Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 496731, 13 pages
doi:10.1155/2012/496731
Research Article
On a Subclass of Harmonic Convex Functions of
Complex Order
N. Magesh1 and S. Mayilvaganan2
1
2
Department of Mathematics, Government Arts College, Krishnagiri 635 001, India
Department of Mathematics, Adhiyamaan College of Engineering (Autonomous),
Hosur 635 109, India
Correspondence should be addressed to N. Magesh, nmagi 2000@yahoo.co.in
Received 21 December 2011; Accepted 30 March 2012
Academic Editor: Aloys Krieg
Copyright q 2012 N. Magesh and S. Mayilvaganan. This is an open access article distributed
under the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
We introduce and study a subclass of harmonic convex functions of complex order. Coefficient
bounds, extreme points, distortion bounds, convolution conditions, and convex combination are
determined for functions in this class. Further, we obtain the closure property of this class under
integral operator.
1. Introduction
A continuous function f uiv is a complex-valued harmonic function in a complex domain
Ω if both u and v are real and harmonic in Ω. In any simply connected domain D ⊂ Ω, we can
write f hg, where h and g are analytic in D. We call h the analytic part and g the coanalytic
part of f. A necessary and sufficient condition for f to be locally univalent and orientation
preserving in D is that |h z| > |g z| in D see 1.
Denote by SH the family of functions f h g, which are harmonic, univalent, and
orientation preserving in the open unit disc U {z : |z| < 1} so that f is normalized by f0 h0 fz 0 − 1 0. Thus, for f h g ∈ SH , the functions h and g analytic in U can be
expressed in the following forms:
hz z ∞
k2
ak zk ,
gz ∞
k1
bk zk
|b1 | < 1,
1.1
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International Journal of Mathematics and Mathematical Sciences
and fz is then given by
fz z ∞
∞
ak zk bk zk
k2
|b1 | < 1.
1.2
k1
We note that the family SH of orientation preserving, normalized harmonic univalent
functions reduces to the well-known class S of normalized univalent functions if the coanalytic part of f is identically zero g ≡ 0.
Also, we denote by T SH the subfamily of SH consisting of harmonic functions of the
form f h g such that h and g are of the form
hz z −
∞
|ak |zk ,
gz k2
∞
|bk |zk .
1.3
k1
In 1, Clunie and Sheil-Small investigated the class SH as well as its geometric subclasses and its properties. Since then, there have been several studies related to the class SH
and its subclasses. Following Clunie and Sheil-Small 1, Frasin 2, Jahangiri et al. 3–6,
Silverman 7, Silverman and Silvia 8, Yalçin and Öztürk 9, and others have investigated
various subclasses of SH and its properties. In particular, Avcı and Złotkiewicz 10 proved
that the coefficient condition
∞
k2 |ak | |bk | ≤ 1 b1 0
1.4
k2
is sufficient for functions f h g ∈ SH to be harmonic convex. Also, Silverman 7 studied
that this coefficient condition is also necessary if ak and bk k 2, 3, . . . in 1.2 are negative.
Further, Jahangiri 3 showed that if f h g is given by 1.2 and if
∞
k k−γ
k1
1−γ
k kγ
|ak | |bk | ≤ 2
1−γ
0 ≤ γ < 1, a1 1 ,
1.5
then, f is harmonic, univalent, and convex of order γ in U. This condition is proved to be
also necessary if h and g are of the form 1.3. Furthermore, Yalçin and Öztürk 11 have
∗
γ of harmonic starlike functions of complex order based on a corresconsidered a class T SH
ponding study of Nasr and Aouf 12 for the analytic case.
Motivated by the earlier works given in the literature 9, 11 now we define the class
of harmonic convex functions of complex order as follows.
International Journal of Mathematics and Mathematical Sciences
3
Definition 1.1. For 0 ≤ γ < 1, 0 ≤ λ ≤ γ/1 γ or λ ≥ 1/1 γ, and b ∈ C \ {0} with |b| < 1,
let SCH b, γ, λ denote the family of harmonic functions f ∈ SH of the form 1.2 that satisfy
the following condition:
⎧
⎪
⎨
⎛
3 3 g z 2λ 1z2 h z
h
z
λ
z
−
z
1⎜
1 ⎝ ⎪
b
⎩
λ z2 h z z2 g z zh z 2λ − 1zg z
⎞⎫
⎪
⎬
zh z 1 −
1 −
⎟
− 1⎠
≥ γ,
⎪
⎭
λ z2 h z z2 g z zh z 2λ − 1zg z
4λz2 g z
2λzg z
1.6
for z ∈ U. Further, we define the subclass T SCH b, γ, λ of SCH b, γ, λ consisting of functions
f h g of the form 1.3.
