J ournal of
Classical
A nalysis
Volume 2, Number 1 (2013), 49–60
doi:10.7153/jca-02-05
CERTAIN SUBCLASSES OF BI–UNIVALENT
FUNCTIONS SATISFYING SUBORDINATE CONDITIONS
E RHAN D ENIZ
Abstract. In this paper, we introduce and investigate each of the following subclasses:
SΣ (λ , γ ; ϕ ), H S Σ (α ), RΣ (η , γ ; ϕ ) and BΣ ( μ ; ϕ )
(0 λ 1; γ ∈ C{0}; α ∈ C; 0 η < 1; μ 0)
of bi-univalent functions, ϕ is an analytic function with positive real part in the unit disk D,
satisfying ϕ (0) = 1, ϕ (0) > 0, and ϕ (D) is symmetric with respect to the real axis. We obtain
coefficient bounds involving the Taylor-Maclaurin coefficients |a2 | and |a3 | of the function
f when f is in these classes. The various results, which are presented in this paper, would
generalize and improve those in related works of several earlier authors.
1. Introduction
Let A be the class of all analytic functions of the form
∞
f (z) = z + ∑ ak zk
(1.1)
k=2
in the open unit disk D := {z ∈ C : |z| < 1} normalized by the conditions f (0) = 0
and f (0) = 1. The well-known Koebe one-quarter theorem [5] ensures that the image
of D under every univalent function f ∈ A contains a disk of radius 1/4 . Thus every
univalent function f has an inverse f −1 satisfying f −1 ( f (z)) = z, (z ∈ D) and
f −1 ( f (w)) = w, (|w| < r0 ( f ), r0 ( f ) 1/4)
where
f −1 (w) = w − a2w2 + (2a22 − a3)w3 − ....
A function f ∈ A is said to be bi-univalent in D if both f and f −1 are univalent
in D. Let Σ denote the class of bi-univalent functions defined in the unit disk D.
A function f ∈ A is called starlike, denoted by f ∈ S ∗ , if tw ∈ f (D) whenever
w ∈ f (D) and t ∈ [0, 1]. A function f ∈ A that maps the unit disk onto a convex
domain is called a convex function. Let C denote the class of all functions f ∈ A
that are convex. Analytically, these geometric properties are, respectively, equivalent
Mathematics subject classification (2010): 30C45; Secondary 30C50, 30C80.
Keywords and phrases: Univalent functions, bi-univalent functions, bi-starlike and bi-convex functions
of complex order, bi-Bazilevič functions, subordination.
c
, Zagreb
Paper JCA-02-05
49
50
E RHAN D ENIZ
to the conditions ℜz f (z)/ f (z) > 0 and 1 + ℜz f (z)/ f (z) > 0. An analytic function
f is subordinate to an analytic function g, written f (z) ≺ g(z), provided there is an
analytic function w defined on D with w(0) = 0 and |w(z)| < 1 satisfying f (z) =
g(w(z)). In terms of subordination, these conditions are, respectively, equivalent to
z f (z)/ f (z) ≺ (1 + z)(1 − z) and 1 + z f (z)/ f (z) ≺ (1 + z)(1 − z). Ma and Minda
[8] gave a unified presentation of various subclasses of starlike and convex functions by
replacing the subordinate function (1 + z)(1 − z) by a more general analytic function
ϕ with positive real part and normalized by the conditions ϕ (0) = 1, ϕ (0) > 0 and ϕ
maps D onto a region starlike with respect to 1 and symmetric with respect to the real
axis. They introduced the following general classes that envelopes several well-known
classes as special cases:
z f (z)
z f (z)
S ∗ (ϕ ) = f ∈ A :
≺ ϕ (z) and C (ϕ ) = f ∈ A : 1 + ≺ ϕ (z) .
