Gen. Math. Notes, Vol. 29, No. 1, July 2015, pp.6-21
c
ISSN 2219-7184; Copyright ICSRS
Publication, 2015
www.i-csrs.org
Available free online at http://www.geman.in
Coefficient Estimates for λ–Bazilevič Functions
of Bi-univalent Functions
Shuhai Li1 , Lina Ma2 and Huo Tang3
1,2,3
School of Mathematics and Statistics, Chifeng University
Chifeng 024001, Inner Mongolia, China
1
E-mail: lishms66@sina.com
2
E-mail: malina00@163.com
3
E-mail: thth2009@163.com
(Received: 26-5-15 / Accepted: 28-6-15)
Abstract
In this paper, we introduce two new subclasses of the function class Σ of
λ–Bazilevič functions of bi-univalent functions defined in the open unit disc.
We find estimates on the coefficients |a2 | and |a3 | for functions in these new
subclasses. The results presented in this paper would generalize some recent
works of Xu et al. and Ali et al.
Keywords: Analytic functions, Univalent functions, Bazilevič functions,
Bi-univalent functions, Coefficient estimates.
1
Introduction
Let A denote the class of functions of the form
f (z) = z +
∞
X
an z n ,
(1)
n=2
which are analytic in the open unit disc U = {z ∈ C : |z| < 1}. We also denote
by S the subclass of the normalized analytic function class A consisting of all
functions in A which are also univalent in U (see [1-4]). Familiar subclasses
of starlike functions of order ξ(0 ≤ ξ < 1) and convex functions of order ξ for
7
Coefficient Estimates for λ–Bazilevič Functions...
which either of the quantity
zf 0 (z)
<
f (z)
(
)
zf 00 (z)
> ξ and < 1 + 0
f (z)
(
)
> ξ.
The class consisting these two functions are given by S ∗ (ξ) and K(ξ) , respectively. For a constant β ∈ (−π/2, π/2), a function f is univalent on U and
satisfies the condition that <{eiθ zf 0 (z)/f (z)} > 0 in U . We denote this class
by T S ∗ (see [2]).
It is well known that every function f (z) ∈ S has an inverse f −1 , which is
defined by
f −1 (f (z)) = z (z ∈ U )
and
1
f (f −1 (ω)) = ω, (|ω| < r0 (f ), r0 (f ) ≥ ).
4
In fact, the inverse function is given by
f −1 (ω) = ω − a2 ω 2 + (2a22 − a3 )ω 3 − (5a32 − 5a2 a3 + a4 )ω 4 + · · ·
(2)
A function f ∈ S is bi-univalent in U if both f and f −1 are univalent
in U . We denote by Σ the class of all bi-univalent functions in U given by
the Taylor-Maclaurin series expansion (1). Lewin [5] investigated the class Σ
of bi-univalent functions and obtained the bound for the second coefficient.
Several authors have subsequently studied similar problems in this direction
(see [6]). Srivastava et al.[7], and Frasin and Aouf [8] introduced subclasses
of bi-univalent functions and obtained bounds for the initial coefficients. Recently, Xu et al. [9], Goyal and Goswami [10] and Ali et al.[11] introduced and
investigated subclasses of bi-univalent functions and obtained bounds for the
initial coefficients.
Let f and g be analytic functions in U , we say that f is subordinate to
g, written as f (z) ≺ g(z) if there exists a Schwarz function ω(z) in U , with
ω(0) = 0 and |ω(z)| < 1(z ∈ U ), such that f (z) = g(ω(z)). In particular,
when g is univalent, then the above subordination is equivalent to f (0) = 0
and f (U ) ⊆ g(U ).
Let
H(U ) = {h : U → C, <{h(z)} > 0 and h(0) = 1, h(z) = h(z)(z ∈ U )}.
Assume that ϕ is an analytic univalent function with positive part in U ,
ϕ(U ) is symmetric with respect to the real axis and starlike with respect to
ϕ(0) = 1, and ϕ0 (0) > 0. Such a function has series expansion of the form
ϕ(z) = 1 + B1 z + B2 z 2 + · · · , (B1 > 0).
