ON A SUBCLASS OF n-STARLIKE FUNCTIONS
MUGUR ACU AND SHIGEYOSHI OWA
Received 21 February 2005
In 1999, Kanas and Rønning introduced the classes of starlike and convex functions,
which are normalized with f (w) = f (w) − 1 = 0 and w a fixed point in U. In 2005,
the authors introduced the classes of functions close to convex and α-convex, which are
normalized in the same way. All these definitions are somewhat similar to the ones for
the uniform-type functions and it is easy to see that for w = 0, the well-known classes
of starlike, convex, close-to-convex, and α-convex functions are obtained. In this paper, we continue the investigation of the univalent functions normalized with f (w) =
f (w) − 1 = 0, where w is a fixed point in U.
1. Introduction
Let Ᏼ(U) be the set of functions which are regular in the unit disc U = {z ∈ C : |z| < 1},
A = { f ∈ Ᏼ(U) : f (0) = f (0) − 1 = 0}, and S = { f ∈ A : f is univalent in U }.
We recall here the definitions of the well-known classes of starlike and convex functions:
S∗ = f ∈ A : Re
z f (z)
> 0, z ∈ U ,
f (z)
(1.1)
z f (z)
> 0, z ∈ U .
S = f ∈ A : Re 1 + f (z)
c
Let w be a fixed point in U and A(w) = { f ∈ Ᏼ(U) : f (w) = f (w) − 1 = 0}.
In [3], Kanas and Rønning introduced the following classes:
S(w) = f ∈ A(w) : f is univalent in U ,
ST(w) = S∗ (w) = f ∈ S(w) : Re
(z − w) f (z)
> 0, z ∈ U ,
f (z)
CV (w) = Sc (w) = f ∈ S(w) : 1 + Re
(z − w) f (z)
> 0, z ∈ U .
f (z)
Copyright © 2005 Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences 2005:17 (2005) 2841–2846
DOI: 10.1155/IJMMS.2005.2841
(1.2)
2842
On a subclass of n-starlike functions
It is obvious that a natural “Alexander relation” exists between the classes S∗ (w) and
Sc (w):
g ∈ Sc (w)
iff f (z) = (z − w)g (z) ∈ S∗ (w).
Denote with ᏼ(w) the class of all functions p(z) = 1 +
ular in U and satisfy p(w) = 1 and Re p(z) > 0 for z ∈ U.
∞
n=1 Bn · (z − w)
(1.3)
n
that are reg-
2. Preliminary results
If is easy to see that a function f(z) ∈ A(w) has the series of expansions:
f (z) = (z − w) + a2 (z − w)2 + ....
(2.1)
In [8], Wald gives the sharp bounds for the coefficients Bn of the function p ∈ ᏼ(w).
Theorem 2.1. If p(z) ∈ ᏼ(w), p(z) = 1 +
Bn ≤
∞
n=1 Bn · (z − w)
2
,
(1 + d)(1 − d)n
n,
then
where d = |w|, n ≥ 1.
(2.2)
Using the above result, Kanas and Rønning obtain the following theorem in [3].
Theorem 2.2. Let f ∈ S∗ (w) and f (z) = (z − w) + b2 (z − w)2 + .... Then
b2 ≤
2
1 − d2
2 (2 + d)(3 + d)
b4 ≤ · 3 ,
3
1 − d2
b3 ≤ 3 + d ,
2
1 − d2
,
1 (2 + d)(3 + d)(3d + 5)
b5 ≤ ·
,
4
6
1 − d2
(2.3)
where d = |w|.
Remark 2.3. It is clear that the above theorem also provides bounds for the coefficients of
functions in Sc (w), due to the relation between Sc (w) and S∗ (w).
In [1], are also defined the following sets:
D(w) = z ∈ U : Re
w
z(1 + z)
< 1, Re
>0
z
(z − w)(1 − z)
s(w) = f : D(w) −→ C ∩ S(w);
for w = 0, D(0) = U;
s∗ (w) = S∗ (w) ∩ s(w),
(2.4)
where w is a fixed point in U.
M. Acu and S. Owa
2843
The authors consider the integral operator La : A(w) → A(w) defined by
f (z) = La F(z) =
1+a
·
(z − w)a
z
w
F(t) · (t − w)a−1 dt,
a ∈ R, a ≥ 0.
(2.5)
The next theorem is a result of the so called “admissible functions method” introduced
by Mocanu and Miller (see [3, 4, 6]).