f (z)
f (z)
In literature, the functions belonging to these classes are called Ma-Minda starlike
and Ma-Minda convex, respectively. A function f is bi-starlike of Ma-Minda type or
bi-convex of Ma-Minda type if both f and f −1 are, respectively, Ma-Minda starlike
or convex. These classes are denoted, respectively, by SΣ∗ (ϕ ) and CΣ (ϕ ). For β
(0 β < 1), let ϕβ : D → C be defined by ϕβ (z) = (1+(1−2β )z)(1−z). Then the
classes S ∗ (ϕβ ) and C (ϕβ ) reduce to, respectively, the familiar classes S ∗ (β ) and
C (β ) of univalent starlike of and convex functions of order β which are, respectively,
characterized by ℜz f (z)/ f (z) > β and 1 + ℜz f (z)/ f (z) > β . Also we say that a
function f ∈ A is Ma-Minda starlike and convex of complex order γ (γ ∈ C{0}) :
1 z f (z)
∗
S (γ ; ϕ ) = f ∈ A : 1 +
− 1 ≺ ϕ (z)
γ
f (z)
and
C (γ ; ϕ ) =
1
f ∈ A : 1+
γ
z f (z)
f (z)
≺ ϕ (z) ,
respectively. A function f is bi-starlike and bi-convex of complex order γ (γ ∈ C{0})
of Ma-Minda type if both f and f −1 are, respectively, Ma-Minda starlike and MaMinda convex of complex order γ . These classes are denoted respectively by SΣ∗ (γ ; ϕ )
1+z
∗
and CΣ (γ ; ϕ ). Especially the classes SΣ∗ (γ ; 1+z
1−z ) = SΣ (γ ) and CΣ (γ ; 1−z ) = C Σ (γ )
are bi-starlike and bi-convex functions of complex order γ (γ ∈ C{0}), respectively.
We note that:
(1) SΣ∗ ((1 − β )e−iλ cos λ ) = SΣ∗ [λ , β ] (|λ | < π 2, 0 β < 1) is class of bi- λ spirallike univalent functions of order β
(2) CΣ ((1 − β )e−iλ cos λ ) = CΣ [λ , β ] (|λ | < π 2, 0 β < 1) is class of bi- λ Robertson univalent functions of order β .
The class Σ of bi-univalent functions was introduced in 1967 by Lewin [7] and
he showed that, for every functions f ∈ Σ of the form (1.1), the second coefficient of
f satisfy the inequality |a2 | < 1.51
√. Subsequently, Brannan and Clunie [4] improved
Lewin’s result by showing |a2 | 2. Later, Netanyahu [9] proved that max f ∈Σ |a2 | =
C ERTAIN SUBCLASSES OF BI - UNIVALENT FUNCTIONS
51
4/3 . Since then, various subclasses of the bi-univalent function class Σ were introduced and non-sharp estimates on the first two coefficients |a2 | and |a3 | in the Taylor–
Maclaurin series expansion (1.1) were found in several recent investigations (see [3],
[16]; see also [1], [6], [13] and [18])
The object of the present paper is to introduce four new subclasses of the function class Σ and find estimates on the coefficients |a2 | and |a3 | for functions in these
new subclasses of the function class Σ. As special cases, estimates on the initial coefficients for bi-starlike and bi-convex of complex order γ (γ ∈ C{0}) of Ma-Minda
type functions are obtained. Several related classes are also considered, and connection
to earlier known results are made.
In order to derive our main results, we have to recall here the following lemma [5].
L EMMA 1.1. If p ∈ P then |bk | 2 for each k, where P is the family of all
functions p analytic in D for which ℜp(z) > 0, p(z) = 1 + b1z + b2 z2 + b3 z3 + ... for
z ∈ D.
2. Coefficient estimates
In the sequel, it is assumed that ϕ is an analytic function with positive real part in
the unit disk D, satisfying ϕ (0) = 1, ϕ (0) > 0, and ϕ (D) is symmetric with respect
to the real axis. Such a function has a series expansion of the form
ϕ (z) = 1 + B1z + B2z2 + B3 z3 + ..., B1 > 0.
(2.1)
Define the functions p1 and p2 in P given by
p1 (z) =
1 + u(z)
= 1 + c1z + c2 z2 + c3 z3 + ...
1 − u(z)
p2 (z) =
1 + v(z)
= 1 + d1z + d2z2 + d3z3 + ....
