General Mathematics Vol. 15, Nr. 2–3(2007), 190-200
On some analytic functions with negative
coefficients
Mugur Acu
Abstract
We will study some classes of analytic functions with negative
coefficients introduced by using a modified Sălăgean operator.
2000 Mathematical Subject Classification: 30C45
1
Introduction and preliminaries
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 [7] 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
190
On some analytic functions with negative coefficients
191
was introduced.
Let Dn be the Sălăgean differential operator (see [6]) 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))
Let n ∈ N and λ ≥ 0. Let denote with Dλn the Al-Oboudi operator (see
[4]) 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) .
Definition 1. [3] 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
∞
P
Theorem 1. [6] If f (z) = z −
aj z j , aj ≥ 0, j = 2, 3, ..., z ∈ U then
j=2
the next assertions are equivalent:
∞
P
(i)
jaj ≤ 1
j=2
(ii) f ∈ T
(iii) f ∈ T ∗ , where T ∗ = T
starlike functions.
T
S ∗ and S ∗ is the well-known class of
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Mugur Acu
Definition 2. [6] Let α ∈ [0, 1) and n ∈ N, then
½
¾
Dn+1 f (z)
Sn (α) = f ∈ A : Re n
> α, z ∈ U
D f (z)
is the set of n-starlike functions of order α.
Definition 3. [5] Let α ∈ [0, 1), β ∈ (0, 1] and let n ∈ N; we define the
class Tn (α, β) of n-starlike functions of order α and type β with negative
coefficients by
Tn (α, β) = {f ∈ A : |Jn (f, α; z)| < β, z ∈ U },
where
Dn+1 f (z)
−1
Dn f (z)
Jn (f, α; z) = n+1
, z∈U
D f (z)
+ 1 − 2α
Dn f (z)
Remark 1. The class Tn (α, 1) is the class of n-starlike functions of order
T
α with negative coefficients i.e. Tn (α, 1) = T Sn (α).
Theorem 2. [5] Let α ∈ [0, 1), β ∈ (0, 1] and n ∈ N. The function f of
the form (1) is in Tn (α, β) if and only if
∞
X
j n [j − 1 + β(j + 1 − 2α)]aj ≤ 2β(1 − α)
j=2
Remark 2. From Remark 1 and Theorem 2, for f (z) of the form (1), we
have f ∈ Tn (α, 1) = Tn (α) iff
∞
X
j=2
j n (j − α)aj ≤ 1 − α, where α ∈ [0, 1)
On some analytic functions with negative coefficients
193
We denote by f ∗ g the modified Hadamard product of two functions
∞
P
aj z j , (aj ≥ 0, j = 2, 3, ...) and g(z) =
f (z), g(z) ∈ T , f (z) = z −
z−
∞
P
j=2
j
bj z , (bj ≥ 0, j=2,3,...), is defined by
j=2
(f ∗ g)(z) = z −
∞
X
aj bj z j .
j=2
We say that an analytic function f is subordinate to an analytic function
g if f (z) = g(w(z)), z ∈ U , for some analytic function w with w(0) = 0 and
|w(z)| < 1(z ∈ U ). We denote the subordination relation by f ≺ g.
2
Main results
Definition 4. [1] , [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 T Lβ (α) if:
Re
Dλβ+1 f (z)
Dλβ f (z)
> α, α ∈ [0, 1), λ ≥ 0, β ≥ 0, z ∈ U.
We say that f is in the class T c Lβ (α) if:
Re
Dλβ+2 f (z)
Dλβ+1 f (z)
> α, α ∈ [0, 1), λ ≥ 0, β ≥ 0, z ∈ U.
Remark 3. We observe that both classes may also be defined, by using the
subordination relation, such that:
(
T Lβ (α) =
Dλβ+1 f (z)
1+z
, α ∈ [0, 1), λ ≥ 0, β ≥ 0, z ∈ U
−α≺
f ∈T :
β
1−z
Dλ f (z)
)
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Mugur Acu
and
(
T c Lβ (α) =
Dλβ+2 f (z)
1+z
−α≺
, α ∈ [0, 1), λ ≥ 0, β ≥ 0, z ∈ U
f ∈ T : β+1
1−z
Dλ f (z)
Theorem 3. [1] , [2] Let α ∈ [0, 1), λ ≥ 0 and β ≥ 0.
The function f ∈ T of the form (1) is in the class T Lβ (α) iff
∞
X
[(1 + (j − 1)λ)β (1 + (j − 1)λ − α)]aj < 1 − α.
(2)
j=2
The function f ∈ T of the form (1) is in the class T c Lβ (α) iff
(3)
∞
X
[(1 + (j − 1)λ)β+1 (1 + (j − 1)λ − α)]aj < 1 − α.
j=2
Proof. Let f ∈ T Lβ (α), with α ∈ [0, 1), λ ≥ 0 and β ≥ 0 . We have
Re
Dλβ+1 f (z)
Dλβ f (z)
> α.
