DOI: 10.1515/auom-2015-0015
An. Şt. Univ. Ovidius Constanţa
Vol. 23(1),2015, 225–236
(α, β, λ, δ, m, Ω)p −Neighborhood for some
families of analytic and multivalent functions
Halit ORHAN
Abstract
In the present investigation, we give some interesting results related
with neighborhoods of p−valent functions. Relevant connections with
some other recent works are also pointed out.
1
Introduction and Definitions
Let A demonstrate the family of functions f (z) of the form
f (z) = z +
∞
X
an z n
n=2
which are analytic in the open unit disk U = {z ∈ C : |z| < 1} .
We denote by Ap (n) the class of functions f (z) normalized by
f (z) = z p +
∞
X
ak+p z k+p
(n, p ∈ N := {1, 2, 3, ...})
(1)
k=n
which are analytic and p-valent in U.
Key Words: p−valent functions, Inclusion relations, Neighborhood properties, Salagean
differential operator, Miller and Mocanu’s lemma.
2010 Mathematics Subject Classification: 30C45.
Received: 2 May, 2014.
Revised: 16 June, 2014.
Accepted: 27 June, 2014.
225
(α, β, λ, δ, m, Ω)p −NEIGHBORHOOD FOR SOME FAMILIES OF ANALYTIC
AND MULTIVALENT FUNCTIONS
226
Upon differentiating both sides of (1) m times with respect to z, we have
∞
f (m) (z) =
X (k + p)!
p!
z p−m +
ak+p z k+p−m
(p − m)!
(k + p − m)!
(2)
k=n
(n, p ∈ N; m ∈ N0 := N ∪ {0}; p > m).
We show by Ap (n, m) the class of functions of the form (2) which are
analytic and p-valent in U.
The concept of neighborhood for f (z) ∈ A was first given by Goodman [7].
The concept of δ-neighborhoods Nδ (f ) of analytic functions f (z) ∈ A was first
studied by Ruscheweyh [8]. Walker [12], defined a neighborhood of analytic
functions having positive real part. Later, Owa et al.[13] generalized the results
given by Walker. In 1996, Altıntaş and Owa [14] gave (n, δ)-neighborhoods for
functions f (z) ∈ A with negative coefficients. In 2007, (n, δ)-neighborhoods
for p-valent functions with negative coefficients were considered by Srivastava
et al. [4], and Orhan [5]. Very recently, Orhan et al.[1], introduced a new
definition of (n, δ)-neighborhood for analytic functions f (z) ∈ A. Orhan et
al.’s [1] results were generalized for the functions f (z) ∈ A and f (z) ∈ Ap (n)
by many author (see, [6, 9, 10, 15]).
In this paper, we introduce the neighborhoods (α, β, λ, m, δ, Ω)p − N (g)
and (α, β, λ, m, δ, Ω)p − M (g) of a function f (m) (z) when f (z) ∈ Ap (n).
Using the Salagean derivative operator [3]; we can write the following equalities for the function f (m) (z) given by
D0 f (m) (z) = f (m) (z)
D1 f (m) (z) =
0
z
f (m) (z)
(p − m)
∞
=
X (k + p − m)(k + p)!
p!
z p−m +
ak+p z k+p−m
(p − m)!
(p − m)(k + p − m)!
k=n
D2 f (m) (z) = D(Df (m) (z))
∞
=
X (k + p − m)2 (k + p)!
p!
z p−m +
ak+p z k+p−m
(p − m)!
(p − m)2 (k + p − m)!
k=n
(α, β, λ, δ, m, Ω)p −NEIGHBORHOOD FOR SOME FAMILIES OF ANALYTIC
AND MULTIVALENT FUNCTIONS
227
.........................................................
DΩ f (m) (z) = D(DΩ−1 f (m) (z))
∞
=
X (k + p − m)Ω (k + p)!
p!
z p−m +
ak+p z k+p−m .
(p − m)!
