ON CERTAIN SUBCLASS OF BAZILEVIČ
FUNCTIONS
Subclass of Bazilevič
Functions
DONG GUO AND MING-SHENG LIU
School of Mathematical Sciences,
South China Normal University,
Guangzhou 510631, Guangdong, China.
EMail: liumsh@scnu.edu.cn
Dong Guo and Ming-Sheng Liu
vol. 8, iss. 1, art. 12, 2007
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Received:
22 September, 2006
Accepted:
26 February, 2007
Communicated by:
H.M. Srivastava
2000 AMS Sub. Class.:
Primary 30C45; Secondary 26A33, 33C05.
Key words:
Starlike function, Bazilevič Function, Subordination relationships, Inclusion relationship, Coefficient estimates, Integral operator, Covering theorem, FeketeSzegö inequalities.
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Abstract:
P∞
Let H be the class of functions f (z) of the form f (z) = z + n=2 an z n ,
which are analytic in the unit disk U = {z : |z| < 1}. In this paper, the
authors introduce a subclass M (α, λ, ρ) of H and study its some properties. The subordination relationships, inclusion relationships, coefficient
estimates, the integral operator and covering theorem are proven here for
each of the function classes. Furthermore, some interesting Fekete-Szegö
inequalities are obtained. Some of the results, presented in this paper, generalize the corresponding results of earlier authors.
Subclass of Bazilevič
Functions
Dong Guo and Ming-Sheng Liu
vol. 8, iss. 1, art. 12, 2007
Acknowledgements: This research is partly supported by the Doctoral Foundation of the Education committee of China (No. 20050574002).
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Contents
1
Introduction
4
2
Preliminaries
7
3
Main Results and Their Proofs
9
Subclass of Bazilevič
Functions
Dong Guo and Ming-Sheng Liu
vol. 8, iss. 1, art. 12, 2007
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1.
Introduction
Let H denote the class of functions f of the form
(1.1)
f (z) = z +
∞
X
an z n ,
n=2
which are analytic in the open unit disk U = {z : |z| < 1}, and let S denote the
class of all functions in H which are univalent in the disk U . Suppose also that S ∗ , K
and α − K denote the familiar subclasses of H consisting of functions which are,
respectively, starlike in U, convex in U and α− convex in U. Thus we have
0 zf (z)
∗
S = f : f ∈ H and R
> 0, z ∈ U ,
f (z)
zf 00 (z)
K = f : f ∈ H and R 1 + 0
> 0, z ∈ U
f (z)
and
zf 0 (z)
zf 00 (z)
α−K = f : f ∈ H and R α 1 + 0
+ (1 − α)
> 0,
f (z)
f (z)
Dong Guo and Ming-Sheng Liu
vol. 8, iss. 1, art. 12, 2007
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z∈U
Subclass of Bazilevič
Functions
.
Let f (z) and F (z) be analytic in U . Then we say that the function f (z) is subordinate to F (z) in U , if there exists an analytic function ω(z) in U such that |ω(z)| ≤ |z|
and f (z) = F (ω(z)), denoted f ≺ F or f (z) ≺ F (z). If F (z) is univalent in U ,
then the subordination is equivalent to f (0) = F (0) and f (U ) ⊂ F (U ) (see [18]).
Assuming that α > 0, λ ≥ 0, ρ < 1, a function p(z) = 1 + p1 z + p2 z 2 + · · ·
is said to be in the class Pρ if and only if p(z) is analytic in the unit disk U and
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Rp(z) > ρ, z ∈ U . A function f (z) ∈ H is said to be in the class B(λ, α, ρ) if and
only if it satisfies
α
α 0
f (z)
zf (z) f (z)
(1.2)
R(1 − λ)
+λ
> ρ, z ∈ U,
z
f (z)
z
α
α f (z)
such
that
= 1. It is
where we choose the branch of the power f (z)
z
z
z=0
obvious that the subclass B(1, α, 0) is the subclass of Bazilevič functions, which is
the subclass of univalent functions S, we set B(α, ρ) ≡ B(1, α, ρ). The function
class B(λ, α, ρ) was introduced and studied by Liu [10]. Some special cases of the
function class B(λ, α, ρ) had been studied by Bazilevič [1], Chichra [2], Ding, Ling
and Bao [3], Liu [9] and Singh [19], respectively.
Liu [11] introduced the following class B(λ, α, A, B, g(z)) of analytic functions,
and studied its some properties.
B(λ, α, A, B, g(z))
α
α
0
f (z)
zf (z) f (z)
1 + Az
zg 0 (z)
= f ∈H: 1−λ
+λ
≺
,
g(z)
g(z)
f (z)
g(z)
1 + Bz
where α > 0, λ ≥ 0, −1 ≤ B < A ≤ 1, g(z) ∈ S ∗ .
Fekete and Szegö [4] showed that for f ∈ S given by (1.1),

