ConvexF Negative Coeff
ACTA UNIVERSITATIS APULENSIS
No 14/2007
ABOUT SOME CONVEX FUNCTIONS WITH NEGATIVE
COEFFICIENTS
Mugur Acu, Irina Dorca and Daniel Breaz
Abstract. Our subject of study in this paper is represented by a class
of convex functions with negative coefficients defined by using a modified
S˘al˘agean operator.
2000 Mathematics Subject Classification: 30C45.
Keywords and phrases: differential opperators, convex, starlike, regular,
subordination.
1. Introduction
Let H(U ) to be the set of functions which are regular in the unit disc U ,
A = {f ∈ H(U ) : f (0) = f ′ (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
f (z) = z −
∞
X
aj z j , aj ≥ 0, j = 2, 3, ..., z ∈ U
(1)
j=2
was introduced.
The purpose of this paper is to define a class of convex functions with
negative coefficients and to give some properties of its by using a modified
S˘al˘agean operator.
97
M. Acu, I. Dorca, D. Breaz - About some convex functions with negative...
2. Preliminary results
Let Dn be the S˘al˘agean differential operator (see [6]) Dn : A → A, n ∈ N,
defined as:
D0 f (z) = f (z)
D1 f (z) = Df (z) = zf ′ (z)
Dn f (z) = D(Dn−1 f (z))
∞
P
Remark 2.1. If f ∈ T, f (z) = z −
aj z j , aj ≥ 0, j = 2, 3, ..., z ∈ U
j=2
then Dn f (z) = z −
∞
P
j n aj z j .
j=2
Definition 2.1. [1] Let β, λ ∈ R, β ≥ 0, λ ≥ 0 and f (z) = z +
∞
X
aj z j .
j=2
We 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
Theorem 2.1. [6] If f (z) = z −
∞
P
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
T
(iii) f ∈ T ∗ , where T ∗ = T S ∗ and S ∗ is the well-known class of starlike
functions.
For α ∈ [0, 1) and n ∈ N, we denote
Dn+2 f (z)
c
> α, z ∈ U
Sn (α) = f ∈ A : Re n+1
D f (z)
the set of n-convex functions of order α.
98
M. Acu, I. Dorca, D. Breaz - About some convex functions with negative...
Definition 2.2 [5] Let Ic : A → A be the integral operator defined by
f = Ic (F ), where c ∈ (−1, ∞), F ∈ A and
c+1
f (z) =
zc
Zz
tc−1 F (t)dt.
(2)
0
We note if F ∈ A is a function of the form (1), then
f (z) = Ic F (z) = z −
∞
X
c+1
c+j
j=2
aj z j .
(3)
We denote by f ∗ g the modified Hadamard product of two functions
∞
P
f (z), g(z) ∈ T , f (z) = z −
aj z j , (aj ≥ 0, j = 2, 3, ...) and g(z) = z −
j=2
∞
P
j
bj z , (bj ≥ 0, j=2,3,...), is defined by
j=2
(f ∗ g)(z) = z −
∞
X
aj b j z j .
j=2
An analytic function f is set to be 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 this subordination by f ≺ g.
Theorem 2.2. [4] If f and g are analytic in U with f ≺ g, then for
µ > 0 and z = reiθ (0 < r < 1), we have
Z2π
|f (z)|µ dθ ≦
Z2π
|g(z)|µ dθ.
0
0
Definition 2.3. [2] We consider the integral operator Ic+δ : A → A,
0 < u ≤ 1, 1 ≤ δ < ∞, 0 < c < ∞, defined by
f (z) = Ic+δ (F (z)) = (c + δ)
Z1
0
99
uc+δ−2 F (uz)du.
(4)
M. Acu, I. Dorca, D. Breaz - About some convex functions with negative...
Remark 2.2. For F (z) = z +
∞
P
aj z j . From (4) we obtain
j=2
f (z) = z +
∞
X
j=2
Also, we notice that 0 <
1 ≤ δ < ∞.
c+δ
aj z j .
c+j+δ−1
c+δ
< 1, where 0 < c < ∞, j ≥ 2,
c+j+δ−1
Remark 2.3. It is easy to prove that for F (z) ∈ T and f (z) = Ic+δ (F (z)),
we have f (z) ∈ T , where Ic+δ is the integral operator defined by (4).
3. Main results
Definition 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 T c Lβ (α) if:
Re
Dλβ+2 f (z)
Dλβ+1 f (z)
> α, α ∈ [0, 1), λ ≥ 0, β ≥ 0, z ∈ U.
