Estimates on Bloch constants for certain log-p-harmonic mappings

Abstract

In this paper, we first provide a brief overview of Landau-type theorems for log-p-harmonic mappings. Next, we establish four new versions of Landau-type theorems for certain bounded p-harmonic mappings F with $J_F(0)=1$. Then, as applications of these results, the corresponding Landau-type theorems for certain log-p-harmonic mappings f with $J_f(0)=1$ are provided. In particular, several sharp results of Landau-type theorems for certain bounded p-harmonic mappings or log-p-harmonic mappings with $J_f(0)=1$ are obtained. Finally, we also establish a Landau-type theorem for a certain bounded log-p-harmonic mappings with $J_f(0)=1$, which improves the corresponding results of different authors.

1 Introduction

Suppose that $F(z)=u(z)+iv(z)$ is a 2p times continuously differentiable complex-valued function in a domain $D\subseteq \mathbb {C}$, where p is a positive integer. For $z=x+iy\in D$, we denote the formal derivatives of f by

$$\begin{aligned} F_z=\frac{1}{2}(F_x-iF_y) \text{ and } F_{\overline{z}}=\frac{1}{2}(F_x-i F_y). \end{aligned}$$

Let $\Delta$ denote the complex Laplacian operator

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

Then we say that F is p-harmonic in D if F satisfies the p-harmonic equation

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

Evidently, when $p=1\, (\mathrm{resp.} \, p=2)$, F is called harmonic (resp. biharmonic) mapping. For $k\in \{1,\ldots ,p\}$, we recall that a mapping F is p-harmonic in a simply connected domain $D\subset \mathbb {C}$ if and only if F has the following representation:

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

(1.1)

where each $G_{p-k+1}$ is harmonic in D. For details and the special case of $p=2$ for biharmonic mappings, we refer to [1, 9], and [25] where one can find characterizations of certain p-harmonic functions.

A mapping f is said to be $\log$-p-harmonic if $\log f$ is a p-harmonic mapping. Then it follows from (1.1) that f is $\log$-p-harmonic in a simply connected domain D if and only if f can be written as

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

(1.2)

where $g_{p-k+1}$ are log-harmonic mappings in D for each $k\in \{1,\ldots ,p\}$. Obviously, when $p=1$, f is log-harmonic; when $p=2$, f is the so-called log-biharmonic (cf. [23, 24]).

By [21], it’s known that a harmonic mapping f is locally univalent in D if and only if the Jacobian of f satisfies $J_{f}(z)=|f_{z}(z)|^{2}-|f_{\overline{z}}(z)|^{2}\ne 0$ for all $z\in D$.

For a continuously differentiable mapping f in D, we may define

$$\begin{aligned}{} & {} \Lambda _{f}=\max _{0\leqslant \theta \leqslant 2\pi }|f_{z}+e^{-2i\theta }f_{\overline{z}}|=|f_{z}|+|f_{\overline{z}}| \text{ and }\\{} & {} \lambda _{f}=\min _{0\leqslant \theta \leqslant 2\pi }|f_{z}+e^{-2i\theta }f_{\overline{z}}|=||f_{z}|-|f_{\overline{z}}||. \end{aligned}$$

Then $J_{f}=\lambda _{f}\Lambda _{f}$ if $J_{f}\geqslant 0$.

Methods of Harmonic mappings have been used to study and solve fluid flow problems (cf. [4, 15]). For example, in 2012, Aleman and Constantin [4] established a connection between harmonic mappings and ideal fluid flows. In fact, they have developed ingenious technique to solve the incompressible two dimensional Euler equations in terms of univalent harmonic mappings (cf. [15]). However, the investigation of harmonic mappings in the context of geometric function theory is a recent one (cf. [7, 8, 16, 17, 19, 27] and the references therein).

The classical Landau’s theorem [20] states that if f is an analytic function on the unit disk $\mathbb {D}:=\{z\in \mathbb {C}:|z|<1\}$ with $f(0)=f'(0)-1=0$ and $|f(z)|<M$ for all $z\in \mathbb {D}$, then f is univalent in the disk $\mathbb {D}_{r_0}:=\{z\in \mathbb {C}:|z|<r_{0}\}$ with $r_{0}=M-\sqrt{M^{2}-1}$, and $f(\mathbb {D}_{r_0})$ contains a disk $\mathbb {D}_{R_{0}}$ with $R_{0}=M r_{0}^{2}$. This result is sharp, with the extremal function $f_0(z)=Mz((1-Mz)/(M-z))$. The Bloch theorem asserts the existence of a positive constant number b such that if f is an analytic function on the unit disk $\mathbb {D}$ with $f^{\prime }(0)=1$, then $f(\mathbb {D} )$ contains a schlicht disk of radius b, that is, a disk of radius b which is the univalent image of some region on $\mathbb {D}$. The supremum of all such constants b is called the Bloch constant (cf. [7]). For the sake of convenience, we say that $f\in S_{z'}(r, R)$ if f is an univalent $\log$-p-harmonic mapping in the disk $\mathbb {D}_{r}$ and $f(\mathbb {D}_{r})$ contains a schlicht disk $\mathbb {D}(z',R):=\{z\in \mathbb {C}:|z-z'|<R\}$. In particular, we denote $S_{0}(r,R)$ by S(r, R).

The study on Landau-type theorems for harmonic mappings has attracted much attention. We may refer the interested readers to [7, 8, 10,11,12, 16, 17, 19, 22, 27,28,29] for more discussions on harmonic mappings, and refer to [2, 3, 9, 24, 26, 31, 34] for more results of biharmonic and p-harmonic mappings. Recently, Chen et al. generalized the results of planar harmonic mappings to several variables (cf. [6, 8]). In 2011, Li and Wang [23] introduced the log-p-harmonic mappings and derived two versions of Landau-type theorems. We recall the following result which actually improves the result of [23].

Theorem A

([30]) 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 all $g_{p-k+1}$ are log-harmonic on $\mathbb {D}$ with $g_{p-k+1}(0)=g_{p}(0)=J_{f}(0)=1$, $|g_{p-k+1}|<M_{p-k+1}^{*}$ for $k\in \{2,\ldots ,p\}$, and $|g_{p}|<M_{p}^{*}$, where $M_{i}^{*}\geqslant 1,\, M_{i}=\log M_{i}^{*}+\pi \, (i=1,\ldots , p)$ and $\lambda _{0}(M)$ is defined by

$$\begin{aligned} \lambda _{0}(M)=\left\{ \begin{array}{ll} \frac{\sqrt{2}}{\sqrt{M^{2}-1}+\sqrt{M^{2}+1}}, &{} {1\leqslant M\leqslant M_{0},} \\ \frac{\pi }{4M}, &{} {M>M_{0}}. \end{array} \right. \end{aligned}$$

(1.3)

Then $f\in S_{z_2}(\rho _2,\sigma _2)$, where $\rho _{2}$ is the unique root in (0, 1) of the equation:

$$\begin{aligned}{} & {} \lambda _{0}(M_{p})-\sum _{k=1}^{p-1}\Big (\frac{4}{\pi (1-r^{2})}+\frac{8k}{\pi (1-r)}\Big )M_{p-k} r^{2k}\\{} & {} -\lambda _{0}(M_{p})\sqrt{(M_{p})^{4}-1}\cdot \frac{r\sqrt{r^{4}-3r^{2}+4}}{(1-r^{2})^{\frac{3}{2}}}=0,\\{} & {} z_{2}=\cosh \left( \frac{\sigma '_{2}}{\sqrt{2}}\right) ,\quad \sigma _{2}=\min \left\{ \sinh \left( \frac{\sigma '_{2}}{\sqrt{2}}\right) ,\cosh \left( \frac{\sigma '_{2}}{\sqrt{2}}\right) \sin \left( \frac{\sigma '_{2}}{\sqrt{2}}\right) \right\} , \end{aligned}$$

and

$$\begin{aligned} \sigma '_{2}=\lambda _{0}(M_{p})\rho _{2}-\lambda _{0}(M_{p})\sqrt{(M_{p})^{4}-1}\cdot \frac{\rho _{2}^{2}}{(1-\rho _{2}^{2})^{\frac{1}{2}}}- \frac{4\rho _{2}}{\pi (1-\rho _{2})}\sum _{k=1}^{p-1}M_{p-k}\rho _{2}^{2k}. \end{aligned}$$

Theorem B

([30]) Let $f(z)=g(z)^{|z|^{2(p-1)}}$ be a log-p-harmonic of $\mathbb {D}$, where $p>1$, g is log-harmonic, $g(0)=J_{g}(0)=1$, $|g(z)|\leqslant M^{*}$ for some $M^{*}\geqslant 1$, and $M=\log M^{*}+\pi$. Then $f\in S_{z_3}(\rho _3,\sigma _3)$, where $\rho _3$ is the unique root in (0, 1) of the following equation:

$$\begin{aligned}{} & {} 1-2\sqrt{M^{4}-1}\cdot \frac{r}{(1-r^{2})^{\frac{1}{2}}}-\sqrt{M^{4}-1}\cdot \frac{r\sqrt{r^{4}-3r^{2}+4}}{(1-r^{2})^{\frac{3}{2}}}=0,\\{} & {} z_3=\cosh \left( \frac{\sigma '_3}{\sqrt{2}}\right) ,\quad \sigma _3=\min \left\{ \sinh \left( \frac{\sigma '_3}{\sqrt{2}}\right) ,\cosh \left( \frac{\sigma '_3}{\sqrt{2}}\right) \sin \left( \frac{\sigma '_3}{\sqrt{2}}\right) \right\} , \end{aligned}$$

and

$$\begin{aligned} \sigma '_3=\rho _3^{2(p-1)}\lambda _{0}(M)\left( \rho _3-\sqrt{M^{4}-1}\cdot \frac{\rho _3^{2}}{(1-\rho _3^{2})^{\frac{1}{2}}}\right) . \end{aligned}$$

