Quasiconformal Extensions of Biharmonic Mappings and Strongly Starlike Biharmonic Mappings

Abstract

As a generalization of conformal mappings and planar harmonic mappings, biharmonic mappings have attracted more and more attention. This paper investigates the properties of biharmonic mappings. First, in terms of coefficients, we give sufficient conditions for quasiconformal extensions of biharmonic mappings in the unit disk. Second, we consider a class of strongly starlike biharmonic mappings called hereditarily strongly starlike biharmonic mappings. We obtain quasiconformal extensions of hereditarily strongly starlike biharmonic mappings under a condition on the dilatations. Based on the coefficients and convolution, respectively, we then give two sufficient conditions for hereditarily strongly starlike biharmonic mappings. Finally, we study quasiconformal extensions of biharmonic mappings in the exterior unit disk.

1 Introduction

Let $\mathbb {C}$ be the complex plane, and $\overline{\mathbb {C}}$ be the extended complex plane, that is $\overline{\mathbb {C}}=\mathbb {C}\cup \{\infty \}$. We denote by $\mathbb {T}$ the unit circle $\{z\in \mathbb {C}:~|z|=1\}$, $\mathbb {D}$ the unit disk $\{z\in \mathbb {C}:~|z|<1\}$, and $\overline{\mathbb {D}}$ the closed unit disk $\{z\in \mathbb {C}:~|z|\le 1\}$. A complex-valued function f is called a harmonic mapping in a simply connected domain D if it satisfies the Laplace equation $\Delta f=4f_{z\overline{z}}=0$ therein, where $\Delta$ represents the complex Laplacian operator. Each harmonic mapping f in D has a decomposition $f=h+\overline{g}$, where h and g are analytic in D and $g(z_0)=0$ for some prescribed point $z_0\in {D}$ (cf. [1]). A function $F=|z|^2G+H$ is said to be biharmonic in D if G and H are both harmonic in D, see [2, 3]. Biharmonic mappings arise in many physical situations, particularly in fluid dynamics and elasticity problems and have many important applications in engineering and biology, see [4,5,6]. The present article mainly concerns about the class of biharmonic mappings which is a special case of p-harmonic mappings. Therefore, it is natural to include the following recent work which concerns univalency of p-harmonic mappings [7, 8]. For more information concerning biharmonic and more general p-harmonic mappings, we refer to [9,10,11,12,13]. Let $\mathcal{B}\mathcal{H}$ denote the class of biharmonic mappings F in $\mathbb {D}$ with the normalization $F(0)=H(0)=H_z(0)-1=0$. The Jacobian of F is given by

$$\begin{aligned} J_F(z)=|F_z(z)|^2-|F_{\overline{z}}(z)|^2. \end{aligned}$$

If $J_F(z)>0$, then F is said to be sense-preserving. A sense-preserving biharmonic mapping F is said to be k-quasiconformal, if its complex dilatation $\mu _F=F_{\overline{z}}/F_z$ satisfies $|\mu _F(z)|\le k<1$ almost everywhere in the given domain. In the most literature, f is called a K-quasiconformal mapping with $K = (1+k)/(1-k) \ge 1$. For the general theory of quasiconformal mappings, the book by Lehto and Virtanen[14] is recommended.

Let $\{\varphi _n\}$ and $\{\psi _n\}$ be two sequences of non-negative real numbers. We denote a class of harmonic mappings $f(z)=\sum _{n=1}^{\infty }a_n z^n + \sum _{n=1}^{\infty }\overline{b_n} \overline{z}^n$ in $\mathbb {D}$ by $\mathcal {H}(\{\varphi _n\},\{\psi _n\})$, satisfying $a_1=1$, $0<|b_1|<1$ and the coefficient condition

$$\begin{aligned} \psi _1|b_1|+\sum _{n=2}^{\infty }(\varphi _n|a_n|+\psi _n|b_n|)\le 1. \end{aligned}$$

Many mathematicians have studied harmonic mappings by certain coefficient conditions. Silverman [15] proved, if $f\in \mathcal {H}(\{n\},~\{n\})$ and $f_{\overline{z}}(0)=0$, then f is univalent and fully starlike in $\mathbb {D}$. Jahangiri [16] obtained the same result for f in $\mathcal {H}(\{n\},~\{n\})$ without the condition $f_{\overline{z}}(0)=0$. Ganczar [17] gave a sufficient coefficient condition for a harmonic mapping $f\in \mathcal {H}(\{\varphi _n\},~\{\varphi _n\})$ to be quasiconformal homeomorphism in $\mathbb {D}$ and to have a quasiconformal extension to $\mathbb {C}$ when $f_{\overline{z}}(0)$ is 0. Hadama et al. [18] obtained the same result when $f_{\overline{z}}(0)$ is not necessary 0. They also gave a sufficient coefficient condition under which a starlike harmonic mapping f of order $\alpha \in [0,1)$ is a quasiconformal homeomorphism in $\mathbb {D}$. Ma [19] considered the sufficient condition for harmonic mappings in $\mathcal {H}(\{\varphi _n\},~\{\psi _n\})$ which can be extended to $\mathbb {C}$ and also investigated the quasiconformal extension of harmonic mappings in the exterior unit disk. Inspired by these results, we consider a class of biharmonic mappings in $\mathbb {D}$. Let

$$\begin{aligned} F=|z|^2G+H\in \mathcal{B}\mathcal{H} \end{aligned}$$

(1.1)

be a biharmonic mapping, where

$$\begin{aligned} G(z)= & {} h_1(z)+\overline{g_1(z)}=\sum _{n=1}^{\infty }a_{1,n}z^n+\sum _{n=1}^{\infty }\overline{b_{1,n}}\overline{z}^n,\\ H(z)= & {} h_2(z)+\overline{g_2(z)}=z+\sum _{n=2}^{\infty }a_{2,n}z^n+\sum _{n=1}^{\infty }\overline{b_{2,n}}\overline{z}^n. \end{aligned}$$

For sequences $\{\psi _n\}$, $\{\varphi _n\}$, $\{\xi _n\}$, $\{\eta _n\}$ of non-negative real numbers, we denote by $\mathcal{B}\mathcal{H}(\{\psi _n\},~\{\varphi _n\},~\{\xi _n\},~\{\eta _n\})$ the set of biharmonic mappings F of the form (1.1) that satisfy the condition $|b_{2,1}|+3(|a_{1,1}|+|b_{1,1,}|)<1$ and the coefficient inequality

$$\begin{aligned}{} & {} (\psi _1|a_{1,1}|+\varphi _1|b_{1,1,}|+\eta _1|b_{2,1}|)\nonumber \\{} & {} \quad +\sum _{n=2}^{\infty }(\psi _n|a_{1,n}|+\varphi _n|b_{1,n}|+\xi _n|a_{2,n}| +\eta _n|b_{2,n}|)\le 1. \end{aligned}$$

(1.2)

Qiao and Wang [20] proved, if $F\in \mathcal{B}\mathcal{H}(\{n+2\},~\{n+2\},~\{n\},~\{n\})$, then F is sense-preserving, univalent and fully starlike. They also obtained that if $F\in \mathcal{B}\mathcal{H}(\{n^2+2\},~\{n^2+2\},~\{n^2\},~\{n^2\})$, then F is sense-preserving, univalent and fully convex. In this paper, we consider biharmonic mappings $F\in \mathcal{B}\mathcal{H}(\{\psi _n\},~\{\varphi _n\},~\{\xi _n\},~\{\eta _n\})$ with four sequences $\{\psi _n\}$, $\{\varphi _n\}$, $\{\xi _n\}$, $\{\eta _n\}$ not exactly same. We give some sufficient coefficient conditions such that $F\in \mathcal{B}\mathcal{H}(\{\psi _n\},~\{\varphi _n\},~\{\xi _n\},~\{\eta _n\})$ can be extended to the whole plane.

Recall that, a Jordan curve $\Gamma$ in $\overline{\mathbb {C}}$ is called a quasicircle if it is the image line of a circle under a quasiconformal automorphism of $\overline{\mathbb {C}}$. A well-known result given by Fait, Kray$\dot{z}$ and Zygmunt [21] shows that any strongly starlike curve of order $\alpha$ $(0<\alpha <1)$ is a K-quasicircle with $K=(1+k)/(1-k)$, $k\le \sin (\pi \alpha /2)$, and any strongly starlike analytic function of order $\alpha$ has an explicit quasiconformal extension in $\mathbb {C}$. It is natural to ask whether the same results can be obtained for strongly starlike biharmonic mappings of order $\alpha$. Ma [19] considered this problem for harmonic mappings. To answer this question for biharmonic mappings but not harmonic mappings, we introduce the class of hereditarily strongly starlike biharmonic mappings of order $\alpha$ $(0<\alpha <1)$ defined in $\mathbb {D}$. In Sect. 3, we consider properties of hereditarily strongly starlike biharmonic mappings.

In this paper, we also consider the class $\Sigma _{BH}$ of sense-preserving univalent biharmonic mappings defined in the exterior unit disk $\mathbb {E}=\{z\in \mathbb {C}:~|z|>1\}$ that maps $\infty$ to $\infty$. These biharmonic mappings have the representation

$$\begin{aligned} F(z)= & {} |z|^2G(z)+H(z)=\alpha z+\beta \overline{z}+(\sum _{n=1}^{\infty }a_{1,n}z^{-n}+\sum _{n=1}^{\infty }\overline{b_{1,n}}\overline{z}^{-n})|z|^2\\{} & {} +\sum _{n=0}^{\infty }a_{2,n}z^{-n}+\sum _{n=1}^{\infty }\overline{b_{2,n}}\overline{z}^{-n}+A\log |z|, \quad z\in \mathbb {E}, \end{aligned}$$

where $0\le |\beta |<|\alpha |$ and $A\in \mathbb {C}$. The subclass $\Sigma _{H}$ of $\Sigma _{BH}$ for harmonic mappings was first introduced by Hengartner and Schober [22]. Jahangiri [16] investigated some properties of starlike harmonic mappings with $A=0$ in this class. Recently, Ma [19] obtained a sufficient condition for harmonic mappings in $\Sigma _{H}$ to be extended to a quasiconformal mapping of $\mathbb {C}$ and considered also the convolution problem of harmonic mappings in $\Sigma _{H}$. In this direction, we generalize these results in the setting of biharmonic mappings.

2 Quasiconformal Extensions of Biharmonic Mappings

We begin this section with the following lemma.

