General Mathematics Vol. 11, No. 1–2 (2003), 53–62
On a subclass of functions with negative
coefficients
Mugur Acu
Dedicated to Professor dr. Gheorghe Micula on his 60th birthday
Abstract
We determine conditions for a function to be n-close to convex of
order α, α ∈ [0, 1), n ∈ N, with negative coefficients.
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 ([4]) 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
53
54
Mugur Acu
was introduced.
The purpose of this paper is to give a condition for f ∈ T to be n-close
to convex of order α, α ∈ [0, 1), n ∈ N, and to determine some properties
of this class.
2
Preliminary results
Let Dn be the Sălăgean differential operator (see [2]) 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))
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.[2]. If f (z) = z −
the next assertions are equivalent:
∞
P
jaj ≤ 1
(i)
∞
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.[2]. 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)
On a subclass of functions with negative coefficients
55
is the set of n-starlike functions of order α.
Remark 2.2. If f ∈ Sn (α) according to the definition of the Sălăgean
differential operator we can write that
Re
z(Dn f (z))0
>α
Dn f (z)
and thus the function F (z) = Dn f (z) ∈ S(α), α ∈ [0, 1), where
½
¾
zh0 (z)
S(α) = h ∈ A : Re
> α, z ∈ U .
h(z)
T
Definition 2.2.[2]. Tn (α) = T Sn (α).
Definition 2.3.[3]. 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)
, z∈U
Jn (f, α; z) = n+1
D f (z)
+ 1 − 2α
Dn f (z)
Remark 2.3. The class T0 (α, 1) is the class of starlike functions of order
α with negative coefficients; T1 (α, 1) is the well-known class of convex functions of order α with negative coefficients; Tn (α, 1) is the class of n-starlike
T
functions of order α with negative coefficients i.e. Tn (α, 1) = T Sn (α).
We also note that the functions in Tn (α, β) are univalent because Tn (α, β) ⊂
Tn (α, 1), β ∈ (0, 1) and Tn (α1 , β) ⊂ Tn (α, β) with 1 > α1 > α ≥ 0,
β ∈ (0, 1].
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Mugur Acu
Theorem 2.2.[3]. 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
The result is sharp and the extremal functions are:
fj (z) = z −
2β(1 − α)
z j , j = 2, 3, ....
j [j − 1 + β(j + 1 − 2α)]
n
From this result we have Tn+1 (α, β) ⊂ Tn (α, β), n ∈ N.
Definition 2.4.[3].
Let Ic : A → A be the integral operator defined by
f = Ic (F ), where c ∈ (−1, ∞), F ∈ A and
c+1
f (z) =
zc
(2)
Zz
tc−1 F (t)dt.
0
We note if F ∈ A is a function of the form (1), then
(3)
f (z) = Ic F (z) = z −
∞
X
c+1
j=2
c+j
aj z j .
Remark 2.4. In [3] is showed that if F ∈ Tn (α, β) then f = Ic (F ) ∈
Tn (α, β).
Definition 2.5.[1].
Let f ∈ A. We say that f is n-close to convex of
order α with respect to a half-plane, and denote by CCn (α) the set of these
functions, if there exists g ∈ Sn (0) = Sn so that
Re
where n ∈ N, α ∈ [0, 1).
Dn+1 f (z)
> α, z ∈ U,
Dn g(z)
On a subclass of functions with negative coefficients
57
Remark 2.5. CC0 (α) = CC(α), where CC(α) is the well-known class of
close to convex functions of order α.
Remark 2.6. In [1] the author show that if n ∈ N and α ∈ [0, 1) then
CCn+1 (α) ⊂ CCn (α) and thus the functions from CCn (α) are univalent.
Remark 2.7. From Remark 2.3 and Theorem 2.2 we have for f of the form
(1) with f ∈ Tn (α, 1) = Tn (α):
∞
X
j n (j − α)aj ≤ 1 − α, where α ∈ [0, 1)
j=2
3
Main results
Definition 3.1. Let f ∈ T , f (z) = z −
∞
P
aj z j , aj ≥ 0, j = 2, 3, ..., z ∈ U .
j=2
We say that f is in the class CCTn (α), α ∈ [0, 1), n ∈ N, with respect to
the function g ∈ Tn (0), if:
Re
Dn+1 f
> α, z ∈ U.
Dn g
Theorem 3.1. Let α ∈ [0, 1) and n ∈ N. The function f ∈ T of the form
∞
P
(1) is in CCTn (α), with respect to the function g ∈ Tn (0), g(z) = z− bj z j ,
bj ≥ 0, j = 2, 3, ..., if and only if
(4)
∞
X
j n [jaj + (2 − α)bj ] < 1 − α
j=2
Proof. Let f ∈ CCTn (α), with α ∈ [0, 1). We have
Re
Dn+1 f (z)
> α.
