coefficient estimates for certain classes of bi univalent functions

Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2013, Article ID 190560, 4 pages
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Research Article
Coefficient Estimates for Certain Classes of
Bi-Univalent Functions
Jay M. Jahangiri1 and Samaneh G. Hamidi2
1
2

Department of Mathematical Sciences, Kent State University, Burton, OH 44021-9500, USA
Institute of Mathematical Sciences, Faculty of Science, University of Malaya, 50603 Kuala Lumpur, Malaysia

Correspondence should be addressed to Jay M. Jahangiri;
Received 29 April 2013; Revised 27 July 2013; Accepted 31 July 2013
Academic Editor: Heinrich Begehr
Copyright © 2013 J. M. Jahangiri and S. G. Hamidi. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
A function analytic in the open unit disk D is said to be bi-univalent in D if both the function and its inverse map are univalent
there. The bi-univalency condition imposed on the functions analytic in D makes the behavior of their coefficients unpredictable.
Not much is known about the behavior of the higher order coefficients of classes of bi-univalent functions. We use Faber polynomial
expansions of bi-univalent functions to obtain estimates for their general coefficients subject to certain gap series as well as providing
bounds for early coefficients of such functions.

1. Introduction
Let A denote the class of functions 𝑓 which are analytic in
the open unit disk D := {𝑧 ∈ C : |𝑧| < 1} and normalized by
∞

𝑓 (𝑧) = 𝑧 + ∑ 𝑎𝑛 𝑧𝑛 .

(1)

𝑛=2

Let S denote the class of functions 𝑓 ∈ A that are
univalent in D and let P be the class of functions 𝑝(𝑧) =
𝑛
1 + ∑∞
𝑛=1 𝑝𝑛 𝑧 that are analytic in D and satisfy the condition
Re(𝑝(𝑧)) > 0 in D. By the Caratheodory lemma (e.g., see [1])
we have |𝑝𝑛 | ≤ 2.
For 0 ≤ 𝛼 < 1 and 𝜆 ≥ 1 we let D(𝛼; 𝜆) denote the family
of analytic functions 𝑓 ∈ A so that
Re ((1 − 𝜆)

𝑓 (𝑧)
+ 𝜆𝑓󸀠 (𝑧)) > 𝛼,
𝑧

𝑧 ∈ D.

(2)

We note that D(0; 1) is the class of bounded boundary
turning functions and also that D(𝛼; 𝜆) ⊂ D(𝛽; 𝜆) if 0 ≤ 𝛽 <
𝛼. For 𝑓 ∈ A, the class D(𝛼; 𝜆) ⊂ S and was first defined and
investigated by Ding et al. [2].
It is well known that every function 𝑓 ∈ S has an inverse

𝑓−1 satisfying 𝑓−1 (𝑓(𝑧)) = 𝑧 for 𝑧 ∈ D and 𝑓(𝑓−1 (𝑤)) = 𝑤

for |𝑤| < 1/4, according to Kobe One Quarter Theorem (e.g.,
see [1]).
A function 𝑓 ∈ A is said to be bi-univalent in D if
both 𝑓 ∈ S and 𝑔 = 𝑓−1 ∈ S. Finding bounds for the
coefficients of classes of bi-univalent functions dates back
to 1967 (see Lewin [3]). But the interest on the bounds for
the coefficients of classes of bi-univalent functions picked
up by the publications of Brannan and Taha [4], Srivastava
et al. [5], Frasin and Aouf [6], Ali et al. [7], and Hamidi et
al. [8]. The bi-univalency condition imposed on the functions
𝑓 ∈ A makes the behavior of their coefficients unpredictable.
Not much is known about the behavior of the higher order
coefficients of classes of bi-univalent functions, as Ali et al.
[7] also remarked that finding the bounds for |𝑎𝑛 | when 𝑛 ≥ 4
is an open problem. Here in this paper we let 𝑓 ∈ D(𝛼; 𝜆) and
𝑔 = 𝑓−1 ∈ D(𝛼; 𝜆) and use the Faber polynomial coefficient
expansions to provide bounds for the general coefficients |𝑎𝑛 |
of such functions with a given gap series. We also obtain
estimates for the first two coefficients |𝑎2 | and |𝑎3 | of these
functions as well as providing an estimate for their coefficient
body (𝑎2 , 𝑎3 ). The bounds provided in this paper prove to be
better than those estimates provided by Srivastava et al. [5]
and Frasin and Aouf [6].


