Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 915625, 10 pages
doi:10.1155/2012/915625
Research Article
Subclass of Multivalent Harmonic Functions with
Missing Coefficients
R. M. El-Ashwah
Department of Mathematics, Faculty of Science (Damietta Branch), Mansoura University,
New Damietta 34517, Egypt
Correspondence should be addressed to R. M. El-Ashwah, r elashwah@yahoo.com
Received 20 March 2012; Accepted 8 July 2012
Academic Editor: Attila Gilányi
Copyright q 2012 R. M. El-Ashwah. 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 have studied subclass of multivalent harmonic functions with missing coefficients in the open
unit disc and obtained the basic properties such as coefficient characterization and distortion
theorem, extreme points, and convolution.
1. Introduction
A continuous function f u iv is a complex-valued harmonic function in a simply
connected complex domain D ⊂ C if both u and v are real harmonic in D. It was shown
by Clunie and Sheil-Small 1 that such harmonic function can be represented by f h g,
where h and g are analytic in D. Also, a necessary and sufficient condition for f to be locally
univalent and sense preserving in D is that |h z| > |g z| see also, 2–4.
Denote by H the family of functions f h g, which are harmonic univalent and
sense-preserving in the open-unit disc U {z ∈ C : |z| < 1} with normalization f0 h0 fz 0 − 1 0.
For m ≥ 1, 0 ≤ β < 1, and γ ≥ 0, let Rm, β, γ denote the class of all multivalent
harmonic functions f h g with missing coefficients that are sense-preserving in U, and
h, g are of the form
hz zm ∞
an1 zn1 ,
nm1
gz ∞
bn1 zn1
nm
m ≥ 1; z ∈ U
1.1
2
International Journal of Mathematics and Mathematical Sciences
and satisfying the following condition:
Re
1 γeiφ
zf z
z fz
− γmeiφ
≥ mβ
m ≥ 1; 0 ≤ β < 1; γ ≥ 0; φ real ,
1.2
where
z ∂ z reiθ ,
∂θ
f z ∂ iθ f re .
∂θ
1.3
We note that:
i Rm, β, 1 Rm, β with am1 ; bm /
0 see Jahangiri et al. 5;
ii R1, β, γ JH α, β, γ see Kharinar and More 6;
iii R1, β, 1 GH β with a2 ; b1 /
0 see Rosy et al. 7 and Ahuja and Jahangiri 2.
Also, the subclass denoted by T m, β, γ consists of harmonic functions f h g, so
that h and g are of the form
hz zm −
∞
an1 zn1 ,
nm1
gz ∞
bn1 zn1
nm
an1 ; bn1 ≥ 0; m ≥ 1; z ∈ U.
1.4
We note that:
i T m, β, 1 T m, β with am1 ; bm / 0 see Jahangiri et al. 5;
ii T 1, β, γ JH α, β, γ see Kharinar and More 6;
iii T 1, β, 1 GH β with a2 ; b1 /
0 see Rosy et al. 7 and Ahuja and Jahangiri 2.
From Ahuja and Jahangiri 2 with slight modification and among other things proved,
if f h g is of the form 1.1 and satisfies the coefficient condition
∞
nm−1
n 1 − mβ
n 1 mβ
|an1 | |bn1 | ≤ 2 am 1; am1 bm 0,
m 1−β
m 1−β
1.5
then the harmonic function f is sense-preserving, harmonic multivalent with missing
coefficients and starlike of order β 0 ≤ β < 1 in U. They also proved that the condition
1.5 is also necessary for the starlikeness of function f h g of the form 1.4.
In this paper, we obtain sufficient coefficient bounds for functions in the class
Rm, β, γ. These sufficient coefficient conditions are shown to be also necessary for functions
in the class T m, β, γ. Basic properties such as distortion theorem, extreme points, and
convolution for the class T m, β, γ are also obtained.
2. Coefficient Characterization and Distortion Theorem
Unless otherwise mentioned, we assume throughout this paper that m ≥ 1, 0 ≤ β < 1, γ ≥
0, and φ is real. We begin with a sufficient condition for functions in the class Rm, β, γ.
International Journal of Mathematics and Mathematical Sciences
3
Theorem 2.1. Let f h g be such that h and g are given by 1.1. Furthermore, let
∞
1 γ n 1 − m γ β
1 γ n 1 m γ β
|an1 | |bn1 | ≤ 2,
m 1−β
m 1−β
nm−1
2.1
where am 1 and am1 bm 0. Then f is sense-preserving, harmonic multivalent in U and
f ∈ Rm, β, γ.