3 − 4u,
if µ ≤ 0,



|a3 − µa22 | ≤
1 + 2e−2/(1−µ) , if 0 ≤ µ < 1,



4 − 3µ,
if µ ≥ 1.
As a result, many authors studied similar problems for some subclasses of H or
S (see [6, 7, 8, 13, 14, 15, 20]), which is popularly referred to as the Fekete-Szegö
Subclass of Bazilevič
Functions
Dong Guo and Ming-Sheng Liu
vol. 8, iss. 1, art. 12, 2007
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inequality or the Fekete-Szegö problem. Li and Liu [12] obtained the Fekete-Szegö
inequality for the function class B(λ, α, ρ).
Recently, Patel [17] introduced the following subclass Mp (λ, µ, A, B) of p−valent
Bazilevič functions, and studied some
its properties.
Pof
∞
p
An analytic function f (z) = z + n=p+1 an z n is said to be in the class Mp (λ, µ, A, B)
P
n
if and only if there exists a p−valent starlike function g(z) = z p + ∞
n=p+1 bn z such
that
µ
0
zf 0 (z) f (z)
zf 00 (z) zf 0 (z)
zf (z) zg 0 (z)
1 + Az
+λ 1 + 0
−
+µ
−
≺p
,
f (z)
g(z)
f (z)
f (z)
f (z)
g(z)
1 + Bz
where µ ≥ 0, λ > 0, −1 ≤ B < A ≤ 1.
In the present paper, we introduce the following subclass of analytic functions,
and obtain some interesting results.
Definition 1.1. Assume that α ≥ 0, λ ≥ 0, 0 ≤ ρ < 1, f ∈ H. We say that
f (z) ∈ M (α, λ, ρ) if and only if f (z) satisfies the following inequality:
0
α
0
zf (z) f (z)
zf 00 (z) zf 0 (z)
zf (z)
R
+λ 1+ 0
−
+α
−1
> ρ, z ∈ U.
f (z)
z
f (z)
f (z)
f (z)
It is evident that M (α, 0, ρ) = B(α, ρ)(α ≥ 0) and M (0, α, 0) = α −K(α ≥ 0).
Subclass of Bazilevič
Functions
Dong Guo and Ming-Sheng Liu
vol. 8, iss. 1, art. 12, 2007
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2.
Preliminaries
To derive our main results, we shall require the following lemmas.
Lemma 2.1 ([16]). If −1 ≤ B < A ≤ 1, β > 0 and the complex number γ satisfies
R(λ) ≥ −β(1−A)
, then the differential equation
1−B
q(z) +
1 + Az
zq 0 (z)
=
,
βq(z) + γ
1 + Bz
z ∈ U,
Dong Guo and Ming-Sheng Liu
has a univalent solution in U given by

z β+γ (1 + Bz)β(A−B)/B


Rz
− βγ , B =
6 0,

β+γ−1
β(A−B)/B

(1 + Bt)
dt
 β 0 t
(2.1)
q(z) =


z β+γ exp(βAz)


