Using subordination, we determine the regions of variability of several
subclasses of harmonic mappings. We also graphically illustrate the regions of variability for
several sets of parameters for certain special cases.

Introduction

A planar harmonic mapping in a simply connected domain D⊂ℂ is a complex-valued function f=u+iv defined in D for which both u and v are real harmonic in D, that is, Δf=4fzz¯=0, where Δ represents the Laplacian operator. The mapping f can be written as a sum of an analytic and antianalytic functions, that is, f=h+g¯. We refer to [1] and the book of Duren [2] for many interesting results on planar harmonic mappings.

We note that the composition f∘ϕ of a harmonic function f with an analytic function ϕ is harmonic, but this is not true for the function ϕ∘f, that is, an analytic function of a harmonic function need not be harmonic. It is known that [2, Theorem 2.4] the only univalent harmonic mappings of ℂ onto ℂ are the affine mappings g(z)=βz+γz¯+η(|β|≠|γ|). Motivated by the work of [3], we say that F is an affine harmonic mapping of a harmonic mapping of f if and only if F has the form

F:=Fα(f)=f+αf¯
for some α∈ℂ with |α|<1. Obviously, an affine transformation applied to a harmonic mapping is again harmonic. The affine harmonic mappings Fα(f) and f share many properties in common (see [4]).

Let ℋ denote the class of analytic functions in the unit disk 𝔻={z∈ℂ:|z|<1}, and 𝒜0={h∈ℋ:h(0)=0}. Also, let 𝒮0 be the subclass of 𝒜0 consisting of functions that are univalent in 𝔻. For a given ϕ∈𝒮0, we will denote by 𝒜0(ϕ) and 𝒮0(ϕ) the subsets defined by {h∈𝒜0:h≺ϕ} and {h∈𝒮0:h≺ϕ}∪{0}, respectively. From now onwards, we use the notation f≺g, or, f(z)≺g(z) in 𝔻 for analytic functions f and g on 𝔻 to mean the subordination, namely there exists ω∈ℬ0 such that f(z)=g(ω(z)). Here ℬ0 denotes the class of analytic maps ψ of the unit disk 𝔻 into itself with the normalization ψ(0)=0. We remark that if g is univalent in 𝔻, then the subordination f≺g is equivalent to the condition that f(0)=g(0) and f(𝔻)⊂g(𝔻). This fact will be used in our investigation. Moreover, the special choices of ϕ have been the subjects of extensive studies; we suggest that the reader to consult the books of Pommerenke [5], Duren [6] and of Miller and Mocanu [7] for general back ground material.

We denote by 𝒜a,b the class of functions f∈ℋ with f(0)=(b-a)/2, and -a<Ref(z)<b for z∈𝔻. We note that if a>0, then each function f∈𝒜a,a obviously satisfy the normalization condition f(0)=0. A function f∈ℋ is called a Bloch function if

∥f∥ℬ=supz∈𝔻(1-|z|2)|f′(z)|<∞.
Then the set of all Bloch functions forms a complex Banach space ℬ with the norm ∥·∥ given by

∥f∥=|f(0)|+∥f∥ℬ,
see [8]. Every bounded function in ℋ is Bloch, but there are unbounded Bloch functions, as can be seen also from the following result which shows that 𝒜a,b⊂ℬ.

Proposition 1.

If f∈𝒜a,b, then ∥f∥ℬ≤2(b+a)/π. The constant 2(b+a)/π is sharp. In particular, if f∈𝒜a,a then ∥f∥ℬ≤4a/π and the constant 4a/π is sharp.

Proof.

Let
P(z)=b+aiπlog(1+z1-z)+b-a2,z∈𝔻.
Then P(0)=(b-a)/2,
P′(z)=2(b+a)iπ(1-z2)
and P maps 𝔻 univalently onto the vertical strip {w:-a<Rew<b}, and ∥P∥ℬ=2(b+a)/π. Consequently, if f∈𝒜a,b, then we have f≺P and so, there exists a Schwarz function ω∈ℬ0 such that f(z)=P(ω(z)). Thus, as ω(0)=0, the Schwarz-Pick lemma gives that
(1-|z|2)|f′(z)|=(1-|z|2)|ω′(z)||P′(ω(z))|≤(1-|ω|2)|P′(ω)|≤∥P∥ℬ
so that ∥f∥ℬ≤2(b+a)/π, with equality for f(z)=Pα(z), where
Pα(z)=b+aiπlog(1+zeiα1-zeiα)+b-a2,α∈ℝ.

