Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 379130, 10 pages
doi:10.1155/2012/379130
Research Article
On Certain Classes of Biharmonic Mappings
Defined by Convolution
J. Chen and X. Wang
Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, China
Correspondence should be addressed to X. Wang, xtwang@hunnu.edu.cn
Received 1 March 2012; Accepted 7 August 2012
Academic Editor: Saminathan Ponnusamy
Copyright q 2012 J. Chen and X. Wang. 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 introduce a class of complex-valued biharmonic mappings, denoted by BH 0 φk ; σ, a, b,
together with its subclass T BH 0 φk ; σ, a, b, and then generalize the discussions in Ali et al. 2010
to the setting of BH 0 φk ; σ, a, b and T BH 0 φk ; σ, a, b in a unified way.
1. Introduction
A four times continuously differentiable complex-valued function F u iv in a domain
D ⊂ C is biharmonic if ΔF, the Laplacian of F, is harmonic in D. Note that ΔF is harmonic in
D if F satisfies the biharmonic equation ΔΔF 0 in D, where Δ represents the Laplacian
operator
Δ4
∂2
∂2
∂2
.
:
2
∂z∂z
∂x
∂y2
1.1
It is known that, when D is simply connected, a mapping F is biharmonic if and only
if F has the following representation:
Fz 2
k1
|z|2k−1 Gk z,
1.2
2
Abstract and Applied Analysis
where Gk are complex-valued harmonic mappings in D for k ∈ {1, 2} cf. 1–6. Also it is
known that Gk can be expressed as the form
Gk hk gk
1.3
for k ∈ {1, 2}, where all hk and gk are analytic in D cf. 7, 8.
Biharmonic mappings arise in a lot of physical situations, particularly, in fluid
dynamics and elasticity problems, and have many important applications in engineering and
biology cf. 9–11. However, the investigation of biharmonic mappings in the context of
geometric function theory is a recent one cf. 1–6.
In this paper, we consider the biharmonic mappings in D {z ∈ C : |z| < 1}. Let
BH 0 D denote the set of all biharmonic mappings F in D with the following form:
Fz 2
|z|2k−1 hk z gk z
k1
2
⎛
|z|2k−1 ⎝
∞
ak,j zj j1
k1
∞
⎞
1.4
bk,j zj ⎠,
j1
with a1,1 1, a2,1 0, b1,1 0, and b2,1 0.
In 12, Qiao and Wang proved that for each F ∈ BH 0 D, if the coefficients of F satisfy
the following inequality:
2 ∞
2k − 1 j
ak,j bk,j ≤ 2,
k1 j1
1.5
then F is sense preserving, univalent, and starlike in D see 12, Theorems 3.1 and 3.2.
Let SH denote the set of all univalent harmonic mappings f in D, where
fz hz gz z ∞
aj zj j2
∞
bj zj ,
j1
1.6
with |b1 | < 1. In particular, we use S0H to denote the set of all mappings in SH with b1 0.
Obviously, S0H ⊂ BH 0 D.
In 1984, Clunie and Sheil-Small 7 discussed the class SH and its geometric subclasses.
Since then, there have been many related papers on SH and its subclasses see 13, 14 and the
references therein. In 1999, Jahangiri 15 studied the class S∗H α consisting of all mappings
f h g such that h and g are of the form
hz z −
∞ aj zj ,
j2
gz ∞ bj zj
j1
1.7
Abstract and Applied Analysis
3
and satisfy the condition
∂ Re
arg f reiθ
∂θ
zh − zg hg
1.8
>α
in D, where 0 ≤ α < 1.