If we take z ∈ [0, 1), β ≥ 0, λ ≥ 0, we have (see Definition 1.1):
∞
X
1−
(1 + (j − 1)λ)β+1 aj z j−1
j=2
∞
X
(4)
1−
> α.
β
(1 + (j − 1)λ) aj z
j−1
j=2
From the above we obtain:
∞
∞
X
X
β+1
j−1
1−
(1 + (j − 1)λ) aj z
>α−
(1 + (j − 1)λ)β αaj z j−1 ,
j=2
j=2
∞
X
(1 + (j − 1)λ)β (1 + (j − 1)λ − α)aj z j−1 < 1 − α.
j=2
)
.
On some analytic functions with negative coefficients
195
Letting z → 1− along the real axis we have:
∞
X
(1 + (j − 1)λ)β (1 + (j − 1)λ − α)aj < 1 − α.
j=2
Conversely, let take f ∈ T for which the relation (2) hold.
The condition Re
(5)
We have
Dλβ+1 f (z)
> α is equivalent with
Dλβ f (z)
Ã
!
β+1
Dλ f (z)
α − Re
− 1 < 1.
Dλβ f (z)
Ã
α − Re
Dλβ+1 f (z)
Dλβ f (z)
!
−1
¯
¯
¯ Dβ+1 f (z)
¯
¯
¯
≤ α + ¯ λβ
− 1¯
¯ Dλ f (z)
¯
¯
¯ ∞
¯
¯X
β
j−1 ¯
¯
(1
+
(j
−
1)λ)
a
[(j
−
1)λ]z
¯
¯
j
¯
¯
¯ Dβ+1 f (z) − Dβ f (z) ¯
¯
¯ j=2
¯ λ
¯
λ
¯
=α+¯
¯ = α + ¯¯
∞
¯
β
X
¯
¯
Dλ f (z)
¯
¯
β
j−1
[1 + (j − 1)λ] aj z
¯
¯ 1−
¯
¯
j=2
∞
X
≤ α+
β
[1 + (j − 1)λ] aj |1 − j|λ|z|
j=2
1−
∞
X
∞
X
[1 + (j − 1)λ]β aj (j − 1)λ|z|j−1
j−1
= α+
β
[1 + (j − 1)λ] aj |z|
j−1
j=2
∞
X
1−
[1 + (j − 1)λ]β aj |z|j−1
j=2
j=2
∞
X
[1 + (j − 1)λ]β aj (j − 1)λ
< α+
j=2
∞
X
1−
[1 + (j − 1)λ]β aj
j=2
Thus
α+
=
∞
X
[1 + (j − 1)λ]β aj [(j − 1)λ − α]
j=2
∞
X
1−
[1 + (j − 1)λ]β aj
j=2
∞
X
[1 + (j − 1)λ]β aj [(j − 1)λ + 1 − α] < 1,
α+
j=2
< 1.
196
Mugur Acu
which is the condition (2).
The proof of the second part of the theorem is similarly with the above
one, so it is omitted.
Remark 4. Using the conditions (2) and (3) it is easy to prove that
T Lβ+1 (α) ⊆ T Lβ (α)
and
T c Lβ+1 (α) ⊆ T c Lβ (α),
where β ≥ 0, α ∈ [0, 1) and λ ≥ 0.
Theorem 4. [1] , [2] If f (z) = z−
g(z) = z −
∞
P
∞
P
aj z j ∈ T Lβ (α), (aj ≥ 0, j = 2, 3, ...),
j=2
j
bj z ∈ T Lβ (α), (bj ≥ 0, j = 2, 3, ...), α ∈ [0, 1), λ ≥ 0, β ≥ 0,
j=2
then f (z) ∗ g(z) ∈ T Lβ (α).
If f (z) = z −
g(z) = z −
∞
P
∞
P
aj z j ∈ T c Lβ (α), (aj ≥ 0, j = 2, 3, ...),
j=2
bj z j ∈ T c Lβ (α), (bj ≥ 0, j = 2, 3, ...), α ∈ [0, 1), λ ≥ 0,
j=2
β ≥ 0, then f (z) ∗ g(z) ∈ T c Lβ (α).
Proof. We have
∞
X
[1 + (j − 1)λ]β [(j − 1)λ + 1 − α]aj < 1 − α
j=2
and
∞
X
[1 + (j − 1)λ]β [(j − 1)λ + 1 − α]bj < 1 − α.
j=2
We know that f (z) ∗ g(z) = z −
Theorem 1 , we have
∞
P
j=2
∞
P
aj bj z j . From g(z) ∈ T , by using
j=2
jbj ≤ 1. We notice that bj < 1, j = 2, 3, ... .
On some analytic functions with negative coefficients
197
Thus,
∞
∞
X
X
β
[1+(j−1)λ] [(j−1)λ+1−α]aj bj <
[1+(j−1)λ]β [(j−1)λ+1−α]aj < 1−α.
j=2
j=2
This means that f (z) ∗ g(z) ∈ T Lβ (α), β ≥ 0, α ∈ [0, 1) and λ ≥ 0.
