ON THE SUBCLASS OF SALAGEAN-TYPE
HARMONIC UNIVALENT FUNCTIONS
SİBEL YALÇIN, METİN ÖZTÜRK AND
MÜMİN YAMANKARADENİZ
Uludaǧ Üniversitesi Fen Edebiyat Fakültesi
Matematik Bölümü, 16059 Bursa, Turkey
EMail: skarpuz@uludag.edu.tr
Salagean-Type Harmonic
Univalent Functions
Sibel Yalçin, Metin Öztürk
and Mümin Yamankaradeniz
vol. 8, iss. 2, art. 54, 2007
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Contents
Received:
10 October, 2006
Accepted:
22 April, 2007
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Communicated by:
H.M. Srivastava
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2000 AMS Sub. Class.:
30C45, 30C50, 30C55.
Page 1 of 17
Key words:
Harmonic, Meromorphic, Starlike, Symmetric, Conjugate, Convex functions.
Abstract:
Using the Salagean derivative, we introduce and study a class of Goodman- Ronning type harmonic univalent functions. We obtain coefficient conditions, extreme points, distortion bounds, convolution conditions, and convex combination
for the above class of harmonic functions.
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1
Introduction
3
2
Main Results
5
Salagean-Type Harmonic
Univalent Functions
Sibel Yalçin, Metin Öztürk
and Mümin Yamankaradeniz
vol. 8, iss. 2, art. 54, 2007
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1.
Introduction
A continuous complex valued function f = u + iv defined in a simply connected
complex domain D is said to be harmonic in D if both u and v are real harmonic in
D. In any simply connected domain D we can write f = h + ḡ, where h and g are
analytic in D. A necessary and sufficient condition for f to be locally univalent and
sense preserving in D is that |h0 (z)| > |g 0 (z)|, z ∈ D.
Denote by SH the class of functions f = h + ḡ that are harmonic univalent and
sense preserving in the unit disk U = {z : |z| < 1} for which f (0) = fz (0) − 1 = 0.
Then for f = h + ḡ ∈ SH we may express the analytic functions h and g as
∞
∞
X
X
n
(1.1)
h(z) = z +
an z ,
g(z) =
bn z n , |b1 | < 1.
n=2
n=1
In 1984 Clunie and Sheil-Small [1] investigated the class SH as well as its geometric
subclasses and obtained some coefficient bounds. Since then, there have been several
related papers on SH and its subclasses. Jahangiri et al. [3] make use of the Alexander integral transforms of certain analytic functions (which are starlike or convex
of positive order) with a view to investigating the construction of sense-preserving,
univalent, and close to convex harmonic functions.
Definition 1.1. Recently, Rosy et al. [4], defined the subclass GH (γ) ⊂ SH consisting of harmonic univalent functions f (z) satisfying the following condition
0
iα
iα zf (z)
−e
≥ γ,
0 ≤ γ < 1, α ∈ R.
(1.2)
Re (1 + e )
f (z)
They proved that if f = h + ḡ is given by (1.1) and if
∞ X
2n − 1 − γ
2n + 1 + γ
|an | +
|bn | ≤ 2,
(1.3)
1−γ
1−γ
n=1
0 ≤ γ < 1,
Salagean-Type Harmonic
Univalent Functions
Sibel Yalçin, Metin Öztürk
and Mümin Yamankaradeniz
vol. 8, iss. 2, art. 54, 2007
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then f is a Goodman-Ronning type harmonic univalent function in U . This condition
is proved to be also necessary if h and g are of the form
∞
∞
X
X
(1.4)
h(z) = z −
|an |z n ,
g(z) =
|bn |z n .
n=2
n=1
Jahangiri et al. [2] has introduced the modified Salagean operator of harmonic
univalent function f as
(1.5)
Dk f (z) = Dk h(z) + (−1)k Dk g(z),
k ∈ N,
vol. 8, iss. 2, art. 54, 2007
where
Dk h(z) = z +
∞
X
n k an z n
and
Dk g(z) =
n=2
∞
X
n k bn z n .
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n=1
We let RSH (k, γ) denote the family of harmonic functions f of the form (1.1)
such that
k+1
f (z)
iα D
iα
(1.6)
Re (1 + e ) k
−e
≥ γ, 0 ≤ γ < 1, α ∈ R,
D f (z)
where Dk f is defined by (1.5).