(p − m)Ω (k + p − m)!
k=n
We define F : Ap (n, m) → Ap (n, m) such that
F(f (m) (z)) = (1 − λ) DΩ f (m) (z) +
λz Ω (m) 0
D f (z)
(p − m)
∞
=
X (k + p)!(k + p − m)Ω (1 + λk(p − m)−1 )
p!
z p−m +
ak+p z k+p−m
(p − m)!
(p − m)Ω (k + p − m)!
k=n
(3)
(0 ≤ λ ≤ 1; Ω, m ∈ N0 ; p > m).
Let F(λ, m, Ω) denote class of functions of the form (3) which are analytic
in U.
For f, g ∈ F(λ, m, Ω), f said to be (α, β, λ, m, δ, Ω)p −neighborhood for g if
it satisfies
iα 0 (m)
e F (f (z)) eiβ F0 (g (m) (z)) < δ (z ∈ U)
−
z p−m−1
z p−m−1
p
p!
for some −π ≤ α − β ≤ π and δ > (p−m−1)!
2[1 − cos(α − β)]. We show this
neighborhood by (α, β, λ, m, δ, Ω)p − N (g).
Also, we say that f ∈ (α, β, λ, m, δ, Ω)p − M (g) if it satisfies
iα
e F(f (m) (z)) eiβ F(g (m) (z)) < δ (z ∈ U)
−
z p−m
z p−m
p
p!
for some −π ≤ α − β ≤ π and δ > (p−m−1)!
2[1 − cos(α − β)].
We give some results for functions belonging to (α, β, λ, m, δ, Ω)p − N (g)
and (α, β, λ, m, δ, Ω)p − M (g).
(α, β, λ, δ, m, Ω)p −NEIGHBORHOOD FOR SOME FAMILIES OF ANALYTIC
AND MULTIVALENT FUNCTIONS
2
228
Main Results
Now we can establish our main results.
Theorem 2.1. If f ∈ F(λ, m, Ω) satisfies
∞
X
(k + p − m)Ω+1 (k + p)!(1 + λk(p − m)−1 ) iα
e ak+p − eiβ bk+p Ω
(p − m) (k + p − m)!
k=n
p
p!
2[1 − cos(α − β)]
(4)
(p − m − 1)!
p
p!
2[1 − cos(α − β)], then
for some −π ≤ α − β ≤ π; p > m and δ > (p−m−1)!
f ∈ (α, β, λ, m, δ, Ω)p − N (g).
≤δ−
Proof. By virtue of (3), we can write
iα 0 (m)
e F (f
(z))
eiβ F0 (g (m) (z)) −
z p−m−1
z p−m−1
=
∞
X
(k + p − m)Ω+1 (k + p)!(1 + λk(p − m)−1 )
p!(p − m) iα
e + eiα
ak+p z k
(p − m)!
(p − m)Ω (k + p − m)!
k=n
∞
X
(k + p − m)Ω+1 (k + p)!(1 + λk(p − m)−1 )
p!(p − m) iβ
iβ
k
−
e −e
bk+p z Ω
(p − m)!
(p − m) (k + p − m)!
k=n
<
p
p!
2[1 − cos(α − β)]
(p − m − 1)!
∞
X
(k + p − m)Ω+1 (k + p)!(1 + λk(p − m)−1 ) iα
iβ
+
e ak+p − e bk+p .
Ω
(p − m) (k + p − m)!
k=n
If
∞
X
(k + p − m)Ω+1 (k + p)!(1 + λk(p − m)−1 ) iα
e ak+p − eiβ bk+p Ω
(p − m) (k + p − m)!
k=n
≤δ−
p
p!
2[1 − cos(α − β)],
(p − m − 1)!
(α, β, λ, δ, m, Ω)p −NEIGHBORHOOD FOR SOME FAMILIES OF ANALYTIC
AND MULTIVALENT FUNCTIONS
then we observe that
iα 0 (m)
e F (f (z)) eiβ F0 (g (m) (z)) <δ
−
z p−m−1
z p−m−1
229
(z ∈ U).
