#### 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.

#### 1. Preliminaries and Main Results

A planar harmonic mapping in a simply connected domain is a complex-valued function defined in such that and are real harmonic in , that is, and . Here represents the complex Laplacian operator

The mapping has a canonical decomposition , where and are analytic (holomorphic) in [1, 2]. Lewy’s theorem tells us that a harmonic mapping is locally univalent in if and only if its Jacobian for each [3].

A four-time continuously differentiable complex-valued function in is *biharmonic* if and only if the Laplacian of is harmonic. Note that is harmonic in if satisfies the biharmonic equation .

It is a simple exercise to see that every biharmonic mapping in a simply connected domain has the representation

where and are some complex-valued harmonic functions in . 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 onto are the affine mappings , where (see [2, Section ]).

We denote the class of analytic functions in the unit disk by , and we think of as a topological vector space endowed with the topology of uniform convergence over compact subsets of .

*Definition 1.1. *A function is said to be an * affine harmonic mapping* of if and only if
for some with . If is biharmonic, then is called the * affine biharmonic mapping* of .

It follows easily that every affine harmonic mapping is harmonic. Also, every affine biharmonic mapping is biharmonic. Definition 1.1 is motivated by the following result.

Theorem 1 A (see [9, Theorem]). * For a harmonic mapping , the following are equivalent: *(1)*The Schwarzian derivative ; *(2)* for some analytic Möbius transformation ; *(3)* takes circles to ellipses.*

Here denotes the Schwarzian derivative of the harmonic mapping defined by

A domain is said to be *-linearly connected* if there exists a positive constant such that any two points are joined by a path with

or equivalently

where denotes the Euclidean length of (cf. [10]). A complex-valued function is -linearly connected if is an -linearly connected domain.

In a recent paper, Chuaqui and Hernández [11] investigated the relation between the univalency of planar harmonic mappings and the linear connectedness of . They proved the following theorem.

Theorem 1 B (see [11, Theorem ]). *Let be a holomorphic univalent map. Then there exists a constant such that every harmonic mapping with dilation is univalent in if and only if is an -linearly connected domain, where can be taken to be .*

We say that a complex-valued function defined on is said to be *convex* in if and only if is a domain. A well-known analytic characterization gives that if is analytic in , then is convex in if and only if

The main of this paper is twofold. We first discuss the close relationships that exist among the local univalency, convexity, and linear connectedness of and its corresponding . One of our main results follows.

Lemma 1.2. *Let . Then *(1)* is locally univalent if and only if is locally univalent; *(2)* is convex if and only if is convex, where is locally univalent; *(3)* is -linearly connected if and only if is -linearly connected, where is univalent and the constants and depend only on each other and . *

Let denote the class of analytic functions in with , and

A function is called a * Bloch function* if

Then the set of all Bloch functions form a complex Banach space with the norm

see [12]. The following result shows that .

Proposition 1.3. *If , then . The constant is sharp.*

*Proof. *Let and let be defined by , where
It follows that and is analytic mapping of the unit disk to with
Also, by the classical Schwarz-Pick lemma, we have
which gives
whence
Thus, we have .

For the proof of sharpness part, we consider
Then with for and
which gives . The proof is complete.

Let

For any fixed and with , we define

Let denote the class of all functions that are biharmonic but not harmonic in with the normalization and . For any fixed , let

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 . 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 , , and . The following are our results.

Theorem 1.4. *The boundary of is the Jordan curve given by
*

Theorem 1.5. *The boundary of is the Jordan curve given by
**
where
*

The following two lemmas are useful for the proofs of Theorems 1.4 and 1.5.

Lemma 1.6. *Both sets and are compact and convex.*

*Proof. *Obviously, both sets and are convex. Since is closed, the compactness of sets and easily follows from Proposition 1.3.

The argument as in the proof of Lemma 1.6 yields the following.

Lemma 1.7. *Both sets and are compact and convex.*

Finally, we have

Theorem 1.8. *The boundary of is an ellipse.*

Proofs of Lemma 1.2, and Theorems 1.4, 1.5, and 1.8 will be presented in Section 2.

#### 2. Proofs of Main Results

*Proof of Lemma 1.2. *
() We see that and the proof of Lemma 1.2() follows.

() Let be analytic and locally univalent in , and , where and . Then for any fixed ,
Since the curvature of is defined by
we see that
Multiplying both sides by
and then simplifying the resulting expression on the right give
The proof of Lemma Lemma 1.2() follows from (1.7).

() For the proof of the sufficiency, let be -linearly connected and
where . For any distinct points , there exists a path joining and such that

Let . Then we have
On the other hand,
Hence
For the proof of necessity, since is univalent, we define
By the chain rule, we have
Differentiating both sides of the equation, yields the following relations:
Solving these two equations yields
It follows from (2.12) that

For any two distinct points , since is -linearly connected, there exists a path joining and such that

Now, we let . Then we see that

On the other hand, by the definition of , we have
from which we obtain
and therefore,
The proof of the theorem is complete.

Before we indicate the proof of Theorem 1.4, we need to determine the region of variability .

Lemma 2.1. *The boundary of is the Jordan curve given by
*

*Proof. * For , let . Then . Using the Schwarz lemma, we have . That is,
or equivalently
This yields that for each ,
where
Consider
Then and, for , the function is a nonconstant analytic function in . Hence it is an open mapping. Thus . Next we will show that for all . An elementary computation shows that
where . Hence
Since
we have
provided . Obviously, is univalent in . By Lemma 1.6, we know that is a simple closed curve. Therefore, the boundary is the Jordan curve given by (2.21). The proof of the lemma is complete.

*Proof of Theorem 1.4. *The proof of Theorem 1.4 is an immediate consequence of Lemmas 1.2, 1.6, and 2.1.

Clearly, for the proof of Theorem 1.5, it suffices to describe the region of variability .

Lemma 2.2. *The boundary of is the Jordan curve given by
**
where
*

*Proof. * Let with . It follows from the classical Schwarz lemma for analytic functions that
which, after some computation, is equivalent to
where
According to the Schwarz lemma, we see that the equality sign occurs in (2.33) if and only if for some , where
Further calculations show that the inequality (2.34) is equivalent to
where
It follows from (2.35) that
so that simplifies to the form
Further,
From (2.37), we have
Equation (2.36) gives
and so, we infer from (2.35) that
As
the above equalities together with (2.39), (2.40), and (2.42) yield that
Substituting by in the last equality, we see from (2.41) that
which implies that
whence
We claim that the closed curve
is simple.

For the proof of this claim, we suppose that there are with such that
By (2.36), we obtain that which is a contradiction and this completes the proof of our claim.

Since is a compact convex subset of and its interior is nonempty, we see that its boundary is a simple closed curve. It follows from (2.40) that the curve
is a subcurve of .

The fact that a simple closed curve cannot contain any simple closed curve other than itself yields that is given by
The proof of Lemma 2.2 is complete.

*Proof of Theorem 1.5. *The proof of Theorem 1.5 follows from Lemmas 1.2, 1.7, and 2.2.

Proof of Theorem 1.8 is a consequence of Theorem A and the following lemma.

Lemma 2.3. *For any fixed , let be defined by
**
and . Then and .*

*Proof. * We write
and so, we have where
Obviously, is a biharmonic mapping and , from which the first assertion in Lemma 2.3 follows.

Since , we see that is a conformal automorphism of the unit disk and the range is itself. By hypotheses, we know that for any , . Hence coincides with .

#### Acknowledgment

The research was partly supported by NSFs of China (no. 10771059).