Lemma 1

Let $\{\psi _n\}$, $\{\varphi _n\}$, $\{\xi _n\}$, $\{\eta _n\}$ be sequences of positive real numbers with the condition

$$\begin{aligned} \frac{\psi _n}{n+2}\ge & {} \frac{\psi _2}{4}>1,\quad \frac{\varphi _n}{n+2}\ge \frac{\varphi _2}{4}>1,\\ \frac{\xi _n}{n}\ge & {} \frac{\xi _2}{2}>1,\quad \frac{\eta _n}{n}\ge \frac{\eta _2}{2}>1 \end{aligned}$$

for $n\ge 2$, $\psi _1\ge 3,~\varphi _1\ge 3,~\xi _1=1,~\eta _1\ge 1$. Suppose $F\in \mathcal{B}\mathcal{H}(\{\psi _n\},~\{\varphi _n\},~\{\xi _n\},~\{\eta _n\})$. Then,

$$\begin{aligned} \Vert \mu _F\Vert _\infty =\sup _{z\in \mathbb {D}}|\mu _F(z)|\le k<1, \end{aligned}$$

where $k=|b_{2,1}|+\sum _{n=1}^{\infty }(n+2)(|a_{1,n}|+|b_{1,n}|)+\sum _{n=2}^{\infty }{n}(|a_{2,n}|+|b_{2,n}|).$

Proof

Elementary computation yields

$$\begin{aligned} |\mu _F(z)|= & {} \Big |\frac{F_{\overline{z}}(z)}{F_z(z)}\Big | \\= & {} \Big |\frac{\sum _{n=1}^{\infty }a_{1,n}z^{n+1}+\sum _{n=1}^{\infty }\overline{b_{1,n}}\overline{z}^nz +\sum _{n=1}^{\infty }n\overline{b_{1,n}}\overline{z}^nz+\sum _{n=1}^{\infty }n\overline{b_{2,n}}\overline{z}^{n-1}}{1 +\sum _{{n=1}}^{\infty }a_{1,n}z^n\overline{z}+\sum _{n=1}^{\infty }\overline{b_{1,n}}\overline{z}^{n+1}+\sum _{n=1}^{\infty }na_{1,n}z^n\overline{z} +\sum _{n=2}^{\infty }na_{2,n}z^{n-1}}\Big |\\\le & {} \frac{\sum _{n=1}^{\infty }(|a_{1,n}|+|b_{1,n}|) +\sum _{n=1}^{\infty }n(|b_{1,n}|+|b_{2,n}|)}{1-\sum _{n=1}^{\infty }(|a_{1,n}|+|b_{1,n}|)-|a_{1,1}| -\sum _{n=2}^{\infty }n(|a_{1,n}|+|a_{2,n}|)} \end{aligned}$$

for $z\in \mathbb {D}$. We set

$$\begin{aligned} t=|a_{1,1}|+\sum _{n=1}^{\infty }(|a_{1,n}|+|b_{1,n}|)+\sum _{n=2}^{\infty }n(|a_{1,n}|+|a_{2,n}|) \end{aligned}$$

and

$$\begin{aligned} k=|b_{2,1}|+\sum _{n=1}^{\infty }(n+2)(|a_{1,n}|+|b_{1,n}|)+\sum _{n=2}^{\infty }n(|a_{2,n}|+|b_{2,n}|). \end{aligned}$$

Then, from the condition (1.2), we obtain

$$\begin{aligned} k\le & {} \eta _1|b_{2,1}|+(\psi _1|a_{1,1}|+\varphi _1|b_{1,1}|)+\frac{4}{\psi _2}\sum _{n=2}^{\infty }\psi _n|a_{1,n}| +\frac{4}{\varphi _2}\sum _{n=2}^{\infty }\varphi _n|b_{1,n}|\\{} & {} +\frac{2}{\xi _2}\sum _{n=2}^{\infty }\xi _n|a_{2,n}|+\frac{2}{\eta _2}\sum _{n=2}^{\infty }\eta _n|b_{2,n}|\\< & {} (\psi _1|a_{1,1}|+\varphi _1|b_{1,1}|+\eta _1|b_{2,1}|)+\sum _{n=2}^{\infty }(\psi _n|a_{1,n}|+\varphi _n|b_{1,n}|+\xi _n|a_{2,n}|+\eta _n|b_{2,n}|)\\\le & {} 1. \end{aligned}$$

Also, we have

$$\begin{aligned} |\mu _F(z)|\le \frac{k-t}{1-t}=1-\frac{1-k}{1-t}\le 1-\frac{1-k}{1-0}=k<1 \end{aligned}$$

(2.1)

for $z\in \mathbb {D}$, since the function $1-\frac{1-k}{1-t}$ is monotone decreasing for $t\in [0,1)$. $\square$

As a consequence of Lemma 1, we obtain the following result.

Corollary 1

Let $\{\psi _n\}$, $\{\varphi _n\}$, $\{\xi _n\}$, $\{\eta _n\}$ be sequences of positive real numbers with the condition

$$\begin{aligned} \frac{\psi _n}{n+2}\ge \frac{\psi _2}{4}>1,\quad \frac{\varphi _n}{n+2}\ge \frac{\varphi _2}{4}>1,\quad \frac{\xi _n}{n}\ge \frac{\xi _2}{2}>1,\quad \frac{\eta _n}{n}\ge \frac{\eta _2}{2}>1 \end{aligned}$$

for $n\ge 3$, $\psi _1\ge 3,~ \varphi _1\ge 3,~ \xi _1=1,~ \eta _1\ge 1$, and let $F\in$ $\mathcal{B}\mathcal{H}(\{\psi _n\},~\{\varphi _n\},~\{\xi _n\},~\{\eta _n\})$. Then, F is a quasiconformal homeomorphism in $\mathbb {D}$ such that the complex dilatation $\mu _F$ satisfies

$$\begin{aligned} |\mu _F(z)|\le k=|b_{2,1}|+\sum _{n=1}^{\infty }(n+2)(|a_{1,n}|+|b_{1,n}|)+\sum _{n=2}^{\infty }n(|a_{2,n}|+|b_{2,n}|)<1. \end{aligned}$$

According to Ahlfors [23], the Jordan curve $\gamma$ in $\overline{\mathbb {C}}$ is a quasicircle if and only if

$$\begin{aligned} K(\gamma )=\sup \frac{|w_1-w_2|\cdot |w_3-w_4|+|w_1-w_4|\cdot |w_2-w_3|}{|w_1-w_3|\cdot |w_2-w_4|} \end{aligned}$$

(2.2)

is finite, where supreme is taken over the set of all ordered quadruples $\{w_1,~w_2,~w_3,~w_4\}$ of points on $\gamma$. A sufficient condition for a quasiconformal mapping f in $\mathbb {D}$ to have a quasiconformal extension to $\overline{\mathbb {C}}$ is that the boundary curve of the image $f(\mathbb {D})$ is quasicircle (see [14, Theorem 8.3]).

In the following, as a generalization of [18, Theorem 3.5], [18, Theorem 3.6], [17, Theorem 1] and [17, Theorem2], we give quasiconformal extensions of biharmonic mappings in terms of coefficients.

Theorem 1

Let $\{\psi _n\}$, $\{\varphi _n\}$, $\{\xi _n\}$, $\{\eta _n\}$ be sequences of positive real numbers with the condition

$$\begin{aligned} \frac{\psi _n}{n+2}\ge \frac{\psi _2}{4}>1,~ \frac{\varphi _n}{n+2}\ge \frac{\varphi _2}{4}>1,~ \frac{\xi _n}{n}\ge \frac{\xi _2}{2}>1, ~\frac{\eta _n}{n}\ge \frac{\eta _2}{2}>1 \end{aligned}$$

for $n\ge 3$, $\psi _1>3,~ \varphi _1>3, ~\xi _1=1,~ \eta _1>1$. Let $F\in \mathcal{B}\mathcal{H}(\{\psi _n\},~ \{\varphi _n\},~ \{\xi _n\},~ \{\eta _n\})$, and

$$\begin{aligned} F^{\star }(z)= \left\{ \begin{array}{ll} F(z) &{} \text{ for } |z|<1,\\ \displaystyle |z|^2G^{\star }(z)+H^{\star }(z) &{} \text{ for } |z|\ge 1 \end{array} \right. \end{aligned}$$

(2.3)

with

$$\begin{aligned} G^{\star }(z)=\sum _{n=1}^{\infty }a_{1,n}\overline{z}^{-n}+\overline{b_{1,n}}z^{-n},~ H^{\star }(z)=z+\sum _{n=2}^{\infty }a_{2,n}\overline{z}^{-n}+\sum _{n=1}^{\infty }\overline{b_{2,n}}z^{-n}. \end{aligned}$$

Then, F is univalent and has a homeomorphic extension $F^{\star }$ in $\overline{\mathbb {D}}$ such that the curve $F^{\star }(\mathbb {T})$ is a quasicircle. Moreover, $F^{\star }$ is bi-Lipschitz in $\mathbb {C}$ and it is also a quasiconformal extension of F onto $\overline{\mathbb {C}}$ such that

$$\begin{aligned} |\mu _{F^\star }(z)|\le k=|b_{2,1}|+\sum _{n=1}^{\infty }(n+2)(|a_{1,n}|+|b_{1,n}|)+\sum _{n=2}^{\infty }n(|a_{2,n}|+|b_{2,n}|)<1. \end{aligned}$$

Proof

By the proof of Lemma 1, we have

$$\begin{aligned} k=|b_{2,1}|+\sum _{n=1}^{\infty }(n+2)(|a_{1,n}|+|b_{1,n}|)+\sum _{n=2}^{\infty }n(|a_{2,n}|+|b_{2,n}|)<1. \end{aligned}$$

For $z_1,~z_2\in \mathbb {D}$ with $z_1\ne z_2$, without loss of generality, let $z_2>0$ and $|z_2|=z_2\le |z_1|$. We obtain

$$\begin{aligned} |F(z_1)-F(z_2)|= & {} \Big |\sum _{n=1}^{\infty }a_{1,n}(z_1^n|z_1|^2-z_2^n|z_2|^2)+\sum _{n=1}^{\infty }\overline{b_{1,n}}(\overline{z_1}^n|z_1|^2 -\overline{z_2}^n|z_2|^2)\\{} & {} +(z_1-z_2)+\sum _{n=2}^{\infty }a_{2,n}(z_1^n-z_2^n)+\sum _{n=1}^{\infty }\overline{b_{2,n}}(\overline{z_1}^n-\overline{z_2}^n)\Big |\\\le & {} |z_1-z_2|\Big (\sum _{n=1}^{\infty }(n+2)(|a_{1,n}|+|b_{1,n}|)+\sum _{n=1}^{\infty }n(|a_{2,n}|+|b_{2,n}|)\Big )\\\le & {} (1+k)|z_1-z_2|, \end{aligned}$$

since

$$\begin{aligned} \begin{aligned} \big ||z_1|^2z_1^n-|z_2|^2z_2^n\big |&=\big ||z_1|^2z_1^n-z_2^{n+2}\big | \le \big ||z_1|^2z_1^n-|z_1|^2z_2^n\big |+\big ||z_1|^2z_2^n-z_2^{n+2}\big |\\&\le n|z_1-z_2|+\big ||z_1|-z_2\big |\big ||z_1|+z_2\big |\le (n+2)|z_1-z_2|. \end{aligned} \end{aligned}$$

Similarly, $|F(z_1)-F(z_2)|\ge (1-k)|z_1-z_2|.$ It follows that F has a homeomorphic extension $F^{\star }$ to $\overline{\mathbb {D}}$ such that

$$\begin{aligned} (1-k)|z_1-z_2|\le |F^{\star }(z_1)- F^{\star }(z_2)|\le (1+k)|z_1-z_2| \end{aligned}$$

(2.4)

for $z_1,~z_2\in \overline{\mathbb {D}}$. Therefore, the image $F^{\star }(\mathbb {T})$ is a Jordan curve. A simple computation yields

$$\begin{aligned} K( F^{\star }(\mathbb {T}))= & {} \sup \Big \{\frac{|F^{\star }(w_1)-F^{\star }(w_2)||F^{\star }(w_3)-F^{\star }(w_4)|}{|F^{\star }(w_1)-F^{\star }(w_3)||F^{\star }(w_2)-F^{\star }(w_4)|}\\{} & {} +\frac{|F^{\star }(w_1)-F^{\star }(w_4)||F^{\star }(w_2)-F^{\star }(w_3)|}{|F^{\star }(w_1)-F^{\star }(w_3)||F^{\star }(w_2)-F^{\star }(w_4)|}\Big \}\nonumber \\\le & {} \left( \frac{1+k}{1-k}\right) ^2K(\mathbb {T}). \end{aligned}$$

Since $\mathbb {T}$ is a quasicircle, it follows that $K(\mathbb {T})<\infty$. This implies $K(F^{\star }(\mathbb {T}))<\infty$. Thus, $F^{\star }(\mathbb {T})$ is a quasicircle.

