Journal of Inequalities in Pure and
Applied Mathematics
ON NEIGHBORHOODS OF ANALYTIC FUNCTIONS HAVING
POSITIVE REAL PART
volume 7, issue 3, article 109,
2006.
SHIGEYOSHI OWA, NIGAR YILDIRIM AND MUHAMMET KAMALİ
Department of Mathematics
Kinki University
Higashi-Osaka, Osaka 577-8502, Japan.
EMail: owa@math.kindai.ac.jp
Kafkas Üniversitesi,Fen-Edebiyat Fakültesi
Matematik Bölümü,
Kars, Turkey.
Atatürk Üniversitesi, Fen-Edebiyat Fakültesi,
Matematik Bölümü,
25240 Erzurum Turkey.
EMail: mkamali@atauni.edu.tr
Received 10 November, 2005;
accepted 15 July, 2006.
Communicated by: G. Kohr
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
334-05
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Abstract
Two subclasses P α−m
and P 0 α−m
of certain analytic functions having posn
n
itive real part in the open unit disk U are introduced. In the present paper,
several properties of the subclass P α−m
of analytic
functions with real part
n
α−m
α−m
greater than n are derived. For p(z) ∈ P n and δ ≥ 0, the δ−neighborhood
0 α−m
Nδ (p(z)) of p(z) is defined. For P α−m
n , P n , and Nδ (p(z)), we prove that
α−m
if p(z) ∈ P 0 α−m
n , then Nβδ (p(z)) ⊂ P
n .
2000 Mathematics Subject Classification: Primary 30C45.
Key words: Function with positive real part, subordinate function, δ−neighborhood,
convolution (Hadamard product).
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3
2
Some Inequalities for the Class P α−m
................ 4
n
3
Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
References
On Neighborhoods of Analytic
Functions having Positive Real
Part
Shigeyoshi Owa, Nigar Yildirim
and Muhammet Kamali
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1.
Introduction
Let T be the class of functions of the form
(1.1)
p(z) = 1 +
∞
X
pk z k ,
k=1
which are analytic in the open unit disk U
= {z ∈ C : |z| < 1}. A function
α−m
p(z) ∈ T is said to be in the class P n if it satisfies
α−m
Re {p(z)} >
n
(z ∈ U)
for some m ≤ α < m + n, m ∈ N0 = 0, 1, 2, 3, . . . , and n ∈ N = 1, 2, 3, . . . .
For any p(z) ∈ P α−m
and δ ≥ 0, we define the δ−neighborhood Nδ (p(z))
n
of p(z) by
(
)
∞
∞
X
X
Nδ (p(z)) = q(z) = 1 +
qk z k ∈ T :
|pk − qk | ≤ δ .
k=1
k=1
The concept of δ−neighborhoods Nδ (f (z)) of analytic functions f (z) in U with
f (0) = f 0 (0) − 1 = 0 was fırst introduced by Ruscheweyh [12] and was studied
by Fournier [4, 6] and by Brown [2]. Walker has studied the δ1 −neighborhood
Nδ1 (p(z)) of p(z) ∈ P1 (0) [13]. Later, Owa et al. [9] extended the result by
Walker.
α−m
In this paper, we give some inequalities for the
class
P
. Furthermore,
n
0 α−m
we define a neighborhood
of p(z) ∈ P
and determine δ > 0 so that
n
m+n−α
Nβδ (p(z)) ⊂ P α−m
,
where
β
=
.
n
n
On Neighborhoods of Analytic
Functions having Positive Real
Part
Shigeyoshi Owa, Nigar Yildirim
and Muhammet Kamali
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α−m
n
α−m
Our first result for functions p(z) in P n is contained in
Theorem 2.1. Let p(z) ∈ P α−m
. Then, for |z| = r < 1, m ≤ α < m +
n
n, m ∈ N0 and n ∈ N,
2r
α−m
0
(2.1)
|zp (z)| ≤
Re p(z) −
.
1 − r2
n
2.
