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Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2008, Article ID 259205, 11 pages
doi:10.1155/2008/259205
Research Article
On Meromorphic Harmonic Functions with
Respect to k-Symmetric Points
K. Al-Shaqsi and M. Darus
School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia,
Bangi, Selangor D. Ehsan 43600, Malaysia
Correspondence should be addressed to M. Darus,
Received 22 May 2008; Revised 20 July 2008; Accepted 23 August 2008
Recommended by Ramm Mohapatra
In our previous work in this journal in 2008, we introduced the generalized derivative operator
D
j
m
for f ∈S
H
. In this paper, we introduce a class of meromorphic harmonic function with
respect to k-symmetric points defined by
D
j
m
.Coefficient bounds, distortion theorems, extreme
points, convolution conditions, and convex combinations for the functions belonging to this class
are obtained.
Copyright q 2008 K. Al-Shaqsi and M. Darus. 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.
1. Introduction
A continuous function f u iv is a complex valued harmonic function in a domain D ⊂ C
if both u and v are real harmonic in D. In any simply connected domain, we write f h
g
where h and g are analytic in D. A necessary and sufficient condition for f to be locally
univalent and orientation preserving in D is that |h
| > |g
| in D see 1. Hengartner and
Schober 2 investigated functions harmonic in the exterior of the unit disk
U {z : |z| > 1}.
They showed that complex valued, harmonic, sense preserving, univalent mapping f must
admit the representation
fzhz
gzA log |z|, 1.1
where hz and gz are defined by
hzαz
∞
n1
a
n
z
−n
,gzβz
∞
n1
b
n
z
−n
, 1.2
for 0 ≤|β| < |α|,A∈ C and z ∈
U.
2 Journal of Inequalities and Applications
For z ∈ U \{0}, let M
H
denote the class of functions:
fzhz
gz
1
z
∞
n1
a
n
z
n
∞
n1
b
n
z
n
, 1.3
which are harmonic in the punctured unit disk U \{0}, where hz and gz are analytic in
U \{0} and U, respectively, and hz has a simple pole at the origin with residue 1 here.
In 3, the authors introduced the operator D
j
m
for f ∈S
H
which is the class of
functions f h
g that are harmonic univalent and sense-preserving in the unit disk
U {z : |z| < 1} for which f0h0f
z
0 − 1 0. For more details about the operator
D
j
m
,see4.
Now, we define D
j
m
for f h g given by 1.3 as
D
j
m
fzD
j
m
hz−1
j
D
j
m
gz, j, m ∈ N
0
N ∪{0}; z ∈ U \{0}, 1.4
where
D
j
m
hz
−1
j
z
∞
n1
n
j
Cm, na
n
z
n
,
D
j
m
gz
∞
n1
n
j
Cm, nb
n
z
n
,
Cm, n
n m − 1
m
n m − 1!
m!n − 1!
.
1.5
A function f ∈M
H
is said to be in the subclass MS
∗
H
of meromorphically harmonic starlike
functions in U \{0} if it satisfies the condition
Re
−
zh
z − zg
z
hzgz
> 0, z ∈ U \{0}. 1.6
Note that the class of harmonic meromorphic starlike functions has been studied by Jahangiri
and Silverman 5, and Jahangiri 6.
Now, we have the following definition.
Definition 1.1. For j, m ∈ N
0
, 0 ≤ α<1andk ≥ 1, let MHS
k
s
j, m, α denote the class of
meromorphic harmonic functions f of the form 1.3 such that
Re
−
D
j1
m
fz
D
j
m
f
k
z
> 0, z ∈ U \{0}, 1.7
where
D
j
m
f
k
zD
j
m
h
k
−1
j
D
j
m
g
k
j, m ∈ N
0
,k ≥ 1, 1.8
h
k
z
−1
j
z
∞
n1
a
n
Φ
n
z
n
,g
k
z
∞
n1
Φ
n
z
n
, 1.9
Φ
n
1
k
k−1
ν0
ε
n−1ν
,
k ≥ 1; ε exp
2πi
k
. 1.10
K. Al-Shaqsi and M. Darus 3
For more details about harmonic functions with respect to k-symmetric points, see
papers 7, 8 given by the authors.