1 − v(z)
and
It follows
and
1
c21 2
p1 (z) − 1 c1
= z+
u(z) =
c2 −
z + ...
p1 (z) + 1
2
2
2
(2.2)
1
p2 (z) − 1 d1
d12 2
= z+
v(z) =
d2 −
z + ....
p2 (z) + 1
2
2
2
(2.3)
Using (2.2) and (2.3) together with (2.1), it is evident that
c2
1
B 1 c1
1
z+
ϕ (u(z)) = 1 +
c2 − 1 B1 + c21 B2 z2 + ...
2
2
2
4
and
ϕ (v(z)) = 1 +
B1 d 1
z+
2
1
d2
1
d2 − 1 B1 + d12 B2 z2 + ...
2
2
4
(2.4)
(2.5)
52
E RHAN D ENIZ
A function f ∈ Σ is said to be in the class SΣ (λ , γ ; ϕ ), 0 λ 1, γ ∈ C{0}
if the following subordinations hold:
z f (z) + λ z2 f (z)
1
−
1
≺ ϕ (z)
1+
γ λ z f (z) + (1 − λ ) f (z)
1
wg (w) + λ w2 g (w)
−
1
≺ ϕ (w), g(w) := f −1 (w) .
and 1 +
γ λ wg (w) + (1 − λ )g(w)
A function in the class SΣ (λ , γ ; ϕ ) is called both bi- λ − convex function and biλ − starlike function of complex order γ of Ma-Minda type. This class introduced in
this paper is motivated by the corresponding class investigated in [17] and [14].
This class unifies the classes SΣ∗ (γ ; ϕ ) and CΣ (γ ; ϕ ). Note that SΣ (0, γ ; ϕ ) ≡
∗
∗
SΣ (γ ; ϕ ) and SΣ (1, γ ; ϕ ) ≡ CΣ (γ ; ϕ ). Also SΣ (0, (1 − α )e−iλ cos λ ; 1+z
1−z ) ≡ SΣ [λ , α ]
and SΣ (1, (1 − α )e−iλ cos λ ; 1+z
1−z ) ≡ CΣ [λ , α ] (|λ | < π 2, 0 α < 1) .
For functions in the class SΣ (λ , γ ; ϕ ), the following coefficient estimation holds.
T HEOREM 2.1. Let f given by (1.1) be in the class SΣ (λ , γ ; ϕ ). Then
√
|γ | B1 B1
|a2 | γ (1 + 2λ − λ 2)B2 + (1 + λ )2(B1 − B2 )
1
and
|a3 | |γ | (B1 + |B1 − B2 |)
.
1 + 2λ − λ 2
(2.6)
(2.7)
Proof. Let f ∈ SΣ (λ , γ ; ϕ ). Then there are analytic functions u, v : D → D, with
u(0) = v(0) = 0 , satisfying
z f (z) + λ z2 f (z)
1
−
1
= ϕ (u(z))
1+
(2.8)
γ λ z f (z) + (1 − λ ) f (z)
wg (w) + λ w2 g (w)
1
− 1 = ϕ (v(w)), g := f −1 .
and 1 +
γ λ wg (w) + (1 − λ )g(w)
Since
1+
= 1+
and
1
1+
γ
= 1−
1
γ
z f (z) + λ z2 f (z)
−1
λ z f (z) + (1 − λ ) f (z)
(1 + λ )a2
2(1 + 2λ )a3 − (1 + λ )2a22 2
z+
z + ...
γ
γ
wg (w) + λ w2 g (w)
−1
λ wg (w) + (1 − λ )g(w)
−2(1 + 2λ )a3 + (3 + 6λ − λ 2)a22 2
(1 + λ )a2
w+
w + ...
γ
γ
C ERTAIN SUBCLASSES OF BI - UNIVALENT FUNCTIONS
53
then (2.4), (2.5) and (2.8) yield
(1 + λ )a2 B1 c1
,
=
γ
2
2(1 + 2λ )a3 − (1 + λ )2a22
1
c2
1
=
c2 − 1 B1 + c21 B2 ,
γ
2
2
4
−
and
(1 + λ )a2 B1 d1
=
γ
2
−2(1 + 2λ )a3 + (3 + 6λ − λ 2 )a22
1
d12
1
=
d2 −
B1 + d12 B2 .
