Journal of Inequalities in Pure and
Applied Mathematics
ON SOME SUBCLASSES OF UNIVALENT FUNCTIONS
MUGUR ACU AND SHIGEYOSHI OWA
University "Lucian Blaga" of Sibiu
Department of Mathematics
Str. Dr. I. Rat.iu, No. 5-7
550012 - Sibiu, Romania.
Department of Mathematics
School of Science and Engineering
Kinki University
Higashi-Osaka, Osaka 577-8502, Japan
EMail: owa@math.kindai.ac.jp
volume 6, issue 3, article 70,
2005.
Received 17 February, 2005;
accepted 01 June, 2005.
Communicated by: N.E. Cho
Abstract
Contents
JJ
J
II
I
Home Page
Go Back
Close
c
2000
Victoria University
ISSN (electronic): 1443-5756
042-05
Quit
Abstract
In 1999, S. Kanas and F. Ronning introduced the classes of functions starlike
and convex, which are normalized with f(w) = f 0 (w) − 1 = 0 and w is a fixed
point in U. The aim of this paper is to continue the investigation of the univalent
functions normalized with f(w) = f 0 (w) − 1 = 0, where w is a fixed point in U.
On Some Subclasses of
Univalent Functions
2000 Mathematics Subject Classification: 30C45.
Key words: Close-to-convex functions, α-convex functions, Briot-Bouquet differential
subordination.
Mugur Acu and Shigeyoshi Owa
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
5
7
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 2 of 14
J. Ineq. Pure and Appl. Math. 6(3) Art. 70, 2005
http://jipam.vu.edu.au
1.
Introduction
Let H(U) be the set of functions which are regular in the unit disc U = {z ∈
C : |z| < 1},
A = {f ∈ H(U) : f (0) = f 0 (0) − 1 = 0}
S = {f ∈ A : f is univalent in U}.
and
We recall here the definitions of the well-known classes of starlike, convex,
close-to-convex and α-convex functions:
0 zf (z)
∗
S = f ∈ A : Re
> 0, z ∈ U ,
f (z)
zf 00 (z)
c
S = f ∈ A : Re 1 + 0
> 0, z ∈ U ,
f (z)
0 zf (z)
∗
CC = f ∈ A : ∃g ∈ S , Re
> 0, z ∈ U ,
g(z)
f (z)f 0 (z)
6= 0, Re J(α, f : z) > 0, z ∈ U ,
Mα = f ∈ A :
z
where
zf 0 (z)
zf 00 (z)
J(α, f ; z) = (1 − α)
+α 1+ 0
.
f (z)
f (z)
Let w be a fixed point in U and A(w) = {f ∈ H(U) : f (w) = f 0 (w) − 1 =
On Some Subclasses of
Univalent Functions
Mugur Acu and Shigeyoshi Owa
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 3 of 14
0}.
J. Ineq. Pure and Appl. Math. 6(3) Art. 70, 2005
http://jipam.vu.edu.au
In [3], S. Kanas and F. Ronning introduced the following classes:
S(w) = {f ∈ A(w) : f is univalent in U}
(z − w)f 0 (z)
∗
ST (w) = S (w) = f ∈ S(w) : Re
> 0, z ∈ U
f (z)
(z − w)f 00 (z)
c
CV (w) = S (w) = f ∈ S(w) : 1 + Re
> 0, z ∈ U .
f 0 (z)
The class S ∗ (w) is defined by the geometric property that the image of any
circular arc centered at w is starlike with respect to f (w) and the corresponding
class S c (w) is defined by the property that the image of any circular arc centered
at w is convex. We observe that the definitions are somewhat similar to the ones
for uniformly starlike and convex functions introduced by A. W. Goodman in
[1] and [2], except that in this case the point w is fixed.
It is obvious that there exists a natural "Alexander relation" between the
classes S ∗ (w) and S c (w):
g ∈ S c (w) if and only if f (z) = (z − w)g 0 (z) ∈ S ∗ (w).
Let P(w) denote the class of all functions
p(z) = 1 +
∞
X
On Some Subclasses of
Univalent Functions
Mugur Acu and Shigeyoshi Owa
Title Page
Contents
JJ
J
II
I
Go Back
Bn (z − w)n
n=1
that are regular in U and satisfy p(w) = 1 and Re p(z) > 0 for z ∈ U.