 R z β+γ−1
− γ,
B = 0.
β 0 t
exp(βAt)dt β
If φ(z) = 1 + c1 z + c2 z 2 + · · · is analytic in U and satisfies
(2.2)
zφ0 (z)
1 + Az
≺
,
φ(z) +
βφ(z) + γ
1 + Bz
then
φ(z) ≺ q(z) ≺
1 + Az
,
1 + Bz
Subclass of Bazilevič
Functions
(z ∈ U),
vol. 8, iss. 1, art. 12, 2007
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(z ∈ U),
and q(z) is the best dominant of (2.2).
Lemma 2.2 ([11]). Suppose that F (z) is analytic and convex in U, and 0 ≤ λ ≤ 1,
f (z) ∈ H, g(z) ∈ H. If f (z) ≺ F (z) and g(z) ≺ F (z). Then
λf (z) + (1 − λ)g(z) ≺ F (z).
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P
n
Lemma 2.3 ([18]). Let p(z) = 1 + ∞
n=1 pn z ∈ P0 . Then
1
2
p2 − p1 ≤ 2 − 1 |p21 |
2 2
and |pn | ≤ 2 for all n ∈ N+ .
Lemma 2.4 ([1]). Let α ≥ 0, f ∈ H and for |z| < R ≤ 1,
0
α zf (z) f (z)
R
> 0,
f (z)
z
Subclass of Bazilevič
Functions
Dong Guo and Ming-Sheng Liu
vol. 8, iss. 1, art. 12, 2007
then f (z) is univalent in |z| < R.
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3.
Main Results and Their Proofs
Theorem 3.1. Let α ≥ 0 and λ > 0. If f (z) ∈ M (α, λ, ρ). Then
α
1 + (1 − 2ρ)z
zf 0 (z) f (z)
≺ q(z) ≺
, (z ∈ U),
(3.1)
f (z)
z
1−z
where
λz 1/λ (1 − z)−2(1−ρ)/λ
q(z) = R z (1−λ)/λ
,
−2(1−ρ)/λ dt
t
(1
−
t)
0
and q(z) is the best dominant of (3.1).
Proof. By applying the method of the proof of Theorem 3.1 in [17] mutatis mutandis,
we can prove this theorem.
Subclass of Bazilevič
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Dong Guo and Ming-Sheng Liu
vol. 8, iss. 1, art. 12, 2007
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With the aid of Lemma 2.4, from Theorem 3.1, we have the following inclusion
relation.
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Corollary 3.2. Let α ≥ 0, 0 ≤ ρ < 1 and λ ≥ 0, then
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M (α, λ, ρ) ⊂ M (α, 0, ρ) ⊂ M (α, 0, 0) ⊂ S.
Theorem 3.3. Let α ≥ 0 and λ2 > λ1 ≥ 0, 1 > ρ2 ≥ ρ1 ≥ 0, then
M (α, λ2 , ρ2 ) ⊂ M (α, λ1 , ρ1 ).
Proof. Suppose that f (z) ∈ M (α, λ2 , ρ2 ). Then, by the definition of M (α, λ2 , ρ2 ),
we have
0
α
0
zf (z) f (z)
zf 00 (z) zf 0 (z)
zf (z)
(3.2) R
+ λ2 1 + 0
−
+α
−1
f (z)
z
f (z)
f (z)
f (z)
> ρ2
(z ∈ U).
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Since α ≥ 0 and λ2 > λ1 ≥ 0, by Theorem 3.1, we obtain
0
α zf (z) f (z)
(3.3)
R
> ρ2 (z ∈ U).
f (z)
z
Setting λ =
λ1
,
λ2
so that 0 ≤ λ < 1, we find from (3.2) and (3.3) that
α
0
zf 00 (z) zf 0 (z)
zf (z)
zf 0 (z) f (z)
R
+ λ1 1 + 0
−
+α
−1
f (z)
z
f (z)
f (z)
f (z)
0
α
0
zf 00 (z) zf 0 (z)
zf (z)
zf (z) f (z)
= λR
+ λ2 [1 + 0
−
+α
−1
f (z)
z
f (z)
f (z)
f (z)
α
zf 0 (z) f (z)
+ (1 − λ)R
> ρ2 ≥ ρ1 (z ∈ U),
f (z)
z
Subclass of Bazilevič
Functions
Dong Guo and Ming-Sheng Liu
vol. 8, iss. 1, art. 12, 2007
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that is, f (z) ∈ M (α, λ1 , ρ1 ). Hence, we have M (α, λ2 , ρ2 ) ⊂ M (α, λ1 , ρ1 ), and
the proof of Theorem 3.3 is complete.
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Remark 1. Theorem 3.3 obviously provides a refinement of Corollary 3.2. Setting
α = 0, ρ2 = ρ1 = 0 in Theorem 3.3, we get Theorem 9.4 of [5].
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With the aid of Lemma 2.2, by using the method of our proof of Theorem 3.3, we
can prove the following inclusion relation.
Theorem 3.4. Let µ ≥ 0, −1 ≤ B < A ≤ 1 and λ2 > λ1 ≥ 0, then
Mp (λ2 , µ, A, B) ⊂ Mp (λ1 , µ, A, B).
By applying the method of the proof of Theorem 3.13, Theorem 3.6 and Theorem
3.11 in [17] mutatis mutandis, we can prove the following three results.
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Theorem 3.5. Let α ≥ 0, λ > 0 and γ > 0. If f (z) ∈ H satisfies
0
α 0
zf (z) f (z)
zf (z)
zf 00 (z) zf 0 (z)
γ
+λ 1 + 0
−
+α
−1
6= it,
f (z)
z
f (z)
f (z)
f (z)
p
where t is a real number satisfying |t| > λ(λ + 2γ), then
α 0
zf (z) f (z)
R
> 0, (z ∈ U).
f (z)
z
Theorem 3.6. Suppose that α > 0 and 0 ≤ ρ < 1. If f (z) ∈ H satisfies
0
α zf (z) f (z)
Re
> ρ, (z ∈ U),
f (z)
z
(z ∈ U),
Subclass of Bazilevič
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Dong Guo and Ming-Sheng Liu
vol. 8, iss. 1, art. 12, 2007
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then f (z) ∈ M (α, λ, ρ) for |z| < R(λ, ρ), where λ > 0, and
√