Theorem 3.1. Let α ∈ [0, 1), λ ≥ 0 and β ≥ 0. The function f ∈ T of
the form (1) is in the class T c Lβ (α) iff
∞
X
[(1 + (j − 1)λ)β+1 (1 + (j − 1)λ − α)]aj < 1 − α.
j=2
Proof. Let f ∈ T c Lβ (α), with α ∈ [0, 1), λ ≥ 0 and β ≥ 0 . We have
Re
Dλβ+2 f (z)
Dλβ+1 f (z)
100
> α.
(5)
M. Acu, I. Dorca, D. Breaz - About some convex functions with negative...
If we take z ∈ [0, 1), β ≥ 0, λ ≥ 0, we have (see Definition 2.1):
1−
1−
∞
X
j=2
∞
X
(1 + (j − 1)λ)β+2 aj z j−1
> α.
β+1
(1 + (j − 1)λ)
aj z
(6)
j−1
j=2
From (6) we obtain:
∞
X
(1 + (j − 1)λ)β+1 (1 + (j − 1)λ − α)aj z j−1 < 1 − α.
j=2
Letting z → 1− along the real axis we have:
∞
X
(1 + (j − 1)λ)β+1 (1 + (j − 1)λ − α)aj < 1 − α.
j=2
Conversely, let take f ∈ T for which the relation (5) hold.
The condition Re
Dλβ+2 f (z)
> α is equivalent with
Dλβ+1 f (z)
!
β+2
Dλ f (z)
α − Re
− 1 < 1.
Dλβ+1 f (z)
We have
α − Re
Dλβ+2 f (z)
Dλβ+1 f (z)
!
−1
Dβ+2 f (z)
λ
≤ α + β+1
− 1
Dλ f (z)
∞
X
β+1
j−1
(1
+
(j
−
1)λ)
a
[(j
−
1)λ]z
j
j=2
= α +
∞
X
β+1
j−1
[1 + (j − 1)λ] aj z
1−
j=2
101
(7)
M. Acu, I. Dorca, D. Breaz - About some convex functions with negative...
≤α+
∞
X
[1 + (j − 1)λ]β+1 aj |1 − j|λ|z|j−1
j=2
1−
∞
X
[1 + (j − 1)λ]β+1 aj |z|j−1
j=2
=α+
∞
X
[1 + (j − 1)λ]β+1 aj (j − 1)λ|z|j−1
j=2
1−
∞
X
[1 + (j − 1)λ]β+1 aj |z|j−1
j=2
0
1−α
z j , j=2,3... . Then
and fj (z) = z −
β+1
[(1 + (j − 1)λ) (1 + (j − 1)λ − α)]
for z = reiθ (0 < r < 1), we have
Z2π
0
µ
|f (z)| dθ ≤
Z2π
0
105
|fj (reiθ )|µ dθ.
M. Acu, I. Dorca, D. Breaz - About some convex functions with negative...
Proof We have to show that
µ
µ
Z 2π
Z 2π
∞
X
1
−
α
j−1
j−1
1 −
z dθ.
aj z dθ ≤
1 −
β+1
[(1 + (j − 1)λ) (1 + (j − 1)λ − α)]
0
0
j=2
From Theorem 2.3 we deduce that it is sufficiently to prove that
1−
∞
X
aj z j−1 ≺ 1 −
j=2
[(1 + (j −
1−α
z j−1 .
+ (j − 1)λ − α)]
1)λ)β+1 (1
Considering
1−
∞
X
aj z j−1 = 1 −
j=2
[(1 + (j −
1−α
w(z)j−1
+ (j − 1)λ − α)]
1)λ)β+1 (1
we find that
∞
{w(z)}j−1 =
[(1 + (j − 1)λ)β+1 (1 + (j − 1)λ − α)] X
aj z j−1
1−α
j=2
which readily yields w(0) = 0.
By using the condition (5), we can write
1 − α > [(1 + λ)β+1 (1 + λ − α)]a2 + [(1 + 2λ)β+1 (1 + 2λ − α)]a3 + ...
+[(1 + (j − 1)λ)β+1 (1 + (j − 1)λ − α)]aj + [(1 + jλ)β+1 (1 + jλ − α)]aj+1 + ...
