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, maslina@ukm.my
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
j
Dm for f ∈ SH . In this paper, we introduce a class of meromorphic harmonic function with
j
respect to k-symmetric points defined by Dm . 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
{z : |z| > 1}.
Schober 2 investigated functions harmonic in the exterior of the unit disk U
They showed that complex valued, harmonic, sense preserving, univalent mapping f must
admit the representation
fz hz gz A log |z|,
1.1
where hz and gz are defined by
hz αz ∞
n1
for 0 ≤ |β| < |α|, A ∈ C and z ∈ U.
an z−n ,
gz βz ∞
bn z−n ,
n1
1.2
2
Journal of Inequalities and Applications
For z ∈ U \ {0}, let MH denote the class of functions:
fz hz gz ∞
∞
1 an zn bn zn ,
z n1
n1
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.
j
In 3, the authors introduced the operator Dm for f ∈ SH 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 f0 h0 fz 0 − 1 0. For more details about the operator
j
Dm , see 4.
j
Now, we define Dm for f h g given by 1.3 as
j
j
j
Dm fz Dm hz −1j Dm gz,
j, m ∈ N0 N ∪ {0}; z ∈ U \ {0},
1.4
where
j
Dm hz j
Dm gz ∞
−1j nj Cm, nan zn ,
z
n1
∞
nj Cm, nbn zn ,
1.5
n1
Cm, n nm−1
m
n m − 1!
.
m!n − 1!
A function f ∈ MH is said to be in the subclass MS∗H of meromorphically harmonic starlike
functions in U \ {0} if it satisfies the condition
zh z − zg z
> 0,
Re −
hz gz
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.
k
Definition 1.1. For j, m ∈ N0 , 0 ≤ α < 1 and k ≥ 1, let MHSs j, m, α denote the class of
meromorphic harmonic functions f of the form 1.3 such that
j1
Dm fz
Re − j
> 0,
Dm fk z
z ∈ U \ {0},
1.7
where
j
j
j
Dm fk z Dm hk −1j Dm gk j, m ∈ N0 , k ≥ 1,
∞
∞
−1j hk z an Φn zn ,
gk z Φn zn ,
z
n1
n1
k−1
1
2πi
.
εn−1ν ,
k ≥ 1; ε exp
Φn k ν0
k
1.8
1.9
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.
2
Also, note that MHSs j, 0, α ⊂ MHS∗S n, α was introduced by Bostancı and Öztürk
9.
k
k
Finally, let MHSs j, m, α denote the subclass of MHSs j, m, α consist of harmonic
functions fj hj gj such that hj and gj are of the form
∞
−1j |an |zn ,
z
n1
hj z gj z −1j
∞
|bn |zn .
1.11
n1
Also, let fkj hkj gkj where hkj and gkj are of the form
hkj z ∞
−1j Φn |an |zn ,
z
n1
gkj z −1j
∞
Φn |bn |zn ,
1.12
n1
where Φn is given by 1.10.
In this paper, we will give a sufficient condition for functions f h g, where h and
k
g given by 1.3 to be in the class MHSs j, m, α. Indeed, it is shown that this coefficient
k
condition is also necessary for functions to be in the class MHSs j, m, α. Also, we obtain
k
distortion bounds and characterize the extreme points for functions in MHSs 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 fk hk gk where hk and gk are given by
1.9. If
∞
j
n − 1k 1 α|an−1k1 | n − 1k 1 − α|bn−1k1 |Ωm n, k
n1
∞
j1
n
2.1
Cm, n|an | |bn | ≤ 1 − α,
n2
n/
lk1
j
where j, m ∈ N0 , 0 ≤ α < 1, k ≥ 1 and Ωm n, k n − 1k 1j Cm, nk 1, then f is harmonic
k
univalent, sense preserving in U \ {0} and f ∈ MHSs j, m, α.
Proof. For 0 < |z1 | ≤ |z2 | < 1, we have
f z1 − f z2 ≥ h z1 − h z2 − g z1 − g z2 ∞ z1 − z2 an bn zn−1 · · · zn−1 ≥ − z1 − z2 2
1
z1 z2 n1
∞
z1 − z2 2 > 1 − z2 n an bn z1 z2 n1
4
Journal of Inequalities and Applications
∞
∞
z1 − z2 2 n − 1k 1 an−1k1 bn−1k1
n an bn > 1 − z2
z1 z2 n1
n1
∞
z1 − z2 j
> 1 −
n − 1k 1 α an−1k1 − n − 1k 1 − α bn−1k1 Ωm n, k
z1 z2 n1
−
∞
nj1 Cm, n an bn .
