# Some Properties and Regions of Variability of Affine Harmonic Mappings and Affine Biharmonic Mappings

Authors: Sh. Chen; S. Ponnusamy; X. Wang
Journal: International Journal of Mathematics and Mathematical Sciences
 (2009)

---

## Abstract

We first obtain the relations of local univalency, convexity, and linear connectedness between analytic functions and their corresponding affine harmonic mappings. In addition, the paper deals with the regions of variability of values of affine harmonic and biharmonic mappings. The regions (their boundaries) are determined explicitly and the proofs rely on Schwarz lemma or subordination.


---

## Body

## 1. Preliminaries and Main Results

A planar harmonic mapping in a simply connected domainD⊂ℂ is a complex-valued function f=u+iv defined in D such that u and v are real harmonic in D, that is,  Δu=0 and Δv=0. Here Δ represents the complex Laplacian operator(1.1)Δ=4∂2∂z∂z¯:=∂2∂x2+∂2∂y2.
The mapping f has a canonical decomposition f=h+g¯, where h and g are analytic (holomorphic) in D  [1, 2]. Lewy’s theorem tells us that a harmonic mapping f is locally univalent in D if and only if its Jacobian Jf(z)≠0 for each z∈D [3].A four-time  continuously differentiable complex-valued functionF=u+iv in D is biharmonic if and only if the Laplacian of F is harmonic. Note that ΔF is harmonic in D if F satisfies the biharmonic equation Δ(ΔF)=0.It is a simple exercise to see that every biharmonic mappingF in a simply connected domain D has the representation(1.2)F=|z|2G+K,
where G and K are some complex-valued harmonic functions in D. Recently, the class of biharmonic mappings has been studied by a number of authors [4–8]. It is known that a harmonic function of an analytic function is harmonic, but an analytic function of a harmonic function needs not be harmonic. Moreover, the only univalent harmonic mappings of ℂ onto ℂ are the affine mappings g(z)=βz+γz¯+η, where |β|≠|γ| (see [2, Section 2.4]).We denote the class of analytic functions in the unit disk𝔻={z∈ℂ:|z|<1} by ℋ, and we think of ℋ as a topological vector space endowed   with the topology of uniform convergence over compact subsets of 𝔻.Definition 1.1.
A functionF is said to be an affine harmonic mapping of f∈ℋ if and only if
(1.3)F:=Fα(f)=f+αf¯
for some α∈ℂ with |α|<1. If f is biharmonic, then Fα(f) is called the affine biharmonic mapping of f.It follows easily that every affine harmonic mapping is harmonic. Also, every affine biharmonic mapping is biharmonic. Definition1.1 is  motivated by the following result.Theorem 1 A (see [9, Theorem]).
For a harmonic mappingf, the following are equivalent:  (1)
The Schwarzian derivativeS(f)=0;(2)
f=Fα(h) for some (analytic) Möbius transformation h;(3)
f takes circles to ellipses.HereS(f) denotes the Schwarzian derivative of the harmonic mapping f=h+g¯ defined by(1.4)S(f)=2{(logρ)zz-((logρ)z)2},ρ=|h′|+|g′|.A domainD⊂ℂ is said to be  M-linearly connected if there exists a positive constant M such that any two points w1,w2∈D are joined by a path γ⊂D with(1.5)ℓ(γ)≤M|w1-w2|
or equivalently(1.6)diam(γ)≤M|w1-w2|,
where ℓ(γ) denotes the Euclidean length of γ (cf. [10]). A complex-valued function f is M-linearly connected if f(𝔻) is an  M-linearly connected domain.In a recent paper, Chuaqui and Hernández [11] investigated the relation between the univalency of planar harmonic mappings f and the linear connectedness of f(𝔻). They proved the following theorem.Theorem 1 B (see [11, Theorem 1]).
