Filomat 27:5 (2013), 831–842
DOI 10.2298/FIL1305831S
Published by Faculty of Sciences and Mathematics,
University of Ni
ˇ
s, Serbia
Available at: http://www.pmf.ni.ac.rs/filomat
Coefficient Estimates for a General Subclass of Analytic and
Bi-Univalent Functions
H. M. Srivastava
a
, Serap Bulut
b
, Murat C¸ a˘glar
c
, Nihat Ya˘gmur
d
a
Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada
b
Kocaeli University, Civil Aviation College, Arslanbey Campus, TR-41285
˙
Izmit-Kocaeli, Turkey
c
Department of Mathematics, Faculty of Science, Atat¨urk University, TR-25240 Yakutiye-Erzurum, Turkey
d
Department of Mathematics, Faculty of Science and Art, Erzincan University, TR-24000 Erzincan, Turkey
Abstract. In this paper, we introduce and investigate an interesting subclass N
h,p
Σ
λ, µ
of analytic and
bi-univalent functions in the open unit disk U. For functions belonging to the class N
h,p
Σ
λ, µ
, we obtain
estimates on the first two Taylor-Maclaurin coefficients
|
a
2
|
and
|
a
3
|
. The results presented in this paper
would generalize and improve some recent works of C¸ a
˘
glar et al. [3], Xu et al. [10], and other authors.
1. Introduction
Let R =
(
−∞, ∞
)
be the set of real numbers, C be the set of complex numbers and
N :=
{
1, 2, 3, . . .
}
= N
0
\
{
0
}
be the set of positive integers.
Let A denote the class of all functions of the form:
f (z) = z +
∞
n=2
a
n
z
n
, (1)
which are analytic in the open unit disk
U =
{
z : z ∈ C and
|
z
|
< 1
}
.
We also denote by Sthe class of all functions in the normalized analytic function class Awhich are univalent
in U.
It is well known that every function f ∈ S has an inverse f
−1
, which is defined by
f
−1
f
(
z
)
= z
(
z ∈ U
)
2010 Mathematics Subject Classification. Primary 30C45; Secondary 30C50
Keywords. Analytic functions; Univalent functions; Bi-univalent functions; Taylor-Maclaurin series expansion; Coefficient bounds
and coefficient estimates; Taylor-Maclaurin coefficients
Received: 26 September 2012; Accepted: 21 March 2013
Communicated by Miodrag Mateljevi
´
c
Email addresses: harimsri@math.uvic.ca (H. M. Srivastava), serap.bulut@kocaeli.edu.tr (Serap Bulut),
mcaglar25@gmail.com (Murat C¸ a
˘
glar), nhtyagmur@gmail.com (Nihat Ya
˘
gmur)
H. M. Srivastava et al. / Filomat 27:5 (2013), 831–842 832
and
f
f
−1
(
w
)
= w
|
w
|
< r
0
f
; r
0
f
=
1
4
.
In fact, the inverse function f
−1
is given by
f
−1
(
w
)
= w − a
2
w
2
+
2a
2
2
− a
3
w
3
−
5a
3
2
− 5a
2
a
3
+ a
4
w
4
+ ··· .
A function f ∈ A is said to be bi-univalent in U if both f and f
−1
are univalent in U. Let Σ denote the
class of bi-univalent functions in U given by
(
1
)
. For a brief history and interesting examples of functions
in the class Σ, see [8] (see also [1]). In fact, the aforecited work of Srivastava et al. [8] essentially revived
the investigation of various subclasses of the bi-univalent function class Σ in recent years; it was followed
by such works as those by Frasin and Aouf [4], Xu et al. [9, 10], Hayami and Owa [6], and others (see, for
example, [5], [7] and [11]).
Recently, C¸ a
˘
glar et al. [3] introduced the following two subclasses of the bi-univalent function class Σ
and obtained non-sharp estimates on the first two Taylor-Maclaurin coefficients
|
a
2
|
and
|
a
3
|
of functions
in each of these subclasses (see also [4] and [10]). It should be mentioned in passing that the functional
expression used in the inequalities in
(
2
)
and
(
7
)
of Definitions 1 and 2 is precisely the same as that used by
Zhu [12] for investigating various extensions, generalizations and improvements of the starlikeness criteria
which were proven by earlier authors (see, for details, Remark 1 below).
