General Mathematics Vol. 13, No. 1 (2005), 91–98
On a subclass of n-starlike functions
associated with some hyperbola
Mugur Acu
Dedicated to Professor Emil C. Popa on his 60th birthday
Abstract
In this paper we define a subclass of n-starlike functions associated with some hyperbola and we obtain some properties regarding
this class.
2000 Mathematics Subject Classification: 30C45
Key words and phrases: n-starlike functions, Libera-Pascu integral
operator, Briot-Bouquet differential subordination
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}
and S = {f ∈ A : f is univalent in U }.
91
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We recall here the definition of the well - known class of starlike func-
tions:
½
∗
S =
zf 0 (z)
f ∈ A : Re
>0, z∈U
f (z)
¾
,
Let consider the Libera-Pascu integral operator La : A → A defined as:
(1)
1+a
f (z) = La F (z) =
za
Zz
F (t) · ta−1 dt ,
a∈C,
Re a ≥ 0.
0
For a = 1 we obtain the Libera integral operator, for a = 0 we obtain
the Alexander integral operator and in the case a = 1, 2, 3, ... we obtain the
Bernardi integral operator.
Let Dn be the Sălăgean differential operator (see [5]) Dn : A → A,
n ∈ N, defined as:
D0 f (z) = f (z)
D1 f (z) = Df (z) = zf 0 (z)
Dn f (z) = D(Dn−1 f (z))
We observe that if f ∈ S , f (z) = z +
n
D f (z) = z +
∞
P
∞
P
aj z j , z ∈ U then
j=2
n
j
j aj z .
j=2
The purpose of this note is to define a subclass of n-starlike functions functions associated with some hyperbola and to obtain some estimations for
the coefficients of the series expansion and some other properties regarding
this class.
On a subclass of n-starlike functions associated with some hyperbola
2
93
Preliminary results
Definition 2.1. [6] A function f ∈ S is said to be in the class SH(α) if it
satisfies
¯ 0
½
¾
´¯¯
³√
´
³√
√ zf 0 (z)
¯ zf (z)
¯
¯
<
Re
2
−
1
2
+
2α
2
−
1
,
−
2α
¯
¯ f (z)
f (z)
for some α (α > 0) and for all z ∈ U .
Remark 2.1. Geometric interpretation. Let
½ 0
¾
zf (z)
Ω(α) =
: z ∈ U , f ∈ SH(α) .
f (z)
Then Ω(α) = {w = u + i · v : v 2 < 4αu + u2 , u > 0} . Note that Ω(α) is
the interior of a hyperbola in the right half-plane which is symmetric about
the real axis and has vertex at the origin.
Theorem 2.1. [6] Let f ∈ SH(α) and f (z) = z + b2 z 2 + b3 z 3 + . . . . Then
|b2 | ≤
1 + 4α
(1 + 4α)(3 + 16α + 24α2 )
, |b3 | ≤
.
1 + 2α
4(1 + 2α)3
The next theorem is result of the so called ”admissible functions method”
due to P.T. Mocanu and S.S. Miller (see [1], [2], [3]).
Theorem 2.2. Let h convex in U and Re[βh(z) + γ] > 0, z ∈ U . If
p ∈ H(U ) with p(0) = h(0) and p satisfied the Briot-Bouquet differential
subordination
zp0 (z)
p(z) +
≺ h(z),
βp(z) + γ
then p(z) ≺ h(z).
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3
Main results
Definition 3.1. Let f ∈ S and α > 0. We say that the function f is in the
class SHn (α), n ∈ N , if
¯ n+1
½
¾
´¯¯
³√
´
³√
√ Dn+1 f (z)
¯ D f (z)
¯
¯
<
Re
2
−
1
2
+2α
2
−
1
,z∈U.
−
2α
¯ Dn f (z)
¯
Dn f (z)
Remark 3.1. Geometric interpretation: If we denote with pα the analytic and univalent functions with the properties pα (0) = 1, p0 α (0) > 0
and pα (U ) = Ω(α) (see Remark 2.1), then f ∈ SHn (α) if and only if
Dn+1 f (z)
≺ pα (z), where the symbol “≺” denotes the subordination in U .