(3)
8
Shuhai Li et al.
Obviously, ϕ(U ) ⊆ H(U ).
Wang et al.[13] (also see Li [14]) introduced and investigated the class of
λ−Bazilevič functions consists of functions f ∈ A satisfying the subordination:
zf 0 (z)
(1 − λ)
f (z)
f (z)
g(z)
!α+iµ
zf 00 (z)
+λ 1+ 0
f (z)
!
f 0 (z)
g 0 (z)
!α+iµ
≺
1 + Az
1 + Bz
(α ≥ 0, λ ≥ 0, µ, A, B ∈ R and A 6= B, −1 ≤ B ≤ 1; g ∈ S ∗ (ξ)).
In this paper, using the subordination, we introduce the following two
classes of λ−Bazilevič functions of bi-univalent functions.
Definition 1.1 Let the function f (z), defined by (1), be in the analytic
function class A. We say that f (z) ∈ Uαp,q (β, b, λ) if the following conditions
are satisfied:
f (z) ∈ Σ
and
eiβ
1
zf 0 (z)
1+
(1 − λ)
cos β
b
f (z)
"
f (z)
z
!α
zf 00 (z)
+λ 1+ 0
(f 0 (z))α − 1
f (z)
!
!#
−i tan β ∈ p(U )
(4)
and
eiβ
1
ωg 0 (ω)
1+
(1 − λ)
cos β
b
g(ω)
"
g(ω)
ω
!α
ωg 00 (ω)
+λ 1+ 0
(g 0 (ω))α − 1
g (ω)
!
!#
−i tan β ∈ q(U )
(5)
(p(z), q(ω) ∈ H(U ); z, ω ∈ U )
where β ∈ (− π2 , π2 ); b ∈ C\{0}; α ≥ 0, λ ≥ 0; n ∈ N0 , the function g(ω) =
f −1 (ω) is given by (2).
Definition 1.2 Let the function f (z) of the form (1), be in the analytic
function class A. We say that f (z) ∈ Lϕα (β, b, λ) if the following conditions
are satisfied:
f (z) ∈ Σ
and
eiβ
1
zf 0 (z) f (z) α
zf 00 (z) 0
1+
(1 − λ)
(
) + λ(1 + 0
)(f (z))α − 1
cos β
b
f (z)
z
f (z)
"
!#
−i tan β ≺ ϕ(z)
9
Coefficient Estimates for λ–Bazilevič Functions...
and
eiβ
1
ωg 0 (ω) g(ω) α
ωg 00 (ω) 0
1+
(1 − λ)
(
) + λ(1 + 0
)(g (ω))α − 1
cos β
b
g(ω)
ω
g (ω)
"
!#
−i tan β ≺ ϕ(ω)
(g(ω) = f −1 (ω); z, ω ∈ U ).
1+z η
) (0 < η ≤ 1), Definition 1.1 readily
For f (z) ∈ Lϕα (β, b, λ) and ϕ = ( 1−z
η
yields the following class Bα (β, b, λ) satisfying:
f (z) ∈ Σ
and
arg
eiβ
zf 00 (z) 0
1
zf 0 (z) f (z) α
(
) + λ(1 + 0
)(f (z))α − 1
1+
(1 − λ)
cos β
b
f (z)
z
f (z)
"
!#
<
ηπ
2
and
arg
1
ωg 0 (ω) g(ω) α
ωg 00 (ω) 0
eiβ
1+
(1 − λ)
(
) + λ(1 + 0
)(g (ω))α − 1
cos β
b
g(ω)
ω
g (ω)
"
!#
<
ηπ
2
(g(ω) = f −1 (ω); 0 < η ≤ 1; z, ω ∈ U )
1+Az
For f (z) ∈ Lϕα (β, b, λ) and ϕ = ( 1+Bz
)(−1 ≤ B < A ≤ 1), Definition 1.2
A,B
readily yields the following class Lα (β, b, λ) satisfying:
f (z) ∈ Σ
and
eiβ
1
zf 0 (z) f (z) α
zf 00 (z) 0
1+
(1 − λ)
(
) + λ(1 + 0
)(f (z))α − 1
cos β
b
f (z)
z
f (z)
"
!#
−i tan β ≺
1 + Az
1 + Bz
and
eiβ
1
ωg 0 (ω) g(ω) α
ωg 00 (ω) 0
1+
(1 − λ)
(
) + λ(1 + 0
)(g (ω))α − 1
cos β
b
g(ω)
ω
g (ω)
"
!#
−i tan β ≺
1 + Aω
1 + Bω
(g(ω) = f −1 (ω); z, ω ∈ U ).
10
Shuhai Li et al.
For suitable choices of p, q and by specializing the parameters b, λ, α, η, β involved in the class Uαp,q (β, b, λ), Lϕα (β, b, λ) and Bαη (β, b, λ), we also obtain the
following subclasses which were studied in many earlier works:
(1)SΣ (ξ) = L1−2ξ,−1
(0, 1, 0)(Bi-Starlike function)(Brannan and Taha [6]);
0
1−2ξ,−1
(2)KΣ (ξ) = L0
(0, 1, 1)(Bi-Starlike function)(Brannan and Taha [6]);
p,q
(3)U (p, q) = U1 (0, 1, 0) (Xu et al. [9]);
(4)MΣ (p, λ) = U0p,p (0, 1, λ)(General Bi-Mocanu-convex function of Ma-Minda)
(Ali et al.[11]);
(5)BΣ (α, ϕ) = Lϕα (0, 1, 0)(Bi-Bazilevič functions of Ma-Minda type [16]).
In this paper, estimates on the initial coefficients for class Uαp,q (β, b, λ),
Bαη (β, b, λ) are obtained. Several related classes are also considered, and a connection to earlier known results is made.
Lϕα (β, b, λ)and
2
Coefficient Bounds for the Function Class
Uαp,q (β, b, λ)
Theorem 2.1 Suppose that f (z) ∈ A of the form (1), be in the class
Uαp,q (β, b, λ). Then
|a2 | ≤ min
q