2

International Journal of Mathematics and Mathematical Sciences

2. Main Results
Using the Faber polynomial expansion of functions 𝑓 ∈ A of
the form (1), the coefficients of its inverse map 𝑔 = 𝑓−1 may
be expressed as, [9],
∞
1 −𝑛
𝑔 (𝑤) = 𝑓−1 (𝑤) = 𝑤 + ∑ 𝐾𝑛−1
(𝑎2 , 𝑎3 , . . .) 𝑤𝑛 ,
𝑛
𝑛=2

(3)

2 (1 − 𝛼)
󵄨󵄨 󵄨󵄨
;
󵄨󵄨𝑎𝑛 󵄨󵄨 ≤
1 + (𝑛 − 1) 𝜆

𝑓 (𝑧)
+ 𝜆𝑓󸀠 (𝑧)
𝑧

(10)

= 1 + ∑ (1 + (𝑛 − 1) 𝜆) 𝑎𝑛 𝑧𝑛−1 ,
𝑛=2

(−𝑛)!

+
𝑎𝑛−3 𝑎
(2 (−𝑛 + 1))! (𝑛 − 3)! 2 3

and for its inverse map, 𝑔 = 𝑓−1 , we have

(−𝑛)!
𝑎𝑛−4 𝑎
(−2𝑛 + 3)! (𝑛 − 4)! 2 4

(1 − 𝜆)

(−𝑛)!
+
𝑎𝑛−5 [𝑎5 + (−𝑛 + 2) 𝑎32 ]
(2 (−𝑛 + 2))! (𝑛 − 5)! 2

(4)

𝑔 (𝑤)
+ 𝜆𝑔󸀠 (𝑤)
𝑤
∞

= 1 + ∑ (1 + (𝑛 − 1) 𝜆) 𝑏𝑛 𝑤𝑛−1
𝑛=2

(11)

∞

(−𝑛)!
+
𝑎𝑛−6 [𝑎6 + (−2𝑛 + 5) 𝑎3 𝑎4 ]
(−2𝑛 + 5)! (𝑛 − 6)! 2
+

(9)

∞

(−𝑛)!
𝑎𝑛−1
(−2𝑛 + 1)! (𝑛 − 1)! 2

+

𝑛 ≥ 4.

Proof. For analytic functions 𝑓 of the form (1) we have
(1 − 𝜆)

where
−𝑛
=
𝐾𝑛−1

Theorem 1. For 0 ≤ 𝛼 < 1 and 𝜆 ≥ 1 let 𝑓 ∈ D(𝛼; 𝜆) and
𝑔 ∈ D(𝛼; 𝜆). If 𝑎𝑘 = 0; 2 ≤ 𝑘 ≤ 𝑛 − 1, then

= 1 + ∑ (1 + (𝑛 − 1) 𝜆)
𝑛=2

𝑛−𝑗
∑ 𝑎2 𝑉𝑗 ,
𝑗≥7

1 −𝑛
× 𝐾𝑛−1
(𝑎2 , 𝑎3 , . . . , 𝑎𝑛 ) 𝑤𝑛−1 .
𝑛

such that 𝑉𝑗 with 7 ≤ 𝑗 ≤ 𝑛 is a homogeneous polynomial
in the variables 𝑎2 , 𝑎3 , . . . , 𝑎𝑛 [10]. In particular, the first three
−𝑛
are
terms of 𝐾𝑛−1
1 −2
𝐾 = −𝑎2 ,
2 1

(1 − 𝜆)

1 −3
𝐾 = 2𝑎22 − 𝑎3 ,
3 2

(5)

In general, for any 𝑝 ∈ N, an expansion of 𝐾𝑛𝑝 is as, [9, page

183],
𝑝 (𝑝 − 1) 2
𝑝!
𝐷3
𝐷𝑛 +
𝐾𝑛𝑝 = 𝑝𝑎𝑛 +
2
(𝑝 − 3)!3! 𝑛
𝑝!
𝐷𝑛 ,
(𝑝 − 𝑛)!𝑛! 𝑛

∞

𝜇

Evidently, 𝐷𝑛𝑛 (𝑎1 , 𝑎2 , . . . , 𝑎𝑠+𝑚 ) = 𝑎1𝑛 , [13].