Thus, f ∈ (α, β, λ, m, δ, Ω)p − N (g). This evidently completes the proof of
Theorem 2.1.
Remark 2.2. In its special case when
m = Ω = λ = α = 0 and β = α,
(5)
in Theorem 2.1 yields a result given earlier by Altuntaş et al. ([9] p.3, Theorem
1).
We give an example for Theorem 2.1.
Example 2.1. For given
∞
g(z) =
X
p!
z p−m +
Bk+p (α, β, λ, m, δ, Ω)z k+p−m ∈ F(λ, m, Ω)
(p − m)!
k=n
(n, p ∈ N = {1, 2, 3, ...} ; p > m; Ω, m ∈ N0 )
we consider
∞
f (z) =
X
p!
z p−m +
Ak+p (α, β, λ, m, δ, Ω)z k+p−m ∈ F(λ, m, Ω)
(p − m)!
k=n
(n, p ∈ N = {1, 2, 3, ...} ; p > m; Ω, m ∈ N0 )
with
Ak+p =
p!
(p − m)Ω {δ− (p−m−1)!
p
2[1 − cos(α − β)]}(k + p − m)!(n + p − 1)
(1 + λk(p − m)−1 )(k + p − m)Ω+1 (k + p − 1)!(k + p)2 (k + p − 1)
e−iα
+ei(β−α) B k+p .
Then we have that
∞
X
(k + p − m)Ω+1 (k + p)!(1 + λk(p − m)−1 ) iα
e Ak+p − eiβ B k+p Ω
(p − m) (k + p − m)!
k=n
(α, β, λ, δ, m, Ω)p −NEIGHBORHOOD FOR SOME FAMILIES OF ANALYTIC
AND MULTIVALENT FUNCTIONS
230
X
∞
p
p!
1
= (n + p − 1) δ−
2[1 − cos(α − β)]
.
(p − m − 1)!
(k + p − 1)(k + p)
k=n
(6)
Finally, in view of the telescopic series, we obtain
∞
X
k=n
1
(k + p − 1)(k + p)
=
=
=
ζ X
1
1
lim
−
ζ−→∞
k+p−1 k+p
k=n
1
1
lim
−
ζ−→∞ n + p − 1
ζ +p
1
.
n+p−1
(7)
Using (7) in (6), we get
∞
X
(k + p − m)Ω+1 (k + p)!(1 + λk(p − m)−1 ) iα
iβ
A
−
e
B
e
k+p
k+p
(p − m)Ω (k + p − m)!
k=n
=δ−
p
p!
2[1 − cos(α − β)].
(p − m − 1)!
Therefore, we say that f ∈ (α, β, λ, m, δ, Ω)p − N (g).
Also, Theorem 2.1 gives us the following corollary.
Corollary 2.3. If f ∈ F(λ, m, Ω) satisfies
∞
X
(k + p − m)Ω+1 (k + p)!(1 + λk(p − m)−1 )
||ak+p | − |bk+p ||
(p − m)Ω (k + p − m)!
k=n
p
p!
2[1 − cos(α − β)]
(p − m − 1)!
p
p!
for some −π ≤ α − β ≤ π and δ > (p−m−1)!
2[1 − cos(α − β)], and
arg(ak+p ) − arg(bk+p ) = β − α (n, p ∈ N = {1, 2, 3, ...} ; m ∈ N0 , p > m), then
f ∈ (α, β, λ, m, δ, Ω)p − N (g).
≤ δ−
Proof. By Theorem 2.1, we see the inequality (4) which implies that f ∈
(α, β, λ, m, δ, Ω)p − N (g).