Leth:𝔻→ℂ be a holomorphic univalent map. Then there exists a constant c>0 such that every harmonic mapping f=h+g¯ with dilation |ω|<c is univalent in 𝔻 if and only if h(𝔻) is an  M-linearly connected domain, where M can be taken to be 1/c.We say that a complex-valued functiong defined on 𝔻 is said to be convex in 𝔻 if and only if g(𝔻) is a convex domain. A well-known analytic characterization gives that if f is analytic in 𝔻, then f is convex in 𝔻 if and only if(1.7)Re(1+zf′′(z)f′(z))>0,z∈𝔻.The main of this paper is twofold. We first discuss the close relationships that exist among the local univalency, convexity, and linear connectedness off and its corresponding Fα(f). One of our main results follows.Lemma 1.2.
Letf∈ℋ. Then  (1)
Fα(f) is locally univalent if and only if f is locally univalent;(2)
Fα(f) is convex if and only if f is convex, where f is locally univalent;(3)
Fα(f) is M2-linearly connected if and only if f is M1-linearly connected, where f is univalent and the constants M1 and M2 depend only on each other and α.Let𝒮0 denote the class of analytic functions f in 𝔻 with f(0)=0, and(1.8)𝒮10={f∈𝒮0:|Re(f)|<1}.A functionf∈ℋ is called a Bloch function if(1.9)∥f∥ℬ=supz∈𝔻(1-|z|2)|f′(z)|<∞.
Then the set of all Bloch functions form a complex Banach space ℬ with the norm(1.10)|f(0)|+∥f∥ℬ;
see [12]. The following result shows that 𝒮10⊂ℬ.Proposition 1.3.
Iff∈𝒮10, then ∥f∥ℬ≤4/π. The constant 4/π is sharp.Proof.
Letf∈𝒮10 and let  P be defined by P=g∘h∘f, where
(1.11)g(z)=1-z1+z,h(z)=exp((iπ2)z).
It follows that P(z)=itan((π/4)f(z)) and P is analytic mapping of the unit disk 𝔻 to 𝔻 with
(1.12)P'(z)=iπ4(1-P2(z))f′(z),z∈𝔻.
Also, by the classical Schwarz-Pick lemma, we have
(1.13)(1-|z|2)|P′(z)|≤1-|P(z)|2,z∈𝔻,
which gives
(1.14)π4(1-|z|2)|1-P2(z)||f′(z)|≤1-|P(z)|2,
whence
(1.15)(1-|z|2)|f′(z)|≤4π1-|P(z)|2|1-P(z)2|≤4π.
Thus, we have ∥f∥ℬ≤4/π.
For the proof of sharpness part, we consider(1.16)f(z)=-2iπlog(1+eiαz1-eiαz),α∈ℝ.
Then f∈𝒮0 with |Re(f(z))|<1 for z∈𝔻 and
(1.17)f′(z)=-i4π(eiα1-e2iαz2),
which gives ∥f∥ℬ=4/π. The proof is complete.LetSu={f:f∈𝒮10andfisunivalent}∪{0}.For any fixedz0∈𝔻∖{0} and λ∈ℂ with 0<|λ|<1, we define(1.18)V(z0)={f(z0):f∈𝒮u},Vℋ(z0)={Fα(f)(z0):f∈𝒮u},V(z0,λ)={f(z0):f∈𝒮10,f′(0)=4λπ},Vℋ(z0,λ)={Fα(f)(z0):f∈𝒮10,f′(0)=4λπ}.Letℬℋ denote the class of all functions f that are biharmonic but not harmonic in 𝔻 with the normalization fz¯(0)=1 and |f(z)|≤1. For any fixed z0∈𝔻, let(1.19)Vℬℋ(z0)={Fα(f)(z0):f∈ℬℋ}.Recently, by using the Herglotz representation formula for analytic functions, Yanagihara [13], and later in a number of papers Ponnusamy and Vasudevarao (see, e.g., [14]), discussed region of variability problems for a number of classical subclasses of univalent and analytic functions in the unit disk 𝔻. Because the class of harmonic univalent mappings includes the class of conformal mappings, it is natural to study the class of harmonic mappings and the class of biharmonic mappings. In this paper, we also discuss the region of variability problems for the class of affine harmonic mappings and affine biharmonic mappings, respectively. By a different method of proof, we will also determine the regions of variability for Vℋ(z0), Vℋ(z0,λ), and Vℬℋ(z0). The following are our results.Theorem 1.4.
The boundary∂Vℋ(z0) of Vℋ(z0) is the Jordan curve given by
(1.20)(-π,π]∋θ↦-i2π(log1+eiθz01-eiθz0+αlog1+eiθz01-eiθz0¯).Theorem 1.5.
The boundary∂Vℋ(z0,λ) of Vℋ(z0,λ) is the Jordan curve given by
(1.21)(-π,π]∋θ↦-i2π(log1+z0iδ(eiθz0,λ)1-z0iδ(eiθz0,λ)+αlog1+z0iδ(eiθz0,λ)1-z0iδ(eiθz0,λ)¯),
where
(1.22)δ(eiθz,λ)=eiθz+λ1+eiθzλ¯.The following two lemmas are useful for the proofs of Theorems1.4 and 1.5.Lemma 1.6.
Both setsV(z0) and Vℋ(z0) are compact and convex.Proof.
Obviously, both setsV(z0) and Vℋ(z0) are convex. Since 𝒮u is closed, the compactness of sets V(z0) and Vℋ(z0) easily follows from Proposition 1.3.The argument as in the proof of Lemma1.6 yields the following.Lemma 1.7.
Both setsV(z0,λ) and Vℋ(z0,λ) are compact and convex.Finally, we haveTheorem 1.8.
The boundary∂Vℬℋ(z0) of Vℬℋ(z0) is an ellipse.Proofs of Lemma1.2, and Theorems 1.4, 1.5, and 1.8 will be  presented in Section 2.