≥
∞
X
[(1 + (j − 1)λ)β+1 (1 + (j − 1)λ − α)]ai
i=2
β+1
= [(1 + (j − 1)λ)
(1 + (j − 1)λ − α)]
∞
X
ai .
i=2
Thus
∞
X
j=2
and
j−1
|{w(z)}|
aj <
[(1 + (j −
1−α
+ (j − 1)λ − α)]
1)λ)β+1 (1
∞
[(1 + (j − 1)λ)β+1 (1 + (j − 1)λ − α)] X
j−1
aj z
=
1−α
j=2
106
M. Acu, I. Dorca, D. Breaz - About some convex functions with negative...
∞
[(1 + (j − 1)λ)β+1 (1 + (j − 1)λ − α)] X
aj |z|j−1 < |z| < 1 .
≤
1−α
j=2
This completes our theorem’s proof.
Remark 3.2. We notice that in the particulary case, obtained for λ = 1
and β ∈ N, we find similarly results for the class Tnc (α) of the n-convex
functions of order α with negative coefficients.
References
[1] 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.
[2] M. Acu, I. Dorca, S. Owa, On some starlike functions with negative
coefficients (to appear).
[4] D. Blezu, On the n-uniform close to convex functions with respect to
a convex domain, General Mathematics, Vol. 9, Nr. 3 - 4, 3 - 14.
[5] J.E. Littlewood, On inequalities in the theory of functions, Proc. London Math. Soc., 23(1995), 481-519.
[6] G. S. S˘al˘agean, On some classes of univalent functions, Seminar of
geometric function theory, Cluj - Napoca, 1983.
[7] G. S. S˘al˘agean, Geometria Planului Complex, Ed. Promedia Plus,
Cluj - Napoca, 1999.
[8] H. Silverman, Univalent functions with negative coefficients, Proc.
Amer. Math. Soc. 5(1975), 109-116.
Authors:
Mugur Acu
”Lucian Blaga” University of Sibiu,
Department of Mathematics,
Str. Dr. I. Rat¸iu, Nr. 5–7, 550012 - Sibiu, Romania.
email: acu mugur@yahoo.com
107
M. Acu, I. Dorca, D. Breaz - About some convex functions with negative...
Irina Dorca
”Lucian Blaga” University of Sibiu,
Department of Mathematics,
Str. Dr. I. Rat¸iu, Nr. 5–7, 550012 - Sibiu, Romania.
email: ira dorca@yahoo.com
Daniel Breaz
”1 Decembrie 1918” University of Alba Iulia,
Department of Mathematics
Alba Iulia, Romania.
email: dbreaz@uab.ro
108
No 14/2007
ABOUT SOME CONVEX FUNCTIONS WITH NEGATIVE
COEFFICIENTS
Mugur Acu, Irina Dorca and Daniel Breaz
Abstract. Our subject of study in this paper is represented by a class
of convex functions with negative coefficients defined by using a modified
S˘al˘agean operator.
2000 Mathematics Subject Classification: 30C45.
Keywords and phrases: differential opperators, convex, starlike, regular,
subordination.
1. Introduction
Let H(U ) to be the set of functions which are regular in the unit disc U ,
A = {f ∈ H(U ) : f (0) = f ′ (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
f (z) = z −
∞
X
aj z j , aj ≥ 0, j = 2, 3, ..., z ∈ U
(1)
j=2
was introduced.
The purpose of this paper is to define a class of convex functions with
negative coefficients and to give some properties of its by using a modified
S˘al˘agean operator.
97
M. Acu, I. Dorca, D. Breaz - About some convex functions with negative...
2. Preliminary results
Let Dn be the S˘al˘agean differential operator (see [6]) Dn : A → A, n ∈ N,
defined as:
D0 f (z) = f (z)
D1 f (z) = Df (z) = zf ′ (z)
Dn f (z) = D(Dn−1 f (z))
∞
P
Remark 2.1. If f ∈ T, f (z) = z −
aj z j , aj ≥ 0, j = 2, 3, ..., z ∈ U
j=2
then Dn f (z) = z −
∞
P
j n aj z j .
j=2
Definition 2.1. [1] Let β, λ ∈ R, β ≥ 0, λ ≥ 0 and f (z) = z +
∞
X
aj z j .
j=2
We 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
Theorem 2.1. [6] If f (z) = z −
∞
P
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
T
(iii) f ∈ T ∗ , where T ∗ = T S ∗ and S ∗ is the well-known class of starlike
functions.
For α ∈ [0, 1) and n ∈ N, we denote
Dn+2 f (z)
c
> α, z ∈ U
Sn (α) = f ∈ A : Re n+1
D f (z)
the set of n-convex functions of order α.
