#### Abstract

Let be a biharmonic mapping of the unit disk , where and are harmonic in . In this paper, the Landau-type theorems for biharmonic mappings of the form are provided. Here represents the linear complex operator defined on the class of complex-valued functions in the plane. The results, presented in this paper, improve the related results of earlier authors.

#### 1. Introduction

Suppose that is a four times continuously differentiable complex-valued function in a domain . If satisfies the biharmonic equation , then we call that is biharmonic, where represents the Laplacian operator:

Biharmonic functions arise in many physical situations, particularly in fluid dynamics and elasticity problems, and have many important applications in engineering (see [1] for details). It is known that a mapping is biharmonic in a simply connected domain if and only if has the following representation: where and are complex-valued harmonic functions in [1]. Also, it is known that and can be expressed as where , , , and are analytic in [2, 3].

For a continuously differentiable mapping in , we define We use to denote the Jacobian of Then if .

In [4], the authors considered the following differential operator defined on the class of complex-valued functions:

Evidently, is a complex linear operator and satisfies the usual product rule: where and are complex constants; and are functions. In addition, the operator possesses a number of interesting properties. For instance, it is easy to see that the operator preserves both harmonicity and biharmonicity. Many other basic properties are stated in [4].

Landau’s theorem states that if is an analytic function on the unit disk with and for , then is univalent in the disk with , and contains a disk with . This result is sharp, with the extremal function . Recently, many authors considered the Landau-type theorems for harmonic mappings [5–9] and biharmonic mappings [1, 4, 10–13]. Chen et al. [10] obtained the Landau-type theorems for biharmonic mappings of the form as follows.

Theorem A (see [10]). *Let be a biharmonic mapping of the unit disk such that and , where and are harmonic in . Assume that both and are bounded by . Then there is a constant such that is univalent in , where satisfies the following equation:
**
where is the minimum value of the function
**
for . The minimum is attained at . Moreover, the range contains a schlicht disk , where
*

Theorem B (see [10]). *Let be a biharmonic mapping in such that , , and , where is harmonic in . Then there is a constant such that is univalent in , where satisfies the following equation:
**
where is defined as in Theorem A. Moreover, contains a disk with
*

However, these results are not sharp. The main object of this paper is to improve Theorems A and B. We get three versions of Landau-type theorems for biharmonic mappings of the form , where belongs to the class of biharmonic mappings, and Theorems 11 and 14 improve Theorems A and B. In order to establish our main results, we need to recall the following lemmas.

Lemma 1 (see [6, 14]). *Suppose that is a harmonic mapping of the unit disk such that for all . Then
**
The inequality is sharp.*

Lemma 2 (see [9, 12, 15]). *Suppose that is a harmonic mapping of the unit disk such that for all with and . Then and for any **
These estimates are sharp.*

Lemma 3 (see [8, 11]). *Suppose that is a harmonic mapping of with . If for ; then
**
These estimates are sharp.*

Lemma 4 (see [11]). *Suppose that is a harmonic mapping of the unit disk such that for all with and . If ; then , where and
*

Lemma 5 (see [13]). *Suppose that is a harmonic mapping of the unit disk with and . If satisfies for all and , then
*

Lemma 6. *Suppose that , . Then the equation
**
has a unique root in .*

*Proof . *It is easy to prove that the function is continuous and strictly decreasing on , , and . Hence, the assertion follows from the mean value theorem. This completes the proof.

Lemma 7. *Suppose that , , and is defined by (16). Then the equation
**
has a unique root in .*

Lemma 8. *Let . Then the equation
**
has a unique root in .*

Lemma 9. *For any in , we have
*

#### 2. Main Results

We first establish a new version of the Landau-type theorem for biharmonic mappings on the unit disk as follows.

Theorem 10. *Let be a biharmonic mapping of the unit disk , with , , and for , where . Then is univalent in the disk , where is the unique root in of the equation
**
and contains a schlicht disk , where
*

*Proof . *Let satisfy the hypothesis of Theorem 10, where
are harmonic in . As is linear and , we may set

Then we have

Note that ; by Lemma 3, we have

Thus, for in , we have

Let

By Lemmas 1, 2, and 3, elementary calculations yield that
Using these estimates and Lemma 6, we obtain
which implies .

For any such that , by Lemmas 2, 4, and 5, we obtain
This completes the proof.

Next we improve Theorem A as follows.

Theorem 11. *Let be a biharmonic mapping of the unit disk , with , , and for , where , . Then is univalent in the disk , where is the unique root in of the equation
**
and contains a schlicht disk , where is defined by (16) and
*

*Proof. *Note that ; by Lemma 4, we have

We adopt the same method in Theorem 10, for in ; by Lemmas 1, 2, and 5, we get
Using these estimates and Lemma 7, by (35), we obtain
which implies .

For any such that , by (35) and Lemmas 2 and 5, we obtain
This completes the proof.

Setting in Theorem 11, we have the following corollary.

Corollary 12. *Let be a biharmonic mapping of the unit disk , with , and both and are bounded by . Then is univalent in the disk , where is the minimum root of the equation
**
and contains a schlicht disk , where
*

In order to show Corollary 12 improves Theorem A, we use Mathematica to compute the approximate values for various choices of as in Table 1.

*Remark 13. *From Table 1 we can see, for the same ,
Finally we improve Theorems B as follows.

Theorem 14. *Let be a biharmonic mapping in such that , and , where and is harmonic in . Then is univalent in the disk , where is the minimum positive root in of the following equation:
**
and contains a schlicht disk with
**
where is defined by (16).*

*Proof. *Let

Let ; then we have

For in , by Lemmas 4, 5, 8, and 9, we get
which implies .

For any such that , by Lemmas 4 and 5, we obtain
This completes the proof of Theorem 14.

In order to show Theorem 14 improves Theorem B, we use Mathematica to compute the approximate values for various choices of as in Table 2.

*Remark 15. *From Table 2 we can see, for the same ,

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The work was financially supported by Foundation for Distinguished Young Talents in Higher Education of Guangdong China (no. 2013LYM0093) and Training plan for the Outstanding Young Teachers in Higher Education of Guangdong (no. Yq 2013159). The authors are grateful to the anonymous referees for making many suggestions that improved the readability of this paper.