Landau-Type Theorems of Polyharmonic Mappings and log-p-Harmonic Mappings

Abstract

In this paper, we first establish three new Landau-type theorems of polyharmonic mappings, which extend the related results of biharmonic mappings of earlier authors. Then three new Landau-type theorems of log-p-harmonic mappings are also provided.

1 Introduction and Preliminaries

A 2p-times continuously differentiable complex-valued function $F(z)=u(z)+iv(z)$ in a domain $D\subseteq \mathbb {C}$ is polyharmonic mapping or p-harmonic if F(z) satisfies the p-harmonic equation

$$\begin{aligned} \Delta ^{p} F=\Delta (\Delta ^{p-1})F=0, \end{aligned}$$

where the Laplacian operator

$$\begin{aligned} \Delta F= 4F_{z\overline{z}}=\frac{\partial ^{2}F}{\partial x^{2}}+\frac{\partial ^{2}F}{\partial y^{2}}. \end{aligned}$$

As we see in Proposition 1 in [2], we know that a mapping F is polyharmonic in a simply connected domain $D\subseteq \mathbb {C}$ if and only if F has the following representation

$$\begin{aligned} F(z)=\sum _{k=1}^{p}|z|^{2(k-1)}G_{p-k+1}(z), \end{aligned}$$

where each $G_{p-k+1}$ is harmonic for $k\in {\{1,\ldots , p}\}$. 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-p-harmonic mapping in a simply connected domain $D\subseteq \mathbb {C}$ if and only if f has the form

$$\begin{aligned} f(z)=\prod _{k=1}^{p}(g_{p-k+1}(z))^{|z|^{2(k-1)}}, \end{aligned}$$

where each $g_{p-k+1}$ is log-harmonic for $k\in {\{1,\ldots , p}\}$.

For a continuously differentiable mapping f in D, we define

$$\begin{aligned} \Lambda _{f}(z)=\max _{0\le \theta \le 2\pi }|f_{z}(z)+{e}^{-2i\theta }f_{\overline{z}}(z)|=|f_{z}(z)|+|f_{\overline{z}}(z)|, \end{aligned}$$

and

$$\begin{aligned} \lambda _{f}(z)=\min _{0\le \theta \le 2\pi }|f_{z}(z)+{e}^{-2i\theta }f_{\overline{z}}(z)|=||f_{z}(z)|-|f_{\overline{z}}(z)||. \end{aligned}$$

We denote the Jacobian of f by $J_{f}$,

$$\begin{aligned} J_{f}(z)=|f_{z}|^{2}-|f_{\overline{z}}|^{2}. \end{aligned}$$

For $r>0$, we let $\mathbb {D}_{r}$ denote the open disk with center at the origin and radius r. The classical Landau theorem states that if f is an analytic function on the unit disk $\mathbb {D}=\{z\in \mathbb {C}:|z|<1\}$ with and $f(0)=f'(0)-1=0$ and $|f(z)|<M$ for $z\in \mathbb {D}$, then f is univalent in the disk $\mathbb {D}_{r_0}$ with $r_{0}=1/(M+\sqrt{M^{2}-1})$, and $f(\mathbb {D}_{r_0})$ contains a disk $\mathbb {D}_{R_{0}}$ with $R_{0}=Mr_{0}^{2}$. This result is sharp, with the extremal function $f(z)=Mz((1-Mz)/(M-z))$.

Recently, many authors considered Landau-type theorems for harmonic mappings, biharmonic mappings and p-harmonic mappings [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]. Li and Wang [10] introduced the log-p-harmonic mappings and derived two versions of Landau-type theorems, however these results are not sharp. Now we recall the following results.

Theorem A

[2] Let $F(z)=\sum _{k=1}^{p}|z|^{2(k-1)}G_{p-k+1}(z)$ be a p-harmonic mapping of $\mathbb {D}$ satisfying $\Delta G_{p-k+1}=G_{p-k+1}(0)=J_{F}(0)-1=0$, $|G_{p-k+1}|< M$ for $k\in \{1, \ldots , p\}$, where $M\ge 1$. Then there exists $\rho _{1}\in (0, 1)$ such that F is univalent in $\mathbb {D}_{\rho _{1}}$, where $\rho _{1}$ satisfies the following equation:

$$\begin{aligned} \lambda (M)-\frac{4M}{\pi (1-\rho _{1})^{2}}\sum _{k=1}^{p-1}\rho _{1}^{2k}-2M\sum _{k=1}^{p-1}k\rho _{1}^{2k-1} -T(M)\frac{\rho _{1}(2-\rho _{1})}{(1-\rho _{1})^{2}}=0, \end{aligned}$$

where $\lambda (M)$ is defined as

$$\begin{aligned} \lambda (M)=\left\{ \begin{array}{ll} \frac{\sqrt{2}}{\sqrt{M^{2}-1}+\sqrt{M^{2}+1}}, &{}\quad {1\le M\le M_{0}=\frac{\pi }{2\root 4 \of {2\pi ^2-16}}\approx 1.1296,} \\ \frac{\pi }{4M}, &{}\quad {M>M_{0}}. \end{array} \right. \end{aligned}$$

(1.1)

Here and in what follows, we always assume that $T(x):=\min \{\sqrt{2x^{2}-2}, 4x/\pi \}$.

Moreover, the range $F(\mathbb {D}_{\rho _{1}})$ contains a univalent disk $\mathbb {D}_{\rho _{1}'}$, where

$$\begin{aligned} \rho _{1}'=\rho _{1}\left[ \lambda _{0}(M)-\frac{T(M)\rho _{1}}{1-\rho _{1}}- \frac{4M}{\pi (1-\rho _{1})}\sum _{k=1}^{p-1}\rho _{1}^{2k}\right] . \end{aligned}$$

Theorem B

[10] Let $f(z)=\prod _{k=1}^{p}(g_{p-k+1}(z))^{|z|^{2(k-1)}}$ be a log-p-harmonic mapping of $\mathbb {D}$, where $g_{p-k+1}$ is log-harmonic with $g_{p-k+1}(0)=g_{p}(0)=J_{f}(0)=1$, $|g_{p-k+1}|<M_{1}$ for $k\in \{2, \ldots , p\}$, and $|g_{p}|<M_{2}$, where $M_{i}\ge 1(i=1, 2)$ are positive constants. Then there exists $\rho _{2}\in (0, 1)$ such that f is univalent in $\mathbb {D}_{\rho _{2}}$, where $\rho _{2}$ satisfies the following equation:

$$\begin{aligned} \lambda _{0}(M_{2}^{*})-\frac{T(M_{2}^{*})\rho _{2}(2-\rho _{2})}{(1-\rho _{2})^{2}}- \frac{4M_{1}^{*}}{\pi {(1-\rho _{2})^{2}}}\sum _{k=1}^{p-1}\rho _{2}^{2k}-2M_{1}^{*}\sum _{k=1}^{p-1}k\rho _{2}^{2k-1}=0, \end{aligned}$$

and $M_{i}^{*}=\log M_{i}+\pi (i=1, 2)$.

