# On Certain Classes of Biharmonic Mappings Defined by Convolution

Authors: J. Chen; X. Wang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We introduce a class of complex-valued biharmonic mappings, denoted byBH0(ϕk;σ,a,b), together with its subclass TBH0(ϕk;σ,a,b), and then generalize the discussions in Ali et al. (2010) to the setting of BH0(ϕk;σ,a,b) and TBH0(ϕk;σ,a,b) in a unified way.


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## Body

## 1. Introduction

A four times continuously differentiable complex-valued functionF=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 Δ(ΔF)=0 in D, where Δ represents the Laplacian operator
(1.1)Δ=4∂2∂z∂z-:=∂2∂x2+∂2∂y2.It is known that, whenD is simply connected, a mapping F is biharmonic if and only if F has the following representation:
(1.2)F(z)=∑k=12‍|z|2(k-1)Gk(z),
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
(1.3)Gk=hk+gk¯
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 in𝔻={z∈ℂ:|z|<1}. Let BH0(𝔻) denote the set of all biharmonic mappings F in 𝔻 with the following form:
(1.4)F(z)=∑k=12‍|z|2(k-1)(hk(z)+gk(z)-)=∑k=12‍|z|2(k-1)(∑j=1∞‍ak,jzj+∑j=1∞‍bk,jzj-),
with a1,1=1, a2,1=0,  b1,1=0, and b2,1=0.In [12], Qiao and Wang proved that for each F∈BH0(𝔻), if the coefficients of F satisfy the following inequality:
(1.5)∑k=12‍∑j=1∞‍(2(k-1)+j)(|ak,j|+|bk,j|)≤2,
then F is sense preserving, univalent, and starlike in 𝔻 (see [12, Theorems 3.1 and 3.2]).LetSH denote the set of all univalent harmonic mappings f in 𝔻, where
(1.6)f(z)=h(z)+g(z)¯=z+∑j=2∞‍ajzj+∑j=1∞‍bjzj¯,
with |b1|<1. In particular, we use SH0 to denote the set of all mappings in SH with b1=0. Obviously, SH0⊂BH0(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 SH*(α) consisting of all mappings f=h+g¯ such that h and g are of the form
(1.7)h(z)=z-∑j=2∞‍|aj|zj,g(z)=∑j=1∞‍|bj|zj
and satisfy the condition
(1.8)∂∂θ(argf(reiθ))=Re{zh'-zg'¯h+g¯}>α
in 𝔻, where 0≤α<1.For two analytic functionsf1 and f2, if
(1.9)f1(z)=∑j=1∞‍ajzj,f2(z)=∑j=1∞‍Ajzj,
then the convolution of f1 and f2 is defined by
(1.10)(f1*f2)(z)=f1(z)*f2(z)=∑j=1∞‍ajAjzj.By using the convolution, in [16], Ali et al. introduced the class SH0(ϕ,σ,α) of harmonic mappings in the form of (1.6) such that
(1.11)Re{z(h*ϕ)′(z)-σz(g*ϕ)′(z)¯(h*ϕ)(z)+σ(g*ϕ)(z)¯}>α
and the class SPH0(ϕ,σ,α) such that
(1.12)Re{(1+eiγ)z(h*ϕ)′(z)-σz(g*ϕ)′(z)¯(h*ϕ)(z)+σ(g*ϕ)(z)¯-eiγ}>α,
where σ∈ℝ and α∈[0,1) are constants, γ∈ℝ and ϕ(z)=z+∑n=2∞‍ϕnzn is analytic in 𝔻.Now we consider a class of biharmonic mappings, denoted byBH0(ϕk;σ,a,b), as follows: F∈BH0(𝔻) with the form (1.4) is said to be in BH0(ϕk;σ,a,b) if and only if
(1.13)Re{aΦ(z)Ψ(z)-b}>0,
where
(1.14)Φ(z)=z[(∑k=12‍|z|2(k-1)(hk*ϕk)(z))′+σ(∑k=12‍|z|2(k-1)(gk*ϕk)(z)¯)′],Ψ(z)=z′∑k=12‍|z|2(k-1)((hk*ϕk)(z)+σ(gk*ϕk)(z)¯),ϕk(z)=z+∑j=2∞‍ϕk,jzj are analytic in 𝔻 for k∈{1,2}, σ∈ℝ is a constant, a=p+ρeiγ, 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, ifϕ2=0, a=1 and b=α, then BH0(ϕk;σ,a,b) reduces to SH0(ϕ,σ,α), and if ϕ2=0, a=1+eiγ and b=α+eiγ, then BH0(ϕk;σ,a,b) reduces to SPH0(ϕ,σ,α).Further, we useTBH0(ϕk;σ,a,b) to denote the class consisting of all mappings F in BH0(ϕk;σ,a,b) with the form
(1.15)F(z)=∑k=12‍|z|2(k-1)(hk(z)+gk(z)¯),
where
(1.16)hk(z)=ak,1z-∑j=2∞‍ak,jzj,ak,j≥0,a1,1=1,a2,1=0,gk(z)=σ∑j=1∞‍bk,jzj,bk,j≥0,b1,1=b2,1=0.The object of this paper is to generalize the discussions in [16] to the setting of BH0(ϕk;σ,a,b) and TBH0(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 BH0(ϕk;σ,a,b). As a corollary, we derive a sufficient coefficient condition for mappings in BH0(𝔻) to belong to BH0(ϕk;σ,a,b). The main results are Theorems 2.1 and 2.3. In Section 3, first, we get a coefficient characterization for TBH0(ϕk;σ,a,b), and then find the extreme points of TBH0(ϕk;σ,a,b). The corresponding results are Theorems 3.1 and 3.6.