We observe that for b 1 the class SCH 1, γ, λ SCH γ, λ was introduced and
studied by Yalçin and Öztürk 9, the class SCH 1, γ, 0 SCH γ is given in 3, 4, and the
class SCH 1, 0, 0 SCH is studied in 10.
In this paper, we investigate coefficient conditions, extreme points, and distortion
bounds for the function class T SCH b, γ, λ. We also examine their convolution and convex
combination properties and the closure property of this class under integral operator. We
remark that the results obtained for these general families can be viewed as extensions and
generalizations for various subclasses of SH as listed previously in this section.
2. Coefficient Inequalities
Our first theorem gives a sufficient condition for functions in SCH b, γ, λ.
Theorem 2.1. Let f h g be so that h and g are given by 1.2. If
∞ kkλ − λ 1
∞ kkλ λ − 1
k − 1 |b| 1 − γ
k 1 − |b| 1 − γ
|ak | |bk | ≤ 2,
1 − γ |b|
1 − γ |b|
k1
k1
2.1
where a1 1, 0 ≤ γ < 1, and b |b| < 1 is a nonzero complex number, 0 ≤ λ ≤ γ/1 γ or
λ ≥ 1/1 γ. Then, f ∈ SCH b, γ, λ and f is sense preserving, univalent, and harmonic in U.
4
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Proof. We show that f ∈ SCH b, γ, λ. We only need to show that if 2.1 holds, then condition
1.6 is satisfied. In view of 1.2, condition 1.6 takes the form
⎛
k ∞ ⎜ 1 − γ k2 kkλ − λ 1 k − 1 b 1 − γ /b |ak | z /z
⎝
k ∞
k
z
1 ∞
kkλ
−
λ
1|a
/z
kkλ
λ
−
1|b
z
/z
|
|
k
k
k2
k1
⎞
k z
/z
kkλ
λ
−
1
1
−
b
1
−
γ
/b
k
|
|b
k
k1
⎟
−
⎠
k ∞
∞
k
1 k2 kkλ − λ 1|ak | z /z k1 kkλ λ − 1|bk | z /z
∞ 2.2
1 Az
.
1 Bz
Setting
1 Az 1 ωz
,
1 Bz 1 − ωz
2.3
we will have 1 Az/1 Bz > 0 if |ωz| < 1,
ωz Az − Bz
2 Az Bz
k − 1 b 1 − γ
− γ kkλ − λ 1
− 1 |ak |zk−1
b
k2
− ∞
k 1 − b 1 − γ /b 1 |bk | zk /z
k1 kkλ λ − 1
2−γ ∞
k − 1 b 1 − γ /b 1 |ak |zk−1
k2 kkλ − λ 1
∞
k 1 − b 1 − γ
zk
− 1 |bk | ,
− kkλ λ − 1
b
z
k1
∞
2.4
so that
∞
γ ∞
k2 kkλ − λ 1A/|b| − 1|ak | k1 kkλ λ − 1B/|b| 1|bk |
∞
,
|ωz| <
2 − γ − k2 kkλ − λ 1A/|b| 1|ak | − ∞
k1 kkλ λ − 1B/|b| − 1|bk |
2.5
where A denotes k − 1 |b|1 − γ and B denotes k 1 − |b|1 − γ.This last expression is
bounded above by 1 if and only if
∞ kkλ − λ 1
∞ kkλ λ − 1
k − 1 |b| 1 − γ
k 1 − |b| 1 − γ
|ak | |bk | ≤ 1 − γ .
|b|
|b|
k2
k1
2.6
International Journal of Mathematics and Mathematical Sciences
5
Or, equivalently
∞ kkλ − λ 1
∞ kkλ λ − 1
k − 1 |b| 1 − γ
k 1 − |b| 1 − γ
|ak | |bk | ≤ 2.
