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 ![](https://www.scirp.org/html/htmlimages\1-7402071x\54d64d54-76f6-4d4c-8a6f-b6c93eff509f.png)
It is well known that every function
has an inverse
defined by
![](https://www.scirp.org/html/htmlimages\1-7402071x\9d9a3db1-f5f3-446d-a250-98c10c8bb3bd.png)
and
![](https://www.scirp.org/html/htmlimages\1-7402071x\1904ff77-03e9-4d09-9d6d-08dfc156f3d2.png)
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
(![](https://www.scirp.org/html/htmlimages\1-7402071x\c3db5f4e-cd3c-44e0-b6c9-d0651f13e1a9.png)
) is still an open problem.
Motivated by the aforecited works (especially [1] ), we introduce the following subclass
of the analytic function class ![](https://www.scirp.org/html/htmlimages\1-7402071x\c1fade43-dd43-4310-9fe8-8d81f663802a.png)
Definition 1 Let
and the functions
be so constrained that
![](https://www.scirp.org/html/htmlimages\1-7402071x\a2070a97-b88f-4c4a-b300-0b085022da7d.png)
and
We say that
if the following conditions are satisfied: ![](https://www.scirp.org/html/htmlimages\1-7402071x\487428e8-de47-4c8a-881e-83820d461436.png)
(1.3)
and
(1.4)
where the function
is the extension of
to ![](https://www.scirp.org/html/htmlimages\1-7402071x\faf6db01-f229-4956-baa0-b1a9aea560a5.png)
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
![](https://www.scirp.org/html/htmlimages\1-7402071x\98a0951c-9d41-4fbe-b3c5-e8bfade90caf.png)
in the class
then we have
Also,
if the following conditions are satisfied:
![](https://www.scirp.org/html/htmlimages\1-7402071x\61c3c658-70f9-474c-9ba8-a1871d808aec.png)
and
![](https://www.scirp.org/html/htmlimages\1-7402071x\495be683-ef20-4d36-9bed-5649c3d6770d.png)
where
is the extension of
to ![](https://www.scirp.org/html/htmlimages\1-7402071x\21226433-93f8-40a5-86d4-21bd8ce41fef.png)
Similarly, if we let
![](https://www.scirp.org/html/htmlimages\1-7402071x\c7757f45-1556-4046-913f-1cba99e706d8.png)
in the class
then we get
Further, we say that
if the following conditions are satisfied:
![](https://www.scirp.org/html/htmlimages\1-7402071x\9b7aefcb-fed2-47b0-ac6e-211badb6dc15.png)
and
![](https://www.scirp.org/html/htmlimages\1-7402071x\6b8770e8-39e5-4c13-98ad-c05fcf3b42b2.png)
where
is the extension of
to ![](https://www.scirp.org/html/htmlimages\1-7402071x\0c5ef752-ee00-4545-a898-485038c37af2.png)
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,
![](https://www.scirp.org/html/htmlimages\1-7402071x\a6b95f6a-b34c-45c2-9c3b-6964571f43c0.png)
and
![](https://www.scirp.org/html/htmlimages\1-7402071x\cb9ce39b-ff53-482c-9d12-7ef75f236569.png)
where
![](https://www.scirp.org/html/htmlimages\1-7402071x\4c67229a-6a83-4969-b131-4293ae49df4c.png)
and
![](https://www.scirp.org/html/htmlimages\1-7402071x\7dfed7e8-9461-4737-9be9-73b4725edd47.png)
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
![](https://www.scirp.org/html/htmlimages\1-7402071x\9d618f3a-3f20-4ba0-9f7e-bfcac1ee8df1.png)
and
![](https://www.scirp.org/html/htmlimages\1-7402071x\f8dfa3a2-3966-48e7-b52c-7d389da583b4.png)
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
![](https://www.scirp.org/html/htmlimages\1-7402071x\11baeff9-0d0a-46c8-8c0e-c480b6b9116a.png)
and
![](https://www.scirp.org/html/htmlimages\1-7402071x\6521c5d8-b237-4faa-a487-9b3e487829d5.png)
respectively. Since
and
we readily get
![](https://www.scirp.org/html/htmlimages\1-7402071x\9ecb99d4-fef9-47c3-a080-c7a16c72d600.png)
and
![](https://www.scirp.org/html/htmlimages\1-7402071x\1f6588d0-bd3e-43ad-9470-488a0be3a1e5.png)
This completes the proof of Theorem 1.
If we choose
![](https://www.scirp.org/html/htmlimages\1-7402071x\7a22285d-6663-4f9f-8fde-391b983056bb.png)
in Theorem 1, we have the following corollary.
Corollary 1 Let
be of the form (1.1) and in the class
Then
![](https://www.scirp.org/html/htmlimages\1-7402071x\30e31095-9a5b-408c-9592-6bd51dcffee4.png)
and
![](https://www.scirp.org/html/htmlimages\1-7402071x\18663f2f-cc3a-48e7-927d-6811402ce96d.png)
If we set
![](https://www.scirp.org/html/htmlimages\1-7402071x\1bc67770-6074-44d1-8106-499468d53549.png)
in Theorem 1, we readily have the following corollary.
Corollary 2 Let
be of the form (1.1) and in the class
Then
![](https://www.scirp.org/html/htmlimages\1-7402071x\558619a8-428c-486b-9ee8-58c712c1c813.png)
and
![](https://www.scirp.org/html/htmlimages\1-7402071x\442c863e-7a25-46c1-af1c-e5d5535fe4e7.png)
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.