EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 10, No. 4, 2017, 638-644
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Coefficient estimates for the generalized subclass of
analytic and bi-univalent functions

Haigen Xiao1, Qinghua Xu2,∗

1 The Binjiang Campus of the High School Attached to Jiangxi Normal University, China
2 School of Science, Zhejiang University of Science and Technology, China

Abstract. In this paper, we introduce and investigate an interesting subclass Bh,pΣ (λ) of analytic

and bi-univalent functions in the open unit disk U. For functions belonging to the class Bh,pΣ (λ),
obtain estimates on the first two coefficients |a2| and |a3|. The results presented in this paper
generalize and improve some recent works of Frasin et al. [B.A.Frasin, M.K.Aouf, New subclasses
of bi-univalent functions, Appl. Math. Lett. 24:1569-1573, 2011] and Srivastava et al. [Qing-Hua
Xu, Ying-Chun Gui, H.M.Srivastava, coefficient estimates for a certain subclass of analytic and
bi-univalent functions, Appl. Math. Lett. 25: 990-994, 2012].

2010 Mathematics Subject Classifications: 30C45

Key Words and Phrases: Univalent functions, Bi-univalent functions, Coefficient bounds

1. Introduction and definitions

Let A denote the class of functions of the form

f(z) = z +
∞∑
n=2

anz
n, (1)

which are analytic in the open unit disk

U = {z ∈ C : |z| < 1}.

We denote by S the subclass of the analytic function class A consisting of all functions in
A which are also univalent in U.

It is well known that every function f ∈ S has an inverse f−1, defined by

f−1(f(z)) = z (z ∈ U)

∗Corresponding author.

Email addresses: haigen2008@sina.com (H.G Xiao), xuqh@mail.ustc.edu.cn (Q.H.Xu)

http://www.ejpam.com 638 c© 2017 EJPAM All rights reserved.



H.G Xiao, Q.H.Xu / Eur. J. Pure Appl. Math, 10 (4) (2017), 638-644 639

and

f−1(f(w)) = w (|w| < r0(f); r0(f) ≥ 1

4
).

In fact, the inverse function f−1 is given by

f−1(w) = w − a2w
2 + (2a2

2 − a3)w3 − (5a3
2 − 5a2a3 + a4)w4 + · · · .

A function f ∈ A is said to be bi-univalent in U if both f(z) and f−1(z) are univalent
in U. Let Σ denote the class of bi-univalent functions in U given (1).

The coefficient bounds for the class Σ have been studied by Lewin [1], Brannan and
Clunie [2], Netanyahu [3]. The coefficient estimate problem for |an| (n ∈ N \ {1, 2}; N :=
{1, 2, 3, · · · }) is presumably still an open problem. In [4](see [5, 6, 7]), certain subclasses
of the bi-univalent function class Σ were introduced, and non-sharp estimates on the first
two coefficients |a2| and |a3| were found.

Recently, Frasin et al.[8] introduced the following subclasses of the bi-univalent function
class Σ and obtained non-sharp estimates on the first two coefficients |a2| and |a3|.
Definition 1(see [8]). A function f(z) given by (1) is said to be in the class BΣ(α, λ) if the
following conditions are satisfied:

f ∈ Σ and

∣∣∣∣arg

(
(1− λ)

f(z)

z
+ λf ′(z)

)∣∣∣∣ ≤ απ

2
(z ∈ U; 0 < α ≤ 1; λ ≥ 1)

and ∣∣∣∣arg

(
(1− λ)

g(w)

w
+ λg′(w)

)∣∣∣∣ ≤ απ

2
(w ∈ U; 0 < α ≤ 1; λ ≥ 1),

g(w) = w − a2w
2 + (2a2

2 − a3)w3 − (5a3
2 − 5a2a3 + a4)w4 + · · · . (2)

Theorem 1(see [8]). Let f(z) given by (1) be in the function class BΣ(α, λ). Then

|a2| ≤
2α√

(λ+ 1)2 + α(1 + 2λ− λ2)

and

|a3| ≤
4α2

(λ+ 1)2
+

2α

2λ+ 1
.