 |b| cos β |p0 (0)|2
v

u
+ |q 0 (0)|2 u
(|p00 (0)| + |q 00 (0)|)|b| cos β 
,t
(6)
(α + 1)(λ + 1)
2(α + 2)[λ + α(1 + 3λ) + 1] 

and
(
|a3 | ≤ min
(|p00 (0)| + |q 00 (0)|)|b| cos β |b|2 cos2 β(|p0 (0)|2 + |q 0 (0)|2 )
+
,
4(α + 2)(2λ + 1)
2(α + 1)2 (λ + 1)2
|b| cos β |p00 (0)|[5λ + α(3λ + 1) + 3] + |q 00 (0)|(3λ + 1)|1 − α|
·
.
4(α + 2)
(2λ + 1)[λ + α(3λ + 1) + 1]
(7)
)
Proof. It follows from the conditions (4) and (5) that
(
iβ
e
1
zf 0 (z) f (z) α
zf 00 (z) 0
1+
(1 − λ)
(
) + λ(1 + 0
)(f (z))α − 1
b
f (z)
z
f (z)
"
#)
= p(z) cos β + i sin β
(8)
and
1
ωg 0 (ω) g(ω) α
ωg 00 (ω) 0
1+
(1 − λ)
(
) + λ(1 + 0
)(g (ω))α − 1
b
g(ω)
ω
g (ω)
(
e
iβ
"
#)
11
Coefficient Estimates for λ–Bazilevič Functions...
= q(ω) cos β + i sin β,
(9)
where p(z) ∈ H(U ), q(ω) ∈ H(U ). Furthermore, the functions p(z) and q(ω)
have the following series expansions
p(z) = 1 + p1 z + p2 z 2 + · · · , pm =
p(m) (0)
m!
(m ∈ N )
(10)
and
q (m) (0)
(m ∈ N )
(11)
m!
respectively. Now, in view of the series expansions (10) and (11), by equating
the coefficients in (8) and (9), we get
q(ω) = 1 + q1 ω + q2 ω 2 + · · · , qm =
eiβ
[(α + 1)(λ + 1)] a2 = p1 cos β,
b
(12)
(α − 1)(α + 2)(3λ + 1) 2
eiβ
(α + 2)(2λ + 1)a3 +
a2 = p2 cos β,
b
2
"
#
−
eiβ
[(α + 1)(λ + 1)] a2 = q1 cos β,
b
(13)
(14)
and
eiβ
(α − 1)(α + 2)(3λ + 1) 2
a2 = q2 cos β. (15)
(α + 2)(2λ + 1)(2a22 − a3 ) +
b
2
"
#
We find from (12) and (14) that
p1 = −q1
and
"
2e
i2β
(α + 1)(λ + 1)
b
(16)
#2
a22 = (p21 + q12 ) cos2 β.
(17)
Also, from (13) and (15), we obtain
eiβ
(α + 2)[λ + α(3λ + 1) + 1] a22 = (p2 + q2 ) cos β.
b
(18)
Therefore, we find from (17) and (18) that
a22 =
(p21 + q12 )b2 cos2 β
2e2iβ (α + 1)2 (λ + 1)2
and
a22 =
eiβ (α
(p2 + q2 )b cos β
.
+ 2)[λ + α(3λ + 1) + 1]
(19)
(20)
12
Shuhai Li et al.
Since p1 = p0 (0), p2 =
that
p00 (0)
, q1
2
= q 0 (0), q2 =
q 00 (0)
,
2
it follows from (19) and (20)
q
|a2 | ≤
and
|a2 | ≤
v
u
u
t
|b| cos β |p0 (0)|2 + |q 0 (0)|2
(α + 1)(λ + 1)
(|p00 (0)| + |q 00 (0)|)|b| cos β
,
2(α + 2)[λ + α(1 + 3λ) + 1]
which gives us the desired estimate on |a2 | as asserted in (6).
Next, in order to find the bound on |a3 |, by subtracting (13) from (15), we