 (1+λ−ρ)− (1+λ−ρ)2 −(1−2ρ) , ρ 6= 1 ,
1−2ρ
2
R(λ, ρ) =
 1
,
ρ = 12 .
1+2λ
The bound R(λ, ρ) is the best possible.
(3.4)
Fα,δ (f ) = Fα,δ (f )(z) =
α+δ
zδ
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For a function f ∈ H, we define the integral operator Fα,δ as follows:
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α1
z
δ−1
α
t f (t) dt
0
where α and δ are real numbers with α > 0, δ > −α.
(z ∈ U),
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Theorem 3.7. Let α and δ be real numbers with α > 0, 0 ≤ ρ < 1, δ > max{−α, −αρ}
and let f (z) ∈ H. If
0
α
π
zf
(z)
f
(z)
arg
− ρ ≤ β (0 ≤ ρ < 1; 0 < β ≤ 1),
f (z)
z
2
then
0
π
zFα,δ (f ) Fα,δ (f ) α
arg
− ρ ≤ β,
Fα,δ (f )
z
2
where Fα,δ (f ) is the operator given by (3.4).
Now we derive the Fekete-Szegö inequality for the function class M (α, λ, ρ).
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Theorem 3.8. Suppose that f (z) = z + ∞
n=2 an z ∈ M (α, λ, ρ). Then
|a2 | ≤
2(1 − ρ)
,
(1 + λ)(1 + α)
and for each µ ∈ C, the following bound is sharp
2(1 − ρ)
|a3 − µa22 | ≤
(1 + 2λ)(2 + α)
(1
−
ρ)[2λ(3
+
α)
−
(2
+
α)(α
−
1
+
2µ
+
4µλ)]
.
× max 1, 1 +
(1 + λ)2 (1 + α)2
Proof.
Since f (z) ∈ M (α, λ, ρ), by Definition 1.1, there exists a function p(z) =
P+∞
1 + k=1 pk z k ∈ P0 , such that
α
0
zf 00 (z) zf 0 (z)
zf (z)
zf 0 (z) f (z)
+λ 1+ 0
−
+α
−1
f (z)
z
f (z)
f (z)
f (z)
= (1 − ρ)p(z) + ρ, z ∈ U.
Subclass of Bazilevič
Functions
Dong Guo and Ming-Sheng Liu
vol. 8, iss. 1, art. 12, 2007
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Equating coefficients, we obtain
1−ρ
p1 ,
(1 + λ)(1 + α)
h
i
(α+2)(α−1)
2
(1
−
ρ)
λ(3
+
α)
−
2
1−ρ
a3 =
p2 +
p2 .
2
2
(1 + 2λ)(2 + α)
(1 + λ) (1 + α) (1 + 2λ)(2 + α) 1
Thus, we have
1−ρ
1 2
2
a3 − µa2 =
p2 − p1
(1 + 2λ)(2 + α)
2
2
(1−ρ) [2λ(3+α)−(2+α)(α−1)−2µ(1+2λ)(2+α)]+(1−ρ)(1+λ)2 (1+α)2 2
+
p1 .
2(1+λ)2 (1+α)2 (1+2λ)(2+α)
a2 =
By Lemma 2.3, we obtain that |a2 | =
1−ρ
|p |
(1+λ)(1+α) 1
|a3 − µa22 | ≤ H(x) = A +
≤
2(1−ρ)
,
(1+λ)(1+α)
Dong Guo and Ming-Sheng Liu
vol. 8, iss. 1, art. 12, 2007
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and
ABx2
,
4
where x = |p1 | ≤ 2,
A=
Subclass of Bazilevič
Functions
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2(1 − ρ)
,
(1 + 2λ)(2 + α)
2
B=
2
|C| − (1 + λ) (1 + α)
,
(1 + λ)2 (1 + α)2
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and
C = (1 + λ)2 (1 + α)2 + (1 − ρ)[2λ(3 + α) − (2 + α)(α − 1 + 2µ + 4µλ)].
So, we have
|a3 − µa22 | ≤

