Journal of Inequalities in Pure and
Applied Mathematics
ON NEIGHBORHOODS OF A CERTAIN CLASS OF COMPLEX
ORDER DEFINED BY RUSCHEWEYH DERIVATIVE OPERATOR
volume 7, issue 3, article 103,
2006.
ÖZNUR ÖZKAN AND OSMAN ALTINTAŞ
Department of Statistics and Computer Sciences
Baskent University
Baglıca, TR 06530
Ankara, Turkey.
EMail: oznur@baskent.edu.tr
Department of Mathematics Education
Baskent University
Baglica, TR 06530
Ankara, Turkey.
EMail: oaltintas@baskent.edu.tr
Received 06 October, 2005;
accepted 09 March, 2006.
Communicated by: G. Kohr
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
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Abstract
In this paper, we introduce the subclass Rbλ (A, B, α, µ) which is defined by concept of subordination. According to this, we obtain a necessary and sufficient
condition which is equivalent to this class. Further, we apply to the δ− neighborhoods for belonging to Rbλ (A, B, α, µ) to this condition.
On Neighborhoods of a Certain
Class of Complex Order Defined
by Ruscheweyh Derivative
Operator
2000 Mathematics Subject Classification: 30C45.
Key words: Analytic function, Hadamard product, δ− neighborhood, Subordination,
Close-to-convex function.
Contents
1
Introduction and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
The Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
Öznur Özkan and Osman Altintaş
3
7
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1.
Introduction and Definitions
Let U= {z : z ∈ C and |z| < 1} and H (U) be the set of all functions analytic
in U, and let
A := {f ∈ H (U) : f (0) = f 0 (0) − 1 = 0} .
Given two functions f and g, which are analytic in U. The function f is said
to be subordinate to g, written
f ≺g
and
f (z) ≺ g (z)
(z ∈ U) ,
if there exists a Schwarz function ω analytic in U, with
ω (0) = 0 and
|ω (z)| < 1
(z ∈ U) ,
On Neighborhoods of a Certain
Class of Complex Order Defined
by Ruscheweyh Derivative
Operator
Öznur Özkan and Osman Altintaş
and such that
f (z) = g (ω (z))
(z ∈ U) .
In particular, if g is univalent in U, then f ≺ g if and only if f (0) = g (0)
and f (U) ⊂ g (U) in [7].
Next, for the functions fj (j = 1, 2) given by
fj (z) = z +
∞
X
ak,j z k
(j = 1, 2) .
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k=2
Let f1 ∗f2 denote the Hadamard product (or convolution) of f1 and f2 , defined
by
(1.1)
(f1 ∗ f2 ) (z) := z +
∞
X
k=2
ak,1 ak,2 z k =: (f2 ∗ f1 ) (z) .
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(a)v denotes the Pochhammer symbol (or the shifted factorial), since
(1)n = n!
n ∈ N0 := N ∪ {0} ,
for
defined (for a, v ∈ C and in terms of the Gamma function) by
(
1;
(v = 0, a ∈ C\ {0}) ,
Γ (a + v)
(a)v :=
=
Γ (a)
a (a + 1) . . . (a + n − 1) ; (v = n ∈ N; a ∈ C) .
The earlier investigations by Goodman [1] and Ruscheweyh [9], we define
the δ− neighborhood of a function f ∈ A by
(
∞
X
ak z k ,
Nδ (f ) := g ∈ A : f (z) = z +
k=2
∞
X
g (z) = z +
bk z k
and
k=2
so that, obviously,
(
Nδ (e) :=
g ∈ A : g (z) = z +
∞
X
k=2
∞
X
and
Öznur Özkan and Osman Altintaş
)
k |ak − bk | ≤ δ
k=2
bk z k
On Neighborhoods of a Certain
Class of Complex Order Defined
by Ruscheweyh Derivative
Operator
∞
X
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)
k |bk | ≤ δ
,
k=2
where e (z) := z.
Ruscheweyh [8] introduced an linear operator Dλ : A −→ A, defined by the
Hadamard product as follows:
z
Dλ f (z) :=
∗ f (z)
(λ > −1; z ∈ U),
(1 − z)λ+1
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which implies that
(n)
z (z n−1 f (z))
D f (z) =
n!
n
(n ∈ N0 := N∪ {0}).
Clearly, we have
D0 f (z) = f (z) ,
D1 f (z) = zf 0 (z)
and
n
D f (z) =
∞
X
(λ + 1)
k=0
(1)k
k
ak+1 z
k+1
=
∞
X
(λ + 1)
k=0
(1)k
!
k k+1
z
∗f
(z) ,
On Neighborhoods of a Certain
Class of Complex Order Defined
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Operator
Öznur Özkan and Osman Altintaş
where f ∈ A.
