Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2008, Article ID 263413, 10 pages
doi:10.1155/2008/263413
Research Article
On Harmonic Functions Defined by
Derivative Operator
K. Al-Shaqsi and M. Darus
School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia,
Bangi 43600 Selangor D. Ehsan, Malaysia
Correspondence should be addressed to M. Darus,
Received 16 September 2007; Revised 20 November 2007; Accepted 26 November 2007
Recommended by Vijay Gupta
Let S
H
denote the class of functions f h
––
g
that are harmonic univalent and sense-preserv-
ing in the unit disk
U {z : |z| < 1}, where hzz
∞
k2
a
k
z
k
,gz
∞
k2
|a
k
|z
k
,g
n
z
−1
n
∞
k1
|b
k
|z
k
and n ∈ N
0
.Coefficient conditions, such as distortion bounds, convolution con-
ditions, convex combination, extreme points, and neighborhood for the class M
––
H
n, λ, α,areob-
tained.
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 complex domain
k1
b
k
z
k
,
b
1
< 1. 1.1
2 Journal of Inequalities and Applications
Observe that S
H
reduces to S, the class of normalized univalent analytic functions, if the coan-
alytic part of f is zero. Also, denote by S
∗
H
the subclasses of S
H
consisting of functions f that
map
U onto starlike domain.
For f h
g given by 1.1, we define the derivative operator introduced by authors
see 1 of f as
D
n
λ
gz
∞
k1
k
n
Cλ, kb
k
z
k
, and Cλ, k
kλ−1
λ
.
We let M
H
n, λ, α denote the family of harmonic functions f of the form 1.1 such that
Re
D
n1
λ
fz
D
n
λ
fz
a
k
z
k
,g
n
z−1
n
∞
k1
b
k
z
k
. 1.4
It is clear that the class M
H
n, λ, α includes a variety of well-known subclasses of S
H
.For
example, M
M
H
n, 0,α is the class of Salagean-type harmonic univalent functions introduced by Jahangiri
et al. 4;andM
H
0,λ,α is the class of Ruscheweyh-type harmonic univalent functions stud-
ied by Murugusundaramoorthy and Vijaya 5.
In 1984, Clunie and Sheil-Small 2 investigated the class S
H
as well as its geometric sub-
classes and obtained some coefficient bounds. Since then, there has been several related papers
on S
H
and its subclasses such that Silverman 6, Silverman and Silvia 7, and Jahangiri 3, 8
studied the harmonic univalent functions. Jahangiri and Silverman 9 prove the following
theorem.
Theorem 1.1. Let f h
g given by 1.1.If
∞
k2
k
a
k
given by 1.1 to be in the class M
H
n, λ, α; and it is shown that this coe fficient condition is
K. Al-Shaqsi and M. Darus 3
also necessary for functions in the class M
H
n, λ, α. Also, we obtain distortion theorems and
characterize the extreme points for functions in M
H
n, λ, α. Closure theorems and application
of neighborhood are also obtained.
2. Coefficient bounds
We begin with a sufficient coefficient condition for functions in M
H
n, λ, α.
Theorem 2.1. Let f h
g be given by 1.1.If
∞
k1
k − α
a
k
k α
f
z
1
− f
z
2
h
z
1
− h
z
2
≥ 1 −
∞
k1
b
k
z
k
1
− z
k
2
z
1
− z
2
∞
k2
a
k
z
k
1
− z
∞
k1
k αk
n
Cλ, k/1 − α
b
k
1−
∞
k2
k − αk
n
Cλ, k/1 − α
a
k
≥0,
1 − α
a
k
≥
∞
k1
k αk
n
Cλ, k
1 − α
b
k
>
∞
k1
k αk
n
Cλ, k
1 − α
Using the fact that Rew>αif and only if |1 − α w|≥|1 α − w|,itsuffices to show that
1 − αD
n
λ
fzD
n1
λ
fz
−
1 αD
n
λ
fz −D
n1
λ
fz
≥ 0.
2.4
4 Journal of Inequalities and Applications
Substituting D
n
λ
2 − αz
∞
k2
k 1 − αk
n
Cλ, ka
k
z
k
− −1
n
∞
k1
k − 1 αk
n
Cλ, kb
k
z
k
−
1−
∞
k2
k−αk
n
Cλ, k
1 − α
a
k
|z|
k−1
∞
k1
kαk
n
Cλ, k
1 − α
b
k
|z|
.