Definition 1. (see [3]) A function f
(
z
)
given by
(
1
)
is said to be in the class N
µ
Σ
(
α, λ
)
if the following
conditions are satisfied:
f ∈ Σ and
arg
(
1 − λ
)
f
(
z
)
z
µ
+ λ f
′
(
z
)
f
(
z
)
z
µ−1
<
απ
2
(2)
0 < α 5 1; λ = 1; µ = 0; z ∈ U
and
arg
(
1 − λ
)
1
(
w
)
w
µ
+ λ1
′
(
w
)
1
(
w
)
w
µ−1
<
απ
2
(3)
0 < α 5 1; λ = 1; µ = 0; w ∈ U
,
where the function 1 is given by
1
(
w
)
= w − a
2
w
2
+
2a
2
2
− a
3
w
3
−
5a
3
2
− 5a
2
a
3
+ a
4
w
4
+ ··· . (4)
Theorem 1. (see [3]) Let the function f
(
z
)
given by the Taylor-Maclaurin series expansion
(
1
)
be in the class
N
µ
Σ
(
α, λ
)
(0 < α 5 1; λ = 1; µ = 0).
Then
|
a
2
|
5
2α
λ + µ
2
+ α
µ + 2λ − λ
2
(5)
and
|
a
3
|
5
4α
2
λ + µ
2
+
2α
2λ + µ
. (6)
H. M. Srivastava et al. / Filomat 27:5 (2013), 831–842 833
Definition 2. (see [3]) A function f
(
z
)
given by
(
1
)
is said to be in the class N
µ
Σ
β, λ
if the following
conditions are satisfied:
f ∈ Σ and ℜ
(
1 − λ
)
f
(
z
)
z
µ
+ λ f
′
(
z
)
f
(
z
)
z
µ−1
> β (7)
0 5 β < 1; λ = 1; µ = 0; z ∈ U
and
ℜ
(
1 − λ
)
1
(
w
)
w
µ
+ λ1
′
(
w
)
1
(
w
)
w
µ−1
> β (8)
0 5 β < 1; λ = 1; µ = 0; w ∈ U
,
where the function 1 is defined by
(
4
)
.
Remark 1. For functions f (z), which are analytic in U and normalized by
f (z) = z +
∞
k=n+1
a
k
z
k
(n ∈ N),
Zhu [12] determined the conditions on the parameters M, α, λ and µ such that the following inequality:
(
1 − λ
)
f
(
z
)
z
µ
+ λ f
′
(
z
)
f
(
z
)
z
µ−1
− 1
< M
implies that the so-normalized function f (z) is in the corresponding class of starlike functions of order
α (0 5 α < 1). Interestingly, the functional expression used by Zhu [12] is precisely the same as that used
in the inequalities in
(
2
)
and
(
7
)
above. The work of Zhu [12] provided extensions, generalizations and
improvements of the various starlikeness criteria which were proven by a number of earlier authors (see,
for details, [12]).
Theorem 2. (see [3]) Let the function f
(
z
)
given by the Taylor-Maclaurin series expansion
(
1
)
be in the class
N
µ
Σ
β, λ
(0 5 β < 1; λ = 1; µ = 0).
Then
|
a
2
|
5 min
4
1 − β
µ + 1
2λ + µ
,
2
1 − β
λ + µ
(9)
and
|
a
3
|
5
min
4
1 − β
µ + 1
2λ + µ
,
4
1 − β
2
λ + µ
2
+
2
1 − β
2λ + µ
(0 5 µ < 1)
2
1 − β
2λ + µ
(µ = 1).
(10)
Remark 2. The following special cases of Definitions 1 and 2 are worthy of note:
(i) For µ = 1, we obtain the bi-univalent function classes
N
1
Σ
(
α, λ
)
= B
Σ
(
α, λ
)
and N
1
Σ
β, λ
= B
Σ
β, λ
H. M. Srivastava et al. / Filomat 27:5 (2013), 831–842 834
introduced by Frasin and Aouf [4].
(ii) For µ = 1 and λ = 1, we have the bi-univalent function classes
N
1
Σ
(
α, 1
)
= H
α
Σ
and N
1
Σ
β, 1
= H
Σ
β
introduced by Srivastava et al. [8].
(iii) For µ = 0 and λ = 1, we get the well-known classes
N
0
Σ
(
α, 1
)
= S
∗
Σ
[α] and N
0
Σ
β, 1
= S
∗
Σ
β
of strongly bi-starlike functions of order α and of bi-starlike functions of order β, respectively.