Also, note that MHS
2
s
j, 0,α ⊂ MHS
∗
S
n, α was introduced by Bostancı and
¨
Ozt
¨
urk
9.
Finally, let
MHS
k
s
j, m, α denote the subclass of MHS
k
s
j, m, α consist of harmonic
functions f
j
h
j
g
j
such that h
j
and g
j
are of the form
h
j
z
−1
j
z
∞
n1
|a
n
|z
n
,g
j
z−1
j
∞
n1
|b
n
|z
n
. 1.11
Also, let f
k
j
h
k
j
g
k
j
where h
k
j
and g
k
j
are of the form
h
k
j
z
−1
j
z
∞
n1
Φ
n
|a
n
|z
n
,g
k
j
z−1
j
∞
n1
Φ
n
|b
n
|z
n
, 1.12
where Φ
n
is given by 1.10.
In this paper, we will give a sufficient condition for functions f h
g, where h and
g given by 1.3 to be in the class MHS
k
s
j, m, α. Indeed, it is shown that this coefficient
condition is also necessary for functions to be in the class
MHS
k
s
j, m, α. Also, we obtain
distortion bounds and characterize the extreme points for functions in
MHS
k
s
j, m, α.
Convolution and closure theorems are also obtained.
2. Coefficient bounds
First, we prove a sufficient coefficient bound.
Theorem 2.1. Let f h
g be of the form 1.3 and f
k
h
k
g
k
where h
k
and g
k
are given by
1.9.If
∞
n1
n − 1k 1 α|a
n−1k1
| n − 1k 1 − α|b
n−1k1
|Ω
j
m
n, k
∞
n2
n
/
lk1
n
j1
Cm, n|a
n
| |b
n
| ≤ 1 − α,
2.1
where j, m ∈ N
0
, 0 ≤ α<1,k≥ 1 and Ω
j
m
n, kn − 1k 1
j
Cm, nk 1,thenf is harmonic
univalent, sense preserving in U \{0} and f ∈ MHS
k
s
j, m, α.
Proof. For 0 < |z
1
|≤|z
2
| < 1, we have
f
z
1
− f
z
2
≥
h
z
1
− h
z
2
−
g
z
1
− g
z
2
≥
z
1
− z
2
z
1
z
2
−
z
1
− z
2
∞
n1
a
n
b
n
z
n−1
1
··· z
n−1
2
>
z
1
− z
2
z
1
z
2
1 −
z
2
2
∞
n1
n
a
n
b
n
4 Journal of Inequalities and Applications
>
z
1
− z
2
z
1
z
2
1 −
z
2
2
∞
n1
n
a
n
b
n
∞
n1
n − 1k 1
a
n−1k1
b
n−1k1
>
z
1
− z
2
z
1
z
2
1 −
∞
n1
n − 1k 1 α
a
n−1k1
− n − 1k 1 − α
b
n−1k1
Ω
j
m
n, k
−
∞
n2
n
/
lk1
n
j1
Cm, n
a
n
b
n
.
2.2
This last expression is nonnegative by 2.1,andsof is univalent in U \{0}. To show that f
is sense preserving in U \{0}, we need to show that |h
z|≥|g
z| in U \{0}. We have
|h
z|≥
1
|z|
2
−
∞
n1
n|a
n
||z|
n−1
1
r
2
−
∞
n1
n|a
n
|r
n−1
> 1 −
∞
n1
n|a
n
|
≥ 1 −
∞
n1
n − 1k 1 α|a
n−1k1
|Ω
j
m
n, k −
∞
n2
n
/
lk1
n
j1
Cm, n|a
n
|
≥
∞
n1
n − 1k 1 − α|b
n−1k1
|Ω
j
m
n, k
∞
n2
n
/
lk1
n
j1
Cm, n|b
n
|
≥
∞
n1
2n|b
2n
|
∞
n1
2n − 1|b
2n−1
|
>
∞
n1
n|b
n
|r
n−1
∞
n1
n|b
n
||z|
n−1
≥|g
z|.