Dn f (z)
r
1 + bz
1 + 4α − 4α2
We have pα (z) = (1 + 2α)
− 2α , b = b(α) =
and the
2
√ 1−z
√ (1 + 2α)
branch of the square root w is chosen so that Im w ≥ 0 . If we consider
1 + 4α
pα (z) = 1 + C1 z + . . . , we have C1 =
.
1 + 2α
Theorem 3.1. Let f ∈ SHn (α), α > 0, and f (z) = z + a2 z 2 + a3 z 3 + . . . ,
then
|a2 | ≤
1 1 + 4α
1 (1 + 4α)(3 + 16α + 24α2 )
·
,
|a
|
≤
·
.
3
2n 1 + 2α
3n
4(1 + 2α)3
n
Proof. If we denote by D f (z) = g(z), g(z) =
∞
X
bj z j , we have:
j=2
f ∈ SHn (α) if and only if g ∈ SH(α) .
1
· |bj | , j ≥ 2 . Using
jn
the estimations from the Theorem 2.1 we obtain the needed results.
From the above series expansions we obtain |aj | ≤
Theorem 3.2. If F (z) ∈ SHn (α), α > 0, n ∈ N , and f (z) = La F (z),
where La is the integral operator defined by (1), then f (z) ∈ SHn (α), α > 0,
n ∈ N.
On a subclass of n-starlike functions associated with some hyperbola
95
Proof. By differentiating (1) we obtain (1 + a)F (z) = af (z) + zf 0 (z) .
By means of the application of the linear operator Dn+1 we obtain
(1 + a)Dn+1 F (z) = aDn+1 f (z) + Dn+1 (zf 0 (z))
or
(1 + a)Dn+1 F (z) = aDn+1 f (z) + Dn+2 f (z)
Similarly, by means of the application of the linear operator Dn we
obtain
(1 + a)Dn F (z) = aDn f (z) + Dn+1 f (z)
Thus
Dn+1 F (z)
=
Dn F (z)
Dn+2 f (z)
·
Dn+1 f (z)
=
(2)
With notation
Dn+2 f (z) + aDn+1 f (z)
=
Dn+1 f (z) + aDn f (z)
Dn+1 f (z)
Dn+1 f (z)
+a·
Dn f (z)
Dn f (z)
Dn+1 f (z)
+a
Dn f (z)
Dn+1 f (z)
= p(z), where p(z) = 1 + p1 z + ..., we have
Dn f (z)
µ
0
zp (z) = z ·
Dn+1 f (z)
Dn f (z)
¶0
=
0
z (Dn+1 f (z)) · Dn f (z) − Dn+1 f (z) · z (Dn f (z))0
=
=
(Dn f (z))2
2
Dn+2 f (z) · Dn f (z) − (Dn+1 f (z))
=
(Dn f (z))2
and
Dn+2 f (z) Dn+1 f (z)
Dn+2 f (z)
1
· zp0 (z) = n+1
−
=
− p(z)
p(z)
D f (z)
Dn f (z)
Dn+1 f (z)
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From the above he have
Dn+2 f (z)
1
= p(z) +
· zp0 (z)
n+1
D f (z)
p(z)
Thus from (2) we obtain
³
p(z) · zp0 (z) ·
n+1
D F (z)
=
Dn F (z)
(3)
1
p(z)
´
+ p(z) + a · p(z)
p(z) + a
= p(z) +
From Remark 3.1 we have
=
1
· zp0 (z)
p(z) + a
Dn+1 F (z)
≺ pα (z) and thus, using (3), we
Dn F (z)
obtain
p(z) +
1
zp0 (z) ≺ pα (z) .
p(z) + a
We have from Remark 3.1 and from the hypothesis Re (pα (z) + a) > 0,
z ∈ U . In this conditions from Theorem 2.2 we obtain p(z) ≺ pα (z) or
Dn+1 f (z)
≺ pα (z) . This means that f (z) = La F (z) ∈ SH(α) .