2.5
This last expression is nonnegative by 2.1, and so the proof is complete.
The harmonic function
fzz
∞
k2
1 − α
k − αk
n
Cλ, k
x
k
z
k
∞
k1
1 − α
k αk
n
Cλ, k
y
k
z
a
k
k α
1 − α
b
k
k
n
Cλ, k1
∞
k2
x
k
∞
k1
a
k
k α
b
k
k
n
Cλ, k ≤ 21 − α,
2.8
where a
1
1,n,λ∈ N
0
,Cλ, k
kλ−1
λ
, and 0 ≤ α<1.
Proof. Since M
H
n, λ, α ⊂ M
H
∞
k2
k
n
Cλ, ka
k
z
k
−1
2n
∞
k1
k
n
Cλ, kb
k
z
k
≥ 0. 2.9
K. Al-Shaqsi and M. Darus 5
The above required condition 2.9 must hold for all values of z in
U. Upon choosing the values
of z on the positive real axis, where 0 ≤ z r<1, we must have
1 − α −
∞
k2
k − αk
n
Cλ, kb
k
r
k−1
≥ 0.
2.10
If the condition 2.8 does not hold, then the numerator in 2.10 is negative for r sufficiently
close to 1. Hence there exist z
0
r
0
in 0, 1 for which the quotient in 2.8 is negative. This
contradicts the required condition for f
n
∈ M
H
n, λ, α and so the proof is complete.
3. Distortion bounds
In this section, we will obtain distortion bounds for functions in M
H
n, λ, α.
Theorem 3.1. Let f
n
∈ M
H
n, λ, α.Thenfor|z| r<1, one has
f
r
2
,
f
n
z
≥
1 −
b
1
r −
1
2
n
λ 1
1 − α
2 − α
z −
∞
k2
a
k
z
k
−1
n
∞
k1
b
k
z
k
≥
1 −
1 −
b
1
r − r
2
∞
k2
a
k
b
k
≥
λ 1
1 − α
b
k
r
2
≥
1−
b
1
r−
1 − α
2−α2
n
λ1
∞
k2
1
r −
1 − α
2 − α2
n
λ 1
1 −
1 α
1 − α
b
1
r
2
.
3.2
The functions
fzz
b
1
z −
1
2
n
λ 1
1 − α
2 − α
−
1 α
2 − α
b
1
z
2
3.3
for |b
1
|≤1 − α/1 α show that the bounds given in Theorem 3.1 are sharp.
6 Journal of Inequalities and Applications
The following covering result follows from the left-hand inequality in Theorem 3.1.
Corollary 3.2. If the function f
2
n
λ 12 − α
b
1
⊂f
n
U.
3.4
4. Convolution, convex combination, and extreme points
In this section, we show that the class M
H
n, λ, α is invariant under convolution and convex
combination of its member.
For harmonic functions f
n
zz −
∞
k2
a
k
z
k
−1
n
is given by
f
n
∗F
n
zf
n
z∗F
n
zz −
∞
k2
a
k
A
k
z
k
−1
n
∞
k1
b
k
B
H
n, λ, β, we note that |A
k
|≤1and|B
k
|≤1. Now, for the convolution
function f
n
∗F
n
,weobtain
∞
k2
k − βk
n
Cλ, k
1 − β
a
k
A
k
a
k
∞
k1
k βk
n
Cλ, k
1 − β
b
k
≤
∞
k2
k − αk
n
Cλ, k
1 − α
a
k
n, λ, β.
We now examine the convex combination of M
H
n, λ, α.
Let the functions f
n
j
z be defined, for j 1, 2, ,by
f
n
j
zz −
∞
k2
a
k,j
z
k
−1
n
∞
k1
≤ 1
4.4
are also in the class M
H
n, λ, α,where
m
j1
c
j
1.
K. Al-Shaqsi and M. Darus 7
Proof. According to the definition of t
j
,wecanwrite
t
j
zz −
∞
k2
m
j1
c
j
a
k,j
k − α
m
j1
c
j
a
k,j
k α
m
j1
c
j
b
k,j
k
n
Cλ, k
≤
m
j1
c
j
21 − α ≤ 21 − α.
4.6
Hence the theorem follows.