For $z_1,~z_2\in \mathbb {E}$, $z_1\ne z_2$, it is harmless to assume that $1<|z_1|\le |z_2|$, then

$$\begin{aligned}{} & {} |F^{\star }(z_1)-F^{\star }(z_2)|\\{} & {} \quad =\Big |\sum _{n=1}^{\infty }a_{1,n}(\overline{z_1}^{-n}|z_1|^2-\overline{z_2}^{-n}|z_2|^2) +\sum _{n=1}^{\infty }\overline{b_{1,n}}(z_1^{-n}|z_1|^2-z_2^{-n}|z_2|^2)\\{} & {} \qquad + (z_1-z_2)+\sum _{n=2}^{\infty }a_{2,n}(\overline{z_1}^{-n}-\overline{z_2}^{-n}) +\sum _{n=1}^{\infty }b_{2,n}(\overline{z_1}^{-n}-\overline{z_2}^{-n})\Big |\\{} & {} \quad \le \sum _{n=1}^{\infty }|a_{1,n}|\big |z_1^{-n}|z_1|^2-z_2^{-n}|z_2|^2\big | +\sum _{n=1}^{\infty }|b_{1,n}|\big |z_1^{-n}|z_1|^2-z_2^{-n}|z_2|^2\big |+|z_1-z_2|\\{} & {} \qquad + \sum _{n=2}^{\infty }|a_{2,n}||z_1^{-n}-z_2^{-n}|+\sum _{n=1}^{\infty }|b_{2,n}||z_1^{-n}-z_2^{-n}|, \end{aligned}$$

since

$$\begin{aligned}{} & {} \big |z_1^{-n}|z_1|^2-z_2^{-n}|z_2|^2\big |\le \big |z_1^{-n}|z_1|^2-z_2^{-n}|z_1|^2\big |+\big |z_2^{-n}|z_1|^2-z_2^{-n}|z_2|^2\big |\\{} & {} \quad \le |z_1|^2\Big |\frac{1}{z_1}-\frac{1}{z_2}\Big |\Big (\frac{1}{|z_1|^{n-1}}+\frac{1}{|z_1|^{n-2}|z_2|}+...+\frac{1}{|z_2|^{n-1}}\Big )\\{} & {} \qquad +\frac{|z_1|+|z_2|}{|z_2|^n}\big (|z_1|-|z_2|\big )\\{} & {} \quad \le n|z_1-z_2|+2|z_1-z_2|. \end{aligned}$$

Thus

$$\begin{aligned}{} & {} |F^{\star }(z_1)-F^{\star }(z_2)|\\{} & {} \quad \le { \Big (\sum _{n=1}^{\infty }(n+2)(|a_{1,n}|+|b_{1,n}|)+\sum _{n=2}^{\infty }n(|a_{2,n}|+|b_{1,n}|)+|b_{2,1}|+1\Big )}|z_1-z_2|\\{} & {} \quad =(k+1)|z_1-z_2|, \end{aligned}$$

and, similarly, $|F^{\star }(z_1)-F^{\star }(z_2)|\ge (1-k)|z_1-z_2|.$ Therefore,

$$\begin{aligned} (1-k)|z_1-z_2|\le |F^{\star }(z_1)- F^{\star }(z_2)|\le (1+k)|z_1-z_2| \end{aligned}$$

(2.5)

for $z_1,~z_2\in \mathbb {E}$, $z_1\ne z_2$.

Now, we will consider the case $z_1\in \mathbb {D}$ and $z_2\in \mathbb {E}$. Let $z_3\in \mathbb {T}\cap [z_1,~z_2]$, where $[z_1,~z_2]$ is the line segment from $z_1$ to $z_2$. By the similar argument as that in the proof of (2.4) and (2.5), we have

$$\begin{aligned} |F^{\star }(z_1)-F^{\star }(z_2)|\le & {} |F^{\star }(z_1)-F^{\star }(z_3)|+|F^{\star }(z_3)-F^{\star }(z_2)|\\\le & {} (1+k)|z_1-z_3|+(1+k)|z_3-z_2|\\= & {} (1+k)|z_1-z_2|, \end{aligned}$$

and, similarly, $|F^{\star }(z_1)-F^{\star }(z_2)|\ge (1-k)|z_1-z_2|.$ Thus, we obtain

$$\begin{aligned} (1-k)|z_1-z_2|\le |F^{\star }(z_1)-F^{\star }(z_2)|\le (1+k)|z_1-z_2| \end{aligned}$$

(2.6)

for $z_1\in \mathbb {D}$ and $z_2\in \mathbb {E}$.

Inequalities (2.4), (2.5) and (2.6) show that

$$\begin{aligned} (1-k)|z_1-z_2|\le |F^{\star }(z_1)-F^{\star }(z_2)|\le (1+k)|z_1-z_2| \end{aligned}$$

for all $z_1,~z_2\in \mathbb {C}$. This implies that $F^{\star }$ is bi-Lipschitz continuous and univalent in $\mathbb {C}$ and $\lim _{z\rightarrow \infty }F^{\star }(z)=\infty$.

By Corollary 1, F is quasiconformal in $\mathbb {D}$ and $|\mu _F(z)|\le k<1$ for $z\in \mathbb {D}$. For $z\in \mathbb {E}$, we obtain

$$\begin{aligned} |\mu _{F^\star }(z)|= & {} \Big |\frac{F_{\overline{z}}^\star (z)}{F_z^\star (z)}\Big |\\= & {} \Big |\frac{\sum _{n=1}^{\infty }(1-n)a_{1,n}\overline{z}^{-n}z+\sum _{n=1}^{\infty }\overline{b_{1,n}}z^{-n+1} -\sum _{n=2}^{\infty }na_{2,n}\overline{z}^{-n-1}}{1 +\sum _{n=1}^{\infty }a_{1,n}\overline{z}^{-n+1}+\sum _{n=1}^{\infty }(1-n)\overline{b_{1,n}}z^{-n}\overline{z} -\sum _{n=1}^{\infty }n\overline{b_{2,n}}z^{-n-1}}\Big |\\\le & {} \frac{\sum _{n=2}^{\infty }n(|a_{1,n}|+|a_{2,n}|)+\sum _{n=2}^{\infty }(|a_{1,n}|+|b_{1,n}|)+2|a_{1,1}|+|b_{1,1}|}{1 -\sum _{n=1}^{\infty }(|a_{1,n}|+|b_{1,n}|)-\sum _{n=1}^{\infty }n(|b_{1,n}|+|b_{2,n}|)}\\\le & {} k<1. \end{aligned}$$

Therefore, $|\mu _{F^{\star }}(z)|\le k<1$ in $\mathbb {C}/\mathbb {T}$. Since $\mathbb {T}$ is a null set, $F^\star$ is quasiconformal in $\overline{\mathbb {C}}$.$\square$

Theorem 2

For given four real numbers $k_1,~ k_2,~ k_3,~k_4$ with $0<k_i<1,~ i=1,~ 2,~ 3,~ 4$, let $\{\psi _n\},~ \{\varphi _n\},~ \{\xi _n\},~ \{\eta _n\}$ be four sequences of positive real numbers which satisfy

$$\begin{aligned} \frac{\psi _n}{n+2}\ge & {} \frac{1}{k_1}(n\ge 1),~ \frac{\varphi _n}{n+2}\ge \frac{1}{k_2}(n\ge 1),\nonumber \\~ \frac{\xi _n}{n}\ge & {} \frac{1}{k_3}(n\ge 2),~ \frac{\eta _n}{n}\ge \frac{1}{k_4}(n\ge 1). \end{aligned}$$

(2.7)

Suppose that $F\in \mathcal{B}\mathcal{H}(\{\psi _n\},~ \{\varphi _n\},~ \{\xi _n\},~ \{\eta _n\})$. Then, F is univalent in $\mathbb {D}$ and has a homeomorphic extension to the boundary. Moreover, the mapping $F^{\star }$ of the form (2.3) is bi-Lipschitz in $\mathbb {C}$. Furthermore, $F^{\star }$ is a quasiconformal extension of F to $\mathbb {C}$ with $|\mu _{F^{\star }}(z)|\le k=\max \{k_1,~ k_2,~ k_3,~ k_4\}$, that is $F^{\star }$ is a K-quasiconformal mapping of $\mathbb {C}$, where $K=(1+k)/(1-k)$.

Proof

Consider a biharmonic mapping $F\in \mathcal{B}\mathcal{H}(\{\psi _n\},~\{\varphi _n\},~\{\xi _n\},~\{\eta _n\})$ of the form (1.1). It follows from the condition (2.7) that

$$\begin{aligned}{} & {} \sum _{n=1}^{\infty }(n+2)\big (|a_{1,n}|+|b_{1,n}|\big )+\sum _{n=2}^{\infty }n\big (|a_{2,n}|+|b_{2,n}|\big )+|b_{2,1}|\\{} & {} \quad \le k_1\sum _{n=1}^{\infty }\psi _n|a_{1,n}|+k_2\sum _{n=1}^{\infty }\varphi _n|b_{1,n}| +k_3\sum _{n=2}^{\infty }\xi _n|a_{2,n}|+k_4\sum _{n=1}^{\infty }\eta _n|b_{2,n}|\\{} & {} \quad \le \max \{k_1,~k_2,~k_3,~k_4\}=k<1. \end{aligned}$$

For any two points $z_1,~z_2\in \mathbb {C}$, $z_1\ne z_2$, by using the similar argument as that in the proof of Theorem 1, we get

$$\begin{aligned} (1-k)|z_1-z_2|\le |F^{\star }(z_1)-F^{\star }(z_2)|\le (1+k)|z_1-z_2|. \end{aligned}$$

(2.8)

Therefore, F is univalent in $\mathbb {D}$ and has a homeomorphic extension to $\overline{\mathbb {D}}$, and $F^{\star }$ is bi-Lipschitz in $\mathbb {C}$.

