Abstract
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 is a complex-valued function defined in for which both and are real harmonic in , that is, where represents the Laplacian operator. The mapping can be written as a sum of an analytic and antianalytic functions, that is, We refer to [1] and the book of Duren [2] for many interesting results on planar harmonic mappings.
We note that the composition of a harmonic function with an analytic function is harmonic, but this is not true for the function , 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 onto are the affine mappings Motivated by the work of [3], we say that is an affine harmonic mapping of a harmonic mapping of if and only if has the form
for some with Obviously, an affine transformation applied to a harmonic mapping is again harmonic. The affine harmonic mappings and share many properties in common (see [4]).
Let denote the class of analytic functions in the unit disk , and . Also, let be the subclass of consisting of functions that are univalent in . For a given , we will denote by and the subsets defined by and , respectively. From now onwards, we use the notation , or, in for analytic functions and on to mean the subordination, namely there exists such that . Here denotes the class of analytic maps of the unit disk into itself with the normalization . We remark that if is univalent in , then the subordination is equivalent to the condition that and . This fact will be used in our investigation. Moreover, the special choices of 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 the class of functions with , and for . We note that if , then each function obviously satisfy the normalization condition . A function is called a Bloch function if
Then the set of all Bloch functions forms a complex Banach space with the norm given by
see [8]. Every bounded function in is Bloch, but there are unbounded Bloch functions, as can be seen also from the following result which shows that .
Proposition 1. If , then The constant is sharp. In particular, if then and the constant is sharp.
Proof. Let Then , and maps univalently onto the vertical strip and . Consequently, if , then we have and so, there exists a Schwarz function such that . Thus, as , the Schwarz-Pick lemma gives that so that with equality for , where
It may be interesting to remark that the function belongs to [9, Theorem 1] is a good example of a Bloch function which is not in -space for any Bloch functions are intimately close with univalent functions (see [5]).
In order to state our main results, we introduce some basics. For given , let be the class of functions and for . Now, we define
We note that each function in has the normalization For any fixed and with we consider the following sets:
We now recall the definition of subordination for the harmonic case from [10, page 162]. Let and be two harmonic functions defined on . We say is subordinate to denoted by , if , where . Obviously, if and are two harmonic functions in , then
Here we see that is the analytic dilatation for both and .
For each fixed , using extreme function theory, it has been shown by Grunsky (see, e. g., Duren [6, Theorem 10.6]) that the region of variability of
is precisely a closed disk, where : 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 (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 , , and , respectively.
Theorem 1. The boundary of is the Jordan curve given by
Proof. We define : In order to determine the set we first recall that each can be written as for some . By the Riemann mapping theorem, is univalent and analytic in with . It follows from the classical Schwarz lemma that for any we have in . Because, in our situation is also univalent in , we easily show that the region of variability coincides with the set : Hence the region of variability is precisely the set : We remark that depends only on , because is preserved under rotation and therefore, we may assume that . Finally, the region of variability follows from . The proof of this theorem is complete.
There are many choices for for which Theorem 1 is applicable. For example, if we choose to be
for some , then we have following result from Theorem 1.
Corollary 1. The boundary of is the Jordan curve given by
Theorem 2. The boundary of is the Jordan curve given by where
Proof. Let such that for some . Because , there exists a Schwarz function with where Therefore, for any fixed and with , it is natural to consider the set First, we determine . Then the determination of the set follows from Now, we define We observe that By the Schwarz lemma, we have . If we set then the region of variability : coincides with the set :. It follows from the two expressions in (19) that coincides with the set The proof of this theorem is complete.
The case of Theorem 2 gives the following result.
Corollary 2. The boundary of is the Jordan curve given by
If is given by (14) for some , then and reduces to
and the corresponding in the proof of the theorem will be precisely of the form
This observation gives the following corollary.
Corollary 3. The boundary of is the Jordan curve given by where and are given by (14) and (17), respectively.
The boundary of is the Jordan curve given by
Theorem 3. The boundary of is the Jordan curve given by
Proof. We define . It suffices to determine as the region of variability follows from . In order to do this, first we consider
Then . We see that the Möbius transformation maps the open unit disk conformally onto the half-plane
and so, we easily obtain that maps conformally onto the vertical strip . This observation shows that and is in fact an extremal function for this class.
