General Mathematics Vol. 12, No. 4 (2004), 39–48
Certain preserving properties of the
generalized Alexander operator
Mugur Acu
Abstract
In this paper we give certain preserving properties of the generalized Alexander operator on some subclasses of n-uniformly functions.
2000 Mathematical Subject Classification: 30C45
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}
and S = {f ∈ A : f is univalent in U }.
In [8] the subfamily T of S consisting of functions f of the form
(1)
f (z) = z −
∞
X
aj z j , aj ≥ 0, j = 2, 3, ..., z ∈ U
j=2
39
40
Mugur Acu
was introduced.
Let define the Alexander operator I p : A → H(U )
I 0 f (z) = f (z)
Z z
f (t)
1
I f (z) = If (z) =
dt
t
0
I p f (z) = I(I p−1 f (z)), p = 1, 2, 3, ...
∞
P
We have for f (z) = z +
aj z j ,
j=2
∞
X
1
j
I f (z) = z +
p aj z , p = 1, 2, 3, ...
j
j=2
p
Now we can define the generalized Alexander operator
∞
X
1
I λ : A → H(U ), I λ f (z) = z +
(2)
a zj ,
λ j
j
j=2
with λ ∈ R, λ ≥ 0, where f (z) = z +
∞
P
aj z j .
j=2
The purpose of this paper is to show that the class of n-uniform starlike functions of type α and order γ with negative coefficients, the class of
n-uniform close to convex functions of type α and order γ with negative coefficients and some subclasses of functions with negative coefficients, which
derive from the above mentioned classes, are preserved by the generalized
Alexander operator.
2
Preliminary results
Let Dn be the Sălăgean differential operator (see [6]) Dn : A → A, n ∈ N,
defined as:
D0 f (z) = f (z)
Certain preserving properties of the generalized Alexander operator
41
D1 f (z) = Df (z) = zf 0 (z)
Dn f (z) = D(Dn−1 f (z))
Remark 2.1 If f ∈ T, f (z) = z −
Dn f (z) = z −
∞
P
∞
P
aj z j , aj ≥ 0, j = 2, 3, ..., z ∈ U then
j=2
j n aj z j .
j=2
Theorem 2.1 [6] If f (z) = z −
the next assertions are equivalent:
∞
P
(i)
jaj ≤ 1
∞
P
aj z j , aj ≥ 0, j = 2, 3, ..., z ∈ U then
j=2
j=2
(ii) f ∈ T
(iii) f ∈ T ∗ , where T ∗ = T
T
S ∗ and S ∗ is the well-known class of
starlike functions.
Definition 2.1 [6] Let γ ∈ [0, 1) and n ∈ N, then
½
¾
Dn+1 f (z)
Sn (γ) = f ∈ A : Re n
>γ, z ∈U
D f (z)
is the set of n-starlike functions of order γ.
We denote by Tn (γ) the subclass T
T
Sn (γ).
Definition 2.2 [3] Let f ∈ A. We say that f is n-close to convex of
order γ with respect to a half-plane, and denote by CCn (γ) the set of these
functions, if there exists g ∈ Sn (0) so that
Re
where n ∈ N, γ ∈ [0, 1).
Dn+1 f (z)
> γ, z ∈ U,
Dn g(z)
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Mugur Acu
Definition 2.3 [1] Let f ∈ T , f (z) = z −
∞
P
aj z j , aj ≥ 0, j = 2, 3, ...,
j=2
z ∈ U . We say that f is in the class CCTn (γ), γ ∈ [0, 1), n ∈ N, with
respect to the function g ∈ Tn (0), if:
Re
Dn+1 f
> γ , z ∈ U.
Dn g
Theorem 2.2 [1] Let γ ∈ [0, 1) and n ∈ N. The function f ∈ T of the form
∞
P
(1) is in CCTn (γ), with respect to the function g ∈ Tn (0), g(z) = z− bj z j ,
bj ≥ 0, j = 2, 3, ..., if and only if
∞
X
(3)
j=2
j n [jaj + (2 − α)bj ] < 1 − γ .
j=2
Definition 2.4 [4] Let f ∈ A, we say that f is n-uniform starlike function
of type α if
µ
Re
Dn+1 f (z)
Dn f (z)
¶
¯ n+1
¯
¯ D f (z)
¯
¯
≥α·¯ n
− 1¯¯ , z ∈ U
D f (z)
where α ≥ 0, n ∈ N. We denote this class with U Sn (α).
T
We denote by U Tn (α) the subclass T U Sn (α).
Definition 2.5 [3] Let f ∈ A, we say that f is n-uniform close to convex
function of type α in respect to the function n-uniform starlike of type α
g(z), where α ≥ 0, n ∈ N, if
µ
Re
Dn+1 f (z)
Dn g(z)
¶
¯
¯ n+1
¯
¯ D f (z)
− 1¯¯ , z ∈ U
≥ α · ¯¯ n
D g(z)
where α ≥ 0, n ∈ N. We denote this class with U CCn (α).