Finally, we shall compute the dilatation of $F^{\star }$. For $k_0=\max \{k_1,~ k_2,~ k_4\}$, clearly,

$$\begin{aligned} \frac{1}{k_0}+(n+1)\le \frac{n+2}{k_1},~ 1+\frac{n+1}{k_0}\le \frac{n+2}{k_2}, ~n\le \frac{n}{k_3},~ \frac{n}{k_0}\le \frac{n}{k_4}. \end{aligned}$$

It follows that

$$\begin{aligned}{} & {} \sum _{n=1}^{\infty }\frac{1+k_0(n+1)}{k_0}|a_{1,n}|+\sum _{n=1}^{\infty }\frac{k_0+(n+1)}{k_0}|b_{1,n}| +\sum _{n=2}^{\infty }n|a_{2,n}|+\sum _{n=1}^{\infty }\frac{n}{k_0}|b_{2,n}|\\{} & {} \quad \le \sum _{n=1}^{\infty }\psi _n|a_{1,n}|+\sum _{n=1}^{\infty }\varphi _n|b_{1,n}| +\sum _{n=2}^{\infty }\xi _n|a_{2,n}|+\sum _{n=1}^{\infty }\eta _n|b_{2,n}|\\{} & {} \quad \le 1, \end{aligned}$$

which shows

$$\begin{aligned} |\mu _F(z)|= & {} |\frac{F_{\overline{z}}(z)}{F_z(z)}|\\\le & {} \frac{\sum _{n=1}^{\infty }|a_{1,n}|+\sum _{n=1}^{\infty }(n+1)|b_{1,n}|+\sum _{n=1}^{\infty }n|b_{2,n}|}{1 -\big (\sum _{n=1}^{\infty }(n+1)|a_{1,n}|+\sum _{n=1}^{\infty }|b_{1,n}|+\sum _{n=2}^{\infty }n|a_{2,n}|\big )} \le k_0 \end{aligned}$$

with $z\in \mathbb {D}$. On the other hand, for $k'=\max \{k_1,~k_2,~k_3\}$, since

$$\begin{aligned} 1+\frac{n-1}{k'}<\frac{n+2}{k_1},~ \frac{1}{k'}+(n-1)<\frac{n+2}{k_2},~ \frac{n}{k'}\le \frac{n}{k_3},~n\le \frac{n}{k_4}, \end{aligned}$$

it follows that

$$\begin{aligned} \sum _{n=1}^{\infty }\left( 1+\frac{n-1}{k'}\right) |a_{1,n}|+\sum _{n=1}^{\infty }\left( \frac{1}{k'}+(n-1)\right) |b_{1,n}| +\sum _{n=2}^{\infty }\frac{n}{k'}|a_{2,n}|+\sum _{n=1}^{\infty }n|b_{2,n}|\le 1. \end{aligned}$$

This implies

$$\begin{aligned} \big |\mu _{F^{\star }}(z)\big |\le & {} \frac{\sum _{n=1}^{\infty }(n-1)|a_{1,n}|+\sum _{n=1}^{\infty }|b_{1,n}| +\sum _{n=2}^{\infty }n|a_{2,n}|}{1-(\sum _{n=1}^{\infty }|a_{1,n}|+\sum _{n=1}^{\infty }(n-1)|b_{1,n}|+\sum _{n=1}^{\infty }n|b_{2,n}|)}\le k' \end{aligned}$$

for $z\in \mathbb {C}/\mathbb {D}$. For any $z\in \mathbb {C}$, we conclude that

$$\begin{aligned} |\mu _{F^{\star }}(z)|\le \max \{k_1,~k_2,~k_3,~k_4\}=\max \{k_0,~k'\}=k. \end{aligned}$$

Therefore, $F^{\star }$ is K-quasiconformal in $\mathbb {C}$, where $K=(1+k)/(1-k)$, since the unit circle $\mathbb {T}$ is removable for quasiconformality.$\square$

Theorem 2 is a generalization of [19, Theorem 2.1].

Corollary 2

Consider that $\{\psi _n\},~ \{\varphi _n\},~ \{\xi _n\},~ \{\eta _n\}$ are four sequences of positive real numbers, satisfying $\psi _n\ge {n+2}$, $\varphi _n\ge {n+2}$, $\xi _n\ge n$, $\eta _n\ge n$, $n=2,~ 3,~...$, $\psi _1>3,~ \varphi _1>3,~ \xi _1=1, ~\eta _1>1$. If $F\in \mathcal{B}\mathcal{H}(\{\psi _n\},~ \{\varphi _n\},~ \{\xi _n\},~ \{\eta _n\})$ and

$$\begin{aligned} \sum _{n=1}^{\infty }\psi _n|a_{1,n}|+\sum _{n=1}^{\infty }\varphi _n|b_{1,n}|+\sum _{n=2}^{\infty }\xi _n|a_{2,n}|+\sum _{n=1}^{\infty }\eta _n|b_{2,n}|\le k_0<1, \end{aligned}$$

(2.9)

then the biharmonic mapping $F^{\star }$ with the form (2.3) is a quasiconformal extension of F to $\mathbb {C}$, and $|\mu _{F^{\star }}(z)|\le k_0<1$ for $z\in \mathbb {C}$.

In the following, by using the similar arguments as that in the proof of Theorems 1 and 2, we obtain the analog of these results in the setting of p-harmonic mappings which are some improvements and supplements.

A 2p $(p\ge 1)$ times continuously differentiable complex-valued function $F=u+iv$ in $\mathbb {D}$ is p-harmonic if F satisfies the p-harmonic equation $\Delta ^p F=\Delta (\Delta ^{p-1})F=0$. In fact, F is p-harmonic in $\mathbb {D}$ if and only if F has the following representation

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

(2.10)

where each $G_{p-k+1}$ is harmonic, i.e., $\Delta G_{p-k+1}(z)=0$ for $k\in \{1,\cdots ,p\}$ (cf. [24, Proposition 1]). Let

$$\begin{aligned} G_{p-k+1}=h_{p-k+1}+\overline{g}_{p-k+1}=z+\sum _{j=2}^{\infty }a_{p-k+1,j}z^j+\sum _{j=1}^{\infty }\overline{b_{p-k+1,j}}\bar{z}^j. \end{aligned}$$

(2.11)

We denote by $\mathcal{P}\mathcal{H}$ the class of p-harmonic mappings F in $\mathbb {D}$ with the form (2.10) and satisfying (2.11). In [20], Qiao and Wang considered geometrical properties of p-harmonic mappings $F\in \mathcal{P}\mathcal{H}$ that satisfy $0\le |b_{p,1}|+\sum _{k=2}^{p}(2k-1)(|a_{p-k+1, 1}|+|b_{p-k+1, 1}|)<1$ and

$$\begin{aligned}{} & {} \sum _{k=1}^{p}\sum _{j=2}^{\infty }\Big (2(k-1)+j\Big )(|a_{p-k+1,j}|+|b_{p-k+1,j}|)\\{} & {} \quad \le 1-|b_{p,1}|-\sum _{k=2}^{p}(2k-1)(|a_{p-k+1,1}|+|b_{p-k+1,1}|). \end{aligned}$$

Let $\{\psi _{1,n}\},~\{\psi _{2,n}\},~..., ~\{\psi _{p,n}\}, ~ \{\varphi _{1,n}\},~ \{\varphi _{2,n}\},~...~, ~ \{\varphi _{p,n}\}$ be sequences of positive real numbers. We denote by $\mathcal{P}\mathcal{H}(\{\psi _{1,n}\},~ \{\varphi _{1,n}\},~ \{\psi _{2,n}\},~ \{\varphi _{2,n}\},~...~,~ \{\psi _{p,n}\},~ \{\varphi _{p,n}\})$ the set of p-harmonic mappings $F\in \mathcal{P}\mathcal{H}$ that satisfy the condition $0\le |b_{p,1}|+\sum _{k=2}^{p}(2k-1)(|a_{p-k+1, 1}|+|b_{p-k+1,1}|)<1$ and

$$\begin{aligned} \varphi _{p,1}|b_{p,1}|+\sum _{i=2}^{p}(\psi _{i,1}|a_{i,1}|+\varphi _{i,1}|b_{i,1}|) +\sum _{i=2}^{p}\sum _{n=2}^{\infty }\Big (\psi _{i,n}|a_{i,n}|+\varphi _{i,n}|b_{i,n}|\Big )\le 1. \end{aligned}$$

Remark 1

Let $F\in \mathcal{P}\mathcal{H}(\{\psi _{1,n}\}, ~\{\varphi _{1,n}\}, ~\{\psi _{2,n}\}, ~\{\varphi _{2,n}\},~..., ~ \{\psi _{p,n}\},~ \{\varphi _{p,n}\})$ be a p-harmonic mapping, and

$$\begin{aligned} F^{\star }(z)= \left\{ \begin{array}{ll} F(z) &{} \text{ for } |z|<1,\\ \displaystyle \sum _{k=1}^{p}|z|^{2(k-1)}G^{\star }_{p-k+1}(z) &{} \text{ for } |z|\ge 1, \end{array} \right. \end{aligned}$$

(2.12)

where

$$\begin{aligned} G^{\star }_{p-k+1}(z)= & {} \sum _{n=1}^{\infty }a_{p-k+1,n}\overline{z}^{-n}+\sum _{n=1}^{\infty }\overline{b_{p-k+1,n}}{z}^{-n}~\text{ for }~ k\in \{2,~3,~...,~p\},\\ G^{\star }_{p}(z)= & {} z+\sum _{n=2}^{\infty }a_{{p,n}}\overline{z}^{-n}+\sum _{n=1}^{\infty }\overline{b_{{p,n}}}{z}^{-n}. \end{aligned}$$

If

$$\begin{aligned} \frac{\psi _{i,n}}{n+2(p-i)}\ge \frac{\psi _{i,2}}{2(p-i+1)}>1,~ \frac{\varphi _{i,n}}{n+2(p-i)}\ge \frac{\varphi _{i,2}}{2(p-i+1)}>1 \end{aligned}$$

for $i=1,2,...,p$, $n\ge 2$, $\psi _{p,1}=1$, $\psi _{i,1}\ge 1+2(p-i)$ with $i\in \{1,~2,~\cdots ,~ p-1\}$, $\varphi _{i,1}\ge 1+2(p-i)$ with $i\in \{1,~2,~\cdots ,~ p\}$, then F has a homeomorphic extension ${F^{\star }}$ in $\overline{\mathbb {D}}$ such that the curve ${F^{\star }}(\mathbb {T})$ is a quasicircle. Moreover, $F^{\star }$ is a quasiconformal extension of F onto $\overline{\mathbb {C}}$ such that $|\mu _{F^{\star }}(z)|\le k<1, z\in \mathbb {C}$, where

$$\begin{aligned} k= & {} |b_{p,1}|+\sum _{k=2}^{p}(2k-1)(|a_{p-k+1,1}|+|b_{p-k+1,1}|)\\{} & {} +\sum _{k=1}^{p}\sum _{j=2}^{\infty }\Big (2(k-1)+j\Big )(|a_{p-k+1,j}|+|b_{p-k+1,j}|). \end{aligned}$$