 H(0) = A,
 H(2) =
A|C|
,
(1+λ)2 (1+α)2
|c| ≤ (1 + λ)2 (1 + α)2 ,
|c| ≥ (1 + λ)2 (1 + α)2 .
Close
Here equality is attained for the function given by
α
zf 0 (z) f (z)
(3.5)
f (z)
z

λz 1/λ (1 − z 2 )(ρ−1)/λ


R
, λ > 0,

z (1−λ)/λ

2 )(ρ−1)/λ dt

t
(1
−
t

0




1 + (1 − 2ρ)z 2



,
λ = 0,

1 − z2
=


λz 1/λ (1 − z)2(ρ−1)/λ


Rz
, λ > 0,


t(1−λ)/λ (1 − t)2(ρ−1)/λ dt


0





 1 + (1 − 2ρ)z ,
λ = 0,
1−z
|c| ≤ (1 + λ)2 (1 + α)2 ,
|c| ≤ (1 + λ)2 (1 + α)2
Dong Guo and Ming-Sheng Liu
vol. 8, iss. 1, art. 12, 2007
|c| ≥ (1 + λ)2 (1 + α)2 ,
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|c| ≥ (1 + λ)2 (1 + α)2 .
Setting λ = 0 in Theorem 3.8, we have the following corollary.
Corollary 3.9. If f (z) ∈ B(α, ρ) given by (1.1), then
|a2 | ≤
Subclass of Bazilevič
Functions
2(1 − ρ)
,
1+α
and for each µ ∈ C, the following bound is sharp
2(1 − ρ)
(1 − ρ)(2 + α)(1 − 2µ − α) 2
|a3 − µa2 | ≤
max 1, 1 +
.
2+α
(1 + α)2
Notice that M (0, α, 0) ≡ α − K, and from Theorem 3.8, we have the following
corollary.
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Corollary 3.10. Let α ≥ 0. If f (z) ∈ α − K given by (1.1). Then
|a2 | ≤
2
,
1+α
and for each µ ∈ C, the following bound is sharp
1
6α
+
2
−
4µ
−
8µα
2
.
|a3 − µa2 | ≤
max 1, 1 +
1 + 2α
(1 + α)2
Theorem 3.11 (Covering Theorem). Let α ≥ 0, λ ≥ 0 and f (z) ∈ M (α, λ, ρ),
then the unit disk U is mapped by f (z) on a domain that contains the disk |ω| < r1 ,
where
(1 + α)(1 + λ)
r1 =
.
2(1 + α)(1 + λ) + 2(1 − ρ)
Proof. Let ω 0 be any complex number such that f (z) 6= ω 0 (z ∈ U), then ω 0 6= 0
and (by Corollary 3.2) the function
1
ω 0 f (z)
= z + a2 +
z2 + · · · ,
ω 0 − f (z)
ω0
is univalent in U, so that
Dong Guo and Ming-Sheng Liu
vol. 8, iss. 1, art. 12, 2007
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a2 + 1 ≤ 2,
ω0 Therefore, according to Theorem 3.8, we obtain
|ω 0 | ≥
Subclass of Bazilevič
Functions
(1 + α)(1 + λ)
= r1 .
2(1 + α)(1 + λ) + 2(1 − ρ)
Thus we have completed the proof of Theorem 3.11.
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Remark 2. Setting α = λ = ρ = 0 in Theorem 3.11, we get the well-known 14 −
covering theorem for the familiar class S ∗ of starlike functions.
If 0 ≤ µ ≤ µ1 and µ is a real number, Theorem 3.8 can be improved as follows.
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Theorem 3.12. Suppose that f (z) = z + ∞
n=2 an z ∈ M (λ, α, ρ) and µ ∈ R.
Then
(3.6) |a3 − µa22 | + µ|a2 |2
2(1 − ρ)