get
eiβ
[2(α + 2)(2λ + 1)] (a3 − a22 ) = (p2 − q2 ) cos β.
(21)
b
Thus, upon substituting the value of a22 from (16) and (19) into (21), it follows
that
(p2 − q2 )b cos β
b2 cos2 β(p21 + q12 )
a3 = iβ
+ 2iβ
,
2e (α + 2)(2λ + 1) 2e (α + 1)2 (λ + 1)2
which yields
|a3 | ≤
(|p00 (0)| + |q 00 (0)|)|b| cos β |b|2 cos2 β(|p0 (0)|2 + |q 0 (0)|2 )
+
.
4(α + 2)(2λ + 1)
(α + 1)2 (λ + 1)2
On the other hand, by using (16) and (10) in (21), we obtain
a3 =
[5λ + α(1 + 3λ) + 3]p2 + (3λ + 1)(1 − α)q2
b cos β
·
,
+ 2)
(2λ + 1)[λ + α(1 + 3λ) + 1]
2eiβ (α
it follows that
|b| cos β [5λ + α(1 + 3λ) + 3]|p00 (0)| + (3λ + 1)|1 − α||q 00 (0)|
|a3 | ≤
·
.
4(α + 2)
(2λ + 1)[λ + α(1 + 3λ) + 1]
This completes the proof of Theorem 2.1.
For b = 1, β = 0, Theorem 2.1 readily yields the following coefficient estimates for Uαp,q (0, 1, λ).
Corollary 2.2 Suppose that f (z) ∈ A of the form (1), be in the class
Uαp,q (0, 1, λ). Then
q
 |p0 (0)|2
v

u

+ |q 0 (0)|2 u
|p00 (0)| + |q 00 (0)|
|a2 | ≤ min
,t
 (α + 1)(λ + 1)
2(α + 2)[λ + α(3λ + 1) + 1] 
and
(
|a3 | ≤ min
|p00 (0)| + |q 00 (0)|
(|p0 (0)|2 + |q 0 (0)|2 )
+
,
4(α + 2)(2λ + 1)
2(α + 1)2 (λ + 1)2
|p00 (0)|[5λ + α(3λ + 1) + 3] + |q 00 (0)|(3λ + 1)|1 − α|
.
4(α + 2)(2λ + 1)[λ + α(3λ + 1) + 1]
)
13
Coefficient Estimates for λ–Bazilevič Functions...
For b = 1, β = 0, α = 0, λ = 0, we obtain the results in [15] by S. Bulut.
For b = 1, β = 0, α = 1, λ = 0, Theorem 2.1 readily improve coefficient
estimates for U (p, q) in [9] as follows.
Corollary 2.3 Suppose that f (z) ∈ A of the form (1), be in the class
U (p, q). Then


s
 |p0 (0)|
|p00 (0)| + |q 00 (0)| 
,
|a2 | ≤ min

 2
12
and
|p00 (0)| + |q 00 (0)| (p0 (0))2 |p00 (0)|
|a3 | ≤ min
+
,
.
12
4
6
(
3
)
Coefficient Bounds for the Function Class
Lϕα(β, b, λ) and Bαη (β, b, λ)
In order to prove our main results, we first recall the following lemmas.
Lemma 3.1 (see [12]) If p(z) ∈ P,then |pk | ≤ 2 for each k, where P is
the family of all functions p(z) analytic in U for which <{p(z)} > 0, p(z) =
1 + p1 z + p2 z 2 + · · · for z ∈ U .
Theorem 3.2 Suppose that f (z) ∈ A of the form (1), be in the class
Then
Lϕα (β, b, λ).
v
u


