DOI: 10.1515/auom-2015-0020
An. Şt. Univ. Ovidius Constanţa
Vol. 23(1),2015, 291–301
On neighborhoods of functions associated with
conic domains
Nihat Yağmur
Abstract
Let k − ST [A, B], k ≥ 0, −1 ≤ B < A ≤ 1 be the class of normalized
analytic functions defined in the open unit disk satisfying


0
0
(z)
(z)
(B − 1) zff (z)
(B − 1) zff (z)
− (A − 1)
− (A − 1)


<
>
k
−1 .
0
0
zf (z)
zf (z)
(B + 1) f (z) − (A + 1)
(B + 1) f (z) − (A + 1)
and let k − U CV [A, B], k ≥ 0, −1 ≤ B < A ≤ 1 be the corresponding
class satisfying


(zf 0 (z))0
(zf 0 (z))0
−
(A
−
1)
(B − 1) f 0 (z) − (A − 1)
(B
−
1)
f 0 (z)


−1 .
<
>k
0 (z))0
0 (z))0
(B + 1) (zff 0 (z)
− (A + 1)
(B + 1) (zff 0 (z)
− (A + 1)
For an appropriate δ > 0, the δ neighborhood of a function f ∈ k −
U CV [A, B] is shown to consist of functions in the class k − ST [A, B].
1
Introduction
Let A denote the family of functions f of the form
f (z) = z +
∞
X
an z n
(1)
n=2
Key Words: Analytic function, k−starlike functions, k−uniformly convex functions.
2010 Mathematics Subject Classification: 30C45.
Received: 8 May, 2014.
Accepted: 30 June, 2014.
291
ON NEIGHBORHOODS OF FUNCTIONS ASSOCIATED WITH CONIC
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292
which are analytic in the open unit disk U = {z : |z| < 1}. The classes S ∗
and C are the well-known classes of starlike and convex univalent functions
respectively; for details, see[4].
Noor and Malik introduced and studied the class k − U CV [A, B] and the
corresponding class k − ST [A, B] in [10] as following:
A function f (z) ∈ A is said to be in the class k − ST [A, B], k ≥ 0, −1 ≤
B < A ≤ 1, if and only if,


0
0
(z)
(z)
(B − 1) zff (z)
− (A − 1)
− (A − 1)
(B − 1) zff (z)

(2)
>
k
−1 .
<
zf 0 (z)
zf 0 (z)
(B + 1) f (z) − (A + 1)
(B + 1) f (z) − (A + 1)
A function f (z) ∈ A is said to be in the class k − U CV [A, B], k ≥ 0, −1 ≤
B < A ≤ 1, if and only if,


0
0
(zf 0 (z))
(zf 0 (z))
(B
−
1)
−
(A
−
1)
(B − 1) f 0 (z) − (A − 1)
f 0 (z)


<
− 1 . (3)
>k
0 (z))0
0 (z))0
(B + 1) (zff 0 (z)
− (A + 1)
(B + 1) (zff 0 (z)
− (A + 1)
It can be easily seen that
f (z) ∈ k − U CV [A, B] ⇐⇒ zf 0 (z) ∈ k − ST [A, B].
(4)
Special cases.
i. k − ST [1, −1] = k − ST, k − U CV [1, −1] = k − U CV, the well-known
classes of k−starlike and k−uniformly convex functions respectively, introduced by Kanas and Wisniowska [6, 7].
ii. k − ST [1 − 2α, −1] = SD(k, α), k − U CV [1 − 2α, −1] = KD(k, α), the
classes introduced by Shams et al. in [17].
iii. 0 − ST [A, B] = S ∗ [A, B], 0 − U CV [A, B] = C[A, B], the well-known
classes of Janowski starlike and Janowski convex functions respectively, introduced by Janowski [5].
Geometrically, if a function f (z) ∈ k − ST [A, B] then
zf 0 (z)
−(A−1)
f (z)
zf 0 (z)
(B+1) f (z) −(A+1)
(B−1)
=
w takes all values from the domain Ωk , k ≥ 0, as
Ωk = {w : <w > k |w − 1|}
(5)
n
o
p
Ωk = u + iv : u > k (u − 1)2 + v 2 .
(6)
or equivalently
ON NEIGHBORHOODS OF FUNCTIONS ASSOCIATED WITH CONIC
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The domain Ωk represents the right half plane for k = 0, a hyperbola for
0 < k < 1, a parabola for k = 1 and an ellipse for k > 1, for more details see
[10].
Given δ ≥ 0, Ruscheweyh [16] defined the δ − neighborhood Nδ (f ) of a
function f (z) ∈ A by
(
)
∞
∞
X
X
n
Nδ (f ) = g(z) : g(z) = z +
bn z , and
n |an −bn | ≤ δ .
(7)
n=2
n=2
Ruscheweyh [16] proved among other results that N1/4 (f ) ⊂ S ∗ for f ∈ C.
Sheil-Small and Silvia [18] introduced more general notions of neighborhood
of an analytic function. These included noncoefficient neighbourhoods as well.
Problems related to the neighborhoods and some other properties of analytic
functions were considered by many authors, for example, see [1, 2, 3, 8, 9, 11,
12, 13, 14].
In this paper, the neighborhood Nδ (f ) for functions f (z) ∈ k − U CV [A, B]
is investigated. It is shown that all functions g ∈ Nδ (f ) are in the class
k − ST [A, B] for certain δ > 0.
2
Main Results
In order to obtain the main results, a characterization of the class k −ST [A, B]
in terms of the functions in another class k − ST 0 [A, B] is needed. For z ∈ U,
a function Ht is said to be in the class k − ST 0 [A, B] if the function Ht is of
the form
Ht (z)
=
1
q
2
2
(B − A) 1 − kt ± t − (kt − 1) i
 q
2
2
 z B − 1 − (B + 1) kt ± t − (kt − 1) i
×