Moreover, the range $f(\mathbb {D}_{\rho _{2}})$ contains a univalent disk $\mathbb {D}(z_{2}, \rho _{2}')$, where

$$\begin{aligned} z_{2}=\cosh \left( \frac{\sigma _{2}}{\sqrt{2}}\right) , \quad \rho _{2}'=\min \left\{ \sinh \left( \frac{\sigma _{2}}{\sqrt{2}}\right) , \cosh \left( \frac{\sigma _{2}}{\sqrt{2}}\right) \sin \left( \frac{\sigma _{2}}{\sqrt{2}}\right) \right\} , \end{aligned}$$

and

$$\begin{aligned} \sigma _{2}=\rho _{2}\left[ \lambda _{0}(M_{2}^{*})-\frac{T(M_{2}^{*})\rho _{2}}{1-\rho _{2}}- \frac{4M_{1}^{*}}{\pi {(1-\rho _{2})}}\sum _{k=1}^{p-1}\rho _{2}^{2k}\right] . \end{aligned}$$

Theorem C

[2] Let $f(z)=|z|^{2(p-1)}G(z)$ be p-harmonic of $\mathbb {D}$, where $p>1,G$ is harmonic, $g(0)=J_{G}(0)-1=0$, and $|G(z)|\le M$ for some $M\ge 1$. Then f is univalent in $\mathbb {D}_{\rho _3}$, where

$$\begin{aligned} \rho _3=\frac{\lambda +2T-\sqrt{4T_{2}+T\lambda }}{\lambda +3T}. \end{aligned}$$

Moreover, the range $f(\mathbb {D}_{\rho _3})$ contains a univalent disk $\mathbb {D}_{\rho _3'}$, where

$$\begin{aligned} \rho _3'=\frac{(\lambda +2T-\sqrt{4T_{2}+T\lambda })^{2p}}{(\lambda +3T)^{2p-1}}. \end{aligned}$$

Here $T=T(M)=\min \{\sqrt{2M^{2}-2}, 4M/\pi \}$ and $\lambda =\lambda (M)$ is defined by (1.1).

Theorem D

[11] Let $F(z)=|z|^2g(z)+h(z)$ be a biharmonic mapping of the unit disk $\mathbb {D}$, with g(z), h(z) are harmonic on $\mathbb {D}$ and $F(0)=g(0)=h(0)=\lambda _{F}(0)-1=0,|g(z)|\le M$ and $\Lambda _{h}\le \Lambda$ for $z\in \mathbb {D}$, where $M\ge 0$, $\Lambda \ge 1$. Then F is univalent in the disk $\mathbb {D}_{\rho _4}$, where $\rho _4$ is the minimum positive root of the equation

$$\begin{aligned} 1-2\rho M-\frac{2M\rho ^2}{(1-\rho )^2}-\frac{(\Lambda ^2-1)\rho }{\Lambda (1-\rho )}=0, \end{aligned}$$

(1.2)

and $F(\mathbb {D}_{\rho _4})$ contains a schlicht disk $\mathbb {D}_{\rho _4'}$ with

$$\begin{aligned} \rho _4'=\left( 1+\Lambda -\frac{1}{\Lambda }\right) \rho _{4}-\frac{2M\rho _{4}^3}{1-\rho _{4}}+\left( \Lambda -\frac{1}{\Lambda }\right) \ln (1-\rho _{4}). \end{aligned}$$

(1.3)

Theorem E

[15] Let $F(z)=|z|^2g(z)+h(z)$ be a biharmonic mapping of the unit disk $\mathbb {D}$ with $g(z),\, h(z)$ are harmonic mappings in $\mathbb {D}$, and $g(0)=h(0)=0, \lambda _{F}(0)=\lambda _{g}(0)=\Lambda _{g}(0)=1$, $\Lambda _{g}(z)\le \Lambda _{1}$ and $\Lambda _{h}(z)\le \Lambda _{2}$ for $z\in \mathbb {D}$. Then $\Lambda _{1}\ge 1,\Lambda _{2}\ge 1$, and F is univalent in the disk $\mathbb {D}_{\rho _5}$, where $\rho _5$ is the minumum positive root in (0, 1) of the equation:

$$\begin{aligned} 1-3r^2-\frac{\Lambda _{2}^2-1}{\Lambda _{2}}\cdot \frac{r}{1-r}+\frac{\Lambda _{1}^2-1}{\Lambda _{1}}\cdot \left[ 2r^2+2r\ln (1-r) -\frac{r^3}{1-r}\right] =0,\qquad \end{aligned}$$

(1.4)

and $F(\mathbb {D}_{\rho _5})$ contains a schlicht disk $\mathbb {D}_{\rho _5'}$, with

$$\begin{aligned} \rho _5'=\rho _5-\rho _5^3+ \left( \rho _5^2\cdot \frac{\Lambda _{1}^2-1}{\Lambda _{1}}+ \frac{\Lambda _{2}^2-1}{\Lambda _{2}}\right) [\rho _5 +\ln (1-\rho _5)]. \end{aligned}$$

(1.5)

When $\Lambda _{1}=\Lambda _{2}=1,\rho _5=\frac{\sqrt{3}}{3}$ and $\rho _5'=\frac{2\sqrt{3}}{9}$ are sharp.

The above Theorems A, B and C didn’t consider the situation when $\Lambda _{G_p}\le \Lambda _{p}$. In this paper, we consider the case of $\Lambda _{G_p}\le \Lambda _{p}$, and establish several new Landau-type theorems for polyharmonic mappings or log-p-harmonic mappings (Theorems 2.1, 2.4, 2.6, 2.8, 2.9 and 2.10). In order to derive our main results, we need the following lemmas.

Lemma 1.1

([4]) Suppose that $f(z)=h(z)+\overline{g(z)}$ is a harmonic mapping of the unit disk $\mathbb {D}$ with $h(z)=\sum _{n=1}^\infty a_{n}z^{n}$ and $g(z)=\sum _{n=1}^\infty b_{n}z^{n}$ are analytic on $\mathbb {D}$. If $|f(z)|\le M$ for all $\mathbb {D}$, then

$$\begin{aligned} \Lambda _{f}(z)\le \frac{4M}{\pi (1-|z|^{2})}. \end{aligned}$$

(1.6)

Lemma 1.2

([14]) Suppose that $f(z)=h(z)+\overline{g(z)}$ is a harmonic mapping of the unit disk $\mathbb {D}$ with $h(z)=\sum _{n=1}^\infty a_{n}z^{n}$ and $g(z)=\sum _{n=1}^\infty b_{n}z^{n}$ are analytic on $\mathbb {D}$. If $|f(z)|\le M$ for all $\mathbb {D}$, then for $|z|=r<1$, we have

$$\begin{aligned} |f(z)|\le \frac{4Mr}{\pi (1-r)}. \end{aligned}$$

(1.7)

Lemma 1.3

([11, 12]) Suppose that $f(z)=h(z)+\overline{g(z)}$ is a harmonic mapping of the unit disk $\mathbb {D}$ with $h(z)=\sum _{n=1}^\infty a_{n}z^{n}$ and $g(z)=\sum _{n=1}^\infty b_{n}z^{n}$ are analytic on $\mathbb {D}$ and $\lambda _{f}(0)=1$. If $\Lambda _{f}(z)\le \Lambda$ for all $z\in \mathbb {D}$, then

$$\begin{aligned} |a_{n}|+|b_{n}|\le \frac{\Lambda ^{2}-1}{n\Lambda }, \, \, \, n=2, 3, \ldots . \end{aligned}$$

(1.8)

When $\Lambda >1$ , the above estimates are sharp for all $n=2, 3, \ldots$, with the extremal functions $f_{n}(z)$ and $\overline{f_{n}(z)}$ as follows

$$\begin{aligned} f_{n}(z)=\Lambda ^{2}z-(\Lambda ^{3}-\Lambda )\int _{0}^{z}\frac{1}{\Lambda +z^{n-1}}dz . \end{aligned}$$

(1.9)

When $\Lambda =1$, then $f(z)=a_{1}z+\overline{b_{1}}\overline{z}$ with $||a_{1}|-|b_{1}||=1$.