It may be interesting to remark that the function f(z)=∑n=1∞z2n belongs to ℬ [9, Theorem 1] is a good example of a Bloch function which is not in Hp-space for any p. Bloch functions are intimately close with univalent functions (see [5]).

In order to state our main results, we introduce some basics. For given a,b>0, let 𝒮a,b be the class of functions f∈𝒜0 and -a<Ref(z)<b for z∈𝔻. Now, we define

𝒮a,b,u={f:f∈𝒮a,bandfisunivalent}∪{0}.
We note that each function in 𝒮a,b has the normalization f(0)=0. For any fixed z0∈𝔻∖{0} and λ∈ℂ with 0<|λ|<1, we consider the following sets:

We now recall the definition of subordination for the harmonic case from [10, page 162]. Let f and F be two harmonic functions defined on 𝔻. We say f is subordinate to F, denoted by f≺F, if f(z)=F(ω(z)), where ω∈ℬ0. Obviously, if f1 and f2 are two harmonic functions in 𝔻, then

f1≺f2⇔Fα(f1)≺Fα(f2).
Here we see that α¯ is the analytic dilatation for both Fα(f1) and Fα(f2).

For each fixed z0∈𝔻, using extreme function theory, it has been shown by Grunsky (see, e. g., Duren [6, Theorem 10.6]) that the region of variability of

V𝒮(z0)={logf(z0)z0:f∈𝒮}
is precisely a closed disk, where 𝒮={f∈𝒮0:f′(0)=1}. Recently, by using the Herglotz representation formula for analytic functions, many authors have discussed region of variability problems for a number of classical subclasses of univalent and analytic functions in the unit disk 𝔻 (see [11, 12] and the references therein). Because the class of harmonic univalent mappings includes the class of conformal mappings, it is natural to study the class of harmonic mappings. In the following, we will use the method of subordination and determine the regions of variability for Vϕ,ℋ(z0), Vϕ,ℋ(z0,λ), Vℋ,𝒮a,b,u(z0) and Vℋ,𝒮a,b(z0,λ), respectively.

Theorem 1.

The boundary ∂Vϕ,ℋ(z0) of Vϕ,ℋ(z0) is the Jordan curve given by
(-π,π]∋θ↦ϕ(eiθz0)+αϕ(eiθz0)¯.

Proof.

We define Vϕ(z0)={f(z0):f∈𝒮0(ϕ)}. In order to determine the set Vϕ(z0), we first recall that each f∈𝒮0(ϕ)∖{0} can be written as f(z)=ϕ(ω(z)) for some ω∈ℬ0∖{0}. By the Riemann mapping theorem, ω=ϕ-1∘f is univalent and analytic in 𝔻 with ω(0)=0. It follows from the classical Schwarz lemma that for any ω∈ℬ0, we have |ω(z)|≤|z| in 𝔻. Because, in our situation ω is also univalent in 𝔻, we easily show that the region of variability
Vℬ(z0)={ω(z0):ω∈(ℬ0∩𝒮0)∪{0}}
coincides with the set {z:|z|≤|z0|}. Hence the region of variability Vϕ(z0) is precisely the set {ϕ(z):|z|≤|z0|}. We remark that Vϕ(z0) depends only on |z0|, because 𝒮0 is preserved under rotation and therefore, we may assume that 0<z0<1. Finally, the region of variability Vϕ,ℋ(z0) follows from Vϕ(z0). The proof of this theorem is complete.

There are many choices for ϕ for which Theorem 1 is applicable. For example, if we choose ϕ to be

ϕ(z)=(1+z1-z)β-1,
for some 0<β≤2, then we have following result from Theorem 1.

Corollary 1.

The boundary ∂Vϕ0,ℋ(z0) of Vϕ0,ℋ(z0) is the Jordan curve given by
(-π,π]∋θ↦(1+eiθz01-eiθz0)β+α(1+eiθz01-eiθz0)β¯-1-α.

Theorem 2.