98
M. Acu, I. Dorca, D. Breaz - About some convex functions with negative...
Definition 2.2 [5] Let Ic : A → A be the integral operator defined by
f = Ic (F ), where c ∈ (−1, ∞), F ∈ A and
c+1
f (z) =
zc
Zz
tc−1 F (t)dt.
(2)
0
We note if F ∈ A is a function of the form (1), then
f (z) = Ic F (z) = z −
∞
X
c+1
c+j
j=2
aj z j .
(3)
We denote by f ∗ g the modified Hadamard product of two functions
∞
P
f (z), g(z) ∈ T , f (z) = z −
aj z j , (aj ≥ 0, j = 2, 3, ...) and g(z) = z −
j=2
∞
P
j
bj z , (bj ≥ 0, j=2,3,...), is defined by
j=2
(f ∗ g)(z) = z −
∞
X
aj b j z j .
j=2
An analytic function f is set to be 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 this subordination by f ≺ g.
Theorem 2.2. [4] If f and g are analytic in U with f ≺ g, then for
µ > 0 and z = reiθ (0 < r < 1), we have
Z2π
|f (z)|µ dθ ≦
Z2π
|g(z)|µ dθ.
0
0
Definition 2.3. [2] We consider the integral operator Ic+δ : A → A,
0 < u ≤ 1, 1 ≤ δ < ∞, 0 < c < ∞, defined by
f (z) = Ic+δ (F (z)) = (c + δ)
Z1
0
99
uc+δ−2 F (uz)du.
(4)
M. Acu, I. Dorca, D. Breaz - About some convex functions with negative...
Remark 2.2. For F (z) = z +
∞
P
aj z j . From (4) we obtain
j=2
f (z) = z +
∞
X
j=2
Also, we notice that 0 <
1 ≤ δ < ∞.
c+δ
aj z j .
c+j+δ−1
c+δ
< 1, where 0 < c < ∞, j ≥ 2,
c+j+δ−1
Remark 2.3. It is easy to prove that for F (z) ∈ T and f (z) = Ic+δ (F (z)),
we have f (z) ∈ T , where Ic+δ is the integral operator defined by (4).
3. Main results
Definition 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 T c Lβ (α) if:
Re
Dλβ+2 f (z)
Dλβ+1 f (z)
> α, α ∈ [0, 1), λ ≥ 0, β ≥ 0, z ∈ U.
Theorem 3.1. Let α ∈ [0, 1), λ ≥ 0 and β ≥ 0. The function f ∈ T of
the form (1) is in the class T c Lβ (α) iff
∞
X
[(1 + (j − 1)λ)β+1 (1 + (j − 1)λ − α)]aj < 1 − α.
j=2
Proof. Let f ∈ T c Lβ (α), with α ∈ [0, 1), λ ≥ 0 and β ≥ 0 . We have
Re
Dλβ+2 f (z)
Dλβ+1 f (z)
100
> α.
(5)
M. Acu, I. Dorca, D. Breaz - About some convex functions with negative...
If we take z ∈ [0, 1), β ≥ 0, λ ≥ 0, we have (see Definition 2.1):
1−
1−
∞
X
j=2
∞
X
(1 + (j − 1)λ)β+2 aj z j−1
> α.
β+1
(1 + (j − 1)λ)
aj z
(6)
j−1
j=2
From (6) we obtain:
∞
X
(1 + (j − 1)λ)β+1 (1 + (j − 1)λ − α)aj z j−1 < 1 − α.
j=2
Letting z → 1− along the real axis we have:
∞
X
(1 + (j − 1)λ)β+1 (1 + (j − 1)λ − α)aj < 1 − α.
j=2
Conversely, let take f ∈ T for which the relation (5) hold.
The condition Re
Dλβ+2 f (z)
> α is equivalent with
Dλβ+1 f (z)
!
β+2
Dλ f (z)
α − Re
− 1 < 1.
Dλβ+1 f (z)
We have
α − Re
Dλβ+2 f (z)
Dλβ+1 f (z)
!
−1
Dβ+2 f (z)
λ
≤ α + β+1
− 1
Dλ f (z)
∞
X
β+1
j−1
(1
+
(j
−
1)λ)
a
[(j
−
1)λ]z
j
j=2
= α +
∞
X
β+1
j−1
[1 + (j − 1)λ] aj z
1−
j=2
101
(7)
M. Acu, I. Dorca, D. Breaz - About some convex functions with negative...
≤α+
∞
X
[1 + (j − 1)λ]β+1 aj |1 − j|λ|z|j−1
j=2
1−
∞
X
[1 + (j − 1)λ]β+1 aj |z|j−1
j=2
=α+
∞
X
[1 + (j − 1)λ]β+1 aj (j − 1)λ|z|j−1
j=2
1−
∞
X
[1 + (j − 1)λ]β+1 aj |z|j−1
j=2
0
1−α
z j , j=2,3... . Then
and fj (z) = z −
β+1
[(1 + (j − 1)λ) (1 + (j − 1)λ − α)]
for z = reiθ (0 < r < 1), we have
Z2π
0
µ
|f (z)| dθ ≤
Z2π
0
105
|fj (reiθ )|µ dθ.