Therefore, we can write the following equality, the easily verified result from
the above definitions:
0
Dλ f (z)
1
λ
(1.2)
(1 − µ)
+ µ D f (z) ∗
z
(1 − z)2
0
00
= Dλ f (z) + µz Dλ f (z) ,
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where f ∈ A, λ (λ > −1) , µ (µ ≥ 0) and for all z ∈ U.
For each A and B such that −1 ≤ B < A ≤ 1 and for all real numbers α
such that 0 ≤ α < 1, we define the function
1 + {(1 − α) A + αB} z
h (A, B, α; z) :=
1 + Bz
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(z ∈ U) .
J. Ineq. Pure and Appl. Math. 7(3) Art. 103, 2006
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Also, let h (α) denote the extremal function of functions with positive real
part of order α (0 ≤ α < 1), defined by
h (α) := h (1, −1, α; z) =
1 + (1 − 2α) z
1−z
(z ∈ U) .
The class Rb (A, B) is studied by Premabai in [3]. According to this, we
introduce the subclass Rλb (A, B, α, µ) which is a generalization of this class, as
follows:
i
0
00
1h λ
(1.3)
1+
D f (z) + µz Dλ f (z) − 1 ≺ h (A, B, α; z) ,
b
where f ∈ A, b ∈ C/ {0}, for some real numbers A, B (−1 ≤ B < A ≤ 1) , λ
(λ > −1) , α (0 ≤ α < 1) , µ (µ ≥ 0) and for all z ∈ U with Rb (A, B, α, µ) :=
R0b (A, B, α, µ) and Rb (A, B) := Rb (A, B, 0, 0) .
We note that the class Rb (µ) := Rb (1, −1, 0, µ) is studied by Altıntaş and
Özkan in [4]. Therefore C (b) := Rb (1, −1, 0, 0) is the class of close-to-convex
functions of complex order b. C (α) := R1−α (1, −1, 0, 0) is the class of closeto-convex functions of order α (0 ≤ α < 1).
Also, let Tb (A, B, α) denote the class of functions φ normalized by
n
o
it
1
1 + 1b (1−z)
−
1
− 1+{(1−α)A+αB}e
2
1+Beit
(1.4) φ (z) :=
(t ∈ (0, 2π)) ,
it
1 − 1+{(1−α)A+αB}e
1+Beit
where b ∈ C/ {0} , for some real numbers A, B (−1 ≤ B < A ≤ 1) , for all
α (0 ≤ α < 1) and for all z ∈ U with Tb (A, B) := Tb (A, B, 0) and T (b) :=
Tb (1, −1, 0) .
On Neighborhoods of a Certain
Class of Complex Order Defined
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Operator
Öznur Özkan and Osman Altintaş
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2.
The Main Results
A theorem that contains the relationship between the above classes is given as
follows:
Theorem 2.1. f ∈ Rλb (A, B, α, µ) if and only if
0
Dλ f (z)
λ
(1 − µ)
+ µ D f (z) ∗ φ (z) 6= 0
z
for all φ ∈ Tb (A, B, α) and for all f ∈ A.
Proof. Firstly, let
zλ (f, µ; z) := (1 − µ)
0
Dλ f (z)
+ µ Dλ f (z)
z
On Neighborhoods of a Certain
Class of Complex Order Defined
by Ruscheweyh Derivative
Operator
Öznur Özkan and Osman Altintaş
and we suppose that
zλ (f, µ; z) ∗ φ (z) 6= 0
(2.1)
for all f ∈ A and for all φ ∈ Tb (A, B, α) . In view of (1.4) , we have
zλ (f, µ; z) ∗ φ (z)
n
o
it
1
1 + 1b zλ (f, µ; z) ∗ (1−z)
−
1
− 1+{(1−α)A+αB}e
2
1+Beit
=
it
1 − 1+{(1−α)A+αB}e
1+Beit
n
o
0
00
it
1
λ
λ
1 + b D f (z) + µz D f (z) − 1 − 1+{(1−α)A+αB}e
it
1+Be
=
1+{(1−α)A+αB}eit
1−
1+Beit
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6= 0.
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From this inequality we find that
1+
o
0
00
1n λ
D f (z) + µz Dλ f (z) − 1 6= h A, B, α; eit ,
b
where t ∈ (0, 2π) .
n
o
0
00
This means that 1 + 1b Dλ f (z) + µz Dλ f (z) − 1 does not take
any value on the image of under
h (A, B, α; z) function of theoboundary of U.
n
0
00
1
λ
Therefore we note that 1+ b D f (z) + µz Dλ f (z) − 1 takes the value
1 for z = 0. Since 0 ≤ α < 1 and B < A, 1 is contained by the image under
h (A, B, α; z) function of U.Thus, we can write
o
0
00
1n λ
λ
1+
D f (z) + µz D f (z) − 1 ≺ h (A, B, α; z) .
b
Hence f ∈ Rλb (A, B, α, µ) .