For two analytic functions f1 and f2 , if
f1 z ∞
aj zj ,
f2 z j1
∞
Aj zj ,
1.9
j1
then the convolution of f1 and f2 is defined by
∞
aj Aj zj .
f1 ∗ f2 z f1 z ∗ f2 z 1.10
j1
By using the convolution, in 16, Ali et al. introduced the class S0H φ, σ, α of harmonic
mappings in the form of 1.6 such that
⎧ ⎫
⎨ z h ∗ φ
z − σzg ∗ φ
z ⎬
Re
>α
⎩ h ∗ φ
z σ g ∗ φ
z ⎭
1.11
0
φ, σ, α such that
and the class SPH
⎧
⎫
⎨
⎬
zh ∗ φ
z − σzg ∗ φ
z
Re
1 eiγ
− eiγ > α,
⎩
⎭
h ∗ φ z σ g ∗ φ z
1.12
n
where σ ∈ R and α ∈ 0, 1 are constants, γ ∈ R and φz z ∞
n2 φn z is analytic in D.
Now we consider a class of biharmonic mappings, denoted by BH 0 φk ; σ, a, b, as
follows: F ∈ BH 0 D with the form 1.4 is said to be in BH 0 φk ; σ, a, b if and only if
Φz
Re a
− b > 0,
Ψz
1.13
where
2
2
2k−1 2k−1 Φz z
,
hk ∗ φk z σ
gk ∗ φk z
|z|
|z|
k1
Ψz z
k1
2
k1
|z|2k−1
hk ∗ φk z σ gk ∗ φk z ,
1.14
4
Abstract and Applied Analysis
j
iγ
φk z z ∞
j2 φk,j z are analytic in D for k ∈ {1, 2}, σ ∈ R is a constant, a p ρe ,
b q ρeiγ , p, q, ρ ∈ 0, ∞ are constants with a − b > 0, γ ∈ R, and z reiθ . Here and in
what follows, “ ” always stands for “∂/∂θ”.
Obviously, if φ2 0, a 1 and b α, then BH 0 φk ; σ, a, b reduces to S0H φ, σ, α, and
0
φ, σ, α.
if φ2 0, a 1 eiγ and b α eiγ , then BH 0 φk ; σ, a, b reduces to SPH
0
Further, we use T BH φk ; σ, a, b to denote the class consisting of all mappings F in
BH 0 φk ; σ, a, b with the form
Fz 2
|z|2k−1 hk z gk z ,
1.15
k1
where
hk z ak,1 z −
∞
ak,j zj ,
ak,j ≥ 0, a1,1 1, a2,1 0,
j2
gk z σ
∞
1.16
j
bk,j z ,
bk,j ≥ 0, b1,1 b2,1 0.
j1
The object of this paper is to generalize the discussions in 16 to the setting of
BH 0 φk ; σ, a, b and T BH 0 φk ; σ, a, b in a unified way. The organization of this paper is
as follows. In Section 2, we get a convolution characterization for BH 0 φk ; σ, a, b. As a
corollary, we derive a sufficient coefficient condition for mappings in BH 0 D to belong
to BH 0 φk ; σ, a, b. The main results are Theorems 2.1 and 2.3. In Section 3, first, we get
a coefficient characterization for T BH 0 φk ; σ, a, b, and then find the extreme points of
T BH 0 φk ; σ, a, b. The corresponding results are Theorems 3.1 and 3.6.
2. A Convolution Characterization
We begin with a convolution characterization for BH 0 φk ; σ, a, b.
Theorem 2.1. Let F ∈ BH 0 D. Then F ∈ BH 0 φk ; σ, a, b if and only if
2
|z|
2k−1 hk ∗ φk z ∗
z ax − a 2b/2a − 2bz2
k1
−σ
2
|z|
2k−1 gk ∗ φk z ∗
1 − z2
ax b/a − bz − ax − a 2b/2a − 2bz2
k1
for all z ∈ D \ {0} and all x ∈ C with |x| 1.
1 − z2
0,
/
2.1
Abstract and Applied Analysis
5
Proof. By definition, a necessary and sufficient condition for a mapping F in BH 0 D to be in
BH 0 φk ; σ, a, b is given by 1.13. Let
Gz Φz
1
a
−b .
a−b
Ψz
2.2
Then G0 1, and so the condition 1.13 is equivalent to
Gz /
x−1
,
x1
2.3
for all z ∈ D \ {0} and all x ∈ C with |x| 1 and x /
− 1. Obviously, 2.3 holds if and only if
ax 1Φz − bx 1Ψz − a − bx − 1Ψz /
0.