The purpose of this note is to define the classes of close to convex and αconvex functions normalized with f (w) = f 0 (w) − 1 = 0, where w is a fixed
point in U, and to obtain some results concerning these classes.
Close
Quit
Page 4 of 14
J. Ineq. Pure and Appl. Math. 6(3) Art. 70, 2005
http://jipam.vu.edu.au
2.
Preliminary Results
It is easy to see that a function f ∈ A(w) has the series expansion:
f (z) = (z − w) + a2 (z − w)2 + · · · .
In [7], J.K. Wald gives the sharp bounds for the coefficients Bn of the function p ∈ P(w) as follows.
Theorem 2.1. If p ∈ P(w),
p(z) = 1 +
∞
X
On Some Subclasses of
Univalent Functions
Bn (z − w)n ,
Mugur Acu and Shigeyoshi Owa
n=1
then
2
|Bn | ≤
,
(1 + d)(1 − d)n
(2.1)
where d = |w| and n ≥ 1.
Using the above result, S. Kanas and F. Ronning [3] obtain the following:
Theorem 2.2. Let f ∈ S ∗ (w) and f (z) = (z − w) + a2 (z − w)2 + · · · . Then
(2.2)
(2.3)
2
|a2 | ≤
,
1 − d2
2 (2 + d)(3 + d)
,
|a4 | ≤
3 (1 − d2 )3
3+d
|a3 | ≤
,
(1 − d2 )2
1 (2 + d)(3 + d)(3d + 5)
|a5 | ≤
6
(1 − d2 )4
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 5 of 14
J. Ineq. Pure and Appl. Math. 6(3) Art. 70, 2005
where d = |w|.
http://jipam.vu.edu.au
Remark 1. It is clear that the above theorem also provides bounds for the coefficients of functions in S c (w), due to the relation between S c (w) and S ∗ (w).
The next theorem is the result of the so called "admissible functions method"
introduced by P.T. Mocanu and S.S. Miller (see [4], [5], [6]).
Theorem 2.3. Let h be convex in U and Re[βh(z)+γ] > 0, z ∈ U. If p ∈ H(U)
with p(0) = h(0) and p satisfies the Briot-Bouquet differential subordination
zp0 (z)
p(z) +
≺ h(z),
βp(z) + γ
then p(z) ≺ h(z), z ∈ U.
z ∈ U,
On Some Subclasses of
Univalent Functions
Mugur Acu and Shigeyoshi Owa
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 6 of 14
J. Ineq. Pure and Appl. Math. 6(3) Art. 70, 2005
http://jipam.vu.edu.au
3.
Main Results
Let us consider the integral operator La : A(w) → A(w) defined by
Z z
1+a
(3.1) f (z) = La F (z) =
F (t)(t − w)a−1 dt, a ∈ R, a ≥ 0.
(z − w)a w
We denote by
w
< 1 and
D(w) = z ∈ U : Re
z
Re
z(1 + z)
(z − w)(1 − z)
>0 ,
On Some Subclasses of
Univalent Functions
Mugur Acu and Shigeyoshi Owa
with D(0) = U, and
s(w) = {f : D(w) → C} ∩ S(w),
Title Page
where w is a fixed point in U. Denoting s∗ (w) = S ∗ (w) ∩ s(w), where w is a
fixed point in U, we obtain
Contents
Theorem 3.1. Let w be a fixed point in U and F (z) ∈ s∗ (w). Then f (z) =
La F (z) ∈ S ∗ (w), where the integral operator La is defined by (3.1).
Proof. By differentiating (3.1), we obtain
(3.2)
(1 + a)F (z) = af (z) + (z − w)f 0 (z).
JJ
J
II
I
Go Back
Close
Quit
Page 7 of 14
From (3.2), we also have
(3.3)
(1 + a)F 0 (z) = (1 + a)f 0 (z) + (z − w)f 00 (z).
J. Ineq. Pure and Appl. Math. 6(3) Art. 70, 2005
http://jipam.vu.edu.au
Using (3.2) and (3.3), we obtain
0
(3.4)
00
(z)
(z)
(1 + a)(z − w) ff (z)
+ (z − w)2 ff (z)
(z − w)F 0 (z)
=
.