Conversely, assume the function f is in the class Rλb (A, B, α, µ) . From the
definition of subordination, we can write the following inequality:
o
0
00
1n λ
λ
1+
D f (z) + µz D f (z) − 1 6= h A, B, α; eit ,
b
where t ∈ (0, 2π) . From (1.2) we can write
1
1
λ
it
1+
z (f, µ; z) ∗
−
1
=
6
h
A,
B,
α;
e
b
(1 − z)2
On Neighborhoods of a Certain
Class of Complex Order Defined
by Ruscheweyh Derivative
Operator
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or equivalently,
zλ (f, µ; z) ∗

 1 + 1b

1
(1−z)2

− 1 − h (A, B, α; eit ) 
1 − h (A, B, α; eit )
.

Thus, from the definition of the function φ, we can write
0
Dλ f (z)
λ
(1 − µ)
+ µ D f (z) ∗ φ (z) 6= 0
z
for all φ ∈ Tb (A, B, α) and for all f ∈ A.
h
i
0
Corollary 2.2. f ∈ Rb (A, B, α, µ) if and only if (1 − µ) f (z)
+
µf
(z)
∗
z
φ (z) 6= 0 for all φ ∈ Tb (A, B, α) and for all f ∈ A.
Proof. By putting λ = 0 in Theorem 2.1.
b
f (z)
z
∗ φ (z) 6= 0 for all φ ∈
Proof. By putting α = 0, µ = 0 in Corollary 2.2. And, we obtain the result of
Theorem 1 in [3].
f (z)
z
Öznur Özkan and Osman Altintaş
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Corollary 2.3. f ∈ R (A, B) if and only if
Tb (A, B) and for all f ∈ A.
Corollary 2.4. f ∈ C (b) if and only if
all f ∈ A.
On Neighborhoods of a Certain
Class of Complex Order Defined
by Ruscheweyh Derivative
Operator
∗ φ (z) 6= 0 for all φ ∈ T (b) and for
Proof. By putting A = 1, B = −1 in Corollary 2.3.
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Theorem 2.5. Let z (z) = f (z)+z
for ∈ C and f ∈ A. If z ∈ Rλb (A, B, α, µ)
1+
for || < δ ∗ , then
λ
0
D
f
(z)
λ
(1 − µ)
≥ δ∗
(2.2)
+
µ
D
f
(z)
∗
φ
(z)
(z ∈ U) ,
z
where φ ∈ Tb (A, B, α) and for all f ∈ A.
Proof. Let φ ∈ Tb (A, B, α) and z ∈ Rλb (A, B, α, µ) . From Theorem 2.1, we
can write
0
Dλ z (z)
λ
(1 − µ)
+ µ D z (z) ∗ φ (z) 6= 0.
z
Using Dλ (z) = z, we find that the following inequality
0
1
Dλ f (z)
λ
(1 − µ)
+ µ D f (z) ∗ φ (z) + 6= 0
1+
z
that is,
0
Dλ f (z)
λ
(1 − µ)
+ µ D f (z) ∗ φ (z) 6= −
z
or equivalently (2.2) .
Lemma 2.6. If φ (z) = 1 +
(2.3)
On Neighborhoods of a Certain
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P∞
k=1 ck z
k
∈ Tb (A, B, α), then we have
(k + 1) (1 + |B|)
|ck | ≤
(1 − α) |b| |B − A|
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Proof. We suppose that φ (z) = 1 +
we have
1+
∞
X
P∞
k=1 ck z
k
∈ Tb (A, B, α) . From (1.4) ,
ck z k
k=1
=
1+
1
b
1+
P∞
k=2
1−
1+{(1−α)A+αB}eit
1+Beit
1+{(1−α)A+αB}eit
1+Beit
kz k−1 − 1 −
(t ∈ (0, 2π)) .
We write the following equality result which is easily verified result from the
above equality:
(k + 1) (1 + Beit )
ck =
.
.
b (1 − α) (B − A) eit
Taking the modulus of both sides, we obtain inequality (2.3) .
Theorem 2.7. If z ∈
Rλb
(A, B, α, µ) for || < δ , then
(1−α)|b||B−A|
δ∗.
(1+λ)(1+µ)(1+|B|)
Proof. Let g ∈ Nδ (f ) for δ =
If we take
(1−α)|b||B−A|
δ∗.