γ
2
2
4
(2.9)
(2.10)
(2.11)
(2.12)
Now, considering (2.9) and (2.11), we get
c1 = −d1.
(2.13)
Also, from (2.10), (2.11), (2.12) and (2.13), we find that
γ 2 B31 (c2 + d2)
a22 = 4 γ (1 + 2λ − λ 2 )B21 + (1 + λ )2(B1 − B2)
which, in view of the inequalities |c2 | 2 and |d2 | 2 from Lemma 1.1, yield
|γ |2 B31
.
|a2 |2 γ (1 + 2λ − λ 2)B21 + (1 + λ )2(B1 − B2 )
Since B1 > 0 , the last inequality gives the desired estimate on |a2 | given in (2.6).
Next, in order to find the bound on |a3 |, by further computations from (2.10),
(2.11), (2.12) and (2.13) lead to
(γ B1 2) (3 + 6λ − λ 2 )c2 + (1 + λ )2d2 + γ (1 + 2λ )d12(B2 − B1 )
a3 =
.
4(1 + 2λ )(1 + 2λ − λ 2 )
Applying Lemma 1.1 for the coefficients d1 , c2 and d2 , we readily get
(|γ | B1 2) 2(3 + 6λ − λ 2) + 2(1 + λ )2 + 4 |γ | (1 + 2λ ) |B2 − B1|
|a3 | 4(1 + 2λ )(1 + 2λ − λ 2 )
|γ | (B1 + |B1 − B2 |)
=
1 + 2λ − λ 2
which is the bound on |a3 | as asserted in (2.7).
C OROLLARY 2.2. Let f given by (1.1) be in the class CΣ (γ ; ϕ ). Then
√
|γ | B1 B1
|γ | (B1 + |B1 − B2 |)
.
|a2 | and |a3 | 2
2 γ B21 + 2(B1 − B2 )
54
E RHAN D ENIZ
C OROLLARY 2.3. Let f given by (1.1) be in the class SΣ∗ (γ ; ϕ ). Then
√
|γ | B1 B1
|a2 | and |a3 | |γ | (B1 + |B1 − B2 |) .
γ B2 + B1 − B2 1
R EMARK 2.4. If the function is bi- λ -spirallike univalent of order β , that is f ∈
SΣ∗ [λ , β ], then
|a2 | 2(1 − β ) cos λ and |a3 | 2(1 − β ) cos λ .
If the function is bi- λ -Robertson of order β , that is f ∈ CΣ [λ , β ], then
|a2 | (1 − β ) cos λ and |a3 | (1 − β ) cos λ .
Let α be a complex number. We say that a function f ∈ Σ belongs to the class
H S Σ (α ), α ∈ C if the functions F and G defined by
1
1−α
α
=
+ F(z)
f (z)
z f (z)
and
1
1−α
α
=
+
,
G(w)
g(w) wg (w)
are in the class SΣ∗ (ϕ ), g(w) := f −1 (w). This class introduced in this paper is motivated by the corresponding class investigated in [10].
Note that H S Σ (0) ≡ SΣ∗ (ϕ ) and H S Σ (1) ≡ CΣ (ϕ ). For functions in the class
H S Σ (α ), the following coefficient estimation holds.
T HEOREM 2.5. Let f given by (1.1) be in the class H S Σ (α ). Then
√
B1 B1
|a2 | (1 + α 2 )B2 + (1 + α )2(B1 − B2 )
1
and
(B1 4) |(α + 3)(α + 1)| + α 2 − 4α − 1 + (1 + 2α ) |B2 − B1 |
|a3 | ,
|1 + 2α ||α 2 + 1|
then
and
(2.14)
(2.15)
Proof. Let f ∈ H S Σ (α ). Simple calculations show that if f is in H S Σ (α ),
z f (z)
z f (z)
z f (z) + (1 − α )z2 f (z)
+ 1+ ≺ ϕ (z)
−
f (z)
f (z)
(1 − α )z f (z) + α f (z)
wg (w)
wg (w)
wg (w) + (1 − α )w2g (w)
+ 1+ ≺ ϕ (w).