2|b| cos β(|B1 | + |B2 − B1 |)
|B1 ||b| cos β u
|a2 | ≤ min
,t
,
 (α + 1)(λ + 1)
(α + 2)[λ + α(3λ + 1) + 1]
q


|B1 | 2|B1 ||b| cos β
q
|B12 b cos β(α + 2)[λ + α(3λ + 1) + 1] − 2(B2 − B1 )eiβ (α + 1)2 (λ + 1)2 | 
(22)
and
|B1 |2 |b|2 cos2 β
|B1 ||b| cos β
+
, Q1 , Q2 ,
|a3 | ≤ min
(α + 2)(2λ + 1) (α + 1)2 (λ + 1)2
(
)
(23)
where
Q1 =
|b| cos β{[5λ + (α + |1 − α|)(1 + 3λ) + 3]|B1 | + 4(2λ + 1)|B2 − B1 |}
,
2(α + 2)(2λ + 1)[λ + α(3λ + 1) + 1]
Q2 =
|B1 b|{B12 |b| cos β(α + 2)[5λ + (α + |1 − α|)(3λ + 1) + 3] + Q3 }
,
2(α + 2)(2λ + 1)|B12 b(α + 2)[λ + α(3λ + 1) + 1] − Q4 |
Q3 = 4|B2 − B1 |(α + 1)2 (λ + 1)2
and
Q4 = 2(B2 − B1 )(1 + i tan β)(1 + α)2 (1 + λ)2 .
14
Shuhai Li et al.
Proof. Let f ∈ Lϕα (β, b, λ), consider the analytic functions u, v : U −→ U ,
with u(0) = v(0) = 0, such that
(
iβ
e
1
zf 0 (z) f (z) α
zf 00 (z) 0
1+
(1 − λ)
(
) + λ(1 + 0
)(f (z))α − 1
b
f (z)
z
f (z)
"
#)
= ϕ(u(z)) cos β + i sin β
(24)
and
1
ωg 0 (ω) g(ω) α
ωg 00 (ω) 0
1+
(1 − λ)
(
) + λ(1 + 0
)(g (ω))α − 1
b
g(ω)
ω
g (ω)
(
e
iβ
"
#)
= ϕ(v(ω)) cos β + i sin β,
(25)
where g := f −1 .
Define the function m and n by
m(z) =
1 + u(z)
= 1 + m1 z + m2 z 2 + · · · ,
1 − u(z)
n(z) =
1 + v(z)
= 1 + n1 z + n2 z 2 + · · · ,
1 − v(z)
and
it follows that
u(z) =
m(z) − 1
m1
1
m2
=
z + (m2 − 1 )z 2 + · · ·
m(z) + 1
2
2
2
(26)
n(z) − 1
n1
1
n2
= z + (n2 − 1 )z 2 + · · ·
n(z) + 1
2
2
2
(27)
and
v(z) =
It is clear that m and n are analytic in U and m(0) = n(0) = 1. Since
u, v : U −→ U , the function m and n have positive real part in U , by virtue
of Lemma 3.1, we have |mi | ≤ 2 and |ni | ≤ 2 (i = 1, 2, · · ·).
From (24),(25),(26) and (27), it follows that
(
eiβ
1
zf 0 (z) f (z) α
zf 00 (z) 0
1+
(1 − λ)
(
) + λ(1 + 0
)(f (z))α − 1
b
f (z)
z
f (z)
"
#)
= ϕ(
m(z) − 1
) cos β + i sin β
m(z) + 1
(28)
and
1
ωg 0 (ω) g(ω) α
ωg 00 (ω) 0
1+
(1 − λ)
(
) + λ(1 + 0
)(g (ω))α − 1
b
g(ω)
ω
g (ω)
(
e
iβ
"
#)
15
Coefficient Estimates for λ–Bazilevič Functions...
= ϕ(
n(ω) − 1
) cos β + i sin β.