2.3
Now, we will show that f ∈ MHS
k
s
j, m, α. According to 1.4 and 1.7,for0≤ α<1, we
have
Re
−
D
j1
m
fz
D
j
m
f
k
z
Re
⎧
⎨
⎩
−
D
j1
m
hz − −1
j
D
j1
m
gz
D
j
m
h
k
z−1
j
D
j
m
g
k
z
⎫
⎬
⎭
≥ α. 2.4
Using the fact that Re{w}≥α if and only if |1 − α w|≥|1 α − w|,itsuffices to show that
1 − α −
D
j1
m
fz
D
j
m
f
k
z
≥
1 α
D
j1
m
fz
D
j
m
f
k
z
, 2.5
which is equivalent to
D
j1
m
fz − 1 − αD
j
m
f
k
z
−
D
j1
m
fz1 αD
j
m
f
k
z
≥ 0. 2.6
K. Al-Shaqsi and M. Darus 5
Substituting D
j
m
fz, D
j1
m
fz, and D
j
m
f
k
z in 2.6 yields
D
j1
m
hz − −1
j
D
j1
m
gz − 1 − α
D
j
m
h
k
z−1
j
D
j
m
g
k
z
−
D
j1
m
hz − −1
j
D
j1
m
gz1 α
D
j
m
h
k
z−1
j
D
j
m
g
k
z
−1
j
z
−
∞
n1
n
j1
Cm, na
n
z
n
−1
j
∞
n1
n
j1
Cm, nb
n
z
n
1 − α
−1
j
z
∞
n1
n
j
Cm, nΦ
n
a
n
z
n
−1
j
∞
n1
n
j
Cm, nΦ
n
b
n
z
n
−
−1
j
z
−
∞
n1
n
j1
Cm, na
n
z
n
−1
j
∞
n1
n
j1
Cm, nb
n
z
n
− 1 α
−1
j
z
∞
n1
n
j
Cm, nΦ
n
a
n
z
n
−1
j
∞
n1
n
j
Cm, nΦ
n
b
n
z
n
2 − α−1
j
z
−
∞
n1
n
j
Cm, nn−1−αΦ
n
a
n
z
n
−1
j
∞
n1
n
j
Cm, nn1 −αΦ
n
b
n
z
n
−
α−1
j
z
−
∞
n1
n
j
Cm, nn 1 αΦ
n
a
n
z
n
−1
j
∞
n1
n
j
Cm, nn − 1 αΦ
n
b
n
z
n
≥
2 − α
|z|
−
∞
n1
n
j
Cm, nn − 1 − αΦ
n
|a
n
||z
n
|−
∞
n1
n
j
Cm, nn 1 − αΦ
n
|b
n
||z
n
|
−
α
|z|
−
∞
n1
n
j
Cm, nn 1 αΦ
n
|a
n
||z
n
|−
∞
n1
n
j
Cm, nn − 1 αΦ
n
|b
n
||z
n
|
21 − α
|z|
1 −
∞
n1
n
j
Cm, nn αΦ
n
1 − α
|a
n
|
z
n1
−
∞
n1
n
j
Cm, nn − αΦ
n
1 − α
|b
n
|
z
n1
≥ 21 − α
1 −
∞
n1
n
j
Cm, nn αΦ
n
1 − α
|a
n
|−
∞
n1
n
j
Cm, nn − αΦ
n
1 − α
|b
n
|
.
2.7
From the definition of Φ
n
, we know that
Φ
n
⎧
⎨
⎩
1,n lk 1,
0,n
/
lk 1,
n ≥ 2,k,l≥ 1. 2.8
6 Journal of Inequalities and Applications
Substituting 2.8 in 2.7, then 2.7 is equivalent to
D
j1
m
fz − 1 − αD
j
m
f
k
z
−
D
j1
m
fz1 αD
j
m
f
k
z
≥ 21 − α
1 −
∞
n1
nk 1
j
Cm, nk 1nk 1 α
1 − α
|a
nk1
|
−
∞
n1
nk 1
j
Cm, nk 1nk 1 − α
1 − α
|b
nk1
|−
∞
n2
n
/
lk1
n
j
Cm, n
1 − α
|a
n
|
−
∞
n2
n
/
lk1
n
j
Cm, n
1 − α
|b
n
|−
1 α
1 − α
|a
1
|−|b
1
|
21 − α
1 −
∞
n1
n − 1k 1 α
1 − α
|a
n−1k1
|−
n − 1k 1 − α
1 − α
|b
n−1k1
|
Ω
j
m
n, k
−
∞
n2
n
/
lk1
n
j1
Cm, n
1 − α
|a
n
| |b
n
|
≥ 0, by 2.6.