The boundary ∂Vϕ,ℋ(z0,λ) of Vϕ,ℋ(z0,λ) is the Jordan curve given by
(-π,π]∋θ↦ϕ(z0δ(eiθz0,λ))+αϕ(z0δ(eiθz0,λ))¯,
where
δ(cz,λ)=cz+λ1+czλ¯(c∈𝔻¯).

Proof.

Let f∈𝒜0 such that f≺ϕ for some ϕ∈𝒮0. Because f≺ϕ, there exists a Schwarz function ω=ϕ-1∘f∈ℬ0 with ω′(0)=f′(0)/ϕ′(0)=λ, where |λ|≤1. Therefore, for any fixed z0∈𝔻∖{0} and λ∈ℂ with 0<|λ|≤1, it is natural to consider the set
Vϕ(z0,λ)={f(z0):f∈𝒜0(ϕ),f′(0)=λϕ′(0)}.
First, we determine Vϕ(z0,λ). Then the determination of the set Vϕ,ℋ(z0,λ) follows from Vϕ(z0,λ). Now, we define
Fω(z)=ω(z)/z-λ1-(λ¯ω(z)/z),i.e.,ω(z)=z(Fω(z)+λ)1+Fω(z)λ¯.
We observe that Fω∈ℬ0. By the Schwarz lemma, we have |Fω(z)|≤|z|. If we set
ℬ0λ={Fω:ω∈ℬ0,ω′(0)=λ}
then the region of variability {ω(z0):ω∈ℬ0λ} coincides with the set {z:|z|≤|z0|}. It follows from the two expressions in (19) that Vϕ(z0,λ) coincides with the set
{ϕ(z0δ(z,λ)):|z|≤|z0|,whereδ(z,λ)=z+λ1+zλ¯}.
The proof of this theorem is complete.

The case λ=0 of Theorem 2 gives the following result.

Corollary 2.

The boundary ∂Vϕ,ℋ(z0,0) of Vϕ,ℋ(z0,0) is the Jordan curve given by
(-π,π]∋θ↦ϕ(z02eiθ)+αϕ(z02eiθz0)¯.

If ϕ0(z) is given by (14) for some 0<β≤2, then ϕ0′(0)=2β and Vϕ0,ℋ(z0,λ) reduces to

Vϕ0,ℋ(z0,λ)={Fα(f)(z0):f∈𝒜0(ϕ0),f′(0)=2βλ}
and the corresponding ω(z) in the proof of the theorem will be precisely of the form

ω(z)=(1+f(z))1/β-1(1+f(z))1/β+1.
This observation gives the following corollary.

Corollary 3.

The boundary ∂Vϕ0,ℋ(z0,λ) of Vϕ0,ℋ(z0,λ) is the Jordan curve given by
(-π,π]∋θ↦(1+z0δ(eiθz0,λ)1-z0δ(eiθz0,λ))β+α(1+z0δ(eiθz0,λ)1-z0δ(eiθz0,λ))β¯-1-α,
where ϕ0(z) and δ(cz,λ) are given by (14) and (17), respectively.

The boundary ∂Vϕ0,ℋ(z0,0) of Vϕ0,ℋ(z0,0) is the Jordan curve given by

The boundary ∂Vℋ,𝒮a,b,u(z0) of Vℋ,𝒮a,b,u(z0) is the Jordan curve given by
(-π,π]∋θ↦a+biπ[log(1-z0eiθe-2πai/(a+b)1-z0eiθ)-αlog(1-z0eiθe-2πai/(a+b)1-z0eiθ)¯].

Proof.

We define V𝒮a,b,u(z0)={f(z0):f∈𝒮a,b,u}. It suffices to determine V𝒮a,b,u(z0) as the region of variability Vℋ,𝒮a,b,u(z0) follows from V𝒮a,b,u(z0). In order to do this, first we consider
T(z)=a+biπlogw(z),w(z)=1-ze-2πai/(a+b)1-z.
Then T(0)=0. We see that the Möbius transformation w(z) maps the open unit disk 𝔻 conformally onto the half-plane
{w=u+iv:usin(πaa+b)+vcos(πaa+b)>0}
and so, we easily obtain that T maps 𝔻 conformally onto the vertical strip {w:-a<Rew<b}. This observation shows that T∈𝒮a,b,u and is in fact an extremal function for this class.