n(ω) + 1
(29)
According to (3), it is evident that
m1 B1
m2 B1 m21 (B2 − B1 ) 2
ϕ(u(z)) = 1 +
z+(
+
)z + · · ·
2
2
4
and
ϕ(v(ω)) = 1 +
n2 B1 n21 (B2 − B1 ) 2
n1 B1
ω+(
+
)ω + · · ·
2
2
4
Since
(
iβ
e
1
zf 0 (z) f (z) α
zf 00 (z) 0
1+
(1 − λ)
(
) + λ(1 + 0
)(f (z))α − 1
b
f (z)
z
f (z)
"
= eiβ +
#)
eiβ
eiβ (α + 1)(λ + 1)
a2 z +
(α + 2)(2λ + 1)a3
b
b
(α − 1)(α + 2)(3λ + 1) 2 2
+
a2 z + · · ·
2
and
ωg 0 (z) g(ω) α
ωg 00 (ω) 0
1
(1 − λ)
(
) + λ(1 + 0
)(g (ω))α − 1
1+
b
g(z)
ω
g (ω)
"
(
e
iβ
#)
eiβ (α + 1)(λ + 1)
eiβ
a2 ω +
(α + 2)(2λ + 1)(2a22 − a3 )
b
b
(α − 1)(α + 2)(3λ + 1) 2 2
+
a2 ω + · · ·
2
By equating the coefficients in (28) and (29), we get
= eiβ −
m1 B1
eiβ (α + 1)(λ + 1)
a2 =
,
b cos β
2
(30)
eiβ
(α − 1)(α + 2)(1 + 3λ) 2
[(α + 2)(1 + 2λ)a3 +
a2 ]
b cos β
2
m2 B1 m21 (B2 − B1 )
+
2
4
iβ
n1 B1
e (α + 1)(λ + 1)
−
a2 =
,
b cos β
2
=
and
eiβ
(α − 1)(α + 2)(3λ + 1) 2
[(α + 2)(2λ + 1)(a22 − a3 ) +
a2 ]
b cos β
2
(31)
(32)
16
Shuhai Li et al.
n2 B1 n21 (B2 − B1 )
+
2
4
We find from (30) and (32) that
=
and
(33)
m1 = −n1
(34)
2e2iβ (α + 1)2 (λ + 1)2 2 B12 2
a2 =
(m1 + n21 ).
b2 cos2 β
4
(35)
Also, from (31) and (33), we obtain
eiβ
(m2 + n2 )B1 (B2 − B1 )(m21 + n21 )
(α+2)[λ+α(1+3λ)+1]a22 =
+
(36)
b cos β
2
4
From (35) and (36), we have
(m2 + n2 )B13 b2 cos2 β
=
2beiβ B12 cos β(α + 2)[λ + α(3λ + 1) + 1] − 4(B2 − B1 )e2iβ (α + 1)2 (λ + 1)2
(37)
Since |mi | ≤ 2 and |ni | ≤ 2 (i = 1, 2), it follows from (35), (36) and (37) that
a22
|a2 | ≤
|a2 | ≤
|B1 ||b| cos β
,
(α + 1)(1 + λ)
v
u
u 2|b| cos β(|B1 | + |B2
t
− B1 |)
,
(α + 2)[1 + λ + α(1 + 3λ)]
and
q
|a2 | ≤ q
|B1 | 2|B1 ||b| cos β
|B12 b cos β(α + 2)[1 + λ + α(1 + 3λ)] − 2(B2 − B1 )eiβ (1 + α)2 (1 + λ)2 |
which yields the desired estimate on |a2 | as asserted in (22).
Next, in order to find the bound on |a3 |, by subtracting (31) from (33), we
get
2eiβ
(m2 − n2 )B1
(α + 2)(2λ + 1)(a3 − a22 ) =
(38)
b cos β
2
Substituting value of a22 from (35), (36) and (37) in (38), we get
a3 =