2.9
Thus, this completes the proof of the t heorem.
We next show that condition 2.1 is also necessary for functions in MHS
k
s
j, m, α.
Theorem 2.2. Let f
j
h
j
g
j
,whereh
j
and g
j
are given by 1.11, and f
k
j
h
k
j
g
k
j
where h
k
j
and g
k
j
are given by 1.12. Then, f
j
∈ MHS
k
s
j, m, α, if and only if the inequality 2.1 holds for
the coefficient of f
j
h
j
g
j
and f
k
j
h
k
j
g
k
j
.
Proof. In view of Theorem 2.1, we need only to show that f
j
/
∈
MHS
k
s
j, m, α if condition
2.1 does not hold. We note that for f
j
∈ MHS
k
s
j, m, α, then by 1.7 the condition 2.4
must be satisfied for all values of z in U \{0}. Substituting for h
j
,g
j
,h
k
j
,
and g
k
j
given by
1.11 and 1.12, respectively, in 2.4 and choosing 0 <z r<1, we are required to have
Re{Ψz/Υz}≥0, where
Ψz−D
j1
m
h
j
z−1
n
D
j1
m
g
j
z − αD
j
m
h
k
j
z − α−1
j
D
j
m
g
k
j
z
1 − α
z
−
∞
n1
n
j
Cm, nn αΦ
n
|a
n
|z
n
∞
n1
n
j
Cm, nn − αΦ
n
|b
n
|z
n
,
ΥzD
j
m
h
k
j
z−1
j
D
j
m
g
k
j
z
1
z
∞
n1
n
j
Cm, nΦ
n
|a
n
|z
n
∞
n1
n
j
Cm, nΦ
n
|b
n
|z
n
.
2.10
Then, the required condition Re{Ψz/Υz}≥0 is equivalent to
1 − α/z −
∞
n1
n
j
Cm, nn αΦ
n
|a
n
|r
n
∞
n1
n
j
Cm, nn − αΦ
n
|b
n
|r
n
1/z
∞
n1
n
j
Cm, nΦ
n
|a
n
|r
n
∞
n1
n
j
Cm, nΦ
n
|b
n
|r
n
. 2.11
K. Al-Shaqsi and M. Darus 7
By using 2.8, and if condition 2.1 does not hold, then the numerator of 2.11 is negative
for r sufficiently close to 1. Thus, there exists a z
0
r
0
in 0, 1 for which the quotient in 2.11
is negative. This contradicts the required condition for f
j
∈ MHS
k
s
j, m, α and so the proof
is complete.
3. Distortion bounds and extreme points
In this section, we will obtain distortion bounds for functions f
j
∈ MHS
k
s
j, m, α and also
provide extreme points for the class
MHS
k
s
j, m, α.
Theorem 3.1. If f
j
h
j
g
j
∈ MHS
k
s
j, m, α and 0 < |z| r<1,then
1
r
−
1 − α
2
j
m 12 − α
r ≤|f
j
z|≤
1
r
1 − α
2
j
m 12 − α
r. 3.1
Proof. We will prove the left side of the inequality. The argument for the right side of the
inequality is similar to the left side, and thus the details will be omitted. Let f
j
h
j
g
j
∈
MHS
k
s
j, m, α. Taking the absolute value of f,weobtain
|f
j
|
−1
j
z
∞
n1
a
n
z
n
−1
n
∞
n1
b
n
z
n
≥
1
r
−
∞
n1
|a
n
| |b
n
|r
n
≥
1
r
−
∞
n1
|a
n
| |b
n
|r
≥
1
r
−
1 − α
2
j
m 12 − αΦ
2
∞
n1
2
j
m 12 − αΦ
2
1 − α
|a
n
| |b
n
|r
≥
1
r
−
1 − α
2
j
m 12 − α
∞
n1
n
j
Cm, nn αΦ
n
1 − α
|a
n
|
n
j
Cm, nn − αΦ
n
1 − α
|b
n
|
r
≥
1
r
−
1 − α
2
j
m 12 − α
r, by 2.7.