2.4
Straightforward computations show that
ax 1Φz − bx 1Ψz − a − bx − 1Ψz
ax 1z
2
⎛
|z|2k−1 ⎝z ∞
jak,j φk,j zj − σ
j2
k1
− ax − a 2bz
2
z
2
|z|
2k−1 ⎛
|z|2k−1 ⎝z hk ∗ φk z ∗
− σz
2
|z|
2k−1 jbk,j φk,j zj ⎠
ak,j φk,j zj σ
∞
⎞
bk,j φk,j zj ⎠
j2
2a − bz ax − a 2bz2
gk ∗ φk z ∗
2.5
1 − z2
k1
∞
j2
⎞
j2
k1
∞
2ax bz − ax − a 2bz2
k1
1 − z2
,
from which we see that 2.3 is true if and only if so is 2.1. The proof is complete.
Remark 2.2. If h2 g2 0, a 1 and b α, then Theorem 2.1 coincides with Theorem 2.1 in
16, and if h2 g2 0, a 1 eiγ , and b α eiγ , then Theorem 2.1 coincides with Theorem
2.3 in 16.
As an application of Theorem 2.1, we derive a sufficient condition for mappings in
BH 0 D to be in BH 0 φk ; σ, a, b in terms of their coefficients.
Theorem 2.3. Let F ∈ BH 0 D. Then F ∈ BH 0 φk ; σ, a, b if
∞
∞
2 2 jamax − bmax jamax bmax φk,j ak,j |σ|
φk,j bk,j ≤ 1,
a−b
a−b
k1 j2
k1 j2
2.6
6
Abstract and Applied Analysis
here and in the following, zmax maxγ∈R {|xyeiγ |} xy, where z xyeiγ , x and y ∈ 0, ∞
are constants.
Proof. For F given by 1.4, we see that
2
z ax − a 2b/2a − 2bz2
2k−1 hk ∗ φk z ∗
Lz |z|
k1
1 − z2
−σ
2
|z|
2k−1 gk ∗ φk z ∗
ax b/a − bz − ax − a 2b/2a − 2bz2
1 − z2
k1
∞ 2
ax − a 2b
z φk,j ak,j zj
j j −1
|z|2k−1
2a
−
2b
j2
k1
−σ
2
|z|
k1
2k−1
∞ ax − a 2b
ax b − j −1
φk,j bk,j zj .
j
a−b
2a − 2b
j2
2.7
If F is the identity, obviously, Lz |z|.
If F is not the identity, then
⎛
⎞
∞
∞
2 2 ja
−
ja
b
b
max
max max
max Lz > |z|⎝1 −
φk,j ak,j −|σ|
φk,j bk,j ⎠.
a
−
b
a
−
b
k1 j2
k1 j2
2.8
Hence the assumption implies that Lz > 0 for all z ∈ D \ {0} and all x ∈ C with |x| 1. It
follows from Theorem 2.1 that F ∈ BH 0 φk ; σ, a, b.
Remark 2.4. If h2 g2 0, a 1 and b α, then Theorem 2.3 coincides with Theorem 2.2 in
16, and if h2 g2 0, a 1 eiγ and b α eiγ , then Theorem 2.3 coincides with Theorem
2.4 in 16.
3. A Coefficient Characterization and Extreme Points
We start with a coefficient characterization for T BH 0 φk ; σ, a, b.
j
Theorem 3.1. Let φk z z ∞
j2 φk,j z with φk,j ≥ 0, and let F be of the form 1.15. Then
0
F ∈ T BH φk ; σ, a, b if and only if
∞
2 jamax − bmax
k1 j2
a−b
φk,j ak,j σ 2
∞
2 jamax bmax
k1 j2
a−b
φk,j bk,j ≤ 1.
