Journal of Inequalities in Pure and
Applied Mathematics
ON THE UNIVALENCY OF CERTAIN ANALYTIC FUNCTIONS
ZHI-GANG WANG, CHUN-YI GAO AND SHAO-MOU YUAN
College of Mathematics and Computing Science
Changsha University of Science and Technology
Changsha, Hunan 410076
People’s Republic of China
volume 7, issue 1, article 9,
2006.
Received 20 April, 2005;
accepted 03 September, 2005.
Communicated by: H.M. Srivastava
EMail: zhigwang@163.com
Abstract
Contents
JJ
J
II
I
Home Page
Go Back
Close
c
2000
Victoria University
ISSN (electronic): 1443-5756
125-05
Quit
Abstract
Let Q(α, β, γ) denote the class of functions of the form f(z) = z + a2 z 2 + · · · ,
which are analytic in the unit disk U = {z : |z| < 1} and satisfy the condition
<{α(f(z)/z) + βf 0 (z)} > γ (α, β > 0; 0 ≤ γ < α + β ≤ 1; z ∈ U).
The extreme points for this class are provided, the coefficient bounds and radius
of univalency for functions belonging to this class are also provided. The results
presented here include a number of known results as their special cases.
On the Univalency of Certain
Analytic Functions
Zhi-Gang Wang, Chun-Yi Gao and
Shao-Mou Yuan
2000 Mathematics Subject Classification: Primary 30C45.
Key words: Univalency; extreme point; bound.
Title Page
This work was supported by the Scientific Research Fund of Hunan Provincial
Education Department and the Hunan Provincial Natural Science Foundation (No.
05JJ30013) of People’s Republic of China.
Contents
JJ
J
Contents
II
I
Go Back
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Extreme Points of the Class Q(α, β, γ) . . . . . . . . . . . . . . . . . .
3
Radius of Univalency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
4
8
Close
Quit
Page 2 of 10
J. Ineq. Pure and Appl. Math. 7(1) Art. 9, 2006
http://jipam.vu.edu.au
1.
Introduction
Let A denote the class of functions of the form
f (z) = z +
∞
X
an z n ,
n=2
which are analytic in the unit disk U = {z : |z| < 1}. Also let S denote the
familiar subclass of A consisting of all functions which are univalent in U.
In the present paper, we consider the following subclass of A:
f (z)
0
+ βf (z) > γ (z ∈ U) ,
(1.1) Q(α, β, γ) = f (z) ∈ A : < α
z
where α, β > 0 and 0 ≤ γ < α + β ≤ 1.
In some recent papers, Saitoh [2] and Owa [3, 4] discussed the related properties of the class Q(1 − β, β, γ). In the present paper, first we determine the
extreme points of the class Q(α, β, γ), then we find the coefficient bounds and
radius of univalency for functions belonging to this class. The results presented
here include a number of known results as their special cases.
On the Univalency of Certain
Analytic Functions
Zhi-Gang Wang, Chun-Yi Gao and
Shao-Mou Yuan
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 3 of 10
J. Ineq. Pure and Appl. Math. 7(1) Art. 9, 2006
http://jipam.vu.edu.au
2.
Extreme Points of the Class Q(α, β, γ)
First we give the following theorem.
Theorem 2.1. A function f (z) ∈ Q(α, β, γ) if and only if f (z) can be expressed
as
"
Z
1
(2.1) f (z) =
(2γ − α − β)z
α + β |x|=1
#
∞
X
(α + β)xn z n+1
+2(α + β − γ)
dµ(x),
(n + 1)β + α
n=0
where µ(x) is the probability measure defined on X = {x : |x| = 1}. For
fixed α, β and γ, Q(α, β, γ) and the probability measures {µ} defined on X
are one-to-one by the expression (2.1).
Proof. By the definition of Q(α, β, γ), we know f (z) ∈ Q(α, β, γ) if and only
if
α(f (z)/z) + βf 0 (z) − γ
∈ P,
α+β−γ
where P denotes the normalized well-known class of analytic functions which
have positive real part. By the aid of Herglotz expressions of functions in P, we
have
Z
α(f (z)/z) + βf 0 (z) − γ
1 + xz
=
dµ(x),
α+β−γ
|x|=1 1 − xz
On the Univalency of Certain
Analytic Functions
Zhi-Gang Wang, Chun-Yi Gao and
Shao-Mou Yuan
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 4 of 10
J. Ineq. Pure and Appl. Math. 7(1) Art. 9, 2006
http://jipam.vu.edu.au
or equivalently,
α f (z)
1
+ f 0 (z) =
β z
β
Z
|x|=1
α + β + (α + β − 2γ)xz
dµ(x).
1 − xz
Thus we have
Z z
α
α f (ζ)
−α
0
β
z
+ f (ζ) ζ β dζ
β ζ
0
Z
Z z
α
1
α
+
β
+
(α
+
β
−
2γ)xζ
−α
ζ β dζ dµ(x),
=
z β
1 − xζ
β |x|=1
0
On the Univalency of Certain
Analytic Functions
Zhi-Gang Wang, Chun-Yi Gao and
Shao-Mou Yuan
that is,
1
f (z) =
α+β
Z
Title Page
"
(2γ − α − β)z
Contents
|x|=1
+2(α + β − γ)
∞
X
(α + β)xn z n+1
n=0
(n + 1)β + α
#
dµ(x).
