General Mathematics Vol. 12, No. 2 (2004), 3–10
Order of certain classes of analytic and
univalent functions using Ruscheweyh
derivative
B. A. Frasin
and
Gheorghe Oros
Abstract
α
Let D f (z) be the Ruscheweyh derivative defined by using the
Hadamard product of f (z) and z/(1−z)α+1 . The object of this paper
is to find the order for certain analytic and univalent functions using
α
the Ruscheweyh derivative D f (z).
2000 Mathematical Subject Classification: 30C45.
Key words and phrases:Univalent, analytic, Ruscheweyh derivative,
starlike, convex, close-to-convex and quasi-convex functions.
1
Introduction and definitions
Let A denote the class of functions of the form :
(1.1)
f (z) = z +
∞
X
an z n ,
n=2
which are analytic in the open unit disk U = {z : |z| < 1}. Further, by S
we shall denote the class of all functions in A which are univalent in U . A
function f (z) belonging to A is said to be starlike in U if it satisfies
Ã
(1.2)
zf 0 (z)
Re
f (z)
3
!
>0
4
Order of certain classes of analytic and univalent functions...
for all z ∈ U . We denote by S ∗ the subclass of A consisting of functions
which are starlike in U. Also, a function f (z) belonging to A is said to be
convex in U if it satisfies
Ã
(1.3)
zf 00 (z)
Re 1 + 0
f (z)
!
>0
for all z ∈ U . We denote by K the subclass of A consisting of functions
which are convex in U. A function f (z) in A is said to be close-to-convex
of order δ if there exists a function g(z) belonging to S ∗ such that
Ã
(1.4)
zf 0 (z)
Re
g(z)
!
>δ
for some δ(0 ≤ δ < 1), and for all z ∈ U. We denote by C(δ) the subclass
of A consisting of functions which are close-to-convex of order δ in U. It is
well known that K ⊂ S ∗ ⊂ C ≡C(0) ⊂ S. A function f (z) belonging to A is
said to be quasi-convex of order δ(0 ≤ δ < 1) if there exists a function g(z)
belonging to C such that
Ã
(1.5)
(zf 0 (z))0
Re
g 0 (z)
!
>δ
for all z ∈ U. Denote the class of quasi-convex of order δ by C ∗ (δ). The
class C ∗ (0) was introduced and studied by Noor [1]. We note that every
quasi-convex function is close-to-convex and hence univalent in U.
Let the function f (z) be defined by (1.1) and the function g(z) be defined
by
(1.6)
g(z) = z +
∞
X
bn z n ,
n=2
then the Hadamard product (or convolution) of the functions f (z) and g(z)
is defined by
(1.7)
f (z) ∗ g(z) = z +
∞
X
a n bn z n .
n=2
Using the convolution (1.5), Ruscheweyh [3 ] introduced what is now reα
ferred to as the Ruscheweyh derivative D f (z) of order α of f (z) ∈ A
by
B. A. Frasin and Gheorghe Oros
α
(1.8)
D f (z) =
5
z
∗ f (z)
(1 − z)α+1
(α ≥ −1).
We note that D0 f (z) = f (z) and D1 f (z) = zf 0 (z).
Owa et al. [2] have introduced and studied the following classes:
α
Sα∗ = {f (z) ∈ A: D f (z) ∈ S ∗ , α ≥ −1}
(1.9)
and
(1.10)
α
Kα = {f (z) ∈ A: D f (z) ∈ K, α ≥ −1}.
Note that S0∗ ≡ S ∗ and S1∗ ≡ K0 ≡ K.
The aim of this paper is to find the order for certain analytic and uniα
valent functions using the Ruscheweyh derivative D f (z).
In order to show our results, we shall need the following lemmas due to
Owa et al. [2].
Lemma 1 . Let the function f (z) be in the class Sα∗ with α ≥ −1 . Then
Ã
(1.11)
α
D f (z)
Re
z
!β−1
>
1
,
2β − 1
z ∈ U,
where 1 < β ≤ 3/2.
Lemma 2. Let the function f (z) be in the class Kα with α ≥ −1 . Then
(1.12)
³
α
Re (D f (z))0
´β−1
>
1
,
2β − 1
z ∈ U,
where 1 < β ≤ 3/2.
2
Main Results
With the aid of Lemma 1, we can prove the following
Theorem 1 . If the function f (z) in A satisfies the condition
"
(2.1)
then
α
#
z(D f (z))00
Re
> −β ,
(Dα f (z))0
z∈U
6
Order of certain classes of analytic and univalent functions...
"
#
α
z(D f (z))0
1
Re
>
,
α
D g(z)
2β − 1
(2.2)
where
α ≥ −1, 1 < β ≤ 3/2
h
α
and
α
D g(z) = z (D f (z))0
(2.3)
z ∈ U,
i1
β
,
z ∈ U.
Proof. From (2.3) by differentiating, we obtain
α
α
z[D g(z)]0
1 z[D f (z)]00
=
1
+
,
Dα g(z)
β [Dα f (z)]0
(2.4)
z ∈ U.
Using (2.1) in (2.4) we have
"
#
α
"
#
α
z(D g(z))0
1 z(D f (z))00
1
Re
= Re 1 +
> 1 + (−β) > 0,
α
α
0
D g(z)
β (D f (z))
β
from which we deduce g(z) ∈ Sα∗ , z ∈ U.
From (2.3) we obtain
"
α
D g(z)
[D f (z)] =
z
α
#β−1
0
α
D g(z)
·
,
z
z ∈ U,
z 6= 0
and we have
"
α
α
z[D f (z)]0
D g(z)
=
α
D g(z)
z
(2.5)
#β−1
,
z ∈ U,
z 6= 0.