## 2. Proofs of Main Results

Proof of Lemma1.2.
(1) We see that JF=(1-|α|2)Jf and the proof of Lemma 1.2(1) follows.
(2) Let f be analytic and locally univalent in 𝔻, and z=reit, where r∈(0,1) and t∈[0,2π). Then for any fixed r∈(0,1),
(2.1)dFα(f)(reit)dt=ireitf′(reit)-iαre-itf′(reit)¯=i(zf'(z)-αzf′(z)¯).
Since the curvature of Fα(f) is defined by
(2.2)𝒦Fα(f)=d(arg(dFα(f)/dt))ds=d(arg(dFα(f)/dt))dt/dsdt,
we see that
(2.3)|dFα(f)dt|𝒦Fα(f)=ddt(argdFα(f)(reit)dt)=Re(zf′+z2f′′+αzf′¯+αz2f′′¯zf′-αzf′¯).
Multiplying both sides by
(2.4)|zf′(z)-αzf′(z)¯|2=(zf′(z)-αzf′(z)¯)(zf′(z)¯-α¯zf′(z)),
and then simplifying the resulting expression on the right give
(2.5)1|z|2|dFα(f)dt|3𝒦Fα(f)=|f′(z)|2(1-|α|2)Re(1+zf′′(z)f′(z)).
The proof of Lemma Lemma 1.2(2) follows from (1.7).
(3) For the proof of the sufficiency, let f(𝔻) be M1-linearly connected and
(2.6)H(w)=Fα(f)(f-1(w))=w+αw¯,
where w∈f(𝔻). For any distinct points w1,w2∈f(𝔻), there exists a path γ⊂f(𝔻) joining w1 and w2 such that
(2.7)ℓ(γ)≤M1|w1-w2|.
LetΓ=Fα(f)(γ). Then we have
(2.8)ℓ(Γ)=∫Γ|dH(w)|=∫γ|Hwdw+Hw¯dw¯|≤ℓ(γ)(1+|α|)≤(1+|α|)M1|w1-w2|.
On the other hand,
(2.9)|H(w1)-H(w2)|=|w1+αw1¯-w2-αw2¯|≥(1-|α|)|w1-w2|.
Hence
(2.10)ℓ(Γ)≤M1(1+|α|)1-|α||H(w1)-H(w2)|.
For the proof of necessity, since Fα(f) is univalent, we define
(2.11)G=f∘Fα(f)-1.
By the chain rule, we have
(2.12)Gw=f′·(Fα(f)-1)w,Gw¯=f′·(Fα(f)-1)w¯.
Differentiating both sides of the equation, Fα(f)-1(Fα(f)(z))=z yields the following relations:
(2.13)(Fα(f)-1)w·f′+α¯(Fα(f)-1)w¯·f′=1,α(Fα(f)-1)w·f′¯+(Fα(f)-1)w¯·f′¯=0.
Solving these two equations yields
(2.14)(Fα(f)-1)w=1(1-|α|2)f′,(Fα(f)-1)w¯=-α(1-|α|2)f′.
It follows from (2.12) that
(2.15)|Gw|+|Gw¯|=11-|α|.
For any two distinct pointsw1,w2∈Fα(f)(𝔻), since Fα(f)(𝔻) is M2-linearly connected, there exists a path γ⊂Fα(f)(𝔻) joining w1 and w2 such that
(2.16)ℓ(γ)≤M2|w1-w2|.
Now, we letΓ=G(γ). Then we see that
(2.17)ℓ(Γ)=∫Γ|dG(w)|=∫γ|Gwdw+Gw¯dw¯|≤M2|w1-w2|1-|α|.
On the other hand, by the definition ofG, we have
(2.18)w1-w2=G(w1)-G(w2)+α(G(w1)¯-G(w2)¯)
from which we obtain
(2.19)|w1-w2|≤|G(w1)-G(w2)|(1+|α|)
and therefore,
(2.20)ℓ(Γ)≤M2(1+|α|)|G(w1)-G(w2)|1-|α|.
The proof of the theorem is complete.Before we indicate the proof of Theorem1.4, we need to determine the region of variability V(z0).Lemma 2.1.
The boundary∂V(z0) of V(z0) is the Jordan curve given by
(2.21)(-π,π]∋θ↦-2iπlog(1+eiθz01-eiθz0).Proof.
Forf∈𝒮u, let P=itan((π/4)f). Then P(𝔻)⊂𝔻. Using the Schwarz lemma, we have |P(z)|≤|z|. That is,