(1 − ρ)[2λ(3 + α) − (2 + α)(α − 1)]
≤
1+
,
(1 + 2λ)(2 + α)
(1 + λ)2 (1 + α)2
0 ≤ µ ≤ µ0 ,
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Dong Guo and Ming-Sheng Liu
vol. 8, iss. 1, art. 12, 2007
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(3.7)
|a3 − µa22 | + (µ1 − µ)|a2 |2 ≤
2(1 − ρ)
,
(1 + 2λ)(2 + α)
Contents
µ0 ≤ µ ≤ µ1 ,
and these inequalities are sharp, where
1
2λ − α(2 + α)
(1 + λ)2 (1 + α)2
µ0 = +
+
,
2 2(1 + 2λ)(2 + α) 2(1 + 2λ)(2 + α)(1 − ρ)
1
2λ − α(2 + α)
(1 + λ)2 (1 + α)2
+
+
.
2 2(1 + 2λ)(2 + α) (1 + 2λ)(2 + α)(1 − ρ)
Proof. From Theorem 3.8, we get
µ1 =
2(1 − ρ)
2(1 − ρ)
(3.8) |a3 − µa22 | ≤
+
(1 + 2λ)(2 + α) (1 + 2λ)(2 + α)
|(1 − ρ)[2λ(3 + α) − (2 + α) [(α − 1) + 2µ(1 + 2λ)]] + (1 + λ)2 (1 + α)2 | 1
− |p1 |2 .
·
2
2
4(1 + λ) (1 + α)
4
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Using (3.8) and a2 =
1−ρ
p,
(1+λ)(1+α) 1
if 0 ≤ µ ≤ µ0 , we obtain
2(1 − ρ)
2(1 − ρ)
+
(1 + 2λ)(2 + α) (1 + 2λ)(2 + α)
(1 − ρ)[2λ(3 + α) − (2 + α)(α − 1) − 2µ(1 + 2λ)(2 + α)]
×
|p1 |2
4(1 + λ)2 (1 + α)2
2(1 − ρ)2 [2λ(3 + α) − (2 + α)(α − 1)]
2(1 − ρ)
=
+
|p1 |2 − µ|a2 |2 .
2
2
(1 + 2λ)(2 + α)
4(1 + 2λ)(2 + α)(1 + λ) (1 + α)
|a3 − µa22 | ≤
Dong Guo and Ming-Sheng Liu
vol. 8, iss. 1, art. 12, 2007
Hence
|a3 − µa22 | + µ|a2 |2
2(1 − ρ)
2(1 − ρ)2 [2λ(3 + α) − (2 + α)(α − 1)]
+
|p1 |2
≤
(1 + 2λ)(2 + α)
4(1 + 2λ)(2 + α)(1 + λ)2 (1 + α)2
2(1 − ρ)
(1 − ρ)[2λ(3 + α) − (2 + α)(α − 1)]
≤
1+
,
(1 + 2λ)(2 + α)
(1 + λ)2 (1 + α)2
Subclass of Bazilevič
Functions
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0 ≤ µ ≤ µ0 .
If µ0 ≤ µ ≤ µ1 , from (3.8), we obtain
|a3 − µa22 |
2(1 − ρ)
2(1 − ρ)
≤
+
(1 + 2λ)(2 + α) (1 + 2λ)(2 + α)
−2(1 + λ)2 (1 + α)2 − (1 − ρ)[2λ(3 + α) − (2 + α)(α − 1 + 2µ + 4µλ)]
×
|p1 |2
2
2
4(1 + λ) (1 + α)
2(1 − ρ)
=
− (µ1 − µ)|a2 |2 .
(1 + 2λ)(2 + α)
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Therefore
2(1 − ρ)
,
µ0 ≤ µ ≤ µ1 .
(1 + 2λ)(2 + α)
Here equality is attained for the function given by (3.5), and the proof of Theorem
3.12 is complete.
|a3 − µa22 | + (µ1 − µ)|a2 |2 ≤
Theorem 3.13. Let f (z) ∈ H, α ≥ 0, λ ≥ 0 and 0 < k ≤ 1. If
0
zf (z) f (z) α
(3.9) f (z)
z
0
zf 00 (z) zf 0 (z)
zf (z)
+λ 1 + 0
−
+α
−1
− 1 < k,
f (z)
f (z)
f (z)
then
k
|a2 | ≤
,
(1 + λ)(1 + α)
and for each µ ∈ C, the following bound is sharp
|a3 −
Subclass of Bazilevič
Functions
Dong Guo and Ming-Sheng Liu
vol. 8, iss. 1, art. 12, 2007
z ∈ U,
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Contents
µa22 |