The proof of the second part of the theorem is similarly with the above
one, so it is omitted.
Theorem 5. [1] , [2] Let f1 (z) = z and
fj (z) = z −
1−α
z j , j = 2, 3, ...
β
(1 + (j − 1)λ) (1 − α + (j − 1)λ)
∞
P
Then f ∈ T Lβ (α) iff it can be expressed in the form f (z) =
where λj ≥ 0 and
∞
P
λj fj (z),
j=1
λj = 1.
j=1
Let f1 (z) = z and
fj (z) = z −
(1 + (j −
1−α
z j , j = 2, 3, ...
− α + (j − 1)λ)
1)λ)β+1 (1
∞
P
Then f ∈ T c Lβ (α) iff it can be expressed in the form f (z) =
where λj ≥ 0 and
∞
P
λj fj (z),
j=1
λj = 1.
j=1
Proof.
Let f (z) =
λj fj (z), λj ≥ 0, j=1,2, ... , with
∞
X
λj fj (z) = λ1 z+
∞
X
j=1
j=2
=
∞
X
j=1
λj z −
∞
X
j=2
∞
P
λj = 1. We
j=1
j=1
obtain
f (z) =
∞
P
λj
µ
λj z −
1−α
zj
β
[1 + (j − 1)λ] [1 − α + (j − 1)λ]
1−α
zj
β
[1 + (j − 1)λ] [1 − α + (j − 1)λ]
¶
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Mugur Acu
=z−
∞
X
j=2
λj
1−α
zj .
β
[1 + (j − 1)λ] [1 − α + (j − 1)λ]
We have
∞
X
[1 + (j − 1)λ]β [1 − α + (j − 1)λ]λj
j=2
= (1 − α)
∞
X
1−α
[1 + (j − 1)λ]β [1 − α + (j − 1)λ]
∞
X
λj = (1 − α)(
λj − λ1 ) < 1 − α
j=2
j=1
which is the condition (2) for f (z) =
∞
P
λj fj (z). Thus f (z) ∈ T Lβ (α).
j=1
Conversely, we suppose that f (z) ∈ T Lβ (α), f (z) = z −
∞
P
aj z j , aj ≥ 0
j=2
[1 + (j − 1)λ]β [1 − α + (j − 1)λ]
and we take λj =
aj ≥ 0, j=2,3, ... , with
1−α
∞
X
λ1 = 1 −
λj .
j=2
Using the condition (2), we obtain
∞
X
∞
1 X
1
λj =
[1 + (j − 1)λ]β [1 − α + (j − 1)λ]aj <
(1 − α) = 1.
1 − α j=2
1−α
j=2
Then f (z) =
∞
P
λj fj , where λj ≥ 0, j=1,2, ... and
∞
P
λj = 1. This
j=1
j=1
completes our proof.
The proof of the second part of the theorem is similarly with the above
one, so it is omitted.
Corollary 1. [1] , [2] The extreme points of T Lβ (α) are f1 (z) = z and
fj (z) = z −
1−α
z j , j = 2, 3, ... .
(1 + (j − 1)λ)β (1 − α + (j − 1)λ)
The extreme points of T c Lβ (α) are f1 (z) = z and
fj (z) = z −
(1 + (j −
1−α
z j , j = 2, 3, ... .
− α + (j − 1)λ)
1)λ)β+1 (1
On some analytic functions with negative coefficients
199
Remark 5. We notice that in the particulary case, obtained for λ = 1 and
β ∈ N, we find similarly results for the class Tn (α) of the n-starlike functions
of order α with negative coefficients (inclusive the necessary and sufficiently
condition presented in Remark 2) and for the class Tnc (α) of the n-convex
functions of order α with negative coefficients .
References
[1] M. Acu and all, On some starlike functions with negative coefficients,
(to appear).
[2] M. Acu and all, About some convex functions with negative coefficients,
(to appear).
[3] M. Acu, S. Owa, Note on a class of starlike functions, Proceeding Of
the International Short Joint Work on Study on Calculus Operators in
Univalent Function Theory - Kyoto 2006, 1-10.
[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] G. S. Sălăgean, On some classes of univalent functions, Seminar of
geometric function theory, Cluj - Napoca, 1983.
[6] G. S. Sălăgean, Geometria Planului Complex, Ed. Promedia Plus, Cluj
- Napoca, 1999.
[7] H. Silverman, Univalent functions with negative coefficients, Proc.
Amer. Math. Soc. 5(1975), 109-116.
200
Mugur Acu
This paper is a result of the joint-work between Department of
Mathematics from ”Lucian Blaga” University, Sibiu, Romania and
Department of Mathematics from School of Sciences and Engineering,
Kinki University, Higashi-Osaka, Japan.
Department of Mathematics
Faculty of Science
University ”Lucian Blaga” of Sibiu
Str. I. Ratiu, no. 5-7,
550012 - Sibiu, Romania
Email address: acu mugur@yahoo.com