Since arg(ak+p )−arg(bk+p ) = β−α, if arg(ak+p ) = αk+p , we see arg(bk+p ) =
αk+p − β + α. Therefore,
eiα ak+p − eiβ bk+p = eiα |ak+p | eiαk+p − eiβ |bk+p | ei(αk+p −β+α)
(α, β, λ, δ, m, Ω)p −NEIGHBORHOOD FOR SOME FAMILIES OF ANALYTIC
AND MULTIVALENT FUNCTIONS
231
= (|ak+p | − |bk+p |)ei(αk+p +α)
implies that
iα
e ak+p − eiβ bk+p = ||ak+p | − |bk+p || .
(8)
Using (8) in (4) the proof of the corollary is complete.
Next, we can prove the following theorem.
Theorem 2.4. If f ∈ F(λ, m, Ω) satisfies
∞
X
(k + p − m)Ω (k + p)!(1 + λk(p − m)−1 ) iα
iβ
e
a
−
e
b
k+p
k+p
(p − m)Ω (k + p − m)!
k=n
≤δ−
p
p!
2[1 − cos(α − β)]
(p − m)!
(z ∈ U) .
p
p!
for some −π ≤ α − β ≤ π; p > m and δ > (p−m)!
2[1 − cos(α − β)] then
f ∈ (α, β, λ, m, δ, Ω)p − M (g).
The proof of this theorem is similar with Theorem 2.1.
Corollary 2.5. If f ∈ F(λ, m, Ω) satisfies
∞
X
(k + p − m)Ω (k + p)!(1 + λk(p − m)−1 )
||ak+p | − |bk+p ||
(p − m)Ω (k + p − m)!
k=n
≤δ−
p
p!
2[1 − cos(α − β)]
(p − m)!
(z ∈ U) .
p
p!
for some −π ≤ α − β ≤ π; p > m and δ > (p−m)!
2[1 − cos(α − β)] and
arg(ak+p ) − arg(bk+p ) = β − α, then f ∈ (α, β, λ, m, δ, Ω)p − M (g).
Our next result as follows.
Theorem 2.6. If f ∈ (α, β, λ, m, δ, Ω)p − N (g), 0 ≤ α < β ≤ π; p > m and
arg(eiα ak+p − eiβ bk+p ) = kφ, then
∞
X
(k + p − m)Ω+1 (k + p)!(1 + λk(p − m)−1 ) iα
e ak+p − eiβ bk+p Ω
(p − m) (k + p − m)!
k=n
≤ δ−
p!
(cos α − cos β).
(p − m − 1)!
(α, β, λ, δ, m, Ω)p −NEIGHBORHOOD FOR SOME FAMILIES OF ANALYTIC
AND MULTIVALENT FUNCTIONS
232
Proof. For f ∈ (α, β, λ, m, δ, Ω)p − N (g), we have
=
eiα F 0 (f (m) (z))
eiβ F 0 (g (m) (z)) −
p−m−1
p−m−1
z
z
∞
Ω+1
p!(eiα − eiβ )
X
(k + p − m)
(k + p)!(1 + λk(p − m)−1 ) iα
iβ
k
+
(e
a
−
e
b
)z
k+p
k+p
Ω (k + p − m)!
(p − m − 1)!
(p
−
m)
k=n
∞
p!(eiα − eiβ )
X
(k + p − m)Ω+1 (k + p)!(1 + λk(p − m)−1 ) iα
iβ
ikφ k =
+
(e ak+p − e bk+p )e
z Ω
(p − m − 1)!
(p − m) (k + p − m)!
k=n
< δ.
k
Let us consider z such that arg z = −φ. Then z k = |z| e−ikφ . For such a
point z ∈ U, we see that
=
"
=
eiα F 0 (f (z))
eiβ F 0 (g(z)) −
z p−m−1
z p−m−1 ∞
p!(eiα − eiβ )
X
(k + p − m)Ω+1 (k + p)!(1 + λk(p − m)−1 ) iα
k
iβ
+
a
−
e
b
|z|
e
k+p
k+p
Ω (k + p − m)!
(p − m − 1)!
(p
−
m)
k=n
∞
X
p!(cos α − cos β)
(k + p − m)Ω+1 (k + p)!(1 + λk(p − m)−1 ) iα
k
iβ
e ak+p − e bk+p |z| +
Ω (k + p − m)!
