General Mathematics Vol. 14, No. 2 (2006), 31–42
Subclasses of starlike functions associated
with some hyperbola1
Mugur Acu
Abstract
In this paper we define some subclasses of starlike functions associated with some hyperbola by using a generalized Sălăgean operator
and we give some properties regarding these classes.
2000 Mathematics Subject Classification: 30C45
Key words and phrases: Starlike functions, Libera-Pascu integral
operator, Briot-Bouquet differential subordination, generalized Sălăgean
operator
1
Introduction
Let H(U ) be the set of functions which are regular in the unit disc U ,
A = {f ∈ H(U ) : f (0) = f 0 (0) − 1 = 0}, Hu (U ) = {f ∈ H(U ) :
f is univalent in U } and S = {f ∈ A : f is univalent in U }.
1
Received April 4, 2006
Accepted for publication (in revised form) April 10, 2006
31
32
Mugur Acu
Let Dn be the Sălăgean differential operator (see [12]) defined as:
Dn : A → A ,
n ∈ N and D0 f (z) = f (z)
D1 f (z) = Df (z) = zf 0 (z) ,
Remark 1.1. If f
∞
P
Dn f (z) = z +
∈
S , f (z)
Dn f (z) = D(Dn−1 f (z)).
=
z +
∞
P
aj z j , z
∈
U then
j=2
j n aj z j .
j=2
We recall here the definition of the well - known class of starlike functions
½
¾
zf 0 (z)
∗
S = f ∈ A : Re
>0, z∈U .
f (z)
Let consider the Libera-Pascu integral operator La : A → A defined as:
Zz
1+a
(1) f (z) = La F (z) =
F (t) · ta−1 dt , a ∈ C , Re a ≥ 0.
a
z
0
Generalizations of the Libera-Pascu integral operator was studied by
many mathematicians such are P.T. Mocanu in [7], E. Drăghici in [6] and
D. Breaz in [5].
Definition 1.1.[4] Let n ∈ N and λ ≥ 0. We denote with Dλn the operator
defined by
Dλn : A → A ,
Dλ0 f (z) = f (z) , Dλ1 f (z) = (1 − λ)f (z) + λzf 0 (z) = Dλ f (z) ,
¡
¢
Dλn f (z) = Dλ Dλn−1 f (z) .
Remark 1.2.[4] We observe that Dλn is a linear operator and for
∞
X
f (z) = z +
aj z j we have
j=2
Dλn f (z)
=z+
∞
X
j=2
(1 + (j − 1)λ)n aj z j .
Subclasses of starlike functions associated with some hyperbola
33
Also, it is easy to observe that if we consider λ = 1 in the above definition
we obtain the Sălăgean differential operator.
The next theorem is result of the so called ”admissible functions method”
introduced by P.T. Mocanu and S.S. Miller (see [8], [9], [10]).
Theorem 1.1. 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
p(z) +
zp0 (z)
≺ h(z),
βp(z) + γ
then p(z) ≺ h(z).
In [1] is introduced the following operator:
Definition 1.2. Let β, λ ∈ R, β ≥ 0, λ ≥ 0 and f (z) = z +
∞
X
aj z j . We
j=2
denote by Dλβ the linear operator defined by
Dλβ : A → A ,
Dλβ f (z)
=z+
∞
X
(1 + (j − 1)λ)β aj z j .
j=2
Remark 1.3. It is easy to observe that for β = n ∈ N we obtain the AlOboudi operator Dλn and for β = n ∈ N , λ = 1 we obtain the Sălăgean
operator Dn .
The purpose of this note is to define some subclasses of starlike functions
associated with some hyperbola by using the operator Dλβ defined above and
to obtain some properties regarding these classes.
34
2
Mugur Acu
Preliminary results
Definition 2.1. [13] A function f ∈ S is said to be in the class SH(α) if
it satisfies
¯ 0
¾
½
³√
³√
´¯¯
´
√ zf 0 (z)
¯ zf (z)
¯
¯
+
2α
−
2α
2
−
1
<
Re
2
2
−
1
,
¯ f (z)
¯
f (z)
for some α (α > 0) and for all z ∈ U .
Remark 2.1. Geometric
interpretation:
½ 0
¾
zf (z)
Let 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.