Certain preserving properties of the generalized Alexander operator
43
Definition 2.6 [5] Let f ∈ A, we say that f is n-uniform starlike function
of order γ and type α if
¯
¯ n+1
µ n+1
¶
¯ D f (z)
¯
D f (z)
Re
− 1¯¯ + γ, z ∈ U
≥ α · ¯¯ n
n
D f (z)
D f (z)
where α ≥ 0, γ ∈ [−1, 1), α + γ ≥ 0, n ∈ N. We denote this class with
U Sn (α, γ).
We denote by U Tn (α, γ) the subclass T
T
U Sn (α, γ).
Theorem 2.3 [5] Let α ≥ 0, γ ∈ [−1, 1), α + γ ≥ 0 and n ∈ N. The
function f of the form (1) is in U Tn (α, γ) if and only if
∞
X
j n [(α + 1)j − (α + γ)]aj ≤ 1 − γ.
j=2
Definition 2.7 [3] Let f ∈ A, we say that f is n-uniform close to convex
function of order γ and type α in respect to the function n-uniform starlike
of order γ and type α, g(α), where α ≥ 0, γ ∈ [−1, 1), α + γ ≥ 0, n ∈ N, if
¯ n+1
¯
¶
µ n+1
¯ D f (z)
¯
D f (z)
¯
≥α·¯ n
− 1¯¯ + γ, z ∈ U
Re
n
D g(z)
D g(z)
where α ≥ 0, γ ∈ [−1, 1), α + γ ≥ 0, n ∈ N. We denote this class with
U CCn (α, γ).
Definition 2.8 [2] Let f ∈ T , f (z) = z −
∞
P
aj z j , aj ≥ 0, j = 2, 3, ...,
j=2
z ∈ U . We say that f is in the class U CCTn (α), α ≥ 0, n ∈ N, with respect
∞
P
to the function g(z) ∈ U Tn (α) g(z) = z −
bj z j , bj ≥ 0, j = 2, 3, ...,
j=2
z ∈ U , if:
µ
Re
Dn+1 f (z)
Dn g(z)
¶
¯
¯ n+1
¯
¯ D f (z)
− 1¯¯ z ∈ U.
> α · ¯¯ n
D g(z)
44
Mugur Acu
∞
P
Definition 2.9 [2] Let f ∈ T, f (z) = z −
aj z j , aj ≥ 0, j = 2, 3, ..., z ∈
j=2
U . We say that f is in the class U CCTn (α, γ), α ≥ 0, γ ∈ [−1, 1), α + γ ≥
∞
P
0, n ∈ N, with respect to the function g(z) ∈ U Tn (α, γ), g(z) = z −
bj z j ,
bj ≥ 0, j = 2, 3, ..., z ∈ U , if
¯ n+1
¯
µ n+1
¶
¯ D f (z)
¯
D f (z)
¯
Re
≥α·¯ n
− 1¯¯ + γ, z ∈ U.
n
D g(z)
D g(z)
j=2
Theorem 2.4 [2] Let n ∈ N, α ≥ 0, γ ∈ [−1, 1), with α + γ ≥ 0. The
function f ∈ T of the form (1) is in U CCTn (α, γ), with respect to the
∞
P
bj z j , bj ≥ 0, j = 2, 3, ..., z ∈ U , if
function g ∈ U Tn (α, γ), g(z) = z −
j=2
and only if
∞
X
(4)
j n [(α + 1)|jaj − bj | + (1 − γ)bj ] ≤ 1 − γ .
j=2
3
Main results
Theorem 3.1 If F ∈ U Tn (α, γ), α ≥ 0, γ ∈ [−1, 1), α + γ ≥ 0, n ∈ N
and f = I λ (F ), where I λ is defined by (2), then f ∈ U Tn (α, γ), α ≥ 0,
γ ∈ [−1, 1), α + γ ≥ 0, n ∈ N .
Proof. From F (z) = z −
∞
P
aj z j , aj ≥ 0, j = 2, 3, ... and f (z) = I λ (F )(z)
j=2
we have:
f (z) = z −
∞
X
j=2
αj z j , where αj =
1
aj ≥ 0, j = 2, 3, ...
jλ
From Theorem 2.3 we need only to show that:
(5)
∞
X
j=2
j n [(α + 1)j − (α + γ)]αj ≤ 1 − γ .
Certain preserving properties of the generalized Alexander operator
45
For λ ∈ R, λ ≥ 0, aj ≥ 0 and j = 2, 3, ... , we have:
αj =
1
aj ≤ aj
jλ
and thus
(6)
∞
X
n
j [(α + 1)j − (α + γ)]αj ≤
j=2
∞
X
j n [(α + 1)j − (α + γ)]aj
j=2
where α ≥ 0, γ ∈ [−1, 1), α + γ ≥ 0 and n ∈ N .