Next, we choose an arbitrary f∈𝒮a,b,u∖{0}. Then we have f≺T and so, there exists a Schwarz function ω∈ℬ0∖{0} such that f(z)=T(ω(z)). Note that both f and T are univalent in 𝔻 and so, ω=T-1∘f is univalent in 𝔻 with ω(0)=0. It follows from the classical Schwarz lemma that |ω(z)|≤|z| in 𝔻. Because ω is also univalent in 𝔻, we obtain that the region of variability of
Vω,u(z0)={ω(z0):ω∈(ℬ0∩𝒮0)∪{0}}
coincides with the set {z:|z|≤|z0|}. Hence the region of variability V𝒮a,b,u(z0) coincides with the set
{a+biπlog(1-ze-2πai/(a+b)1-z):|z|≤|z0|}.
The proof of Theorem 3 is complete.

Theorem 4.

The boundary ∂Vℋ,𝒮a,b(z0,λ) of Vℋ,𝒮a,b(z0,λ) is the Jordan curve given by
(-π,π]∋θ↦a+biπ[log(1-z0δ(z0eiθ,λ)e-2πai/(a+b)1-z0δ(z0eiθ,λ))-αlog(1-z0δ(z0eiθ,λ)e-2πai/(a+b)1-z0δ(z0eiθ,λ))¯],
where δ(cz,λ) is given by (17).

Proof.

For convenience, we let p=(a+b)/(iπ) and q=e-2πai/(a+b) and consider
V𝒮a,b(z0,λ)={f(z0):f∈𝒮a,b,f′(0)=p(1-q)λ}.
As before, it suffices to prove the theorem for V𝒮a,b(z0,λ). Let f∈𝒮a,b with f′(0)=p(1-q)λ. Define
g(z)=f(z)p,h(z)=ez,ϕ(z)=z-1z-q.
Then, by the mapping properties of these functions, it can be easily seen that the composed mapping
ωf(z)=(ϕ∘h∘g)(z)=ef(z)/p-1ef(z)/p-q
is analytic in 𝔻 and maps unit disk 𝔻 into 𝔻 such that ωf(0)=0 and ωf′(0)=λ. Next, we introduce Qf:𝔻→𝔻 by
Qf(z)=ωf(z)/z-λ1-λ¯(ωf(z)/z).
Clearly, Qf∈ℬ0. If we let
Sa,b,ωf,λ={Qf:ωf∈ℬ0,ωf′(0)=λ},VQf(z0)={ωf(z0):ωf∈Sa,b,ωf,λ},
then, by the Schwarz lemma, we have |Qf(z)|≤|z|. The region of variability VQf(z0) coincides with the set {z:|z|≤|z0|}. Equation (36) implies that
ωf(z)=z(Qf(z)+λ)1+Qf(z)λ¯.
It follows from (35) and (38) that V𝒮a,b(z0,λ) coincides with the set
{plog1-z0δ(z,λ)q1-z0δ(z,λ):|z|≤|z0|,whereδ(z,λ)=z+λ1+zλ¯}.
The proof of Theorem 4 is complete.

Geometric View of the Jordan Curves: (<xref ref-type="disp-formula" rid="EEq2">15</xref>), (<xref ref-type="disp-formula" rid="EEq5">26</xref>), and (<xref ref-type="disp-formula" rid="EEq6">27</xref>)

Table 1 gives the list of these parameter values corresponding to Figures 1–8 which concern the regions of variability for ∂Vϕ0,ℋ(z0), ∂Vϕ0,ℋ(z0,0), and ∂Vℋ,𝒮a,a,u(z0), respectively.

Using Mathematica (see [13]), we describe the boundary sets ∂Vϕ0,ℋ(z0), ∂Vϕ0,ℋ(z0,0), and ∂Vℋ,𝒮a,a,u(z0) described by the Jordan curve given by (15), (26), and (27), respectively. In the program below, “z0 stands for z0,” “[Alpha] for α,” and “[Beta] for β.”

In Table 1, the parameter values of z0 and α are common for all the three cases, namely, ∂Vϕ0,ℋ(z0), ∂Vϕ0,ℋ(z0,0), and ∂Vℋ,𝒮a,a,u(z0), whereas the β value is applicable only for the first two cases and the a=b values listed in the last column is meant only for the last case.

Clear[the, z0, ∖[Alpha], ∖[Beta], a, myf1, myf2, myf3];

Acknowledgment

The research was partly supported by NSFs of china (No. 10771059).

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