Lemma 1.4

Suppose that $f(z)=h(z)+\overline{g(z)}$ is a harmonic mapping of the unit disk $\mathbb {D}$ with $h(z)=\sum _{n=1}^\infty a_{n}z^{n}$ and $g(z)=\sum _{n=1}^\infty b_{n}z^{n}$ are analytic on $\mathbb {D}$. If $\Lambda _{f}(z)\le \Lambda$ for all $z\in \mathbb {D}$, then for each $z\in \mathbb {D}$,

$$\begin{aligned} |f(z)|\le \Lambda |z|. \end{aligned}$$

Proof

Since $\Lambda _{f}(z)\le \Lambda$ for all $z\in \mathbb {D}$, we have

$$\begin{aligned} |f(z)|=\left| \int _{[0,z]}f_{z}(z)dz+f_{\overline{z}}(z)d\overline{z}\right| \le \int _{[0,z]}\Lambda _{f}(z)|dz|\le \Lambda |z|, \end{aligned}$$

which completes the proof. $\square$

Lemma 1.5

([14]) Let p be a positive integer. Then for any $z_{1}\ne z_{2}$ in $\mathbb {D}_{r}(0<r<1)$, we have

$$\begin{aligned}\int _{0}^{1}|(z_1-z_2)t+z_2|^{2(p-1)}dt\ge \frac{1}{2p-1}\cdot \frac{|z_1|^{2p-1}+|z_2|^{2p-1}}{|z_1|+|z_2|}>0.\end{aligned}$$

2 Main Results

We first prove a new version of Landau-type theorem of polyharmonic mappings as follows.

Theorem 2.1

Let $F(z)=\sum _{k=1}^{p}|z|^{2(k-1)}G_{p-k+1}(z)$ be a p-harmonic mapping of $\mathbb {D}$ satisfying $F(0)=\lambda _{F}(0)-1=0$. Suppose that for each $k\in {\{1\dots p}\}$, we have

• (i) $G_{p-k+1}(z)$ is harmonic in $\mathbb {D}$, and $G_{p-k+1}(0)=0$;

• (ii) $|G_{p-k+1}(z)|\le M_{p-k+1}$, and $\Lambda _{G_p}(z)\le {\Lambda _{p}}$, where $M_{p-k+1}\ge 0, \Lambda _{p}\ge 1$.

Then there is a positive number $r_{0}$ such that F(z) is univalent in $\mathbb {D}_{r_{0}}$, where $r_{0}$ is a unique root in (0, 1) of the equation $A(r)=0$ and

$$\begin{aligned} A(r)= & {} 1-\frac{4}{\pi (1-r^{2})}\sum _{k=1}^{p-1}r^{2k}M_{p-k}-\frac{8}{\pi (1-r)}\sum _{k=1}^{p-1}kM_{p-k}r^{2k}\nonumber \\&- \frac{\Lambda _{p}^2-1}{\Lambda _{p}}\frac{r}{1-r}. \end{aligned}$$

(2.1)

Moreover, the range $F(\mathbb {D}_{r_0})$ contains a univalent disk $\mathbb {D}_{R_0}$, where

$$\begin{aligned} R_{0}=r_{0}+\frac{\Lambda _{p}^{2}-1}{\Lambda _{p}}[r_{0}+\ln (1-r_{0})]-\frac{4r_{0}}{\pi (1-r_{0})}\sum _{k=1}^{p-1}M_{p-k}r_{0}^{2k}. \end{aligned}$$

(2.2)

Proof

For each $k\in {\{1,2,\ldots , p}\}$, we may represent the harmonic functions $G_{p-k+1}(z)$ in series form as

$$\begin{aligned} G_{p-k+1}(z)= & {} \sum _{j=1}^\infty a_{j, p-k+1}z^{j}+\sum _{j=1}^\infty \overline{b_{j, p-k+1}}\overline{z}^{j},\\ ||a_{1, p}|-|b_{1, p}||= & {} \lambda _{F}(0)=||(G_p)_{z}(0)|-|(G_p)_{\overline{z}}(0)||=\lambda _{G_p}(0)=1. \end{aligned}$$

Now we fix r with $0<r<1$, to prove the univalence of F, we choose two points $z_{1},z_{2}$ in $\mathbb {D}_{r}$. Let $\Gamma =\{(z_{1}-z_{2})t+z_2:\, 0\le t \le 1\}$, then

$$\begin{aligned} |F(z_{1})-F(z_{2})|= & {} \left| \int _{\Gamma }F_{z}(z)dz+F_{\overline{z}}(z)d\overline{z}\right| \\\ge & {} I_{1}-I_{2}-I_{3}-I_{4}, \end{aligned}$$

where

$$\begin{aligned} I_{1}= & {} \left| \int _{\Gamma }(G_p)_{z}(0)dz+(G_p)_{\overline{z}}(0)d\overline{z}\right| , \\ I_{2}= & {} \left| \sum _{k=1}^{p-1} \int _{\Gamma }|z|^{2k}[(G_{p-k})_{z}(z)dz+(G_{p-k})_{\overline{z}}(z)d\overline{z}]\right| , \\ I_{3}= & {} \left| \sum _{k=1}^{p-1}\int _{\Gamma }k G_{p-k}(z)(\overline{z}^{k}z^{k-1}dz+ \overline{z}^{k-1}z^{k}d\overline{z})\right| , \\ I_{4}= & {} \left| \int _{\Gamma }[(G_{p})_{z}(z)-(G_{p})_{z}(0)]dz+[(G_p)_{\overline{z}}(z)- (G_p)_{\overline{z}}(0)]d\overline{z}\right| . \end{aligned}$$

By a simple calculation, we have

$$\begin{aligned} I_{1}\ge \int _{\Gamma }\lambda _{G_p}(0)|dz|=\lambda _{G_p}(0)|z_{1}-z_{2}|= |z_{1}-z_{2}|. \end{aligned}$$

By Lemma 1.1, we have

$$\begin{aligned} I_{2}\le & {} \sum _{k=1}^{p-1}\int _{\Gamma }|z|^{2k}[|(G_{p-k})_{z}(z)||dz|+|(G_{p-k})_{\overline{z}}(z)||d\overline{z}|]\\\le & {} \sum _{k=1}^{p-1}r^{2k}\int _{\Gamma }\Lambda _{G_{p-k}}(z)|dz|\\\le & {} |z_{1}-z_{2}|\sum _{k=1}^{p-1}r^{2k}\frac{4M_{p-k}}{\pi (1-r^{2})}. \end{aligned}$$

By Lemma 1.2, we have

$$\begin{aligned} I_{3}\le & {} \sum _{k=1}^{p-1}\int _{\Gamma }|k G_{p-k}(z)|[|\overline{z}^{k}z^{k-1}||dz|+ |\overline{z}^{k-1}z^{k}||d\overline{z}|]\\\le & {} |z_{1}-z_{2}|\frac{8}{\pi (1-r)}\sum _{k=1}^{p-1}k M_{p-k}r^{2k}. \end{aligned}$$

By Lemma 1.3, we have

$$\begin{aligned} I_{4}\le & {} \int _{\Gamma }|(G_{p})_{z}(z)-(G_{p})_{z}(0)||dz|+|(G_{p})_{\overline{z}}(z)- (G_{p})_{\overline{z}}(0)||d\overline{z}|\\\le & {} |z_{1}-z_{2}|\sum _{n=2}^\infty n(|a_{n, p}|+|b_{n, p}|)r^{n-1}\\\le & {} |z_{1}-z_{2}|\sum _{n=2}^\infty r^{n-1}\frac{\Lambda _{p}^{2}-1}{\Lambda _{p}}=|z_{1}-z_{2}|\frac{\Lambda _{p}^{2}-1}{\Lambda _{p}}\frac{r}{1-r}. \end{aligned}$$

Using these estimates, we obtain that

$$\begin{aligned} |F(z_{1})-F(z_{2})|\ge I_{1}-I_{2} -I_{3}-I_{4}\ge |z_{1}-z_{2}|A(r), \end{aligned}$$

where

$$\begin{aligned} A(r)= & {} 1-\frac{4}{\pi (1-r^{2})}\sum _{k=1}^{p-1}M_{p-k}r^{2k}-\frac{8}{\pi (1-r)}\sum _{k=1}^{p-1}k M_{p-k}r^{2k}\\&- \frac{\Lambda _{p}^2-1}{\Lambda _{p}}\frac{r}{1-r}. \end{aligned}$$

It is not difficult to verify that A(r) is strictly decreasing in (0, 1), and

$$\begin{aligned} \lim _{r \rightarrow 0}A(r)=1,\quad \lim _{r \rightarrow 1}A(r)=-\infty . \end{aligned}$$

Hence there exists a unique root $r_{0}$ in (0, 1) of the equation $A(r)=0$. This shows that

$$\begin{aligned} |F(z_{1})-F(z_{2})|>0 \end{aligned}$$

for any two distinct points $z_{1},z_{2}$ in $|z|<r_{0}$, which proves the univalency of F in the disk $\mathbb {D}_{r_{0}}$.

