Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2010, Article ID 821531, 12 pages
doi:10.1155/2010/821531
Research Article
A New Subclass of Salagean-Type Harmonic
Univalent Functions
Khalifa Al-Shaqsi,1 Maslina Darus,2
and Olubunmi Abidemi Fadipe-Joseph3
1
Department of Applied Science, Ministry of Education, P.O. Box 75, P.C. 612, Bahla, Oman
School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia,
Selangor Darul Ehsan, Bangi 43600, Malaysia
3
Department of Mathematic, University of Ilorin, P.M.B. 1515, Ilorin, Nigeria
2
Correspondence should be addressed to Maslina Darus, maslina@ukm.my
Received 5 November 2010; Accepted 31 December 2010
Academic Editor: Paul Eloe
Copyright q 2010 Khalifa Al-Shaqsi et al. 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.
We define and investigate a new subclass of Salagean-type harmonic univalent functions.
We obtain coefficient conditions, extreme points, distortion bounds, convolution, and convex
combination for the above subclass of harmonic functions.
1. Introduction
Let A denote the class of functions of the form:
fz z ∞
ak zk
1.1
k2
which are analytic in the open unit disk {z ∈ : |z| < 1}.
We denote the subclass of A consisting of analytic and univalent functions fz in the
unit disk by S.
The following classes of functions and many others are well known and have been
studied repeatedly by many authors, namely, Sălăgean 1, Abdul Halim 2, and Darus 3
to mention but a few.
i S0 {fz ∈ A : Re{fz/z} > 0, z ∈ }.
ii Bα {fz ∈ A : Re{fz/z} > α, 0 ≤ α < 1, z ∈ }.
2
Abstract and Applied Analysis
iii δα {fz ∈ A : Re{f z} > α, 0 ≤ α < 1, z ∈ }.
iv Bn β {fz ∈ A : Re{Dn fzβ /zβ } > 0, z ∈ U, n ∈ 0 N ∪ {0}, β > 0}.
β
In 1994, Opoola defined the class Tn α to be a subclass of A consisting of analytic functions
satisfying the condition
Dn fzβ
Re
z ∈ , n ∈ 0 , 0 ≤ α < 1, β > 0,
> α,
zβ
1.2
where Dn is the Salagean differential operator defined as follows:
D0 fz fz,
D1 fz Dfz zf z,
Dn fz D Dn−1 fz z Dn−1 fz .
1.3
β
We note that Tn α is a generalization of the classes of functions S0 , Bα, δα, and Bn β.
Some properties of this class of functions were established by Opoola 4 namely,
β
i Tn α is a subclass of univalent functions;
β
β
ii Tn1 α ⊂ Tn α;
β
iii if fz ∈ Tn α, then the integral operator
β
Fc z βc
zβ
c
tβ−1 fzβ dt
c ≥ 0
1.4
0
β
is also in Tn α.
Now, by Binomial expansion, we have
β β−1
a32 zβ2
βa3 2!
β
fz z βa2 z
β
β1
β β−1
β β − 1 β − 2 3 β3
2a2 a3 a2 z · · · .
βa4 2!
3!
1.5
Hence, we define
fzβ zβ ∞
βak zβk−1 ,
β > 0,
k2
Dn fzβ zβ ∞
k2
1.6
βkn ak zβk−1 ,
n ∈ 0 .
Abstract and Applied Analysis
3
2. Preliminaries
A continuous function f u iv is a complex-valued harmonic function in a domain D ⊂ if both u and v are real harmonic in D. In any simply connected domain, we can write
f h g,
2.1
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 |h | > |g | in D.
Denote by SH the class of functions f of the form 2.1 that are harmonic univalent
and sense-preserving in the unit disk . The subclasses of harmonic univalent functions have
been studied by some authors for different purposes and different properties see examples
5–12. In this work, we may express the analytic functions h and g as
hzβ zβ ∞
βak zβk−1 ,
gzβ k2
∞
βbk zβk−1 ,
|b1 | < 1.
2.2
k1
Therefore,
fzβ hzβ gzβ .
2.3
We define the modified Salagean operator of f as
Dn fzβ Dn hzβ −1n Dn gzβ ,
2.4
where
Dn hzβ zβ ∞
βkn ak zβk−1 ,
k2
Dn gzβ ∞
βkn bk zβk−1 .