(1 − z)2

q
2
2
z A − 1 − (A + 1) (kt ± t − (kt − 1) i) 

−

1−z
where
t≥
1
k+1
1
k+1 ,
≤ t≤
1
k−1 ,
if 0 ≤ k ≤ 1,
if k > 1.
(8)
(9)
ON NEIGHBORHOODS OF FUNCTIONS ASSOCIATED WITH CONIC
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294
Recall that for any two functions f (z) and g(z) given by
f (z) = z +
∞
X
an z n ,
g(z) = z +
n=2
∞
X
bn z n
n=2
the Hadamard product (or convolution) of f and g is defined by
(f ∗ g)(z) = z +
∞
X
an bn z n =(g ∗ f )(z)
(10)
n=2
Lemma 2.1. A function f is in the class k − ST [A, B] if and only if
1
(f ∗ Ht )(z) 6= 0,
z
z∈U
(11)
for all Ht ∈ k − ST 0 [A, B].
Proof. Let
f
∈
k − ST [A, B].
Then
the
image
of
zf 0 (z)
−(A−1)
f (z)
zf 0 (z)
(B+1) f (z) −(A+1)
U
under
(B−1)
lies in the conic type regions
= u + iv =
n
o
p
Ωk = u + iv : u > k (u − 1)2 + v 2 so that for z ∈ U
w
0
(z)
− (A − 1)
(B − 1) zff (z)
0
(z)
(B + 1) zff (z)
− (A + 1)
where
t≥
1
k+1
1
k+1 ,
≤ t≤
q
2
6 kt ± t2 − (kt − 1) i
=
1
k−1 ,
if 0 ≤ k ≤ 1,
if k > 1.
Thus f ∈ k − ST [A, B] if and only if
q
2
B − 1 − (B + 1) (kt ± t2 − (kt − 1) i) zf 0 (z)
q
2
− A − 1 − (A + 1) (kt ± t2 − (kt − 1) i) f (z)
q
2
/ z (B − A) 1 − kt ± t2 − (kt − 1) i
6= 0
or equivalently
1
(f ∗ Ht )(z) 6= 0,
z
for all Ht ∈ k − ST 0 [A, B].
z∈U
(12)
(13)
(14)
ON NEIGHBORHOODS OF FUNCTIONS ASSOCIATED WITH CONIC
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Lemma 2.2. If
Ht (z) = z +
∞
X
z∈U
hn (t)z n ∈ k − ST 0 [A, B],
(15)
n=2
then
|hn (t)| <
2n(t + 1)
|B − A| t
(16)
where t is given in (13).
Proof. Writing Ht (z) = z +
∞
P
hn (t)z n , and comparing coefficients of z n in
n=2
(8) we obtain
hn (t)
=
[n (B − 1) − (A − 1) − (n (B + 1) − (A + 1))kt
q
2
±(n (B + 1) − (A + 1)) t2 − (kt − 1) i]
q
2
2
/ (B − A) 1 − kt ± t − (kt − 1) i
(17)
Thus
2
|hn (t)|
[|n (B − 1) − (A − 1) − (n (B + 1) − (A + 1))kt
q
2
±(n (B + 1) − (A + 1)) t2 − (kt − 1) i 2 ]
2
q
2
2
/ (B − A) 1 − kt ± t − (kt − 1) i
h
2
= {n (B − 1) − (A − 1) − [n (B + 1) − (A + 1)]kt}
i
2
+(n (B + 1) − (A + 1))2 t2 − (kt − 1)
h
i
2
2
2
/ (B − A) (1 − kt) + t2 − (kt − 1)
=
2
=
4(n − 1) [(nB − A) (kt − 1) + (n − 1) kt] + [nB − A + n − 1] t2
2
(B − A) t2
.
If −1 ≤ B < A ≤ 1, n ≥ 2 then nB − A < n − 1, and from (13), kt − 1 < t.
So
2
|hn (t)| ≤