In 2019, Bai and Liu established two new versions of Landau-type theorems as follows:

Theorem C

([5]) Let $f(z)=\prod _{k=1}^{p}(g_{p-k+1}(z))^{|z|^{2(k-1)}}$ be a $\log$-p-harmonic mapping in $\mathbb {D}$ 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 $G_{p}(z):=\log \, g_{p}(z)$;

• (ii) $|g_{p-k+1}(z)|\leqslant {M_{p-k+1}^*}$, and $\Lambda _{G_p}(z)\leqslant {\Lambda _{p}}$ for all $z\in \mathbb {D}$, where $M_{p-k+1}^*\geqslant 1$, $\Lambda _{p}\geqslant {1}$. Then $f\in S_{z_4}(\rho _4,\sigma _4)$, where $\rho _4\in (0, 1)$ satisfies the following equation:

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

  (1.4)

  where $M_{p-k+1}=\log \, M_{p-k+1}^*+\pi$, $k=2,3,\ldots ,p$,

  $$\begin{aligned} z_4=\cosh (\frac{\sigma '_4}{\sqrt{2}}),~~\sigma _4=\min {\{\sinh (\frac{\sigma '_4}{\sqrt{2}})}, {\cosh (\frac{\sigma '_4}{\sqrt{2}})\sin (\frac{\sigma '_4}{\sqrt{2}})}\}, \end{aligned}$$

  (1.5)

  and

  $$\begin{aligned} \sigma '_4=\rho _4+\frac{\Lambda _{p}^{2}-1}{\Lambda _{p}}[\rho _4+\textrm{ln}(1-\rho _4)]-\sum _{k=1}^{p-1} \frac{4M_{p-k}\rho _4^{2k+1}}{\pi (1-\rho _4)}. \end{aligned}$$

  (1.6)

In 2021, Liu and Luo obtained the following sharp forms of [5, Theorem 2.9].

Theorem D

([32]) Suppose that p is a positive integer, $\Lambda _1,\ \Lambda _2,\ \ldots ,\ \Lambda _{p-1}\geqslant 0,\ \Lambda _p>1$. 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 that

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

• (ii) for each $k\in \{2,\ldots , p\}$, $\Lambda _{G_{p-k+1}}(z)\leqslant \Lambda _{p-k+1}$, and $\Lambda _{G_p}(z)<\Lambda _p$ for all $z\in \mathbb {D}$. Then $f\in S_{z_6}(\rho _6,\sigma _6)$, where $\rho _6$ is the unique root in (0, 1) of the equation

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

  and $z_6=\cosh \sigma _6',\, \sigma _6=\sinh \sigma _6'$, and $\sigma _6'=\Lambda _p^2\rho _6-\sum _{k=1}^{p-1}\Lambda _{p-k}\rho _6^{2k+1}+\left( \Lambda _p^3-\Lambda _p\right) \ln \left( 1-\frac{\rho _6}{\Lambda _p}\right)$. Both of the radii, $\rho _6$ and $\sigma _6=\sinh \sigma _6'$ are sharp.

Theorem E

([32]) Suppose that p is a positive integer, $p\geqslant 2,\ \Lambda _1,\ \Lambda _2,\ \ldots ,\ \Lambda _{p-1}\geqslant 0$. Let $f(z)=\prod _{k=1}^{p}(g_{p-k+1}(z))^{|z|^{2(k-1)}}$ be a log-p-harmonic mapping in $\mathbb {D}$ satisfying $f(0)=\lambda _{f}(0)=1$. Suppose that for each $k\in \{1,\ldots , p\}$, we have that

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

• (ii) for each $k\in \{2,\ldots , p\}$, $\Lambda _{G_{p-k+1}}(z)\leqslant \Lambda _{p-k+1}$, and $\Lambda _{G_p}(z)\leqslant 1$ for all $z\in \mathbb {D}$. Then (1) $F(z):=\log f(z)=\sum _{k=1}^{p}|z|^{2(k-1)}G_{p-k+1}(z)$ is a p-harmonic mapping in $\mathbb {D}$, and $F(z)\in S(\rho _7, \sigma _7')$, where $\sigma _7'= \rho _7-\sum _{k=1}^{p-1}\Lambda _{p-k}\rho _7^{2k+1}$,

  $$\begin{aligned} \rho _7=\left\{ \begin{array}{lll} 1, &{} \text{ if } \sum \limits _{k=1}^{p-1}(2k+1)\Lambda _{p-k}\leqslant 1,\\ \rho _7', &{} \text{ if } \sum \limits _{k=1}^{p-1}(2k+1)\Lambda _{p-k}> 1 \,, \end{array} \right. \end{aligned}$$

  and $\rho _7'$ is the unique root in (0, 1) of the equation: $1-\sum _{k=1}^{p-1}(2k+1)\Lambda _{p-k}r^{2k}=0$. Both of the radii, $\rho _7$ and $\sigma _7'\pounds \lnot$ are sharp, with an extremal function given by

  $$\begin{aligned} F_1(z)=z-\sum _{k=1}^{p-1}\Lambda _{p-k}|z|^{2k}z,\ z\in U. \end{aligned}$$

  (1.7)

  (2) $f\in S_{z_7}(\rho _7,\sigma _7)$, where $z_7=\cosh \sigma _7',\, \sigma _7=\sinh \sigma _7'$. Both of the radii, $\rho _7$ and $\sigma _7$, are sharp.

Remark F

The number $\sigma _6=\sinh \sigma _6'$ is the accurate value of the Bloch constant of the subclass of $\log$-p-harmonic mappings in the unit disk $\mathbb {D}$ satisfying the hypotheses of Theorem D, and $\sigma _7=\sinh \sigma _7'$ is the accurate value of the Bloch constant of the subclass of $\log$-p-harmonic mappings in the unit disk $\mathbb {D}$ satisfying the hypotheses of Theorem E.

It is natural to raise the following problems.

Problem 1:   If the condition $\lambda _F(0)=1$ is replaced by $J_F(0)=1$ in Theorems D or E, can we obtain the sharp versions of Landau-type theorems for such bounded and normalized log-p-harmonic mappings?

Problem 2:   Can we improve the radii in Theorem A?

The paper is organized as follows. In Sect. 2, we recall several lemmas, and prove five new lemmas which play crucial role in the proofs of our Theorems, where Lemma 2.6 is sharp. In Sect. 3, we establish four new versions of Landau-type theorems for certain bounded p-harmonic mappings with $J_f(0)=1$. In Sect. 4, using these estimates, we present four versions of Landau-type theorems for log-p-harmonic mappings (see Theorems 4.1,  4.3,  4.4 and  4.5), which are the analogues versions of Theorems D, C and B respectively. We also establish a sharp version of Landau-type theorem for a certain log-p-harmonic mappings (see Theorems 4.2), which gives a part of answer to Problem 1. Finally, we improve Theorem A by establishing Theorem 4.6, which gives an affirmative answer to Problem 2.

2 Preliminaries

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

Lemma 2.1

([14]) Suppose that f(z) is a harmonic mapping of the unit disk $\mathbb {D}$ such that $|f(z)|\leqslant M$ for all $\mathbb {D}$. Then

$$\begin{aligned} \Lambda _{f}(z)\leqslant \frac{4M}{\pi (1-|z|^{2})},\quad z\in \mathbb {D}. \end{aligned}$$

The inequality is sharp.

Lemma 2.2

([18]) Suppose that f(z) is a harmonic mapping of the unit disk $\mathbb {D}$ with $f(0)=0$ and $f(\mathbb {D})\subset \mathbb {D}$. Then

$$\begin{aligned} |f(z)|\leqslant \frac{4}{\pi }\mathrm {\arctan }\vert z\vert \le \frac{4}{\pi }|z|\, \text{ for } \, z\in \mathbb {D}. \end{aligned}$$

Lemma 2.3

([13]) Suppose $\Lambda >1$. Let f(z) be a harmonic mapping of the unit disk $\mathbb {D}$ with $J_{f}(0)=1$ and $\Lambda _{f}(z)<\Lambda$ for all $z \in \mathbb {D}$. Then:

• (1) for all $z_{1}, z_{2} \in$ $\mathbb {D}_{r}\left( 0<r<1, z_{1} \ne z_{2}\right)$, we have

  $$\begin{aligned} |f(z_2)-f(z_1)|=\left| \int _{\overline{z_{1} z_{2}}} f_{z}(z) d z+f_{\bar{z}}(z) d \bar{z}\right| \geqslant \Lambda \frac{\lambda _{f}(0)-\Lambda r}{\Lambda -\lambda _{f}(0)r}\left| z_{1}-z_{2}\right| . \end{aligned}$$

