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.