Finally, we consider any z with $|z|=r_{0}$. Then we have

$$\begin{aligned} |F(z)|= & {} \left| \sum _{n=1}^{\infty }(a_{n, p}z^{n}+\overline{b_{n, p}}\overline{z}^{n})+\sum _{k=1}^{p-1}|z|^{2k}G_{p-k}(z)\right| \\\ge & {} |a_{1, p}z+\overline{b_{1, p}}\overline{z}|-\left| \sum _{n=2}^{\infty }(a_{n, p}z^{n}+\overline{b_{n, p}}\overline{z}^{n})\right| - \left| \sum _{k=1}^{p-1}|z|^{2k}G_{p-k}(z)\right| \\\ge & {} r_{0}-\frac{\Lambda _{p}^{2}-1}{\Lambda _{p}}\sum _{n=2}^{\infty }\frac{r_{0}^{n}}{n}-\sum _{k=1}^{p-1}r_{0}^{2k} \frac{4M_{p-k}r_{0}}{\pi (1-r_{0})}\\= & {} r_{0}+\frac{\Lambda _{p}^{2}-1}{\Lambda _{p}}[r_{0}+\ln (1-r_{0})]-\frac{4r_{0}}{\pi (1-r_{0})}\sum _{k=1}^{p-1}M_{p-k}r_{0}^{2k}\\= & {} R_{0}. \end{aligned}$$

So, the range $F(\mathbb {D}_{r_0})$ contains a univalent disk $\mathbb {D}_{R_0}$. The proof of this theorem is complete. $\square$

The Eq. (2.1) cannot be solved explicitly. The Computer Algebra System Mathematica has been manipulated to obtain the numerical solutions to Eqs. (2.1) and (2.2). Table 1 shows the approximate values of $r_0, R_0$ that correspond to different choice of the constants $M_1,\, M_2$ and $\Lambda _3$.

Table 1 The values of $r_0,R_0$ are in Theorem 2.1 when $p=3$

Setting $p=2$ in Theorem 2.1, we have the following corollary, which is an improvement of Theorem D or Theorem 2.6 in [11].

Corollary 2.2

Let $F(z)=|z|^2g(z)+h(z)$ be a biharmonic mapping of the unit disk $\mathbb {D}$, with g(z), h(z) are harmonic on $\mathbb {D}$ and $F(0)=g(0)=h(0)=\lambda _{F}(0)-1=0$, $|g(z)|\le M$ and $\Lambda _{h}\le \Lambda$ for $z\in \mathbb {D}$, where $M\ge 0,\Lambda \ge 1$. Then F is univalent in the disk $\mathbb {D}_{t_0}$, where $t_0$ is the minimum positive root of the equation

$$\begin{aligned} 1-\frac{4M(3+2r)r^2}{\pi (1-r^2)}-\frac{(\Lambda ^2-1)r}{\Lambda (1-r)}=0, \end{aligned}$$

(2.3)

and $F(\mathbb {D}_{t_0})$ contains a schlicht disk $\mathbb {D}_{t_0'}$ with

$$\begin{aligned} t_0'=\left( 1+\Lambda -\frac{1}{\Lambda }\right) t_0-\frac{4Mt_0^3}{\pi (1-t_0)}+\left( \Lambda -\frac{1}{\Lambda }\right) \ln (1-t_0). \end{aligned}$$

(2.4)

Remark 2.3

Setting $M=0$ in Corollary 2.2, we get Theorem 2.2 in [12].

The Eq. (2.3) cannot be solved explicitly. The Computer Algebra System Mathematica has calculated the numerical solutions to Eqs. (2.3), (2.4), (1.2) and (1.3). Table 2 shows the approximate values of $t_0, \rho _4$ and $t_0', \rho _4'$ that correspond to different choice of the constants M and $\Lambda$, which shows that $t_0>\rho _4$ and $t_0'>\rho _4'$, that is, Corollary 2.2 is an improvement of Theorem D.

Table 2 The values of $t_0,t_0'$ are in Corollary 2.2 and the values of $\rho _4,\rho _4'$ are in Theorem D

Next, we derive another two new versions of Landau-type theorems of polyharmonic mappings as follows.

Theorem 2.4

Let $F(z)=\sum _{k=1}^{p}|z|^{2(k-1)}G_{p-k+1}(z)$ be a polyharmonic mapping of $\mathbb {D}$ satisfying $F(0)=\lambda _{F}(0)-1=0$. Suppose that for each $k\in \{1, 2, \ldots , p\}$, we have

• (i) $G_{p-k+1}(z)$ is harmonic in $\mathbb {D}$, and $G_{p-k+1}(0)=0$;

• (ii) $\Lambda _{G_{p-k+1}}(z)\le \Lambda _{p-k+1}$ for all $z\in \mathbb {D}$, where $\Lambda _{p-k+1}\ge 0,\, k=2, 3, \ldots , p$, and $\Lambda _{p}\ge 1$.

Then there exists a positive number $r_1$ such that F(z) is univalent in $\mathbb {D}_{r_1}$, where $r_1$ is a unique root in (0, 1) of the equation $A_1(r)=0$ and

$$\begin{aligned} A_1(r)=1-\sum _{k=1}^{p-1}(2k+1) \Lambda _{p-k}r^{2k}-\frac{\Lambda _{p}^2-1}{\Lambda _{p}}\frac{r}{1-r}. \end{aligned}$$

(2.5)

Moreover, the range $F(\mathbb {D}_{r_1})$ contains a univalent disk $\mathbb {D}_{R_1}$, where

$$\begin{aligned} R_1=r_1+\frac{\Lambda _{p}^{2}-1}{\Lambda _{p}}[r_1+\ln (1-r_1)]-\sum _{k=1}^{p-1}\Lambda _{p-k}r_1^{2k+1}. \end{aligned}$$

(2.6)

Proof

For each $k\in \{1,2,\ldots , p\}$, we may represent the harmonic functions $G_{p-k+1}(z)$ in series form as

$$\begin{aligned} G_{p-k+1}(z)= & {} \sum _{j=1}^\infty a_{j, p-k+1}z^{j}+\sum _{j=1}^\infty \overline{b_{j, p-k+1}}\overline{z}^{j}, \\ ||a_{1, p}|-|b_{1, p}||= & {} \lambda _{F}(0)=||(G_p)_{z}(0)|-|(G_p)_{\overline{z}}(0)||=\lambda _{G_p}(0)=1. \end{aligned}$$