Remark 2

For given real numbers $0<k_j<1,~ j=1,~ 2,~...,~ 2p$, let $\{\psi _{1,n}\}$, $\{\varphi _{1,n}\}$, $\{\psi _{2,n}\}$, $\{\varphi _{2,n}\}$,..., $\{\psi _{p,n}\}$, $\{\varphi _{p,n}\}$ be sequences of positive real numbers which satisfy

$$\begin{aligned} \frac{\psi _{i,n}}{n+2(p-i)}\ge \frac{1}{k_{2i-1}}, ~\frac{\varphi _{i,n}}{n+2(p-i)}\ge \frac{1}{k_{2i}}, ~ i=1,~2,~...,~p,~ n\ge 2, \end{aligned}$$

$\psi _{p,1}=1$, $\frac{\psi _{i,1}}{1+2(p-i)}\ge \frac{1}{k_{2i-1}}$ with $i\in \{1,~2,~\cdots ,~ p-1\}$, $\frac{\varphi _{i,1}}{1+2(p-i)}\ge \frac{1}{k_{2i}}$ with $i\in \{1,~ 2,~ \cdots ,~ p\}$. Suppose that

$$\begin{aligned} F\in \mathcal{P}\mathcal{H}(\{\psi _{1,n}\},~ \{\varphi _{1,n}\},~ \{\psi _{2,n}\},~ \{\varphi _{2,n}\},~...~, ~ \{\psi _{p,n}\}, ~\{\varphi _{p,n}\}). \end{aligned}$$

Then, F is univalent in $\mathbb {D}$ and has a homeomorphic extension to the boundary. Moreover, the mapping $F^{\star }$ of the form (2.12) is bi-Lipschitz in $\mathbb {C}$. Furthermore, $F^\star$ is a quasiconformal extension of F to $\mathbb {C}$, with $|\mu _{F^{\star }}(z)|\le k=\max \{k_1,~ k_2,~...,~ k_{2p}\}$, that is $F^{\star }$ is a K-quasiconformal mapping of $\mathbb {C}$, where $K=(1+k)/(1-k)$.

3 Properties of Hereditarily Starlike Biharmonic Mappings

A curve in the complex plane of the form $w=w_0\exp (te^{i\lambda })$, $t\in \mathbb {R}$, is called a $\lambda$-spiral (about the origin), where $w_0\in \mathbb {C}/\{0\}$ and the real number $|\lambda |<\pi /2$. A domain $\Omega$ containing zero is called $\lambda$-spirallike (with respect to the origin) if the $\lambda$-spiral segment $\{w\exp (te^{i\lambda }):~ t\le 0\}\cup \{0\}$ is contained in $\Omega$ for all $w\in \Omega$, cf. [25]. For $0< \alpha < 1$, let $V_{\alpha }$ be the Jordan domain bounded by the two logarithmic spiral segments $\{e^{(-\tau +i)\theta }:\, 0\le \theta \le \pi \}$ and $\{e^{(\tau +i)\theta }:\, -\pi \le \theta \le 0\}$, where $\tau =\tan (\pi \alpha /2)$. Set $w_0 V_{\alpha }=\{w_0 w:\, w\in V_{\alpha }\}$. Note that $V_{\alpha }$ contains the disk $|w|< e^{-\pi \tau }$. We remark that $w_0 V_\alpha$ shrinks to the segment $[0, w_0)$ as $\alpha \rightarrow 1$. A domain $\Omega$ is called strongly starlike of order $\alpha$ (with respect to the origin) if $w_0 V_{\alpha }\subset \Omega$ for all $w_0\in \Omega$, see [26] and [27]. Fully starlikeness is a reasonable notion because (as is already known) starlikeness is not a hereditary property of univalent harmonic mappings, cf. [28]. A biharmonic mapping F is fully starlike if F maps each $|z|=r$ $(0<r<1)$ injectively onto a starlike curve with respect to the origin. We call $F\in \mathcal{B}\mathcal{H}$ hereditarily starlike if F is a fully starlike biharmonic mapping.

Definition 1

Let $\lambda$ and $\alpha$ be real numbers with $|\lambda |<\pi /2$ and $0<\alpha <1$. A biharmonic mapping $F\in \mathcal{B}\mathcal{H}$ is called hereditarily $\lambda$-spirallike if F is sense-preserving and univalent in $\mathbb {D}$, and $F(\mathbb {D}_r)$ with $\mathbb {D}_r=\{z:\, |z|<r\}$ is $\lambda$-spirallike for each $0<r<1$. The class of such mappings will be denoted by $\mathcal{S}\mathcal{P}_{BH}(\lambda )$. Similarly, a biharmonic mapping $F\in \mathcal{B}\mathcal{H}$ is called hereditarily strongly starlike of order $\alpha$ if it is sense-preserving and univalent in $\mathbb {D}$, and $F(\mathbb {D}_r)$ is a strongly starlike domain of order $\alpha$ for each $0<r<1$. We denote by $\mathcal{S}\mathcal{S}_{BH}(\alpha )$ the class of such mappings.

Let $\mathcal{S}\mathcal{S}(\alpha )$ and $\mathcal{S}\mathcal{S}_{H}(\alpha )$ be the subclasses of $\mathcal{S}\mathcal{S}_{BH}(\alpha )$ for analytic functions and harmonic mappings, respectively.

We observe that a domain $\Omega$ with $0\in \Omega$ is strongly starlike of order $\alpha$ if and only if $\Omega$ is $\pm \pi (1-\alpha )/2$-spirallike at the same time. Therefore, we have

$$\begin{aligned} \mathcal{S}\mathcal{S}_{BH}(\alpha )=\mathcal{S}\mathcal{P}_{BH}\left( \frac{\pi (1-\alpha )}{2}\right) \cap \mathcal{S}\mathcal{P}_{BH}\left( -\frac{\pi (1-\alpha )}{2}\right) . \end{aligned}$$

(3.1)

For continuously differentiable functions $F\in C^1(\mathbb {D})$, we define the differential operator D by

$$\begin{aligned} D F=zF_z(z)-\overline{z}F_{\overline{z}}(z), \end{aligned}$$

where $F_z=\frac{F_x-iF_y}{2}$ and $F_{\overline{z}}=\frac{F_x+iF_y}{2}$. Here, $F_x$ and $F_y$ are the partial derivatives of F with respect to $x=\Re z$ and $y=\Im z$, respectively.

Lemma 2

([29, Lemma 2.1]) Let $\lambda$ be a real number with $|\lambda |<\frac{\pi }{2}$. Suppose that a function $F\in C^1(\mathbb {D})$ satisfies the conditions that $F(0)=0$ if and only if $z=0$, and that $J_F(z)>0$ in $\mathbb {D}$. Then, F is injective in $\mathbb {D}$ and $F(\mathbb {D}_r)$ is $\lambda$-spirallike for each $0<r<1$ if and only if

$$\begin{aligned} \Re (e^{-i\lambda }\frac{DF(z)}{F(z)})>0,\quad z\in \mathbb {D}/\{0\}. \end{aligned}$$

(3.2)

In the setting of biharmonic mappings, we have the following conclusion.

Corollary 3

Let $\lambda$ be a real number with $|\lambda |<\frac{\pi }{2}$. Suppose that a biharmonic mapping $F\in \mathcal{B}\mathcal{H}$ satisfies the conditions that $F(z)\ne 0$ for $0<|z|<1$ and that $J_F(z)>0$ in $\mathbb {D}$. Then, $F\in \mathcal{S}\mathcal{P}_{BH}(\lambda )$ if and only if the inequality (3.2) holds.

The following result gives a characterization of hereditarily strongly starlike biharmonic mappings of order $\alpha$.

Corollary 4

Let $\alpha$ be a real number such that $0<\alpha <1$. Suppose that a biharmonic mapping $F\in \mathcal{B}\mathcal{H}$ satisfies the conditions that $F(z)\ne 0$ for $0<|z|<1$ and that $J_F(z)>0$ in $\mathbb {D}$. Then, $F\in \mathcal{S}\mathcal{S}_{BH}(\alpha )$ if and only if

$$\begin{aligned} |\arg \frac{DF(z)}{F(z)}|<\frac{\pi \alpha }{2},\quad z\in {\mathbb {D}}/\{0\}. \end{aligned}$$

Brannan and Kirwan [26] showed that an analytic function $f\in { \mathcal{S}\mathcal{S}(\alpha )}$ is bounded. Recently, the authors of [29] generalized this result to the case of harmonic mappings; they obtained that a harmonic mapping $f\in { \mathcal{S}\mathcal{S}_{H}(\alpha )}$ satisfies

$$\begin{aligned} |f(z)|\le \frac{\pi }{2}\exp \left\{ \pi \tan \left( \frac{\pi \alpha }{2}\right) \right\} , \quad z\in \mathbb {D}. \end{aligned}$$

We will prove that this holds for biharmonic mappings in $F\in \mathcal{S}\mathcal{S}_{BH}(\alpha )$ with the form $F=|z|^2\,G$. To this end, we give the following lemma due to Hall, see, for example, [1, §6.2].

Lemma 3

Let f be a univalent harmonic mapping with $f(0)=f_z(0)-1=0$. Then, there is a point $w_0\in \mathbb {C}$ with $|w_0|\le \frac{\pi }{2}$ such that $w_0\notin f(\mathbb {D})$. The bound $\frac{\pi }{2}$ is sharp.

Theorem 3

Let $\alpha$ be a real number with $0<\alpha <1$. For each $F=|z|^2G\in \mathcal{S}\mathcal{S}_{BH}(\alpha )$, the inequality $|F(z)|\le N(\alpha )$ with $z\in \mathbb {D}$ holds, where

$$\begin{aligned} N(\alpha )=\frac{\pi }{2}\exp \left\{ \pi \tan \left( \frac{\pi \alpha }{2}\right) \right\} . \end{aligned}$$

Proof

We define $G^{*}_r$ by $G^{*}_r(z)=r^2G(z)$, where $0<r<1$. Then, $G^{*}_r$ is harmonic and univalent on $|z|=r$ which shows G is harmonic and univalent on $|z|=r$. Since $F\in \mathcal{S}\mathcal{S}_{BH}(\alpha )$, it follows that F is sense-preserving. By using [3, Lemma 6], we have that G is sense-preserving. It follows from the argument principle of harmonic mappings that G is a univalent harmonic mapping. Let $G_r(z)=\frac{G(rz)}{r}$, $D_r= G_r(\mathbb {D})$. By Lemma 3, there is a point $w_0\in \mathbb {C}/D_r$ with $|w_0|<\frac{\pi }{2}$.