Also, we let the subclass RS H (k, γ) consist of harmonic functions fk = h + ḡk
in RSH (k, γ) so that h and gk are of the form
(1.7)
Salagean-Type Harmonic
Univalent Functions
Sibel Yalçin, Metin Öztürk
and Mümin Yamankaradeniz
h(z) = z −
∞
X
n=2
n
|an |z ,
k
gk (z) = (−1)
∞
X
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n
|bn |z .
n=1
In this paper, the coefficient condition given in [4] for the class GH (γ) is extended to the class RSH (k, γ) of the form (1.6). Furthermore, we determine extreme
points, a distortion theorem, convolution conditions and convex combinations for the
functions in RS H (k, γ).
Close
2.
Main Results
In our first theorem, we introduce a sufficient coefficient bound for harmonic functions in RSH (k, γ).
Theorem 2.1. Let f = h + ḡ be given by (1.1). If
∞
X
(2.1)
nk [(2n − 1 − γ)|an | + (2n + 1 + γ)|bn |] ≤ 2(1 − γ),
n=1
where a1 = 1 and 0 ≤ γ < 1, then f is sense preserving, harmonic univalent in U ,
and f ∈ RSH (k, γ).
Proof. If the inequality (2.1) holds for the coefficients of f = h + ḡ, then by (1.3),
f is sense preserving and harmonic univalent in U. According to the condition (1.5)
we only need to show that if (2.1) holds then
k+1
f (z)
iα D
iα
Re (1 + e ) k
−e
D f (z)
(
)
k+1
k k+1
D
h(z)
−
(−1)
D
g(z)
= Re (1 + eiα )
− eiα ≥ γ,
k
k
k
D h(z) + (−1) D g(z)
where 0 ≤ γ < 1.
Using the fact that Re w ≥ γ if and only if |1 − γ + w| ≥ |1 + γ − w|, it suffices
to show that
(2.2) (1 − γ)Dk f (z) + (1 + eiα )Dk+1 f (z) − eiα Dk f (z)
− (1 + γ)Dk f (z) − (1 + eiα )Dk+1 f (z) + eiα Dk f (z) ≥ 0.
Substituting for Dk f and Dk+1 f in (2.2) yields
(1 − γ)Dk f (z) + (1 + eiα )Dk+1 f (z) − eiα Dk f (z)
Salagean-Type Harmonic
Univalent Functions
Sibel Yalçin, Metin Öztürk
and Mümin Yamankaradeniz
vol. 8, iss. 2, art. 54, 2007
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− (1 + γ)Dk f (z) − (1 + eiα )Dk+1 f (z) + eiα Dk f (z)
∞
X
= (2 − γ)z +
(1 − γ − eiα + n + neiα )nk an z n
n=2
∞
X
−(−1)k
(n + neiα − 1 + γ + eiα )bn z n n=1
∞
X
− γz −
(n + neiα − 1 − γ − eiα )nk an z n
n=2
∞
X
k
iα
iα
n
+(−1)
(n + ne + 1 + γ + e )bn z ≥ (2 − γ)|z| −
n=1
∞
X
k
∞
X
n=2
n=1
n (2n − γ)|an ||z|n −
− γ|z| −
∞
X
= 2(1 − γ)|z| − 2
(
∞
X
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nk (2n + γ + 2)|bn ||z|n
n=1
∞
X
nk (2n − γ − 1)|an ||z|n − 2
n=2
∞
X
vol. 8, iss. 2, art. 54, 2007
nk (2n + γ)|bn ||z|n
nk (2n − γ − 2)|an ||z|n −
n=2
∞
X
Salagean-Type Harmonic
Univalent Functions
Sibel Yalçin, Metin Öztürk
and Mümin Yamankaradeniz
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Page 6 of 17
nk (2n + γ + 1)|bn ||z|n
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n=1
∞
X
nk (2n − γ − 1)
nk (2n + γ + 1)
n−1
= 2(1 − γ)|z| 1 −
|an ||z|
−
|bn ||z|n−1
1
−
γ
1
−
γ
n=1
n=2
!)