• (2) Set $\gamma =f^{-1}(\overline{o w^{\prime }})$ with $w^{\prime } \in f(\partial \mathbb {D}_{r})(0<r \leqslant 1)$ and $\overline{o w^{\prime }}$ denotes the closed line segment joining the origin and $w^{\prime }$, then

  $$\begin{aligned} \left| \int _{\gamma } f_{\zeta }(\zeta ) d \zeta +f_{\bar{\zeta }}(\zeta ) d \bar{\zeta }\right| \geqslant \Lambda \int _{0}^{r} \frac{\lambda _{f}(0)-\Lambda t}{\Lambda -\lambda _{f}(0)t} d t. \end{aligned}$$

Lemma 2.4

Suppose f(z) is a harmonic mapping of the unit disk $\mathbb {D}$. If $J_f(0)=1$ and $\Lambda _f(z)<\Lambda \, (\Lambda >1)$ for all $z\in \mathbb {D}$, then

• (i) for all $z_1,z_2\in \mathbb {D}_r\, (0<r<1, z_1\ne z_2)$, we have

  $$\begin{aligned} \vert f(z_1)-f(z_2)\vert \ge \frac{\Lambda (1-\Lambda ^2r)}{\Lambda ^2-r}\vert z_1-z_2\vert . \end{aligned}$$

• (ii) for $w\in \partial f(\mathbb {D}_r)\, (0<r\leqslant 1)$, $\gamma =f^{-1}(\overline{ow})$ and $\overline{ow}$ denotes the closed line segment joining the origin and w, we have

  $$\begin{aligned} \left| \int _\gamma f_\xi (\xi )d\xi +f_{\overline{\xi }}(\xi )d{\overline{\xi }}\right| \geqslant \Lambda ^3r+(\Lambda ^5-\Lambda )\ln \, (1-\frac{r}{\Lambda ^2}). \end{aligned}$$

Proof

For any $z_1,z_2\in \mathbb {D}_r$ with $z_1\ne z_2$, by Lemma 2.3, we have

$$\begin{aligned} \vert f(z_1)-f(z_2)\vert \ge \Lambda \frac{\lambda _f(0)-\Lambda r}{\Lambda -\lambda _f(0)r}\vert z_1-z_2\vert ,\\ \left| \int _\gamma f_\xi (\xi )d\xi +f_{\overline{\xi }}(\xi )d{\overline{\xi }}\right| \geqslant \Lambda \int _{0}^{r} \frac{\lambda _{f}(0)-\Lambda t}{\Lambda -\lambda _{f}(0)t} dt. \end{aligned}$$

Since $J_f(0)=\Lambda _f(0)\lambda _f(0)=1$, it follows that

$$\begin{aligned} \lambda _f(0)=\frac{1}{\Lambda _f(0)}>\frac{1}{\Lambda }. \end{aligned}$$

As $\Lambda \frac{x-\Lambda r}{\Lambda -xr}$ is an increasing function of x, we obtain that

$$\begin{aligned} \vert f(z_1)-f(z_2)\vert \ge \frac{\Lambda (1-\Lambda ^2r)}{\Lambda ^2-r}\vert z_1-z_2\vert , \end{aligned}$$

and for $w\in \partial f(\mathbb {D}_r)(0<r\leqslant 1)$ and $\gamma =f^{-1}(\overline{ow})$,

$$\begin{aligned} \hspace{1cm}\left| \int _\gamma f_\xi (\xi )d\xi +f_{\overline{\xi }}(\xi )d{\overline{\xi }}\right|\geqslant & {} \Lambda \int _0^r\frac{\lambda _f(0)-\Lambda t}{\Lambda -\lambda _f(0)t}dt\ge \Lambda \int _0^r\frac{1-\Lambda ^2t}{\Lambda ^2-t}dt\\= & {} \Lambda ^3r+(\Lambda ^5-\Lambda )\textrm{ln}(1-\frac{r}{\Lambda ^2}). \end{aligned}$$

$\square$

Lemma 2.5

([13]) Suppose that $f(z)=h(z)+\overline{g(z)}$ is a harmonic mapping in $\mathbb {D}$ with $h(z)=\sum _{n=1}^\infty a_{n}z^{n}$, $g(z)=\sum _{n=1}^\infty b_{n}z^{n}$ and $f(0)=J_{f}(0)-1=0$. Then $|f(z)|\leqslant 1$ for all $z\in \mathbb {D}$ if and only if $\Lambda _f(z)\leqslant 1$ for all $z\in \mathbb {D}$.

Moreover, if $\Lambda _{f}(z)\leqslant \Lambda$ for all $z\in \mathbb {D}$, then $\Lambda \ge 1$, $\vert a_1\vert +\vert b_1\vert \leqslant \Lambda$, and

$$\begin{aligned} |a_{n}|+|b_{n}|\leqslant \frac{\Lambda ^{4}-1}{n\Lambda ^3},\,\,\,n=2,3,\ldots ,\quad \frac{1}{\Lambda }\leqslant \lambda _f(0)\leqslant 1. \end{aligned}$$

(2.1)

When $\Lambda =1$, then $f(z)=a_{1}z$ with $|a_{1}|=1$, and $\lambda _f(0)=1$.

Now we establish several sharp coefficient inequalities for harmonic mappings with bounded dilation and $J_{f}(0)=1$, which has independent interest.

Lemma 2.6

Suppose that $f(z)=h(z)+\overline{g(z)}$ is a harmonic mapping in $\mathbb {D}$ with $h(z)=\sum _{n=1}^\infty a_{n}z^{n}$, $g(z)=\sum _{n=1}^\infty b_{n}z^{n}$, $f(0)=J_{f}(0)-1=0$ and $\Lambda _{f}(0)\leqslant \Lambda$ for some $\Lambda >1$. Then we have the following sharp inequalities:

$$\begin{aligned} |a_1|\leqslant \frac{1}{2}\Big (\Lambda +\frac{1}{\Lambda }\Big ),\quad |b_1|\leqslant \frac{1}{2}\Big (\Lambda -\frac{1}{\Lambda }\Big ),\quad |a_1|+|b_1|\leqslant \Lambda \, \text{ and } \, \lambda _f(0)\geqslant \frac{1}{\Lambda }.\nonumber \\ \end{aligned}$$

(2.2)

Proof

By the assumption, the proofs of the inequalities for $|a_1|$ and $|b_1|$ follow trivially by solving $x + y\leqslant \Lambda$ and $x^2-y^2 = 1$, where $x = |a_1|$ and $y = |b_1|$. The sharpness follow from the affine mapping

$$\begin{aligned} f_0(z)=a z+b\overline{z},\quad a=\frac{1}{2}\Big (\Lambda +\frac{1}{\Lambda }\Big ), b=\frac{1}{2}\Big (\Lambda -\frac{1}{\Lambda }\Big ). \end{aligned}$$

$\square$

Remark 2.1

It is natural to raise an open problem: under the assumptions of Lemma 2.6, and $\Lambda _{f}(z)\leqslant \Lambda$ for all $z\in \mathbb {D}$, what is the sharp upper bound of $|a_{n}|+|b_{n}|$ for each $n\geqslant 2$?

Lemma 2.7

([26]) Suppose that $f(z)=h(z)+\overline{g(z)}$ is a harmonic mapping in $\mathbb {D}$ such that $|f(z)|\leqslant M$ for all $z\in \mathbb {D}$ with $h(z)=\sum _{n=0}^{\infty }a_{n}z^{n}$ and $g(z)=\sum _{n=1}^{\infty }b_{n}z^{n}$. If $J_f(0)=1$, then $\lambda _{f}(0)\geqslant \lambda _{0}(M)$, where $\lambda _{0}(M)$ is defined by (1.3).

Lemma 2.8

Suppose that $F(z)=|z|^{2k}G(z)$ is p-harmonic in $\mathbb {D}$, where $k\geqslant 0$ is an integer, and G(z) is harmonic in $\mathbb {D}$ with $G(0) = 0$.

• (1) If $\Lambda _{G}(z)\leqslant \Lambda$ for all $z\in \mathbb {D}$. Then for $z_1,\, z_2\in \mathbb {D}_r\, (0<r<1)$, we have

  $$\begin{aligned} |F(z_{1})-F(z_{2})|\leqslant |z_1-z_2|(2k+1)\Lambda r^{2k}. \end{aligned}$$

  (2.3)

• (2) If $|G(z)|\leqslant M$ for all $z\in \mathbb {D}$. Then for $z_1,\, z_2\in \mathbb {D}_r\, (0<r<1)$, we have

  $$\begin{aligned} |F(z_{1})-F(z_{2})|\leqslant |z_1-z_2|\bigg (\frac{4}{\pi (1-r^2)}+\frac{8k}{\pi }\bigg )M\, r^{2k}. \end{aligned}$$

  (2.4)

Proof

The case $k = 0$ is trivial. For $k\geqslant 1$, we choose two distinct points $z_1,z_2\in \mathbb {D}_r$, and let $\Gamma =\{(z_{2}-z_{1})t+z_{1}:t\in [0, 1]\}$. Elementary calculations show that

$$\begin{aligned} \left| F(z_{1})-F(z_{2})\right|\leqslant & {} \int _{\Gamma }|z|^{2k}[|G_{z}(z)||dz|+|G_{\overline{z}}(z)||d\overline{z}|]\nonumber \\{} & {} +\int _{\Gamma }k|G(z)|[|\overline{z}^{k}z^{k-1}||dz|+ |\overline{z}^{k-1}z^{k}||d\overline{z}|]\nonumber \\\leqslant & {} \int _\Gamma \Lambda _{G}(z)|z|^{2k}|dz|+\int _\Gamma 2k|G(z)||z|^{2k-1}|dz|, \end{aligned}$$

(2.5)

and

$$\begin{aligned} |G(z)|\leqslant \left| \int _{[0,z]}G_z dz+G_{{\overline{z}}}d\overline{z}\right| \leqslant \int _{[0,z]}\Lambda _{G}(z) |dz|. \end{aligned}$$

(2.6)

• (1) Since $\Lambda _{G}(z)\leqslant \Lambda$ for all $z\in \mathbb {D}$, by (2.5) and (2.6), we have

  $$\begin{aligned} \left| F(z_{1})-F(z_{2})\right|\leqslant & {} \int _\Gamma \Lambda _{G}(z)|z|^{2k}|dz|+\int _\Gamma 2k\Lambda |z|\cdot |z|^{2k-1}|dz|\\\leqslant & {} |z_1-z_2|(2k+1)\Lambda r^{2k}. \end{aligned}$$

  This implies that the inequality (2.3) holds.