Now we fix r with $0<r<1$, to prove the univalence of F, we choose two points $z_{1},z_{2}$ in $\mathbb {D}_{r}$. Let $\Gamma =\{(z_{1}-z_{2})t+z_2:\, 0\le t \le 1\}$. Then as in the proof of Theorem 2.1, we have

$$\begin{aligned} |F(z_{1})-F(z_{2})|= & {} \left| \int _{\Gamma }F_{z}(z)dz+F_{\overline{z}}(z)d\overline{z}\right| \\\ge & {} I_{1}-I_{2}-I_{3}-I_{4}, \end{aligned}$$

where

$$\begin{aligned} I_{1}\ge \int _{\Gamma }\lambda _{G_p}(0)|dz|=\lambda _{G_p}(0)|z_{1}-z_{2}|= |z_{1}-z_{2}|, \end{aligned}$$

and

$$\begin{aligned} I_{2}\le & {} \sum _{k=1}^{p-1}\int _{\Gamma }|z|^{2k}[|(G_{p-k})_{z}(z)||dz|+|(G_{p-k})_{\overline{z}}(z)||d\overline{z}|]\\\le & {} \sum _{k=1}^{p-1}r^{2k}\int _{\Gamma }\Lambda _{G_{p-k}}(z)|dz|\\\le & {} |z_{1}-z_{2}|\sum _{k=1}^{p-1}\Lambda _{p-k}r^{2k}. \end{aligned}$$

By the condition (ii) of Theorem 2.4 and Lemma 1.4, we have

$$\begin{aligned} I_{3}\le & {} \sum _{k=1}^{p-1}\int _{\Gamma }|k G_{p-k}(z)|[|\overline{z}^{k}z^{k-1}||dz|+ |\overline{z}^{k-1}z^{k}||d\overline{z}|]\\\le & {} |z_{1}-z_{2}|\sum _{k=1}^{p-1}2k \Lambda _{p-k}r^{2k}. \end{aligned}$$

By Lemma 1.3, we have

$$\begin{aligned} I_{4}\le & {} \int _{\Gamma }|(G_{p})_{z}(z)-(G_{p})_{z}(0)||dz|+|(G_{p})_{\overline{z}}(z)- (G_{p})_{\overline{z}}(0)||d\overline{z}|\\\le & {} |z_{1}-z_{2}|\sum _{n=2}^\infty n(|a_{n, p}|+|b_{n, p}|)r^{n-1}\\\le & {} |z_{1}-z_{2}|\sum _{n=2}^\infty r^{n-1}\frac{\Lambda _{p}^{2}-1}{\Lambda _{p}}=|z_{1}-z_{2}|\frac{\Lambda _{p}^{2}-1}{\Lambda _{p}}\frac{r}{1-r}. \end{aligned}$$

Using these estimates, we obtain that

$$\begin{aligned}&|F(z_{1})-F(z_{2})|\ge I_{1}-I_{2} -I_{3}-I_{4}\\&\quad \ge |z_{1}-z_{2}|\left[ 1-\sum _{k=1}^{p-1}\Lambda _{p-k}r^{2k}-\sum _{k=1}^{p-1}2k \Lambda _{p-k}r^{2k}-\frac{\Lambda _{p}^2-1}{\Lambda _{p}}\frac{r}{1-r}\right] \\&\quad =|z_{1}-z_{2}|\left[ 1-\sum _{k=1}^{p-1}(2k+1) \Lambda _{p-k}r^{2k}-\frac{\Lambda _{p}^2-1}{\Lambda _{p}}\frac{r}{1-r}\right] . \end{aligned}$$

Let

$$\begin{aligned} A_1(r)=1-\sum _{k=1}^{p-1}(2k+1) \Lambda _{p-k}r^{2k}-\frac{\Lambda _{p}^2-1}{\Lambda _{p}}\frac{r}{1-r}. \end{aligned}$$

It is not difficult to verify that $A_1(r)$ is strictly decreasing in (0, 1), and

$$\begin{aligned} \lim _{r \rightarrow 0}A_1(r)=1,\quad \lim _{r \rightarrow 1}A_1(r)=-\infty . \end{aligned}$$

Hence there exists a unique root $r_1$ in (0, 1) of the equation $A_1(r)=0$. This shows that

$$\begin{aligned} |F(z_{1})-F(z_{2})|>0 \end{aligned}$$

for any two distinct points $z_{1},z_{2}$ in $|z|<r_1$, which proves the univalence of F in the disk $\mathbb {D}_{r_1}$.

Finally, we now consider any z with $|z|=r_1$. Then we have

$$\begin{aligned} |F(z)|= & {} \left| \sum _{n=1}^{\infty }(a_{n, p}z^{n}+\overline{b_{n, p}}\overline{z}^{n})+\sum _{k=1}^{p-1}|z|^{2k}G_{p-k}(z)\right| \\\ge & {} |a_{1, p}z+\overline{b_{1, p}}\overline{z}|-\left| \sum _{n=2}^{\infty }(a_{n, p}z^{n}+\overline{b_{n, p}}\overline{z}^{n})\right| - \left| \sum _{k=1}^{p-1}|z|^{2k}G_{p-k}(z)\right| \\\ge & {} r_1-\frac{\Lambda _{p}^{2}-1}{\Lambda _{p}}\sum _{n=2}^{\infty }\frac{r_1^{n}}{n}-\sum _{k=1}^{p-1}\Lambda _{p-k}r_1^{2k+1}\\= & {} r_1+\frac{\Lambda _{p}^{2}-1}{\Lambda _{p}}[r_1+\ln (1-r_1)]-\sum _{k=1}^{p-1}\Lambda _{p-k}r_1^{2k+1}\\= & {} R_1. \end{aligned}$$

So, the range $F(\mathbb {D}_{r_1})$ contains a univalent disk $\mathbb {D}_{R_1}$. The proof of this theorem is complete. $\square$

By Lemma 1.4, we have

$$\begin{aligned} |G_{p-k+1}(z)|\le \Lambda _{p-k+1} |z|\le \Lambda _{p-k+1}. \end{aligned}$$

Thus applying Theorem 2.1 directly, we can get the following corollary.

Corollary 2.5

Let $F(z)=\sum _{k=1}^{p}|z|^{2(k-1)}G_{p-k+1}(z)$ be a polyharmonic mapping of $\mathbb {D}$ satisfying the hypothesis of Theorem 2.4. Then there exists a positive number $r_1'$ such that F(z) is univalent in $\mathbb {D}_{r_1'}$, where $r_1'$ is a unique root in (0, 1) of the equation

$$\begin{aligned} A_1^{*}(r)= & {} 1-\frac{4}{\pi (1-r^{2})}\sum _{k=1}^{p-1}r^{2k}\Lambda _{p-k}-\frac{8}{\pi (1-r)}\sum _{k=1}^{p-1}k\Lambda _{p-k}r^{2k}\nonumber \\&- \frac{\Lambda _{p}^2-1}{\Lambda _{p}}\frac{r}{1-r}=0. \end{aligned}$$

(2.7)

Moreover, the range $F(\mathbb {D}_{r_1'})$ contains a univalent disk $\mathbb {D}_{R_1'}$, where

$$\begin{aligned} R_1'=r_1'+\frac{\Lambda _{p}^{2}-1}{\Lambda _{p}}[r_1'+\ln (1-r_1')]-\frac{4r_1'}{\pi (1-r_1')}\sum _{k=1}^{p-1}\Lambda _{p-k}r_1'^{2k}. \end{aligned}$$

(2.8)

The Eqs. (2.5) and (2.7) cannot be solved explicitly. The Computer Algebra System Mathematica has solved the numerical solutions to Eqs. (2.5)–(2.8) for $p=2$. Table 3 shows the approximate values of $r_1, r_1'$ and $R_1, R_1'$ that correspond to different choice of the constants $\Lambda _1$ and $\Lambda _2$, which shows that $r_1>r_1'$ and $R_1>R_1'$, that is, Theorem 2.4 is an improvement of Corollary 2.5.