B12 b2 cos2 β(m21 + n21 )
(m2 − n2 )B1 b cos β
+
,
a3 = iβ
4e (α + 2)(2λ + 1) 8e2iβ (α + 1)2 (λ + 1)2
(39)
b cos β{([5λ + α(3λ + 1) + 3]m2 + (3λ + 1)(1 − α)n2 )B1 + W3 }
4eiβ (α + 2)(2λ + 1)[λ + α(1 + 3λ) + 1]
(40)
,
17
Coefficient Estimates for λ–Bazilevič Functions...
and
a3 =
B1 b cos β(W1 m2 + W2 n2 )
, (41)
4eiβ (α + 2)(2λ + 1){B12 b cos β(α + 2)[λ + α(1 + 3λ) + 1] − W4 }
where
W1 = (α + 2)[5λ + α(1 + 3λ) + 3]B12 b cos β − 2(B2 − B1 )eiβ (α + 1)2 (λ + 1)2 ,
W2 = (α + 2)(3λ + 1)(1 − α)B12 b cos β + 2(B2 − B1 )eiβ (α + 1)2 (λ + 1)2 ,
W3 = 2(2λ + 1)(B2 − B1 )m21
and
W4 = 2(B2 − B1 )eiβ (α + 1)2 (λ + 1)2 .
Using (39), (40) and (41), we have
|a3 | ≤
|a3 | ≤
|B1 |2 |b|2 cos2 β
|B1 ||b| cos β
+
,
(α + 2)(2λ + 1) (α + 1)2 (λ + 1)2
(42)
|b| cos β{[5λ + (α + |1 − α|)(3λ + 1) + 3]|B1 | + 4(2λ + 1)|B2 − B1 |}
2(α + 2)(2λ + 1)[λ + α(1 + 3λ) + 1]
(43)
and
|a3 | ≤
|B1 ||b| cos β{B12 |b|(α + 2)[5λ + (α + |1 − α|)(3λ + 1) + 3] + W5 }
,
2(α + 2)(2λ + 1)|B12 b(α + 2)[λ + α(1 + 3λ) + 1] − W6 |
(44)
where
W5 = 4|B2 − B1 |(α + 1)2 (λ + 1)2
and
W6 = 2(B2 − B1 )(1 + i tan β)(1 + α)2 (1 + λ)2 .
From (42), (43) and (44), we obtain the desired estimate on |a3 | given in
(23). This is the end of Theorem 3.2.
Let b = 1, β = 0, λ = 0, Theorem 3.2 improves Theorem 2.8 in [6] by E.
Deniz as follows.
Corollary 3.3 Suppose that f ∈ A of the form (1), be in the class BΣ (α, ϕ).
Then
v
u
|B1 | u
2(|B1 | + |B2 − B1 )|)
|a2 | ≤ min
,t
,
α+1
(α + 2)(α + 1)
q
|B1 | 2|B1 |
q
|B12 (α + 2)(α + 1) − 2(B2 − B1 )(α + 1)2 |
18
Shuhai Li et al.
and
|B1 |
|B1 |2 (α + |1 − α| + 3)|B1 | + 4|B2 − B1 |
|a3 | ≤ min
+
,
,
α + 2 (1 + α)2
2(α + 2)(α + 1)
(
|B1 |{(α + 2)(α + |1 − α| + 3)B12 + 4|B2 − B1 |(α + 1)2 }
.
2(α + 2)|B12 (α + 2)(α + 1) − 2(B2 − B1 )(α + 1)2 |
)
Also, let b = 1, β = 0, α = 0 in Theorem 3.2, we obtain the following
Corollary, which improves Theorem 2.3 in [11].
Corollary 3.4 Suppose that f (z) ∈ A of the form (1), be in the class
MΣ (λ, ϕ). Then
q
v
u