For $F_r=|z|^2G_r$, $F_r\in \mathcal{S}\mathcal{S}_{BH}(\alpha )$ for each $0<r<1$. Let $\Omega _r=F_r(\mathbb {D})$ for $0<r<1$. Then, $\Omega _r$ is a strongly starlike domain of order $\alpha$. Obviously, $\Omega _r\subseteq D_r$, so $w_0\in \mathbb {C}/\Omega _r$. For an arbitrary point $w\in \Omega _r/\{0\}$, we have $wV_\alpha \subset \Omega _r$. Since

$$\begin{aligned} V_\alpha \supset \left\{ w:~|w|<\exp \left\{ \pi \tan \left( \frac{\pi \alpha }{2}\right) \right\} \right\} , \end{aligned}$$

we have

$$\begin{aligned} |w|\exp \left\{ -\pi \tan \left( \frac{\pi \alpha }{2}\right) \right\} \le |w_0|\le \frac{\pi }{2} \end{aligned}$$

for $w\in \Omega _r$, which implies $|w|\le N(\alpha )$. Since $0<r<1$ is arbitrary, we obtain the expected conclusion.$\square$

It is well known that $f_1\circ f_2$ is $K_1\, K_2$-quasiconformal whenever $f_j$ is $K_j$- quasiconformal (cf. [30]). Fait, Frzy$\dot{z}$ and Zygmunt [21] showed the following result.

Lemma 4

Let $0<\alpha <1$. A strongly starlike analytic function in $\mathcal{S}\mathcal{S}(\alpha )$ extend to a $\cot ^2{\frac{\pi (1-\alpha )}{4}}$-quasiconformal endomorphism of $\mathbb {C}$. In particular, a strongly starlike domain of order $\alpha$ is a $\cot ^2{\frac{\pi (1-\alpha )}{4}}$-quasidisk.

Ma, Ponnusamy and Sugawa [29] extended this result to the case of harmonic mappings in $\mathcal{S}\mathcal{S}_{H}(\alpha )$. We will prove that this result holds in the class $\mathcal{S}\mathcal{S}_{BH}(\alpha )$.

Theorem 4

Let $F\in \mathcal{S}\mathcal{S}_{BH}(\alpha )$ for some $0<\alpha <1$. Suppose that the dilatation $\mu _F$ of F satisfies the inequality $|\mu _F(z)|\le (K-1)/(K+1)$ in $\mathbb {D}$ for a constant $K\ge 1$. Then, F extends to a $K\cot ^2{\frac{\pi (1-\alpha )}{4}}$-quasiconformal endomorphism of $\mathbb {C}$.

Proof

Let $\Omega =F(\mathbb {D})$. It follows from the assumption that $\Omega$ is a strongly starlike domain of order $\alpha$. Since $|\mu _F(z)|\le (K-1)/(K+1)<1$, by the measurable Riemann mapping theorem (see, for example, [30]), there is a quasiconformal homeomorphism $w:~\mathbb {D}\rightarrow \mathbb {D}$ such that $w(0)=0,~w(1)=1$ and ${w_{\overline{z}}}/{w_z}=\mu _F$ a.e. in $\mathbb {D}$. Moreover, the mapping w extends to a K-quasiconformal mapping of $\mathbb {C}$ with the property ${1}/{w(\frac{1}{z})}=w(z)$ in $\mathbb {D}$. Hence, $F^{\star }=F\circ w^{-1}:~\mathbb {D}\rightarrow \Omega$ is a 1-quasiconformal mapping, that is a conformal function, and then, it is a univalent analytic function and satisfies $F^{\star }(0)=0$. Let $a={F^{\star }}'(0)$ and $G={F^{\star }}/{a}$. Since the image $G(\mathbb {D})={\Omega }/{a}$ is strongly starlike of order $\alpha$, we observe that $G\in SS_{BH}(\alpha )$. Now Lemma 4 implies that G extends to a $\cot ^2{\frac{\pi (1-\alpha )}{4}}$-quasiconformal endomorphism of $\mathbb {C}$. Hence, $F=F^{\star }\circ w$ also extends to a $K\cot ^2{\frac{\pi (1-\alpha )}{4}}$-quasiconformal endomorphism of $\mathbb {C}$.$\square$

In the following, we give simple sufficient conditions such that $F\in \mathcal{S}\mathcal{S}_{BH}(\alpha )$ in terms of the coefficients by employing the ideas due to Silverman [15].

We first observe that the condition $\Re (e^{-i\lambda }\frac{DF(z)}{F(z)})>0,~ z\in {\mathbb {D}/\{0\}}$, means that the quantity DF(z)/F(z) lies in the half-plane $H_\lambda =\{w\in \mathbb {C}:~\Re (e^{-i\lambda }w)>0\}$. Let

$$\begin{aligned} {c=-e^{i2\lambda }=-\cos {2\lambda }-i\sin {2\lambda }.} \end{aligned}$$

Then, a point $w\in \mathbb {C}$ lies in the half-plane $H_\lambda$ if and only if $|w-1|<|w-c|$. We apply this idea to our results.

For $0<\alpha <1$, we introduce the following quantities for integers $n\ge 1$:

$$\begin{aligned} A_{1,n}(\alpha )= & {} n-1+|n-e^{-i\pi \alpha }|+4\sin \frac{\pi \alpha }{2}\\= & {} n-1+\sqrt{n^2-2n\cos (\pi \alpha )+1}+4\sin \frac{\pi \alpha }{2},\\ B_{1,n}(\alpha )= & {} n+1+|n+e^{i\pi \alpha }|+4\sin \frac{\pi \alpha }{2}\\= & {} n+1+\sqrt{n^2+2n\cos (\pi \alpha )+1}+4\sin \frac{\pi \alpha }{2},\\ A_{2,n}(\alpha )= & {} A_{1,n}-4\sin \frac{\pi \alpha }{2},~ B_{2,n}(\alpha )=B_{1,n}-4\sin \frac{\pi \alpha }{2}. \end{aligned}$$

Lemma 5

([29, Lemma 4.1]) For $n\ge 2$, $0<\alpha <1$, the following inequalities hold:

$$\begin{aligned} 2(n+2)\sin \frac{\pi \alpha }{2}<A_{1,n}(\alpha )<B_{1,n}(\alpha ),\,\, 2n\sin \frac{\pi \alpha }{2}<A_{2,n}(\alpha )<B_{2,n}(\alpha ). \end{aligned}$$

Theorem 5

Let $F=|z|^2G+H\in \mathcal{B}\mathcal{H}$ with $G(z)=\sum _{n=1}^{\infty }a_{1,n}z^n+\sum _{n=1}^{\infty }\overline{b_{1,n}}\overline{z}^n$ and $H(z)=z+\sum _{n=2}^{\infty }a_{2,n}z^n+\sum _{n=1}^{\infty }\overline{b_{2,n}}\overline{z}^n$. Suppose that the inequality

$$\begin{aligned}{} & {} \sum _{n=1}^{\infty }A_{1,n}(\alpha )|a_{1,n}|+\sum _{n=1}^{\infty }B_{1,n}(\alpha )|b_{1,n}| +\sum _{n=2}^{\infty }A_{2,n}(\alpha )|a_{2,n}|\nonumber \\{} & {} \quad +\sum _{n=1}^{\infty }B_{2,n}(\alpha )|b_{2,n}|\le 2\sin {\frac{\pi \alpha }{2}} \end{aligned}$$

(3.3)

holds. Then, $F\in \mathcal{S}\mathcal{S}_{BH}(\alpha )$.

Proof

Obviously, $F(z)=z\in \mathcal{S}\mathcal{S}_{BH}(\alpha )$. In the following, we can assume that F is not identically z. We first show that F is sense-preserving. Indeed, by Lemma 5 and (3.3), it follows

$$\begin{aligned} |F_z(z)|-|F_{\overline{z}}(z)| \ge 1-\sum _{n=1}^{\infty }(n+2)(|a_{1,n}|+|b_{1,n}|)-\sum _{n=2}^{\infty }n|a_{2,n}|-\sum _{n=1}^{\infty }n|b_{2,n}| >0 \end{aligned}$$

for $z\in \mathbb {D}$. This implies $J_F(z)=|F_z(z)|^2-|F_{\overline{z}}(z)|^2>0$, which means that F is sense-preserving. Let $\lambda$ be a real number with $|\lambda |<\pi /2$ and let $c=-e^{2i\lambda }$ as above. Note that $DF/F\in H_\lambda$ if and only if $|DF(z)-F(z)|<|DF(z)-cF(z)|$.

Since

$$\begin{aligned}{} & {} \sum _{n=1}^{\infty }\big (n-1+|n-c|\big )|a_{1,n}|+\sum _{n=1}^{\infty }\big (n+1+|n+c|\big )|b_{1,n}|\\{} & {} \quad +\sum _{n=2}^{\infty }\big (n-1+|n-c|\big )|a_{2,n}|\\{} & {} \quad +\sum _{n=1}^{\infty }\big (n+1+|n+c|\big )|b_{2,n}|\le |1-c|=2\cos \lambda , \end{aligned}$$

it follows that

$$\begin{aligned}{} & {} \Big |\sum _{n=1}^{\infty }(n-1)a_{1,n}z^{n+1}\overline{z}\Big |+\Big |\sum _{n=1}^{\infty }(n+1)\overline{b_{1,n}}\overline{z}^{n+1}z\Big |+\Big |\sum _{n=2}^{\infty }(n-1)a_{2,n}z^n\Big |\\{} & {} \qquad +\Big |\sum _{n=1}^{\infty }(n+1)\overline{b_{2,n}}\overline{z}^n\Big |\\{} & {} \quad <|1-c||z|-\Big |\sum _{n=1}^{\infty }(n-c)a_{1,n}z^{n+1}\overline{z}\Big |-\Big |\sum _{n=1}^{\infty }(n+c)\overline{b_{1,n}}\overline{z}^{n+1}z\Big |\\{} & {} \qquad -\Big |\sum _{n=2}^{\infty }(n-c)a_{2,n}z^n\Big |-\Big |\sum _{n=1}^{\infty }(n+c)\overline{b_{2,n}}\overline{z}^n\Big |, \end{aligned}$$

which implies $|DF(z)-F(z)|<|DF(z)-cF(z)|$. This shows $DF/F\in H_\lambda$. Note that the last inequality remains invariant when $\lambda$ is replaced by $-\lambda$. Therefore, for $\lambda ={\pi (1-\alpha )}/2$, we obtain $\Big |\arg \frac{DF(z)}{F(z)}\Big |<\frac{\pi \alpha }{2}$ in $\mathbb {D}$. The required conclusion follows from Corollary 4.$\square$