Some Inequalities for the Class P
For each m ≤ α < m + n, the equality is attained at z = r for the function
α−m
α−m 1−z
2
p(z) =
+ 1−
= 1 − (n − α + m) z + · · · .
n
n
1+z
n
Proof. Let us consider the case of p(z) ∈ P(0). Then the function k(z) defined
by
1 − p(z)
k(z) =
= η 1 z + η2 z 2 + · · ·
1 + p(z)
is analytic in U and |k(z)| < 1 (z ∈ U). Hence k(z) = zΦ(z), where Φ(z) is
analytic in U and |Φ(z)| ≤ 1 (z ∈ U). For such a function Φ(z), we have
1 − |Φ(z)|2
0
(2.2)
|Φ (z)| ≤
(z ∈ U).
1 − |z|2
From zΦ(z) =
(i)
1−p(z)
,
1+p(z)
we obtain
On Neighborhoods of Analytic
Functions having Positive Real
Part
Shigeyoshi Owa, Nigar Yildirim
and Muhammet Kamali
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1
|Φ(z)| = 2
r
2
1 − p(z) 2
1 + p(z) ,
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(ii)
1
|Φ (z)| = 2
r
0
2zp0 (z) + (1 − p2 (z)) ,
(1 + p(z))2
where |z| = r. Substituting (i) and (ii) into (2.2), and then multiplying by
|1 + p(z)|2 we obtain
2
2
2
0
2zp (z) + (1 − p2 (z)) ≤ r |1 + p(z)| − |1 − p(z)| ,
1 − r2
which implies that
Shigeyoshi Owa, Nigar Yildirim
and Muhammet Kamali
r2 |1 + p(z)|2 − |1 − p(z)|2
|2zp0 (z)| ≤ (1 − p2 (z)) +
.
1 − r2
Thus, to prove (2.1) (with α = m), it is sufficient to show that
2
(2.3)
2
2
(1 − p2 (z)) + r |1 + p(z)| − |1 − p(z)| ≤ 4r Re p(z) .
1 − r2
1 − r2
Now we express |1 + p(z)|2 , |1 − p(z)|2 and Re p(z) in terms of |1 − p2 (z)|.
1−p(z)
From zΦ(z) = 1+p(z)
we obtain that
2
On Neighborhoods of Analytic
Functions having Positive Real
Part
2
(iii) |1 − p(z)| = |1 − p (z)| |zΦ(z)|
and
(iv) |1 + p(z)|2 |zΦ(z)| = 1 − Re2 (z) .
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From (iii) and (iv) we have
(v)
"
#
1 − |zΦ(z)|2
2
4 Re p(z) = |1 + p(z)| − |1 − p(z)| = 1 − p (z)
.
|zΦ(z)|
2
2
Substituting (iii), (iv), and (v) into (2.3), and then cancelling |1 − p2 | we obtain
r2
(1 − p2 (z)) +
|1−p2 (z)|
|zΦ(z)|
− |1 − p2 (z)| |zΦ(z)|
1 − r2
4 Re p(z) + (1 − r2 ) |1 − p2 (z)| 1 −
=
≤
On Neighborhoods of Analytic
Functions having Positive Real
Part
1
|zΦ(z)|
Shigeyoshi Owa, Nigar Yildirim
and Muhammet Kamali
1 − r2
4r Re p(z)
,
1 − r2
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which gives us that the inequality (2.1) holds true when α = m. Further, considering the function w(z) defined by
p(z) − ( α−m
)
n
w(z) =
,
α−m
1−( n )
in the case of α 6= m, we complete the proof of the theorem.
Remark 1. The result obtained from Theorem 2.1 for n = 1 and m = 0 coincides with the result due to Bernardi [1].
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Lemma 2.2. The function w(z) defined by
w(z) =
1−
1
n
{2α − (2m + n)} z
1−z
for m < α < m + n, m ∈ N0 ,
is univalent in U, w(0) = 1, and Re w(z) > α−m
n
and n ∈ N for U.