𝑛

(𝑐1 , 𝑐2 , . . . , 𝑐𝑛 ) 𝑧 ,

𝑔 (𝑤)
+ 𝜆𝑔󸀠 (𝑤)
𝑤
∞

(13)

𝑛=1

(6)

Comparing the corresponding coefficients of (10) and (12)
yields
(14)

and similarly, from (11) and (13) we obtain

𝜇

(7)

while 𝑎1 = 1, and the sum is taken over all nonnegative
integers 𝜇1 , . . . , 𝜇𝑛 satisfying

𝜇1 + 2𝜇2 + ⋅ ⋅ ⋅ + 𝑛𝜇𝑛 = 𝑛.

𝛼) ∑ 𝐾𝑛1
𝑛=1

1
(𝑐1 , 𝑐2 , . . . , 𝑐𝑛−1 ) ,
(1 + 𝜆 (𝑛 − 1)) 𝑎𝑛 = (1 − 𝛼) 𝐾𝑛−1

𝑚!(𝑎1 ) 1 ⋅ ⋅ ⋅ (𝑎𝑛 ) 𝑛
,
𝜇1 ! ⋅ ⋅ ⋅ 𝜇𝑛 !
𝑚=1

𝜇1 + 𝜇2 + ⋅ ⋅ ⋅ + 𝜇𝑛 = 𝑚,

(1 − 𝜆)

(12)

∞

= 1 + (1 − 𝛼) ∑ 𝐾𝑛1 (𝑑1 , 𝑑2 , . . . , 𝑑𝑛 ) 𝑤𝑛 .

where 𝐷𝑛𝑝 = 𝐷𝑛𝑝 (𝑎2 , 𝑎3 , . . .) and by [11] or [12],
𝐷𝑛𝑚 (𝑎1 , 𝑎2 , . . . , 𝑎𝑛 ) = ∑

𝑓 (𝑧)
+ 𝜆𝑓󸀠 (𝑧)
𝑧

= 1 + (1 −

1 −4
𝐾 = − (5𝑎23 − 5𝑎2 𝑎3 + 𝑎4 ) .
4 3

+ ⋅⋅⋅ +

On the other hand, since 𝑓 ∈ D(𝛼; 𝜆) and 𝑔 = 𝑓−1 ∈
D(𝛼; 𝜆), by definition, there exist two positive real part
−𝑛
−𝑛
and 𝑞(𝑤) = 1 + ∑∞

functions 𝑝(𝑧) = 1 + ∑∞
𝑛=1 𝑐𝑛 𝑧
𝑛=1 𝑑𝑛 𝑤
where Re 𝑝(𝑧) > 0 and Re 𝑞(𝑤) > 0 in D so that

(8)

1
−𝑛
(𝑏0 , 𝑏1 , . . . , 𝑏𝑛 )
(1 + (𝑛 − 1) 𝜆) 𝐾𝑛−1
𝑛
= (1 −

1
𝛼) 𝐾𝑛−1

(15)

(𝑑1 , 𝑑2 , . . . , 𝑑𝑛−1 ) .

Note that for 𝑎𝑘 = 0; 2 ≤ 𝑘 ≤ 𝑛 − 1 we have 𝑏𝑛 = −𝑎𝑛 and so
(1 + (𝑛 − 1) 𝜆) 𝑎𝑛 = (1 − 𝛼) 𝑐𝑛−1 ,
− (1 + (𝑛 − 1) 𝜆) 𝑎𝑛 = (1 − 𝛼) 𝑑𝑛−1 .