Definition 2(see [8]). A function f(z) given by (1) is said to be in the class BΣ(β, λ) if the
following conditions are satisfied:

f ∈ Σ and <
(

(1− λ)
f(z)

z
+ λf ′(z)

)
> β (z ∈ U; 0 ≤ β < 1;λ ≥ 1)

and

<
(

(1− λ)
g(w)

z
+ λg′(w)

)
> β (w ∈ U; 0 ≤ β < 1; λ ≥ 1),



H.G Xiao, Q.H.Xu / Eur. J. Pure Appl. Math, 10 (4) (2017), 638-644 640

where the function g is defined by (2).
Theorem 2(see [8]). Let f(z) given by (1) be in the function class BΣ(β, λ) . Then

|a2| ≤
√

2(1− β)

2λ+ 1

and

|a3| ≤
4(1− β)2

(λ+ 1)2
+

2(1− β)

2λ+ 1
.

Here, in our present sequel to some of the aforecited works (especially [7, 8]), we
introduce the following subclass of analytic functions.
Definition 3. Let h, p : U→ C be functions such that

min{<(h(z)), <(p(z))} > 0, (z ∈ U) and h(0) = p(0) = 1,

Also let f be an analytic function in U defined by (1). We say that f ∈ Bh,pΣ (λ) if the
following conditions are satisfied:

f ∈ Σ and (1− λ)
f(z)

z
+ λf ′(z) ∈ h(U) (z ∈ U; λ ≥ 1) (3)

and

(1− λ)
g(w)

w
+ λg′(w) ∈ p(U) (w ∈ U; λ ≥ 1), (4)

where the function g is given by (2).

We note that for λ = 1, the class Bh,pΣ (λ) reduces to the class Hh,pΣ introduced and
studied by Xu et al.[7].
Remark 1. There are many choices of the functions h and p which would provide
interesting subclasses of analytic functions. For example, if we let

h(z) = p(z) =

(
1 + z

1− z

)α
(z ∈ U; 0 < α ≤ 1)

or

h(z) = p(z) =
1 + (1− 2β)z

1− z
(z ∈ U; 0 ≤ β < 1),

it is easy to verify that h(z) and p(z) satisfy the hypotheses of Definition 3. If f ∈ Bh,pΣ (λ),
then

f ∈ Σ and

∣∣∣∣arg

(
(1− λ)

f(z)

z
+ λf ′(z)

)∣∣∣∣ ≤ απ

2
(z ∈ U; 0 < α ≤ 1; λ ≥ 1)



H.G Xiao, Q.H.Xu / Eur. J. Pure Appl. Math, 10 (4) (2017), 638-644 641

and ∣∣∣∣arg

(
(1− λ)

g(w)

w
+ λg′(w)

)∣∣∣∣ ≤ απ

2
(w ∈ U; 0 < α ≤ 1; λ ≥ 1),

or

f ∈ Σ and <
(

(1− λ)
f(z)

z
+ λf ′(z)

)
> β (z ∈ U; 0 ≤ β < 1)

and

<
(

(1− λ)
g(w)

w
+ λg′(w)

)
> β (w ∈ U; 0 ≤ β < 1; 0 ≤ β < 1),

where the function g is given by (2).
This means that

f ∈ BΣ(α, λ) or f ∈ BΣ(β, λ).

In this paper, stimulated by [7, 8], we introduce the following subclass of the bi-
univalent function class Σ and obtain estimates on the first two coefficients |a2| and |a3|.
Our results would generalize and improve the related works of Frasin et al.[8] and Xu et
al.[7] .

2. Main results and their proofs

In this section, we state and prove our results involving the bi-univalent function class
Bh,pΣ (λ) given by Definition 3.

Theorem 3. Let f(z) given by (1) be in the function class f ∈ Bh,pΣ (λ). Then

|a2| ≤

√
|h′′(0)|+ |p′′(0)|

4(1 + 2λ)
and |a3| ≤

|h′′(0)|
2(1 + 2λ)

. (5)

Proof. It follows from (3) and (4) that

(1− λ)
f(z)

z
+ λf ′(z) = h(z) (z ∈ U) (6)

and

(1− λ)
g(w)

w
+ λg′(w) = p(w) (w ∈ U), (7)

where h and p satisfy the conditions of Definition 3, Furthermore, the functions h(z) and
p(w) have the following series expansions:

h(z) = 1 + h1z + h2z
2 + · · ·



H.G Xiao, Q.H.Xu / Eur. J. Pure Appl. Math, 10 (4) (2017), 638-644 642

and

p(w) = 1 + p1w + p2w
2 + · · · ,

respectively. Now, equating the coefficients in (6) and (7), we get

(1 + λ)a2 = h1, (8)

(1 + 2λ)a3 = h2, (9)

−(1 + λ)a2 = p1 (10)

and

(1 + 2λ)(2a2
2 − a3) = p2. (11)

From (8) and (10), we get

h1 = −p1 2(1 + λ)2a2
2 = h2

1 + p2
1. (12)

Also, from (9) and (11), we find that

2(1 + 2λ)a2
2 = h2 + p2, (13)

which gives us the desired estimate on |a2| as asserted in (5).
Next, in order to find the bound on |a3|, by subtracting (11) from (9), we get

2(1 + 2λ)a3 − 2(1 + 2λ)a2
2 = h2 − p2. (14)

Upon substituting the value of a2
2 from (13)into (14), it follows that

a3 =
h2

1 + 2λ
, (15)

as claimed. This completes the proof of Theorem 1.