• (2) Since $|G(z)|\leqslant M$ for all $z\in \mathbb {D}$, by Lemmas 2.1, 2.2, (2.5) and (2.6), we have that

  $$\begin{aligned} \left| F(z_{1})-F(z_{2})\right|\leqslant & {} \int _\Gamma \Lambda _{G}(z)|z|^{2k}|dz|+\int _\Gamma 2k\cdot \frac{4M |z|}{\pi }\cdot |z|^{2k-1}|dz|\\\leqslant & {} |z_1-z_2|\bigg (\frac{4}{\pi (1-r^2)}+\frac{8k}{\pi }\bigg )M\, r^{2k}. \end{aligned}$$

  This implies the inequalities (2.4) hold. The proof of Lemma 2.8 is complete.

$\square$

Lemma 2.9

([34]) Suppose that $f(z)=h(z)+\overline{g(z)}$ is a harmonic mapping in $\mathbb {D}$ with $h(z)=\sum _{n=0}^{\infty }a_{n}z^{n}$ and $g(z)=\sum _{n=1}^{\infty }b_{n}z^{n}$. If $|f(z)|\leqslant M$ for all $z\in \mathbb {D}$ and $|J_f(0)|=1$, then

$$\begin{aligned} \left( \sum _{n=2}^{\infty }(|a_{n}|+|b_{n}|)^{2}\right) ^{\frac{1}{2}}\leqslant \sqrt{M^{4}-1}\cdot \lambda _{f}(0). \end{aligned}$$

Lemma 2.10

Suppose $0<t<1$, then $\min \{\sinh t, \cosh t\cdot \sin t\}=\sinh t$.

Proof

Set $g(t)=\cosh t\cdot \sin t-\sinh t$, then $g(0)=0$, and a direct computation yields

$$\begin{aligned} g'(t)=\sinh t\cdot \sin t+\cosh t\, (\cos t-1),\quad g'(0)=0, \end{aligned}$$

and

$$\begin{aligned} g''(t)=\sinh t\cdot (2\cos t-1)\geqslant \sinh t\cdot (2\cos 1-1)\geqslant 0, \quad t\in \, [0,1]. \end{aligned}$$

We conclude that $g'(t)$ is an increasing function of t in [0, 1] and therefore, $g'(t)\geqslant g'(0)=0$ for $t\in [0, 1]$, which shows that g(t) is an increasing function of t in [0, 1]. Hence we obtain that $g(t)\geqslant g(0)=0$ for $t\in [0, 1]$, which completes the proof. $\square$

By means of Lemma 2.10, using arguments similar to those in the proof of [32, Lemma 2.4], we may prove the following lemma, and so we omit the details.

Lemma 2.11

Suppose that p is a positive integer, $0<t<1$ and $0<\rho \leqslant 1$. Let f(z) be a log-p-harmonic mapping in $\mathbb {D}$ with $f(0)=J_{f}(0)=1$. Suppose that f(z) is univalent in $\mathbb {D}_{\rho }$ and $F(\mathbb {D}_{\rho })\supset \mathbb {D}_{t}$, where $F(z)=\log f(z)$. Then the range $f(\mathbb {D}_{\rho })$ contains a schlicht disk $\mathbb {D}(w_1, r_1)=\{w\in \mathbb {C}||w-w_1|<r_1\}$, where

$$\begin{aligned} w_1=\cosh t,\quad r_1=\sinh t. \end{aligned}$$

(2.7)

Moreover, if $\rho$ is the biggest univalent radius of f(z) and t is the biggest radius of the schlicht disk of F(z), then the radius $r_1=\sinh t$ is sharp.

Remark 2.2

In Sect. 4, we mainly focus on Landau-type theorems for log-p-harmonic mappings when $p\ge 2$. As for the case of  $p=1$, we may recall [33, Theorem 3.2] as follows:

Suppose $F=H\overline{G}$ is a logharmonic mapping in $\mathbb {D}$. If $F(0)=J_{F}(0)=1$ and $M_1\le |F(z)|\le M_2$, where $M_1$ and $M_2$ are positive constants, $M^*=\max \{-\log M_1,\log M_2\}+\pi$ and

$$\begin{aligned} m=\min _{r\in (0,1)}\frac{2-r^2}{r(1-r^2)}\approx 4.20. \end{aligned}$$

Then F is univalent in the disk $\mathbb {D}_{\xi _0}$ and $F(\mathbb {D}_{\xi _0})$ contains a schlicht disk $\mathbb {D}(z_0, r_0)$, where

$$\begin{aligned} \xi _0=\frac{\pi ^3}{64m{M^*}^2},\, z_0=\cosh (\eta _0/\sqrt{2}),\, \eta _0=\frac{\pi }{8M^*}\xi _0, \end{aligned}$$

and

$$\begin{aligned} r_0=\min \Big \{\sinh (\eta _0/\sqrt{2}),\cosh (\eta _0/\sqrt{2})\sin (\eta _0/\sqrt{2})\Big \}. \end{aligned}$$

It is worth mentioning that the radius $r_0$ in this theorem is not sharp, while Lemma 2.11 provides a new way to obtain the accurate value of the Bloch constant for log-p-harmonic mappings.

Lemma 2.12

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

$$\begin{aligned} \int _{0}^{1}|tz_1+(1-t)z_2|^{2(p-1)}dt\ge \frac{1}{2p-1}\cdot \frac{|z_1|^{2p-1}+|z_2|^{2p-1}}{|z_1|+|z_2|}>0. \end{aligned}$$

3 Landau-type theorems of certain bounded p-harmonic mappings

We establish four new versions of Landau-type theorems for bounded p-harmonic mappings.

Theorem 3.1

Let $F(z)=\sum _{k=1}^{p}|z|^{2(k-1)}G_{p-k+1}(z)$ be a p-harmonic mapping in $\mathbb {D}$ with $F(0)=J_{F}(0)-1=0$. Suppose that for each $k\in \{2,\ldots ,p\}$, $\Lambda _{p-k+1}\geqslant 0$, $\Lambda _{p}>{1}$ and

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

• (ii) for each $k\in \{2,\ldots ,p\}$,  $\Lambda _{G_{p-k+1}}(z)\leqslant \Lambda _{p-k+1}$ and $\Lambda _{G_{p}}(z)< \Lambda _{p}$ for all $z\in \mathbb {D}$. Then $F(z)\in S(r_1,R_1)$, where $r_{1}$ is the unique root in (0, 1) of the equation

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

  (3.1)

  and

  $$\begin{aligned} R_1=\Lambda _p^3r_1+(\Lambda _p^5-\Lambda _p)\textrm{ln}(1-\frac{r_1}{\Lambda _p^2})-\sum _{k=1}^{p-1}\Lambda _{p-k}r_1^{2k+1}. \end{aligned}$$

  (3.2)

Proof

We first observe that

$$\begin{aligned} J_{F}(0)=|F_{z}(0)|^{2}-|F_{\overline{z}}(0)|^{2}=|(G_{p})_{z}(0)|^{2}-|(G_{p})_{\overline{z}}(0)|^{2}=J_{G_{p}}(0)=1. \end{aligned}$$

Next, to prove that F is univalent in $\mathbb {D}_{r_1}$, we choose two distinct points $z_1,z_2\in \mathbb {D}_r\, (0<r<r_1)$, and let $\Gamma =\{z_{1}+t ~(z_{2}-z_{1}):~t\in [0, 1]\}$. By Lemmas 2.4 and  2.8(1), we find that

$$\begin{aligned} |F(z_{1})-F(z_{2})|\geqslant & {} \left| G_p(z_1)-G_p(z_2)\right| -\sum _{k=1}^{p-1}\left| G_{p-k}(z_1)|z_1|^{2k}-G_{p-k}(z_2)|z_2|^{2k}\right| \\\geqslant & {} \bigg [\frac{\Lambda _p(1-\Lambda _p^2r)}{\Lambda _p^2-r}-\sum _{k=1}^{p-1}(2k+1)\Lambda _{p-k}r^{2k}\bigg ]|z_1-z_2|. \end{aligned}$$

Moreover, an easy calculation shows that the function

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

is continuous and strictly decreasing on [0, 1], and

$$\begin{aligned} \mu (0)=\frac{1}{\Lambda _p}>0, ~\mu (1)=-\sum _{k=0}^{p-1}(2k+1)\Lambda _{p-k}<0. \end{aligned}$$

Then it follows from the intermediate value theorem that the equation $\mu (r)=0$ has a unique root $r_1$ in (0, 1). Hence

$$\begin{aligned} |F(z_{1})-F(z_{2})|\geqslant & {} \bigg (\frac{\Lambda _p(1-\Lambda _p^2r)}{\Lambda _p^2-r}-\sum _{k=1}^{p-1}(2k+1)\Lambda _{p-k}r^{2k}\bigg )\vert z_1-z_2\vert \\> & {} \vert z_1-z_2\vert \mu (r_1)=0, \end{aligned}$$

which shows that F(z) is univalent in $\mathbb {D}_{r_1}$.