Proof. To prove f ∈ Rm, β, γ, by definition, we only need to show that the condition 2.1
holds for f. Substituting h g for f in 1.2, it suffices to show that
Re
⎫
⎧
⎨ 1 γeiθ zh z − zg z − m β γeiθ hz gz ⎬
⎩
hz gz
⎭
≥ 0,
2.2
where h z ∂/∂zhz and g z ∂/∂zgz. Substituting for h, g, h , and g in 2.2,
and dividing by m1 − βzm , we obtain ReAz/Bz ≥ 0, where
∞
n 1 1 γeiθ − m β γeiθ
an1 zn−m1
Az 1 m
1
−
β
nm1
m ∞
n 1 1 γe−iθ m β γe−iθ
z
bn1 zn−m1 ,
−
z
m
1
−
β
nm1
Bz 1 ∞
an1 zn−m1 nm1
2.3
m ∞
z
bn1 zn−m1 .
z
nm1
Using the fact that Rew ≥ 0 if and only if |1w| > |1−w| in U, it suffices to show that |Az
Bz| − |Az − Bz| ≥ 0. Substituting for Az and Bz gives
|Az Bz| − |Az − Bz|
∞
n 1 1 γeiθ − m −1 2β γeiθ
an1 zn−m1
2 m
1
−
β
nm1
m ∞
n 1 1 γe−iθ m −1 2β γe−iθ
z
n−m1 bn1 z
−
z
m 1−β
nm
∞
n 1 1 γeiθ − m 1 γeiθ
−
an1 zn−m1
nm1
m 1−β
z ∞ n 1 1 γe−iθ m 1 γe−iθ
n−m1 bn1 z
−
z nm
m 1−β
4
International Journal of Mathematics and Mathematical Sciences
∞
n 1 1 γ − m 2β γ − 1
≥2−
|an1 | |z|n−m1
m
1
−
β
nm1
∞
n 1 1 γ m 2β γ − 1
−
|bn1 | |z|n−m1
m
1
−
β
nm
∞
n 1 1 γ − m 1 γ
−
|an1 | |z|n−m1
m 1−β
nm1
∞
n 1 1 γ m 1 γ
−
|bn1 | |z|n−m1
m 1−β
nm
∞
n 1 1 γ − m β γ
≥2 1−
|an1 |
m 1−β
nm1
∞
n 1 1 γ m β γ
−
|bn1 |
m 1−β
nm
≥ 0 by 2.1
2.4
The harmonic functions
m 1−β
fz z xn zn1
1
1
γ
−
m
β
γ
n
nm1
∞
m 1−β
yn zn1 ,
1
1
γ
m
β
γ
n
nm
m
∞
2.5
∞
where ∞
nm1 |xn | nm |yn | 1, show that the coefficient boundary given by 2.1 is sharp.
The functions of the form 2.5 are in the class Rm, β, γ because
∞
∞ 1γ
1 γ n 1 − m β γ
n 1 m β γ
|an1 | |bn1 |
m 1−β
m 1−β
nm
nm1
∞
∞ yn 1.
|xn | nm1
2.6
nm1
This completes the proof of Theorem 2.1.
In the following theorem, it is shown that the condition 2.1 is also necessary for
functions f h g, where h and g are of the form 1.4.
International Journal of Mathematics and Mathematical Sciences
5
Theorem 2.2. Let f h g be such that h and g are given by 1.4. Then f ∈ T m, β, γ if and only
if
∞
1 γ n 1 − m γ β
1 γ n 1 m γ β
an1 bn1 ≤ 2.
m 1−β
m 1−β
nm−1
2.7
Proof. Since Rm, β, γ ⊂ T m, β, γ, we only need to prove the “only if” part of the theorem.
To this end, for functions f of the form 1.4, we notice that the condition Re{1 γeiθ zf z/z fz − γmeiθ } ≥ mβ is equivalent to
Re
⎫
⎧
⎨ 1 γeiθ zh z − zg z − m β γeiθ hz gz ⎬
⎩
hz gz
⎭
> 0,
2.8
which implies that
iθ
m 1 γeiθ − m β γeiθ zm − ∞
n 1 − m β γeiθ an1 zn1
nm1 1 γe
Re
∞
n1
n1 zm − ∞
nm1 an1 z
nm bn1 z
∞ iθ
n 1 m β γeiθ bn1 zn1
nm 1 γe
−
∞
n1
n1 zm − ∞
nm1 an1 z
nm bn1 z
2.9
iθ
m 1−β − ∞
n 1 − m β γeiθ an1 zn−m1
nm1 1 γe
Re
∞
n−m1
n−m1 1− ∞
nm1 an1 z
nm bn1 z
∞ iθ
n 1 m β γeiθ bn1 zn−m1
nm 1 γe
> 0.
