AbstractIn this paper, we introduce two new subclasses of the function class Σ of bi-univalent functions defined in the open unit disc. Furthermore, we find estimates on the coefficients |a2| and |a3| for functions in these new subclasses

Aouf, M.K.

Frasin, B.A.

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Applied Mathematics Letters 24 (2011) 1569–1573
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Applied Mathematics Letters
journal homepage: www.elsevier.com/locate/aml
New subclasses of bi-univalent functions
B.A. Frasin a, M.K. Aouf b,∗
a Department of Mathematics, Al al-Bayt University, P.O. Box: 130095 Mafraq, Jordan
b Department of Mathematics, Mansoura University, Mansoura 35516, Egypt
a r t i c l e i n f o
Article history:
Received 2 October 2010
Received in revised form 24 March 2011
Accepted 28 March 2011
Keywords:
Analytic and univalent functions
Bi-univalent functions
Starlike and convex functions
Coefficients bounds
a b s t r a c t
In this paper, we introduce two new subclasses of the function class Σ of bi-univalent
functions defined in the open unit disc. Furthermore, we find estimates on the coefficients
|a2| and |a3| for functions in these new subclasses.
© 2011 Elsevier Ltd. All rights reserved.
1. Introduction and definitions
LetA denote the class of functions of the form:
f (z) = z +
∞−
n=2
anzn (1.1)
which are analytic in the open unit discU ={z : |z| < 1}. Further, by S we shall denote the class of all functions inAwhich
are univalent inU.
Ding et al. [1] introduced the following class Qλ(β) of analytic functions defined as follows:
Qλ(β) =

f ∈ A : Re

(1− λ) f (z)
z
+ λf ′(z)

> β, β < 1, λ ≥ 0

.
It is easy to see that Qλ1(β) ⊂ Qλ2(β) for λ1 > λ2 ≥ 0. Thus, for λ ≥ 1, 0 ≤ β < 1,Qλ(β) ⊂ Q1(β) = {f ∈ A :
Ref ′(z) > β, 0 ≤ β < 1} and hence Qλ(β) is univalent class (see [2–4]).
It is well known that every function f ∈ S has an inverse f −1, defined by
f −1(f (z)) = z (z ∈ U)
and
f (f −1(w)) = w