α

k(1
+
2λ)(2
+
α)
+
1 − 2µ − 1+2λ
k
≤
max 1,

(1 + 2λ)(2 + α)
2(1 + λ)2 (1 + α)2
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

2λ
(1+2λ)(2+α) Proof. By (3.9), there exists a function p(z) ∈ P0 such that for all z ∈ U
α
0
zf 0 (z) f (z)
zf 00 (z) zf 0 (z)
zf (z)
+λ 1+ 0
−
+α
−1
f (z)
z
f (z)
f (z)
f (z)
2k
+ 1 − k.
=
1 + p(z)

.
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Equating the coefficients, we obtain
k
p1 ,
2(1 + λ)(1 + α)
h
i
k 2 λ(3 + α) − (2+α)(α−1)
2
k
1
(1 + 2λ)(2 + α)a3 = −
p2 − p21 +
p21 .
2
2
2
4(1 + λ) (1 + α)2
a2 = −
Thus, we have
a3 −
µa22
Subclass of Bazilevič
Functions
Dong Guo and Ming-Sheng Liu
1 2
k
=−
p2 − p1
2(1 + 2λ)(2 + α)
2
i
h
k 2 λ(3 + α) − (2+α)(α−1)
−
µ(1
+
2λ)(2
+
α)
2
+
p21 ,
4(1 + λ)2 (1 + α)2 (1 + 2λ)(2 + α)
so that, by Lemma 2.3, we get that |a2 | =
k
|p |
2(1+λ)(1+α) 1
≤
k
,
(1+λ)(1+α)
and
vol. 8, iss. 1, art. 12, 2007
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2
|a3 − µa22 | ≤ H(x) = A +
Bx
,
4
Page 19 of 23
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where x = |p1 | ≤ 2,
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k
A=
,
(1 + 2λ)(2 + α)
and
C=
k 2 |C|
k
B=
−
2
2
[(1 + λ) (1 + α) ] [(1 + 2λ)(2 + α)]
1 − 2µ
α
λ
−
+
.
2
2(1 + 2λ) (1 + 2λ)(2 + α)
Close
Therefore
|a3 − µa22 | ≤

 H(0) = A,
 H(2) =
Ak(1+2λ)(2+α)|C|
,
(1+λ)2 (1+α)2
Here equality is attained for the function given by

2
λz 1/λ exp(−kz )/(2λ)