## 2. A Convolution Characterization

We begin with a convolution characterization forBH0(ϕk;σ,a,b).Theorem 2.1.
LetF∈BH0(𝔻). Then F∈BH0(ϕk;σ,a,b) if and only if
(2.1)∑k=12‍|z|2(k-1)(hk*ϕk)(z)*(z+((ax-a+2b)/(2a-2b))z2(1-z)2)-σ∑k=12‍|z|2(k-1)(gk*ϕk)(z)¯*(((ax+b)/(a-b))z¯-((ax-a+2b)/(2a-2b))z¯2(1-z¯)2)≠0,
for all z∈𝔻∖{0} and all x∈ℂ with |x|=1.Proof.
By definition, a necessary and sufficient condition for a mappingF in BH0(𝔻) to be in BH0(ϕk;σ,a,b) is given by (1.13). Let
(2.2)G(z)=1a-b(aΦ(z)Ψ(z)-b).
Then G(0)=1, and so the condition (1.13) is equivalent to
(2.3)G(z)≠x-1x+1,
for all z∈𝔻∖{0} and all x∈ℂ with |x|=1 and x≠-1. Obviously, (2.3) holds if and only if
(2.4)a(x+1)Φ(z)-b(x+1)Ψ(z)-(a-b)(x-1)Ψ(z)≠0.
Straightforward computations show that(2.5)a(x+1)Φ(z)-b(x+1)Ψ(z)-(a-b)(x-1)Ψ(z)=a(x+1)z'∑k=12‍|z|2(k-1)(z+∑j=2∞‍jak,jϕk,jzj-σ∑j=2∞‍jbk,jϕk,jzj¯)-(ax-a+2b)z'∑k=12‍|z|2(k-1)(z+∑j=2∞‍ak,jϕk,jzj+σ∑j=2∞‍bk,jϕk,jzj¯)=z′∑k=12‍|z|2(k-1)(hk*ϕk)(z)*(2(a-b)z+(ax-a+2b)z2(1-z)2)-σz'∑k=12‍|z|2(k-1)(gk*ϕk)(z)¯*(2(ax+b)z¯-(ax-a+2b)z¯2(1-z¯)2),
from which we see that (2.3) is true if and only if so is (2.1). The proof is complete.Remark 2.2.
Ifh2=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 Theorem2.1, we derive a sufficient condition for mappings in BH0(𝔻) to be in BH0(ϕk;σ,a,b) in terms of their coefficients.Theorem 2.3.
LetF∈BH0(𝔻). Then F∈BH0(ϕk;σ,a,b) if
(2.6)∑k=12‍∑j=2∞‍j∥a∥max-∥b∥maxa-b|ϕk,jak,j|+|σ|∑k=12‍∑j=2∞‍j∥a∥max+∥b∥maxa-b|ϕk,jbk,j|≤1,
here and in the following, ∥z∥max=maxγ∈R{|x+yeiγ|}=x+y, where z=x+yeiγ, x and y∈[0,+∞) are constants.Proof.
ForF given by (1.4), we see that