|w| < r0(f ); r0(f ) ≥ 14

where
f −1(w) = w − a2w2 + (2a22 − a3)w3 − (5a32 − 5a2a3 + a4)w4 + · · · .
∗ Corresponding author.
E-mail addresses: bafrasin@yahoo.com (B.A. Frasin), mkaouf127@yahoo.com (M.K. Aouf).
0893-9659/$ – see front matter© 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.aml.2011.03.048
1570 B.A. Frasin, M.K. Aouf / Applied Mathematics Letters 24 (2011) 1569–1573
A function f ∈ A is said to be bi-univalent inU if both f (z) and f −1(z) are univalent inU.
Let Σ denote the class of bi-univalent functions inU given by (1.1). For a brief history and interesting examples in the
classΣ , see [5].
Brannan and Taha [6] (see also [7]) introduced certain subclasses of the bi-univalent function class Σ similar to the
familiar subclasses S∗(α) andK(α) of starlike and convex functions of order α(0 ≤ α < 1), respectively (see [8]). Thus,
following Brannan and Taha [6] (see also [7]), a function f ∈ A is in the class S∗Σ [α] of strongly bi-starlike functions of order
α(0 < α ≤ 1) if each of the following conditions is satisfied:
f ∈ Σ and
arg zf ′(z)f (z)
 < απ2 (0 < α ≤ 1, z ∈ U)
and arg zg ′(w)g(w)
 < απ2 (0 < α ≤ 1, w ∈ U),
where g is the extension of f −1 toU. The classesS∗Σ (α) andKΣ (α) of bi-starlike functions of orderα and bi-convex functions
of order α, corresponding (respectively) to the function classes S∗(α) andK(α), were also introduced analogously. For each
of the function classes S∗Σ (α) and KΣ (α), they found non-sharp estimates on the first two Taylor–Maclaurin coefficients|a2| and |a3| (for details, see [6,7]).
The object of the present paper is to introduce two new subclasses of the function class Σ and find estimates on the
coefficients |a2| and |a3| for functions in these new subclasses of the function classΣ employing the techniques used earlier
by Srivastava et al. [5].
In order to derive our main results, we have to recall here the following lemma [9].
Lemma 1.1. If h ∈ P then |ck| ≤ 2 for each k, where P is the family of all functions h analytic in U for which Reh(z) > 0
h(z) = 1+ c1z + c2z2 + c3z3 + · · · for z ∈ U.
2. Coefficient bounds for the function classBΣ(α, λ)
Definition 2.1. A function f (z) given by (1.1) is said to be in the classBΣ (α, λ) if the following conditions are satisfied:
f ∈ Σ and
arg(1− λ) f (z)z + λf ′(z)
 < απ2 (0 < α ≤ 1, λ ≥ 1, z ∈ U) (2.1)
and arg(1− λ)g(w)w + λg ′(w)
 < απ2 (0 < α ≤ 1, λ ≥ 1, w ∈ U), (2.2)
where the function g is given by
g(w) = w − a2w2 + (2a22 − a3)w3 − (5a32 − 5a2a3 + a4)w4 + · · · . (2.3)
We note that for λ = 1, the classBΣ (α, λ) reduces to the classHαΣ introduced and studied by Srivastava et al. [5].
We begin by finding the estimates on the coefficients |a2| and |a3| for functions in the classBΣ (α, λ).
Theorem 2.2. Let f (z) given by (1.1) be in the classBΣ (α, λ), 0 < α ≤ 1 and λ ≥ 1. Then
|a2| ≤ 2α
(λ+ 1)2 + α(1+ 2λ− λ2) (2.4)
and
|a3| ≤ 4α
2
(λ+ 1)2 +
2α
2λ+ 1 . (2.5)
Proof. It follows from (2.1) and (2.2) that
(1− λ) f (z)
z
+ λf ′(z) = [p(z)]α (2.6)
and
(1− λ)g(w)
w
+ λg ′(w) = [q(w)]α (2.7)
B.A. Frasin, M.K. Aouf / Applied Mathematics Letters 24 (2011) 1569–1573 1571
where p(z) and q(w) in P and have the forms
p(z) = 1+ p1z + p2z2 + p3z3 + · · · (2.8)
and
q(w) = 1+ q1w + q2w2 + q3w3 + · · · . (2.9)
Now, equating the coefficients in (2.6) and (2.7), we get
(λ+ 1)a2 = αp1, (2.10)
(2λ+ 1)a3 = αp2 + α(α − 1)2 p
2
1, (2.11)
−(λ+ 1)a2 = αq1 (2.12)
and
(2λ+ 1)(2a22 − a3) = αq2 +
α(α − 1)
2
q21. (2.13)
From (2.10) and (2.12), we get
p1 = −q1 (2.14)
and
2(λ+ 1)2a22 = α2(p21 + q21). (2.15)
Now from (2.11), (2.13) and (2.15), we obtain
2(2λ+ 1)a22 = α(p2 + q2)+
α(α − 1)
2
(p21 + q21)
= α(p2 + q2)+ α(α − 1)2
2(λ+ 1)2a22
α2
.
Therefore, we have
a22 =
α2(p2 + q2)
(λ+ 1)2 + α(1+ 2λ− λ2) .
Applying Lemma 1.1 for the coefficients p2 and q2, we immediately have
|a2| ≤ 2α
(λ+ 1)2 + α(1+ 2λ− λ2) .
This gives the bound on |a2| as asserted in (2.4).
Next, in order to find the bound on |a3|, by subtracting (2.13) from (2.11), we get
2(2λ+ 1)a3 − 2(2λ+ 1)a22 = αp2 +
α(α − 1)
2
p21 −

αq2 + α(α − 1)2 q
2
1

. (2.16)
It follows from (2.14)–(2.16) that
2(2λ+ 1)a3 = α
2(2λ+ 1)(p21 + q21)
(λ+ 1)2 + α(p2 − q2)
or, equivalently,
a3 = α
2(p21 + q21)
2(λ+ 1)2 +
α(p2 − q2)
2(2λ+ 1) .
Applying Lemma 1.1 once again for the coefficients p1, p2, q1 and q2, we readily get
|a3| ≤ 4α
2
(λ+ 1)2 +
2α
2λ+ 1 .
This completes the proof of Theorem 2.2. 
Putting λ = 1 in Theorem 2.2, we have
1572 B.A. Frasin, M.K. Aouf / Applied Mathematics Letters 24 (2011) 1569–1573
Corollary 2.3 ([5]). Let f (z) given by (1.1) be in the classHαΣ (0 < α ≤ 1). Then
|a2| ≤ α