Dn g(z)
j=2
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Mugur Acu
If we take z ∈ [0, 1), we have (see Remark 2.1):
1−
∞
X
j n+1 aj z j−1
j=2
∞
X
(5)
1−
>α
n
j bj z
j−1
j=2
∞
P
From g ∈ Tn (0) = Tn (0, 1), g(z) = z −
bj z j , bj ≥ 0, j = 2, 3, ..., we
j=2
have (see Remark 2.7):
∞
X
(6)
j n+1 bj ≤ 1.
j=2
We have:
∞
P
∞
P
j n bj z j−1 ≤
j=2
From (6) we obtain:
∞
P
j n+1 bj z j−1 <
j=2
∞
P
j n+1 bj .
j=2
j n bj z j−1 < 1 and thus 1 −
j=2
∞
P
j n bj z j−1 > 0.
j=2
In this condition from (5) we obtain:
"
#
∞
∞
X
X
1−
j n+1 aj z j−1 > α 1 −
j n bj z j−1
j=2
j=2
Letting z → 1− along the real axis we have:
1−
∞
X
j n+1 aj > α −
j=2
and thus:
∞
X
∞
X
j n αbj ,
j=2
j n [jaj − αbj ] < 1 − α.
j=2
From
∞
P
j n [jaj + (2 − α)bj ] >
j=2
(7)
∞
P
j n [jaj − αbj ] we have that from
j=2
∞
X
j=2
j n [jaj + (2 − α)bj ] < 1 − α
On a subclass of functions with negative coefficients
59
Dn+1 f (z)
> α.
Dn g(z)
Now let take f ∈ T and g ∈ Tn (0) for which the relation (4) hold.
Dn+1 f (z)
The condition Re n
> α is equivalent with
D f (z)
we obtain Re
µ
(8)
α − Re
¶
Dn+1 f (z)
−1 <1
Dn g(z)
We have
µ
α − Re
Dn+1 f (z)
−1
Dn g(z)
¶
¯ n+1
¯
¯ D f (z)
¯
≤ α + ¯¯ n
− 1¯¯ =
D g(z)
¯
¯
∞
∞
X
X
¯
¯
n+1
j−1
¯1 −
¯
j
a
z
j n |bj − jaj | · |z|j−1
j
¯
¯
¯
¯
j=2
j=2
= α + ¯¯
− 1¯¯ ≤ α +
≤
∞
∞
X
X
¯
¯
n
j−1
n
j−1
j bj z
1−
j bj |z|
¯ 1−
¯
¯
¯
j=2
j=2
∞
X
≤α+
∞
X
n
j |bj − jaj |
j=2
1−
≤α+
∞
X
j=2
n
j bj
1−
j=2
α+
j n (bj + jaj )
∞
X
j=2
∞
X
j n [jaj + (1 − α)bj ]
j=2
=
1−
∞
X
j n bj
j=2
Using (8) we obtain:
α+
∞
X
j=2
that is the condition (4).
j n [jaj + (2 − α)bj ] < 1
=
n
j bj
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Mugur Acu
Remark 3.1. If we take f ≡ g we obtain from Theorem 3.1
∞
X
j n aj [jaj + 2 − α] < 1 − α
j=2
From
∞
P
j n aj [j + 2 − α] >
j=2
∞
P
jaj (j − α) we obtain:
j=2
∞
X
jaj (j − α) < 1 − α
j=2
Thus we obtain the result from Remark 2.7.
Remark 3.2. From the proof of the Theorem 3.1 we obtain a necessary
∞
P
condition for a function f ∈ T , f (z) = z −
aj z j to be in the class
j=2
CCTn (α), α ∈ [0, 1), n ∈ N, with respect to the function g ∈ Tn (0),
∞
P
bj z j :
g(z) = z −
j=2
∞
X
j n [jaj − αbj ] < 1 − α.
j=2
Theorem 3.2. If F ∈ CCTn (α), α ∈ [0, 1), n ∈ N, with respect to the
function G ∈ Tn (0) and f = Ic (F ), g = Ic (F ) where Ic is defined by (2),
then f ∈ CCTn (α) with respect to the function g ∈ Tn (0) (see Remark 2.4)
Proof. From F (z) = z −
∞
P
aj z j , aj ≥ 0, j = 2, 3, ... and f (z) = Ic (F )(z)
j=2
we have (see (3)):
f (z) = z −
∞
X
j=2
αj z j , where αj =
c+1
aj , j = 2, 3, ...
c+j
On a subclass of functions with negative coefficients
∞
P
From G(z) = z −
61
bj z j , bj ≥ 0, j = 2, 3, ... and g(z) = Ic (G)(z) we
j=2
have:
g(z) = z −
∞
X
βj z j , where βj =
j=2
c+1
bj , j = 2, 3, ...
c+j
From F ∈ CCTn (α) with respect to the function G ∈ Tn (0) we have (see
Theorem 3.1):
∞
X
(9)
j n [jaj + (2 − α)bj ] < 1 − α.
j=2
From Theorem 3.1 we need only to show that:
∞
X
j n [jαj + (2 − α)βj ] < 1 − α.
j=2
We have for c ∈ (−1, ∞) and j = 2, 3, ...:
∞
X
j n [jαj + (2 − α)βj ] =
j=2
=
∞
X
c+1
j=2
c+j
n
j [jaj + (2 − α)bj ] <
∞
X
j n [jaj + (2 − α)bj ]
j=2
From (9) we have:
∞
X
j n [jαj + (2 − α)βj ] < 1 − α.
j=2
References
[1] D. Blezu, On the n-uniform close to convex functions with respect to a
convex domain, General Mathematics, Vol. 9, Nr. 3 - 4, 2001, 3 - 14.
62
Mugur Acu
[2] G. S. Sălăgean, Geometria Planului Complex, Ed. Promedia Plus, Cluj
- Napoca, 1999.
[3] G. S. Sălăgean, On some classes of univalent functions, Seminar of
geometric function theory, Cluj - Napoca, 1983.
[4] H. Silverman, Univalent functions with negative coefficients, Proc.
Amer. Math. Soc., 5 (1975), 109 - 116.
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