Definition 2.2. [3] Let f ∈ S and α > 0. We say that the function f is in
the class SHn (α), n ∈ N , if
¯ n+1
¾
½
³√
´¯¯
³√
´
√ Dn+1 f (z)
¯ D f (z)
¯
¯
−
2α
2
−
1
2
+2α
2
−
1
,z∈U.
<
Re
¯ Dn f (z)
¯
Dn f (z)
Remark 2.2. 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 .
Theorem 2.1. [3] 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.
Subclasses of starlike functions associated with some hyperbola
Theorem 2.2. [3] Let n ∈ N and α > 0.
35
If f ∈ SHn+1 (α) then
f ∈ SHn (α) .
3
Main results
r
1 + bz
− 2α ,
1−z
√
1 + 4α − 4α2
where b = b(α) =
and
the
branch
of
the
square
root
w is cho2
(1
+
2α)
√
sen so that Im w ≥ 0 . We say that a function f (z) ∈ S is in the class
Definition 3.1. Let β ≥ 0, λ ≥ 0, α > 0 and pα (z) = (1 + 2α)
SHβ,λ (α) if
Dλβ+1 f (z)
Dλβ f (z)
≺ pα (z) , z ∈ U .
Remark 3.1. Geometric interpretation: f (z) ∈ SHβ,λ (α) if and only if
Dλβ+1 f (z)
take all values in the domain Ω(α) which is the interior of a
Dλβ f (z)
hyperbola in the right half-plane which is symmetric about the real axis and
has vertex at the origin (see Remark 2.1 and Remark 2.2).
Remark 3.2. It is easy to observe that for β = n ∈ N and λ = 1 we obtain
in the above definition we obtain the class SHn (α) studied in [3] and for
λ = 1 , β = 0 we obtain the class SH(α) studied in [13].
Theorem 3.1. Let β ≥ 0, α > 0 and λ > 0 . We have
SHβ+1,λ (α) ⊂ SHβ,λ (α) .
Proof. Let f (z) ∈ SHβ+1,λ (α).
With notation
p(z) =
Dλβ+1 f (z)
Dλβ f (z)
, p(0) = 1 ,
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Mugur Acu
we obtain
Dλβ+2 f (z)
1
Dλβ+2 f (z)
=
·
=
·
p(z) Dλβ f (z)
Dλβ+1 f (z)
Dλβ f (z) Dλβ+1 f (z)
(2)
Dλβ+2 f (z)
Dλβ f (z)
Also, we have
Dλβ+2 f (z)
Dλβ f (z)
z+
∞
X
(1 + (j − 1)λ)β+2 aj z j
j=2
∞
X
=
z+
(1 + (j − 1)λ)β aj z j
j=2
and
³
zp0 (z) =
z
Ã
z
´0
Dλβ+1 f (z)
∞
X
·
z
´0
Dλβ f (z)
Dλβ f (z)
!
(1 + (j − 1)λ)β+1 jaj z j−1
j=2
=
z
1+
!
(1 + (j − 1)λ)β jaj z j−1
Dλβ f (z)
or
z+
∞
X
j (1 + (j − 1)λ)β+1 aj z j
j=2
zp0 (z) =
−
Dλβ f (z)
z+
−p(z) ·
z+
∞
X
j=2
−p(z) ·
We have
−
Dλβ f (z)
Ã
(3)
−
Dλβ f (z)
1+
³
Dλβ+1 f (z)
Dλβ f (z)
∞
X
j=2
∞
X
j (1 + (j − 1)λ)β aj z j
j=2
Dλβ f (z)
j (1 + (j − 1)λ)β+1 aj z j =
.
=
Subclasses of starlike functions associated with some hyperbola
=z+
∞
X
37
((j − 1) + 1) (1 + (j − 1)λ)β+1 aj z j =
j=2
=z+
∞
X
β+1
(1 + (j − 1)λ)
∞
X
aj z +
(j − 1) (1 + (j − 1)λ)β+1 aj z j =
j
j=2
j=2
=z+
Dλβ+1 f (z)
∞
X
−z+
(j − 1) (1 + (j − 1)λ)β+1 aj z j =
j=2
∞
=
Dλβ+1 f (z)
1X
+
((j − 1)λ) (1 + (j − 1)λ)β+1 aj z j =
λ j=2
∞
=
Dλβ+1 f (z)
1X
+
(1 + (j − 1)λ − 1) (1 + (j − 1)λ)β+1 aj z j =
λ j=2
∞
=
Dλβ+1 f (z)
∞
1X
1X
(1 + (j − 1)λ)β+1 aj z j +
(1 + (j − 1)λ)β+2 aj z j =
−
λ j=2
λ j=2
´ 1³
´
1 ³ β+1
Dλ f (z) − z +
Dλβ+2 f (z) − z =
λ
λ
1
z
1
z
= Dλβ+1 f (z) − Dλβ+1 f (z) + + Dλβ+2 f (z) − =
λ
λ λ
λ
λ − 1 β+1
1
=
Dλ f (z) + Dλβ+2 f (z) =
λ
λ
³
´
1
=
(λ − 1)Dλβ+1 f (z) + Dλβ+2 f (z) .