n2
n/
lk1
2.2
This last expression is nonnegative by 2.1, and so f 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
−
n|an ||z|n−1
|z|2 n1
∞
∞
1 n−1
−
n|a
|r
>
1
−
n|an |
n
r 2 n1
n1
≥1−
∞
∞
j
n − 1k 1 α|an−1k1 |Ωm n, k −
nj1 Cm, n|an |
n2
n/
lk1
n1
≥
∞
n − 1k 1 −
j
α|bn−1k1 |Ωm n, k
∞
2n|b2n | n1
>
∞
j1
n
2.3
Cm, n|bn |
n2
n/
lk1
n1
≥
∞
∞
2n − 1|b2n−1 |
n1
n|bn |r n−1 n1
∞
n|bn ||z|n−1 ≥ |g z|.
n1
k
Now, we will show that f ∈ MHSs j, m, α. According to 1.4 and 1.7, for 0 ≤ α < 1, we
have
⎧
⎫
j1
⎨ Dj1 hz − −1j Dj1 gz ⎬
Dm fz
m
m
≥ α.
2.4
Re −
Re − j
⎩
⎭
j
j j
Dm fk z
Dm hk z −1 Dm gk z
Using the fact that Re{w} ≥ α if and only if |1 − α w| ≥ |1 α − w|, it suffices to show that
j1
j1
1 − α − Dm fz ≥ 1 α Dm fz ,
j
j
Dm fk z
Dm fk z
2.5
j1
j
Dm fz − 1 − αDjm fk z − Dj1
m fz 1 αDm fk z ≥ 0.
2.6
which is equivalent to
K. Al-Shaqsi and M. Darus
j
5
j1
j
Substituting Dm fz, Dm fz, and Dm fk z in 2.6 yields
j1
j1
j
j j
Dm hz − −1j Dm
gz − 1 − α Dm hk z −1 Dm gk z j1
j1
j
j
− Dm hz − −1j Dm gz 1 α Dm hk z −1j Dm gk z ∞
∞
−1j − nj1 Cm, nan zn −1j nj1 Cm, nbn zn
z
n1
n1
∞
∞
−1j nj Cm, nΦn an zn −1j nj Cm, nΦn bn zn
1 − α
z
n1
n1
∞
∞
−1j −
− nj1 Cm, nan zn −1j nj1 Cm, nbn zn
z
n1
n1
∞
∞
−1j nj Cm, nΦn an zn −1j nj Cm, nΦn bn zn
− 1 α
z
n1
n1
∞
∞
2 − α−1j j
j
n
j
n
− n Cm, nn−1−αΦn an z −1
n Cm, nn1−αΦn bn z z
n1
n1
∞
∞
α−1j j
j
n
j
−
− n Cm, nn 1 αΦn an z −1
n Cm, nn − 1 αΦn bn zn z
n1
n1
≥
∞
∞
2 − α − nj Cm, nn − 1 − αΦn |an ||zn | − nj Cm, nn 1 − αΦn |bn ||zn |
|z|
n1
n1
−
∞
∞
α − nj Cm, nn 1 αΦn |an ||zn | − nj Cm, nn − 1 αΦn |bn ||zn |
|z| n1
n1
∞ j
∞ j
n1 n1 n Cm, nn αΦn n
Cm,
nn
−
αΦ
21 − α
n
1−
|an |z −
|bn |z |z|
1−α
1−α
n1
n1
≥ 21 − α 1 −
∞ j
n Cm, nn αΦn n1
1−α
|an | −
∞ j
n Cm, nn − αΦn n1
1−α
|bn | .
2.7
From the definition of Φn , we know that
Φn ⎧
⎨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
j1
j
Dm fz − 1 − αDjm fk z − Dj1
m fz 1 αDm fk z
∞
nk 1j Cm, nk 1nk 1 α
|ank1 |
≥ 21 − α 1 −
1−α
n1
−
∞
nk 1j Cm, nk 1nk 1 − α
1−α
n1
−
∞
nj Cm, n
1α
|bn | −
|a1 | − |b1 |
1
−
α
1−α
n2
∞
nj Cm, n
|an |
1−α
n2
n
/ lk1
n/
lk1
21 − α 1 −
∞ n − 1k 1 α
n1
−
|bnk1 | −
1−α
n − 1k 1 − α
j
|an−1k1 | −
|bn−1k1 | Ωm n, k
1−α
∞
nj1 Cm, n
|an | |bn |
1−α
n2
≥ 0,
by 2.6.
n
/ lk1
2.9
Thus, this completes the proof of the theorem.
k
We next show that condition 2.1 is also necessary for functions in MHSs j, m, α.