3.2
The bounds given in Theorem 3.1 hold for functions f
j
h g
j
of the form 1.11. And it is
also discovered that the bounds hold for functions of the form 1.3, if the coefficient condition
2.1 is satisfied.
The following covering result follows from the left-hand side of the inequality in
Theorem 3.1.
Corollary 3.2. If f
j
∈ MHS
k
s
j, m, α,then
f
j
U \{0} ⊂
w : |w| <
2
j
m 12 − α − 1 − α
2
j
m 12 − α
. 3.3
8 Journal of Inequalities and Applications
Next, we determine the extreme points of closed convex hulls of
MHS
k
s
j, m, α
denoted by clco
MHS
k
s
j, m, α.
Theorem 3.3. Let f
j
h
j
g
j
where h
j
and g
j
are given by 1.11. Then, f
j
∈ MHS
k
s
j, m, α if
and only if
f
j,n
z
∞
n0
x
n
h
j
n
zy
n
g
j
n
z, 3.4
where h
j,0
g
j,0
z−1
j
/z, h
j,n
z−1
j
/z 1 − α/n
j
Cm, nn αΦ
n
z
n
n
1, 2, 3, ,g
j,n
z−1
j
/z−1
j
1− α/n
j
Cm, nn− αΦ
n
z
k
n 1, 2, 3, ,
∞
n0
x
n
y
n
1,x
n
≥ 0,y
n
≥ 0. In particular, the extreme points of MHS
k
s
j, m, α are {h
j,n
} and {g
j,n
}.
Proof. For functions f
j
h
j
g
j
, where h
j
and g
j
are given by 1.11, we have
f
j,n
z
∞
n0
x
n
h
j,n
zy
n
g
j,n
z
∞
n0
x
n
y
n
−1
j
z
∞
n1
1 − α
n
j
Cm, nn αΦ
n
x
n
z
n
−1
j
∞
n1
1 − α
n
j
Cm, nn − αΦ
n
y
n
z
k
.
3.5
Now, the first part of the proof is complete, and Theorem 2.2 gives
∞
n1
1 − α
n
j
Cm, nn αΦ
n
n
j
Cm, nn αΦ
n
1 − α
x
n
∞
n1
1 − α
n
j
Cm, nn − αΦ
n
n
j
Cm, nn − αΦ
n
1 − α
y
n
∞
n0
x
n
y
n
− x
0
y
0
1 − x
0
y
0
≤ 1.
3.6
Conversely, suppose that f
j
∈ clcoMHS
k
s
j, m, α. For n 1, 2, 3, ,set
x
n
n
j
Cm, nn αΦ
n
1 − α
|a
n
| 0 ≤ x
n
≤ 1,
y
n
n
j
Cm, nn − αΦ
n
1 − α
|b
n
| 0 ≤ y
n
≤ 1,
3.7
K. Al-Shaqsi and M. Darus 9
x
0
1 −
∞
n1
x
n
y
n
. Therefore, f can be written as
f
j,n
z
−1
j
z
∞
n1
|a
n
|z
n
−1
j
∞
n1
|b
n
|z
n
−1
j
z
∞
n1
1 − αx
n
n
j
Cm, nn αΦ
n
z
n
−1
j
∞
n1
1 − αy
n
n
j
Cm, nn − αΦ
n
z
n
−1
j
z
∞
n1
h
j,n
z −
−1
j
z
x
n
∞
n1
g
j,n
z −
−1
j
z
y
n
∞
n1
h
j,n
zx
n
∞
n1
g
j,n
zy
n
−1
j
z
1 −
∞
n1
x
n
−
∞
n1
y
n
∞
n0
h
j,n
zx
n
g
j,n
zy
n
, as required.
3.8
4. Convolution and convex combination
In this section, we show that the class
MHS
k
s
j, m, α is invariant under convolution and
convex combination of its member.