λ
= Dλβ+1 f (z) −
Similarly we have
z+
∞
X
j (1 + (j − 1)λ)β aj z j =
j=2
´
1³
(λ − 1)Dλβ f (z) + Dλβ+1 f (z) .
λ
From (3) we obtain
1
=
λ
Ã
zp0 (z) =
(λ − 1)Dλβ+1 f (z) + Dλβ+2 f (z)
Dλβ f (z)
− p(z)
(λ − 1)Dλβ f (z) + Dλβ+1 f (z)
Dλβ f (z)
!
=
38
Mugur Acu
1
=
λ
Ã
(λ − 1)p(z) +
!
Dλβ+2 f (z)
− p(z) ((λ − 1) + p(z))
Dλβ f (z)
Ã
!
1 Dλβ+2 f (z)
2
=
− p(z)
λ
Dλβ f (z)
Thus
or
λzp0 (z) =
Dλβ+2 f (z)
Dλβ+2 f (z)
= p(z)2 + λzp0 (z) .
Dλβ f (z)
Dλβ f (z)
=
− p(z)2
From (2) we obtain
Dλβ+2 f (z)
Dλβ+1 f (z)
=
¢
1 ¡
zp0 (z)
p(z)2 + λzp0 (z) = p(z) + λ
,
p(z)
p(z)
where λ > 0.
From f (z) ∈ SHβ+1,λ (α) we have
p(z) + λ
zp0 (z)
≺ pα (z) ,
p(z)
with p(0) = pα (0) = 1, α > 0, β ≥ 0, λ > 0, and Re pα (z) > 0 from here
construction. In this conditions from Theorem 1.1, we obtain
p(z) ≺ pα (z)
or
Dλβ+1 f (z)
Dλβ f (z)
≺ pα (z) .
This means f (z) ∈ SHβ,λ (α) .
Theorem 3.2. Let β ≥ 0, α > 0 and λ ≥ 1 . If F (z) ∈ SHβ,λ (α) then
f (z) = La F (z) ∈ SHβ,λ (α), where La is the Libera-Pascu integral operator
defined by (1).
Subclasses of starlike functions associated with some hyperbola
39
Proof. From (1) we have
(1 + a)F (z) = af (z) + zf 0 (z)
and, by using the linear operator Dλβ+1 , we obtain
Ã
(1 + a)Dλβ+1 F (z) = aDλβ+1 f (z) + Dλβ+1 z +
∞
X
!
jaj z j
=
j=2
= aDλβ+1 f (z) + z +
∞
X
(1 + (j − 1)λ)β+1 jaj z j
j=2
We have (see the proof of the above theorem)
z+
∞
X
j (1 + (j − 1)λ)β+1 aj z j =
j=2
´
1³
(λ − 1)Dλβ+1 f (z) + Dλβ+2 f (z)
λ
Thus
(1 + a)Dλβ+1 F (z) = aDλβ+1 f (z) +
µ
=
λ−1
a+
λ
¶
´
1³
(λ − 1)Dλβ+1 f (z) + Dλβ+2 f (z) =
λ
1
Dλβ+1 f (z) + Dλβ+2 f (z)
λ
or
λ(1 + a)Dλβ+1 F (z) = ((a + 1)λ − 1) Dλβ+1 f (z) + Dλβ+2 f (z) .
Similarly, we obtain
λ(1 + a)Dλβ F (z) = ((a + 1)λ − 1) Dλβ f (z) + Dλβ+1 f (z) .
Then
Dλβ+2 f (z) Dλβ+1 f (z)
Dλβ+1 f (z)
·
+ ((a + 1)λ − 1) ·
Dλβ+1 F (z)
Dλβ+1 f (z) Dλβ f (z)
Dλβ f (z)
=
.