Finally, for any $z\in \partial \mathbb {D}_{r_1}$, by Lemmas 2.4,  2.8 and (2.6), we have

$$\begin{aligned} |F(z)-F(0)|= & {} \left| G_p(z)+\sum _{k=1}^{p-1}G_{p-k}(z)\vert z\vert ^{2k}\right| \geqslant \left| G_p(z)\right| -\sum _{k=1}^{p-1}\vert G_{p-k}(z)\vert \vert z\vert ^{2k}\\\ge & {} \Lambda _p^3r_1+(\Lambda _p^5-\Lambda _p)\ln \, (1-\frac{r_1}{\Lambda _p^2})-\sum _{k=1}^{p-1}\Lambda _{p-k}r_1^{2k+1}=R_1. \end{aligned}$$

This completes the proof of Theorem 3.1. $\square$

Using Computer Algebra System, we list some numerical solutions in Table 1 to Eqs. (3.1)–(3.2).

Table 1 For the case $p=2$. The values of $r_{1}$, $R_{1}$ are in Theorem 3.1

Remark 3.1

Note that for harmonic mapping $G_{p}(z)$ of $\mathbb {D}$ with $G_{p}(0)=J_{G_{p}}(0)-1=0$, and $\Lambda _{G_p}(z)\leqslant 1$ or $|G_p(z)|\leqslant 1$ for all $z\in \mathbb {D}$, it follows from Lemma 2.5 that $G_{p}(z)=a_{1}z$ with $|a_{1}|=1$ and $\lambda _{G_p}(0)=1$, then by Theorem E(1), we have the following theorem.

Theorem 3.2

Let $F(z)=\sum _{k=1}^{p}|z|^{2(k-1)}G_{p-k+1}(z)$ be a p-harmonic mapping in $\mathbb {D}$ with $F(0)=J_{F}(0)-1=0$. Suppose that for each $k\in \{2,\ldots ,p\}$, $\Lambda _{p-k+1}\geqslant 0$ and

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

• (ii) for each $k\in \{2,\ldots ,p\}$,  $\Lambda _{G_{p-k+1}}(z)\leqslant \Lambda _{p-k+1}$, and $\Lambda _{G_{p}}(z)\leqslant 1$ or $|G_p(z)|\leqslant 1$ for all $z\in \mathbb {D}$. Then $F(z)\in S(r_2,R_2)$, where

  $$\begin{aligned} r_2=\left\{ \begin{array}{lll} 1, &{} \text{ if } \sum \limits _{k=1}^{p-1}(2k+1)\Lambda _{p-k}\leqslant 1,\\ r_2', &{} \text{ if } \sum \limits _{k=1}^{p-1}(2k+1)\Lambda _{p-k}> 1 \,, \end{array} \right. \end{aligned}$$

  (3.3)

  and $r_2'$ is the unique root in (0, 1) of the equation

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

  (3.4)

  and $R_2= r_2-\sum _{k=1}^{p-1}\Lambda _{p-k}r_2^{2k+1}$. Moreover, both of radii, $r_2$ and $R_2$ are sharp, with an extremal function given by (1.7).

Theorem 3.3

Let $F(z)=\sum _{k=1}^{p}|z|^{2(k-1)}G_{p-k+1}(z)$ be a p-harmonic mapping in $\mathbb {D}$ with $F(0)=J_{F}(0)-1=0$. Suppose that for each $k\in \{2,\ldots ,p\}$, $M_{p-k+1}\geqslant {0}$, $\Lambda _{p}>1$ and

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

• (ii) for each $k\in \{2,\ldots ,p\}$,  $|G_{p-k+1}(z)|\leqslant {M}_{p-k+1}$ and $\Lambda _{G_p}(z)<\Lambda _{p}$ for all $z\in \mathbb {D}$. Then $F(z)\in S(r_3,R_3)$, where $r_3$ is the unique root in (0, 1) of the equation

  $$\begin{aligned} \frac{\Lambda _p(1-\Lambda _p^2r)}{\Lambda _p^2-r}-\sum _{k=1}^{p-1}\bigg (\frac{4}{\pi (1-r^2)}+\frac{8k}{\pi }\bigg )M_{p-k}r^{2k}=0, \end{aligned}$$

  (3.5)

  and

  $$\begin{aligned} R_3=\Lambda _p^3r_3+(\Lambda _p^5-\Lambda _p)\ln \, (1-\frac{r_3}{\Lambda _p^2})-\sum _{k=1}^{p-1}\frac{4M_{p-k}}{\pi }r_3^{2k+1}. \end{aligned}$$

  (3.6)

Proof

We first prove that F(z) is univalent in $\mathbb {D}_{r_3}$. In fact, for any two distinct points $z_1,z_2\in \mathbb {D}_r\, (0<r<r_3)$, let $\Gamma =\{z_{1}+ t~(z_{2}-z_{1}):~t\in [0, 1]\}$. Then we have

$$\begin{aligned} |F(z_{1})-F(z_{2})|\geqslant & {} \left| G_p(z_1)-G_p(z_2)\right| -\sum _{k=1}^{p-1}\left| G_{p-k}(z_1)|z_1|^{2k}-G_{p-k}(z_2)|z_2|^{2k}\right| . \end{aligned}$$

Note that $J_{F}(0)=1$ implies $J_{G_{p}}(0)=1$, and thus, by using Lemma 2.4(i), we obtain

$$\begin{aligned} \vert G_p(z_1)-G_p(z_2)\vert \ge \frac{\Lambda _p(1-\Lambda _p^2r)}{\Lambda _p^2-r}\vert z_1-z_2\vert . \end{aligned}$$

Consequently, by the hypotheses of Theorem 3.3 and Lemma 2.8(2), we have

$$\begin{aligned} \vert F(z_1)-F(z_2)\vert\ge & {} \bigg (\frac{\Lambda _p\, (1-\Lambda _p^2r)}{\Lambda _p^2-r}-\sum _{k=1}^{p-1}\bigg (\frac{4}{\pi (1-r^2)}+\frac{8k}{\pi }\bigg )M_{p-k}r^{2k}\bigg )\vert z_1-z_2\vert \\> & {} 0, \end{aligned}$$

which shows that F(z) is univalent in $\mathbb {D}_{r_3}$.

Next, we denote any $z\in \partial \mathbb {D}_{r_3}$ by $r_3e^{i\theta }$. By Lemmas 2.2 and 2.4(ii), we have

$$\begin{aligned} |F(z)-F(0)|=|F(z)|= & {} \left| G_p(z)+\sum _{k=1}^{p-1}G_{p-k}(z)\vert z\vert ^{2k}\right| \geqslant \left| G_p(z)\right| -\sum _{k=1}^{p-1}|G_{p-k}(z)||z|^{2k}\\\geqslant & {} \Lambda _p^3r_3+(\Lambda _p^5-\Lambda _p)\ln \, (1-\frac{r_3}{\Lambda _p^2})-\sum _{k=1}^{p-1}\frac{4M_{p-k}}{\pi }r_3^{2k+1}=R_3. \end{aligned}$$

This completes the proof. $\square$

Using Computer Algebra System, we list some numerical solutions in Table 2 to Eqs. (3.5)–(3.6).

Table 2 For the case $p=2$. The values of $r_{3}$, $R_{3}$ are in Theorem 3.3

By means of Remark 3.1, using the same method as in our proof of Theorem 3.3, we have the following theorem.

Theorem 3.4

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 $F(0)=J_{F}(0)-1=0$. Suppose that for each $k\in \{2,\ldots ,p\}$, $M_{p-k+1}\geqslant {0}$, and

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

• (ii) for each $k\in \{2,\ldots ,p\}$,  $|G_{p-k+1}(z)|\leqslant {M}_{p-k+1}$, and $\Lambda _{G_p}(z)\leqslant 1$ or $|G_p(z)|\leqslant 1$ for all $z\in \mathbb {D}$. Then $F(z)\in S(r_4,R_4)$, where $r_4=1$ for $M_{p-k+1}=0\, (k=2, \cdots , p)$ and $r_4'$, otherwise. Here $r_4'$ is the unique root in (0, 1) of the equation

  $$\begin{aligned}{} & {} 1-\sum _{k=1}^{p-1}\bigg (\frac{4}{\pi (1-r^2)}+\frac{8k}{\pi }\bigg )M_{p-k}r^{2k}=0, \end{aligned}$$

  (3.7)
  $$\begin{aligned}{} & {} R_4=r_4-\sum _{k=1}^{p-1}\frac{4M_{p-k}}{\pi } r_4^{2k+1}. \end{aligned}$$

  (3.8)

  When $M_{p-k+1}=0\, (k=2, \cdots , p)$, $R_4=r_4=1$, which is sharp.

4 Landau-type theorems for log-p-harmonic mappings

By means of Theorem 3.1 and Lemma 2.11, we may establish the following Landau-type theorem of log-p-harmonic mappings, which is the analogues version of Theorem D.