(16)


International Journal of Mathematics and Mathematical Sciences
Now taking the absolute values of either of the above two

equations and applying the Caratheodory lemma, we obtain
󵄨 󵄨
󵄨
󵄨
(1 − 𝛼) 󵄨󵄨󵄨𝑐𝑛−1 󵄨󵄨󵄨
(1 − 𝛼) 󵄨󵄨󵄨𝑑𝑛−1 󵄨󵄨󵄨
2 (1 − 𝛼)
󵄨󵄨 󵄨󵄨
=
≤
.
󵄨󵄨𝑎𝑛 󵄨󵄨 ≤
|1 + (𝑛 − 1) 𝜆| |1 + (𝑛 − 1) 𝜆| 1 + (𝑛 − 1) 𝜆
(17)

3
Dividing (20) by (1 + 2𝜆), taking the absolute values of both
sides, and applying the Caratheodory lemma yield
󵄨 󵄨
󵄨󵄨 󵄨󵄨 (1 − 𝛼) 󵄨󵄨󵄨𝑐2 󵄨󵄨󵄨 2 (1 − 𝛼)
≤
.
󵄨󵄨𝑎3 󵄨󵄨 =
1 + 2𝜆
1 + 2𝜆

Dividing (22) by (1 + 2𝜆), taking the absolute values of both
sides, and applying the Caratheodory lemma, we obtain
󵄨 2 (1 − 𝛼)
󵄨󵄨

󵄨󵄨𝑎3 − 2𝑎22 󵄨󵄨󵄨 ≤
.
󵄨
󵄨
1 + 2𝜆

Theorem 2. For 0 ≤ 𝛼 < 1 and 𝜆 ≥ 1 let 𝑓 ∈ D(𝛼; 𝜆) and
𝑔 ∈ D(𝛼; 𝜆). Then one has the following

(i)

2 (1 − 𝛼)
1 + 2𝜆 − 𝜆2
{
√
;
,
0
≤
𝛼
<
{
󵄨󵄨 󵄨󵄨 { 1 + 2𝜆
2 (1 + 2𝜆)
󵄨󵄨𝑎2 󵄨󵄨 ≤ {
2
1 + 2𝜆 − 𝜆
{
{ 2 (1 − 𝛼) ,
≤ 𝛼 < 1.

2 (1 + 2𝜆)
{ 1+𝜆

(ii)

󵄨󵄨 󵄨󵄨 2 (1 − 𝛼)
.
󵄨󵄨𝑎3 󵄨󵄨 ≤
1 + 2𝜆

(iii)

󵄨 2 (1 − 𝛼)
󵄨󵄨
󵄨󵄨𝑎3 − 2𝑎22 󵄨󵄨󵄨 ≤
.
󵄨
󵄨
1 + 2𝜆

(28)

(29)

Corollary 3. For 0 ≤ 𝛼 < 1 let 𝑓 ∈ D(𝛼; 1) and 𝑔 ∈ D(𝛼; 1).
Then one has the following
(18)
(i)

Proof. Replacing 𝑛 by 2 and 3 in (14) and (15), respectively, we

deduce

(ii)

{ 2 (1 − 𝛼) ,
󵄨󵄨 󵄨󵄨 {√
3
󵄨󵄨𝑎2 󵄨󵄨 ≤ {
{
1 − 𝛼,
{

1
0≤𝛼< ;
3
1
≤ 𝛼 < 1.
3

(30)

󵄨󵄨 󵄨󵄨 2 (1 − 𝛼)
.
󵄨󵄨𝑎3 󵄨󵄨 ≤
3

(1 + 𝜆) 𝑎2 = (1 − 𝛼) 𝑐1 ,

(19)

(1 + 2𝜆) 𝑎3 = (1 − 𝛼) 𝑐2 ,

(20)

Remark 4. The above two estimates for |𝑎2 | and |𝑎3 | show that
the bounds given in Theorem 2 are better than those given
by Srivastava et al. ([5, page 1191, Theorem 2] and Frasin and
Aouf [6, page 1572, Theorem 3.2]).