−
g(w)
g (w)
(1 − α )wg (w) + α g(w)
55
C ERTAIN SUBCLASSES OF BI - UNIVALENT FUNCTIONS
Consider the analytic functions u, v : D → D, with u(0) = v(0) = 0 , satisfying
z f (z)
z f (z)
z f (z) + (1 − α )z2 f (z)
+ 1+ = ϕ (u(z)) (2.16)
−
f (z)
f (z)
(1 − α )z f (z) + α f (z)
wg (w)
wg (w)
wg (w) + (1 − α )w2 g (w)
+ 1+ = ϕ (v(w))
and
−
g(w)
g (w)
(1 − α )wg (w) + α g(w)
where g := f −1 . Since
z f (z)
z f (z)
z f (z) + (1 − α )z2 f (z)
+ 1+ −
f (z)
f (z)
(1 − α )z f (z) + α f (z)
= 1 + (1 + α )a2 z + 2(1 + 2α )a3 + (α 2 − 4α − 1)a22 z2 + ...
and
wg (w)
wg (w)
wg (w) + (1 − α )w2g (w)
+ 1+ −
g(w)
g (w)
(1 − α )wg (w) + α g(w)
= 1 − (1 + α )a2 w + −2(1 + 2α )a3 + (α + 3)(α + 1)a22 w2 + ..
then (2.4), (2.5) and (2.16) yield
B 1 c1
,
2
1
c2
1
2(1 + 2α )a3 + (α 2 − 4α − 1)a22 =
c2 − 1 B1 + c21 B2 ,
2
2
4
B1 d 1
−(1 + α )a2 =
2
(1 + α )a2 =
(2.17)
(2.18)
(2.19)
and
− 2(1 + 2α )a3 + (α + 3)(α + 1)a22 =
d2
1
1
d2 − 1 B1 + d12 B2 .
2
2
4
(2.20)
Now, considering (2.17) and (2.19), we get
c1 = −d1.
Also, from (2.18), (2.19), (2.20) and (2.21), we find that
B31 (c2 + d2)
a22 = 4 (1 + α 2 )B21 + (1 + α )2(B1 − B2 )
which, in view of the inequalities |c2 | 2 and |d2 | 2 from Lemma 1.1, yield
B31
.
|a2 |2 (1 + α 2)B2 + (1 + α )2(B1 − B2)
1
(2.21)
56
E RHAN D ENIZ
Since B1 > 0 , the last inequality gives the desired estimate on |a2 | given in (2.14).
Next, in order to find the bound on |a3 |, by further computations from (2.18),
(2.19), (2.20) and (2.21) lead to
(B1 2) (α + 3)(α + 1)c2 − α 2 − 4α − 1 d2 + (1 + 2α )d12 (B2 − B1 )
a3 =
.
4 (1 + 2α )(α 2 + 1)
Applying Lemma 1.1 for the coefficients d1 , c2 and d2 , we readily get
(B1 2) 2 |(α + 3)(α + 1)| + 2 α 2 − 4α − 1 + 4 |1 + 2α ||B2 − B1|
|a3 | 4 |1 + 2α ||α 2 + 1|
(B1 4) |(α + 3)(α + 1)| + α 2 − 4α − 1 + |1 + 2α ||B2 − B1|
=
|1 + 2α ||α 2 + 1|
which is the bound on |a3 | as asserted in (2.15).
Let 0 η < 1 and γ ∈ C{0}. A function f ∈ Σ is in the class RΣ (η , γ ; ϕ ) if
1+
and
1+
1 f (z) + η z f (z) − 1 ≺ ϕ (z)
γ
1 g (w) + η wg (w) − 1 ≺ ϕ (w),
γ
g(w) := f −1 (w). The class RΣ (0, 1; ϕ ) ≡ HΣ (ϕ ) was studied by Ali et al [1]. This
class introduced in this paper is motivated by the corresponding class investigated in
[15].
For functions in the class RΣ (η , γ ; ϕ ), the following coefficient estimation holds.