Theorem 4.1

Suppose that p is a positive integer, $p\geqslant 2,\ \Lambda _1,\ \Lambda _2,\ \ldots ,\ \Lambda _{p-1}\geqslant 0$ and $\Lambda _p>{1}$. Let $f(z)=\prod _{k=1}^{p}(g_{p-k+1}(z))^{|z|^{2(k-1)}}$ be a log-p-harmonic mapping of $\mathbb {D}$ with $J_{f}(0)=1$. Assume that for each $k\in \{1,\ldots ,p\}$, we have

• (i) $g_k(z)$ is $\log$-harmonic in $\mathbb {D}$ and $g_k(0)=1$, and  $G_k(z):=\log \, g_k(z)$;

• (ii) for each $k\in \{1,\ldots ,p-1\}$, $\Lambda _{G_k}(z)\leqslant \Lambda _k$ and $\Lambda _{G_{p}}(z)<\Lambda _{p}$ for all $z\in \mathbb {D}$. Then $f(z)\in S_{w_1}(r_1,R_1')$, where $r_1$ is the unique root in (0, 1) of Eq. (3.1), $R_1$ is defined by (3.2), $w_1=\cosh R_1$ and $R_1'=\sinh R_1$.

Proof

Since $G_k(z)=\log \, g_k(z)$ for each $k\in \{1,\cdots , p\}$, we obtain that

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

is p-harmonic in $\mathbb {D}$. A direct computation yields

$$\begin{aligned} J_{f}(0)=|f_{z}(0)|^{2}-|f_{\overline{z}}(0)|^{2}=|f(0)|^{2}(|F_{z}(0)|^{2}-|F_{\overline{z}}(0)|^{2})=J_{F}(0). \end{aligned}$$

Thus the condition $g_{p}(0)=J_{f}(0)=1$ leads to $G_{p}(0)=J_{F}(0)-1=J_f(0)-1=0$.

Therefore, for $z_{1}\ne z_{2}$ in $\mathbb {D}_{r}\, (0<r<r_1)$, by Theorem 3.1, we have

$$\begin{aligned} |\log f(z_{1})-\log f(z_{2})|= & {} |F(z_{1})-F(z_{2})|\\\geqslant & {} |z_{1}-z_{2}|\bigg (\frac{\Lambda _p(1-\Lambda _p^2r)}{\Lambda _p^2-r}-\sum _{k=1}^{p-1}(2k+1)\Lambda _{p-k}r^{2k}\bigg )>0. \end{aligned}$$

This implies that f is univalent in $\mathbb {D}_{r_1}$.

For any $z\in \partial \mathbb {D}_{r_1}$, it also follows from Theorem 3.1 that

$$\begin{aligned} |\log f(z)| =|F(z)|\geqslant & {} \Lambda _p^3r_1+(\Lambda _p^5-\Lambda _p)\ln \, (1-\frac{r_1}{\Lambda _p^2})-\sum _{k=1}^{p-1}\Lambda _{p-k}r_1^{2k+1}=R_1. \end{aligned}$$

Hence, by Lemma 2.11, we get that the range $f(\mathbb {D}_{r_1})$ contains a schlicht disk $\mathbb {D}(w_1, R_1')=\{w\in \mathbb {C}||w-w_1|<R_1'\}$, where $w_1=\cosh R_1,\, R_1'=\sinh R_1.$ This completes the proof. $\square$

Remark 4.1

Note that for harmonic mapping $G_{p}(z)$ of $\mathbb {D}$ with $G_{p}(0)=J_{G_{p}}(0)-1=0$, and $\Lambda _{G_p}(z)\leqslant \Lambda _{p}$ for all $z\in \mathbb {D}$, it follows from Lemma 2.5 that $\Lambda _{p}\geqslant 1$. Theorem 4.1 provides a version of the Landau-type theorem of certain log-p-harmonic mappings with $J_{f}(0)=1$ for the cases $\Lambda _1,\ \ldots ,\ \Lambda _{p-1}\geqslant 0$ and $\Lambda _p>1$. If $\Lambda _1,\ \ldots ,\ \Lambda _{p-1}\geqslant 0$, and $\Lambda _p=1$, then we will prove the following precise form of the Landau-type theorem of certain log-p-harmonic mappings by using Lemma 2.5 and Theorem E(2).

Theorem 4.2

Suppose that p is a positive integer, $p\geqslant 2,\ \Lambda _1,\ \Lambda _2,\ \ldots ,\ \Lambda _{p-1}\geqslant 0$. 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 $f(0)=J_{f}(0)=1$. Suppose that for each $k\in \{1,\ldots , p\}$, we have that

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

• (ii) for each $k\in \{2,\ldots , p\}$, $\Lambda _{G_{p-k+1}}(z)\leqslant \Lambda _{p-k+1}$, and $\Lambda _{G_p}(z)\leqslant 1$ or $|G_p(z)|\leqslant 1$ for all $z\in \mathbb {D}$. Then $f(z)\in S_{w_2}(r_2,R_2')$, where $r_2$ is defined by (3.3), $R_1$ is defined by (3.4) and

  $$\begin{aligned} w_2=\cosh R_2,\quad R_2'=\sinh R_2, \end{aligned}$$

  (4.1)

  Both of the radii, $r_2$ and $R_2'=\sinh R_2$, are sharp.

Proof

Since $f(0)=J_{f}(0)=1$, it is easy to verify that $J_{G_p}(0)=J_{f}(0)=1$.

Since $G_p(0)=0$, and $\Lambda _{G_p}(z)\leqslant 1$ or $|G_p(z)|\leqslant 1$ for all $z\in \mathbb {D}$, it follows from Lemma 2.5 that $\lambda _{G_p}(0)=1$. Hence $\lambda _f(0)=\lambda _{G_p}(0)=1$, and the conclusion of Theorem 4.2 follows from Theorem E(2). $\square$

By means of Theorem 3.3 and Lemma 2.11, we may establish the following Landau-type theorem of log-p-harmonic mappings, which is the analogues version of Theorem C.

Theorem 4.3

Suppose that p is a positive integer, $p\geqslant 2,\ M_1^{*},\ M_2^{*},\ \ldots ,\ M_{p-1}^{*}\geqslant 1$ and $\Lambda _p>{1}$. Let $f(z)=\prod _{k=1}^{p}(g_{p-k+1}(z))^{|z|^{2(k-1)}}$ be a log-p-harmonic mapping of $\mathbb {D}$ such that $J_{f}(0)=1$. Suppose that for each $k\in \{1,\ldots ,p\}$, we have

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

• (ii) for each $k\in \{2,\ldots ,p\}$, $|g_{p-k+1}(z)|\leqslant {M}_{p-k+1}^{*}$ and $\Lambda _{G_p}(z)< {\Lambda _{p}}$ for all $z\in \mathbb {D}$. Then $f(z)\in S_{w_3}(r_3,R_3')$, where $M_{i}=\log M_{i}^{*}+\pi \, (i=1, \ldots , p-1)$, and $r_{3}$ is the unique root in (0, 1) of Eq.(3.5), $R_3$ is defined by (3.6), $w_3=\cosh R_3$ and $R_3'=\sinh R_3$.

Proof

Since $G_k(z)=\log \, g_k(z)$ for each $k\in \{1,\ldots , p\}$, we obtain that

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

is p-harmonic in $\mathbb {D}$. Also, it is easy to see that

$$\begin{aligned} J_{f}(0)=|f_{z}(0)|^{2}-|f_{\overline{z}}(0)|^{2}=|f(0)|^{2}(|F_{z}(0)|^{2}-|F_{\overline{z}}(0)|^{2})=J_{F}(0). \end{aligned}$$

Thus the condition $g_{p}(0)=J_{f}(0)=1$ leads to $G_{p}(0)=J_{F}(0)-1=0$. For $k\in \{1,\ldots , p-1\}$, we have $\arg g_{p-k+1}\in (-\pi ,\pi ]$ and

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

This implies that $|G_{p-k+1}|\leqslant \log M_{p-k+1}^{*}+\pi :=M_{p-k+1}$, $k\in \{1, \ldots , p-1\}$.

Therefore, for $z_{1}\ne z_{2}$ in $\mathbb {D}_{r}\, (0<r<r_{3})$, by Theorem 3.3, we have

$$\begin{aligned} |\log f(z_{1})-\log f(z_{2})|= & {} |F(z_{1})-F(z_{2})|=\Big |\int _{\Gamma }F_{z}(z)dz+F_{\overline{z}}(z)d\overline{z}\Big |\\\geqslant & {} |z_{1}-z_{2}|\bigg (\frac{\Lambda _p(1-\Lambda _p^2r)}{\Lambda _p^2-r}-\sum _{k=1}^{p-1}\bigg (\frac{4}{\pi (1-r^2)}+\frac{8k}{\pi }\bigg )M_{p-k}r^{2k}\bigg )>0. \end{aligned}$$

This implies that f is univalent in $\mathbb {D}_{r_{3}}$.