(
∞
∞
X nk (2n + γ + 1)
X nk (2n − γ − 1)
|an | +
|bn |
.
> 2(1 − γ) 1 −
1−γ
1−γ
n=1
n=2
This last expression is non-negative by (2.1), and so the proof is complete.
)
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The harmonic function
(2.3)
f (z) = z +
∞
X
n=2
where
∞
X
1−γ
1−γ
n
xn z +
ȳn z̄ n ,
k
k
n (2n − γ − 1)
n
(2n
+
γ
+
1)
n=1
∞
X
n=2
|xn | +
∞
X
|yn | = 1
n=1
shows that the coefficient bound given by (2.1) is sharp.
The functions of the form (2.3) are in RSH (k, γ) because
∞
∞ k
∞
X
X
X
n (2n − γ − 1)
nk (2n + γ + 1)
|an | +
|bn | = 1 +
|xn | +
|yn | = 2.
1−γ
1−γ
n=2
n=1
n=1
In the following theorem, it is shown that the condition (2.1) is also necessary for
functions fk = h + ḡk , where h and gk are of the form (1.7).
Theorem 2.2. Let fk = h + ḡk be given by (1.7). Then fk ∈ RS H (k, γ) if and only
if
(2.4)
∞
X
nk [(2n − γ − 1)|an | + (2n + γ + 1)|bn |] ≤ 2(1 − γ).
n=1
Proof. Since RS H (k, γ) ⊂ RSH (k, γ), we only need to prove the “only if” part
of the theorem. To this end, for functions fk of the form (1.7), we notice that the
condition (1.6) is equivalent to
P
(1 − γ)z − ∞
nk [n − γ + (n − 1)eiα ]|an |z n
P∞ kn=2 n
P
Re
k
n
z − n=2 n |an |z + (−1)2k ∞
n=1 n |bn |z̄
Salagean-Type Harmonic
Univalent Functions
Sibel Yalçin, Metin Öztürk
and Mümin Yamankaradeniz
vol. 8, iss. 2, art. 54, 2007
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P
(−1)2k ∞
nk [n + γ + (n + 1)eiα ]|bn |z̄ n
n=1
P∞ k
P
−
k |a |z n + (−1)2k
n
z− ∞
n
n
n=1 n |bn |z̄
n=2
P
1−γ− ∞
nk [n − γ + (n − 1)eiα ]|an |z n−1
P∞ k n=2 n−1 z̄
P
= Re
k
n−1
1 − n=2 n |an |z
+ z (−1)2k ∞
n=1 n |bn |z̄
P
z̄
k
iα
n−1 (−1)2k ∞
n=1 n [n + γ + (n + 1)e ]|bn |z̄
z P
P∞ k
−
k
n−1 + z̄ (−1)2k
n−1
1− ∞
n=2 n |an |z
n=1 n |bn |z̄
z
≥ 0.
The above condition must hold for all values of z, |z| = r < 1. Upon choosing the
values of z on the positive real axis, where 0 ≤ z = r < 1, we must have
P
P
k
n−1
1−γ− ∞
nk (n − γ)|an |rn−1 − ∞
n=2P
n=1 n (n + γ)|bn |r
P
Re
∞
k
n−1 +
k
n−1
1− ∞
n=2 n |an |r
n=1 n |bn |r
P∞ k
P
∞
− 1)|an |rn−1 + n=1 nk (n + 1)|bn |rn−1
iα
n=2 n (n
P
P∞ k
−e
≥ 0.
k
n−1 +
n−1
1− ∞
n=2 n |an |r
n=1 n |bn |r
Since Re(−eiα ) ≥ −|eiα | = −1, the above inequality reduces to
P
P
k
k
n−1
1−γ − ∞
− γ − 1)|an |rn−1 − ∞
n=2 n (2n
n=1 n (2n + γ + 1)|bn |r
P∞ k
P
(2.5)
≥ 0.
k
n−1
1 − n=2 n |an |rn−1 + ∞
n=1 n |bn |r
If the condition (2.4) does not hold then the numerator in (2.5) is negative for r
sufficiently close to 1. Thus there exists a z0 = r0 in (0, 1) for which the quotient in
(2.5) is negative. This contradicts the condition for f ∈ RS H (k, γ) and hence the
result.