0
F (z)
a + (z − w) f (z)
f (z)
Letting
p(z) =
(z − w)f 0 (z)
,
f (z)
On Some Subclasses of
Univalent Functions
where p ∈ H(U) and p(0) = 1, we have
(z − w)p0 (z) = p(z) + (z − w)2 ·
f 00 (z)
− [p(z)]2
f (z)
Title Page
and thus
(3.5)
f 00 (z)
(z − w)2
= (z − w)p0 (z) − p(z)[1 − p(z)].
f (z)
Using (3.4) and (3.5), we obtain
(3.6)
Mugur Acu and Shigeyoshi Owa
(z − w)p0 (z)
(z − w)F 0 (z)
= p(z) +
.
F (z)
a + p(z)
Contents
JJ
J
II
I
Go Back
Close
Quit
Since F ∈ s∗ (w), from (3.6), we have
p(z) +
z−w 0
1+z
p (z) ≺
≡ h(z)
a + p(z)
1−z
Page 8 of 14
J. Ineq. Pure and Appl. Math. 6(3) Art. 70, 2005
http://jipam.vu.edu.au
or
1 − wz
1+z
zp0 (z) ≺
.
p(z) +
a + p(z)
1−z
From the hypothesis, we have
1
Re
1−
a
w h(z) +
1−
z
>0
w
z
and thus from Theorem 2.3, we obtain
p(z) ≺
or
Re
1+z
,
1−z
(z − w)f 0 (z)
f (z)
On Some Subclasses of
Univalent Functions
z∈U
Mugur Acu and Shigeyoshi Owa
> 0,
Title Page
z ∈ U.
Contents
∗
This means that f ∈ S (w).
Definition 3.1. Let f ∈ S(w) where w is a fixed point in U. We say that f is
w-close-to-convex if there exists a function g ∈ S ∗ (w) such that
(z − w)f 0 (z)
Re
> 0, z ∈ U.
g(z)
JJ
J
II
I
Go Back
Close
Quit
We denote this class by CC(w).
Page 9 of 14
∗
∗
Remark 2. If we consider f = g, g ∈ S (w), then we have S (w) ⊂ CC(w).
If we take w = 0, then we obtain the well-known close-to-convex functions.
J. Ineq. Pure and Appl. Math. 6(3) Art. 70, 2005
http://jipam.vu.edu.au
Theorem 3.2. Let w be a fixed point in U and f ∈ CC(w), where
f (z) = (z − w) +
∞
X
bn (z − w)n ,
n=2
with respect to the function g ∈ S ∗ (w), where
g(z) = (z − w) +
∞
X
an (z − w)n .
n=2
Then
On Some Subclasses of
Univalent Functions
"
#
n−1
X
2
1
|ak | ·
|an | +
,
|bn | ≤
n−k
n
(1
+
d)(1
−
d)
k=1
Mugur Acu and Shigeyoshi Owa
where d = |w|, n ≥ 2 and a1 = 1.
Title Page
Proof. Let f ∈ CC(w) with respect to the function g ∈ S ∗ (w). Then there
exists a function p ∈ P(w) such that
Contents
(z − w)f 0 (z)
= p(z),
g(z)
II
I
Go Back
where
p(z) = 1 +
∞
X
Close
Bn (z − w)n .
Quit
n=1
n
Using the hypothesis through identification of (z − w) coefficients, we obtain
(3.7)
JJ
J
nbn = an +
n−1
X
k=1
ak Bn−k ,
Page 10 of 14
J. Ineq. Pure and Appl. Math. 6(3) Art. 70, 2005
http://jipam.vu.edu.au
where a1 = 1 and n ≥ 2. From (3.7), we have
#
"
n−1
X
1
|ak | · |Bn−k | ,
|bn | ≤
|an | +
n
k=1
a1 = 1, n ≥ 2.
Applying the above and the estimates (2.1), we obtain the result.
Remark 3. If we use the estimates (2.2), we obtain the same estimates for the
coefficients bn , n = 2, 3, 4, 5.
Definition 3.2. Let α ∈ R and w be a fixed point in U. For f ∈ S(w), we define
(z − w)f 0 (z)
(z − w)f 00 (z)
J(α, f, w; z) = (1 − α)
+α 1+
.
f (z)
f 0 (z)
We say that f is w − α−convex function if
f (z)f 0 (z)
6= 0,
z−w
On Some Subclasses of
Univalent Functions
Mugur Acu and Shigeyoshi Owa
Title Page
Contents
z∈U
and Re J(α, f, w; z) > 0, z ∈ U. We denote this class by Mα (w).