T HEOREM 2.6. Let f given by (1.1) be in the class RΣ (η , γ ; ϕ ). Then
√
|γ | B1 B1
|a2 | 3γ (1 + 2η )B2 + 4(1 + η )2(B1 − B2 )
1
and
|a3 | |γ | B1
|γ | B1
1
+
4(1 + η )2 3(1 + 2η )
(2.22)
(2.23)
Proof. Let f ∈ RΣ (η , γ ; ϕ ). Consider the analytic functions u, v : D → D, with
u(0) = v(0) = 0 , satisfying
1 f (z) + η z f (z) − 1 = ϕ (u(z))
γ
1
and 1 + g (w) + η wg (w) − 1 = ϕ (v(w)), g := f −1 .
γ
1+
(2.24)
C ERTAIN SUBCLASSES OF BI - UNIVALENT FUNCTIONS
57
Since
1+
1 2(1 + η )
3(1 + 2η ) 2
a2 z +
a3 z + ...
f (z) + η z f (z) − 1 = 1 +
γ
γ
γ
and
1+
6(1 + 2η )a22 − 3(1 + 2η )a3 2
1 2(1 + η )
a2 w+
w +...
g (w) + η wg (w) − 1 = 1−
γ
γ
γ
then (2.4), (2.5) and (2.24) yield
B 1 c1
2(1 + η )
,
a2 =
γ
2
1
c21
1
3(1 + 2η )
a3 =
c2 −
B1 + c21 B2 ,
γ
2
2
4
−
and
B1 d 1
2(1 + η )
a2 =
γ
2
6(1 + 2η )a22 − 3(1 + 2η )a3 1
d2
1
=
d2 − 1 B1 + d12 B2 .
γ
2
2
4
(2.25)
(2.26)
(2.27)
(2.28)
Now, considering (2.25) and (2.27), we get
c1 = −d1.
(2.29)
Also, from (2.26), (2.27), (2.28) and (2.29), we find that
γ 2 B31 (c2 + d2)
a22 = 4 3γ (1 + 2η )B21 + 4(1 + η )2(B1 − B2 )
which, in view of the inequalities |c2 | 2 and |d2 | 2 from Lemma 1.1, yield
|γ |2 B31
.
|a2 |2 2
3γ (1 + 2η )B + 4(1 + η )2(B1 − B2 )
1
Since B1 > 0 , the last inequality gives the desired estimate on |a2 | given in (2.22).
Next, in order to find the bound on |a3 |, by further computations from (2.26),
(2.27), (2.28) and (2.29) lead to
γ B1 d12
(c2 − d2)
a 3 = γ B1
+
.
16(1 + η )2 12(1 + 2η )
Applying Lemma 1.1 for the coefficients d1 , c2 and d2 , we readily get
|γ | B1
1
|a3 | |γ | B1
+
4(1 + η )2 3(1 + 2η )
58
E RHAN D ENIZ
which is the bound on |a3 | as asserted in (2.15).
For the class of strongly starlike functions and starlike functions of order κ , respectively, the function ϕ is given by
ϕ (z) =
1+z
1−z
β
= 1 + 2β z + 2β 2z2 + ... (0 < β 1)
which gives B1 = 2β , B2 = 2β 2 and
ϕ (z) =
1 + (1 − 2κ )z
= 1 + 2(1 − κ )z + 2(1 − κ )z2 + ... (0 κ < 1)
1−z
which gives B1 = B2 = 2(1 − κ ).
R EMARK 2.7. i. Putting γ = 1 and η = 0 in Theorem 2.6, we obtain the corresponding result given earlier by Ali et al [1].
β 1+(1−2κ )z
ii. Putting γ = 1 , η = 0 and ϕ (z) = 1+z
ϕ
(z)
=
also
in Theorem
1−z
1−z
2.6, we obtain the corresponding results given earlier
by
Srivastava
et
al
[13].
β 1+(1−2κ )z
also
and γ = 1 ,
ϕ
(z)
=
iii. Putting γ = 1 , λ = 0 , ϕ (z) = 1+z
1−z
1−z
κ )z
λ = 1 , ϕ (z) = 1+(1−2
in Theorem 2.1, we obtain the corresponding results given
1−z
earlier by Brannan and Taha [3].