Lemma 2.3. Let p(z) ∈ P α−m
. Then the disk |z| ≤ r < 1 is mapped by
n
p(z) onto the disk |p(z) − η(A)| ≤ ξ(A), where
1 + Ar2
η(A) =
,
1 − r2
2m + n − 2α
A=
.
n
Now, we give general inequalities for the class P α−m
.
n
Theorem 2.4. Let the function p(z) be in the class P α−m
, k ≥ 0, and r =
n
|z| < 1. Then we have
zp0 (z)
(2.4) Re p(z) +
p(z) + k
2
α−m
(k + 1) + 2 2 − α−m
r
+
(1
−
k)
−
2(
)
r
α−m
n n
>
+
α−m
α−m
n
(k + 1) − 2 1 − n r + (1 − k) − 2( n ) r2
α−m
× Re p(z) −
.
n
r(A + 1)
ξ(A) =
,
1 − r2
Proof. With the help of Lemma 2.3, we observe that
1 + Ar2
r (A + 1)
|p(z) + k| ≥ |η(A) + k| − ξ(A) =
+k−
.
2
1−r
1 − r2
On Neighborhoods of Analytic
Functions having Positive Real
Part
Shigeyoshi Owa, Nigar Yildirim
and Muhammet Kamali
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Therefore, an application of Theorem 2.1 yields that
zp0 (z)
Re p(z) +
p(z) + k
zp0 (z) ≥ Re {p(z)} − p(z) + k 2r
α−m
1−r 2
≥ Re {p(z)} − 1+Ar2 +k(1−r2 )−r(A+1) Re p(z) −
n
1−r 2
(
)
2r
α−m
α−m
1−r 2
>
− 1 − 1+Ar2 +k(1−r2 )−r(A+1) Re p(z) −
,
n
n
2
1−r
On Neighborhoods of Analytic
Functions having Positive Real
Part
Shigeyoshi Owa, Nigar Yildirim
and Muhammet Kamali
which proves the assertion (2.4).
Remark 2. The result obtained from this theorem for n = 1, and m = 0
coincides with the result by Pashkouleva [10].
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3.
Preliminary Results
Let the functions f (z) and g(z) be analytic in U. Then f (z) is said to be subordinate to g(z), written f (z) ≺ g(z), if there exists an analytic function w(z)
in U with w(0) = 0 and |w(z)| ≤ |z| < 1 such that f (z) = g(w(z)). If g(z) is
univalent in U, then the subordination f (z) ≺ g(z) is equivalent to f (0) = g(0)
and
f (U) ⊂ g(U) (cf. [11, p. 36, Lemma 2.1]).
For f (z) and g(z) given by
f (z) =
∞
X
ak z
k
and
g(z) =
k=0
∞
X
bk z k ,
k=0
On Neighborhoods of Analytic
Functions having Positive Real
Part
Shigeyoshi Owa, Nigar Yildirim
and Muhammet Kamali
the Hadamard product (or convolution) of f (z) and g(z) is defined by
(3.1)
(f ∗ g) (z) =
∞
X
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ak b k z k .
k=0
Further, let P 0 α−m
be the subclass of T consisting of functions p(z) defined
n
by (1.1) which satisfy
α−m
(z ∈ U)
n
for some m ≤ α < m + n,
m ∈ N0 , and n ∈ N. It follows from the definitions
α−m
0 α−m
of P n and P
that
n
1 − n1 {2α − (2m + n)}z
α−m
(3.3) p(z) ∈ P
⇔ p(z) ≺
(z ∈ U)
n
1−z
(3.2)
Re {(zp(z))0 } >
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and that
(3.4)
p(z) ∈P
0
α−m
n
1−
1
n
{2α − (2m + n)} z
1−z
1
0
1 − n {2α − (2m + n)} z
(zp(z))
≺
⇔
0
(z)
1−z
⇔ (zp(z))0 ≺
(z ∈ U)
(z ∈ U).
Applying the result by Millerand Mocanu [7, p. 301, Theorem 10] for (3.4),
we see that if p(z) ∈ P 0 α−m
, then
n
(3.5)
p(z) ≺
which implies that P 0
1−
1
n
{2α − (2m + n)} z
(z ∈ U),
1−z
α−m
α−m
⊂
P
. Noting that the function
n
n
1−
1
n
{2α − (2m + n)} z
1−z
is univalent in U, we have that q(z) ∈ P α−m
if and only if
n
(3.6)
q(z) 6=
1−
1
n
{2α − (2m + n)} eiθ
1 − eiθ
(0 < θ < 2π; z ∈ U)
or
(3.7)
iθ
1−e
1
iθ
q(z) − 1 − (2α − (2m + n)) e
6= 0
n
(0 < θ < 2π; z ∈ U).