|B1 | |B1 |
|B1 | u
(|B1 | + |B2 − B1 |)
|a2 | ≤ min 
,t
,q
(λ + 1)
(λ + 1)
|B12 (λ + 1) − (B2 − B1 )(1 + λ)2 | 
and
|a3 | ≤ min
|B1 |2 |B1 | + |B2 − B1 |
|B1 |
+
,
,
2(2λ + 1) (λ + 1)2
λ+1
|B1 |[2(2λ + 1)B12 + (λ + 1)2 |B2 − B1 |]
.
2(2λ + 1)(λ + 1)|B12 − (B2 − B1 )(λ + 1)|
1+z η
) (0 < η ≤ 1) in Theorem 3.2, we get the following
By setting ϕ(z) = ( 1−z
Corollary:
Corollary 3.5 Suppose that f (z) ∈ A of the form (1), be in the class
Then
Bαη (β, b, λ).


v
u
u
η|b| cos β(2 − η)
2η|b| cos β
|a2 | ≤ min
, 2t
,
 (α + 1)(λ + 1)
(α + 2)[λ + α(1 + 3λ) + 1]
2η|b| cos β
q


|bη cos β(α + 2)[λ + α(3λ + 1) + 1] + eiβ (1 − η)[(α + 1)2 (λ + 1)]2 | 
and
2η|b| cos β
4η 2 |b|2 cos2 β
|a3 | ≤ min
+
, Nλ (α, β, η, b)), Qλ (α, β, η, b) ,
(α + 2)(2λ + 1) (α + 1)2 (λ + 1)2
(
)
where
Nλ (α, β, η, b) =
η|b| cos β [5λ + (α + |1 − α|)(3λ + 1) + 3] + 4(2λ + 1)(1 − η)
·
,
(α + 2)
(2λ + 1)[λ + α(3λ + 1) + 1]
19
Coefficient Estimates for λ–Bazilevič Functions...
Qλ (α, β, η, b) =
η|b|{η|b| cos β(α + 2)[5λ + (α + |1 − α|)(3λ + 1) + 3] + W7 }
,
(α + 2)(2λ + 1)|bη(α + 2)[λ + α(3λ + 1) + 1] + W8 |
W7 = 2(1 − η)(α + 1)2 (λ + 1)2
and
W8 = (1 − η)(1 + i tan β)(α + 1)2 (λ + 1)2 .
Especially, for b = 1, β = 0, α = 0 , Corollary 3.5 readily yields the following
coefficient estimates for B0η (0, 1, λ),
Corollary 3.6 Suppose that f (z) ∈ A of the form (1), be in the class
Then
B0η (0, 1, λ).
|a2 | ≤ min


2η
 (λ + 1)
v
u
u 2η(2 − η)
,t
,q
(λ + 1)
and
|a3 | ≤ min


2η
(λ + 1)|2η + (1 − η)(λ + 1)| 
4η 2
2η(2 − η)
η
+
,
,
2
(2λ + 1) (λ + 1)
(λ + 1)
η{4η(2λ + 1) + (1 − η)(λ + 1)2 }
.
(2λ + 1)(λ + 1)|2η + (1 − η)(λ + 1)|
By setting ϕ(z) =
Corollary:
1+(1−2γ)z
(0
1−z
< γ ≤ 1) in Theorem 3.2, we get the following
Corollary 3.7 Suppose that f (z) ∈ A of the form (1), be in the class
(β, b, λ). Then
L1−2γ,−1
α
v
u
2(1 − γ)|b| cos β u
4|b| cos β(1 − γ)
|a2 | ≤ min
,t
,
(α + 1)(λ + 1)
(α + 2)[λ + α(3λ + 1) + 1]
q
2 (1 − γ)
q
|(α + 2)[λ + α(1 + 3λ) + 1]|
and
2(1 − γ)|b| cos β 4(1 − γ)2 |b|2 cos2 β
|a3 | ≤ min
+
, M1 , M2 ,
(α + 2)(2λ + 1)
(α + 1)2 (λ + 1)2
(
where
M1 =
and
M2 =
(1 − γ)|b| cos β[5λ + (α + |1 − α|)(3λ + 1) + 3]
(α + 2)(2λ + 1)[λ + α(1 + 3λ) + 1]
(1 − γ)|b| cos β{[5λ + (α + |1 − α|)(3λ + 1) + 3]}
.
(α + 2)(2λ + 1)|[λ + α(3λ + 1) + 1]|
)
20
Shuhai Li et al.
Especially, for b = 1, β = 0, α = 1, λ = 0 , Corollary 3.7 readily improves
the result in [9].
Corollary 3.8 Suppose that f (z) ∈ A of the form (1), be in the class
(0, 1, 0). Then
L1−2γ,−1
1


|a2 | ≤ min (1 − γ),
and
(
s

2(1 − γ) 

3
2(1 − γ)
2(1 − γ)
|a3 | ≤ min
+ (1 − γ)2 ,
3
3
)
=
2(1 − γ)
.
3
Acknowledgements: This paper was partly supported by the Natural
Science Foundation of Inner Mongolia of the People’s Republic of China under
Grant 2014MS0101.
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