(1+λ)(1+µ)(1+|B|)
0
Dλ g (z)
z (g, µ; z) := (1 − µ)
+ µ Dλ g (z) ,
z
λ
Öznur Özkan and Osman Altintaş
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∗
Nδ (f ) ⊂ Rλb (A, B, α, µ) ,
where δ :=
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then
λ
z (g, µ; z) ∗ φ (z)
λ
D
(f
+
g
−
f
)
(z)
0
+ µ Dλ (f + g − f ) (z) ∗ φ (z)
= (1 − µ)
z
λ
≥ z (f, µ; z) ∗ φ (z) − zλ (g − f, µ; z) ∗ φ (z)
and using Theorem 2.5 we can write
∞
X
(2.4)
≥ δ∗ − Ψ (k) (bk − ak ) ck−1 z k−1 ,
On Neighborhoods of a Certain
Class of Complex Order Defined
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Operator
k=2
Öznur Özkan and Osman Altintaş
where
Ψ (k) =
(λ + 1)(k−1)
(1)k
(µk − µ + 1) .
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We know that Ψ (k) is an increasing function of k and
0 < Ψ (2) = (1 + λ) (1 + µ) ≤ Ψ (k)
µ ≥ 0; k ∈ N; λ ≥
Contents
−µk
(1 + µk)
.
Since z ∈ Rλb (A, B, α, µ) for || < δ ∗ and using Lemma 2.6 in (2.4) we have
∞
X
λ
z (g, µ; z) ∗ φ (z) > δ ∗ − Ψ (2) |z|
|ak − bk |
k=2
k (1 + |B|)
(1 − α) |b| |B − A|
∞
(1 + λ) (1 + µ) (1 + |B|) X
>δ −
k |ak − bk |
(1 − α) |b| |B − A|
k=2
∗
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> δ∗ − δ
(1 + λ) (1 + µ) (1 + |B|)
> 0.
(1 − α) |b| |B − A|
That is, we can write
0
Dλ g (z)
λ
(1 − µ)
+ µ D g (z) ∗ φ (z) 6= 0
z
(z ∈ U) .
Thus, from Theorem 2.1 we can find that g ∈ Rλb (A, B, α, µ) .
Corollary 2.8. If z ∈ Rb (A, B, α, µ) for || < δ ∗ , then
Nδ (f ) ⊂ Rb (A, B, α, µ) ,
where δ :=
(1−α)|b||B−A| ∗
δ .
(1+µ)(1+|B|)
Proof. By putting λ = 0 in Theorem 2.7.
Corollary 2.9. If z ∈ Rb (A, B) for || < δ ∗ , then
Nδ (f ) ⊂ Rb (A, B)
where δ :=
|b||B−A| ∗
δ .
(1+|B|)
Proof. By putting α = 0, µ = 0 in Corollary 2.8. Thus, we obtain the result of
Theorem 2.7 in [3].
Corollary 2.10. If z ∈ C (b) for || < δ ∗ , then
Nδ (f ) ⊂ C (b) ,
where δ := |b| δ ∗ .
On Neighborhoods of a Certain
Class of Complex Order Defined
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Proof. By putting A = 1, B = −1 in Corollary 2.9.
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References
[1] A.W. GOODMAN, Univalent functions and non analytic curves, Proc.
Amer. Math. Soc., 8 (1957), 598–601.
[2] K.K. DIXIT AND S.K. PAL, On some class of univalent functions related
to complex order, Indian J. Pure and Appl. Math., 26(9) (1995), 889–896.
[3] M. PREMABAI, On the neighborhoods of a subclass of univalent functions
related to complex order, South. Asian Bull. Math., 26 (2002), 71–75.
[4] O. ALTINTAŞ AND Ö. ÖZKAN, Starlike, convex and close-to-convex
functions of complex order, Hacettepe Bull. of Natur.Sci. and Eng., 28
(1999), 37–46.
[5] O. ALTINTAŞ, Ö. ÖZKAN AND H.M. SRIVASTAVA, Neighborhoods of
a class of analytic functions with negative coefficients, App. Math. Lett.,
13(3) (2000), 63–67.
[6] O. ALTINTAŞ, Ö. ÖZKAN AND H.M. SRIVASTAVA, Neighborhoods of
a certain family of multivalent functions with negative coefficients, Comp.
Math. Appl., 47 (2004), 1667–1672.
On Neighborhoods of a Certain
Class of Complex Order Defined
by Ruscheweyh Derivative
Operator
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[7] S.S. MILLER AND P.T. MOCANU, Differential Subordinations: Theory
and Applications, Series on Monographs and Textbooks in Pure and Applied Mathematics, Vol. 225, Marcel Dekker, New York, (2000).
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[8] St. RUSCHEWEYH, New criteria for univalent fınctions, Proc. Amer. Math.
Soc., 49 (1975), 109–115.
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[9] St. RUSCHEWEYH, Neighborhoods of univalent functions, Proc. Amer.
Soc., 81 (1981), 521–527.
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Class of Complex Order Defined
by Ruscheweyh Derivative
Operator
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