Next we determine a representation theorem for functions in RS H (k, γ).
Salagean-Type Harmonic
Univalent Functions
Sibel Yalçin, Metin Öztürk
and Mümin Yamankaradeniz
vol. 8, iss. 2, art. 54, 2007
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Theorem 2.3. Let fk be given by (1.7). Then fk ∈ RS H (k, γ) if and only if
fk (z) =
(2.6)
∞
X
(Xn hn (z) + Yn gkn (z))
n=1
where h1 (z) = z,
1−γ
z n (n = 2, 3, . . . ),
hn (z) = z − k
n (2n − γ − 1)
1−γ
z̄ n (n = 1, 2, . . . ),
gkn (z) = z + (−1)k k
n (2n + γ + 1)
∞
X
(Xn + Yn ) = 1,
Salagean-Type Harmonic
Univalent Functions
Sibel Yalçin, Metin Öztürk
and Mümin Yamankaradeniz
vol. 8, iss. 2, art. 54, 2007
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n=1
Contents
Xn ≥ 0, Yn ≥ 0. In particular, the extreme points of RS H (k, γ) are {hn } and
{gkn }.
Proof. For functions fk of the form (2.6) we have
fk (z) =
∞
X
=
(Xn + Yn )z −
n=1
n=2
+ (−1)k
∞
X
n=1
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(Xn hn (z) + Yn gkn (z))
∞
X
II
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n=1
∞
X
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1−γ
Xn z n
− γ − 1)
nk (2n
1−γ
Yn z̄ n .
nk (2n + γ + 1)
Close
Then
∞
X
nk (2n − γ − 1)
n=2
1−γ
|an | +
∞
X
nk (2n + γ + 1)
n=1
1−γ
|bn | =
∞
X
Xn +
n=2
∞
X
Yn
n=1
= 1 − X1 ≤ 1
and so fk ∈ RS H (k, γ).
Conversely, suppose that fk ∈ RS H (k, γ). Setting
nk (2n − γ − 1)
an ,
1−γ
nk (2n + γ + 1)
bn ,
Yn =
1−γ
Xn =
where
P∞
n=1 (Xn
(n = 2, 3, . . . ),
(n = 1, 2, . . . ),
vol. 8, iss. 2, art. 54, 2007
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+ Yn ) = 1, we obtain
fk (z) =
∞
X
Contents
(Xn hn (z) + Yn gkn (z))
n=1
as required.
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The following theorem gives the distortion bounds for functions in RS H (k, γ)
which yields a covering result for this class.
Theorem 2.4. Let fk ∈ RS H (k, γ). Then for |z| = r < 1 we have
1 1−γ 3+γ
|fk (z)| ≤ (1 + |b1 |)r + k
−
|b1 | r2
2
3−γ 3−γ
and
Salagean-Type Harmonic
Univalent Functions
Sibel Yalçin, Metin Öztürk
and Mümin Yamankaradeniz
1
|fk (z)| ≥ (1 − |b1 |)r − k
2
1−γ 3+γ
−
|b1 | r2 .
3−γ 3−γ
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Proof. We only prove the right hand inequality. The proof for the left hand inequality
is similar and will be omitted. Let fk ∈ RS H (k, γ). Taking the absolute value of fk
we obtain
|fk (z)| ≤ (1 + |b1 |)r +
∞
X
(|an | + |bn |)rn
n=2
≤ (1 + |b1 |)r +
∞
X
(|an | + |bn |)r
2
n=2
∞
≤ (1 + |b1 |)r +
≤ (1 + |b1 |)r +
≤ (1 + |b1 |)r +
≤ (1 + |b1 |)r +
1 − γ X 2k (3 − γ)
(|an | + |bn |)r2
k
2 (3 − γ) n=2 1 − γ
∞ 1 − γ X nk (2n − γ − 1)
nk (2n + γ + 1)
|an | +
|bn | r2
2k (3 − γ) n=2
1−γ
1−γ
3+γ
1−γ
1−
|b1 | r2
2k (3 − γ)
1−γ
1 1−γ 3+γ
−
|b1 | r2 .
