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Advances in Decision Sciences
Volume 2012, Article ID 610406, 8 pages
doi:10.1155/2012/610406
Research Article
A Subclass of Harmonic Univalent Functions
with Varying Arguments Defined by Generalized
Derivative Operator
E. A. Eljamal and M. Darus
School of Mathematical Sciences, Faculty of Science and Technology,
Universiti Kebangsaan Malaysia Selangor, 43600 Bangi, Malaysia
Correspondence should be addressed to M. Darus, maslina@ukm.my
Received 21 November 2011; Revised 1 January 2012; Accepted 16 January 2012
Academic Editor: Shelton Peiris
Copyright q 2012 E. A. Eljamal 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.
Making use of the generalized derivative operator, we introduce a new class of complex valued
harmonic functions which are orientation preserving and univalent in the open unit disc and are
related to uniformly convex functions. We investigate the coefficient bounds, neighborhood, and
extreme points for this generalized class of functions.
1. Introduction
A continuous complex-valued function f u iv defined in a simply connected complex
domain D is said to be harmonic in D if both u and v are real harmonic in D. Such functions
can be expressed as
1.1
f h g,
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 sense preserving in D is
that |hz| > |gz| for all z in D see 1. Let H be the class of functions of the form 1.1
that are harmonic univalent and sense preserving in the unit disk U {z : |z| < 1} for which
f0 fz 0 − 1 0. Then for f h g ∈ H, we may express the analytic functions h and g as
hz z ∞
k2
ak zk ,
gz z ∞
k1
bk zk ,
z ∈ U, |b1 | < 1.
1.2
2
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In 1984, Clunie and Sheil-Small 1 investigated the class SH as well as its geometric subclasses and obtained some coefficient bounds. Since then, there have been several related
papers on SH and its subclasses. Now we will introduce a generalized derivative operator
for f h g given by 1.2. For fixed positive natural m and λ2 ≥ λ1 ≥ 0,
Dλm,k
fz Dλm,k
hz Dλm,k
gz,
1 ,λ2
1 ,λ2
1 ,λ2
z ∈ U,
1.3
where
Dλm,k
hz z 1 ,λ2
Dλm,k
gz
1 ,λ2
∞ 1 λ1 λ2 k − 1 m
1 λ2 k − 1
k2
∞ k1
1 λ1 λ2 k − 1
1 λ2 k − 1
ak zk ,
1.4
m
k
ak z .
reduces to
We note that by specializing the parameters, especially when λ1 λ2 0, Dλm,k
1 ,λ2
Dm which is introduced by Salagean in 2.
Now we will introduce the following definition.
Definition 1.1. For 0 ≤ < 1, let GH , m, k, λ1 , λ2 denote the subfamily of starlike harmonic
functions f ∈ H of the form 1.1 such that
⎧
⎫
⎪
⎪
⎨
⎬
fz
z Dλm,k
1 ,λ2
iψ
iψ
1e
≥
Re
−e
⎪
⎪
⎩
⎭
z Dm,k fz
1.5
λ1 ,λ2
for a suitable real ψ and z ∈ U where Dλm,k
fz d/dθDλm,k
freiθ , d/dθz reiθ .
1 ,λ2
1 ,λ2
We also let VH , m k, λ1 , λ2 GH , m k, λ1 , λ2 ∩ VH where VH is the class of
harmonic functions with varying arguments introduced by Jahangiri and Silverman 3
consisting of functions f of the form 1.1 in H for which there exists a real number φ such
that
ηk k − 1φ ≡ πmod2π,
δk k − 1φ ≡ 0 k ≥ 2,
1.6
where ηk argak and δk argbk . The same class introduced in 4 with different
differential operator.
In this paper, we obtain a sufficient coefficient condition for functions f given by 1.2
to be in the class GH , m, k, λ1 , λ2 . It is shown that this coefficient condition is necessary also
for functions belonging to the class VH , m, k, λ1 , λ2 . Further, extreme points for functions
in VH , m, k, λ1 , λ2 are also obtained.
2. Main Result
We begin deriving a sufficient coefficient condition for the functions belonging to the class
GH , m, k, λ1 , λ2 . This result is contained in the following.