1 − γ |b|
1 − γ |b|
k1
k1
2.7
In order to show that f is univalent in U, we show that fz1 /
fz2 whenever z1 /
z2 .
Since U is simply connected and convex, we have zt 1 − tz1 tz2 ∈ U, where 0 ≤ t ≤ 1,
and if z1 , z2 ∈ U so that z1 /
z2 . Then, we write
fz2 − fz1 1
z2 − z1 h zt z2 − z1 g zt dt.
2.8
0
Dividing the above equation by z2 − z1 /
0 and taking the real part, we obtain
fz2 − fz1 z2 − z1
1
h zt 0
1
>
!
z2 − z1 g zt dt
z2 − z1
"
"
h zt − "g zt" dt.
2.9
0
On the other hand, for |b| < 1, λ ≥ 1/1 γ or 0 ≤ λ ≤ γ/1 γ, we have
∞
"
"
h z − "g z" ≥ h z − k|bk |
k1
≥1−
∞
k2
k|ak | −
∞
k|bk |
k1
∞ kkλ − λ 1
k − 1 |b| 1 − γ
|ak |
1 − γ |b|
k2
∞ kkλ λ − 1
k 1 − |b| 1 − γ
−
|bk |
1 − γ |b|
k1
≥1−
≥ 0, using 2.1.
2.10
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International Journal of Mathematics and Mathematical Sciences
This along with inequality 2.9 leads to the univalence of f. Note that f is sense preserving
in U, for 0 ≤ λ ≤ γ/1 γ or λ ≥ 1/1 γ. This is because
∞
∞
" "
"h z" ≥ 1 − k|ak | |z|k−1 > 1 − k|ak |
k2
k2
∞ kkλ − λ 1
k − 1 |b| 1 − γ
≥1−
|ak |
1 − γ |b|
k2
∞ kkλ λ − 1
k 1 − |b| 1 − γ
≥
|bk |
1 − γ |b|
k1
∞ kkλ λ − 1
k 1 − |b| 1 − γ
>
|bk | |z|k−1
1 − γ |b|
k1
≥
∞
2.11
"
"
k|bk | |z|k−1 ≥ "g z".
k1
The function
1 − γ |b|
xk zk
kkλ
−
λ
1
−
1
1
−
γ
k
|b|
k2
∞
1 − γ |b|
yk zk ,
k1 kkλ λ − 1 k 1 − |b| 1 − γ
fz z ∞
2.12
∞
where ∞
k2 |xk | k1 |yk | 1, shows that the coefficient bound given by 2.1 is sharp. The
functions of the form 2.12 are in SCH b, γ, λ because
∞
kkλ λ − 1 k 1 − |b| 1 − γ
kkλ − λ 1 k − 1 |b| 1 − γ
|ak | |bk |
1 − γ |b|
1 − γ |b|
k1
1
∞
∞ " "
|xk | "yk " 2.
k2
k1
2.13
The next theorem shows that condition 2.1 is necessary for f ∈ T SCH b, γ, λ.
Theorem 2.2. Let f h g be so that h and g are given by 1.3. Then, f ∈ T SCH b, γ, λ if and
only if
∞ kkλ − λ 1
∞ kkλ λ − 1
k − 1 |b| 1 − γ
k 1 − |b| 1 − γ
|ak | |bk | ≤ 2,
1 − γ |b|
1 − γ |b|
k1
k1
2.14
where a1 1, 0 ≤ γ < 1, 0 ≤ λ ≤ γ/1 γ or λ ≥ 1/1 γ, and b ∈ C \ {0}.
International Journal of Mathematics and Mathematical Sciences
7
Proof. The “if part” follows from Theorem 2.1 upon noting that T SCH b, γ, λ ⊂ SCH b, γ, λ.