Table 3 The values of $r_1,R_1$ are in Theorem 2.4 and the values of $r_1',R_1'$ are in Corollary 2.5

The next theorem is different with Theorems 2.1 and 2.4, since $\lambda _F(0)=0$.

Theorem 2.6

Let $p\, (>1)$ be an integer and $F(z)=|z|^{2(p-1)}G(z)$, where $F(0)=G(0)=\lambda _{G}(0)-1=0$. Suppose that G(z) satisfies

• (i) G(z) is harmonic in D, and

• (ii) $\Lambda _{G}(z)\le \Lambda$ for all $z\in \mathbb {D}$, where $\Lambda \ge {1}$.

Then there is a positive number $r_2$ such that F is univalent in $\mathbb {D}_{r_2}$, where $r_2$ is a unique root in (0, 1) of the equation

$$\begin{aligned} B(r)=1+2\frac{\Lambda ^{2}-1}{\Lambda } \left[ \frac{\ln (1-r)}{r}+1\right] -\frac{\Lambda ^{2}-1}{\Lambda }\frac{r}{1-r}=0 \end{aligned}$$

(2.9)

for $\Lambda >1$, and $r_2=1$ for $\Lambda =1$. Moreover, the range $F(\mathbb {D}_{r_2})$ contains a univalent disk $\mathbb {D}_{R_2}$, where

$$\begin{aligned} R_2=\left\{ \begin{array}{lll} r_2^{2p-1}\left[ 1+\frac{\Lambda ^{2}-1}{\Lambda }\left( \frac{\ln (1-r_2)}{r_2}+1\right) \right] , &{} \Lambda >1,\\ 1, &{}\Lambda =1. \end{array} \right. \end{aligned}$$

(2.10)

When $\Lambda =1$, the above results are sharp.

Proof

We may represent the harmonic functions G(z) in the form as

$$\begin{aligned} G(z)=\sum _{n=1}^\infty a_{n}z^{n}+\sum _{n=1}^\infty \overline{b_{n}}\overline{z}^{n}. \end{aligned}$$

When $\Lambda >1$, we fix r with $0<r<1$, to prove the univalency of F, we choose two points $z_{1},z_{2}$ in $D_{r}$. Let $\Gamma =\{z(t)=(z_{1}-z_{2})t+z_2:0\le t\le 1\}$. Then

$$\begin{aligned} |F(z_{1})-F(z_{2})|= & {} \left| \int _{\Gamma }F_{z}(z)dz+F_{\overline{z}}(z)d\overline{z}\right| \\\ge & {} I_{1}-I_{2}-I_{3}, \end{aligned}$$

where

$$\begin{aligned} I_{1}= & {} \left| \int _{\Gamma }(p-1)|z|^{2(p-1)}(G_{z}(0)dz+G_{\overline{z}}(0)d\overline{z})\right| , \\ I_{2}= & {} \left| \int _{\Gamma }(p-1)|z|^{2(p-1)} \left[ \frac{G(z)-z G_{z}(0)}{z}dz+\frac{G(z)-\overline{z}G_{\overline{z}}(0)}{\overline{z}}d \overline{z}\right] \right| , \\ I_{3}= & {} \left| \int _{\Gamma }(p-1)|z|^{2(p-1)} [(G_{z}-G_{z}(0))dz+(G_{\overline{z}}-G_{\overline{z}}(0))d\overline{z}]\right| . \end{aligned}$$

By a simple calculation, we have

$$\begin{aligned} I_{1}\ge & {} \int _{\Gamma }|(p-1)|z|^{2(p-1)}\lambda _{G}(0)||dz|\\= & {} \lambda _{G}(0)|z_{1}-z_{2}|\int _{0}^{1}(p-1)|z|^{2(p-1)}dt\\= & {} |z_{1}-z_{2}|\int _{0}^{1}(p-1)|z|^{2(p-1)}dt. \end{aligned}$$

By Lemma 1.3, we have

$$\begin{aligned} I_{2}\le & {} |z_{1}-z_{2}|\int _{0}^{1}(p-1)|z|^{2(p-1)}dt\,\left( 2\sum _{n=2}^\infty (|a_{n}|+|b_{n}|)r^{n-1}\right) \\\le & {} |z_{1}-z_{2}|\int _{0}^{1}(p-1)|z|^{2(p-1)}dt\, \left( 2\sum _{n=2}^\infty \frac{\Lambda ^{2}-1}{n\Lambda }r^{n-1}\right) \\\le & {} |z_{1}-z_{2}|\int _{0}^{1}(p-1)|z|^{2(p-1)}dt\, 2\frac{\Lambda ^{2}-1}{\Lambda }\left[ -\left( \frac{\ln (1-r)}{r}+1\right) \right] ,\\ I_{3}\le & {} |z_{1}-z_{2}|\int _{0}^{1}(p-1)|z|^{2(p-1)}dt\left( \sum _{n=2}^\infty n(|a_{n}|+|b_{n}|)r^{n-1}\right) \\\le & {} |z_{1}-z_{2}|\int _{0}^{1}(p-1)|z|^{2(p-1)}dt\cdot \frac{\Lambda ^{2}-1}{\Lambda }\sum _{n=2}^\infty r^{n-1}\\= & {} |z_{1}-z_{2}|\int _{0}^{1}(p-1)|z|^{2(p-1)}dt\cdot \frac{\Lambda ^{2}-1}{\Lambda }\frac{r}{1-r}. \end{aligned}$$

Using these estimates, we obtain:

$$\begin{aligned} |F(z_{1})-F(z_{2})|\ge & {} I_{1}-I_{2}-I_{3}\\\ge & {} |z_{1}-z_{2}|\int _{0}^{1}(p-1)|z(t)|^{2(p-1)}dt\\&\cdot \left[ 1+2\frac{\Lambda ^{2}-1}{\Lambda } \left( \frac{\ln (1-r)}{r}+1\right) -\frac{\Lambda ^{2}-1}{\Lambda }\frac{r}{1-r}\right] . \end{aligned}$$

Let

$$\begin{aligned} B(r)=1+2\frac{\Lambda ^{2}-1}{\Lambda } \left( \frac{\ln (1-r)}{r}+1\right) -\frac{\Lambda ^{2}-1}{\Lambda }\frac{r}{1-r}. \end{aligned}$$

(2.11)

It is not difficult to verify that B(r) is strictly decreasing in (0, 1), and

$$\begin{aligned} \lim _{r \rightarrow 0}B(r)=1,\quad \lim _{r \rightarrow 1}B(r)=-\infty \end{aligned}$$

(2.12)

Hence there exists a unique root $r_2$ in (0, 1) of the equation $B(r)=0$.

On the other hand, by Lemma 1.5, we have

$$\begin{aligned} \int _{0}^{1}(p-1)|z(t)|^{2(p-1)}dt\ge \frac{p-1}{2p-1}\cdot \frac{|z_1|^{2p-1}+|z_2|^{2p-1}}{|z_1|+|z_2|}>0 \end{aligned}$$

for any two distinct points $z_{1},z_{2}$ in $|z|<r_2$. This shows that

$$\begin{aligned} |F(z_{1})-F(z_{2})|>0 \end{aligned}$$

for any two distinct points $z_{1},z_{2}$ in $|z|<r_2$, which proves the univalency of F in the disk $\mathbb {D}_{r_2}$.

