-
q
1
2
2
+
1
4
B
2
q
1
2
w
2
+
⋯
.
Now, upon equating the coefficients in (42)-(43), we get
(44)
λ
+
μ
b
2
a
2
=
1
2
B
1
p
1
,
(45)
2
λ
+
μ
b
3
a
3
+
μ
-
1
λ
+
μ
2
b
2
2
a
2
2
=
1
2
B
1
p
2
-
p
1
2
2
+
1
4
B
2
p
1
2
,
(46)
-
λ
+
μ
b
2
a
2
=
1
2
B
1
q
1
,
(47)
-
2
λ
+
μ
b
3
a
3
+
μ
+
3
λ
+
μ
2
b
2
2
a
2
2
=
1
2
B
1
q
2
-
q
1
2
2
+
1
4
B
2
q
1
2
.
From (44) and (46), it follows that
(48)
p
1
=
-
q
1
,
(49)
2
λ
+
μ
2
b
2
2
a
2
2
=
1
4
B
1
2
p
1
2
+
q
1
2
.
Now (45), (47), and (48) yield
(50)
a
2
2
=
B
1
p
2
+
q
2
+
B
2
-
B
1
p
1
2
2
μ
+
1
2
λ
+
μ
b
2
2
.
Also from (45), (47), and (49), we get
(51)
a
2
2
=
B
1
3
p
2
+
q
2
2
B
1
2
μ
+
1
2
λ
+
μ
-
2
B
2
-
B
1
λ
+
μ
2
b
2
2
.
Applying Lemma5 for (44), (50), and (51), we get the desired estimate on the coefficient |
a
2
| as asserted in (36).
Next, in order to find the bound on the coefficient|
a
3
|, by subtracting (47) from (45), we obtain
(52)
2
2
λ
+
μ
b
3
a
3
-
2
2
λ
+
μ
b
2
2
a
2
2
=
1
2
B
1
p
2
-
q
2
.
Upon substituting the value of a
2
2 from (44), (50), and (51) into (52), it follows that
(53)
a
3
=
B
1
2
p
1
2
4
λ
+
μ
2
b
3
+
B
1
p
2
-
q
2
4
2
λ
+
μ
b
3
,
(54)
a
3
=
B
1
p
2
+
q
2
+
B
2
-
B
1
p
1
2
2
μ
+
1
2
λ
+
μ
b
3
+
B
1
p
2
-
q
2
4
2
λ
+
μ
b
3
=
B
1
μ
+
3
p
2
+
1
-
μ
q
2
+
2
B
2
-
B
1
p
1
2
4
μ
+
1
2
λ
+
μ
b
3
,
(55)
a
3
=
B
1
3
p
2
+
q
2
2
B
1
2
μ
+
1
2
λ
+
μ
-
2
B
2
-
B
1
λ
+
μ
2
b
3
+
B
1
p
2
-
q
2
4
2
λ
+
μ
b
3
,
respectively. Applying Lemma 5 for (53), (54), and (55), we get the desired estimate on the coefficient a
3 as asserted in (37).If we take(56)
Θ
z
=
z
1
-
z
in Theorem 6, then we have the following corollary.Corollary 7.
Let the functionf
z given by the Taylor-Maclaurin series expansion (1) be in the function class
(57)
H
Σ
μ
λ
,
φ
λ
≥
1
,
μ
≥
0
.
Then
(58)
a
2
≤
min
⁡
⁡
B
1
2
B
1
B
1
2
λ
+
μ
2
B
1
λ
+
μ
,
2
B
1
+
B
2
-
B
1
μ
+
1
2
λ
+
μ
,
B
1
2
B
1
B
1
2
μ
+
1
2
λ
+
μ
-
2
B
2
-
B
1
λ
+
μ
2
,
a
3
≤
min
⁡
B
1
2
λ
+
μ
2
+
B
1
2
λ
+
μ
,
min
⁡
B
1
μ
+
3
+
1
-
μ
2
μ
+
1
2
λ
+
μ
+
2
B
2
-
B
1
μ
+
1
2
λ
+
μ
,
min
⁡
2
B
1
3
B
1
2
μ
+
1
2
λ
+
μ
-
2
B
2
-
B
1
λ
+
μ
2
min
⁡
+
B
1
2
λ
+
μ
B
1
2
λ
+
μ
2
.Remark 8.
Corollary7 is an improvement of the estimates which were given by Tang et al. [4, Theorem 2.1].Remark 9.