2k 3 − γ 3 − γ
Salagean-Type Harmonic
Univalent Functions
Sibel Yalçin, Metin Öztürk
and Mümin Yamankaradeniz
vol. 8, iss. 2, art. 54, 2007
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The following covering result follows from the left hand inequality in Theorem
2.4.
Corollary 2.5. Let fk of the form (1.7) be so that fk ∈ RS H (k, γ). Then
3.2k − 1 − (2k − 1)γ 3(2k − 1) − (2k + 1)γ
w : |w| <
−
|b1 | ⊂ fk (U ).
2k (3 − γ)
2k (3 − γ)
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For our next theorem, we need to define the convolution of two harmonic functions. For harmonic functions of the form
∞
X
fk (z) = z −
n
k
|an |z + (−1)
n=2
and
Fk (z) = z −
∞
X
∞
X
|bn |z̄ n
n=1
n
k
|An |z + (−1)
n=2
∞
X
|Bn |z̄
n
n=1
vol. 8, iss. 2, art. 54, 2007
we define the convolution of fk and Fk as
(2.7)
(fk ∗ Fk )(z) = fk (z) ∗ Fk (z)
∞
∞
X
X
n
k
|bn ||Bn |z̄ n .
=z−
|an ||An |z + (−1)
n=1
n=2
Theorem 2.6. For 0 ≤ β ≤ γ < 1, let fk ∈ RS H (k, γ) and Fk ∈ RS H (k, β). Then
the convolution
fk ∗ Fk ∈ RS H (k, γ) ⊂ RS H (k, β).
Proof. Then the convolution fk ∗ Fk is given by (2.7). We wish to show that the
coefficients of fk ∗ Fk satisfy the required condition given in Theorem 2.2. For
Fk ∈ RS H (k, β) we note that |An | ≤ 1 and |Bn | ≤ 1. Now, for the convolution
function fk ∗ Fk , we obtain
∞
X
nk (2n − β − 1)
n=2
1−β
|an ||An | +
∞
X
nk (2n + β + 1)
n=1
Salagean-Type Harmonic
Univalent Functions
Sibel Yalçin, Metin Öztürk
and Mümin Yamankaradeniz
1−β
|bn ||Bn |
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≤
≤
∞
X
nk (2n − β − 1)
n=2
∞
X
n=2
1−β
|an | +
∞
X
nk (2n + β + 1)
1−β
n=1
∞
X
|bn |
nk (2n − γ − 1)
nk (2n + γ + 1)
|an | +
|bn | ≤ 1
1−γ
1−γ
n=1
since 0 ≤ β ≤ γ < 1 and fk ∈ RS H (k, γ). Therefore fk ∗ Fk ∈ RS H (k, γ) ⊂
RS H (k, β).
Next we discuss the convex combinations of the class RS H (k, γ).
Salagean-Type Harmonic
Univalent Functions
Sibel Yalçin, Metin Öztürk
and Mümin Yamankaradeniz
vol. 8, iss. 2, art. 54, 2007
Theorem 2.7. The family RS H (k, γ) is closed under convex combination.
Title Page
Proof. For i = 1, 2, . . . , suppose that fki ∈ RS H (k, γ), where
fki (z) = z −
∞
X
n
k
|ain |z + (−1)
n=2
∞
X
Contents
n
|bin |z̄ .
n=1
Then by Theorem 2.2,
∞
X
nk (2n − γ − 1)
(2.8)
n=2
For
1−γ
|ain | +
∞
X
nk (2n + γ + 1)
n=1
1−γ
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|bin | ≤ 1.
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P∞
= 1, 0 ≤ ti ≤ 1, the convex combination of fki may be written as
!
!
∞
∞
∞
∞
∞
X
X
X
X
X
ti fki (z) = z −
ti |ain | z n + (−1)k
ti |bin | z̄ n .
i=1 ti
i=1
n=2
i=1
n=1
i=1
Close
Then by (2.8),
∞
X
nk (2n − γ − 1)
n=2
∞
X
1−γ
=
≤
!
ti |ain |
i=1
∞
X
i=1
∞
X
+
∞
X
nk (2n + γ + 1)
1−γ
n=1
ti
∞
X
nk (2n − γ − 1)
n=2
!
ti |bin |
i=1
|ain | +
1−γ
∞
X
∞
X
nk (2n + γ + 1)
n=1
1−γ
!
|bin |
ti = 1
vol. 8, iss. 2, art. 54, 2007
i=1
and therefore
Salagean-Type Harmonic
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Sibel Yalçin, Metin Öztürk
and Mümin Yamankaradeniz
∞
X
Title Page
ti fki (z) ∈ RS H (k, γ).