This paper is essentially a sequel to some of the aforecited works (especially see [3] and [10]). Here we
introduce and investigate the general subclass N
h,p
Σ
λ, µ
λ = 1; µ = 0
of the analytic function class A,
which is given by Definition 3 below.
Definition 3. Let the functions h, p : U → C be so constrained that
min
ℜ
h
(
z
)
, ℜ
p
(
z
)
> 0
(
z ∈ U
)
and h
(
0
)
= p
(
0
)
= 1.
Also let the function f , defined by
(
1
)
, be in the analytic function class A. We say that
f ∈ N
h,p
Σ
λ, µ
λ = 1; µ = 0
if the following conditions are satisfied:
f ∈ Σ and
(
1 − λ
)
f
(
z
)
z
µ
+ λ f
′
(
z
)
f
(
z
)
z
µ−1
∈ h
(
U
) (
z ∈ U
)
(11)
and
(
1 − λ
)
1
(
w
)
w
µ
+ λ1
′
(
w
)
1
(
w
)
w
µ−1
∈ p
(
U
) (
w ∈ U
)
, (12)
where the function 1 is defined by
(
4
)
.
We note that the class N
h,p
Σ
λ, µ
reduces to the function classes B
h,p
Σ
(
λ
)
and H
h,p
Σ
given by
B
h,p
Σ
(
λ
)
= N
h,p
Σ
(
λ, 1
)
,
B
h,p
Σ
= N
h,p
Σ
(
1, 0
)
and
H
h,p
Σ
= N
h,p
Σ
(
1, 1
)
,
respectively, each of which was introduced and studied recently by Xu et al. [10], Bulut [2] and Xu et al. [9],
respectively.
Remark 3. There are many choices of the functions h(z) and p(z) which would provide interesting subclasses
of the analytic function class A. For example, if we let
h
(
z
)
= p
(
z
)
=
1 + z
1 − z
α
(
0 < α 5 1; z ∈ U
)
(13)
H. M. Srivastava et al. / Filomat 27:5 (2013), 831–842 835
or
h
(
z
)
= p
(
z
)
=
1 +
1 − 2β
z
1 − z
0 5 β < 1; z ∈ U
, (14)
it is easy to verify that the functions h(z) and p(z) satisfy the hypotheses of Definition 3. If f ∈ N
h,p
Σ
λ, µ
,
then
f ∈ Σ and
arg
(
1 − λ
)
f
(
z
)
z
µ
+ λ f
′
(
z
)
f
(
z
)
z
µ−1
<
απ
2
(15)
0 < α 5 1; λ = 1; µ = 0; z ∈ U
and
arg
(
1 − λ
)
1
(
w
)
w
µ
+ λ1
′
(
w
)
1
(
w
)
w
µ−1
<
απ
2
(16)
0 < α 5 1; λ = 1; µ = 0; w ∈ U
or
f ∈ Σ and ℜ
(
1 − λ
)
f
(
z
)
z
µ
+ λ f
′
(
z
)
f
(
z
)
z
µ−1
> β (17)
0 5 β < 1; λ = 1; µ = 0; z ∈ U
and
ℜ
(
1 − λ
)
1
(
w
)
w
µ
+ λ1
′
(
w
)
1
(
w
)
w
µ−1
> β (18)
0 5 β < 1; λ = 1; µ = 0; w ∈ U
,
where the function 1 is defined by
(
4
)
. This means that
f ∈ N
µ
Σ
(
α, λ
)
0 < α 5 1; λ = 1; µ = 0
or
f ∈ N
µ
Σ
β, λ
0 5 β < 1; λ = 1; µ = 0
.
Our paper is motivated and stimulated especially by the works of C¸ a
˘
glar et al. [3] and Xu et al. [10]. Here
we propose to investigate the bi-univalent function class N
h,p
Σ
λ, µ
introduced in Definition 3 and derive
coefficient estimates on the first two Taylor-Maclaurin coefficients
|
a
2
|
and
|
a
3
|
for a function f ∈ N
h,p
Σ
λ, µ
given by
(
1
)
. Our results for the bi-univalent function class N
h,p
Σ
λ, µ
would generalize and improve the
related works of C¸ a
˘
glar et al. [3] and Xu et al. [10] (see also [4] and [8]).
2. A Set of General Coefficient Estimates
In this section, we state and prove our general results involving the bi-univalent function class N
h,p
Σ
λ, µ
given by Definition 3.