Theorem 2.4. Let h be convex in U and Re[βh(z) + γ] > 0, z ∈ U. If p ∈ Ᏼ(U) with p(0) =
h(0) and p satisfied the Briot-Bouquet differential subordination
p(z) +
zp (z)
≺ h(z),
βp(z) + γ
(2.6)
then p(z) ≺ h(z).
3. Main results
Deffinition 3.1. Let w be a fixed point in U, n ∈ N. Dwn denotes the differential operator:
Dwn : A(w) −→ A(w) with ,
Dw0 f (z) = f (z),
Dw1 f (z) = Dw f (z) = (z − w) · f (z),
(3.1)
Dwn f (z) = Dw Dwn−1 f (z) .
Remark 3.2. For f ∈ A(w), f (w) = (z − w) +
Dwn f (z) = (z − w) +
∞
∞
j =2 a j (z − w)
j,
we have
j n · a j · (z − w) j .
(3.2)
j =2
It easy to see that if we take w = 0, we obtain the Sălăgean differential operator (see [7]).
Deffinition 3.3. Let w be a fixed point in U, n ∈ N and f ∈ S(w). f is said to be an
n-w-starlike function if
Re
Dwn+1 f (z)
> 0,
Dwn f (z)
z ∈ U.
(3.3)
The class of all these functions is denoted by S∗n (w).
Remark 3.4. (1) S∗0 (w) = S∗ (w) and S∗n (0) = S∗n , where S∗n is the class of n-starlike functions introduced by Sălăgean in [7].
(2) If f (z) ∈ S∗n (w) and we denote Dwn f (z) = g(z), we obtain g(z) ∈ S∗ (w).
(3) Using the class s(w), we obtain s∗n (w) = S∗n (w) ∩ s(w).
Theorem 3.5. Let w be a fixed point in U and n ∈ N. If f (z) ∈ s∗n+1 (w) then f (z) ∈ s∗n (w).
This means
s∗n+1 (w) ⊂ s∗n (w).
(3.4)
2844
On a subclass of n-starlike functions
Proof. From f (z) ∈ s∗n+1 (w), we have Re(Dwn+2 f (z)/Dwn+1 f (z))>0, z ∈ U. We denote p(z) =
(Dwn+1 f (z)/Dwn f (z)), where p(0) = 1 and p(z) ∈ Ᏼ(U). We obtain
(z − w) Dwn+1 f (z)
Dwn+1 f (z)
Dwn+2 f (z) Dw Dwn+1 f (z)
=
=
= ,
n+1
n
n
Dw f (z)
Dw Dw f (z)
(z − w) Dw f (z)
Dwn f (z)
Dwn+1 f (z) · Dwn f (z) − Dwn+1 f (z) · Dwn f (z)
p (z) =
Dwn f (z)
2
(3.5)
Dn+1 f (z)
Dwn f (z)
Dwn f (z)
= w
−
p(z)
·
.
·
Dwn f (z)
Dwn f (z)
Dwn f (z)
Thus we have
Dn+1 f (z)
(z − w) · Dwn f (z)
(z − w) · p (z) = w
·
Dwn f (z)
Dwn f (z)
− p(z) ·
(z − w) · Dwn f (z)
,
Dwn f (z)
2
Dn+1 f (z)
(z − w) · p (z) = w
· p(z) − p(z) ,
n
Dw f (z)
Dwn+1 f (z)
1
· (z − w) · p (z).
= p(z) +
p(z)
Dwn f (z)
(3.6)
From Re(Dwn+2 f (z)/Dwn+1 f (z)) > 0 we obtain p(z) + (1/ p(z)) · (z − w) · p (z) ≺ ((1 +
z)/(1 − z)) or
p(z) +
zp (z)
1+z
≺
≡ h(z),
1/ 1 − (w/z) · p(z) 1 − z
with h(0) = 1.
(3.7)
By hypothesis, we have Re[1/(1 − (w/z)) · h(z)] > 0, and thus from Theorem 2.4 we ob
tain p(z) ≺ h(z) or Re p(z) > 0. This means f ∈ s∗n (w).
Remark 3.6. From Theorem 3.5, we obtain s∗n (w) ⊂ s∗0 (w) ⊂ S∗ (w), n ∈ N.
Theorem 3.7. If F(z) ∈ s∗n (w) then f (z) = La F(z) ∈ S∗n (w), where La is the integral operator defined by (2.5).
Proof. From (2.5) we obtain
(1 + a) · F(z) = a · f (z) + (z − w) · f (z).
