Normality of the p-Harmonic and Log-p-Harmonic Mappings

Abstract

In this paper, the concepts of p-harmonic mappings and log-p-harmonic mappings in the unit disk have been introduced and studied by many researchers. We proved the normality of the p-harmonic mappings and log-p-harmonic mappings, which extend the related results of harmonic mappings of earlier authors.

1. Introduction and Preliminaries

For real-valued harmonic functions defined in , Lappan [1] established that is normal if where is the gradient vector of . In [2], the authors also proved geometric properties of real-valued harmonic normal functions. Namely, a real-valued harmonic function with the property is normal.

Recently, many authors considered the properties of the complex-valued harmonic mappings and harmonic quasiconformal mappings in [3–13]. We are motivated to establish the topic of normality for complex-valued p-harmonic mappings and log-p-harmonic mappings defined in the unit disk. An important concept related with normal harmonic functions is the Bloch function, which was studied by Colonna in [14]. It is a classical result of Lewy [15] that a harmonic mapping is locally univalent in a domain if and only if its Jacobian does not vanish. In terms of the canonical decomposition, the Jacobian of harmonic mappings is given by , and thus, a locally univalent harmonic mapping in a simply connected domain will be sense-preserving if .

Following the above ideas, particularly the definition of Bloch harmonic function given by Colonna [14], we will prove that the polyharmonic mapping and log-p-harmonic mapping defined in the unit disk are normal if they satisfy a Lipschitz type condition. Further, for the complex-valued polyharmonic mappings and log-p-harmonic mappings, we give out some additional conditions for which are normal. These conditions cannot be omitted. A 2p-times continuously differentiable complex-valued function in a domain is polyharmonic mapping or p-harmonic if satisfies the p-harmonic equation where the Laplacian operator

As we see in Proposition 1 in [16], we know that a mapping is polyharmonic in a simply connected domain if and only if has the following representation where each is harmonic for . When , the mapping is called harmonic. When , the mapping is called biharmonic. is called -p-harmonic mapping if is p-harmonic mapping. When , the mapping is called -harmonic. When , the mapping is called -biharmonic, which can be regarded as generalizations of holomorphic functions. So we say that is called -p-harmonic mapping in a simply connected domain if and only if has the form where each is -harmonic for .

For a continuously differentiable mapping in , we define

Recently, many authors considered Landau-type theorems for harmonic mappings, biharmonic mappings, and p-harmonic mappings [16–23]. Li and Wang [24] introduced the log-p-harmonic mappings and derived two versions of Landau-type theorems. However, in virtue of being inspired by these results, we establish the normality of polyharmonic mappings and log-p-harmonic mappings.

2. Necessary Lemmas

In order to derive our main results, we need the following lemmas.

Lemma 1. [14]. Suppose that is a harmonic mapping of the unit disk with and are analytic on . If for all , then

Lemma 2. [22]. Suppose that is a harmonic mapping of the unit disk with and are analytic on . If for all , then for , we have

Lemma 3. [25]. Suppose that is a harmonic mapping of the unit disk with and are analytic on and . If for all , then When , the above estimates are sharp for all , with the extremal functions and as follows When , then with .

Lemma 4. [19]. Suppose that is a harmonic mapping of the unit disk with and are analytic on . If for all , then for each , We recall that the chordal distance on the generalized complex plane , which is defined by If are the two points on the Riemann sphere, under stereographic projection, corresponding to and , respectively, we have Therefore, where is the spherical distance of and , is any rectifiable curve in with endpoints , and is the spherical length of . On the basis of the paper, given , denotes the hyperbolic distance between . Therefore, if denotes the hyperbolic geodesic joining to , then More explicitly, where With these notations, a polyharmonic mapping or log-p-harmonic mapping is called a normal polyharmonic mapping or normal log-p-harmonic mapping, if

The following lemma provides an alternative method for deciding when a polyharmonic mapping or log-p-harmonic mapping is normal.

Lemma 5. Let be a polyharmonic mapping or log-p-harmonic mapping in the unit disk , then is normal if

Proof. Suppose that and let . If is the hyperbolic geodesic with endpoints and , where stands for the differential of . From here and (21), we have Hence, we obtain So it implies that is normal.

3. Main Results and Their Proofs

In this section, we prove the normality of the polyharmonic mappings and log-p-harmonic mappings as follows.

Theorem 6. Let be a polyharmonic mapping in the unit disk satisfying . Suppose that for each , we have Then, is normal polyharmonic mapping of the unit disk .

Proof. We may represent the harmonic functions in series form as Firstly, we calculate the boundedness of the derivative of . where By a simple calculation, we have Using Lemma 1, we have By Lemma 2, we have Using Lemma 3, we have By the above estimates, we obtain the following result where Now, differentiating , we have In view of and , after a simple calculation, it shows that . It is simple to verify that is strictly increasing in . Obviously, has only one pole . In other words, is bounded in the interval .
Finally, we consider the boundedness of for any , then we have So , where By the similar approach for differentiating , we have the following one By elementary calculations, we get . It implies that is increasing for and . It is simple to verify that is bounded in . Combined with Lemma 5 and Estimation (33), we conclude ultimately that is normal polyharmonic mapping in the unit disk . The proof of this theorem is complete.

Theorem 7. Let, be a polyharmonic mapping of satisfying . Suppose that for , we have Then, is normal polyharmonic mapping in the unit disk .

Proof. We represent the harmonic functions in series form as for each , and To prove the normality of , we first determine that the derivative of is bounded in . Then, as in the proof of Theorem 6, we have where By the condition (2) of Theorem 7 and Lemma 4, we have Applying Lemma 3, we have By above estimates, we obtain that Set For differentiating , we have By a simple calculation, we obtain . It shows that is strictly increasing in , and This shows that has only one pole , and it says that is bounded in . So is a finite value in the unit disk .
Now, we consider any with . Then, we have Differentiating , we have After elementary calculations, we have that . It implies that is strictly increasing. It is simple to verify that is finite for . It says that is bounded in . Using these above estimates and Condition (21), we conclude that is normal polyharmonic mapping in . This proof of Theorem 7 is complete.

Finally, we establish the normality of log-p-harmonic mappings as follows.

Theorem 8. Let be a -p-harmonic mapping in the unit disk satisfying . Suppose that for each we have

Then, is a normal -p-harmonic mapping in the unit disk .

Proof. Let , for each . We may represent the harmonic functions in series form as Then, is a polyharmonic mapping in . We know that and , so it follows from , we have .
Obviously, so we have In order to prove the normality of , it follows from Theorem 6 that we have where So we obtain Differentiating , we have By elementary calculations, we get . It follows that is strictly increasing in , and It implies that is bounded in . Hence, there exists a finite value such that Finally, we consider any with ; then, we have Differentiating , have the following result It is not difficult to verify that , which means that is strictly increasing. It is also easily seen that is bounded in . Hence, there is a finite value such that Applying Lemma 5 and (63), (66), we obtain is normal in . The proof of Theorem 8 is complete.

Data Availability

The data used to support the findings of this study are included with the article.

Conflicts of Interest

We declare that we have no competing interests.

Acknowledgments

The first author would like to thank the Department of Mathematics, University of Houston, USA, for providing a good environment during the preparation of this work. The corresponding authors would like to thank the visiting scholar program of Shiing-Shen Chern Institute of Mathematics at Nankai University. The authors are also thankful to the referees for their invaluable comments and suggestions. This research was supported by the National Natural Science Foundation of China under Grant 11701111, 12031003, and 11771347.