4(n − 1)2 (t + 1)
2
(B − A) t2
and
|hn (t)| <
2
<
2n(t + 1)
.
|B − A| t
4n2 (t + 1)
2
2
(B − A) t2
(18)
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Lemma 2.3. For a function f ∈ A and ε ∈ C, define the function Fε by
f (z) + εz
.
1+ε
Fε =
(19)
If for every ε, |ε| < δ we have Fε ∈ k − ST [A, B], then for every Ht ∈
k − ST 0 [A, B]
1
(20)
(f ∗ Ht )(z) ≥ δ, z ∈ U.
z
Proof. If Fε ∈ k − ST [A, B] for every ε, |ε| < δ, then by Lemma 2.1 for all
Ht ∈ k − ST 0 [A, B]
1
(Fε ∗ Ht )(z) 6= 0, z ∈ U
z
or equivalently
(f ∗ Ht )(z) + εz
6= 0.
(21)
(1 + ε)z
Since |ε| < δ, it is easily follows that
1
(f ∗ Ht )(z) ≥ δ.
z
Theorem 2.1. Let f ∈ A , ε ∈ C and δ > 0. If for every ε, |ε| < δ, we have
Fε ∈ k − ST [A, B] then
Nδ0 (f ) ⊂ k − ST [A, B]
(22)
for
δ0 =
where −1 ≤ B < A ≤ 1 and
∞
P
Proof. Let g(z) = z +
t≥
1
k+1
|B − A| t
δ
2(t + 1)
1
k+1 ,
≤ t≤
1
k−1 ,
(23)
if 0 ≤ k ≤ 1,
if k > 1.
bn z n ∈Nδ0 (f ). For any Ht ∈ k − ST 0 [A, B],
n=2
1
(g ∗ Ht )(z)
z
=
≥
1
1
(f ∗ Ht )(z) + ((g − f ) ∗ Ht ) (z)
z
z
1
1
(f ∗ Ht )(z) − ((g − f ) ∗ Ht ) (z)
z
z
ON NEIGHBORHOODS OF FUNCTIONS ASSOCIATED WITH CONIC
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From lemma 2.3,
1
(g ∗ Ht )(z)
z
≥ δ−
≥ δ−
∞
X
(bn − an ) hn (t)z n
z
n=2
∞
X
|(bn − an )| |hn (t)|
n=2
≥ δ−
∞
2(t + 1) X
n |(bn − an )| .
|B − A| t n=2
Using Lemma 2.2 and noting that g∈Nδ0 (f ) and whence
∞
P
n |(bn − an )| < δ 0 ,
n=2
thus
Therefore
2(t + 1) 0
1
(g ∗ Ht )(z) ≥ δ −
δ =0
z
|B − A| t
1
z (g
∗ Ht )(z) 6= 0 in U for all Ht ∈ k − ST 0 [A, B] if
δ0 =
|B − A| t
δ
2(t + 1)
By Lemma 2.1 g ∈ k − ST [A, B]. This proves that Nδ0 (f ) ⊂ k − ST [A, B].
Lemma 2.4. ([15])If φ is a convex univalent function with φ(0) = 0 = φ0 (0)−
1 in U and g is starlike univalent in U, then for each analytic function F in
U,
(φ ∗ F g)(z)
⊂ coF (U), (z ∈ U)
(24)
(φ ∗ g)(z)
where co stands for the closed convex hull.
Lemma 2.5. If φ(z) ∈ C, f (z) ∈ k − ST [A, B] and g(z) = (B + 1)zf 0 (z) −
(A + 1)f (z) ∈ S ∗ then (φ ∗ f )(z) ∈ k − ST [A, B].
Proof. Assume F (z) =
0
0
zG (z) = φ(z) ∗ zf (z).
zf 0 (z)
−(A−1)
f (z)
zf 0 (z)
(B+1) f (z) −(A+1)
(B−1)
. Then for G(z) = (φ ∗ f )(z), we have
ON NEIGHBORHOODS OF FUNCTIONS ASSOCIATED WITH CONIC