If we setλ
=
1 and μ
=
1 in Corollary 7, then we have an improvement of the estimates which were given by Ali et al. [16, Theorem 2.1].If we take(59)
φ
z
=
1
+
z
1
-
z
α
0
<
α
≤
1
in Theorem 6, then we have the following corollary.Corollary 10.
Let the functionf
z given by the Taylor-Maclaurin series expansion (1) be in the function class
(60)
H
Σ
λ
,
μ
α
;
Θ
λ
≥
1
,
μ
≥
0
,
0
<
α
≤
1
with
(61)
Θ
∈
Σ
,
Θ
z
=
z
+
∑
n
=
2
∞
‍
b
n
z
n
b
n
>
0
.
Then
(62)
a
2
≤
1
b
2
min
⁡
⁡
2
α
λ
+
μ
2
+
α
μ
+
2
λ
-
λ
2
2
α
λ
+
μ
,
4
α
2
-
α
μ
+
1
2
λ
+
μ
,
1
b
2
min
⁡
2
α
λ
+
μ
2
+
α
μ
+
2
λ
-
λ
2
,
a
3
≤
1
b
3
min
⁡
⁡
4
α
2
λ
+
μ
2
+
2
α
2
λ
+
μ
,
1
b
3
min
⁡
2
α
μ
+
3
+
1
-
μ
2
μ
+
1
2
λ
+
μ
+
4
α
1
-
α
μ
+
1
2
λ
+
μ
,
1
b
3
min
⁡
4
α
2
λ
+
μ
2
+
α
μ
+
2
λ
-
λ
2
+
2
α
2
λ
+
μ
.Remark 11.
In Corollary10,(i)
if we setμ
=
1, then we have an improvement of the estimates which were given by El-Ashwah [5, Theorem 1];
(ii)
if we setΘ
z
=
z
/
(
1
-
z
), then we have an improvement of the estimates which were given by Çağlar et al. [10, Theorem 2.1];
(iii)
if we setΘ
z
=
z
/
(
1
-
z
) and μ
=
1, then we have an improvement of the estimates which were given by Frasin and Aouf [6, Theorem 2.2];
(iv)
if we setΘ
z
=
z
+
∑
n
=
2
∞
‍
(
(
(
a
1
)
n
⋯
(
a
q
)
n
/
(
b
1
)
n
⋯
(
b
s
)
n
)
(
1
/
n
!
)
z
n
) and μ
=
1, then we have an improvement of the estimates which were given by Aouf et al. [7, Theorem 4].If we take(63)
φ
z
=
1
+
1
-
2
β
z
1
-
z
0
≤
β
<
1
in Theorem 6, then we have the following corollary.Corollary 12.
Let the functionf
z given by the Taylor-Maclaurin series expansion (1) be in the function class
(64)
B
Σ
Θ
,
β
,
λ
λ
≥
1
,
0
≤
β
<
1
with
(65)
Θ
∈
Σ
,
Θ
z
=
z
+
∑
n
=
2
∞
‍
b
n
z
n
b
n
>
0
.
Then
(66)
a
2
≤
1
b
2
min
⁡
2
1
-
β
λ
+
μ
,
4
1
-
β
μ
+
1
2
λ
+
μ
,
a
3
≤
1
b
3
min
⁡
⁡
4
1
-
β
2
λ
+
μ
2
+
2
1
-
β
2
λ
+
μ
,
4
1
-
β
μ
+
1
2
λ
+
μ
,
0
≤
μ
<
1
,
2
1
-
β
2
λ
+
μ
b
3
,
μ
≥
1
.Remark 13.
In Corollary12,(i)
if we setμ
=
1, then we have an improvement of the estimates which were given by El-Ashwah [5, Theorem 2];
(ii)
if we setΘ
z
=
z
/
(
1
-
z
), then we have the estimates which were given by Çağlar et al. [10, Theorem 3.1];
(iii)
if we setΘ
z
=
z
/
(
1
-
z
) and μ
=
1, then we have an improvement of the estimates which were given by Frasin and Aouf [6, Theorem 3.2];
(iv)
if we setΘ
z
=
z
+
∑
n
=
2
∞
‍
(
(
(
a
1
)
n
⋯
(
a
q
)
n
/
(
b
1
)
n
⋯
(
b
s
)
n
)
(
1
/
n
!
)
z
n
) and μ
=
1, then we have an improvement of the estimates which were given by Aouf et al. [7, Theorem 8].



---

*Source: 302019-2014-11-10.xml*