From F ∈ U Tn (α, γ), using Theorem 2.3, we have
∞
X
j n [(α + 1)j − (α + γ)]aj ≤ 1 − γ
j=2
and thus from (6) we obtain the condition (5).
Theorem 3.2 If F ∈ U CCTn (α, γ), α ≥ 0, γ ∈ [−1, 1), α + γ ≥ 0, n ∈ N,
with respect to the function G ∈ U Tn (α, γ) and f = I λ (F ), g = I λ (G),
where I λ is defined by (2), then f ∈ U CCTn (α, γ), α ≥ 0, γ ∈ [−1, 1),
α + γ ≥ 0, n ∈ N, with respect to the function g ∈ U Tn (α, γ) .
Proof. From the above Theorem we have g ∈ U Tn (α, γ) .
∞
∞
P
P
For F (z) = z− aj z j , aj ≥ 0, j = 2, 3, ... and G(z) = z− bj z j , bj ≥
j=2
∞
P
j=2
1
aj ≥
jλ
j=2
∞
P
1
0 , j = 2, 3, ... and g(z) = I λ (G)(z) = z −
βj z j , where βj = λ bj ≥
j
j=2
0 , j = 2, 3, ... .
0, j = 2, 3, ... we have f (z) = I λ (F )(z) = z −
αj z j , where αj =
From Theorem 2.4 we need only to show that:
(7)
∞
X
j=2
j n [(α + 1) |jαj − βj | + (1 − γ)βj ] ≤ 1 − γ .
46
Mugur Acu
It is easy to see that
(8) (α + 1) |jαj − βj | + (1 − γ)βj =
1
[(α + 1) |jaj − bj | + (1 − γ)bj ] ≤
jλ
≤ (α + 1) |jaj − bj | + (1 − γ)bj
for λ ∈ R, λ ≥ 0, aj ≥ 0, bj ≥ 0, j = 2, 3, ... , α ≥ 0, γ ∈ [−1, 1) and
α + γ ≥ 0.
From F ∈ U CCTn (α, γ), with respect to the function G ∈ U Tn (α, γ),
we have (see Theorem 2.4):
∞
X
j n [(α + 1) |jaj − bj | + (1 − γ)bj ] ≤ 1 − γ
j=2
and thus from (8) we obtain the condition (7).
If we take γ = 0 in Theorem 3.1 and Theorem 3.2 we obtain:
Theorem 3.3 If F ∈ U Tn (α), α ≥ 0, n ∈ N and f = I λ (F ), where I λ is
defined by (2), then f ∈ U Tn (α), α ≥ 0, n ∈ N .
Theorem 3.4 If F ∈ U CCTn (α), α ≥ 0, n ∈ N, with respect to the function G ∈ U Tn (α) and f = I λ (F ), g = I λ (G), where I λ is defined by (2), then
f ∈ U CCTn (α), α ≥ 0, n ∈ N, with respect to the function g ∈ U Tn (α) .
If we take γ ∈ [0, 1) and α = 0 in Definition 2.6 we have U Tn (0, γ) =
Tn (γ) and thus from Theorem 3.1, with γ ∈ [0, 1) and α = 0, we obtain:
Theorem 3.5 If F ∈ Tn (γ), γ ∈ [0, 1), n ∈ N and f = I λ (F ), where I λ is
defined by (2), then f ∈ Tn (γ), γ ∈ [0, 1), n ∈ N .
Certain preserving properties of the generalized Alexander operator
47
Remark 3.1 From Theorem 3.2 with γ ∈ [0, 1) and α = 0, we obtain the
preserving property of the generalized Alexander operator on the subclass
U CCTn (0, γ), γ ∈ [0, 1), which is not the same with the class CCTn (γ) .
In a similarly way with the proof of the Theorem 3.2, using the condition
(3) instead of the condition (4), we obtain:
Theorem 3.6 If F ∈ CCTn (γ), γ ∈ [0, 1), n ∈ N, with respect to the
function G ∈ Tn (0) and f = I λ (F ), g = I λ (G), where I λ is defined by (2),
then f ∈ CCTn (γ), γ ∈ [0, 1), n ∈ N, with respect to the function g ∈ Tn (0) .
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X(2003), 57-68.
[3] D. Blezu, On the n-uniform close to convex functions with respect to a
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coefficients, Studia Mathematica (to appear)
48
Mugur Acu
[6] G. S. Sălăgean, Geometria Planului Complex, Ed. Promedia Plus, ClujNapoca, 1997.
[7] G. S. Sălăgean, On some classes of univalent functions, Seminar of
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University ”Lucian Blaga” of Sibiu
Department of Mathematics
Str. Dr. I. Raţiu, Nr. 5–7,
550012 - Sibiu, Romania.
E-mail address: acu mugur@yahoo.com