2.5
k1
We let Hn, β, α be the family of harmonic functions of the form 2.3 such that
Re
Dn1 fzβ
Dn fzβ
> α,
β ≥ 1, 0 ≤ α < 1, n ∈ 0 ,
2.6
where Dn fzβ is defined by 2.4.
It is clear that the class Hn, β, α includes a variety of well-known subclasses of
SH . For example, H0, 1, α ≡ S∗H α is the class of sense-preserving, harmonic univalent
functions f which are starlike of order α in , that is, ∂/∂θ{argfreiθ } > α, and H1, 1, α ≡
HKα is the class of sense-preserving, harmonic univalent functions f which are convex of
order α in , that is ∂/∂θ{arg∂/∂θfreiθ } > α. Note that the classes S∗H α and HKα
4
Abstract and Applied Analysis
were introduced and studied by Jahangiri 5. Also note that the class Hn, 1, α ≡ HKα is
the class of Salagean-type harmonic univalent functions introduced by Jahangiri et al. 13.
We let the subclass Hn, β, α consist of harmonic functions fn h gn in Hn, β, α so
h and g are of the form
hβ z zβ −
∞
|ak | zβk−1 ,
∞
β
gn z −1n
k2
|bk | zβk−1 .
2.7
k1
In 1984, Clunie and Sheil-Small 14 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 such that Silverman 15, Silverman and Silvia 16, and
Jahangiri 5, 17 studied the harmonic univalent functions. Jahangiri 5 proved the following
theorem.
Theorem 2.1. Let f h g where h z ∞
k−α
k1
1−α
|ak | ∞
k2
ak zk and g kα
|bk | ≤ 2,
1−α
then f is sense-preserving, harmonic, and univalent in
necessary if f ∈ THα ≡ H0, 1, α.
∞
k1
bk zk . If
0 ≤ α < 1,
2.8
and f ∈ S∗H α. The condition 2.8 is also
In this paper, we will give the sufficient condition for functions f β hβ g β where hβ
and g are given by 2.2 to be in the class Hn, β, α and it is shown that these coefficient
conditions are also necessary for functions in the class Hn, β, α. Also, we obtain distortion
theorems and characterize the extreme points for functions in Hn, β, α. Convolution and
convex combination are also obtained.
β
3. Main Results
In this section, we prove the main results.
3.1. Coefficient Estimates
Theorem 3.1. Let f β hβ g β , where hβ and g β are given by 2.2. If
∞
k − α|ak | k α|bk |βkn ≤ 1 β 1 − α,
3.1
k1
where a1 1, n ∈ 0 , β ≥ 1, and 0 ≤ α < 1, then f β is sense-preserving, harmonic univalent in U,
and f ∈ Hn, β, α.
Abstract and Applied Analysis
β
5
β
z2 , then
Proof. If z1 /
kβ−1
∞
kβ−1
fz β − fz β gz β − gz β z
βb
−
z
k
k1
2
1
1
2 1
2 kβ−1
≥ 1−
1 − β
β
β
β
β
β
kβ−1
hz1 − hz2 hz1 − hz2 z − z ∞ βa z
− z2
1
k
k2
2
1
∞ >1−
∞
n
k1 k β − 1 bk
k1 k αβk /1 − α|bk |
≥1−
≥ 0,
∞ ∞
1 − k2 k − αβkn /1 − α|ak |
1 − k2 k β − 1 ak
3.2
which proves univalence. Note that f is sense-preserving in . This is because
∞ ∞
k − αβkn
β
β−1
kβ−2
k β − 1 |ak ||z|
>β 1−
|ak |
h z ≥ β |z| −
1−α
k2
k2
≥β
∞
k1
k αβk
|bk |
1−α
n
≥
∞
β k β − 1 |bk ||z|kβ−2
≥ g zβ .
3.3
k1
By 2.6,
Re
Dn1 fzβ
Dn fzβ
Re
⎧
⎫
⎨ Dn1 hzβ −1n1 Dn1 gzβ ⎬
⎩
Dn hzβ −1n Dn gzβ
⎭
> α.
3.4
Using the fact that Rew > α if and only if |1 − α w| ≥ |1 α − w|, it suffices to show that
Dn1 fzβ Dn1 fzβ 1 − α − 1 α −
≥ 0,
Dn fzβ Dn fzβ n1
D fzβ 1 − αDn fzβ − Dn1 fzβ − 1 αDn fzβ ≥ 0.