(2.7)L(z)≜|∑k=12‍|z|2(k-1)(hk*ϕk)(z)*(z+((ax-a+2b)/(2a-2b))z2(1-z)2)-σ∑k=12‍|z|2(k-1)(gk*ϕk)(z)¯*(((ax+b)/(a-b))z¯-((ax-a+2b)/(2a-2b))z¯2(1-z¯)2)|=|z+∑k=12‍|z|2(k-1)∑j=2∞‍(j+(j-1)ax-a+2b2a-2b)ϕk,jak,jzj-σ∑k=12‍|z|2(k-1)∑j=2∞‍(jax+ba-b-(j-1)ax-a+2b2a-2b)ϕk,jbk,jzj¯|.IfF is the identity, obviously, L(z)=|z|.IfF is not the identity, then
(2.8)L(z)>|z|(1-∑k=12‍∑j=2∞‍j∥a∥max-∥b∥maxa-b|ϕk,jak,j|-|σ|∑k=12‍∑j=2∞‍j∥a∥max+∥b∥maxa-b|ϕk,jbk,j|).Hence the assumption implies that L(z)>0 for all z∈𝔻∖{0} and all x∈ℂ with |x|=1. It follows from Theorem 2.1 that F∈BH0(ϕk;σ,a,b).Remark 2.4.
Ifh2=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 forTBH0(ϕk;σ,a,b).Theorem 3.1.
Letϕk(z)=z+∑j=2∞‍ϕk,jzj with ϕk,j≥0, and let F be of the form (1.15). Then F∈TBH0(ϕk;σ,a,b) if and only if
(3.1)∑k=12‍∑j=2∞‍j∥a∥max-∥b∥maxa-bϕk,jak,j+σ2∑k=12‍∑j=2∞‍j∥a∥max+∥b∥maxa-bϕk,jbk,j≤1.Proof.
By similar arguments as in the proof of Theorem2.3, we see that it suffices to prove the “only if” part. For F∈TBH0(ϕk;σ,a,b), obviously, (1.13) is equivalent to
(3.2)Re{P(z)-Q(z)z-∑k=12‍|z|2(k-1)(∑j=2∞‍ak,jϕk,jzj-σ2∑j=2∞‍bk,jϕk,jz¯j)}>0
in 𝔻, where
(3.3)P(z)=(a-b)z-∑k=12‍|z|2(k-1)∑j=2∞‍(aj-b)ak,jϕk,jzj,Q(z)=σ2∑k=12‍|z|2(k-1)∑j=2∞‍(aj+b)bk,jϕk,jz¯j.
Letting z→1- through real values leads to the desired inequality. So the proof is complete.Remark 3.2.
Ifh2=g2=0, a=1, and b=α, then Theorem 3.1 coincides with Theorem 3.1 in [16].It follows from Theorem3.1 that we have the following.Corollary 3.3.
Letϕk(z)=z+∑j=2∞‍ϕk,jzj with ϕk,j≥ϕ1,2>0(k∈{1,2},j≥2) and |σ|≥(2∥a∥max-∥b∥max)/(2∥a∥max+∥b∥max). If F∈TBH0(ϕk;σ,a,b), then for |z|=r<1, one has
(3.4)r-a-b(2∥a∥max-∥b∥max)ϕ1,2r2≤|F(z)|≤r+a-b(2∥a∥max-∥b∥max)ϕ1,2r2.
The result is sharp with equality for mappings
(3.5)F(z)=z-a-b(2∥a∥max-∥b∥max)ϕ1,2z2.Theorem3.1 and Corollary 3.3 imply the followingCorollary 3.4.