(p
−
m)
(p − m − 1)!
k=n
+
p!(sin α − sin β)
(p − m − 1)!
!2
2 # 12
< δ.
This implies that
∞
X
(k + p − m)Ω+1 (k + p)!(1 + λk(p − m)−1 ) iα
p!(cos α − cos β)
k
iβ
e ak+p − e bk+p |z| +
Ω (k + p − m)!
(p
−
m)
(p − m − 1)!
k=n
!2
2
<δ ,
or
∞
X
(k + p − m)Ω+1 (k + p)!(1 + λk(p − m)−1 ) iα
p!
k
iβ
(cos α−cos β)+
e ak+p − e bk+p |z|
Ω (k + p − m)!
(p − m − 1)!
(p
−
m)
k=n
<δ
for z ∈ U. Letting |z| −→ 1 , we have that
−
∞
X
(k + p − m)Ω+1 (k + p)!(1 + λk(p − m)−1 ) iα
iβ
a
−
e
b
e
k+p
k+p
(p − m)Ω (k + p − m)!
k=n
≤δ−
p!
(cos α − cos β).
(p − m − 1)!
(α, β, λ, δ, m, Ω)p −NEIGHBORHOOD FOR SOME FAMILIES OF ANALYTIC
AND MULTIVALENT FUNCTIONS
233
Remark 2.7. Applying the parametric substitutions listed in (5), Theorem
2.4 and 2.6 would yield a set of known results due to Altuntaş et al. ([9] p.5,
Theorem 4; p.6, Theorem 7).
Theorem 2.8. If f ∈ (α, β, λ, m, δ, Ω)p − M (g), 0 ≤ α < β ≤ π and
arg(eiα ak+p − eiβ bk+p ) = kφ, then
∞
X
(k + p − m)Ω (k + p)!(1 + λk(p − m)−1 ) iα
e ak+p − eiβ bk+p (p − m)Ω (k + p − m)!
k=n
p!
(cos β − cos α).
(p − m − 1)!
The proof of this theorem is similar with Theorem 2.6.
≤δ+
Remark 2.9. Taking λ = α = Ω = m = 0, β = α and p = 1,in Theorem 2.8,
we arrive at the following theorem due to Orhan et al.[1].
Theorem 2.10. If f ∈ (α, δ) − N (g) and arg(an − eiα bn ) = (n − 1)ϕ (n =
2, 3, 4, ...), then
∞
X
n an −eiα bn ≤ δ + cos α − 1.
n=2
We give an application of following lemma due to Miller and Mocanu [2]
(see also, [11]).
Lemma 2.1. Let the function
w(z) = bn z n + bn+1 z n+1 + bn+2 z n+2 + ...
(z ∈ U)
be regular in U with w(z) 6= 0, (n ∈ U). If z0 = r0 eiθ0 (r0 < 1) and |w(z0 )| =
max|z|≤r0 |w(z)| , then z0 w0 (z0 ) = qw(z0 ) where q is real and q ≥ n ≥ 1.
Applying the above lemma, we derive
Theorem 2.11. If f ∈ F(λ, m, Ω) satisfies
iα 0 (m)
p
e F (f
(z))
eiβ F0 (g (m) (z)) p!
−
< δ(p + n − m) −
2[1 − cos(α − β)]
p−m−1
p−m−1
z
z
(p − m − 1)!
p
p!
for some −π ≤ α − β ≤ π; p > m and δ > (p+n−m)(p−m−1)!
2[1 − cos(α − β)],
then
iα
p
e F(f (m) (z))
eiβ F(g (m) (z)) p!
−
<δ+
2[1 − cos(α − β)] (z ∈ U) .
p−m
p−m
z
z
(p − m)!
(α, β, λ, δ, m, Ω)p −NEIGHBORHOOD FOR SOME FAMILIES OF ANALYTIC
AND MULTIVALENT FUNCTIONS
234
Proof. Let us define w(z) by
eiα F(f (m) (z)) eiβ F(g (m) (z))
p!