On Neighborhoods of Analytic
Functions having Positive Real
Part
Shigeyoshi Owa, Nigar Yildirim
and Muhammet Kamali
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Further, using the convolutions, we obtain that
1
iθ
iθ
(3.8)
1 − e q(z) − 1 − (2α − (2m + n)) e
n
1
1
iθ
iθ
= 1−e
∗ q(z) − 1 − [2α − (2m + n)] e
∗ q(z)
1−z
n
1 − eiθ
1
iθ
=
− 1 − (2α − (2m + n))e
∗ q(z).
1−z
n
Therefore, if we define the function hθ (z) by
1 − eiθ
1
n
iθ
(3.9) hθ (z) =
− 1 − (2α − (2m + n))e
,
2(α − m − n)eiθ 1 − z
n
then hθ (0) = 1 (0 < θ < 2π). This gives us that
α−m
(3.10) q(z) ∈ P
n
2
(3.11)
⇔ (α − m − n) eiθ {hθ (z) ∗ q(z)} =
6 0 (0 < θ < 2π; z ∈ U)
n
(3.12)
⇔ hθ (z) ∗ q(z) 6= 0(0 < θ < 2π; z ∈ D).
On Neighborhoods of Analytic
Functions having Positive Real
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Shigeyoshi Owa, Nigar Yildirim
and Muhammet Kamali
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4.
Main Results
In order to derive our main result, we need the following lemmas.
Lemma 4.1. If p(z) ∈ P 0 α−m
with m ≤ α < m + n; m ∈ N0 , n ∈ N, then
n
z(p(z) ∗ hθ (z)) is univalent for each θ (0 < θ < 2π) .
Proof. For fixed θ (0 < θ < 2π), we have
[z(p(z) ∗ hθ (z))]0
0
1 − eiθ
zn
1
iθ
=
− 1 − (2α − (2m + n))e
∗ p(z)
2(α − m − n)eiθ 1 − z
n
0
zn
1
iθ
iθ
=
(1 − e )p(z) − 1 − (2α − (2m + n))e
2(α − m − n)eiθ
n
"
!#0
1
iθ
1
−
(2α
−
(2m
+
n))e
zn
n
(1 − eiθ ) p(z) −
=
1 − eiθ
2(α − m − n)eiθ
"
!#0
1 − n1 (2α − (2m + n))eiθ
(1 − eiθ )
n
=
zp(z) −
z
eiθ
2(α − m − n)
1 − eiθ
(
)
1
iθ
1
−
(2α
−
(2m
+
n))e
1 − eiθ
n
n
.
=
(zp(z))0 −
1 − eiθ
eiθ
2(α − m − n)
By the definition of P 0 ( α−m
), the range of (zp(z))0 for |z| < 1 lies in Re(w) >
n
α−m
. On the other hand
n
1 + n1 {2α − (2m + n)}
1 − n1 {2α − (2m + n)} eiθ
=
Re
.
1 − eiθ
2
On Neighborhoods of Analytic
Functions having Positive Real
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and Muhammet Kamali
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Thus, we write
(4.1)
[z(p(z) ∗ hθ (z))]0
n
e−iφ
=
·
2(α − m − n) K
where
(
1 − n1 (2α − (2m + n))eiθ
0
(zp(z)) −
1 − eiθ
)
,
iθ e
1
= p
K = iθ
e −1
2(1 − cos θ)
and
φ = arg
eiθ
eiθ − 1
= θ − tan
−1
sin θ
cos θ − 1
On Neighborhoods of Analytic
Functions having Positive Real
Part
.
Consequently, we obtain that
Re Keiφ (z(p(z) ∗ hθ (z)))0 > 0
(z ∈ U),
because p(z) ∈ P 0 α−m
. An application of the Noshiro-Warschawski theorem
n
(cf. [3, p. 47]) gives that z(p(z) ∗ hθ (z)) is univalent for each θ (0 < θ < 2π) .
Lemma 4.2. If p(z) ∈ P 0 α−m
with m ≤ α < m + n, m ∈ N0 , and n ∈ N,
n
then
{z(p(z) ∗ hθ (z))}0 ≥ 1 − r
(4.2)
1+r
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for |z| = r < 1 and 0 < θ < 2π.