−
∞
n−m1
n−m1 1− ∞
nm1 an1 z
nm bn1 z
Since Reeiθ ≤ |eiθ | 1, the required condition is that 2.9 is equivalent to
1 γ n 1 − m β γ /m 1 − β an1 r n−m1
∞
n−m1 n−m1
1− ∞
nm1 an1 r
nm bn1 r
∞ 1 γ n 1 m β γ /m 1 − β bn1 r n−m1
nm
≥ 0.
−
∞
n−m1 n−m1
1− ∞
nm1 an1 r
nm bn1 r
1−
∞
nm1
2.10
If the condition 2.7 does not hold, then the numerator in 2.10 is negative for z r
sufficiently close to 1. Hence there exists z0 r0 in 0, 1 for which the quotient in 2.10
is negative. This contradicts the required condition for f ∈ T m, β, γ, and so the proof of
Theorem 2.2 is completed.
Corollary 2.3. The functions in the class T m, β, γ are starlike of order γ β/1 γ.
6
International Journal of Mathematics and Mathematical Sciences
Proof. The proof follows from 1.5, by putting 2.7 in the form
∞
nm−1
n 1 − m γ β / 1 γ
n 1 m γ β / 1 γ
an1 bn1 ≤ 2.
m 1− γ β / 1γ
m 1− γ β / 1γ
2.11
Theorem 2.4. Let f ∈ T m, β, γ. Then for |z| r < 1, we have
m 1 2γ β 1 γ
m 1−β
−
bm1 r m2 ,
m 1−β 2 1γ
m 1−β 2 1γ
m 1 2γ β 1 γ
m 1−β
m
fm z ≥ 1 − bm1 rr −
−
bm1 r m2 .
m 1−β 2 1γ
m 1−β 2 1γ
2.12
fz ≤ 1 bm1 rr m Proof. We prove the left-hand-side inequality for |f|. The proof for the right-hand-side
inequality can be done by using similar arguments.
Let f ∈ T m, β, γ, then we have
∞
∞
m
n1
n1
fz z −
an1 z bn1 z nm
nm1
≥ r m − bm1 r m1 −
∞
an1 bn1 r m2
nm1
≥ r m − bm1 r m1
∞
m 1−β
1 γ m 2 − m γ β
−
an1 bn1 r n1
m 1−β
1 γ m 2 − m γ β nm1
≥ r m − bm1 r m1
∞
m 1−β
1 γ n 1 − m γ β
an1
−
m 1−β
1 γ m 2 − m γ β nm1
1 γ n 1 m γ β
bn1 r n1
m 1−β
International Journal of Mathematics and Mathematical Sciences
7
≥ 1 − bm1 rr m
m 1−β
1 γ m 1 m γ β
bm1 r m2
−
1−
m 1−β
1 γ m 2 − m γ β
≥ 1 − bm1 rr m
m 1 2γ β 1 γ
m 1−β
−
−
bm1 r m2 .
m 1−β 2 1γ
m 1−β 2 1γ
2.13
This completes the proof of Theorem 2.4.
The following covering result follows from the left-side inequality in Theorem 2.4.
Corollary 2.5. Let f ∈ T m, β, γ, then the set
2 1γ
1 γ − 2m γ β
w : |w| < − bm1
m 1−β 2 1γ
m 1−β 2 1γ
2.14
is included in fU.
3. Extreme Points
Our next theorem is on the extreme points of convex hulls of the class T m, β, γ, denoted by
clco T m, β, γ.
Theorem 3.1. Let f h g be such that h and g are given by 1.4. Then f ∈ clco T m, β, γ if and
only if f can be expressed as
fz ∞
nm
Xn1 hn1 z Yn1 gn1 z ,
3.1
where
hm z zm ,
m 1−β
m
hn1 z z − zn1 n m 1, m 2, ...,
1 γ n 1 − m γ β
m 1−β
m
gn1 z z zn1 n m, m 1, m 2, ...,
1 γ n 1 m γ β
Xn1 ≥ 0,
Yn1 ≥ 0,
∞
Xn1 Yn1 1.
nm
In particular, the extreme points of the class T m, β, γ are {hn1 } and {gn1 }, respectively.
3.2
8
International Journal of Mathematics and Mathematical Sciences
Proof. For functions fz of the form 3.1, we have
fz ∞
m
Xn1 Yn1 z −
nm
∞
nm
∞
nm
m 1−β
m 1−β
Xn1 zn1
1 γ n 1 − m γ β
n1
Yn1 z
1 γ n 1 m γ β
3.3
.
Then
m 1−β
1 γ n 1 − m γ β
Xn1
m 1−β
1 γ n 1 − m γ β
nm1
∞ 1γ
m 1−β
n 1 m γ β
Yn1
m 1−β
1 γ n 1 m γ β
nm
∞
∞
Xn1 ∞
3.4
Yn1 1 − Xm ≤ 1,
nm
nm1
and so fz ∈ clco T m, β, γ. Conversely, suppose that fz ∈ clco T m, β, γ. Set
1 γ n 1 − m γ β
Xn1 an1 n m 1, ...,
m 1−β
1 γ n 1 m γ β
bn1 n m, m 1, ...,
Yn1 m 1−β
3.5
then note that by Theorem 2.2, 0 ≤ Xn1 ≤ 1 n m1, ... and 0 ≤ Yn1 ≤ 1 n m, m1, ....