Contents
i=1
Following Ruscheweyh [5], we call the δ−neighborhood of f the set
(
∞
∞
X
X
n
k
Nδ (fk ) = Fk : Fk (z) = z −
|An |z + (−1)
|Bn |z̄ n and
n=2
∞
X
n=2
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.
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Theorem 2.8. Assume that
fk (z) = z −
J
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)
n=2
∞
X
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n=1
n(|an − An | + |bn − Bn |) + |b1 − B1 | ≤ δ
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n
k
|an |z + (−1)
∞
X
n=1
|bn |z̄ n
belongs to RS H (k, γ). If
1
1
1
δ≤
(1 − γ) 1 − k − 3 + 2γ − k (3 + γ) |b1 | ,
3
2
2
then Nδ (fk ) ⊂ RS H (0, γ).
Proof. Let
Fk (z) = z −
∞
X
|An |z n + (−1)k
n=2
∞
X
|Bn |z̄ n
n=1
Salagean-Type Harmonic
Univalent Functions
Sibel Yalçin, Metin Öztürk
and Mümin Yamankaradeniz
vol. 8, iss. 2, art. 54, 2007
belong to Nδ (fk ). We have
∞
X
2n − γ − 1
n=2
≤
1−γ
|An | +
n=1
∞
X
2n − γ − 1
n=2
1−γ
+
3
1−γ
∞
X
n=2
1−γ
|an − An | +
1−γ
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|Bn |
∞
X
2n + γ + 1
1−γ
n=1
∞
X
2n − γ − 1
n=2
≤
∞
X
2n + γ + 1
|an | +
Contents
|bn − Bn |
∞
X
2n + γ + 1
n=1
n(|an − An | + |bn − Bn |) +
1−γ
|bn |
3
γ
|b1 | +
|b1 |
1−γ
1−γ
∞
∞
1 X nk (2n − γ − 1)
1 X nk (2n − γ − 1)
3+γ
+ k
|an | + k
|bn | +
|b1 |
2 n=2
1−γ
2 n=2
1−γ
1−γ
3δ
γ
1
3+γ
3+γ
≤
+
|b1 | + k 1 −
|b1 | +
|b1 | ≤ 1.
1−γ 1−γ
2
1−γ
1−γ
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Thus Fk ∈ RS H (0, γ) for
1
1
1
δ≤
(1 − γ) 1 − k − 3 + 2γ − k (3 + γ) |b1 | .
3
2
2
Salagean-Type Harmonic
Univalent Functions
Sibel Yalçin, Metin Öztürk
and Mümin Yamankaradeniz
vol. 8, iss. 2, art. 54, 2007
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References
[1] J. CLUNIE AND T. SHEIL-SMALL, Harmonic univalent functions, Ann. Acad.
Sci. Fenn. Ser. A I Math., 9 (1984), 3–25.
[2] J.M. JAHANGIRI, G. MURUGUSUNDARAMOORTHY AND K. VIJAYA,
Salagean-type harmonic univalent functions, South. J. Pure and Appl. Math.,
Issue 2 (2002), 77–82.
[3] J.M. JAHANGIRI, CHAN KIM YONG AND H.M. SRIVASTAVA, Construction
of a certain class of harmonic close to convex functions associated with the
Alexander integral transform, Integral Transform. Spec. Funct., 14(3) (2003),
237–242.
[4] T. ROSY, B. ADOLPH STEPHEN AND K.G. SUBRAMANIAN, Goodman
Ronning type harmonic univalent functions, Kyungpook Math. J., 41 (2001),
45–54.
[5] St. RUSCHEWEYH, Neighborhoods of univalent functions, Proc. Amer. Math.
Soc., 81 (1981), 521–527.
Salagean-Type Harmonic
Univalent Functions
Sibel Yalçin, Metin Öztürk
and Mümin Yamankaradeniz
vol. 8, iss. 2, art. 54, 2007
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