3. Corollaries and consequences

In view of Remark 1, if we set

h(z) = p(z) =

(
1 + z

1− z

)α
(z ∈ U; 0 < α ≤ 1)

and

h(z) = p(z) =
1 + (1− 2β)z

1− z
(z ∈ U; 0 ≤ β < 1)



REFERENCES 643

in Theorem 3, respectively, we can readily deduce the following two corollaries, which we
merely state here without proof.
Corollary 1. Let f(z) given by (1) be in the bi-univalent function class BΣ(α, λ) . Then

|a2| ≤
√

2

2λ+ 1
α and |a3| ≤

2α2

2λ+ 1
. (16)

Remark 2. It is easy to prove that√
2

1 + 2λ
α ≤ 2α√

(λ+ 1)2 + α(1 + 2λ− λ2)
(0 < α ≤ 1; λ ≥ 1)

and

2α2

1 + 2λ
≤ 4α2

(λ+ 1)2
+

2α

2λ+ 1
(0 < α ≤ 1; λ ≥ 1),

which, in conjunction with Corollary 1, would obviously yield an improvement of Theorem
1.
Corollary 2. Let f(z) given by (1) be in the bi-univalent function class BΣ(β, λ). Then

|a2| ≤
√

2(1− β)

2λ+ 1
and |a3| ≤

2(1− β)

2λ+ 1
. (17)

Remark 3. It is obvious that

2(1− β)

2λ+ 1
≤ 4(1− β)2

(λ+ 1)2
+

2(1− β)

2λ+ 1
(0 ≤ β < 1; λ ≥ 1),

which, in conjunction with Corollary 2, would lead us to an improvement of Theorem 2.
Setting λ = 1 in Theorem 3, we get the following estimate, which was obtained by Xu

et al. [7].

Corollary 3 (see [7]). Let f(z) given by (1) be in the bi-univalent function class Hh,pΣ .
Then

|a2| ≤
√
|h′′(0)|+ |p′′(0)|

12
and |a3| ≤

|h′′(0)|
6

. (18)

References

[1] M.Lewin, On a coefficient problem for bi-univalent functions. Proc. Amer. Math. Soc.
18: 63-68, 1967.

[2] D.A. Brannan, J.G. Clunie (Eds.), Aspects of Contemporary Complex Analysis (Pro-
ceedings of the NATO Advanced Study Institute held at the University of Durham,
Durham; July 1-20, 1979), Academic Press, New York and London, 1980.



REFERENCES 644

[3] E. Netanyahu, The minimal distance of the image boundary from the origin and the
second coefficient of a univalent function in |z| < 1. Arch. Rational Mech. Anal. 32 :
100-112, 1969.

[4] D.A. Brannan, T.S. Taha, On some classes of bi-univalent functions, in: S.M. Mazhar,
A. Hamoui, N.S. Faour (Eds.), Mathematical Analysis and Its Applications, Kuwait;
February 18-21, 1985, in: KFAS Proceedings Series, vol. 3, Pergamon Press (Elsevier
Science Limited), Oxford, 1988, pp. 53-60; see also Studia Univ. Babeş-Bolyai Math.
31 (2): 70-77, 1986.

[5] T.S. Taha, Topics in Univalent Function Theory, Ph.D. Thesis, University of London,
1981.

[6] H.M. Srivastava, A.K. Mishra, P.Gochhayat, Certain subclasses of analytic and bi-
univalent functions. Appl. Math. Lett. 23: 1188-1192, 2010.

[7] Qing-Hua Xu, Ying-Chun Gui, H.M.Srivastava, Coefficient estimates for a certain
subclass of analytic and bi-univalent functions. Appl. Math. Lett. 25(6): 990-994,
2012.

[8] B.A.Frasin, M.K.Aouf, New subclasses of bi-univalent functions. Appl. Math. Lett.
24: 1569-1573, 2011.