Let $F_1$ and $F_2$ be two biharmonic mappings in $\mathbb {D}$, where

$$\begin{aligned} F_1(z)= & {} |z|^2\Big (\sum _{n=0}^{\infty }a_{1,n}z^n+\sum _{n=1}^{\infty }\overline{b_{1,n}}\overline{z}^n\Big ) +\sum _{n=1}^{\infty }a_{2,n}z^n+\sum _{n=1}^{\infty }\overline{b_{2,n}}\overline{z}^n,\\ F_2(z)= & {} |z|^2\Big (\sum _{n=0}^{\infty }A_{1,n}z^n+\sum _{n=1}^{\infty }\overline{B_{1,n}}\overline{z}^n\Big ) +\sum _{n=1}^{\infty }A_{2,n}z^n+\sum _{n=1}^{\infty }\overline{B_{2,n}}\overline{z}^n. \end{aligned}$$

Then, the (biharmonic) convolution $F_1*F_2$ of $F_1$ and $F_2$ is defined as

$$\begin{aligned} F_1(z)*F_2(z)= & {} |z|^2\Big (\sum _{n=0}^{\infty }a_{1,n}A_{1,n}z^n+\sum _{n=1}^{\infty }\overline{b_{1,n}B_{1,n}}\overline{z}^n\Big )\\{} & {} +\sum _{n=1}^{\infty }a_{2,n}A_{2,n}z^n+\sum _{n=1}^{\infty }\overline{b_{2,n}B_{2,n}}\overline{z}^n. \end{aligned}$$

In other words, for

$$\begin{aligned} F_1(z)= & {} |z|^2G_1(z)+H_1(z)=|z|^2(h_1(z)+\overline{g_1(z)})+h_2(z)+g_2(z),\\ F_2(z)= & {} |z|^2G_2(z)+H_2(z)=|z|^2(h_3(z)+\overline{g_3(z)})+h_4(z)+\overline{g_4(z)}, \end{aligned}$$

the convolution is defined by

$$\begin{aligned} F_1(z)*F_2(z)= & {} |z|^2G_1(z)*G_2(z)+H_1(z)*H_2(z)\\= & {} |z|^2(h_1(z)*h_3(z)+\overline{g_1(z)}*\overline{g_3(z)})+h_2(z)*h_4(z) +\overline{g_2(z)}*\overline{g_4(z)}. \end{aligned}$$

Inspired by [29], we have the following convolution theorem for spirallike biharmonic mappings. It is useful below to note that $h(z)=h(z)*\frac{z}{1-z}$ and $zh'(z)=h(z)*\frac{z}{(1-z)^2}$ for an analytic function h in $\mathbb {D}$ with $h(0)=0$.

Lemma 6

Let $|\lambda |<\pi /2$, $F\in \mathcal{B}\mathcal{H}$ be a sense-preserving biharmonic mapping which satisfies $F(z)\ne 0$ for $0<|z|<1$. Suppose that $F=|z|^2G+H$, where $G=h_1+\overline{g_1},~ H=h_2+\overline{g_2}$, $h_1,~g_1,~h_2,~g_2$ are analytic and take zero at point $z=0$. Then, $F\in \mathcal{S}\mathcal{P}_{BH}(\lambda )$ if and only if

$$\begin{aligned} (F*\Phi _{\lambda ,\zeta })(z)\ne 0 \quad for ~z\in \mathbb {D}/\{0\},~\zeta \in \mathbb {T}/\{-1\}, \end{aligned}$$

(3.4)

where $\Phi _{\lambda ,\zeta }=|z|^2\phi _{\lambda ,\zeta }+\phi _{\lambda ,\zeta }$ with

$$\begin{aligned} \phi _{\lambda ,\zeta }(z)=\frac{(1+e^{2i\lambda })z+(\zeta -e^{2i\lambda })z^2}{(1-z)^2} +\frac{(-1+e^{2i\lambda }-2\zeta )\overline{z}+(\zeta -e^{2i\lambda })\overline{z}^2}{(1-\overline{z})^2}. \end{aligned}$$

(3.5)

Proof

By Corollary 3, F belongs to $\mathcal{S}\mathcal{P}_{BH}(\lambda )$ if and only if F satisfies the inequality (3.2). It may be expressed in the form

$$\begin{aligned} \Re \frac{1}{\cos \lambda }\Big (e^{-i\lambda }\frac{DF(z)}{F(z)}+i\sin \lambda \Big )>0 \end{aligned}$$

for $z\in \mathbb {D}/\{0\}$. The above condition is equivalent to

$$\begin{aligned} \frac{1}{\cos \lambda }\Big (e^{-i\lambda }\frac{DF(z)}{F(z)}+i\sin \lambda \Big )\ne \frac{\zeta -1}{\zeta +1}, \quad \zeta \in \mathbb {T}/\{-1\}. \end{aligned}$$

(3.6)

We can represent (3.6) as

$$\begin{aligned}{} & {} \big [DF(z)+ie^{i\lambda }(\sin \lambda )F(z)\big ](\zeta +1)-e^{i\lambda }(\cos \lambda )F(z)(\zeta -1)\\{} & {} \quad =(\zeta +1)DF(z)+(e^{2i\lambda }-\zeta )F(z)\\{} & {} \quad =|z|^2\Big [(\zeta +1)(zh'_1(z)-\overline{z}\overline{g'_1(z)})+(e^{2i\lambda }-\zeta )(h_1(z)+\overline{g_1(z)})\Big ]\\{} & {} \qquad +(\zeta +1)\big (zh'_2(z)-\overline{z}\overline{g'_2(z)}\big )+(e^{2i\lambda }-\zeta )\big (h_2(z)+\overline{g_2(z)}\big )\\{} & {} \quad =|z|^2\Big [h_1(z)*\Big (\frac{(\zeta +1)z}{(1-z)^2}+\frac{(e^{2i\lambda }-\zeta )z}{1-z}\Big )-\overline{g_1(z)}*\Big (\frac{(\zeta +1)\overline{z}}{(1-\overline{z})^2}+\frac{(\zeta -e^{2i\lambda })\overline{z}}{1-\overline{z}}\Big )\Big ]\\{} & {} \qquad +h_2(z)*\Big (\frac{(\zeta +1)z}{(1-z)^2}+\frac{(e^{2i\lambda }-\zeta )z}{1-z}\Big )-\overline{g_2(z)}* \Big (\frac{(\zeta +1)\overline{z}}{(1-\overline{z})^2}+\frac{(\zeta -e^{2i\lambda })\overline{z}}{1-\overline{z}}\Big )\\{} & {} \quad =|z|^2G(z)*\phi _{\lambda ,\zeta }(z)+H(z)*\phi _{\lambda ,\zeta }(z)\\{} & {} \quad =F(z)*\Phi _{\lambda ,\zeta }(z)\ne 0. \end{aligned}$$

The proof is complete.$\square$

Taking $\pm \pi (1-\alpha )/2$ as $\lambda$ in (3.4), with the help of (3.1), we obtain the following result.

Theorem 6

Let F be a sense-preserving biharmonic mapping in $\mathcal{B}\mathcal{H}$ satisfying the condition $F(z)\ne 0$ for $0<|z|<1$. For $0<\alpha <1$, $F\in \mathcal{S}\mathcal{S}_{BH}(\alpha )$ if and only if

$$\begin{aligned} F(z)*\Phi _{\frac{\pi \alpha }{2},\zeta }(z)\ne 0\quad and \quad F(z)*\Phi _{-\frac{\pi \alpha }{2},\zeta }(z)\ne 0 \end{aligned}$$

for all $z\in \mathbb {D}/\{0\}$ and $\zeta \in \mathbb {T}/\{-1\}$.

4 A Class of Sense-Preserving Univalent Biharmonic Mappings in the Exterior Unit Disk

In this section, we consider biharmonic mappings in the class $\Sigma _{BH}$. Let $F\in \Sigma _{BH}$ with the form

$$\begin{aligned} F(z)= & {} |z|^2\Big (\sum _{n=1}^{\infty }a_{1,n}z^{-n}+\sum _{n=1}^{\infty }\overline{b_{1,n}}\overline{z}^{-n}\Big )+\alpha z +\sum _{n=1}^{\infty }a_{2,n}z^{-n}\nonumber \\{} & {} +\beta \overline{z}+\sum _{n=1}^{\infty }\overline{b_{2,n}}\overline{z}^{-n}+A\log |z|, \end{aligned}$$

(4.1)

where $0\le |\beta |<|\alpha |$ and $A\in \mathbb {C}$.

A simple computation shows

$$\begin{aligned} |\mu _F(z)|= & {} \Big |\frac{F_{\overline{z}}(z)}{F_z(z)}\Big |\\= & {} \Big |\frac{\beta +\sum _{n=1}^{\infty }a_{1,n}z^{-n+1}+\sum _{n=1}^{\infty }(1-n)\overline{b_{1,n}}\overline{z}^{-n}z -\sum _{n=1}^{\infty }n\overline{b_{2,n}}\overline{z}^{-n-1}+\frac{A}{2}\overline{z}^{-1}}{\alpha -\sum _{n=1}^{\infty }(n-1)a_{1,n}z^{-n}\overline{z}+\sum _{n=1}^{\infty }\overline{b_{1,n}}\overline{z}^{-n+1} -\sum _{n=1}^{\infty }na_{2,n}z^{-n-1}+\frac{A}{2}z^{-1}}\Big |\\\le & {} \frac{|\beta |+\sum _{n=1}^{\infty }|a_{1,n}|+\sum _{n=1}^{\infty }(n+1)|b_{1,n}|+\sum _{n=1}^{\infty }n|b_{2,n}|+\frac{|A|}{2}}{|\alpha | -\sum _{n=1}^{\infty }|b_{1,n}|-\sum _{n=1}^{\infty }(n+1)|a_{1,n}|-\sum _{n=1}^{\infty }n|a_{2,n}|-\frac{|A|}{2}}, \end{aligned}$$

which implies that a sufficient condition for $|\mu _F(z)|\le k<1$ is

$$\begin{aligned} |\beta |+|A|+\sum _{n=1}^{\infty }(n+2)\big (|a_{1,n}|+|b_{1,n}|\big )+\sum _{n=1}^{\infty }n\big (|a_{2,n}|+|b_{2,n}|\big )\le k|\alpha |. \end{aligned}$$

(4.2)

Let $\Sigma _{BH}(k)~(0<k<1)$ be the class of biharmonic mappings F which belong to $\Sigma _{BH}$ and satisfy the condition (4.2). We write $\Sigma _{H}(k)$ for the subclass of $\Sigma _{BH}(k)$ for harmonic mappings.

Theorem 7

Let $F\in \Sigma _{BH}(k)$ be of the form (4.1) for some $k\in (0,1)$, and

$$\begin{aligned} F^{\star }(z)= \left\{ \begin{array}{ll} F(z) &{} \text{ for } |z|>1,\\ \displaystyle \alpha z+\beta \overline{z}+|z|^2G^{\star }(z)+H^{\star }(z) &{} \text{ for } |z|\le 1, \end{array} \right. \end{aligned}$$

(4.3)

where $G^{\star }(z)=\sum _{n=1}^{\infty }(a_{1,n}\overline{z}^n+\overline{b_{1,n}}z^n)$ and $H^{\star }(z)=\sum _{n=1}^{\infty }(a_{2,n}\overline{z}^n+\overline{b_{2,n}}z^n)$. Then, F has a homeomorphic extension to the unit circle. Moreover, the mapping $F^{\star }$ is a quasiconformal extension of F with the dilatation $|\mu _{F^{\star }}(z)|\le k$ for $z\in \mathbb {C}$.