For any $z\in \partial \mathbb {D}_{r_{3}}$, it also follows from Theorem 3.3 that

$$\begin{aligned} |\log f(z)| =|F(z)|\geqslant & {} \Lambda _p^3r_3+(\Lambda _p^5-\Lambda _p)\textrm{ln}(1-\frac{r_3}{\Lambda _p^2})-\sum _{k=1}^{p-1}\frac{4M_{p-k}}{\pi }r_3^{2k+1}=R_3. \end{aligned}$$

Hence, by Lemma 2.11, we get that the range $f(\mathbb {D}_{r_3})$ contains a schlicht disk $\mathbb {D}(w_3, R_3)=\{w\in \mathbb {C}||w-w_3|<R_3\}$, where

$$\begin{aligned} w_3=\cosh R_3,\quad R_3'=\sinh R_3. \end{aligned}$$

This completes the proof of Theorem 4.3. $\square$

By means of Theorem 3.4 and Lemma 2.11, using the same method as in our proof of Theorem 4.3, we have the following theorem.

Theorem 4.4

Suppose that p is a positive integer, $p\geqslant 2,\ M_1^{*},\ M_2^{*},\ \ldots ,\ M_{p-1}^{*}\geqslant 1$. Let $f(z)=\prod _{k=1}^{p}(g_{p-k+1}(z))^{|z|^{2(k-1)}}$ be a log-p-harmonic mapping of $\mathbb {D}$ such that $J_{f}(0)=1$. Suppose that for each $k\in \{1,\ldots ,p\}$, we have

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

• (ii) for each $k\in \{2,\ldots ,p\}$, $|g_{p-k+1}(z)|\leqslant {M}_{p-k+1}^{*}$, and $\Lambda _{G_p}(z)\leqslant 1$ or $|G_p(z)|\leqslant 1$ for all $z\in \mathbb {D}$. Then $f(z)\in S_{w_4}(r_4,R_4')$, where $M_{i}=\log M_{i}^{*}+\pi \, (i=1, \ldots , p-1)$, and $r_{4}$ is the unique root in (0, 1) of Eq. (3.7), $R_4$ is defined by (3.8), $w_4=\cosh R_4$ and $R_4'=\sinh R_4$. Next, we establish the following result, which is the analogues version of Theorem B.

Theorem 4.5

Suppose that $f(z)=g(z)^{|z|^{2(p-1)}}$ is a log-p-harmonic of $\mathbb {D}$, where $p>1$, g is log-harmonic and $g(0)=J_{g}(0)=1$. Let $G(z)=\log \,g(z)$, and $\Lambda _G\le \Lambda$. Then $f(z)\in S_{z_5}(r_5,R_5')$, where $r_5=1$ for $\Lambda =1$ and $r_5'$, otherwise, and $r_{5}'$ is the unique root in (0, 1) of the equation

$$\begin{aligned}{} & {} \frac{1}{\Lambda }+2\frac{\Lambda ^4-1}{\Lambda ^3}\Big [\frac{\textrm{ln}(1-r)}{r}+1\Big ]-\frac{\Lambda ^4-1}{\Lambda ^3}\frac{r}{1-r}=0,\\{} & {} z_{5}=\cosh \left( R_{5}\right) ,\, R_{5}'=\sinh \left( R_{5}\right) \text{ and } \nonumber \\{} & {} R_{5}=r_{5}^{2p-1}\left\{ \frac{1}{\Lambda }+\frac{\Lambda ^4-1}{\Lambda ^3}\Big [\frac{\textrm{ln}(1-r_5)}{r_5}+1\Big ]\right\} .\nonumber \end{aligned}$$

(4.2)

When $\Lambda =1$, the radii $r_5=1$ and $R_5'=\sinh 1$ are sharp.

Proof

Let

$$\begin{aligned} F(z):=\log f(z)=|z|^{2(p-1)}\log g(z)=|z|^{2(p-1)}G(z). \end{aligned}$$

Then F is p-harmonic in $\mathbb {D}$, $G(z)=\log g(z)$ is harmonic in $\mathbb {D}$, and it has the series expansion:

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

Note that

$$\begin{aligned} g(0)= & {} J_{g}(0)=|g_{z}(0)|^{2}-|g_{\overline{z}}(0)|^{2}=|g(0)|^{2}(|G_{z}(0)|^{2}-|G_{\overline{z}}(0)|^{2})=J_{G}(0)=1. \end{aligned}$$

So, when $\Lambda >1$, for $z_{1}\ne z_{2}$ in $\mathbb {D}_{r}(0<r<r_{5}')$, we adopt the same method as in [9]. By Lemmas 2.5 and  2.11, we have

$$\begin{aligned}{} & {} |\log f(z_{1})-\log f(z_{2})|=|F(z_{1})-F(z_{2})|\\\geqslant & {} |z_{1}-z_{2}|\Big (\int _{0}^{1}|tz_1+(1-t)z_2|^{2(p-1)}dt\Big )\\{} & {} \times \Big [||a_{1}|-|b_{1}||-2\sum _{n=2}^{\infty }(|a_{n}|+|b_{n}|)r^{n-1}-\sum _{n=2}^{\infty }n(|a_{n}|+|b_{n}|)r^{n-1}\Big ]\\\geqslant & {} |z_{1}-z_{2}|\left( \int _{0}^{1}|tz_1+(1-t)z_2|^{2(p-1)}dt\right) \\{} & {} \times \Big (\lambda _{G}(0)-2\sum _{n=2}^{\infty }\frac{\Lambda ^4-1}{n\Lambda ^3}r^{n-1}-\sum _{n=2}^{\infty }\frac{\Lambda ^4-1}{\Lambda ^3}r^{n-1}\Big )\\\geqslant & {} |z_{1}-z_{2}|\Big (\int _{0}^{1}|tz_1+(1-t)z_2|^{2(p-1)}dt\Big )\\{} & {} \times \left\{ \frac{1}{\Lambda }+2\frac{\Lambda ^4-1}{\Lambda ^3}\Big [\frac{\ln (1-r)}{r}+1\Big ]-\frac{\Lambda ^4-1}{\Lambda ^3}\frac{r}{1-r}\right\} \\> & {} |z_{1}-z_{2}|\Big (\int _{0}^{1}|tz_1+(1-t)z_2|^{2(p-1)}dt\Big )\\{} & {} \times \left\{ \frac{1}{\Lambda }+2\frac{\Lambda ^4-1}{\Lambda ^3}\Big [\frac{\ln (1-r_5')}{r_5'}+1\Big ]-\frac{\Lambda ^4-1}{\Lambda ^3}\frac{r_5'}{1-r_5'}\right\} =0. \end{aligned}$$

This implies f is univalent in $\mathbb {D}_{r_5'}$.

When $\Lambda =1$, it follows from Lemmas 2.5 that $G(z)=a_{1}z$ with $|a_{1}|=1$. Thus, $F(z)=a_{1}|z|^{2(p-1)}z$, and for $z_{1}\ne z_{2}$ in $\mathbb {D}$, by Lemma 2.12, we have that

$$\begin{aligned} |\log f(z_{1})-\log f(z_{2})|= & {} |F(z_{1})-F(z_{2})|=\Big |\int _{\overline{z_{1} z_{2}}} F_{z}(z) d z+F_{\bar{z}}(z) d \bar{z}\Big |\\= & {} \Big |\int _{[z_1,z_2]}p|z|^{2(p-1)}dz+(p-1)|z|^{2(p-2)}z^2d\overline{z}\Big |\\\geqslant & {} |z_{1}-z_{2}|\Big (\int _{0}^{1}|tz_1+(1-t)z_2|^{2(p-1)}dt\Big )>0. \end{aligned}$$

This implies f is univalent in $\mathbb {D}$, and the radius $r_5=1$ is sharp.

Finally, for any $z\in \partial \mathbb {D}_{r_{5}}$, by Lemma 2.5, we obtain

$$\begin{aligned} |\log f(z)|=|F(z)|\geqslant & {} r_{5}^{2(p-1)}\Big (|a_{1}z+\overline{b}_{1}\overline{z}|-|\sum _{n=2}^{\infty }(a_{n}z^{n}+\overline{b}_{n}\overline{z}^{n})|\Big )\\\geqslant & {} r_{5}^{2p-1}\left\{ \frac{1}{\Lambda }+\frac{\Lambda ^4-1}{\Lambda ^3}\Big [\frac{\textrm{ln}(1-r_5)}{r_5}+1\Big ]\right\} =R_{5}. \end{aligned}$$

Thus it follows from Lemma 2.11 that $f(\mathbb {D}_{r_{5}})$ contains a schlicht disk $\mathbb {D}(z_{5},R_{5}')$, where Table 3

$$\begin{aligned} z_{5}=\cosh \left( R_{5}\right) \text{ and } R_{5}'=\sinh \left( R_{5}\right) . \end{aligned}$$

In particular, when $\Lambda =1$, it follows from Lemma 2.11 and the sharpness of $r_5=1$ that the radius $R_{5}'=\sinh 1$ is sharp. This completes the proof of Theorem 4.5. $\square$

Table 3 For the case $p=2$. The values of $r_{5}$, $R_{5}'$ are in Theorem 4.5

Finally, we establish the following result, which improves Theorem A.