Theorem 2.2. Let fj hj gj , where hj and gj are given by 1.11, and fkj hkj gkj where hkj
k
and gkj are given by 1.12. Then, fj ∈ MHSs j, m, α, if and only if the inequality 2.1 holds for
the coefficient of fj hj gj and fkj hkj gkj .
k
∈MHSs j, m, α if condition
Proof. In view of Theorem 2.1, we need only to show that fj /
k
2.1 does not hold. We note that for fj ∈ MHSs j, m, α, then by 1.7 the condition 2.4
must be satisfied for all values of z in U \ {0}. Substituting for hj , gj , hkj , and gkj 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
j1
j1
j
j
Ψz −Dm hj z −1n Dm gj z − αDm hkj z − α−1j Dm gkj z
∞
∞
1−α − nj Cm, nn αΦn |an |zn nj Cm, nn − αΦn |bn |zn ,
z
n1
n1
j
j
2.10
Υz Dm hkj z −1j Dm gkj z
∞
∞
1 nj Cm, nΦn |an |zn nj Cm, nΦn |bn |zn .
z n1
n1
Then, the required condition Re{Ψz/Υz} ≥ 0 is equivalent to
∞ j
j
n
n
1 − α/z − ∞
n1 n Cm, nn αΦn |an |r n1 n Cm, nn − αΦn |bn |r
.
∞ j
∞
j
1/z n1 n Cm, nΦn |an |r n n1 n Cm, nΦn |bn |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 z0 r0 in 0, 1 for which the quotient in 2.11
k
is negative. This contradicts the required condition for fj ∈ MHSs j, m, α and so the proof
is complete.
3. Distortion bounds and extreme points
k
In this section, we will obtain distortion bounds for functions fj ∈ MHSs j, m, α and also
k
provide extreme points for the class MHSs j, m, α.
k
Theorem 3.1. If fj hj gj ∈ MHSs j, m, α and 0 < |z| r < 1, then
1−α
1−α
1
1
− j
r ≤ |fj z| ≤ j
r.
r 2 m 12 − α
r 2 m 12 − α
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 fj hj gj ∈
k
MHSs j, m, α. Taking the absolute value of f, we obtain
∞
∞
−1j n
n
n
an z −1
bn z |fj | z
n1
n1
≥
∞
1 − |an | |bn |r n
r n1
≥
∞
1 − |an | |bn |r
r n1
∞ j
1−α
2 m 12 − αΦ2 1
|an | |bn |r
≥ − j
r 2 m 12 − αΦ2 n1
1−α
3.2
∞ j
1−α
nj Cm, nn − αΦn n Cm, nn αΦn 1
|an | |bn | r
≥ − j
r 2 m 12 − α n1
1−α
1−α
≥
1−α
1
−
r,
r 2j m 12 − α
by 2.7.
The bounds given in Theorem 3.1 hold for functions fj h gj 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.
k
Corollary 3.2. If fj ∈ MHSs j, m, α, then
2j m 12 − α − 1 − α
fj U \ {0} ⊂ w : |w| <
.
2j m 12 − α
3.3
8
Journal of Inequalities and Applications
k
Next, we determine the extreme points of closed convex hulls of MHSs j, m, α
k
denoted by clcoMHSs j, m, α.
k
Theorem 3.3. Let fj hj gj where hj and gj are given by 1.11. Then, fj ∈ MHSs j, m, α if
and only if
fj,n z ∞
xn hjn z yn gjn z,
3.4
n0
where hj,0 gj,0 z −1j /z, hj,n z −1j /z 1 − α/nj Cm, nn αΦn zn n 1, 2, 3, . . ., gj,n z −1j /z −1j 1 − α/nj Cm, nn − αΦn zk n 1, 2, 3, . . ., ∞
n0 xn k
yn 1, xn ≥ 0, yn ≥ 0. In particular, the extreme points of MHSs j, m, α are {hj,n } and {gj,n }.
Proof. For functions fj hj gj , where hj and gj are given by 1.11, we have
fj,n z ∞
xn hj,n z yn gj,n z
n0
∞
∞
1−α
−1j xn zn
xn yn j
z
n
Cm,
nn
αΦ
n
n0
n1
−1j
3.5
∞
1−α
y zk .
j Cm, nn − αΦ n
n
n
n1
Now, the first part of the proof is complete, and Theorem 2.2 gives
∞
1−α
nj Cm, nn αΦn xn
j
1−α
n1 n Cm, nn αΦn ∞
1−α
n1
nj Cm, nn
nj Cm, nn − αΦn yn
1−α
− αΦn 3.6
∞
xn yn − x0 y0 1 − x0 y0 ≤ 1.