For the “only if” part, we show that f ∈ T SCH b, γ, λ. Then, for z reiθ in U, we obtain
⎧
⎪
⎨
⎛
λ z3 h z − z3 g z 2λ 1z2 h z
1⎜
1 ⎝ ⎪
b λ z2 h z z2 g z zh z 2λ − 1zg z
⎩
⎞
⎫
⎪
⎬
1 − 4λz2 g z zh z 1 − 2λzg z
⎟
− 1⎠ − γ
⎪
2
2
⎭
λ z h z z g z zh z 2λ − 1zg z
#
1−γ z− ∞
/b |ak |zk
k2 kkλ − λ 1 k − 1 b 1 − γ
∞
k
z − k2 kkλ − λ 1|ak |zk ∞
k1 kkλ λ − 1|bk |z
$
∞ /b |bk |zk
k1 kkλ λ − 1 k 1 − b 1 − γ
−
∞
k
k
z− ∞
k2 kkλ − λ 1|ak |z k1 kkλ λ − 1|bk |z
/b |ak |r k−1
1−γ − ∞
k2 kkλ − λ 1 k − 1 b 1 − γ
≥
∞
k−1
z − k2 kkλ − λ 1|ak |r k−1 ∞
k1 kkλ λ − 1|bk |r
∞ /b |bk |r k−1
k1 kkλ λ − 1 k 1 − b 1 − γ
−
∞
k−1
z − k2 kkλ − λ 1|ak |r k−1 ∞
k1 kkλ λ − 1|bk |r
2.15
> 0.
The above inequality must hold for all z ∈ U. In particular, letting z r → 1− yields the
required condition.
As special cases of Theorem 2.2, we obtain the following two corollaries.
Corollary 2.3. A function f h g belongs to T SCH b, γ, 0 if and only if
∞ k
∞ k
k − 1 |b| 1 − γ
k 1 − |b| 1 − γ
|ak | |bk | ≤ 2.
1 − γ |b|
1 − γ |b|
k1
k1
2.16
Corollary 2.4. A function f h g belongs to T SCH b, γ, 1 if and only if
∞ k2
∞ k2
k − 1 |b| 1 − γ
k 1 − |b| 1 − γ
|ak | |bk | ≤ 2.
1 − γ |b|
1 − γ |b|
k1
k1
2.17
3. Extreme Points and Distortion Bounds
In this section, our first theorem gives the extreme points of the closed convex hulls of
T SCH b, γ, λ.
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International Journal of Mathematics and Mathematical Sciences
Theorem 3.1. A function f h g belongs to T SCH b, γ, λ if and only if f can be expressed as
fz ∞
Xk hk z Yk gk z ,
z ∈ U,
3.1
k1
where h1 z z, hk z z − 1 − γ|b|/kkλ − λ 1k − 1 |b|1 − γzk k 2, 3, . . ., and
gk z z 1 − γ|b|/kkλ λ − 1k 1 − |b|1 − γzk k 1, 2, 3, . . ., ∞
k1 Xk Yk 1,
Xk ≥ 0, Yk ≥ 0. In particular, the extreme points of T SCH b, γ, λ are {hk } and {gk }.
Proof. For functions f of the form 3.1, we have
fz ∞
Xk hk z Yk gk z
k1
1 − γ |b|
Xk zk
kkλ
−
λ
1
−
1
1
−
γ
k
|b|
k2
∞
∞
Xk Yk z −
k1
3.2
1 − γ |b|
Yk zk .
kkλ
λ
−
1
1
−
1
−
γ
k
|b|
k1
∞
Then,
∞ kkλ − λ 1
1 − γ |b|
k − 1 |b| 1 − γ
Xk
kkλ − λ 1 k − 1 |b| 1 − γ
1 − γ |b|
k2
∞ kkλ λ − 1
1 − γ |b|
k 1 − |b| 1 − γ
Yk
kkλ λ − 1 k 1 − |b| 1 − γ
1 − γ |b|
k1
∞
k2
Xk ∞
Yk
k1
1 − X1 ≤ 1,
3.3
and so f ∈ cl coT SCH b, γ, λ.
Conversely, suppose that f ∈ cl coT SCH b, γ, λ. Letting
X1 1 −
∞
k2
Xk −
∞
k1
Yk ,
3.4
International Journal of Mathematics and Mathematical Sciences
9
where Xk kkλ − λ 1k − 1 |b|1 − γ/1 − γ|b||ak |, k 2, 3, . . ., and Yk kkλ λ −
1k 1 − |b|1 − γ/1 − γ|b||bk |, k 1, 2, . . ., we obtain the required representation since
fz z −
∞
|ak |zk k2
∞
|bk |zk
k1
∞
1 − γ |b|Xk
1 − γ |b|Yk
k
z−
z zk
kkλ
−
λ
1
−
1
1
−
γ
kkλ
λ
−
1
1
−
1
−
γ
k
k
|b|
|b|
k2
k1
∞
z−
z − hk zXk −
k2
∞
1−
∞
k2
∞
Xk −
∞
∞
z − gk z Yk
k1
Yk z k1
∞
hk zXk k2
∞
gk zYk
k1
Xk hk z Yk gk z .
k1
3.5
The following theorem gives the distortion bounds for functions in T SCH b, γ, λ,
which yields a covering result for this family.