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Hence
0
(z)
(B − 1) zG
G(z) − (A − 1)
0
(z)
(B + 1) zG
G(z) − (A + 1)
=
=
=
0
(z)
(B − 1) φ(z)∗zf
(φ∗f )(z) − (A − 1)
=
0
(z)
(B + 1) φ(z)∗zf
(φ∗f )(z) − (A + 1)
φ(z) ∗ [(B − 1)zf 0 (z) − (A − 1)f (z)]
φ(z) ∗ [(B + 1)zf 0 (z) − (A + 1)f (z)]
zf 0 (z)
(B−1) f (z) −(A−1)
φ(z) ∗
[(B + 1)zf 0 (z) − (A + 1)f (z)]
0
zf (z)
(B+1)
−(A+1)
f (z)
φ(z) ∗ [(B + 1)zf 0 (z) − (A + 1)f (z)]
φ(z) ∗ (F g)(z)
.
φ(z) ∗ g(z)
(25)
g)(z)
By Lemma 2.4 the image of U under φ(z)∗(F
φ(z)∗g(z) is a subset of the convex hull
of F (U). That means (φ ∗ f )(z) ∈ k − ST [A, B].
Theorem 2.2. If f ∈ k−U CV [A, B] and g(z) = (B +1)zf 0 (z)−(A+1)f (z) ∈
C then Fε = f (z)+εz
∈ k − ST [A, B] for |ε| < 14 .
1+ε
∞
P
Proof. Let f (z) = z +
an z n ∈ k − U CV [A, B]. Then
n=2
Fε
=
f (z) + εz
1+ε
z(1 + ε) +
=
1+ε
∞
P
f (z) ∗ z(1 + ε) +
zn
n=2
1+ε
= f (z) ∗
ε
z− 1+ε
z2
.
1−z
an z n
n=2
=
where h(z) =
∞
P
z−
ε
2
1+ε z
1−z
|ε|
1−|ε|
<
<
= f (z) ∗ h(z)
Now,
zh0 (z)
1 − 2ρz + ρz 2
=
,
h(z)
(1 − ρz) (1 − z)
Hence |ρ| =
1
3
gives |ε| < 41 . Thus
zh 0 (z)
h(z)
where ρ =
ε
.
1+ε
2
≥
1 − 2 |ρ| |z| − |ρ| |z|
> 0.
(1 − |ρ| |z|) (1 + |z|)
(26)
ON NEIGHBORHOODS OF FUNCTIONS ASSOCIATED WITH CONIC
DOMAINS
This inequality holds for all |ρ| <
starlike in U and so
Zz
1
3
299
which is true for |ε| < 14 . Therefore h is
∞
X
h(t)
cn n
1
dt = z +
z = h(z) ∗ log
t
n
1−z
n=2
0
is convex for |ε| <
(27)
1
4
1
(f ∗ h) (z) = zf (z) ∗ h(z) ∗ log
1−z
0
(28)
0
f (z) ∈ k−U CV [A, B] =⇒ zf (z) ∈ k−ST [A, B] and h(z) ∗ log
1
1−z
∈ C.
Also g(z) ∈ C implies that zg 0 (z) ∈ S ∗ . By Lemma 2.5, we have
1
zf 0 (z) ∗ h(z) ∗ log
∈ k − ST [A, B].
1−z
Thus
(f ∗ h) (z) =
f (z) + εz
∈ k − ST [A, B].
1+ε
Theorem 2.3. Let f ∈ k − U CV [A, B] then
Nδ0 (f ) ⊂ k − ST [A, B]
(29)
for
δ0 =
where −1 ≤ B < A ≤ 1 and
t≥
1
k+1
|B − A| t
8(t + 1)
1
k+1 ,
≤ t≤
1
k−1 ,
(30)
if 0 ≤ k ≤ 1,
if k > 1.
Proof. Let f ∈ k − U CV [A, B] then we get the result by taking δ =
Theorem 2.1 and using Theorem 2.2.
1
4
in
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Nihat Yağmur,
Department of Mathematics,
Faculty of Science and Art,
Erzincan University,
24000, Erzincan, Turkey.
Email: nhtyagmur@gmail.com
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