 R z t(1−λ)/λ
,
2 )/(2λ)
(−kt

exp
dt

0



α 
 1 − kz 2 ,
zf 0 (z) f (z)
=
1/λ exp−kz/λ

f (z)
z

R z λz

(1−λ)/λ exp−kt/λ dt ,

t
0




 1 − kz,
|c| ≤
(1+λ)2 (1+α)2
,
k(1+2λ)(2+α)
|c| ≥
(1+λ)2 (1+α)2
.
k(1+2λ)(2+α)
λ > 0, |c| ≤
(1+λ)2 (1+α)2
,
k(1+2λ)(2+α)
λ = 0, |c| ≤
(1+λ)2 (1+α)2
,
k(1+2λ)(2+α)
λ > 0, |c| ≥
(1+λ)2 (1+α)2
,
k(1+2λ)(2+α)
λ = 0, |c| ≥
(1+λ)2 (1+α)2
.
k(1+2λ)(2+α)
Subclass of Bazilevič
Functions
Dong Guo and Ming-Sheng Liu
This completes the proof of Theorem 3.13.
Setting λ = 0, we get the following corollary.
Corollary 3.14. Let f (z) ∈ H, α ≥ 0 and 0 < k ≤ 1. If
0
zf (z) f (z) α
< k, z ∈ U,
−
1
f (z)
z
vol. 8, iss. 1, art. 12, 2007
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then
k
,
(1 + α)
and for each µ ∈ C, the following bound is sharp
k(2 + α)
k
2
max 1,
|1 − 2µ − α| .
|a3 − µa2 | ≤
2+α
2(1 + α)2
|a2 | ≤
Close
Corollary 3.15. Let f (z) ∈ H, α ≥ 0 and 0 < k ≤ 1. If
0
00
zf
(z)
zf
(z)
(1 − α)
< k, z ∈ U,
+
α
1
+
−
1
f (z)
f 0 (z)
then
k
,
1+α
and for each µ ∈ C the following bound is sharp
(
1 − 2µ +
k(1
+
2α)
k
|a3 − µa22 | ≤
max 1,
2(1 + 2α)
(1 + α)2
|a2 | ≤
Subclass of Bazilevič
Functions
Dong Guo and Ming-Sheng Liu
α
1+2α
)
vol. 8, iss. 1, art. 12, 2007
.
Title Page
Setting α = 1 in Corollary 3.15, we have the following corollary.
Corollary 3.16. Let f (z) ∈ H and 0 < k ≤ 1. If
00 zf (z) f 0 (z) < k, z ∈ U,
then
k
,
2
and for each µ ∈ C the following bound is sharp
k|4 − 6µ|
k
2
.
|a3 − µa2 | ≤ max 1,
6
4
|a2 | ≤
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References
[1] I.E. BAZILEVIĆ, On a case of integrability in quadratures of the LoewnerKuarev equation, Mat. Sb., 37 (1955), 471–476.
[2] P.N. CHICHRA, New subclass of the class of close-to-convex functions, Proc.
Amer. Math. Soc., 62(1) (1977), 37–43.
[3] S.S. DING, Y. LING AND G.J. BAO, Some properties of a class of analytic
functions, J. of Math. Anal. Appl., 195 (1995), 71–81.
[4] M. FEKETE AND G. SZEGÖ, Eine Bermerkung über ungerade schlichte Functionen, J. London Math. Soc., 8 (1933), 85–89.
[5] A.W. GOODMAN, Univalent Functions, Vol. 1, Florida, Mariner Publishing
Co., 1983.
[6] CHUNYI GAO, Fekete-Szegö problem for strongly Bazilevic̆ functions, Northeast Math. J., 12(4) (1996), 469–474.
[7] F.R. KEOGH AND E.P. MERKS, A coefficient inequality for certain classes of
analytic functions, Proc. Amer. Math. Soc., 20(1) (1969), 8–12.
[8] W. KOEPF, On the Fekete-Szegö problem for close-to-convex functions, Proc.
Amer. Math. Soc., 101 (1987), 89–95.
[9] M.S. LIU, The Radius of univalence for certain class of analytic functions, In:
Lu Jianke and Wen Guochun eds, Boundary Value Problems, Integral Equations and Related Problems, Singapore: World Scientific Publishing, 2000,
122–128.
[10] M.S. LIU, Properties for some subclasses of analytic functions, Bulletin of the
Institute of Math. Acdaemia Sinica, 30(1) (2002), 9–26.
Subclass of Bazilevič
Functions
Dong Guo and Ming-Sheng Liu
vol. 8, iss. 1, art. 12, 2007
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[11] M.S. LIU, On certain subclass of analytic functions, J. of South China Normal
Univ., 4 (2002), 15–20 (Chinese).
[12] Z.T. LI AND M.S. LIU, The functional of coefficients for a class of analytic
functions, J. of South China Normal Univ., (3) (2005), 86–91 (Chinese).
[13] M.S. LIU, The Fekete-Szegö inequality for certain class of analytic functions,
Acta Math. Scientia, 22A(1) (2002), 8–14 (Chinese).
[14] R.R. LONDON, Fekete-Szegö inequalities for close-to-convex functions, Proc.
Amer. Math. Soc., 117(4) (1993), 947–950.
[15] M. DARUS, The Fekete-Szegö theorem for close-to-convex functions of the
class KSH (α, β)[J], Acta Math. Academiae Paedagogicae Nyíregyháziensis, 18
(2002), 13–18.
[16] S.S. MILLER AND P.T. MOCANU, Univalent solutions of Briot-Bouquet differential subordination, J. Differential Eqns., 58 (1985), 297–309.
[17] J. PATEL, On certain subclass of p−valently Bazilevic functions, J. of Inequalities in Pure and Applied Math., 6(1) (2005), Art. 16.
[18] C. POMMERENK, Univalent Functions. With a chapter on quadratic differentials by Gerd Jensen. Studia Mathematica/Mathematische Lehrbücher. Band
XXV. Göttingen: Vandenhoeck & Ruprecht,1975.
[19] R. SINGH, On Bazilevic functions, Proc. Amer. Math. Sci., 38 (1973), 261–
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[20] H.M. SRIVASTAVA, A.K. MISHRA AND M.K. DAS, The Fekete-Szegö problem for a subclass of close-to-convex functions, Complex Variables Theory
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Subclass of Bazilevič
Functions
Dong Guo and Ming-Sheng Liu
vol. 8, iss. 1, art. 12, 2007
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