Applying Lemma 1 to (2.5) we obtain
"
α
#
"
α
z(D f (z))0
D g(z)
= Re
Re
α
D g(z)
z
#β−1
>
1
,
2β − 1
z ∈ U,
z 6= 0.
Letting α = 0 in Theorem 1, we obtain:
Corollary 1. If the function f (z) in A satisfies the condition
Ã
(2.6)
zf 00 (z)
Re
f 0 (z)
!
> −β ,
z∈U
B. A. Frasin and Gheorghe Oros
7
then
Ã
zf 0 (z)
Re
g(z)
!
1
,
2β − 1
>
z ∈ U.
Function f (z) belongs to the class C(δ) , where δ = 1/(2β − 1) and
1 < β ≤ 3/2. Therefore f (z) is close-to-convex of order δ.
Letting β = 3/2 in Corollary 1, we have:
Corollary 2. If the function f (z) in A satisfies the condition
Ã
(2.7)
then
!
zf 00 (z)
Re
+ 1 > −1/2 ,
f 0 (z)
Ã
zf 0 (z)
Re
g(z)
!
>
z∈U
1
, z ∈ U , i.e. f (z) is in C(1/2).
2
Letting α = 1 in Theorem 1, we obtain:
Corollary 3. If the function f (z) in A satisfies the condition
"
(2.8)
#
z(zf 0 (z))00
Re
> −β ,
(zf 0 (z))0
z∈U
then
#
"
(2.9)
(zf 0 (z))0
1
Re
>
,
g 0 (z)
2β − 1
z ∈ U,
1
where 1 < β ≤ 3/2. Therefore f (z) is in C ∗ ( 2β−1
).
Letting β = 3/2 in Corollary 3, we have:
Corollary 4. If the function f (z) in A satisfies the condition
"
(2.10)
#
z(zf 0 (z))00
Re
+ 1 > −1/2 ,
(zf 0 (z))0
z∈U
then
"
(2.11)
#
(zf 0 (z))0
> 1/2 ,
Re
g 0 (z)
Therefore f (z) is in C ∗ (1/2).
Next, we prove:
z ∈ U.
8
Order of certain classes of analytic and univalent functions...
Theorem 2. If the function f (z) in A satisfies the condition
"
#
α
z(D f (z))0
Re
>1−β ,
Dα f (z)
(2.12)
z∈U
then
"
(2.13)
#
α
D f (z)
1
Re
>
,
α
z(D g(z))0
2β − 1
z ∈ U,
z 6= 0
where α ≥ −1, 1 < β ≤ 3/2 and
"
(2.14)
α
D f (z)
[D g(z)] =
z
#1
β
0
α
,
z ∈ U,
z 6= 0.
Proof. From (2.12) we obtain
"
#
α
1 z(D f (z))0
1
Re
> −1 ,
α
β
D f (z)
β
z∈U
which is equivalent to
"
(2.15)
#
α
1 z(D f (z))0
Re
− 1 > −1 ,
β
Dα f (z)
z ∈ U.
From (2.14), by differentiating we have
"
α
α
[D g(z)]00
1 (D f (z))0 1
=
−
[Dα g(z)]0
β Dα f (z)
z
#
,
z∈U
which is equivalent to
"
α
(2.16)
#
α
z[D g(z)]00
1 z(D f (z))0
=
−1 ,
[Dα g(z)]0
β
Dα f (z)
z ∈ U.
Using (2.15) in (2.16) we have
"
α
#
z(D g(z))00
Re
+1 >0 ,
(Dα g(z))0
from which g(z) ∈ Kα .
z∈U
B. A. Frasin and Gheorghe Oros
9
From (2.14) we obtain
α
iβ
D f (z) h α
= (D g(z))0 ,
z
z ∈ U,
z 6= 0
from which we obtain
α
h α
i
D f (z)
0 β−1
,
=
(D
g(z))
z[Dα g(z)]0
(2.17)
z ∈ U,
z 6= 0.
Applying Lemma 2 in (2.17) we obtain
"
α
0 β−1
Re [(D g(z)) ]
#
α
D f (z)
1
= Re
>
,
α
z(D g(z))0
2β − 1
z ∈ U,
Letting α = 0 in Theorem 2, we obtain:
Corollary 5. If the function f (z) in A satisfies the condition
Ã
zf 0 (z)
Re
f (z)
(2.18)
!
>1−β ,
z∈U
then
Ã
f (z)
Re
zg 0 (z)
!
>
1
,
2β − 1
z ∈ U,
z 6= 0
where 1 < β ≤ 3/2.
Letting β = 3/2 in Corollary 5, we have:
Corollary 6. If the function f (z) in A satisfies the condition
Ã
(2.19)
zf 0 (z)
Re
f (z)
!
> −1/2 ,
z∈U
then
Ã
zf 0 (z)
Re
g 0 (z)
!
> 1/2 ,
z ∈ U.
z 6= 0.
10
Order of certain classes of analytic and univalent functions...
References
[1] K.I. Noor, On quasi-convex functions and related topics, Internat.J.
Math. & Math. Sci. 10 (2) (1987) 241-258.
[2] S. Owa, S. Fukui, X. Sakaguchi and S. Ogawa, An application of the
Ruscheweyh derivatives, Internat.J. Math. & Math. Sci. 9 (4) (1986)
721-730.
[3] S. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math.
Soc. 49 (1975), 109-115.
Department of Mathematics,
Al al-Bayt University,
Mafraq, Jordan.
E-mail address: bafrasin@yahoo.com.
Department of Mathematics,
University of Oradea
Str. Armatei Române 3-5
410087 Oradea, Romania
E-mail address: gh oros@yahoo.com