Substituting for Dn1 fzβ , Dn fzβ in 3.6, we have
n1
D hzβ −1n1 Dn1 gzβ 1 − α Dn hzβ −1n Dn gzβ n1
β
n1 n1
β
β
n n
β n
− D hz −1 D gz − 1 α D hz −1 D gz 3.5
3.6
6
Abstract and Applied Analysis
∞
∞
βkn1 ak zβk−1 −1n1
βkn1 bk zβk−1
zβ k2
k1
1 − α z β
∞
n
βk−1
βk ak z
∞
n
−1
k2
n
βk bk
zβk−1
k1
∞
∞
− zβ βkn1 ak zβk−1 −1n1
βkn1 bk zβk−1
k2
k1
−1 α z β
∞
n
βk−1
βk ak z
k2
n
−1
∞
n
βk bk
zβk−1
k1
∞
∞
n
β
n
βk−1
n
βk−1 βk 1 − αk ak z
− −1
βk − 1 αk bk z
2 − αz k2
k1
∞
∞
n
β
n
βk−1
n
βk−1 − −αz βk − 1 − αk ak z
− −1
βk 1 αk bk z
k2
k1
≥ 21 − α|z|β −
∞
∞
2βkn k − α|ak |zβk−1 −
2βkn k − α|bk |zβk−1 k2
21 − α 1 −
∞
k1
βk
k2
∞
− α
n k α
βk
|ak | −
|bk | .
1−α
1−α
k1
n k
3.7
This last expression is nonnegative by 3.1, and so the proof is complete.
The harmonic function
fzβ zβ ∞
β
k2
∞
1−α
1−α
xk zkβ−1 β
yk zkβ−1 ,
n
αβkn
k − αβk
k
k1
3.8
∞
where n ∈ 0 , β ≥ 1, 0 ≤ α < 1, and ∞
k2 |xk | k1 |yk | 1, shows that the
coefficient bound given by 3.1 is sharp. The functions of the form 3.8 are in Hn, β, α
because
∞ k−α
k1
1−α
|ak | ∞
∞ kα
|bk | βkn β |xk | yk β 1.
1−α
k2
k1
3.9
In the following theorem, it is shown that the condition 3.1 is also necessary for
β
β
β
functions fn hβ gn where hβ and gn are of the form 2.7.
Abstract and Applied Analysis
β
7
β
β
Theorem 3.2. Let fn hβ gn be given by 2.7. Then fn ∈ Hn, β, α, if and only if
∞
k − α|ak | k α|bk |βkn ≤ 1 β 1 − α,
3.10
k1
where a1 1, n ∈ 0 , β ≥ 1, and 0 ≤ α < 1.
Proof. Since Hn, β, α ⊂ Hn, β, α, we only need to prove the “only if” part of the theorem.
β
To this end, for functions fn of the form 2.7, we notice that the condition 2.6 is equivalent
to
⎧
⎫
⎨ 1 − αzβ − ∞ k − αβkn a zkβ−1 − −12n ∞ k αβkn b zkβ−1 ⎬
k
k
k2
k1
Re
≥ 0.
∞
n ∞
⎩
⎭
β
kβ−1
kβ−1
n
n
z − k2 βk ak z
−1
βk
b
z
k
k1
3.11
The above required condition 3.11 must hold for all values of z in . Upon choosing the
values of z on the positive real axis where 0 ≤ z r < 1, we must have
1−α−
∞
n
k−1
n
k−1
− ∞
k2 k − αβk ak r
k1 k αβk bk r
∞
n
k−1
1 − k2 βkn ak r k−1 ∞
k1 βk bk r
≥ 0.
3.12
If the condition 3.10 does not hold, then the numerator in 3.12 is negative for r sufficiently
close to 1. Hence there exist z0 r0 in 0, 1 for which the quotient in 3.12 is negative. This
β
contradicts the required condition for fn ∈ Hn, λ, α and so the proof is complete.
3.2. Distortion Bounds and Extreme Points
In this section, first we will obtain distortion bounds for functions in Hn, β, α.
β
Theorem 3.3. Let fn ∈ Hn, β, α. Then for |z| r < 1, we have
1 1−α 1α
−
β|b1 | r β1 ,
fn zβ ≤ 1 |b1 |r β n
β2 2 − α 2 − α
1 1−α 1α
β
β
β1
−
β|b
−
.