−
=
(eiα −eiβ ) + δw(z).
z p−m
z p−m
(p − m)!
(9)
Then w(z) is analytic in U and w(0) = 0. By logarithmic differentiation, we
get from (9) that
eiα F0 (f (m) (z)) − eiβ F0 (g (m) (z)) p − m
−
=
z
eiα F(f (m) (z)) − eiβ F(g (m) (z))
δw0(z)
p!
iβ
iα
(p−m)! (e −e )
+ δw(z)
.
Since
p−m
eiα F0 (f (m) (z)) − eiβ F0 (g (m) (z))
=
+
p!
z
z p−m (p−m)!
(eiα −eiβ ) + δw(z)
δw0(z)
p!
iβ
iα
(p−m)! (e −e )
+ δw(z)
,
we see that
eiα F0 (f (m) (z)) eiβ F0 (g (m) (z))
zw0(z)
p!
iα
iβ
(e −e )+δw(z) p − m +
−
=
.
z p−m−1
z p−m−1
(p − m − 1)!
w(z)
This implies that
eiα F 0 (f (m) (z))
eiβ F 0 (g (m) (z)) zw0(z) p!
iα
iβ
−
(e −e ) + δw(z) p − m +
.
= z p−m−1
z p−m−1
(p − m − 1)!
w(z) We claim that
iα 0 (m)
p
e F (f
eiβ F0 (g (m) (z)) (z))
p!
−
< δ(p − m + n) − (p − m − 1)! 2[1 − cos(α − β)]
z p−m−1
z p−m−1
in U.
Otherwise, there exists a point z0 ∈ U such that z0 w0 (z0 ) = qw(z0 ) (by
Miller and Mocanu’s Lemma) where w(z0 ) = eiθ and q ≥ n ≥ 1.
Therefore, we obtain that
iα 0 (m)
e F (f
eiβ F0 (g (m) (z)) (z))
−
z0p−m−1
z0p−m−1
p!
iα
iβ
iθ
(e
−e
)
+
δe
(p
−
m
+
q)
(p − m − 1)!
p!
≥δ (p + q − m) − (eiα −eiβ )
(p − m − 1)!
p
p!
≥δ (p + n − m) −
2[1 − cos(α − β)].
(p − m − 1)!
=
This contradicts our condition in Theorem 2.11.
(α, β, λ, δ, m, Ω)p −NEIGHBORHOOD FOR SOME FAMILIES OF ANALYTIC
AND MULTIVALENT FUNCTIONS
235
Hence, there is no z0 ∈ U such that |w(z0 )| = 1. This means that |w(z)| < 1
for all z ∈ U.
Thus, have that
iα
e F(f (m) (z)) eiβ F(g (m) (z)) p!
iα
iβ
= −
(e
−e
)
+
δw(z)
p−m
p−m
z
z
(p − m)!
iα iβ p!
e −e + δ |w(z)|
≤
(p − m)!
p
p!
< δ+
2[1 − cos(α − β)].
(p − m)!
Upon setting m = 0, α = ϕ, ℘ = F and β = α in Theorem 2.11, we have
the following corollary given by Sağsöz et al.[6].
Corollary 2.12. If f ∈ ℘(Ω, λ) satisfies
iα 0
p
e ℘ (f (z)) eiβ ℘0 (g(z)) < δ(p + n) − p 2[1 − cos(ϕ − α)]
−
p−1
p−1
z
z
p
p
for some −π ≤ α − β ≤ π;and δ > (p+n)
2[1 − cos(α − β)], then
iα
p
e ℘(f (z)) eiβ ℘(g(z)) < δ + 2[1 − cos(ϕ − α)]
−
p
p
z
z
(z ∈ U) .
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Halit ORHAN,
Department of Mathematics,
Faculty of Science,
Ataturk University,
25240, Erzurum, Turkey.
Email: orhanhalit607@gmail.com