Coefficient Estimates for a Certain General Subclass of Analytic and Bi-Univalent Functions ()
1. Introduction
Let 
denote the class of functions of the form
(1.1)
which are analytic in the open unit disc 
Further, by 
we shall denote the class of all functions in 
which are univalent in 
Some of the important and well-investigated subclasses of the univalent function class 
include (for example) the class 
of starlike functions of order 
in 
and the class 
of strongly starlike functions of order 
in 
It is well known that every function 
has an inverse 
defined by
and
where
(1.2)
A function 
is said to be bi-univalent in 
if both 
and 
are univalent in 
We denote by 
the class of all bi-univalent functions in 
For a brief history and interesting examples of functions in the class 
see [2] and the references therein.
In fact, the study of the coefficient problems involving bi-univalent functions was revived recently by Srivastava et al. [2] . Various subclasses of the bi-univalent function class 
were introduced and non-sharp estimates on the first two Taylor-Maclaurin coefficients 
and 
of functions in these subclasses were found in several recent investigations (see, for example, [3] -[13] ). The aforecited all these papers on the subject were motivated by the pioneering work of Srivastava et al. [2] . But the coefficient problem for each of the following Taylor-Maclaurin coefficients 
(
) is still an open problem.
Motivated by the aforecited works (especially [1] ), we introduce the following subclass 
of the analytic function class 
Definition 1 Let 
and the functions 
be so constrained that
and 
We say that 
if the following conditions are satisfied: 
(1.3)
and
(1.4)
where the function 
is the extension of 
to 
We note that, for the different choices of the functions 
and
, we get interesting known and new subclasses of the analytic function class 
For example, if we set
in the class 
then we have 
Also, 
if the following conditions are satisfied:
and
where 
is the extension of 
to 
Similarly, if we let
in the class 
then we get 
Further, we say that 
if the following conditions are satisfied:
and
where 
is the extension of 
to 
The classes 
and 
were introduced and studied by Murugusundaramoorthy et al. [12] , Definition 1.1 and Definition 1.2]. The classes 
and 
are strongly bi-starlike functions of order 
and bi-starlike functions of order 
respectively. The classes 
and 
were introduced and studied by Brannan and Taha [14] , Definition 1.1 and Definition 1.2]. In addition, we note that, 
was introduced and studied by Bulut [4] , Definition 3].
Motivated and stimulated by Bulut [4] and Xu et al. [1] (also [10] ), in this paper, we introduce a new subclass 
and obtain the estimates on the coefficients 
and 
for functions in aforementioned class, employing the techniques used earlier by Xu et al. [1] .
2. A Set of General Coefficient Estimates
In this section we state and prove our general results involving the bi-univalent function class 
given by Definition 1.
Theorem 1 Let 
be of the form (1.1). If 
then
(1.5)
and
(1.6)
Proof 1 Since 
From (1.3) and (1.4), we have,
and
where
and
satisfy the conditions of Definition 1. Now, upon equating the coefficients of 
with those of 
and the coefficients of 
with those of
, we get
(1.7)
(1.8)
(1.9)
and
(1.10)
From (1.7) and (1.9), we get
(1.11)
and
(1.12)
From (1.8) and (1.10), we obtain
(1.13)
Therefore, we find from (1.12) and (1.13) that
(1.14)
and
(1.15)
Since 
and 
we immediately have
and
respectively. So we get the desired estimate on 
as asserted in (1.5).
Next, in order to find the bound on
, by subtracting (1.10) from (1.8), we get
(1.16)
Upon substituting the values of 
from (1.14) and (1.15) into (1.16), we have
and
respectively. Since 
and 
we readily get
and
This completes the proof of Theorem 1.
If we choose
in Theorem 1, we have the following corollary.
Corollary 1 Let 
be of the form (1.1) and in the class 
Then
and
If we set
in Theorem 1, we readily have the following corollary.
Corollary 2 Let 
be of the form (1.1) and in the class 
Then
and
Remark 1 The estimates on the coefficients 
and 
of Corollaries 1 and 2 are improvement of the estimates obtained in [10] , Theorems 4 and 5]. Taking 
in Corollaries 1 and 2, the estimates on the coefficients 
and 
are improvement of the estimates in [14] , Theorems 2.1 and 4.1]. When 
the results discussed in this article reduce to results in [4] . Similarly, various other interesting corollaries and consequences of our main result can be derived by choosing different 
and
.
Acknowledgements
The authors would like to record their sincere thanks to the referees for their valuable suggestions.
Funding
The work is supported by UGC, under the grant F.MRP-3977/11 (MRP/UGC-SERO) of the first author.