3.1
Abstract and Applied Analysis
7
Proof. By similar arguments as in the proof of Theorem 2.3, we see that it suffices to prove the
“only if” part. For F ∈ T BH 0 φk ; σ, a, b, obviously, 1.13 is equivalent to
⎧
⎨
Re
⎩ z − 2
k1 |z|
2k−1
⎫
⎬
P z − Qz
∞
j2
ak,j φk,j zj − σ 2
>0
j ⎭
b
φ
z
k,j
k,j
j2
∞
3.2
in D, where
P z a − bz −
2
|z|2k−1
Qz σ 2
|z|
aj − b ak,j φk,j zj ,
j2
k1
2
∞
∞
2k−1
3.3
aj b bk,j φk,j zj .
j2
k1
Letting z → 1− through real values leads to the desired inequality. So the proof is complete.
Remark 3.2. If h2 g2 0, a 1, and b α, then Theorem 3.1 coincides with Theorem 3.1 in
16.
It follows from Theorem 3.1 that we have the following.
j
Corollary 3.3. Let φk z z ∞
j2 φk,j z with φk,j ≥ φ1,2 > 0 k ∈ {1, 2}, j ≥ 2 and |σ| ≥
2amax − bmax /2amax bmax . If F ∈ T BH 0 φk ; σ, a, b, then for |z| r < 1, one has
r−
a−b
a−b
r 2 ≤ |Fz| ≤ r r2.
2amax − bmax φ1,2
2amax − bmax φ1,2
3.4
The result is sharp with equality for mappings
Fz z −
a−b
z2 .
2amax − bmax φ1,2
3.5
Theorem 3.1 and Corollary 3.3 imply the following
Corollary 3.4. Under the hypotheses of Corollary 3.3, one has that T BH 0 φk ; σ, a, b is closed under
the convex combination.
Definition 3.5. Let X be a topological vector space over the field of complex numbers, and let
E be a subset of X. A point x ∈ E is called an extreme point of E if it has no representation of
the form x ty 1 − tz 0 < t < 1 as a proper convex combination of two distinct points y
and z in E cf. 17.
We now determine the extreme points of T BH 0 φk ; σ, a, b.
8
Abstract and Applied Analysis
Theorem 3.6. Let
1 h11 z z,
2 h21 z g11 z g21 z 0,
3 hkj z z − |z|2k−1 a − b/jamax − bmax φk,j zj for k ∈ {1, 2} and all j ≥ 2,
4 gkj z z |z|2k−1 a − b/σjamax bmax φk,j zj for k ∈ {1, 2} and all j ≥ 2.
Under the hypotheses of Corollary 3.3, one has that F ∈ T BH 0 φk ; σ, a, b if and only if it can be
expressed as
∞
2 xkj hkj z ykj gkj z ,
Fz 3.6
k1 j1
where x21 y11 y21 0, all other xkj and ykj are nonnegative, and
2
k1
∞
j1 xkj
ykj 1.
In particular, the extreme points of T BH φk ; σ, a, b are all mappings hkj and gkj listed in
1, 3, and 4 above.
0
Proof. It follows from the assumptions that
Fz ∞
2 xkj hkj z ykj gkj z
k1 j1
z−
∞
2 a−b
xkj zj
jamax − bmax φk,j
|z|2k−1 k1 j2
σ
∞
2 |z|2k−1
k1 j2
σ2
3.7
a−b
ykj zj ,
jamax bmax φk,j
whence
∞
2 jamax − bmax
a−b
φk,j · xkj
a−b
jamax − bmax φk,j
k1 j2
∞
2 jamax bmax
a−b
φk,j · 2 σ
ykj
a−b
σ jamax bmax φk,j
k1 j2
2
2 ∞
k1 j2
xkj 2 ∞
ykj
k1 j2
≤ 1,
and so Theorem 3.1 implies that F ∈ T BH 0 φk ; σ, a, b.