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Proof. Using the expression (4.1) for {z(p(z) ∗ hθ (z))}0 , we define
1 + n1 (2m + n − 2α)eiθ
−iθ
iθ
F (w) = e (1 − e )
−w ,
1 − eiθ
where
1
n
[2m + n − 2α] reit
(0 ≤ t ≤ 2π).
1 − reit
Then the function F (w) may be rewritten as
1
iθ
iθ
−iθ
F (w) = e
1 + (2m + n − 2α)e − (1 − e )w
n
1
−iθ
=e
(1 − w) +
(2m + n − 2α) + w eiθ
n
1
1−w
−iθ
iθ
+e
=
(2m + n − 2α) + w e
1
n
(2m + n − 2α) + w
n
w=
1+
for 0 < θ < 2π. Thus we see that
1
1
−
w
iθ
|F (w)| = (2m + n − 2α) + w 1
+ e n
(2m + n − 2α) + w
n
1
iθ
= (2m + n − 2α) + w e − reit n
1
= (2m + n − 2α) + w 1 − rei(t−θ) n
1
≥ (2m + n − 2α) + w (1 − r) .
n
On Neighborhoods of Analytic
Functions having Positive Real
Part
Shigeyoshi Owa, Nigar Yildirim
and Muhammet Kamali
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Since
1
it 1
(2m + n − 2α) + w = 1 (2m + n − 2α) + 1 + n (2m + n − 2α)re n
n
1 − reit
1 + n1 (2m + n − 2α) = 1 − reit
1 + n1 (2m + n − 2α)
≥
,
1+r
it is clear that
(1 − r)
1
|F (w)| ≥
1 + (2m + n − 2α) .
(1 + r)
n
Since p ∈ P 0 α−m
and (4.1) holds, by letting w = [zp(z)]0 , we get the desired
n
inequality, That is,
1+
n
·
2(m + n − α)
(1 − r)
=
.
(1 + r)
[zp(z)]0 ≥
1
(2m
n
+ n − 2α)
(1 − r)
1+r
Further, we need the following lemma.
Lemma 4.3. If p(z) ∈ P 0 ( α−m
) with m ≤ α < m + n, m ∈ N0 , and n ∈ N,
n
then
|p(z) ∗ hθ (z)| ≥ δ
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Therefore, the lemma is proved.
(4.3)
On Neighborhoods of Analytic
Functions having Positive Real
Part
(0 < θ < 2π; z ∈ U),
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where
Z
δ=
0
1
2
dt − 1 = 2 ln 2 − 1.
1+t
Proof. Since Lemma 4.1 shows that z(p(z) ∗ hθ (z)) is univalent for each θ
(0 < θ < 2π) for p(z) belonging to the class P 0 α−m
, we can choose a point
n
z0 ∈ U with |z0 | = r < 1 such that
min |z(p(z) ∗ hθ (z))| = |z0 (p(z0 ) ∗ hθ (z0 ))|
|z|=r
for fixed r (0 < r < 1). Then the pre-image γ of the line segment from 0 to
z0 (p(z0 ) ∗ hθ (z0 )) is an arc inside |z| ≤ r. Hence, for |z| ≤ r, we have that
|z(p(z) ∗ hθ (z))| ≥ |z0 (p(z0 ) ∗ hθ (z0 ))|
Z
= |(z(p(z) ∗ hθ (z)))0 | |dz| .
γ
An application of Lemma 4.2 leads us to
Z
Z
1 r 2
1 r 1−t
dt =
dt − 1.
|p(z) ∗ hθ (z)| ≥
r 0 1+t
r 0 1+t
Note that the function Ω(r) defined by
Z
1 r 2
Ω(r) =
dt − 1
r 0 1+t
is decreasing for r (0 < r < 1). Therefore, we have
Z 1
2
|p(z) ∗ hθ (z)| ≥ δ =
dt − 1 = 2 ln 2 − 1,
0 1+t
On Neighborhoods of Analytic
Functions having Positive Real
Part
Shigeyoshi Owa, Nigar Yildirim
and Muhammet Kamali
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which completes the proof of Lemma 4.3.
Now, we give the statement and the proof of our main result.
