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

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 Ω if and only if its Jacobian does not vanish. In terms of the canonical decomposition, the Jacobian of harmonic mappings f = h + g is given by J f = jh′j 2 − jg′j, and thus, a locally univalent harmonic mapping in a simply connected domain Ω will be sense-preserving if ∣h′ ∣ >∣g′∣. Following the above ideas, particularly the definition of Bloch harmonic function given by Colonna [14], we will prove that the polyharmonic mapping F and log-pharmonic mapping f defined in the unit disk D are normal if they satisfy a Lipschitz type condition. Further, for the complex-valued polyharmonic mappings and log-pharmonic mappings, we give out some additional conditions for which are normal. These conditions cannot be omitted. A 2p-times continuously differentiable complex-valued function FðzÞ = uðzÞ + ivðzÞ in a domain D ⊆C is polyharmonic mapping or p-harmonic if FðzÞ satisfies the p-harmonic equation


Introduction and Preliminaries
For real-valued harmonic functions defined in D, Lappan [1] established that φ is normal if where grad φ is the gradient vector of φ. In [2], the authors also proved geometric properties of real-valued harmonic normal functions. Namely, a real-valued harmonic function φ with the property is normal. Recently, many authors considered the properties of the complex-valued harmonic mappings and harmonic quasiconformal mappings in [3][4][5][6][7][8][9][10][11][12][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 Ω if and only if its Jacobian does not vanish. In terms of the canonical decomposition, the Jacobian of harmonic mappings f = h + g is given by J f = jh ′ j 2 − jg ′ j 2 , and thus, a locally univalent har-monic mapping in a simply connected domain Ω will be sense-preserving if |h ′ | >|g ′ |. Following the above ideas, particularly the definition of Bloch harmonic function given by Colonna [14], we will prove that the polyharmonic mapping F and log-pharmonic mapping f defined in the unit disk D are normal if they satisfy a Lipschitz type condition. Further, for the complex-valued polyharmonic mappings and log-pharmonic mappings, we give out some additional conditions for which are normal. These conditions cannot be omitted. A 2p-times continuously differentiable complex-valued function FðzÞ = uðzÞ + ivðzÞ in a domain D ⊆ ℂ is polyharmonic mapping or p-harmonic if FðzÞ satisfies the p-harmonic equation where the Laplacian operator As we see in Proposition 1 in [16], we know that a mapping F is polyharmonic in a simply connected domain D ⊆ ℂ if and only if F has the following representation where each G p−k+1 is harmonic for k ∈ f1,⋯,pg. When p = 1, the mapping F is called harmonic. When p = 2, the mapping F is called biharmonic. f is called log-p-harmonic mapping if log f is p-harmonic mapping. When p = 1, the mapping f is called log-harmonic. When p = 2, the mapping f is called log-biharmonic, which can be regarded as generalizations of holomorphic functions. So we say that f is called log-pharmonic mapping in a simply connected domain D ⊆ ℂ if and only if f has the form where each g p−k+1 is log-harmonic for k ∈ f1,⋯,pg. For a continuously differentiable mapping f in D, we define Recently, many authors considered Landau-type theorems for harmonic mappings, biharmonic mappings, and p-harmonic mappings [16][17][18][19][20][21][22][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.

Necessary Lemmas
In order to derive our main results, we need the following lemmas. Lemma 1. [14]. Suppose that f ðzÞ = hðzÞ + gðzÞ is a harmonic mapping of the unit disk D with hðzÞ = ∑ ∞ n=1 a n z n and gðzÞ = ∑ ∞ n=1 b n z n are analytic on D . If |f ðzÞ | <M for all z ∈ D, then Lemma 2. [22]. Suppose that f ðzÞ = hðzÞ + gðzÞ is a harmonic mapping of the unit disk D with hðzÞ = ∑ ∞ n=1 a n z n and gðzÞ = ∑ ∞ n=1 b n z n are analytic on D . If |f ðzÞ | <M for all z ∈ D, then for |z | = r < 1, we have Lemma 3. [25]. Suppose that f ðzÞ = hðzÞ + gðzÞ is a harmonic mapping of the unit disk D with hðzÞ = ∑ ∞ n=1 a n z n and gðzÞ = ∑ ∞ n=1 b n z n are analytic on D and When Λ > 1, the above estimates are sharp for all n = 2, 3, ⋯, with the extremal functions f n ðzÞ and f n ðzÞ as follows Lemma 4. [19]. Suppose that f ðzÞ = hðzÞ + gðzÞ is a harmonic mapping of the unit disk D with hðzÞ = ∑ ∞ n=1 a n z n and gðzÞ We recall that the chordal distance on the generalized complex plane b ℂ, which is defined by If P z 1 , P z 2 are the two points on the Riemann sphere, under stereographic projection, corresponding to z 1 and z 2 , respectively, we have Therefore, where ϱðz 1 , z 2 Þ is the spherical distance of z 1 and z 2 , Γ is any rectifiable curve in ℂ with endpoints z 1 , z 2 , and is the spherical length of Γ. On the basis of the paper, given z 1 , z 2 ∈ D, ρðz 1 , z 2 Þ denotes the hyperbolic distance between z 1 , z 2 . Therefore, if τ denotes the hyperbolic geodesic joining z 1 to z 2 , then More explicitly, Journal of Function Spaces With these notations, a polyharmonic mapping or log-pharmonic mapping f : D → ℂ 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 f ðzÞ be a polyharmonic mapping or log-pharmonic mapping in the unit disk D, then f is normal if Proof. Suppose that ∥f ∥<∞ and let z 1 , z 2 ∈ D. If τ : ½0, 1 → D is the hyperbolic geodesic with endpoints z 1 and z 2 , where df stands for the differential of f . From here and (21), we have Hence, we obtain So it implies that f is normal.

Main Results and Their Proofs
In this section, we prove the normality of the polyharmonic mappings and log-p-harmonic mappings as follows.
Proof. We may represent the harmonic functions G p−k+1 ðzÞ in series form as Firstly, we calculate the boundedness of the derivative of F. where By a simple calculation, we have

Journal of Function Spaces
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 S 1 ðrÞ, we have In view of Λ p ≥ 1 and r ∈ ð0, 1Þ, after a simple calculation, it shows that S ′ 1 ðrÞ > 0. It is simple to verify that S 1 ðrÞ is strictly increasing in ð0, 1Þ.
Finally, we consider the boundedness of F for any |z | = r 1 , then we have So |FðzÞ | ≤S 2 ðr 1 Þ, where By the similar approach for differentiating S 2 ðr 1 Þ, we have the following one By elementary calculations, we get S 2 ′ ðr 1 Þ > 0. It implies that S 2 ðr 1 Þ is increasing for Λ p ≥ 1 and r 1 ∈ ð0, 1Þ. It is simple to verify that S 2 ðr 1 Þ is bounded in ð0, 1Þ. Combined with Lemma 5 and Estimation (33), we conclude ultimately that F is normal polyharmonic mapping in the unit disk D. The proof of this theorem is complete.
Then, F is normal polyharmonic mapping in the unit disk D.

Journal of Function Spaces
Proof. We represent the harmonic functions G p−k+1 ðzÞ in series form as for each k ∈ f1, 2,⋯,pg, and To prove the normality of F, we first determine that the derivative of F is bounded in D. Then, as in the proof of Theorem 6, we have