Advances in Decision Sciences
3
Theorem 2.1. Let f h g given by 1.2. Furthermore, let
∞ 2k − 1 − 1−
k2
3
2k 1 1 λ1 λ2 k − 1 m
b1 ,
≤1−
|ak | |bk |
1−
1 λ2 k − 1
3−
2.1
where 0 ≤ < 1, then f ∈ GH , m, k, λ1 , λ2 .
Proof. We first show that if the inequality 2.1 holds for the coefficients of f h g, then the
required condition 1.5 is satisfied. Using 1.3 and 1.5, we can write
⎫
⎧
⎪
⎪
m,k
⎪
⎪
⎬
⎨
hz
−
z
D
gz
z Dλm,k
λ1 ,λ2
Az
1 ,λ2
iψ
iψ
−
e
,
Re
1e
Re
⎪
⎪
Bz
⎪
⎪
⎭
⎩
Dm,k hz Dm,k gz
λ1 ,λ2
2.2
λ1 ,λ2
where
m,k
m,k
m,k
Az 1 e
,
z Dλ1 ,λ2 hz − z Dλ1 ,λ2 gz
D
hz
gz
− eiψ Dλm,k
λ1 ,λ2
1 ,λ2
iψ
m,k
.
Bz Dλm,k
D
hz
gz
λ1 ,λ2
1 ,λ2
2.3
In view of the simple assertion that Rew ≥ if and only if |1 − w| ≥ |1 − w|, it
is sufficies to show that
|Az 1 − Bz| − |Az − 1 Bz| ≥ 0.
Substituting for Az and Bz the appropriate expressions in 2.4, we get
|Az 1 − Bz| − |Az − 1 Bz|
∞
1 λ1 λ2 k − 1 m
≥ 2 − |z| − 2k − |ak z|k
1
λ
−
1
k
2
k2
−
∞
2k k2
− |z| −
1 λ1 λ2 k − 1
1 λ2 k − 1
m
|bk z|k
∞
1 λ1 λ2 k − 1 m
2k − 2 − |ak z|k
1
λ
−
1
k
2
k2
2.4
4
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1 λ1 λ2 k − 1 m
|bk z|k
1
λ
−
1
k
2
k2
∞
3
2k − 1 − 1 λ1 λ2 k − 1 m
b1 −
≥ 21 − |z| 1 −
|ak |
1−
1−
1 λ2 k − 1
k2
2k 1 1 λ1 λ2 k − 1 m
≥0
|bk |
1−
1 λ2 k − 1
−
∞
2k 2 2.5
by virtue of the inequality 2.1. This implies that f ∈ GH , m, k, λ1 , λ2 .
Now we obtain the necessary and sufficient condition for function f h g be given
with condition 1.6.
Theorem 2.2. Let f h g be given by 1.2. Then f ∈ VH , m, k, λ1 , λ2 if and only if
∞ 2k − 1 − 1−
k2
|ak | 3
2k 1 1 λ1 λ2 k − 1 m
b1 ,
≤1−
|bk |
1−
1 λ2 k − 1
3−
2.6
where 0 ≤ < 1.
Proof. Since VH , m, k, λ1 , λ2 ⊂ GH , m, k, λ1 , λ2 , we only need to prove the necessary part
of the theorem. Assume that f ∈ VH , m, k, λ1 , λ2 , then by virtue of 1.3 to 1.5, we obtain
⎧
⎪
⎪
⎨
⎡ ⎤⎫
⎪
m,k
m,k
⎬
⎢ z Dλ1 ,λ2 hz − z Dλ1 ,λ2 gz
⎥⎪
iψ ⎢
iψ
⎥ ≥ 0.