3.8
Abstract and Applied Analysis
9
Conversely, assume F ∈ T BH 0 φk ; σ, a, b, and let
x21 y11 y21 0,
x11 1 −
∞
2 xkj −
k1 j2
∞
2 ykj ,
k1 j2
jamax − bmax φk,j ak,j
,
xkj a−b
σ 2 jamax bmax φk,j bk,j
ykj ,
a−b
3.9
for k ∈ {1, 2} and all j ≥ 2. Then
Fz z −
2
|z|2k−1
k1
∞
j2
ak,j zj σ
2
k1
|z|2k−1
∞
bk,j zj .
j2
3.10
The proof of the theorem is complete.
Remark 3.7. If h2 g2 0, a 1 and b α, then Theorem 3.6 coincides with Theorem 3.2 in
16.
Acknowledgment
The research was partly supported by NSFs of China No. 11071063.
References
1 Z. Abdulhadi and Y. Abu Muhanna, “Landau’s theorem for biharmonic mappings,” Journal of
Mathematical Analysis and Applications, vol. 338, no. 1, pp. 705–709, 2008.
2 Z. Abdulhadi, Y. Abu Muhanna, and S. Khuri, “On univalent solutions of the biharmonic equation,”
Journal of Inequalities and Applications, no. 5, pp. 469–478, 2005.
3 Z. Abdulhadi, Y. Abu Muhanna, and S. Khuri, “On some properties of solutions of the biharmonic
equation,” Applied Mathematics and Computation, vol. 177, no. 1, pp. 346–351, 2006.
4 S. Chen, S. Ponnusamy, and X. Wang, “Landau’s theorem for certain biharmonic mappings,” Applied
Mathematics and Computation, vol. 208, no. 2, pp. 427–433, 2009.
5 S. Chen, S. Ponnusamy, and X. Wang, “Compositions of harmonic mappings and biharmonic
mappings,” Bulletin of the Belgian Mathematical Society, vol. 17, no. 4, pp. 693–704, 2010.
6 S. Chen, S. Ponnusamy, and X. Wang, “Bloch constant and Landau’s theorem for planar p-harmonic
mappings,” Journal of Mathematical Analysis and Applications, vol. 373, no. 1, pp. 102–110, 2011.
7 J. G. Clunie and T. Sheil-Small, “Harmonic univalent functions,” Annales Academiae Scientiarum
Fennicae. Series A I., vol. 9, pp. 3–25, 1984.
8 P. Duren, Harmonic Mappings in the Plane, vol. 156 of Cambridge Tracts in Mathematics, Cambridge
University Press, New York, NY, USA, 2004.
9 J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics with Special Applications to Particulate
Media, Prentice-Hall, Englewood Cliffs, NJ, USA, 1965.
10 S. A. Khuri, “Biorthogonal series solution of Stokes flow problems in sectorial regions,” SIAM Journal
on Applied Mathematics, vol. 56, no. 1, pp. 19–39, 1996.
11 W. E. Langlois, Slow Viscous Flow, Macmillan, New York, NY, USA, 1964.
12 J. Qiao and X. Wang, “On p-harmonic univalent mappings,” Acta Mathematica Scientia, vol. 32, pp.
588–600, 2012 Chinese.
10
Abstract and Applied Analysis
13 A. Ganczar, “On harmonic univalent mappings with small coefficients,” Demonstratio Mathematica,
vol. 34, no. 3, pp. 549–558, 2001.
14 H. Silverman, “Harmonic univalent functions with negative coefficients,” Journal of Mathematical
Analysis and Applications, vol. 220, no. 1, pp. 283–289, 1998.
15 J. M. Jahangiri, “Harmonic functions starlike in the unit disk,” Journal of Mathematical Analysis and
Applications, vol. 235, no. 2, pp. 470–477, 1999.
16 R. M. Ali, B. A. Stephen, and K. G. Subramanian, “Subclasses of harmonic mappings defined by
convolution,” Applied Mathematics Letters, vol. 23, no. 10, pp. 1243–1247, 2010.
17 P. L. Duren, Univalent Functions, vol. 259, Springer, New York, NY, USA, 1983.
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