Theorem 3.2. If f ∈ T SCH b, γ, λ then for z reiθ , one has
1 − γ |b|
2λ − 1 2 − b 1 − γ
−
|b1 | ,
2λ 1 1 b 1 − γ
2λ 1 1 b 1 − γ
"
"
1 − γ |b|
2λ − 1 2 − b 1 − γ
"fz" ≥ 1 − |b1 |r − r 2
−
|b1 | .
2λ 1 1 b 1 − γ
2λ 1 1 b 1 − γ
"
"
"fz" ≤ 1 |b1 |r r 2
3.6
Proof. Let f ∈ T SCH b, γ, λ. Taking the absolute value of f and then by Theorem 2.2, we
obtain
∞
"
"
"fz" ≤ 1 |b1 |r |ak | |bk |r k
k2
∞ kkλ − λ 1
1 − γ |b|
k − 1 |b| 1 − γ
≤ 1 |b1 |r |ak |
2λ 1 1 b 1 − γ
1 − γ |b|
k2
kkλ λ − 1 k 1 − |b| 1 − γ
|bk | r 2
1 − γ |b|
1 − γ |b|
2λ − 1 2 − b 1 − γ
≤ 1 |b1 |r −
|b1 | r 2 .
2λ 1 1 b 1 − γ
2λ 1 1 b 1 − γ
3.7
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International Journal of Mathematics and Mathematical Sciences
Similarly,
"
"
"fz" ≥ 1 − |b1 |r −
1 − γ |b|
2λ − 1 2 − b 1 − γ
−
|b1 | r 2 .
2λ 1 1 b 1 − γ
2λ 1 1 b 1 − γ
3.8
The upper and lower bounds given in Theorem 3.2 are, respectively, attained for the
following functions
1 − γ |b|
2λ − 1 2 − b 1 − γ
−
|b1 | z2 ,
2λ 1 1 b 1 − γ
2λ 1 1 b 1 − γ
1 − γ |b|
2λ − 1 2 − b 1 − γ
1
fz 1 − |b1 |z −
−
|b1 | z2 .
Γ2 2λ 1 1 b 1 − γ
2λ 1 1 b 1 − γ
1
fz z |b1 |z Γ2
3.9
4. Convolution and Convex Combinations
In this section we show that the class T SCH b, γ, λ is closed under convolution and convex
combinations. Now we need the following definition of convolution of two harmonic func∞
∞
∞
k
k
k
k
tions. For fz z − ∞
k2 |ak |z k1 |bk |z and Fz z −
k2 |Ak |z k1 |Bk |z , we
define the convolution of two harmonic functions f and F as
∞
∞
f ∗ F z fz ∗ Fz z − |ak | |Ak |zk |bk | |Bk |zk .
k2
4.1
k1
Using the definition, we show that the class T SCH b, γ, λ is closed under convolution.
Theorem 4.1. For 0 ≤ δ < γ < 1, let f ∈ T SCH b, γ, λ and F ∈ T SCH b, δ, λ. Then, f ∗ F ∈
T SCH b, γ, λ ⊂ T SCH b, δ, λ.