Under the hypotheses of Corollary3.3, one has that TBH0(ϕk;σ,a,b) is closed under the convex combination.Definition 3.5.
LetX 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-t)z (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 ofTBH0(ϕk;σ,a,b).Theorem 3.6.
Let(1)
h11(z)=z,(2)
h21(z)=g11(z)=g21(z)=0,(3)
hkj(z)=z-|z|2(k-1)((a-b)/(j∥a∥max-∥b∥max)ϕk,j)zj for k∈{1,2} and all j≥2,(4)
gkj(z)=z+|z|2(k-1)((a-b)/σ(j∥a∥max+∥b∥max)ϕk,j)z¯j for k∈{1,2} and all j≥2. Under the hypotheses of Corollary 3.3, one has that F∈TBH0(ϕk;σ,a,b) if and only if it can be expressed as
(3.6)F(z)=∑k=12‍∑j=1∞‍(xkjhkj(z)+ykjgkj(z)),
where x21=y11=y21=0, all other xkj and ykj are nonnegative, and ∑k=12‍∑j=1∞‍(xkj+ykj)=1.
In particular, the extreme points ofTBH0(ϕk;σ,a,b) are all mappings hkj and gkj listed in (1), (3), and (4) above.Proof.
It follows from the assumptions that(3.7)F(z)=∑k=12‍∑j=1∞‍(xkjhkj(z)+ykjgkj(z))=z-∑k=12‍∑j=2∞‍|z|2(k-1)a-b(j∥a∥max-∥b∥max)ϕk,jxkjzj+σ∑k=12‍∑j=2∞‍|z|2(k-1)a-bσ2(j∥a∥max+∥b∥max)ϕk,jykjz¯j,
whence
(3.8)∑k=12‍∑j=2∞‍(j∥a∥max-∥b∥max)a-bϕk,j·a-b(j∥a∥max-∥b∥max)ϕk,jxkj+σ2∑k=12‍∑j=2∞‍(j∥a∥max+∥b∥max)a-bϕk,j·a-bσ2(j∥a∥max+∥b∥max)ϕk,jykj=∑k=12‍∑j=2∞‍xkj+∑k=12‍∑j=2∞‍ykj≤1,
and so Theorem 3.1 implies that F∈TBH0(ϕk;σ,a,b).
Conversely, assumeF∈TBH0(ϕk;σ,a,b), and let
(3.9)x21=y11=y21=0,x11=1-∑k=12‍∑j=2∞‍xkj-∑k=12‍∑j=2∞‍ykj,xkj=(j∥a∥max-∥b∥max)ϕk,jak,ja-b,ykj=σ2(j∥a∥max+∥b∥max)ϕk,jbk,ja-b,
for k∈{1,2} and all j≥2. Then
(3.10)F(z)=z-∑k=12‍|z|2(k-1)∑j=2∞‍ak,jzj+σ∑k=12‍|z|2(k-1)∑j=2∞‍bk,jz¯j.
The proof of the theorem is complete.Remark 3.7.
Ifh2=g2=0, a=1 and b=α, then Theorem 3.6 coincides with Theorem 3.2 in [16].



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