(2.22)|exp(π2if(z))-1|≤|z||exp(π2if(z))+1|
or equivalently
(2.23)|exp(π2if(z))-1+|z|21-|z|2|≤2|z|1-|z|2.
This yields that for each z0∈𝔻,
(2.24)exp(π2if(z0))∈𝔻¯(C(z0),R(z0))={w∈ℂ:|w-C(z0)|≤R(z0)},
where
(2.25)C(z0)=1+|z0|21-|z0|2,R(z0)=2|z0|1-|z0|2.
Consider
(2.26)fz0(z)=-2iπlog1+z0z1-z0z.
Then fz0∈𝒮u and, for z0∈𝔻∖{0}, the function fz0 is a nonconstant analytic function in 𝔻. Hence it is an open mapping. Thus fz0(𝔻)⊂V(z0). Next we will show that fz0(z)∈∂V(z0) for all z∈∂𝔻. An elementary computation shows that
(2.27)exp(π2ifz0(z))-1+|z0|21-|z0|2=1+zz01-zz0-1+|z0|21-|z0|2=1z(z-z0¯z¯-z0)2z01-|z0|2,
where |z|=1. Hence
(2.28)exp(π2ifz0(z))∈∂𝔻(C(z0),R(z0)).
Since
(2.29)exp(π2ifz0(z))∈exp(iπ2V(z0))⊂𝔻¯(C(z0),R(z0)),exp(π2iV(z0))={exp(π2iξ):ξ∈V(z0)},
we have
(2.30)exp(π2ifz0(z))∈∂exp(iπ2V(z0)),
provided z∈∂𝔻. Obviously, G=exp((π/2)ifz0) is univalent in ∂𝔻. By Lemma 1.6, we know that ∂exp((iπ/2)V(z0)) is a simple closed curve. Therefore, the boundary ∂V(z0) is the Jordan curve given by (2.21). The proof of the lemma is complete.Proof of Theorem1.4.
The proof of Theorem1.4 is an immediate consequence of Lemmas 1.2,   1.6, and 2.1.Clearly, for the proof of Theorem1.5, it suffices to describe the region of variability V(z0,λ).Lemma 2.2.
The boundary∂V(z0,λ) of V(z0,λ) is the Jordan curve given by
(2.31)(-π,π]∋θ↦-2iπlog(1+z0iδ(eiθz0,λ)1-z0iδ(eiθz0,λ)),
where
(2.32)δ(cz,λ)=cz+λ1+czλ¯.Proof.
Letf∈𝒮10 with f'(0)=4λ/π. It follows from the classical Schwarz lemma for analytic functions that
(2.33)|tan((π/4)f(z))/z-λ1-λ¯(tan((π/4)f(z))/z)|≤|z|,
which, after some computation, is equivalent to
(2.34)|e(π/2)if(z)-A(z,λ)e(π/2)if(z)+B(z,λ)|≤|z||T(z,λ)|,
where
(2.35)A(z,λ)=1+iλz1-iλz,B(z,λ)=λ¯+iziz-λ¯,T(z,λ)=iz-λ¯1-iλz.
According to the Schwarz lemma, we see that the equality sign occurs in (2.33) if and only if f=Qeiθ,λ for some θ∈ℝ, where
(2.36)Qc,λ(z)=-2iπlog1+ziδ(cz,λ)1-ziδ(cz,λ).
Further calculations show that the inequality (2.34) is equivalent to
(2.37)|exp(π2if(z))-C(z,λ)|≤r(z,λ),
where
(2.38)C(z,λ)=A(z,λ)+|z|2|T(z,λ)|2B(z,λ)1-|z|2|T(z,λ)|2,r(z,λ)=|z||T(z,λ)||A(z,λ)+B(z,λ)|1-|z|2|T(z,λ)|2.
It follows from (2.35) that
(2.39)1-|z|2|T(z,λ)|2=(1-|z|2)[1+|z|2-2Re(iλz)]|1-iλz|2,(2.40)A(z,λ)+B(z,λ)=2iz(1-|λ|2)(1-iλz)(iz-λ¯),
so that r(z,λ) simplifies to the form
(2.41)r(z,λ)=2|z|2(1-|λ|2)|1-iλz|2(1-|z|2|T(z,λ)|2).
Further,
(2.42)A(z,λ)+|z|2|T(z,λ)|2B(z,λ)=|z|4-2|λz|2+1+2i(1-|z|2)Re(λz)|1-iλz|2.
