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
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
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
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
∞
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
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
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.
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
h
z
1
− h
z
2
−
g
z
1
− g
z
2
≥
n1
a
n
b
n
z
n−1
1
··· z
n−1
2
>
n
a
n
b
n
4 Journal of Inequalities and Applications
>
z
1
− z
2
z
b
n
∞
n1
n − 1k 1
a
n−1k1
b
n−1k1
a
n−1k1
− n − 1k 1 − α
b
n−1k1
Ω
j
m
n, k
−
∞
n2
n
/
lk1
n
j1
Cm, n
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
n
j1
Cm, n|b
n
|
≥
∞
n1
2n|b
2n
|
∞
n1
2n − 1|b
2n−1
|
>
∞
n1
n|b
n
|r
n−1
∞
n1
⎩
−
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
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
D
j1
m
gz − 1 − α
D
j
m
h
k
z−1
j
D
j
m
g
k
z
−
D
j1
−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
−
−1
j
z
−
∞
n1
n
j1
Cm, na
n
z
n
−1
j
∞
n1
n
j1
Cm, nb
n
z
n
2 − α−1
j
z
−
∞
n1
n
j
Cm, nn−1−αΦ
n
a
n
z
n
−1
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
|
21 − α
|z|
1 −
∞
n1
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
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
/
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
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, 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
∞
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
∞
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
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
−1
j
z
∞
n1
a
n
z
n
−1
n
∞
n1
b
n
z
n
≥
1
r
−
∞
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
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
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
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
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
∞
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
|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
x
n
∞
n1
g
j,n
z −
−1
j
z
y
n
∞
n1
h
j,n
zx
n
∞
n1
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
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
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
|
∞
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
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
, we can write
ξ
t
z
−1
j
z
∞
n1
ρ
t1
c
t
a
n,t
z
n
−1
j
∞
n1
ρ
n,t
|
n − αΦ
n
ρ
t1
c
t
|b
n,t
|
n
j
Cm, n
ρ
t1
c
t
∞
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
References
1 J. Clunie and T. Sheil-Small, “Harmonic univalent functions,” Annales Academiae Scientiarum Fennicae.
Series A I. Mathematica, vol. 9, pp. 3–25, 1984.
2 W. Hengartner and G. Schober, “Univalent harmonic functions,” Transactions of the American
Mathematical Society, vol. 299, no. 1, pp. 1–31, 1987.
3 K. Al-Shaqsi and M. Darus, “On harmonic functions defined by derivative operator,” Journal of
Inequalities and Applications, vol. 2008, Article ID 263413, 10 pages, 2008.
4 K. Al-Shaqsi and M. Darus, “An operator defined by convolution involving the polylogarithms
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