−
e
Re
1e ⎣ ⎦⎪
⎪
m,k
⎪
⎪
⎩
⎭
Dλm,k
D
hz
gz
,λ
,λ
λ
1 2
1 2
2.7
The above inequality is equivalent to
⎧
⎪
⎪
⎪
⎪
⎪
∞ ⎨
1 λ λ k − 1 m
1
2
k 1 eiψ − − eiψ
Re
z
|ak |zk
⎪
1
λ
−
1
k
2
⎪
k2
⎪
⎪
⎪
⎩
∞ 1 λ λ k − 1 m
1
2
k 1 eiψ eiψ
−
|bk |zk
1
λ
−
1
k
2
k2
× z
∞ 1 λ1 λ2 k − 1 m
k2
1 λ2 k − 1
|ak |zk ,
∞ 1 λ1 λ2 k − 1 m
k2
1 λ2 k − 1
|bk |zk
−1 ⎫
⎬
⎭
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⎧
⎪
⎪
⎪
⎪
⎪
∞ ⎨
1 λ λ k − 1 m
1
2
k 1 eiψ − − eiψ
Re
1 − |ak |zk−1
⎪
1
λ
−
1
k
2
⎪
k2
⎪
⎪
⎪
⎩
5
∞ 1 λ λ k − 1 m
z
1
2
iψ
iψ
k 1e
e
−
|bk |zk−1
z k2
1 λ2 k − 1
× 1
∞ 1λ1 λ2 k−1 m
1 λ2 k − 1
k2
∞
z
|ak |zk−1 z k2
1λ1 λ2 k−1
1 λ2 k − 1
m
|bk |zk−1
−1 ⎫
⎬
⎭
≥ 0.
2.8
This condition must hold for all values of z, such that |z| r < 1. Upon choosing φ
according to 1.6 and noting that Re−eiψ ≥ −|eiψ | −1, the above inequality reduces to
1 − − 1 − b1 −
∞
2k − 1 − k2
1 λ1 λ2 k − 1
1 λ2 k − 1
m
|ak |r k−1
1 λ1 λ2 k − 1 m
k−1
2k 1 |bk |r
1 λ2 k − 1
× 1−
∞ 1 λ1 λ2 k − 1 m
k2
1 λ2 k − 1
|ak |r
k−1
∞ 1 λ1 λ2 k − 1 m
k1
1 λ2 k − 1
−1
|bk |r
k−1
≥ 0.
2.9
If 2.6 does not hold, then the numerator in 2.9 is negative for r sufficiently close to
1. Therefore, there exists a point z0 r0 in 0, 1 for which the quotient in 2.9 is negative.
This contradicts our assumption that f ∈ VH , m, k, λ1 , λ2 . We thus conclude that it is
both necessary and sufficient that the coefficient bound inequality 2.6 holds true when
f ∈ VH , m, k, λ1 , λ2 . This completes the proof of Theorem 2.2.
Theorem 2.3. The closed convex hull of f ∈ VH , m, k, λ1 , λ2 (denoted by clco VH , m, k, λ1 , λ2 is
∞
∞
∞
k
k|ak | |bk | < 1 − b1 .
2.10
fz z |ak |z |bk |zk :
k2
k1
k2
By setting λk 1 − /2k − 1 − 1 λ1 λ2 k − 1/1 λ2 k − 1m and μk 1 /2k 1 1 λ1 λ2 k − 1/1 λ2 k − 1m , then for b1 fixed, the extreme points for
clco VH , m, k, λ1 , λ2 are
2.11
z λk xzk b1 z ∪ z b1 z μk xzk ,
where k ≥ 2 and |x| 1 − |b1 |.
6
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Proof. Any function f in clco VH , m, k, λ1 , λ2 may be expressed as
fz z ∞
|ak |eiηk zk b1 z k2
∞
|bk |eiδk zk ,
2.12
k2
where the coefficients satisfy the inequality 2.1. Set h1 z z, g1 z b1 z, hk z z λk eiηk zk , gk z b1 z μk eiδk zk for k 2, 3, . . .. Writing χk |ak |/λk , Yk |bk |/μk , k 2, 3, . . .
∞
and χ1 1 − ∞
k2 χk ; Y1 1 −
k2 Yk , we get
fz ∞
!
"
χk hk z Yk gk z .
2.13
k1
In particular, setting
f1 z z b1 z,
fkz z λk xzk b1 z μkyzk ,
# #
!
"
k ≥ 2, |x| #y# 1 − |b1 | .
2.14
We see that extreme points of clco f ∈ VH , m, k λ1 , λ2 ⊂ {fkz }.
To see that f1 z is not in extreme point, note that f1 z may be written as
f 1 z 1
1
f1 z λ2 1 − |b1 |z2 f1 z − λ2 1 − |b1 |z2 ,
2
2
2.15
a convex linear combination of functions in clco VH , m, k, λ1 , λ2 .
0 and |y| /
0, we will show that it can
To see that fm is not an extreme point if both |x| /
then also be expressed as a convex linear combinations of functions in clco VH , m, k, λ1 , λ2 .