From (2.37), we have
(2.43)exp(iπ2V(z0,λ))⊂𝔻¯(C(λ,z0),r(λ,z0))≜{w∈ℂ:|w-C(λ,z0)|≤r(λ,z0)}.
Equation (2.36) gives
(2.44)exp(π2iQc,λ(z))=1+z(λ¯c+iλ)+ciz21+z(λ¯c-iλ)-ciz2,
and so, we infer from (2.35) that
(2.45)exp(π2iQc,λ(z))-A(z,λ)=2icz2(1-|λ|2)(1-ziλ)[1+z(λ¯c-iλ)-ciz2],exp(π2iQc,λ(z))+B(z,λ)=2iz(1-|λ|2)(zi-λ¯)[1+z(λ¯c-iλ)-ciz2].
As
(2.46)w-C(z,λ)=11-|z|2|T(z,λ)|2{(w-A(z,λ))-|z|2|T(z,λ)|2(w+B(z,λ))},
the above equalities together with (2.39), (2.40), and (2.42) yield that
(2.47)exp(π2iQc,λ(z))-C(z,λ)=2iz(1-|λ|2)[cz+icλ¯|z|2+λ|z|2+iz¯|z|2][1+z(cλ¯-iλ)-icz2]|1-ziλ|2(1-|z|2|T(z,λ)|2).
Substituting c by eiθ in the last equality, we see from (2.41) that
(2.48)|exp(π2iQeiθ,λ(z))-C(z,λ)|=r(z,λ)|1+z¯(e-iθλ+iλ¯)+ie-iθz¯2||1+z(eiθλ¯-iλ)-ieiθz2|
which implies that
(2.49)exp(π2iQeiθ,λ(z0))∈exp(iπ2V(z0,λ))⊂𝔻¯(C(λ,z0),r(λ,z0)),
whence
(2.50)exp(π2iQeiθ,λ(z0))∈∂exp(iπ2V(z0,λ)).
We claim that the closed curve
(2.51)(-π,π]∋θ↦exp(π2iQeiθ,λ(z0))
is simple.
For the proof of this claim, we suppose that there areθ1,θ2∈(-π,π] with θ1≠θ2 such that
(2.52)exp(π2iQeiθ1,λ(z0))=exp(π2iQeiθ2,λ(z0)).
By (2.36), we obtain that θ1=θ2 which is a contradiction and this completes the proof of our claim.
SinceV(z0,λ) is a compact convex subset of ℂ and its interior is nonempty, we see that its boundary ∂V(z0,λ) is a simple closed curve. It follows from (2.40) that the curve
(2.53)(-π,π]∋θ↦exp(π2iQeiθ,λ(z0))
is a subcurve of ∂exp((iπ/2)V(z0,λ)).
The fact that a simple closed curve cannot contain any simple closed curve other than itself yields that∂V(z0,λ) is given by
(2.54)(-π,π]∋θ↦Qeiθ,λ(z0).
The proof of Lemma 2.2 is complete.Proof of Theorem1.5.
The proof of Theorem1.5 follows from Lemmas 1.2, 1.7, and 2.2.Proof of Theorem1.8 is a consequence of Theorem A and the following lemma.Lemma 2.3.
For any fixeda∈𝔻¯, let ϕa:𝔻→𝔻¯ be defined by
(2.55)ϕa(z)=z¯-a1-az,z∈𝔻,ℳ={ϕa:a∈𝔻¯},
and V0(z0)={ϕa(z0):ϕa∈ℳ}. Then ℳ⊂ℬℋ and V0(z0)=𝔻¯.Proof.
We write(2.56)ϕa(z)=z¯-a1-az=|z|2G(z,a)+H(z,a),
and so, we have ϕz¯(0)=1, where
(2.57)G(z,a)=a∑n=0∞(az)n,H(z,a)=z¯-a∑n=0∞(az)n.
Obviously, ϕa(z)=(z¯-a)/(1-az) is a biharmonic mapping and ϕa(𝔻)⊂𝔻, from which the first assertion in Lemma 2.3 follows.
Sincez0∈𝔻∖{0}, we see that gz0(w)=(z0¯-w)/(1-z0w) is a conformal automorphism of the unit disk 𝔻 and the range gz0(𝔻¯) is 𝔻¯ itself. By hypotheses, we know that for any q∈ℳ,  q(z0)∈𝔻¯. Hence V0(z0) coincides with 𝔻¯.



---

*Source: 834215-2010-02-10.xml*