This deductive process can be converse, so we have proved the first part of the
theorem. We know that both probability measures {µ} and class P, class P
and Q(α, β, γ) are one-to-one, so the second part of the theorem is true. This
completes the proof of Theorem 2.1.
JJ
J
II
I
Go Back
Close
Quit
Page 5 of 10
J. Ineq. Pure and Appl. Math. 7(1) Art. 9, 2006
http://jipam.vu.edu.au
Corollary 2.2. The extreme points of the class Q(α, β, γ) are
"
1
(2.2) fx (z) =
(2γ − α − β)z
α+β
+2(α + β − γ)
∞
X
(α + β)xn z n+1
n=0
#
(n + 1)β + α
(|x| = 1).
Proof. Using the notation fx (z), (2.1) can be written as
Z
fµ (z) =
fx (z)dµ(x).
|x|=1
By Theorem 2.1, the map µ → fµ is one-to-one, so the assertion follows (see
[1]).
P
n
Corollary 2.3. If f (z) = z + ∞
n=2 an z ∈ Q(α, β, γ), then for n ≥ 2, we have
|an | ≤
2(α + β − γ)
.
nβ + α
The results are sharp.
Proof. The coefficient bounds are maximized at an extreme point. Now from
(2.2), fx (z) can be expressed as
(2.3)
∞
X
xn−1 z n
fx (z) = z + 2(α + β − γ)
(|x| = 1),
nβ + α
n=2
and the result follows.
On the Univalency of Certain
Analytic Functions
Zhi-Gang Wang, Chun-Yi Gao and
Shao-Mou Yuan
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 6 of 10
J. Ineq. Pure and Appl. Math. 7(1) Art. 9, 2006
http://jipam.vu.edu.au
Corollary 2.4. If f (z) = z +
we have
P∞
n=2
an z n ∈ Q(α, β, γ), then for |z| = r < 1,
|f (z)| ≤ r + 2(α + β − γ)
∞
X
n=2
rn
.
nβ + α
This result follows from (2.3).
On the Univalency of Certain
Analytic Functions
Zhi-Gang Wang, Chun-Yi Gao and
Shao-Mou Yuan
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 7 of 10
J. Ineq. Pure and Appl. Math. 7(1) Art. 9, 2006
http://jipam.vu.edu.au
3.
Radius of Univalency
In this section, we shall provide the radius of univalency for functions belonging
to the class Q(α, β, γ).
Theorem 3.1. Let f (z) ∈ Q(α, β, γ), then f (z) is univalent in |z| < R(α, β, γ),
where
1
n−1
nβ + α
R(α, β, γ) = inf
.
n
2n(α + β − γ)
This result is sharp.
Proof. It suffices to show that
(3.1)
|f 0 (z) − 1| < 1.
For the left hand side of (3.1) we have
∞
∞
X
X
n−1 nan z ≤
n |an | |z|n−1 .
n=2
n=2
This last expression is less than 1 if
|z|n−1 <
nβ + α
.
2n(α + β − γ)
To show that the bound R(α, β, γ) is best possible, we consider the function
f (z) ∈ A defined by
f (z) = z −
2(α + β − γ) n
z .
nβ + α
On the Univalency of Certain
Analytic Functions
Zhi-Gang Wang, Chun-Yi Gao and
Shao-Mou Yuan
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 8 of 10
J. Ineq. Pure and Appl. Math. 7(1) Art. 9, 2006
http://jipam.vu.edu.au
If δ > R(α, β, γ), then there exists n ≥ 2 such that
nβ + α
2n(α + β − γ)
1
n−1
< δ.
Since f 0 (0) = 1 > 0 and
f 0 (δ) = 1 −
2n(α + β − γ) n−1
δ
< 0.
nβ + α
Thus, there exists δ0 ∈ (0, δ) such that f 0 (δ0 ) = 0, which implies that f (z) is
not univalent in |z| < δ. This completes the proof of Theorem 3.1.
On the Univalency of Certain
Analytic Functions
Zhi-Gang Wang, Chun-Yi Gao and
Shao-Mou Yuan
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 9 of 10
J. Ineq. Pure and Appl. Math. 7(1) Art. 9, 2006
http://jipam.vu.edu.au
References
[1] D.J. HALLENBECK, Convex hulls and extreme points of some families of
univalent functions, Trans. Amer. Math. Soc., 192 (1974), 285–292.
[2] H. SAITOH, On inequalities for certain analytic functions, Math. Japon.,
35 (1990), 1073–1076.
[3] S. OWA, Some properties of certain analytic functions, Soochow J. Math.,
13 (1987), 197–201.
[4] S. OWA, Generalization properties for certain analytic functions, Internat.
J. Math. Math. Sci., 21 (1998), 707–712.
On the Univalency of Certain
Analytic Functions
Zhi-Gang Wang, Chun-Yi Gao and
Shao-Mou Yuan
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 10 of 10
J. Ineq. Pure and Appl. Math. 7(1) Art. 9, 2006
http://jipam.vu.edu.au