≥
−
fn z 1 |b1 |r
1| r
β2n 2 − α 2 − α
3.13
8
Abstract and Applied Analysis
Proof. We only prove the right-hand inequality. The proof for the left-hand inequality is
β
β
similar and will be omitted. Let fn ∈ Hn, β, α. Taking the absolute value of fn , we obtain
∞
∞
β β
n
kβ−1
kβ−1
ak z
−1
bk z
fn z z k2
k1
≤ 1 |b1 |r β ∞
|ak | |bk |r kβ−1
k2
≤ 1 |b1 |r β r β1
∞
|ak | |bk |
k2
1−α
≤ 1 |b1 |r β 2 − αβ2n
1−α
≤ 1 |b1 |r 2 − αβ2n
k2
β
≤ 1 |b1 |r β ∞
2 − αβ2n
1−α
∞
k − αβkn
k2
1−α
2 − αβ2n
|ak | |bk | r β1
1−α
3.14
k αβkn
|ak | |bk | r β1
1−α
1−α
1α
β|b
1
−
r β1 ,
|
1
1−α
2 − αβ2n
for |b1 | < 1. This shows that the bounds given in Theorem 3.3 are sharp.
The following covering result follows from the left-hand inequality in Theorem 3.3.
β
Corollary 3.4. If function fn hβ g β , where hβ and g β are given by 2.7, is in Hn, β, α, then
β2n1 − 1 − β2n − 1 α
2n1 1
− n
w : |w| <
|b1 | ⊂ fn .
β2n 2 − α
2 2 − α
3.15
Next we determine the extreme points of closed convex hulls of Hn, β, α denoted by
clco Hn, β, α.
β
β
Theorem 3.5. Let fn hβ g β , where hβ and g β are given by 2.7. Then fn ∈ Hn, β, α if and
only if
fn zβ ∞ Xk hk z Yk gnk z ,
3.16
k1
where h1 zβ zβ , hk zβ zβ − 1 − α/k − αkn zkβ−1 k 2, 3, . . ., gnk zβ zβ −1n 1 −
α/k αkn zkβ−1 k 1, 2, 3, .., and ∞
k1 Xk Yk 1, Xk ≥ 0, Yk ≥ 0. In particular, the
extreme points of Hn, β, α are {hk } and {gnk }.
Abstract and Applied Analysis
9
β
Proof. For functions fn hβ g β , where hβ and g β are given by 3.16, we have
fn zβ ∞ Xk hk z Yk gnk z
k1
∞
Xk Yk zβ −
k1
∞
k2
∞
1−α
1−α
n
kβ−1
X
z
Y zkβ−1 .
−1
k
n
n k
αk
k − αk
k
k1
3.17
Then
∞
k − αβkn
1−α
k2
∞
k αβkn
|ak | 1−α
k1
|bk | ∞
Xk k2
∞
Yk 1 − X1 ≤ 1,
3.18
k1
β
and so fn ∈ clco Hn, β, α.
β
Conversely, suppose that fn ∈ clco Hn, β, α. Setting
Xk k − αβkn
|ak |
1−α
0 ≤ Xk ≤ 1 k 2, 3, . . .,
3.19
k αβkn
Yk |bk | 0 ≤ Yk ≤ 1 k 1, 2, 3, . . .,
1−α
and X1 1 −
∞
k2
Xk −
∞
β
k1
fn zβ zβ −
Yk ; therefore, fn can be written as
∞
β|ak |zkβ−1 −1n
k2
zβ −
z k − αk
∞ zkβ−1 −1n
n
β
hk z − z
β
Xk k2
∞
∞ ∞ ∞
1 − αYk kβ−1
z
k αkn
k1
β
gnk z − z
β
3.20
Yk
k1
β
hk z Xk k2
β|bk |zkβ−1
k1
∞
1 − αXk
k2
β
∞
∞
β
gnk z Yk z
k1
β
1−
∞
k2
Xk −
∞
Yk
k1
hk zβ Xk gnk zβ Yk , as required.
k1
3.3. Convolution and Convex Combination
In this section, we show that the class Hn, β, α is invariant under convolution and convex
combination of its member.