M. Acu, I. Dorca, D. Breaz - About some convex functions with negative...
Proof We have to show that
µ
µ
Z 2π
Z 2π
∞
X
1
−
α
j−1
j−1
1 −
z dθ.
aj z dθ ≤
1 −
β+1
[(1 + (j − 1)λ) (1 + (j − 1)λ − α)]
0
0
j=2
From Theorem 2.3 we deduce that it is sufficiently to prove that
1−
∞
X
aj z j−1 ≺ 1 −
j=2
[(1 + (j −
1−α
z j−1 .
+ (j − 1)λ − α)]
1)λ)β+1 (1
Considering
1−
∞
X
aj z j−1 = 1 −
j=2
[(1 + (j −
1−α
w(z)j−1
+ (j − 1)λ − α)]
1)λ)β+1 (1
we find that
∞
{w(z)}j−1 =
[(1 + (j − 1)λ)β+1 (1 + (j − 1)λ − α)] X
aj z j−1
1−α
j=2
which readily yields w(0) = 0.
By using the condition (5), we can write
1 − α > [(1 + λ)β+1 (1 + λ − α)]a2 + [(1 + 2λ)β+1 (1 + 2λ − α)]a3 + ...
+[(1 + (j − 1)λ)β+1 (1 + (j − 1)λ − α)]aj + [(1 + jλ)β+1 (1 + jλ − α)]aj+1 + ...
≥
∞
X
[(1 + (j − 1)λ)β+1 (1 + (j − 1)λ − α)]ai
i=2
β+1
= [(1 + (j − 1)λ)
(1 + (j − 1)λ − α)]
∞
X
ai .
i=2
Thus
∞
X
j=2
and
j−1
|{w(z)}|
aj <
[(1 + (j −
1−α
+ (j − 1)λ − α)]
1)λ)β+1 (1
∞
[(1 + (j − 1)λ)β+1 (1 + (j − 1)λ − α)] X
j−1
aj z
=
1−α
j=2
106
M. Acu, I. Dorca, D. Breaz - About some convex functions with negative...
∞
[(1 + (j − 1)λ)β+1 (1 + (j − 1)λ − α)] X
aj |z|j−1 < |z| < 1 .
≤
1−α
j=2
This completes our theorem’s proof.
Remark 3.2. We notice that in the particulary case, obtained for λ = 1
and β ∈ N, we find similarly results for the class Tnc (α) of the n-convex
functions of order α with negative coefficients.
References
[1] 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.
[2] M. Acu, I. Dorca, S. Owa, On some starlike functions with negative
coefficients (to appear).
[4] D. Blezu, On the n-uniform close to convex functions with respect to
a convex domain, General Mathematics, Vol. 9, Nr. 3 - 4, 3 - 14.
[5] J.E. Littlewood, On inequalities in the theory of functions, Proc. London Math. Soc., 23(1995), 481-519.
[6] G. S. S˘al˘agean, On some classes of univalent functions, Seminar of
geometric function theory, Cluj - Napoca, 1983.
[7] G. S. S˘al˘agean, Geometria Planului Complex, Ed. Promedia Plus,
Cluj - Napoca, 1999.
[8] H. Silverman, Univalent functions with negative coefficients, Proc.
Amer. Math. Soc. 5(1975), 109-116.
Authors:
Mugur Acu
”Lucian Blaga” University of Sibiu,
Department of Mathematics,
Str. Dr. I. Rat¸iu, Nr. 5–7, 550012 - Sibiu, Romania.
email: acu mugur@yahoo.com
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M. Acu, I. Dorca, D. Breaz - About some convex functions with negative...
Irina Dorca
”Lucian Blaga” University of Sibiu,
Department of Mathematics,
Str. Dr. I. Rat¸iu, Nr. 5–7, 550012 - Sibiu, Romania.
email: ira dorca@yahoo.com
Daniel Breaz
”1 Decembrie 1918” University of Alba Iulia,
Department of Mathematics
Alba Iulia, Romania.
email: dbreaz@uab.ro
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