2
2+ α (2.17)
and
|a3| ≤ α(3α + 2)3 . (2.18)
3. Coefficient bounds for the function classBΣ(β, λ)
Definition 3.1. A function f (z) given by (1.1) is said to be in the classBΣ (β, λ) if the following conditions are satisfied:
f ∈ Σ and Re

(1− λ) f (z)
z
+ λf ′(z)

> β (0 ≤ β < 1, λ ≥ 1, z ∈ U) (3.1)
and
Re

(1− λ)g(w)
w
+ λg ′(w)

> β (0 ≤ β < 1, λ ≥ 1, w ∈ U), (3.2)
where the function g is defined by (2.3).
We note that for λ = 1, the classBΣ (β, λ) reduces to the classHΣ (β) introduced and studied by Srivastava et al. [5].
Theorem 3.2. Let f (z) given by (1.1) be in the classBΣ (β, λ), 0 ≤ β < 1 and λ ≥ 1. Then
|a2| ≤

2(1− β)
2λ+ 1 (3.3)
and
|a3| ≤ 4(1− β)
2
(λ+ 1)2 +
2(1− β)
2λ+ 1 . (3.4)
Proof. It follows from (3.1) and (3.2) that there exist p and q ∈ P such that
(1− λ) f (z)
z
+ λf ′(z) = β + (1− β)p(z) (3.5)
and
(1− λ)g(w)
w
+ λg ′(w) = β + (1− β)q(w) (3.6)
where p(z) and q(w) have the forms (2.8) and (2.9), respectively. Equating coefficients in (3.5) and (3.6) yields
(λ+ 1)a2 = (1− β)p1, (3.7)
(2λ+ 1)a3 = (1− β)p2, (3.8)
−(λ+ 1)a2 = (1− β)q1 (3.9)
and
(2λ+ 1)(2a22 − a3) = (1− β)q2. (3.10)
From (3.7) and (3.9), we get
p1 = −q1 (3.11)
and
2(λ+ 1)2a22 = (1− β)2(p21 + q21). (3.12)
Also, from (3.8) and (3.10), we find that
2(2λ+ 1)a22 = (1− β)(p2 + q2).
B.A. Frasin, M.K. Aouf / Applied Mathematics Letters 24 (2011) 1569–1573 1573
Thus, we have
|a22| ≤
(1− β)
2(2λ+ 1) (|p2| + |q2|) =
2(1− β)
2λ+ 1
which is the bound on |a2| as given in (3.3).
Next, in order to find the bound on |a3|, by subtracting (3.10) from (3.8), we get
2(2λ+ 1)a3 − 2(2λ+ 1)a22 = (1− β)(p2 − q2)
or, equivalently,
a3 = a22 +
(1− β)(p2 − q2)
2(2λ+ 1) .
Upon substituting the value of a22 from (3.12), we obtain
a3 = (1− β)
2(p21 + q21)
2(λ+ 1)2 +
(1− β)(p2 − q2)
2(2λ+ 1) .
Applying Lemma 1.1 for the coefficients p1, p2, q1 and q2, we readily get
|a3| ≤ 4(1− β)
2
(λ+ 1)2 +
2(1− β)
2λ+ 1
which is the bound on |a3| as asserted in (3.4). 
Putting λ = 1 in Theorem 3.2, we have the following corollary.
Corollary 3.3 ([5]). Let f (z) given by (1.1) be in the classHΣ (β), (0 ≤ β < 1). Then
|a2| ≤

2(1− β)
3
(3.13)
and
|a3| ≤ (1− β)(5− 3β)3 . (3.14)
Acknowledgements
The authors would like to thank the referees for their helpful comments and suggestions.
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New subclasses of bi-univalent functions 

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New subclasses of bi-univalent functions

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