Dn f (z)
Theorem 3.3. Let a ∈ C, Re a ≥ 0, α > 0, and n ∈ N . If F (z) ∈ SHn (α),
∞
∞
X
X
F (z) = z +
aj z j , and f (z) = La F (z), f (z) = z +
bj z j , where La is
j=2
j=2
the integral operator defined by (1), then
¯
¯
¯
¯
¯ a + 1 ¯ 1 1 + 4α
¯ a + 1 ¯ 1 (1 + 4α)(3 + 16α + 24α2 )
¯·
¯·
|b2 | ≤ ¯¯
·
, |b3 | ≤ ¯¯
·
.
a + 2 ¯ 2n 1 + 2α
a + 3 ¯ 3n
4(1 + 2α)3
Proof. From f (z) = La F (z) we have (1 + a)F (z) = af (z) + zf 0 (z) . Using
the above series expansions we obtain
∞
∞
∞
X
X
X
j
j
jbj z j
abj z + z +
(1 + a)aj z = az +
(1 + a)z +
j=2
j=2
j=2
97
On a subclass of n-starlike functions associated with some hyperbola
and thus
bj ¯(a + j) = (1 + a)aj , j ≥ 2 . From the above we have
¯
¯a + 1¯
¯ · |aj | , j ≥ 2 . Using the estimations from Theorem 3.1 we
|bj | ≤ ¯¯
a + j¯
obtain the needed results.
For a = 1, when the integral operator La become the Libera integral operator, we obtain from the above theorem:
Corollary 3.1. Let α > 0 and n ∈ N. If F (z) ∈ SHn (α), F (z) = z +
∞
X
∞
X
aj z j ,
j=2
and f (z) = L (F (z)), f (z) = z +
bj z j , where L is Libera integral operaj=2
Z
2 z
F (t)dt , then
tor defined by L (F (z)) =
z 0
|b2 | ≤
1
2n−1
·
1 + 4α
1 (1 + 4α)(3 + 16α + 24α2 )
, |b3 | ≤ n ·
.
3 + 6α
3
8(1 + 2α)3
Theorem 3.4. Let n ∈ N and α > 0. If f ∈ SHn+1 (α) then f ∈ SHn (α) .
Proof.
With notation
Theorem 3.2):
Dn+1 f (z)
= p(z) we have (see the proof of the
Dn f (z)
Dn+2 f (z)
1
= p(z) +
· zp0 (z) .
n+1
D f (z)
p(z)
1
· zp0 (z) ≺
p(z)
pα (z) . Using the definition of the function pα (z) we have Re pα (z) > 0 and
From f ∈ SHn+1 (α) we obtain (see Remark 3.1) p(z) +
from Theorem 2.2 we obtain p(z) ≺ pα (z) or f ∈ SHn (α) .
Remark 3.2. From the above theorem we obtain SHn (α) ⊂ SH0 (α) =
SH(α) ⊂ S ∗ for all n ∈ N .
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Mugur Acu
References
[1] S. S. Miller and P. T. Mocanu, Differential subordonations and univalent
functions, Mich. Math. 28 (1981), 157 - 171.
[2] S. S. Miller and P. T. Mocanu, Univalent solution of Briot-Bouquet
differential equations, J. Differential Equations 56 (1985), 297 - 308.
[3] S. S. Miller and P. T. Mocanu, On some classes of first-order differential
subordinations, Mich. Math. 32(1985), 185 - 195.
[4] W. Rogosinski, On the coefficients of subordinate functions, Proc. London Math. Soc. 48(1943), 48-82.
[5] Gr. Sălăgean, Subclasses of univalent functions, Complex Analysis. Fifth
Roumanian-Finnish Seminar, Lectures Notes in Mathematics, 1013,
Springer-Verlag, 1983, 362-372.
[6] J. Stankiewicz, A. Wisniowska, Starlike functions associated with some
hyperbola, Folia Scientiarum Universitatis Tehnicae Resoviensis 147,
Matematyka 19(1996), 117-126.
University ”Lucian Blaga” of Sibiu
Department of Mathematics
Str. Dr. I. Rat.iu, No. 5-7
550012 - Sibiu, Romania
E-mail address: acu mugur@yahoo.com