Let 0 μ . A function f ∈ Σ is in the class BΣ (μ ; ϕ ) if
z1−μ f (z)
≺ ϕ (z)
[ f (z)]1−μ
and
w1−μ f (w)
≺ ϕ (w),
[ f (w)]1−μ
g(w) := f −1 (w). A function in the class BΣ (μ ; ϕ ) is called bi-Bazilevič function of
Ma-Minda type. This class introduced in this paper is motivated by the corresponding
class investigated in [11] (also see [2], [12]). This class unifies the classes SΣ∗ (γ ; ϕ ).
For functions in the class BΣ (μ ; ϕ ), the following coefficient estimation holds.
T HEOREM 2.8. Let f given by (1.1) be in the class BΣ (μ ; ϕ ). Then
√
B1 2B1
|a2 | √
μ + 1 (μ + 2)B21 + 2(μ + 1)(B1 − B2)
and
|a3 | 2B1 +2|B1 −B2 |
( μ +2)( μ +1) ,
( μ +1)B1 +2|B1 −B2 |
,
( μ +2)( μ +1)
if 0 μ < 1
if
μ 1
(2.30)
(2.31)
59
C ERTAIN SUBCLASSES OF BI - UNIVALENT FUNCTIONS
Proof. Let f ∈ BΣ (μ ; ϕ ). Consider the analytic functions u, v : D → D, with
u(0) = v(0) = 0 , satisfying
z1−μ f (z)
w1−μ f (w)
= ϕ (u(z)) and
= ϕ (v(w)), g := f −1 .
1−
μ
1−
μ
[ f (z)]
[ f (w)]
Since
and
(2.32)
z1−μ f (z)
(μ − 1)(μ + 2) 2 2
a
=
1
+
(
μ
+
1)a
z
+
(
μ
+
2)a
+
2
3
2 z + ...
[ f (z)]1−μ
2
(μ + 3)(μ + 2) 2 2
w1−μ f (w)
a
=
1
−
(
μ
+
1)a
w
+
−(
μ
+
2)a
+
2
3
2 w + ...
[ f (w)]1−μ
2
then (2.4), (2.5) and (2.32) yield
B 1 c1
,
2
(μ − 1)(μ + 2) 2 1
c21
1
(μ + 2)a3 +
a2 =
c2 −
B1 + c21 B2 ,
2
2
2
4
(μ + 1)a2 =
− (μ + 1)a2 =
and
− (μ + 2)a3 +
B1 d 1
2
(μ + 3)(μ + 2) 2 1
d2
1
a2 =
d2 − 1 B1 + d12 B2 .
2
2
2
4
(2.33)
(2.34)
(2.35)
(2.36)
Now, considering (2.33) and (2.35), we get
c1 = −d1.
(2.37)
Also, from (2.34), (2.35), (2.36) and (2.37), we find that
a22 =
B31 (c2 + d2)
2(μ + 1) (μ + 2)B21 + 2(μ + 1)(B1 − B2)
which, in view of the inequalities |c2 | 2 and |d2 | 2 from Lemma 1.1, yield
|a2 |2 2B31
.
(μ + 1) (μ + 2)B21 + 2(μ + 1)(B1 − B2 )
Since B1 > 0 , the last inequality gives the desired estimate on |a2 | given in (2.30).
Proceeding similarly as in the earlier proof, using (2.34), (2.35), (2.36) and (2.37)
it follows that
a3 =
(μ + 2)B1 [(μ + 3)c2 + (1 − μ )d2] − 2d12(B1 − B2 )(μ + 2)
.
4(μ + 1)(μ + 2)2
60
E RHAN D ENIZ
Applying Lemma 1.1 for the coefficients d1 , c2 and d2 , we readily get
|a3 | B1 [|μ + 3| + |1 − μ |] + 4 |B1 − B2|
2(μ + 1)(μ + 2)
which yields the estimate (2.31).
Acknowledgement. This Project was supported by the Commission for the Scientific Research Projects of Kafkas University. The Project number: 2012-FEF-30.
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(Received September 28, 2012)
Journal of Classical Analysis
www.ele-math.com
[email protected]
Erhan Deniz
Department of Mathematics, Faculty of Science and Letters
Kafkas University
Kars, Turkey
e-mail: [email protected]