Dλβ F (z)
Dλβ+1 f (z)
+ ((a + 1)λ − 1)
Dλβ f (z)
40
Mugur Acu
With notation
Dλβ+1 f (z)
Dλβ f (z)
= p(z) , p(0) = 1 ,
we obtain
Dλβ+2 f (z)
· p(z) + ((a + 1)λ − 1) · p(z)
Dλβ+1 f (z)
=
.
p(z) + ((a + 1)λ − 1)
Dλβ F (z)
Dλβ+1 F (z)
(4)
We have (see the proof of the above theorem)
λzp0 (z) =
=
Thus
Dλβ+2 f (z) Dλβ+1 f (z)
·
− p(z)2 =
β+1
β
Dλ f (z) Dλ f (z)
Dλβ+2 f (z)
Dλβ+1 f (z)
Dλβ+2 f (z)
Dλβ+1 f (z)
· p(z) − p(z)2 .
¢
1 ¡
· p(z)2 + λzp0 (z) .
p(z)
=
Then, from (4), we obtain
Dλβ+1 F (z)
Dλβ F (z)
=
p(z)2 + λzp0 (z) + ((a + 1)λ − 1) p(z)
=
p(z) + ((a + 1)λ − 1)
zp0 (z)
,
p(z) + ((a + 1)λ − 1)
where a ∈ C, Re a ≥ 0, β ≥ 0, and λ ≥ 1 .
= p(z) + λ
From F (z) ∈ SHβ,λ (α) we have
p(z) +
zp0 (z)
≺ pα (z) ,
1
(p(z)
+
((a
+
1)λ
−
1))
λ
where a ∈ C, Re a ≥ 0, α > 0, β ≥ 0, λ ≥ 1, and from her construction, we
have Re pα (z) > 0. In this conditions we have from Theorem 1.1 we obtain
p(z) ≺ pα (z)
Subclasses of starlike functions associated with some hyperbola
or
Dλβ+1 f (z)
Dλβ f (z)
41
≺ pα (z) .
This means f (z) = La F (z) ∈ SHβ,λ (α) .
Remark 3.3. If we consider β = n ∈ N in the previously results we obtain
the Theorem 3.1 and Theorem 3.2 from [2].
References
[1] M. Acu and S. Owa, Note on a class of starlike functions, Proceedings
of the International Short Point Research Work on Study on Calculus
Operators in Univalent Function Theory - kyoto 2006 (to appear).
[2] M. Acu and S. Owa, On n-starlike functions associated with some hyperbola, IJMS (to appear).
[3] M. Acu, On a subclass of n-starlike functions associated with some hyperbola, General Mathematics, Vol. 13, No. 1(2005), 91-98.
[4] F.M. Al-Oboudi, On univalent funtions defined by a generalized Sălăgean
operator, Ind. J. Math. Math. Sci. 2004, no. 25-28, 1429-1436.
[5] D. Breaz, Operatori integrali pe spaţii de funcţii univalente, Editura
Academiei Române, Bucureşti 2004.
[6] E. Drăghici, Elemente de teoria funcţiilor cu aplicaţii la operatori integrali univalenţi, Editura Constant, Sibiu 1996.
[7] P.T. Mocanu, Classes of univalent integral operators, J. Math. Anal.
Appl. 157, 1(1991), 147-165.
42
Mugur Acu
[8] S. S. Miller and P. T. Mocanu, Differential subordonations and univalent
functions, Mich. Math. 28 (1981), 157 - 171.
[9] S. S. Miller and P. T. Mocanu, Univalent solution of Briot-Bouquet
differential equations, J. Differential Equations 56 (1985), 297 - 308.
[10] S. S. Miller and P. T. Mocanu, On some classes of first-order differential
subordinations, Mich. Math. 32(1985), 185 - 195.
[11] W. Rogosinski, On the coefficients of subordinate functions, Proc. London Math. Soc. 48(1943), 48-82.
[12] Gr. Sălăgean, Subclasses of univalent functions, Complex Analysis.
Fifth Roumanian-Finnish Seminar, Lectures Notes in Mathematics,
1013, Springer-Verlag, 1983, 362-372.
[13] J. Stankiewicz and 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: acu mugur@yahoo.com