Without loss of generality, assume |x| ≥ |y|. Choose > 0 small enough so that > |x|/|y| .
Set A 1 and B 1 − |x/y|. We then see that both
t1 z z λk Axzk b1 z μk yBzk ,
t2 z z λk 2 − Axzk b1 z μk y2 − Bzk
2.16
are in clco VH , m, k, λ1 , λ2 and that
fk z 1
{t1 z t2 z}.
2
2.17
The extremal coefficient bounds show that functions of the form 2.11 are the extreme points
for clco VH , m, k, λ1 , λ2 , and so the proof is complete.
Advances in Decision Sciences
7
Following Avici and Zlotkiewicz 5 and 6, we refer to the δ-neighborhood of the
functions fz defined by 1.2 to be the set of functions F for which
! "
Nδ f Fz z ∞
Ak z k
k2
∞
Bk
zk ,
∞
k|ak − Ak | |bk − Bk | |b1 − B1 | ≤ δ .
k1
k2
2.18
In our case, let us define the generalized δ-neighborhood of f to be the set
∞ ! "
1 λ1 λ2 k − 1 m
Nδ f Fz :
2k − 1 − |ak − Ak | 2k 1 |bk − Bk |
1 λ2 k − 1
k2
1 − |b1 − B1 | ≤ 1 − δ .
2.19
Theorem 2.4. Let f be given by 1.2. If f satisfies the conditions
∞
∞
1 λ1 λ2 k − 1 m 1 λ1 λ2 k − 1 m
k2k − 1 − |ak |
k2k 1 |bk |
1 λ2 k − 1
1 λ2 k − 1
k2
k1
≤ 1 − ,
2.20
where 0 ≤ < 1, and
δ
1−
3
1−
|b1 | ,
3−
1−
2.21
then Nf ⊂ GH , m, k, λ1 , λ2 .
Proof. Let f satisfy 2.20 and Fz be given by
Fz z B1 z ∞ Ak zm Bk zk
k2
which belong to Nf. We obtain
∞
1 λ1 λ2 k − 1 m
3 |B1 | 2k − 1 − |Ak | 2k 1 |Bk |
1 λ2 k − 1
k2
≤ 3 |B1 − b1 | 3 |b1 |
∞ 1 λ1 λ2 k − 1 m
2k − 1 − |Ak − ak | 2k 1 |Bk − bk |
1 λ2 k − 1
k2
2.22
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∞ 1 λ1 λ2 k − 1 m
k2
1 λ2 k − 1
2k − 1 − |ak | 2k 1 |bk |
≤ 1 − δ 3 |b1 |
∞ 1 1 λ1 λ2 k − 1 m
k
2k − 1 − |ak | 2k 1 |bk |
3 − k2
1 λ2 k − 1
≤ 1 − δ 3 |b1 | 1
1 − − 3 |b1 | ≤ 1 − .
3−
2.23
Hence for δ 1 − /3 − 1 − 3 /1 − |b1 |, we infer that Fz ∈ GH , m, k, λ1 , λ2 which concludes the proof of Theorem 2.4.
Acknowledgment
The work here was supported by UKM-ST-06-FRGS0244-2010.
References
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vol. 9, pp. 3–25, 1984.
2 G. S. Salagean, “Subclasses of univalent functions,” in Complex Analysis, vol. 1013 of Lecture Notes in
Math, pp. 362–372, Springer, Berlin, Germany, 1983.
3 J. M. Jahangiri and H. Silverman, “Harmonic univalent functions with varying arguments,” International Journal of Applied Mathematics, vol. 8, no. 3, pp. 267–275, 2002.
4 G. Murugusundaramoorthy, K. Vijaya, and R. K. Raina, “A subclass of harmonic functions with
varying arguments defined by Dziok-Srivastava operator,” Archivum Mathematicum, vol. 45, no. 1, pp.
37–46, 2009.
5 Y. Avici and E. Zotkiewicz, “On harmonic univalent mappings,” Annales Universitatis Mariae CurieSkłodowska A, vol. 44, pp. 1–7, 1990.
6 S. Ruscheweyh, “Neighborhoods of univalent functions,” Proceedings of the American Mathematical Society, vol. 81, no. 4, pp. 521–527, 1981.
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