Theorem 4.6

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

• (i) $g_k(z)$ is $\log$-harmonic in $\mathbb {D}$, $g_k(0)=1$, and $G_k(z):=\log \, g_k (z)$;

• (ii) $|g_k (z)|\leqslant {M_k^{*}}$ for all $z\in \mathbb {D}$, where $M_k^{*}>1$ and $M_k:=\log M_k^{*}+\pi$. Then $f(z)\in S_{z_6}(r_6,R_6')$, where $r_{6}$ is the unique root in (0, 1) of the equation

  $$\begin{aligned}{} & {} \lambda _{0}(M_{p})-\sum _{k=1}^{p-1}\Big (\frac{4}{\pi (1-r^{2})}+\frac{8k}{\pi }\Big ) M_{p-k} r^{2k}\nonumber \\{} & {} \quad -\lambda _{0}(M_{p})\sqrt{M_{p}^{4}-1}\cdot \frac{r\sqrt{r^{4}-3r^{2}+4}}{(1-r^{2})^{\frac{3}{2}}}=0, \end{aligned}$$

  (4.3)
  $$\begin{aligned}{} & {} z_{6}=\cosh \left( R_{6}\right) ,\quad R_{6}'=\sinh \left( R_{6}\right) , \end{aligned}$$

  (4.4)

  and

  $$\begin{aligned} R_{6}=\lambda _{0}(M_{p})r_{6}-\lambda _{0}(M_{p})\sqrt{M_{p}^{4}-1}\cdot \frac{r_{6}^{2}}{(1-r_{6}^{2})^{\frac{1}{2}}}- \frac{4}{\pi }\sum _{k=1}^{p-1}M_{p-k} r_{6}^{2k+1}. \nonumber \\ \end{aligned}$$

  (4.5)

Proof

For all $k\in \{1,\ldots ,p\}$, assume $G_k=\log g_k$ have the following series expansions:

$$\begin{aligned} G_k(z)=\sum _{n=1}^{\infty }a_{n,k}z^n+\sum _{n=1}^{\infty }\overline{b}_{n,k}\overline{z}^n. \end{aligned}$$

Then $F(z)=\log f(z)=\sum _{k=1}^{p}|z|^{2(k-1)}G_{p-k+1}(z)$ is p-harmonic in $\mathbb {D}$. By the proof of Theorem 4.1, we can prove that $G_{p}(0)=J_{F}(0)-1=J_{f}(0)-1=0$, and for any $k\in \{1,\ldots ,p\}$, we have

$$\begin{aligned} |G_{p-k+1}|\leqslant |\log |g_{p-k-1}||+\pi . \end{aligned}$$

This implies $|G_{p-k+1}|\leqslant \log M_{p-k+1}^{*}+\pi =M_{p-k+1}$ for $k\in \{1,\ldots ,p\}$.

Now, we prove that f is univalent in $\mathbb {D}_{r_{6}}$. To this end, for any $z_{1}\ne z_{2}$ in $\mathbb {D}_{r}\, (0<r<r_6)$, let

$$\begin{aligned} I_{1}= & {} \Big |G_{p}(z_1)-G_{p}(z_2)\Big | \text{ and } I_2= \left| \sum _{k=1}^{p-1}G_{p-k}(z_1)|z_1|^{2k}-\sum _{k=1}^{p-1}G_{p-k}(z_2)|z_2|^{2k}\right| . \end{aligned}$$

By Lemmas 2.2, 2.9 and 2.8(2), simple calculation yields

$$\begin{aligned} I_1\geqslant & {} \Big |\int _{[z_{1},z_{2}]}(G_{p})_{z}(0)dz+(G_{p})_{\overline{z}}(0)d\overline{z}\Big |\\{} & {} -\int _{[z_{1},z_{2}]}\Big (|(G_{p})_{z}(z)-(G_{p})_{z}(0)||dz|+|(G_{p})_{\overline{z}}(z)-(G_{p})_{\overline{z}}(0)||d\overline{z}|\Big )\\\geqslant & {} \int _{[z_{1},z_{2}]}\lambda _{G_{p}}(0)|dz|-|z_{1}-z_{2}|\sum _{n=2}^{\infty }n(|a_{n,p}|+|b_{n,p}|)r^{n-1}\\\geqslant & {} |z_{1}-z_{2}|\bigg (\lambda _{G_{p}}(0)-\left( \sum _{n=2}^{\infty }(|a_{n,p}|+|b_{n,p}|)^{2}\right) ^{\frac{1}{2}}\cdot \left( \sum _{n=2}^{\infty }n^{2}r^{2(n-1)}\right) ^{\frac{1}{2}}\bigg )\\\geqslant & {} |z_{1}-z_{2}|\lambda _{G_{p}}(0)\bigg (1-\sqrt{M_{p}^{4}-1}\cdot \frac{r\sqrt{r^{4}-3r^{2}+4}}{(1-r^{2})^{\frac{3}{2}}}\bigg )\\\geqslant & {} |z_{1}-z_{2}|\lambda _0(M_p)\bigg (1-\sqrt{M_{p}^{4}-1}\cdot \frac{r\sqrt{r^{4}-3r^{2}+4}}{(1-r^{2})^{\frac{3}{2}}}\bigg ),\\ I_2\leqslant & {} \sum _{k=1}^{p-1}\left| G_{p-k}(z_1)|z_1|^{2k}-G_{p-k}(z_2)|z_2|^{2k}\right| \\\leqslant & {} |z_1-z_2|\sum _{k=1}^{p-1} \bigg (\frac{4}{\pi (1-r^2)}+\frac{8k}{\pi }\bigg )M_{p-k}\, r^{2k}. \end{aligned}$$

Thus, we have

$$\begin{aligned} |\log f(z_{1})-\log f(z_{2})|= & {} |F(z_{1})-F(z_{2})|\geqslant I_1-I_2\\> & {} |z_1-z_2|\bigg [\lambda _{0}(M_{p})-\sum _{k=1}^{p-1}\Big (\frac{4}{\pi (1-r_6^{2})}+\frac{8k}{\pi }\Big ) M_{p-k} r_6^{2k}\\{} & {} -\lambda _{0}(M_{p} )\sqrt{M_{p}^{4}-1}\cdot \frac{r_{6}\sqrt{r_{6}^{4}-3r_{6}^{2}+4}}{(1-r_{6}^{2})^{\frac{3}{2}}}\bigg ]=0. \end{aligned}$$

This implies that f is univalent in $\mathbb {D}_{r_{6}}$.

For any $z\in \partial \mathbb {D}_{r_6}$, again by Lemmas 2.2 and 2.9, we obtain

$$\begin{aligned} |\log f(z)|= & {} |F(z)|=\Big |\sum _{n=1}^{\infty }(a_{n,p}z^{n}+\overline{b}_{n,p}\overline{z}^{n})+\sum _{k=1}^{p-1}|z|^{2k}G_{p-k}(z)\Big |\\\geqslant & {} |a_{1,p}z+\overline{b}_{1,p}\overline{z}| -\Big |\sum _{n=2}^{\infty }(a_{n,p}z^{n}+\overline{b}_{n,p}\overline{z}^{n})\Big |-\Big |\sum _{k=1}^{p-1}|z|^{2k}G_{p-k}(z)\Big |\\\geqslant & {} \lambda _{G_{p}}(0)r_{6}-\Big (\sum _{n=2}^{\infty }(|a_{n,p}|+|b_{n,p}|)^{2}\Big )^{\frac{1}{2}}\cdot \Big (\sum _{n=2}^{\infty }r_{6}^{2n}\Big )^{\frac{1}{2}} -\frac{4r_{6}}{\pi }\sum _{k=1}^{p-1}M_{p-k}r_{6}^{2k}\\\geqslant & {} \lambda _{G_{p}}(0)\bigg (r_{6}-\sqrt{M_{p}^{4}-1}\cdot \frac{r_{6}^{2}}{(1-r_{6}^{2})^{\frac{1}{2}}}\bigg )-\frac{4r_{6}}{\pi }\sum _{k=1}^{p-1}M_{p-k}r_{6}^{2k}\\\geqslant & {} \lambda _{0}(M_p)r_{6}-\lambda _{0}(M_p)\sqrt{M_{p}^{4}-1}\cdot \frac{r_{6}^{2}}{(1-r_{6}^{2})^{\frac{1}{2}}}- \frac{4}{\pi }\sum _{k=1}^{p-1}M_{p-k}r_{6}^{2k+1}=R_{6}. \end{aligned}$$

Also, this together with Lemma 2.11 imply that $f(\mathbb {D}_{r_{6}})$ contains a schlicht disk $\mathbb {D}(z_{6},R_{6}')$, where

$$\begin{aligned} z_{6}=\cosh R_{6} \text{ and } R_{6}'=\sinh R_{6}. \end{aligned}$$

This completes the proof of Theorem 4.6. $\square$

Remark 4.2

Note that for $r=r_{6}$, we have

$$\begin{aligned}{} & {} \sum _{k=1}^{p-1}\Big (\frac{4}{\pi (1-r^{2})}+\frac{8k}{\pi }\Big ) M_{p-k} r^{2k}<\sum _{k=1}^{p-1}\Big (\frac{4}{\pi (1-r^{2})}+\frac{8k}{\pi (1-r)}\Big ) M_{p-k} r^{2k} \text{ and } \\{} & {} \frac{4}{\pi }\sum _{k=1}^{p-1}M_{p-k}r^{2k+1}<\frac{4}{\pi (1-r)}\sum _{k=1}^{p-1}M_{p-k}r^{2k+1}. \end{aligned}$$

It is easy to verify that $r_{6}>\rho _{2}$ and $R_{6}>\sigma _{2}$, where $\rho _{2}, \sigma _{2}$ are given in Theorem A.

Using Computer Algebra System, we list some numerical solutions to Eqs. (4.3)–(4.4). By Table 4, we know that the results of Theorem 4.6 are better than that of Theorem A.

Table 4 For the case $p=2$. The values of $r_{6}$, $R_{6}$ are in Theorem 4.6 and the values of $\rho _{2}$, $\sigma _{2}$ are in Theorem A