Theorem 4.4. If p(z) ∈ P 0 α−m
with m ≤ α < m + n, m ∈ N0 , and n ∈ N,
n
then
α−m
Nβδ (p(z)) ⊂ P
,
n
where β = m+n−α
and
n
Z 1
2
(4.4)
δ=
dt − 1 = 2 ln 2 − 1.
0 1+t
The result is sharp.
P
k
Proof. Let q(z) = 1+ ∞
k=1 qk z . Then, by the definition ofneighborhoods, we
have to prove that ifq(z) ∈ Nβδ (p(z)) for p(z) ∈ P 0 α−m
, then q(z) belongs
n
to the class P α−m
.
Using
Lemma
4.3
and
the
inequality
n
∞
X
|pk − qk | ≤ δ,
k=1
we get
|hθ (z) ∗ q(z)| ≥ |hθ (z) ∗ p(z)| − |hθ (z) ∗ (p(z) − q(z))|
∞
X
n(1 − eiθ )
k
≥δ−
(pk − qk )z 2(α − m − n)eiθ
On Neighborhoods of Analytic
Functions having Positive Real
Part
Shigeyoshi Owa, Nigar Yildirim
and Muhammet Kamali
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k=1
∞
X
n
|pk − qk |
>δ−
m + n − α k=1
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n
>δ−
m+n−α
≥ δ − δ = 0.
m+n−α
n
δ
Since hθ (z) ∗ q(z) 6= 0 for0 < θ < 2π and z ∈ U, we conclude that q(z)
, that is, that Nβδ (p(z)) ⊂ P ( α−m
).
belongs to the class P α−m
n
n
Further, taking the function p(z) defined by
(zp(z))0 =
1−
1
n
{2α − (2m + n)} z
,
1−z
On Neighborhoods of Analytic
Functions having Positive Real
Part
we have
2
(m + n − α)
1
p(z) = (2α − (2m + n)) + n
n
z
Z
0
z
Shigeyoshi Owa, Nigar Yildirim
and Muhammet Kamali
1
dt .
1−t
Title Page
If we define the function q(z) by
q(z) = p(z) +
m+n−α
n
Contents
δz,
then q(z) ∈ Nβδ (p(z)). Letting z = eiπ , we see that q(z) = q(eiπ ) =
This implies that if
Z 1
2
δ>
dt − 1,
0 1+t
α−m
.
n
then q(eiπ ) < α−m
. Therefore, Re {q(z)} < α−m
for z near eiπ , which contran
n
dicts q(z) ∈ P( α−m
) (otherwise Re {q(z)} > α−m
; z ∈ U). Consequently, the
n
n
result of the theorem is sharp.
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References
[1] S.D. BERNARDI, New distortion theorems for functions of positive real
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[2] J.E. BROWN, Some sharp neighborhoods of univalent functions, Trans.
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[3] P.L. DUREN, Univalent Functions, Springer-Verlag, New York, 1983.
[4] R. FOURNIER, A note on neighborhoods of univalent functions, Proc.
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[5] R. FOURNIER, On neighborhoods of univalent starlike functions, Ann.
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[6] R. FOURNIER, On neighborhoods of univalent convex functions, Rocky
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[8] Z. NEHARI, Conformal Mapping, McGraw-Hill, New York, 1952.
On Neighborhoods of Analytic
Functions having Positive Real
Part
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and Muhammet Kamali
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[9] S. OWA, H. SAITOH AND M. NUNOKAWA, Neighborhoods of certain
analytic functions, Appl. Math. Lett., 6 (1993), 73–77.
Close
[10] D.Z. PASHKOULEVA, The starlikeness and spiral-convexity of certain
subclasses of analytic functions, Current Topics in Analytic Function The-
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ory (H.M. Srivastava and S. Owa (Editors)), World Scientific, Singapore,
New Jersey, London and Hong Kong (1992), 266–273.
[11] Ch. POMMERENKE, Univalent Functions, Vandenhoeck and Ruprecht,
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[12] St. RUSCHEWEYH, Neighborhoods of univalent functions, Proc. Amer.
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[13] J.B. WALKER, A note on neighborhoods of analytic functions having positive real part, Internat. J. Math. Math. Sci., 13 (1990), 425–430.
On Neighborhoods of Analytic
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and Muhammet Kamali
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