Remark 4. It is easy to observe that Mα (0) is the well-known class of α-convex
functions.
Theorem 3.3. Let w be a fixed point in U, α ∈ R, α ≥ 0 and mα (w) =
Mα (w) ∩ s(w). Then we have
1. If f ∈ mα (w) then f ∈ S ∗ (w). This means mα (w) ⊂ S ∗ (w).
JJ
J
II
I
Go Back
Close
Quit
Page 11 of 14
J. Ineq. Pure and Appl. Math. 6(3) Art. 70, 2005
http://jipam.vu.edu.au
2. If α, β ∈ R, with 0 ≤ β/α < 1, then mα (w) ⊂ mβ (w).
Proof. From f ∈ mα (w), we have Re J(α, f, w; z) > 0, z ∈ U. Putting
p(z) =
(z − w)f 0 (z)
,
f (z)
with p ∈ H(U) and p(0) = 1, we obtain
(z − w)p0 (z)
Re J(α, f, w; z) = Re p(z) + α
> 0,
p(z)
z∈U
On Some Subclasses of
Univalent Functions
Mugur Acu and Shigeyoshi Owa
or
w
p(z) +
α 1− z
1+z
zp0 (z) ≺
≡ h(z).
p(z)
1−z
In particular, for α = 0, we have
p(z) ≺
Contents
1+z
,
1−z
JJ
J
z ∈ U.
Using the hypothesis, we have for α > 0,
1
h(z)
α 1 − wz
> 0,
z∈U
Close
Quit
Page 12 of 14
and from Theorem 2.3, we obtain
p(z) ≺
II
I
Go Back
!
Re
Title Page
1+z
,
1−z
z ∈ U.
J. Ineq. Pure and Appl. Math. 6(3) Art. 70, 2005
http://jipam.vu.edu.au
This means that
Re
(z − w)f 0 (z)
f (z)
> 0,
z∈U
for α ≥ 0 or f ∈ S ∗ (w).
If we denote by A = Re p(z) and B = Re ((z − w)p0 (z)/p(z)), then we have
A > 0 and A + Bα > 0, where α ≥ 0. Using the geometric interpretation of
the equation y(x) = A + Bx, x ∈ [0, α], we obtain
y(β) = A + Bβ > 0 for every β ∈ [0, α].
This means that
On Some Subclasses of
Univalent Functions
Mugur Acu and Shigeyoshi Owa
(z − w)p0 (z)
Re p(z) + β
p(z)
> 0,
z∈U
Title Page
Contents
or f ∈ mβ (w).
Remark 5. From Theorem 3.3, we have
m1 (w) ⊆ sc (w) ⊆ mα (w) ⊆ s∗ (w),
where 0 ≤ α ≤ 1 and sc (w) = S c (w) ∩ s(w).
JJ
J
II
I
Go Back
Close
Quit
Page 13 of 14
J. Ineq. Pure and Appl. Math. 6(3) Art. 70, 2005
http://jipam.vu.edu.au
References
[1] A.W. GOODMAN, On Uniformly Starlike Functions, J. Math. Anal. Appl.,
155 (1991), 364–370.
[2] A.W. GOODMAN, On uniformly convex functions, Ann. Polon. Math., 56
(1991), 87–92.
[3] S. KANAS AND F. RONNING, Uniformly starlike and convex functions
and other related classes of univalent functions, Ann. Univ. Mariae Curie Sklodowska Section A, 53 (1999), 95–105.
[4] S.S. MILLER AND P.T. MOCANU, Differential subordonations and univalent functions, Michigan Math. J., 28 (1981), 157–171.
[5] S.S. MILLER AND P.T. MOCANU, Univalent solutions of Briot-Bouquet
differential equations, J. Diff. Eqns., 56 (1985), 297–309.
[6] S.S. MILLER AND P.T. MOCANU, On some classes of first-order differential subordinations, Michigan Math. J., 32 (1985), 185–195.
[7] J.K. WALD, On Starlike Functions, Ph. D. thesis, University of Delaware,
Newark, Delaware (1978).
On Some Subclasses of
Univalent Functions
Mugur Acu and Shigeyoshi Owa
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 14 of 14
J. Ineq. Pure and Appl. Math. 6(3) Art. 70, 2005
http://jipam.vu.edu.au