For harmonic functions f
j
z−1
j
/z
∞
n1
|a
n
|z
n
−1
j
∞
n1
|b
n
|z
n
and F
j
z
−1
j
/z
∞
n1
|A
n
|z
n
−1
j
∞
n1
|B
n
|z
n
, the convolution of f
j
and F
j
is given by
f
j
∗F
j
zf
j
z∗F
j
z
−1
j
z
∞
n1
|a
n
||A
n
|z
n
−1
j
∞
n1
|b
n
||B
n
|z
n
. 4.1
Theorem 4.1. For 0 ≤ β ≤ α<1,letf
j
∈ MHS
k
s
j, m, α and F
j
∈ MHS
k
s
j, m, β. Then,
f
j
∗F
j
∈ MHS
k
s
j, m, α ⊂ MHS
k
s
j, m, β.
Proof. We wish to show that the coefficients of f
j
∗F
j
satisfy the required condition given in
Theorem 2.2. For F
j
∈ MHS
k
s
j, m, β, we note that |A
n
|≤1and|B
n
|≤1. Now, for the
convolution function f
j
∗F
j
,weobtain
∞
n1
n
j
Cm, nn βΦ
n
1 − β
|a
n
||A
n
|
∞
n1
n
j
Cm, nn − βΦ
n
1 − β
|b
n
||B
n
|
≤
∞
n1
n
j
Cm, nn βΦ
n
1 − β
|a
n
|
∞
n1
n
j
Cm, nn − βΦ
n
1 − β
|b
n
|
≤
∞
n1
n
j
Cm, nn − αΦ
n
1 − α
|a
n
|
∞
n1
n
j
Cm, nn − αΦ
n
1 − α
|b
n
|≤1,
4.2
since 0 ≤ β ≤ α<1andf
j
∈ MHS
k
s
j, m, α. Therefore f
j
∗F
j
∈ MHS
k
s
j, m, α ⊂
MHS
k
s
j, m, β.
10 Journal of Inequalities and Applications
We now examine the convex combination of
MHS
k
s
j, m, α.
Let the functions f
j,t
be defined, for t 1, 2, ,ρ,by
f
j,t
z
−1
j
z
∞
n1
|a
n,t
|z
n
−1
j
∞
n1
|b
n,t
|z
n
. 4.3
Theorem 4.2. Let the functions f
j,t
defined by 4.3 be in the class MHS
k
s
j, m, α for every t
1, 2, ,ρ. Then, the functions ξ
t
z defined by
ξ
t
z
ρ
t1
c
t
f
j
n
z, 0 ≤ c
t
≤ 1, 4.4
are also in the class
MHS
k
s
j, m, α, where
ρ
t1
c
t
1.
Proof. According to the definition of ξ
t
, we can write
ξ
t
z
−1
j
z
∞
n1
ρ
t1
c
t
a
n,t
z
n
−1
j
∞
n1
ρ
t1
c
t
b
n,t
z
n
. 4.5
Further, since f
j,t
z are in MHS
k
s
j, m, α for every t 1, 2, ,ρ. Then by 2.7, we have
∞
n1
n αΦ
n
ρ
t1
c
t
|a
n,t
|
n − αΦ
n
ρ
t1
c
t
|b
n,t
|
n
j
Cm, n
ρ
t1
c
t
∞
n1
n αΦ
n
|a
n,t
| n − αΦ
n
|b
n,t
|n
j
Cm, n
≤
ρ
t1
c
t
1 − α ≤ 1 − α.
4.6
Hence, the theorem follows.
Corollary 4.3. The class MHS
k
s
j, m, α is close under convex linear combination.
Proof. Let the functions f
j,t
zt 1, 2 defined by 4.3 be in the class MHS
k
s
j, m, α. Then,
the function ψz defined by
ψzμf
j,1
z1 − μf
j,2
z, 0 ≤ μ ≤ 1, 4.7
is in the class
MHS
k
s
j, m, α. Also, by taking ρ 2,ξ
1
μ, and ξ
2
1 − μ in Theorem 4.2,
we have the corollary.
Acknowledgment
The work here was fully supported by Fundamental Research Grant SAGA: STGL-012-2006,
Academy of Sciences, Malaysia.
K. Al-Shaqsi and M. Darus 11
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