(3.8)
By means of the application of the operator Dwn+1 we obtain
(1 + a) · Dwn+1 F(z) = a · Dwn+1 f (z) + Dwn+1 (z − w) · f (z)
(3.9)
M. Acu and S. Owa
2845
or
(1 + a) · Dwn+1 F(z) = a · Dwn+1 f (z) + Dwn+2 f (z).
(3.10)
Similarly, by means of the application of the operator Dwn , we obtain
(1 + a) · Dwn F(z) = a · Dwn f (z) + Dwn+1 f (z).
(3.11)
Thus
Dwn+2 f (z)/Dwn+1 f (z) · Dwn+1 f (z)/Dwn f (z) + a · Dwn+1 f (z)/Dwn f (z)
Dwn+1 F(z)
.
=
n
Dw F(z)
Dwn+1 f (z)/Dwn f (z) + a
(3.12)
Using the notation Dwn+1 f (z)/Dwn f (z) = p(z), with p(0) = 1, we have
(z − w) · p (z) Dwn+2 f (z)
= n+1
− p(z)
p(z)
Dw f (z)
(3.13)
Dwn+2 f (z)
(z − w) · p (z)
= p(z) +
.
n+1
Dw f (z)
p(z)
(3.14)
or
Thus
Dwn+1 F(z) p(z) p(z) + (z − w)p (z)/ p(z) + a
=
Dwn F(z)
p(z) + a
zp (z)
.
= p(z) + 1/ 1 − (w/z) p(z) + a/ 1 − (w/z)
(3.15)
From F(z) ∈ s∗n (w) we obtain (Dwn+1 F(z)/Dwn F(z)) ≺ ((1 + z)/(1 − z)) ≡ h(z) or
zp (z)
≺ h(z).
p(z) + 1/ 1 − (w/z) p(z) + a/ 1 − (w/z)
(3.16)
By hypothesis, we have Re[(1/(1 − (w/z))) · h(z)+(a/(1 − (w/z)))] > 0 and from Theorem
2.4 we obtain p(z) ≺ h(z) or Re{Dwn+1 f (z)/Dwn f (z)} > 0, z ∈ U. This means f (z) =
La F(z) ∈ S∗n (w).
Remark 3.8. If we consider w = 0 in Theorem 3.7 we obtain that the integral operator
defined by (2.5) preserves the class of n-starlike functions, and if we consider w = 0 and
n = 0 in the above theorem we obtain that the integral operator defined by (2.5) preserves
the well-known class of starlike functions.
2846
On a subclass of n-starlike functions
Theorem 3.9. Let w be a fixed point in U and f ∈ S∗n (w) with f (z) = (z − w) +
(z − w) j . Then
a2 ≤
1
,
2 n −1 · 1 − d 2
a3 ≤
3+d
2 ,
3n · 1 − d 2
a4 ≤
(2 + d)(3 + d)
3 ,
22n−1 · 3 · 1 − d2
∞
j =2 a j
·
(3.17)
(2 + d)(3 + d)(3d + 5)
a5 ≤
4 ,
5n · 6 · 1 − d 2
where d = |w|.
Proof. From Remark 3.4 for f ∈ S∗n (w) we obtain
Dwn f (z) = g(z) ∈ S∗ (w).
(3.18)
j
If we consider g(z) = (z − w) + ∞
j =2 b j · (z − w) , using Remark 3.2, from (3.18) we
n
obtain j · a j = b j , j = 2,3,....
Thus we have a j = 1/ j n · b j , j = 2,3,..., and from the estimates (2.3) we get the result.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
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univalent functions, Ann. Univ. Mariae Curie-Skłodowska Sect. A 53 (1999), 95–105.
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Math. J. 28 (1981), no. 2, 157–172.
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no. 2, 185–195.
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(1985), no. 3, 297–309.
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Seminar, Part 1 (Bucharest, 1981), Lecture Notes in Math., vol. 1013, Springer, Berlin, 1983,
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J. K. Wald, On starlike functions, Ph.D. thesis, University of Delaware, Delaware, 1978.
Mugur Acu: Department of Mathematics, “Lucian Blaga” University of Sibiu, 5–7 Ion Rat.iu Street,
550012 Sibiu, Romania
E-mail address: acu mugur@yahoo.com
Shigeyoshi Owa: Department of Mathematics, School of Science and Engineering, Kinki University, Higashiosaka, Osaka 577-8502, Japan
E-mail address: owa@math.kindai.ac.jp