Finally, we consider any z with $|z|=r_2$.Then we have

$$\begin{aligned} |F(z)|= & {} ||z|^{2(p-1)}G(z)|\\= & {} r_2^{2p-2}\left| \sum _{n=1}^\infty a_{n}z^{n}+\sum _{n=1}^\infty \overline{b_{n}}\overline{z}^{n}\right| \\\ge & {} r_2^{2p-2}\left[ |a_{1}z+\overline{b_{1}}\overline{z}|-\right| \sum _{n=2}^\infty a_{n}z^{n}+\sum _{n=2}^\infty \overline{b_{n}}\overline{z}^{n}\left| \right] \\\ge & {} r_2^{2p-1}\left[ 1+\frac{\Lambda ^{2}-1}{\Lambda }\left( \frac{\ln (1-r_2)}{r_2}+1\right) \right] . \end{aligned}$$

So, the range $F(\mathbb {D}_{r_2})$ contains a univalent disk $\mathbb {D}_{R_2}$, where $R_2$ is defined by (2.10).

When $\Lambda =1$, by Lemma 1.3, we know that $G(z)=a_{1}z+\overline{b_{1}}\overline{z}$ with $||a_{1}|-|b_{1}||=1$. From $1=\lambda _{G}(0)\le \Lambda _{G}(0)\le \Lambda =1$, we obtain that $\Lambda _{G}(0)=|a_{1}|+|b_{1}|=1$, so that $||a_{1}|-|b_{1}||=|a_{1}|+|b_{1}|=1$, thus we have $|a_{1}|=1$, $b_{1}=0$, or $|b_{1}|=1,a_{1}=0$.

When $F(z)=|z|^{2(p-1)}a_{1}z$ with $|a_{1}|=1$, we choose two distinct points $z_{1},z_{2}$ in $\mathbb {D}_{r}\, (0<r<1)$, we have

$$\begin{aligned} |F(z_{1})-F(z_{2})|= & {} ||z_{1}|^{2(p-1)}a_{1}z_{1}-|z_{2}|^{2(p-1)}a_{1}z_{2}|\\= & {} |a_{1}|||z_{1}|^{2(p-1)}z_{1}-|z_{2}|^{2(p-1)}z_{2}|\\= & {} ||z_{1}|^{2(p-1)}z_{1}-|z_{2}|^{2(p-1)}z_{2}|. \end{aligned}$$

We split into two cases to verify that $F(z_{1})\ne F(z_{2})$.

• (i) If $|z_{1}|\ne |z_{2}|$, then we have

  $$\begin{aligned} |F(z_{1})-F(z_{2})|=||z_{1}|^{2(p-1)}z_{1}-|z_{2}|^{2(p-1)}z_{2}|\ge |||z_{1}|^{2p-1}-|z_{2}|^{2p-1}||>0, \end{aligned}$$

• (ii) If $|z_{1}|=|z_{2}|, -\pi <\arg z_{1},\arg z_{2}\le \pi ,\, \arg z_{1}\ne \arg z_{2}$, then we have

  $$\begin{aligned} |F(z_{1})-F(z_{2})|= & {} ||z_{1}|^{2(p-1)}z_{1}-|z_{2}|^{2(p-1)}z_{2}|\\= & {} ||z_{1}|^{2p-1}e^{i\arg z_{1}}-|z_{2}|^{2p-1}e^{i\arg z_{2}}|\\= & {} |z_{1}|^{2p-1}\cdot |e^{i(\arg z_{1}-\arg z_2)}-1|>0. \end{aligned}$$

So $F(z_{1})\ne F(z_{2})$, hence F(z) is univalent on $\mathbb {D}$, and $F(\mathbb {D})=\mathbb {D}$.

Similarly, when $F(z)=|z|^{2(p-1)}\overline{b_{1}}\overline{z}$ with $|b_{1}|=1$, we may prove that F(z) is univalent on $\mathbb {D}$, and $F(\mathbb {D})=\mathbb {D}$. That is, $r_2=R_2=1$ for $\Lambda =1$. It is evident that these results of this case are sharp. The proof of this theorem is complete. $\square$

Remark 2.7

Setting $p=2$ in Theorem 2.6, we get Theorem 3.3 of [15], when $p\ge 3$, the results in Theorem 2.6 are new.

Finally, we establish three new versions of Landau-type theorems of log-p-harmonic mappings as follows.

Theorem 2.8

Let $f(z)=\prod _{k=1}^{p}(g_{p-k+1}(z))^{|z|^{2(k-1)}}$ be a log-p-harmonic mapping of $\mathbb {D}$ satisfying $f(0)=g_{p}(0)=\lambda _{f}(0)=1$. Suppose that for each $k\in \{1,\ldots , p\}$, we have

• (i) $g_{p-k+1}(z)$ is log-harmonic in $\mathbb {D}$, and

• (ii) $|g_{p-k+1}(z)|\le {M_{p-k+1}}$, let $G_{p}=\log g_{p}$, and $\Lambda _{G_{p}}\le {\Lambda _{p}}$, where $M_{p-k+1}\ge {1}$, $\Lambda _{p}\ge {1}$.

Then there is a positive number $r_3$ such that f is univalent in $\mathbb {D}_{r_3}$, where $r_3$$(0<r_3<1)$ satisfies the following equation

$$\begin{aligned}&1-\frac{4}{\pi (1-r^{2})}\sum _{k=1}^{p-1}r^{2k}M_{p-k}^{*}-\sum _{k=1}^{p-1}kM_{p-k}^{*}r^{2k}\frac{8}{\pi (1-r)}\nonumber \\&\quad - \frac{\Lambda _{p}^{2}-1}{\Lambda _{p}}\frac{r}{1-r}=0, \end{aligned}$$

(2.13)

where $M_{p-k+1}^{*}=\log M_{p-k+1}+\pi ,\, k=2, 3, \ldots , p$. Moreover, the range $F(\mathbb {D}_{r_3})$ contains a univalent disk $\mathbb {D}(z_3, R_3)$, where

$$\begin{aligned} z_3= & {} \cosh \bigg (\frac{R_3'}{\sqrt{2}}\bigg ),\quad \rho _{0}=\min {\bigg \{\sinh \bigg (\frac{R_3'}{\sqrt{2}}\bigg )}, {\cosh \bigg (\frac{R_3'}{\sqrt{2}}\bigg )\sin \bigg (\frac{R_3'}{\sqrt{2}}\bigg )}\bigg \},\qquad \end{aligned}$$

(2.14)

$$\begin{aligned} R_3'= & {} r_3+\frac{\Lambda _{p}^{2}-1}{\Lambda _{p}}[r_3+\ln (1-r_3)]-\sum _{k=1}^{p-1}r_3^{2k} \frac{4M_{p-k}^{*}r_3}{\pi (1-r_3)}. \end{aligned}$$

(2.15)

Proof

Let $F(z)=\sum _{k=1}^{p}|z|^{2(k-1)}G_{p-k+1}(z)$, for each $k\in \{1,\ldots , p\}$. We may represent the harmonic functions $G_{p-k+1}(z)=\log g_{p-k+1}$ in series form as

$$\begin{aligned} G_{p-k+1}(z)=\sum _{n=1}^\infty a_{j, p-k+1}z^{j}+\sum _{n=1}^\infty \overline{b_{j, p-k+1}}\overline{z}^{j}. \end{aligned}$$

Then $F=\log f$ is a polyharmonic mapping in $\mathbb {D}$.

We know that

$$\begin{aligned} \lambda _{f}(0)=||f_{z}(0)|-|f_{\overline{z}}(0)||=|f(0)|||F_{z}(0)|-|F_{\overline{z}}(0)||, \end{aligned}$$

and $f(0)=1$, so it follows from $g_{p}(0)=\lambda _{f}(0)=1$, we have $G_{p}(0)=\lambda _{F}(0)-1=0$.