Proof

Suppose that F belongs to the class $\Sigma _{BH}(k)$, and F takes the form (4.1). For any different points $z_1,~z_2$ in $\mathbb {E}$, it is harmless to assume that $|z_1|\ge |z_2|>1$. Then, we obtain that

$$\begin{aligned} |F(z_1)-F(z_2)|\ge & {} |z_1-z_2|\big (|\alpha |-|\beta |\big )-\sum _{n=1}^{\infty }\big (|a_{2,n}|+|b_{2,n}|\big )|z_1^{-n}-z_2^{-n}|\\{} & {} -|z_1-z_2|\sum _{n=1}^{\infty }\Big [(n+2)|b_{1,n}|+(n+2)|a_{1,n}|\Big ]-|A|\int _{|z_2|}^{|z_1|}\frac{dt}{t}\\\ge & {} |z_1-z_2|\Big [|\alpha |-|\beta |-\sum _{n=1}^{\infty }(n+2)\big (|a_{1,n}|+|b_{1,n}|\big )\\{} & {} -\sum _{n=1}^{\infty }n\big (|a_{2,n}|+|b_{2,n}|\big )\Big ]- |A|\int _{|z_2|}^{|z_1|}dt\\\ge & {} |z_1-z_2|(1-k)|\alpha |. \end{aligned}$$

Similarly, we obtain $|F(z_1)-F(z_2)|\le |z_1-z_2|(1+k)\alpha .$ Therefore, we conclude that F satisfies bi-Lipschitz condition

$$\begin{aligned} (1-k)|\alpha ||z_1-z_2|\le |F(z_1)-F(z_2)|\le (1+k)|\alpha ||z_1-z_2|,\quad z\in \mathbb {E}. \end{aligned}$$

It means that F has a homeomorphic extension to the unit circle $\mathbb {T}$.

Next we will show the extension function $F^{\star }$ of the form (4.3) is a K-quasiconformal mapping of the whole plane, where $K=(1+k)/(1-k)$. For $z\in \mathbb {D}$, obviously,

$$\begin{aligned} |\mu _{F^{\star }}(z)|\le \frac{|\beta |+\sum _{n=1}^{\infty }(n+1)|a_{1,n}|+\sum _{n=1}^{\infty }|b_{1,n}|+\sum _{n=1}^{\infty }n|a_{2,n}|}{|\alpha | -\sum _{n=1}^{\infty }|a_{1,n}|-\sum _{n=1}^{\infty }(n+1)|b_{1,n}|-\sum _{n=1}^{\infty }n|b_{2,n}|}\le k. \end{aligned}$$

For $z\in \mathbb {E}$, using (4.3), we obtain

$$\begin{aligned} |\mu _{F^{\star }}(z)|= & {} \Big |\frac{\beta +\frac{A}{2\overline{z}}+\sum _{n=1}^{\infty }a_{1,n}z^{-n+1} +\sum _{n=1}^{\infty }(1-n)\overline{b_{1,n}}\overline{z}^{-n}z-\sum _{n=1}^{\infty }n\overline{b_{2,n}}\overline{z}^{-n-1}}{\alpha +\frac{A}{2z}+\sum _{n=1}^{\infty }(1-n)a_{1,n}z^{-n}\overline{z}+\sum _{n=1}^{\infty }\overline{b_{1,n}}\overline{z}^{-n+1} -\sum _{n=1}^{\infty }na_{2,n}z^{-n-1}}\Big |\\\le & {} \frac{|\beta |+\frac{|A|}{2}+\sum _{n=1}^{\infty }|a_{1,n}|+\sum _{n=1}^{\infty }(n+1)|b_{1,n}| +\sum _{n=1}^{\infty }n|b_{2,n}|}{|\alpha |-\frac{|A|}{2}-(\sum _{n=1}^{\infty }(n+1)|a_{1,n}|+\sum _{n=1}^{\infty }|b_{1,n}| +\sum _{n=1}^{\infty }n|a_{2,n}|)}\\\le & {} \frac{2k|\alpha |-|A|}{2|\alpha |-|A|}\le k. \end{aligned}$$

Hence, we conclude that $|\mu _{F^{\star }}(z)|\le k$ for $z\in \mathbb {C}$.$\square$

The authors of [19] proved that, for harmonic mappings $f_1\in \Sigma _{H}(k_1)$ and $f_2\in \Sigma _{H}(k_2)$, the convolution $f_1*f_2\in \Sigma _{H}(\sqrt{k_1k_2})$. Similarly, we can establish the following theorem.

Theorem 8

If $F_1\in \Sigma _{BH}(k_1)$ and $F_2\in \Sigma _{BH}(k_2)$, then the convolution $F_1*F_2\in \Sigma _{BH}(\sqrt{k_1k_2})$.

Proof

Suppose that

$$\begin{aligned} F_1(z)= & {} |z|^2\Big (\sum _{n=1}^{\infty }a_{1,n}z^{-n}+\sum _{n=1}^{\infty }\overline{b_{1,n}}\overline{z}^{-n}\Big )+\alpha _1 z+\beta _1\overline{z}\\{} & {} +\sum _{n=1}^{\infty }a_{2,n}z^{-n}+\sum _{n=1}^{\infty }\overline{b_{2,n}}\overline{z}^{-n}+c\log |z| \end{aligned}$$

and

$$\begin{aligned} F_2(z)= & {} |z|^2\Big (\sum _{n=1}^{\infty }A_{1,n}z^{-n}+\sum _{n=1}^{\infty }\overline{B_{1,n}}\overline{z}^{-n}\Big )+\alpha _2z+\beta _2\overline{z}\\{} & {} +\sum _{n=1}^{\infty }A_{2,n}z^{-n}+\sum _{n=1}^{\infty }\overline{B_{2,n}}\overline{z}^{-n}+C\log |z| \end{aligned}$$

for $z\in \mathbb {E}$. Then, the convolution of $F_1$ and $F_2$ is

$$\begin{aligned} F_1*F_2(z)= & {} |z|^2\Big (\sum _{n=1}^{\infty }A_{1,n}a_{1,n}z^{-n}+\sum _{n=1}^{\infty }\overline{B_{1,n}b_{1,n}}\overline{z}^{-n}\Big ) +\alpha _1\alpha _2z+\beta _1\beta _2\overline{z}\nonumber \\{} & {} +\sum _{n=0}^{\infty }A_{2,n}a_{2,n}z^{-n}+\sum _{n=1}^{\infty }\overline{B_{2,n}}\overline{b_{2,n}}\overline{z}^{-n}+Cc\log |z|,\quad z\in \mathbb {E}. \end{aligned}$$

(4.4)

Since $F_1\in \Sigma _{BH}(k_1)$ and $F_2\in \Sigma _{BH}(k_2)$, it is obvious that $F_1*F_2\in \Sigma _{BH}$. Now by the condition (4.2), it suffices to show

$$\begin{aligned} M= & {} \frac{1}{|\alpha _1||\alpha _2|}\Big [|\beta _1\beta _2|+\sum _{n=1}^{\infty }(n+2)\big (|A_{1,n}a_{1,n}|+|B_{1,n}b_{1,n}|\big )\\{} & {} +\sum _{n=1}^{\infty }n\big (|A_{2,n}a_{2,n}|+|B_{2,n}b_{2,n}|\big )+|Cc|\Big ] \le \sqrt{k_1k_2}. \end{aligned}$$

The quantity M can be written as $\sum _{m=0}^{\infty }x_mX_m$, where

$$\begin{aligned} |\alpha _1|x_m= \left\{ \begin{array}{rcl} \sqrt{(n+2)|a_{1,n}|^2}, \, &{}{m=4n-1,}\\ \displaystyle \sqrt{(n+2)|b_{1,n}|^2}, \, &{}{ m=4n-2,}\\ \displaystyle \sqrt{n|a_{2,n}|^2}, \, &{}{ m=4n-3,}\\ \displaystyle \sqrt{n|b_{2,n}|^2}, \, &{}{ m=4n,}\\ \displaystyle |\beta _1|+|c|, \, &{}{ m=0,} \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} |\alpha _2|X_m= \left\{ \begin{array}{rcl} \sqrt{(n+2)|A_{1,n}|^2}, \, &{}{m=4n-1,}\\ \displaystyle \sqrt{(n+2)|B_{1,n}|^2}, \, &{}{m=4n-2,}\\ \displaystyle \sqrt{n|A_{2,n}|^2}, \, &{}{m=4n-3,}\\ \displaystyle \sqrt{n|B_{2,n}|^2}, \, &{}{m=4n,}\\ \displaystyle |\beta _2|+|C|, \, &{}{m=0.} \end{array} \right. \end{aligned}$$

Hence by Cauchy-Schwarz inequality, we have

$$\begin{aligned} M\le & {} \Bigg (\sum _{m=0}^{\infty }x_m^2\Bigg )^{\frac{1}{2}}\Bigg (\sum _{m=0}^{\infty }X_m^2\Bigg )^{\frac{1}{2}}\\= & {} \Bigg (\frac{(|\beta _1|+|c|)^2+\sum _{n=1}^{\infty }(n+2)(|a_{1,n}|^2+|b_{1,n}|^2)+\sum _{n=1}^{\infty }n(|a_{2,n}|^2+|b_{2,n}|^2)}{|\alpha _1|^2}\Bigg )^{\frac{1}{2}}\cdot \\{} & {} \Bigg (\frac{(|\beta _2|+|C|)^2+\sum _{n=1}^{\infty }(n+2)(|A_{1,n}|^2+|B_{1,n}|^2)+\sum _{n=1}^{\infty }n(|A_{2,n}|^2+|B_{2,n}|^2)}{|\alpha _2|^2}\Bigg )^{\frac{1}{2}}\\\le & {} \Bigg (\frac{(|\beta _1|+|c|)+\sum _{n=1}^{\infty }(n+2)(|a_{1,n}|+|b_{1,n}|)+\sum _{n=1}^{\infty }n(|a_{2,n}|+|b_{2,n}|)}{|\alpha _1|}\Bigg )^{\frac{1}{2}}\cdot \\{} & {} \Bigg (\frac{(|\beta _2|+|C|)+\sum _{n=1}^{\infty }(n+2)(|A_{1,n}|+|B_{1,n}|)+\sum _{n=1}^{\infty }n(|A_{2,n}|+|B_{2,n}|)}{|\alpha _2|}\Bigg )^{\frac{1}{2}}\\\le & {} \sqrt{k_1k_2}. \end{aligned}$$

Thus, the convolution $F_1*F_2$ of the form (4.4) satisfies the condition (4.2), and then $F_1*F_2\in \Sigma _{BH}(\sqrt{k_1k_2})$. The proof is complete.$\square$