Consequently, we obtain
fz ∞
nm
Xn1 hn1 z Yn1 gn1 z .
3.6
Using Theorem 2.2 it is easily seen that the class T m, β, γ is convex and closed, and so
clco T m, β, γ T m, β, γ.
4. Convolution Result
For harmonic functions of the form
fz zm −
Gz zm −
∞
an1 zn1 ∞
bn1 zn1 ,
nm1
nm
∞
∞
An1 zn1 nm1
nm
Bn1 zn1 ,
4.1
4.2
International Journal of Mathematics and Mathematical Sciences
9
we define the convolution of two harmonic functions f and G as
f ∗ G z fz ∗ Gz
∞
zm −
an1 An1 zn1 ∞
bn1 Bn1 zn1 .
4.3
nm
nm1
Using this definition, we show that the class T m, β, γ is closed under convolution.
Theorem 4.1. For 0 ≤ β < 1, let fz ∈ T m, β, γ and Gz ∈ T m, β, γ. Then fz ∗ Gz ∈
T m, β, γ.
Proof. Let the functions fz defined by 4.1 be in the class T m, β, γ, and let the functions
Gz defined by 4.2 be in the class T m, β, γ. Obviously, the coefficients of f and G must
satisfy a condition similar to the inequality 2.7. So for the coefficients of f ∗ G we can write
1 γ n 1 − m γ β
1 γ n 1 m γ β
an1 An1 bn1 Bn1
m 1−β
m 1−β
nm−1
∞
1 γ n 1 − m γ β
1 γ n 1 m γ β
an1 bn1 ,
≤
m 1−β
m 1−β
nm−1
∞
4.4
where the right hand side of this inequality is bounded by 2 because f ∈ T m, β, γ. Then,
fz ∗ Gz ∈ T m, β, γ.
Finally, we show that T m, β, γ is closed under convex combinations of its members.
Theorem 4.2. The class T m, β, γ is closed under convex combination.
Proof. For i 1, 2, 3, .... let fi ∈ T m, β, γ, where the functions fi are given by
fi z zm −
∞
an1,i zn1 ∞
i1 ti
bn1,i zn1
nm
nm1
For
∞
an1,i ; bn1,i ≥ 0; m ≥ 1.
4.5
1; 0 ≤ ti ≤ 1, the convex combination of fi may be written as
∞
∞
ti fi z zm −
i1
∞
∞
∞
n1
ti an1,i z ti bn1,i zn1
nm1
i1
nm
i1
4.6
10
International Journal of Mathematics and Mathematical Sciences
Then by 2.7, we have
∞
∞
1 γ n 1 − m γ β 1 γ n 1 m γ β ti an1,i ti bn1,i
m 1−β
m 1−β
nm−1
i1
i1
∞
∞
1 γ n 1 − m γ β
1 γ n 1 m γ β
4.7
an1,i bn1,i
ti
m 1−β
m 1−β
i1
nm−1
∞
∞
≤ 2 ti 2.
i1
This is the condition required by 2.7, and so
proof of Theorem 4.2.
∞
i1 ti fi z
∈ T m, β, γ. This completes the
Remark 4.3. Our results for m 1 correct the results obtained by Kharinar and More 6.
Acknowledgment
The author thanks the referees for their valuable suggestions which led to the improvement
of this study.
References
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Curie-Skłodowska A, vol. 55, pp. 1–13, 2001.
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coefficients,” Annales Universitatis Mariae Curie-Skłodowska A, vol. 52, no. 2, pp. 57–66, 1998.
4 H. Silverman and E. M. Silvia, “Subclasses of harmonic univalent functions,” New Zealand Journal of
Mathematics, vol. 28, no. 2, pp. 275–284, 1999.
5 J. M. Jahangiri, G. Murugusundaramoorthy, and K. Vijaya, “On starlikeness of certain multivalent
harmonic functions,” Journal of Natural Geometry, vol. 24, no. 1-2, pp. 1–10, 2003.
6 S. M. Kharinar and M. More, “Certain class of harmonic starlike functions with some missing
coefficients,” Acta Mathematica, vol. 24, no. 3, pp. 323–332, 2008.
7 T. Rosy, B. A. Stephen, K. G. Subramanian, and J. M. Jahangiri, “Goodman-Rønning-type harmonic
univalent functions,” Kyungpook Mathematical Journal, vol. 41, no. 1, pp. 45–54, 2001.
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