10
Abstract and Applied Analysis
kβ−1
kβ−1 and F zβ −1n ∞
For harmonic functions fn zβ zβ − ∞
n
k2 |ak |z
k1 |bk |z
n ∞
kβ−1
kβ−1 .
zβ − ∞
|A
|z
−1
|B
|z
k
k
k2
k1
β
β
The convolution of fn and Fn is given by
∞
∞
β
β
fn ∗ Fn z fn zβ ∗ Fn zβ zβ − |ak ||Ak |zkβ−1 −1n |bk ||Bk |zkβ−1 .
k2
3.21
k1
β
β
β
β
Theorem 3.6. For 0 ≤ λ ≤ α < 1, let fn ∈ Hn, β, α and Fn ∈ Hn, β, β. Then fn ∗ Fn ∈
Hn, β, α ⊂ Hn, β, λ.
kβ−1 be in the class
|ak |zkβ−1 −1n ∞
k1 |bk |z
kβ−1
kβ−1 be in the
Hn, β, α and let the functions Fn zβ z − ∞
−1n ∞
k2 |Ak |z
k1 |Bk |z
Proof. Let the functions fn zβ zβ −
∞
k2
β
β
β
class Hn, β, λ. Then the convolution fn ∗ Fn is given by 3.21. We wish to show that the
β
β
β
coefficients of fn ∗ Fn satisfy the required condition given in Theorem 3.2. For Fn ∈ Hn, β, λ,
we note that |Ak | ≤ 1 and |Bk | ≤ 1. Now, for the convolution function
β
fn
∗
β
Fn ,
we obtain
∞
∞
k − β βkn
k β βkn
|ak ||Ak | |bk ||Bk |
1−β
1−β
k2
k1
∞
∞
k − β βkn
k β βkn
≤
|ak | |bk |
1−β
1−β
k2
k1
≤
∞
k − αβkn
1−α
k2
|ak | β
∞
k αβkn
1−α
k1
β
3.22
|bk | ≤ 1,
β
since 0 ≤ λ ≤ α < 1 and fn ∈ Hn, β, α. Therefore, fn ∗ Fn ∈ Hn, β, α ⊂ Hn, β, λ.
We now examine the convex combination of Hn, β, α.
Let the functions fnj zβ be defined, for j 1, 2, . . ., by
fnj zβ zβ −
∞ ∞ ak,j zkβ−1 −1n bk,j zkβ−1 .
k2
3.23
k1
Theorem 3.7. Let the functions fnj zβ defined by 3.23 be in the class Hn, β, α for every j 1, 2, . . . , m. Then the functions tj zβ defined by
tj zβ m
cj fnj z
j1
are also in the class Hn, β, α where
m
j1 cj
1.
0 ≤ cj ≤ 1
3.24
Abstract and Applied Analysis
11
Proof. According to the definition of tβ , we can write
tzβ zβ −
∞
⎛
⎝
k2
m
⎞
⎛
⎞
∞
m
cj ak,j ⎠zkβ−1 −1n ⎝ cj bn,j ⎠zkβ−1 .
j1
k1
3.25
j1
Further, since fnj zβ are in Hn, β, α for every j 1, 2, . . ., then by 3.1 we have
⎧⎡
⎫
⎛
⎞
⎞⎤
⎛
∞ ⎨
m
m
⎬
⎣k − α⎝ cj ak,j ⎠ k α⎝ cj bk,j ⎠⎦βkn
⎩
⎭
j1
j1
k1
m
∞ "
# n
cj
k − αan,j k αbn,j βk
j1
≤
m
3.26
k1
cj 21 − α ≤ 21 − α.
j1
Hence the theorem follows.
Corollary 3.8. The class Hn, β, α is close under convex linear combination.
Proof. Let the functions fnj zβ j 1, 2 defined by 3.23 be in the class MH n, λ, α. Then
the function Ψzβ defined by
Ψzβ μfn1 zβ 1 − μ fn2 zβ
0≤μ≤1
3.27
is in the class MH n, λ, α. Also, by taking m 2, t1 μ, and t2 1 − μ in Theorem 3.7, we
have the above corollary.
Acknowledgments
The first author was an ex-postgraduate student under the supervision of Professor Maslina
Darus. Both the first and second authors are supported by UKM-ST-06-FRGS0107-2009. The
third author made her contribution to the work while she was a Visiting Researcher at
the African Institute for Mathematical Sciences, South Africa where she enjoyed numerous
stimulating discussions with Professor Alan Beardon. The authors also would like to thank
the anonymous referees for the informative and creative comments given to the paper.
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Abstract and Applied Analysis
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