∞
∞
∞
k
k
k
k
Proof. Let fz z − ∞
k2 |ak |z k1 |bk |z and Fz z −
k2 |Ak |z k1 |Bk |z be in
T SCH b, δ, λ. Then, f ∗ F ∈ T SCH b, δ, λ. We note that |Ak | ≤ 1 and |Bk | ≤ 1. In view of
Theorem 2.2 and the inequality 0 ≤ δ ≤ γ < 1, we have
∞
kkλ − λ 1k − 1 |b|1 − δ
1 − δ|b|
k2
≤
|ak | |Ak | ∞
kkλ − λ 1k − 1 |b|1 − δ
∞
kkλ λ − 1k 1 − |b|1 − δ
k1
|ak | 1 − δ|b|
|bk | |Bk |
∞
kkλ λ − 1k 1 − |b|1 − δ
|bk |
1 − δ|b|
1 − δ|b|
k1
∞ kkλ − λ 1
∞ kkλ λ − 1
k − 1 |b| 1 − γ
k 1 − |b| 1 − γ
≤
|ak | |bk |
1 − γ |b|
1 − γ |b|
k2
k1
k2
≤ 1,
4.2
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11
by Theorem 2.2, f ∈ T SCH b, γ, λ. By the same token, we then conclude that f ∗ F ∈
T SCH b, γ, λ ⊂ T SCH b, δ, λ.
Next, we show that the class T SCH b, γ, λ is closed under convex combination of its
members.
Theorem 4.2. The class T SCH b, γ, λ is closed under convex combinations.
Proof. Suppose that fi z ∈ T SCH b, γ, λ, where fi is given by
fi z z −
∞
∞
|ai,k |zk |bi,k |zk ,
k2
For
∞
i1 ti
i 1, 2, 3, . . . .
4.3
k1
1, 0 ≤ ti ≤ 1, the convex combinations of fi may be written as
∞
∞
∞
∞
∞
k
ti fi z z −
ti |ai,k | z ti |bi,k | zk .
i1
k2
i1
k1
4.4
i1
Since,
∞ kkλ − λ 1
∞ kkλ λ − 1
k − 1 |b| 1 − γ
k 1 − |b| 1 − γ
|ai,k | |bi,k | ≤ 1,
1 − γ |b|
1 − γ |b|
k2
k1
4.5
from the above equation we obtain
∞
∞ kkλ − λ 1
k − 1 |b| 1 − γ ti |ai,k |
1 − γ |b|
i1
k2
∞
∞ kkλ λ − 1
k 1 − |b| 1 − γ ti |bi,k |
1 − γ |b|
i1
k1
#
∞
∞ kkλ − λ 1
k − 1 |b| 1 − γ
ti
|ai,k |
1 − γ |b|
i1
k2
$
∞ kkλ λ − 1
k 1 − |b| 1 − γ
|bi,k |
1 − γ |b|
k1
≤
∞
ti 1.
i1
This is the required condition by 2.14 and so
∞
i1 ti fi z
∈ T SCH b, γ, λ.
4.6
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5. Class-Preserving Integral Operator
In this section, we consider the closure property of the class T SCH b, γ, λ under the Bernardi
integral operator Lc fz, which is defined by
c1
Lc fz c
z
z
ξc−1 fξdξ
c > −1.
5.1
0
Theorem 5.1. Let fz ∈ T SCH b, γ, λ; then Lc fz ∈ T SCH b, γ, λ.
Proof. From the representation of Lc fz, it follows that
c1
Lc fz c
z
c1
c
z
z−
z
ξc−1 hξdξ 0
z
ξ
c−1
ξ−
0
∞
c1
zc
z
ξc−1 gξdξ
0
|ak |ξ
k2
k
c1
dξ c
z
z
ξc−1
0
∞
|bk
|ξk
dξ
5.2
k1
∞
∞
Ak zk Bk zk ,
k2
k1
where Ak c 1/c k|ak | and Bk c 1/c k|bk |. Hence,
%
&
∞ kkλ − λ 1
k − 1 |b| 1 − γ
c1
|ak |
ck
1 − γ |b|
k2
%
&
∞ k|kλ λ − 1|
k 1 − |b| 1 − γ
c1
|bk |
ck
1 − γ |b|
k1
∞ kkλ − λ 1
k − 1 |b| 1 − γ
≤
|ak |
1 − γ |b|
k2
∞ k|kλ λ − 1|
k 1 − |b| 1 − γ
|bk | ≤ 1,
1 − γ |b|
k1
5.3
and since f ∈ T SCH b, γ, λ, by Theorem 2.2, Lc fz ∈ TSCH b, γ, λ.
Acknowledgments
The authors would like to thank the referees for their insightful suggestions and comments.
The authors would like to thank Professor S. Yalçin for his suggestions during the preparation
of the paper.
International Journal of Mathematics and Mathematical Sciences
13
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