Corollary 4.3. The class M
H
n, λ, α is closed under convex linear combination.
Proof. Let the functions f
n
j
zj 1, 2 defined by 4.1 be in the class M
H
n, λ, α. Then the
function Ψz defined by
Ψzμf
n
1
z1 − μf
n
2
z, 0 ≤ μ ≤ 1
k
h
k
zY
k
g
n
k
z
,
4.8
where h
1
zz, h
k
zz − 1 − α/k − αk
n
Cλ, kz
k
,k 2, 3, , g
n
k
zz
−1
n
1 − α/k αk
n
Cλ, kz
k
n
of the form 4.8,wehave
f
n
z
∞
k1
X
k
h
k
zY
k
g
n
k
z
∞
k1
X
k
Y
k
k2
k − αk
n
Cλ, k
1 − α
a
k
∞
k1
k αk
n
Cλ, k
1 − α
b
k
∞
k2
X
a
k
, 0 ≤ X
k
≤ 1,k 2, 3, ,
Y
k
k αk
n
Cλ, k
1 − α
b
k
, 0 ≤ Y
k
≤ 1,k 1, 2, 3, ,
4.11
and X
1
1 −
∞
k1
b
k
z
k
z −
∞
k2
1 − αX
k
k − αk
n
Cλ, k
z
k
−1
n
∞
k1
1 − αY
k
k αk
n
Cλ, k
k
zX
k
∞
k1
g
n
k
zY
k
z
1 −
∞
k2
X
k
−
∞
k1
Y
k
∞
f
Fzz −
∞
k2
A
k
z
k
−
∞
k1
B
k
z
k
,
∞
k2
k
a
k
− A
k
λ 1 − 1 α −
2 − α2
n
λ 1 − 1 − α
b
1
2 − α2
n
λ 1
.
5.2
Then N
δ
M
H
n, λ, α ⊂TH.
K. Al-Shaqsi and M. Darus 9
Proof. Suppose f
n
∈ M
H
n, λ, α.LetF
n
H G
F
n
satisfies the condition TF
∞
k2
k|A
k
| |B
k
||B
1
|≤1. We observe that
TF
∞
k2
k
A
k
B
k
k
− b
k
b
k
B
1
− b
1
b
1
∞
k2
k
A
k
− a
B
1
− b
1
b
1
∞
k2
k
A
k
a
k
b
k
b
1
δ
b
1
∞
∞
k2
2 − α
1 − α
a
k
2 α
1 − α
b
k
2
n
λ 1
≤ δ
b
1
n
Cλ, k
≤ δ
b
1
1 − α
2 − α2
n
λ 1
1 −
1 α
1 − α
b
1
.
5.3
Now this last expression is never greater than one if
δ ≤ 1 −
b
1
2 − α2
n
λ 1
.
5.4
Acknowledgment
The w ork presented here was supported by Fundamental Research Grant Scheme UKM-ST-01-
FRGS0055-2006.
References
1 K. Al-Shaqsi and M. Darus, “An operator defined by convolution involving the polylogarithms func-
tions,” submitted.
2 J. Clunie and T. Sheil-Small, “Harmonic univalent functions,” Annales Academiae Scientiarum Fennicae.
Series A I, vol. 9, pp. 3–25, 1984.
3 J. M. Jahangiri, “Harmonic functions starlike in the unit disk,” Journal of Mathematical Analysis and
Applications, vol. 235, no. 2, pp. 470–477, 1999.
4 J. M. Jahangiri, G. Murugusundaramoorthy, and K. Vijaya, “Salagean-type harmonic univalent func-
tions,” Southwest Journal of Pure and Applied Mathematics, no. 2, pp. 77–82, 2002.
10 Journal of Inequalities and Applications
5 G. Murugusundaramoorthy and K. Vijaya, “On certain classes of harmonic functions involving
Ruscheweyh derivatives,” Bulletin of the Calcutta Mathematical Society, vol. 96, no. 2, pp. 99–108, 2004.
6 H. Silverman, “Harmonic univalent functions with negative coefficients,” Journal of Mathematical Anal-
ysis and Applications, vol. 220, no. 1, pp. 283–289, 1998.
7 H. Silverman and E. M. Silvia, “Subclasses of harmonic univalent functions,” New Zealand Journal of
Mathematics, vol. 28, no. 2, pp. 275–284, 1999.
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