− (1 + 𝜆) 𝑎2 = (1 − 𝛼) 𝑑1 ,

(21)

References

(22)

[1] P. L. Duren, Univalent Functions, vol. 259 of Grundlehren der
Mathematischen Wissenschaften, Springer, New York, NY, USA,
1983.
[2] S. S. Ding, Y. Ling, and G. J. Bao, “Some properties of a class
of analytic functions,” Journal of Mathematical Analysis and
Applications, vol. 195, no. 1, pp. 71–81, 1995.
[3] M. Lewin, “On a coefficient problem for bi-univalent functions,”
Proceedings of the American Mathematical Society, vol. 18, pp.
63–68, 1967.
[4] D. A. Brannan and T. S. Taha, “On some classes of bi-univalent
functions,” Studia Universitatis Babes¸-Bolyai. Mathematica, vol.
31, no. 2, pp. 70–77, 1986.
[5] H. M. Srivastava, A. K. Mishra, and P. Gochhayat, “Certain

subclasses of analytic and bi-univalent functions,” Applied
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[6] B. A. Frasin and M. K. Aouf, “New subclasses of bi-univalent
functions,” Applied Mathematics Letters, vol. 24, no. 9, pp. 1569–
1573, 2011.
[7] R. M. Ali, S. K. Lee, V. Ravichandran, and S. Supramaniam,
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[9] H. Airault and A. Bouali, “Differential calculus on the Faber
polynomials,” Bulletin des Sciences Math´ematiques, vol. 130, no.
3, pp. 179–222, 2006.

(1 +

2𝜆) (2𝑎22

− 𝑎3 ) = (1 − 𝛼) 𝑑2 .

Dividing (19) or (21) by (1 + 𝜆), taking their absolute
values, and applying the Caratheodory lemma, we obtain
󵄨 󵄨
󵄨 󵄨
󵄨󵄨 󵄨󵄨 (1 − 𝛼) 󵄨󵄨󵄨𝑐1 󵄨󵄨󵄨 (1 − 𝛼) 󵄨󵄨󵄨𝑑1 󵄨󵄨󵄨 2 (1 − 𝛼)
=
≤
.
󵄨󵄨𝑎2 󵄨󵄨 ≤

1+𝜆
1+𝜆
1+𝜆

(23)

Adding (20) to (22) implies
2 (1 + 2𝜆) 𝑎22 = (1 − 𝛼) (𝑐2 + 𝑑2 )

(24)

or
𝑎22 =

(1 − 𝛼) (𝑐2 + 𝑑2 )
.
2 (1 + 2𝜆)

(25)

An application of Caratheodory lemma followed by
taking the square roots yields
󵄨 󵄨 󵄨 󵄨
󵄨󵄨 󵄨󵄨 √2 (1 − 𝛼) (󵄨󵄨󵄨𝑐2 󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑑2 󵄨󵄨󵄨) √2 2 (1 − 𝛼)
≤
.
󵄨󵄨𝑎2 󵄨󵄨 ≤
2 (1 + 2𝜆)
1 + 2𝜆

(26)

Now the bounds given in Theorem 2 (i) for |𝑎2 | follow
upon noting that if (1 + 2𝜆 − 𝜆2 )/2(1 + 2𝜆) ≤ 𝛼 < 1, then
2 (1 − 𝛼) √2 2 (1 − 𝛼)
≤
.
1+𝜆
1 + 2𝜆

(27)


4
[10] H. Airault and J. Ren, “An algebra of differential operators and
generating functions on the set of univalent functions,” Bulletin
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[11] P. G. Todorov, “On the Faber polynomials of the univalent
functions of class Σ,” Journal of Mathematical Analysis and
Applications, vol. 162, no. 1, pp. 268–276, 1991.
[12] H. Airault, “Symmetric sums associated to the factorization of
Grunsky coefficients,” in Conference, Groups and Symmetries,
Montreal, Canada, April 2007.
[13] H. Airault, “Remarks on Faber polynomials,” International
Mathematical Forum. Journal for Theory and Applications, vol.
3, no. 9-12, pp. 449–456, 2008.

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