Next, we choose an arbitrary . Then we have and so, there exists a Schwarz function such that Note that both and are univalent in and so, is univalent in with . It follows from the classical Schwarz lemma that in . Because is also univalent in , we obtain that the region of variability of
coincides with the set Hence the region of variability coincides with the set
The proof of Theorem 3 is complete.
Theorem 4. The boundary of is the Jordan curve given by where is given by (17).
Proof. For convenience, we let and and consider As before, it suffices to prove the theorem for . Let with . Define Then, by the mapping properties of these functions, it can be easily seen that the composed mapping is analytic in and maps unit disk into such that and . Next, we introduce by Clearly, . If we let then, by the Schwarz lemma, we have . The region of variability coincides with the set . Equation (36) implies that It follows from (35) and (38) that coincides with the set The proof of Theorem 4 is complete.
Geometric View of the Jordan Curves: (15), (26), and (27)
Table 1 gives the list of these parameter values corresponding to Figures 1–8 which concern the regions of variability for , , and , respectively.
Using Mathematica (see [13]), we describe the boundary sets , , and described by the Jordan curve given by (15), (26), and (27), respectively. In the program below, “z0 stands for ” “[Alpha] for ,” and “[Beta] for ”
In Table 1, the parameter values of and are common for all the three cases, namely, , , and , whereas the value is applicable only for the first two cases and the values listed in the last column is meant only for the last case.
(a)
(b)
(c)
(a)
(b)
(c)
(a)
(b)
(c)
(a)
(b)
(c)
(a)
(b)
(c)
(a)
(b)
(c)
(a)
(b)
(c)
(a)
(b)
(c)
(* Geometric view the main Theorem.... *)
Remove["Global`"];
z0 = Random Exp[I Random[Real, -Pi, Pi]]
[Alpha] = RandomExp[I Random[Real, -Pi, Pi]]
[Beta] = Random[Real, 0, 2]
a = Random[Real, 0,100]
Print["z0=", z0]
Print["[Alpha]=", [Alpha]]
Print["[Beta]=", [Beta]]
Print["a", a]
myf1[the_, [Alpha]_, [Beta]_, z0_]:=
((1+Exp[Ithe]z0)/(1-Exp[Ithe]z0))[Beta] +
[Alpha]Conjugate[((1+Exp[Ithe]z0)/(1-Exp[Ithe]z0))[Beta]]
- 1-[Alpha];
myf2[the_, [Alpha]_, [Beta]_,z0_]:=
((1+Exp[Ithe]z0z0)/(1-Exp[Ithe]z0z0))[Beta]+
[Alpha]Conjugate[((1+Exp[Ithe]z0z0)/(1-Exp[Ithe]z0z0))[Beta]]
-1-[Alpha];
myf3[the_, [Alpha]_, a_,z0_]:=
(2a)/(IPi)(Log((1-Exp[Ithe]Exp[-IPi]z0)/(1-Exp[Ithe]z0))-
[Alpha]Conjugate[Log((1-Exp[Ithe]Exp[-IPi]z0)/(1-Exp[Ithe]z0))])
image1 = ParametricPlot[Re[myf1[the, [Alpha], [Beta], z0]],
Im[myf1[the, [Alpha], [Beta], z0]], the, -Pi, Pi,
AspectRatio - Automatic,DisplayFunction - $DisplayFunction,
TextStyle -FontFamily - "Times", FontSize - 14,
AxesStyle -Thickness[0.0035]];
image2 =ParametricPlot[Re[myf2[the, [Alpha], [Beta], z0]],
Im[myf2[the, [Alpha], [Beta], z0]], the, -Pi, Pi,
AspectRatio - Automatic,DisplayFunction -DisplayFunction,
TextStyle -FontFamily - "Times", FontSize - 14,
AxesStyle -Thickness[0.0035]];
image3 =ParametricPlot[Re[myf3[the, [Alpha], a, z0]],
Im[myf3[the, [Alpha], a, z0]], the, -Pi, Pi,
AspectRatio - Automatic,DisplayFunction - $DisplayFunction,
TextStyle -FontFamily - "Times", FontSize - 14,
AxesStyle -Thickness[0.0035]];
Clear[the, z0, [Alpha], [Beta], a, myf1, myf2, myf3];
Acknowledgment
The research was partly supported by NSFs of china (No. 10771059).