It is obvious that

$$\begin{aligned} |G_{p-k+1}|=|\log g_{p-k+1}|=|\log |g_{p-k+1}|+i \arg g_{p-k+1}|\le |\log |g_{p-k+1}||+\pi , \end{aligned}$$

so we have

$$\begin{aligned} |G_{p-k+1}|\le \log M_{p-k+1}+\pi =M_{p-k+1}^{*}. \end{aligned}$$

In order to prove the univalence of f, we fix r with $0<r<1$, we choose two points $z_{1},z_{2}$ in $\mathbb {D}_{r}$. Let $\Gamma =\{(z_{1}-z_{2})t+z_2:0\le t\le 1\}$, then it follows from Theorem 2.1, we have

$$\begin{aligned} |\log f(z_{1})-\log f(z_{2})|= & {} |F(z_{1})-F(z_{2})|\\= & {} \left| \int _{\Gamma }F_{z}(z)dz+F_{\overline{z}}(z)d\overline{z}\right| \\\ge & {} |z_{1}-z_{2}|\left[ 1-\frac{4}{\pi (1-r^{2})}\sum _{k=1}^{p-1}r^{2k}M_{p-k}^{*}\right. \\&\left. \quad -\frac{8}{\pi (1-r)}\sum _{k=1}^{p-1}k M_{p-k}^{*}r^{2k}- \frac{\Lambda _{p}-1}{\Lambda _{p}}\frac{r}{1-r}\right] . \end{aligned}$$

We know from the proof of Theorem 2.1 that there is a unique $r_3 \in (0, 1)$ satisfying Eq. (2.13), such that

$$\begin{aligned}|\log f(z_{1})-\log f(z_{2})|> 0\end{aligned}$$

for any two distinct points $z_{1},z_{2}$ in $|z|<r_3$, which proves the univalency of f in the disk $\mathbb {D}_{r_{2}}$.

Finally, we consider any z with $|z|=r_3$, then we have

$$\begin{aligned} |\log f(z)|= & {} |F(z)|=\left| \sum _{n=1}^{\infty }(a_{n, p}z^{n}+\overline{b_{n, p}}\overline{z}^{n})+\sum _{k=1}^{p-1}|z|^{2k}G_{p-k}(z)\right| \\\ge & {} |a_{1, p}z+b_{1, p}\overline{z}|-\left| \sum _{n=2}^{\infty }(a_{n, p}z^{n}+\overline{b_{n, p}}\overline{z}^{n})\right| - \left| \sum _{k=1}^{p-1}|z|^{2k}G_{p-k}(z)\right| \\\ge & {} r_3-\frac{\Lambda _{p}^{2}-1}{\Lambda _{p}}\sum _{n=2}^{\infty }\frac{r_3^{n}}{n}-\sum _{k=1}^{p-1}r_3^{2k} \frac{4M_{p-k}^{*}r_3}{\pi (1-r_3)}\\= & {} R_3', \end{aligned}$$

where

$$\begin{aligned} R_3'=r_3+\frac{\Lambda _{p}^{2}-1}{\Lambda _{p}}[r_3+\ln (1-r_3)]-\sum _{k=1}^{p-1}r_3^{2k} \frac{4M_{p-k}^{*}r_3}{\pi (1-r_3)}, \end{aligned}$$

the properties of the exponential function $e^{z}$ shows the $f(\mathbb {D}_{r_3})$ contains a disk $\mathbb {D}(z_3, R_3)$, where $z_3$ and $R_3$ are defined by (2.14) and (2.15). The proof of this theorem is complete. $\square$

By means of Theorem 2.4, applying the same method as in our proof of Theorem 2.8, we have the following theorem.

Theorem 2.9

Let $f(z)=\prod _{k=1}^{p}(g_{p-k+1}(z))^{|z|^{2(k-1)}}$ be a log-p-harmonic mapping of $\mathbb {D}$ satisfying $f(0)=\lambda _{f}(0)=1$. Suppose that for each $k\in \{1,\ldots , p\}$, we have

• (i) $g_{p-k+1}(z)$ is log-harmonic in $\mathbb {D}$, and $g_{p-k+1}(0)=1$;

• (ii) let $G_{p-k+1}=\log g_{p-k+1}$, and $\Lambda _{G_{p-k+1}}(z)\le {\Lambda _{p-k+1}}$ for all $z\in \mathbb {D}$, where $\Lambda _{p-k+1}\ge 1$.

Then there is a positive number $r_{2}$ such that f is univalent in $\mathbb {D}_{r_4}$, where $r_4$$(0<r_4<1)$ satisfies the following equation

$$\begin{aligned} 1-\sum _{k=1}^{p-1}(2k+1) \Lambda _{p-k}r^{2k}-\frac{\Lambda _{p}^2-1}{\Lambda _{p}}\frac{r}{1-r}=0. \end{aligned}$$

(2.16)

Moreover, the range $F(\mathbb {D}_{r_4})$ contains a univalent disk $\mathbb {D}(z_4, R_4)$, where

$$\begin{aligned} z_4= & {} \cosh \bigg (\frac{R_4'}{\sqrt{2}}\bigg ),\nonumber \\ R_4= & {} \min {\bigg \{\sinh \bigg (\frac{R_4'}{\sqrt{2}}\bigg )}, {\cosh \bigg (\frac{R_4'}{\sqrt{2}}\bigg )\sin \bigg (\frac{R_4'}{\sqrt{2}}\bigg )}\bigg \}, \end{aligned}$$

(2.17)

$$\begin{aligned} R_4'= & {} r_4+\frac{\Lambda _{p}^{2}-1}{\Lambda _{p}}[r_4+\ln (1-r_4)]-\sum _{k=1}^{p-1}\Lambda _{p-k}r_4^{2k+1}. \end{aligned}$$

(2.18)

By means of Theorem 2.6, applying the same method as in our proof of Theorem 2.8, we have the following theorem.

Theorem 2.10

Let $f(z)=g(z)^{|z|^{2(p-1)}}$ be a log-p-harmonic mapping of $\mathbb {D}$ satisfying $f(0)=g(0)=\lambda _{g}(0)=1$. Suppose that

• (i) g(z) is log-harmonic in $\mathbb {D}$;

• (ii) let $G(z)=\log g(z)$, and $\Lambda _{G}\le \Lambda$ . Then there is a positive number $r_5$ such that f is univalent in $\mathbb {D}_{r_5}$, where $r_5$ is a unique root in (0, 1) of the equation

  $$\begin{aligned} B(r)=1+2\frac{\Lambda ^{2}-1}{\Lambda }\left[ \frac{\ln (1-r)}{r}+1\right] -\frac{\Lambda ^{2}-1}{\Lambda }\frac{r}{1-r}=0 \end{aligned}$$

  (2.19)

  for $\Lambda >1$, and $r_5=1$ for $\Lambda =1$. Moreover, the range $f(\mathbb {D}_{r_5})$ contains a univalent disk $\mathbb {D}(z_5, R_5')$, where

  $$\begin{aligned} z_5= & {} \cosh \left( \frac{R_5}{\sqrt{2}}\right) ,\nonumber \\ R_5'= & {} \min {\bigg \{\sinh \bigg (\frac{R_5}{\sqrt{2}}\bigg )}, {\cosh \bigg (\frac{R_5}{\sqrt{2}}\bigg )\sin \bigg (\frac{R_5}{\sqrt{2}}\bigg )}\bigg \}, \end{aligned}$$

  (2.20)
  $$\begin{aligned} R_5= & {} \left\{ \begin{array}{lll} r_5^{2p-1}\left[ 1+\frac{\Lambda ^{2}-1}{\Lambda }\left( \frac{\ln (1-r_2)}{r_2}+1\right) \right] , &{} \Lambda >1,\\ 1, &{}\Lambda =1. \end{array} \right. \end{aligned}$$

  (2.21)