n0
k
Conversely, suppose that fj ∈ clcoMHSs j, m, α. For n 1, 2, 3, . . ., set
nj Cm, nn αΦn |an |
1−α
0 ≤ xn ≤ 1,
nj Cm, nn − αΦn yn |bn |
1−α
0 ≤ yn ≤ 1,
xn 3.7
K. Al-Shaqsi and M. Darus
x0 1 −
∞
n1 xn
fj,n z 9
yn . Therefore, f can be written as
∞
∞
−1j |an |zn −1j |bn |zn
z
n1
n1
∞
∞
1 − αyn
1 − αxn
−1j j
n
z
−1
zn
j
j
z
n
Cm,
nn
αΦ
n
Cm,
nn
−
αΦ
n
n
n1
n1
∞ ∞ −1j −1j
−1j
xn yn
hj,n z −
gj,n z −
z
z
z
n1
n1
3.8
∞
∞
∞
∞
−1j
hj,n zxn gj,n zyn 1 − xn − yn
z
n1
n1
n1
n1
∞
hj,n zxn gj,n zyn , as required.
n0
4. Convolution and convex combination
k
In this section, we show that the class MHSs j, m, α is invariant under convolution and
convex combination of its member.
n
j ∞
n
For harmonic functions fj z −1j /z ∞
n1 |an |z −1
n1 |bn |z and Fj z n
j
∞
n
−1j /z ∞
n1 |An |z −1
n1 |Bn |z , the convolution of fj and Fj is given by
fj ∗Fj z fj z∗Fj z ∞
∞
−1j |an ||An |zn −1j |bn ||Bn |zn .
z
n1
n1
k
4.1
k
Theorem 4.1. For 0 ≤ β ≤ α < 1, let fj ∈ MHSs j, m, α and Fj ∈ MHSs j, m, β. Then,
k
k
fj ∗Fj ∈ MHSs j, m, α ⊂ MHSs j, m, β.
Proof. We wish to show that the coefficients of fj ∗Fj satisfy the required condition given in
k
Theorem 2.2. For Fj ∈ MHSs j, m, β, we note that |An | ≤ 1 and |Bn | ≤ 1. Now, for the
convolution function fj ∗Fj , we obtain
∞ j
n Cm, nn βΦn 1−β
n1
≤
1−β
1−β
1−α
|an | |an | 1−β
∞ j
n Cm, nn − αΦn n1
k
|bn ||Bn |
∞ j
n Cm, nn − βΦn n1
∞ j
n Cm, nn − αΦn n1
∞ j
n Cm, nn − βΦn n1
∞ j
n Cm, nn βΦn n1
≤
|an ||An | 1−α
|bn |
4.2
|bn | ≤ 1,
k
since 0 ≤ β ≤ α < 1 and fj ∈ MHSs j, m, α. Therefore fj ∗Fj ∈ MHSs j, m, α ⊂
k
MHSs j, m, β.
10
Journal of Inequalities and Applications
k
We now examine the convex combination of MHSs j, m, α.
Let the functions fj,t be defined, for t 1, 2, . . . , ρ, by
∞
∞
−1j |an,t |zn −1j |bn,t |zn .
z
n1
n1
fj,t z 4.3
k
Theorem 4.2. Let the functions fj,t defined by 4.3 be in the class MHSs j, m, α for every t 1, 2, . . . , ρ. Then, the functions ξt z defined by
ξt z ρ
ct fjn z,
0 ≤ ct ≤ 1,
4.4
t1
k
are also in the class MHSs j, m, α, where
ρ
t1 ct
1.
Proof. According to the definition of ξt , we can write
ρ
ρ
∞
∞
−1j j
n
ξt z ct an,t z −1
ct bn,t zn .
z
n1
t1
n1
t1
4.5
k
Further, since fj,t z are in MHSs j, m, α for every t 1, 2, . . . , ρ. Then by 2.7, we have
ρ
ρ
∞
j
n αΦn ct |an,t | n − αΦn ct |bn,t | n Cm, n
n1
t1
t1
ρ
∞
j
ct
n αΦn |an,t | n − αΦn |bn,t |n Cm, n
t1
≤
4.6
n1
ρ
ct 1 − α ≤ 1 − α.
t1
Hence, the theorem follows.
k
Corollary 4.3. The class MHSs j, m, α is close under convex linear combination.
k
Proof. Let the functions fj,t z t 1, 2 defined by 4.3 be in the class MHSs j, m, α. Then,
the function ψz defined by
ψz μfj,1 z 1 − μfj,2 z,
0 ≤ μ ≤ 1,
4.7
k
is in the class MHSs j, m, α. Also, by taking ρ 2, ξ1 μ, and ξ2 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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