Several Properties of a Class of Generalized Harmonic Mappings

Abstract

We call the solution of a kind of second order homogeneous partial differential equation as real kernel $\alpha$-harmonic mappings. In this paper, the representation theorem, the Lipschitz continuity, the univalency and the related problems of the real kernel $\alpha$-harmonic mappings are explored.

1 Introduction

Let ${\mathbb {D}}$ be the open unit disk and ${\mathbb {T}}$ the unit circle. For $\alpha \in {\mathbb {R}}$ and $z\in {\mathbb {D}}$, let

$$\begin{aligned} T_{\alpha }=-\frac{\alpha ^{2}}{4}(1-|z|^{2})^{-\alpha -1} +\frac{\alpha }{2}(1-|z|^{2})^{-\alpha -1}\left( z\frac{\partial }{\partial z} +{\bar{z}}\frac{\partial }{\partial {\bar{z}}}\right) +(1-|z|^{2})^{-\alpha }\triangle \end{aligned}$$

be the second order elliptic partial differential operator, where $\triangle$ is the usual complex Laplacian operator

$$\begin{aligned} \triangle :=4\frac{\partial ^{2}}{\partial z \partial {\bar{z}}} =\frac{\partial ^{2}}{\partial x^{2}}+\frac{\partial ^{2}}{\partial y^{2}}. \end{aligned}$$

The corresponding partial differential equation is

$$\begin{aligned} T_{\alpha }(u)=0 \quad \text{ in }\,{\mathbb {D}}. \end{aligned}$$

(1.1)

The associated Dirichlet boundary value problem is

$$\begin{aligned} \left\{ \begin{array}{ll} T_{\alpha }(u)=0 &{}\quad \text{ in }\,{\mathbb {D}},\\ u=u^{*} &{}\quad \text{ on } \,{\mathbb {T}}. \end{array}\right. \end{aligned}$$

(1.2)

Here, the boundary data $u^{*}\in {\mathfrak {D}}^{\prime }({\mathbb {T}})$ is a distribution on the boundary of ${\mathbb {D}}$, and the boundary condition in (1.2) is interpreted in the distributional sense that $u_{r}\rightarrow u^{*}$ in ${\mathfrak {D}}^{\prime }({\mathbb {T}})$ as $r\rightarrow 1^{-}$, where

$$\begin{aligned} u_{r}(e^{i\theta })=u(re^{i\theta }), \quad e^{i\theta }\in {\mathbb {T}}, \end{aligned}$$

for $r\in [0,1)$. In [24], Olofsson proved that, for the parameter $\alpha >-1$, if a function $u\in {\mathcal {C}}^{2}({\mathbb {D}})$ satisfies (1.1) with $\lim _{r\rightarrow 1^{-}}u_{r}=u^{*}\in {\mathfrak {D}}^{\prime }({\mathbb {T}})$, then it has the form of Poisson type integral

$$\begin{aligned} u(z)=\frac{1}{2\pi }\int _{0}^{2\pi }K_{\alpha }(ze^{-i\tau })u^{*}(e^{i\tau })d\tau , \quad \text{ for }\,z\in {\mathbb {D}}, \end{aligned}$$

(1.3)

where

$$\begin{aligned} K_{\alpha }(z)=c_{\alpha }\frac{(1-|z|^{2})^{\alpha +1}}{|1-z|^{\alpha +2}}, \end{aligned}$$

(1.4)

$c_{\alpha }=\Gamma ^{2}(\alpha /2+1)/\Gamma (1+\alpha )$ and $\Gamma (s)=\int _{0}^{\infty }t^{s-1}e^{-t}dt$ for $s>0$ is the standard Gamma function. If $\alpha \le -1$, $u\in {\mathcal {C}}^{2}({\mathbb {D}})$ satisfies (1.1), and the boundary limit $u^{*}=\lim _{r\rightarrow 1^{-}}u_{r}$ exists in ${\mathfrak {D}}^{\prime }({\mathbb {T}})$, then $u(z)=0$ for all $z\in {\mathbb {D}}$. So, in the following of this paper, we always assume that $\alpha >-1$.

For $c\ne 0, -1, -2,\ldots$, the Gauss hypergeometric function is defined by the series

$$\begin{aligned} F(a,b;c; x)=\sum _{n=0}^{\infty }\frac{(a)_{n}(b)_{n}}{(c)_{n}} \frac{x^{n}}{n!} \end{aligned}$$

for $|x|<1$, and has a continuation to the complex plane with branch points at 1 and $\infty$, where $(a)_{0}=1$ and $(a)_{n}=a(a+1)\ldots (a+n-1)$ for $n=1,2,\ldots$ are the Pochhammer symbols. Obviously, for $n=0, 1,2,\ldots$, $(a)_{n}=\Gamma (a+n)/\Gamma (a)$. It is easily to verified that

$$\begin{aligned} \frac{d}{dx}F(a,b;c;x)=\frac{ab}{c}F(a+1,b+1;c+1; x). \end{aligned}$$

(1.5)

Furthermore, it holds that (cf. [3])

$$\begin{aligned} \lim _{x\rightarrow 1}F(a,b;c; x)=\frac{\Gamma (c)\Gamma (c-a-b)}{\Gamma (c-a)\Gamma (c-b)} \end{aligned}$$

(1.6)

if $Re(c-a-b)>0$.

The following Lemma 1.1 involves the determination of monotonicity of Gauss hypergeometric functions.

Lemma 1.1

[24] Let $c>0$, $a\le c$, $b\le c$ and $ab\le 0$ $(ab\ge 0)$. Then the function F(a, b; c; x) is decreasing (increasing) on $x\in (0, 1)$.

The following result of [24] is the homogeneous expansion of solutions of (1.1).

Theorem 1.2

[24] Let $\alpha \in {\mathbb {R}}$ and $u\in {\mathcal {C}}^{2}({\mathbb {D}})$. Then u satisfies (1.1) if and only if it has a series expansion of the form

$$\begin{aligned}{} & {} u(z)=\sum _{k=0}^{\infty }c_{k}F\left( -\frac{\alpha }{2},k-\frac{\alpha }{2}; k+1; |z|^{2}\right) z^{k}\nonumber \\{} & {} +\sum _{k=1}^{\infty }c_{-k}F\left( -\frac{\alpha }{2},k-\frac{\alpha }{2}; k+1; |z|^{2}\right) {\bar{z}}^{k}, \quad z\in {\mathbb {D}}, \end{aligned}$$

(1.7)

for some sequence $\{c_{k}\}_{-\infty }^{\infty }$ of complex number satisfying

$$\begin{aligned} \lim _{|k|\rightarrow \infty }\sup |c_{k}|^{\frac{1}{|k|}}\le 1. \end{aligned}$$

(1.8)

In particular, the expansion (1.7), subject to (1.8), converges in ${\mathcal {C}}^{\infty }({\mathbb {D}})$, and every solution u of (1.1) is ${\mathcal {C}}^{\infty }$-smooth in ${\mathbb {D}}$.

Let

$$\begin{aligned} v(z)=\sum _{k=0}^{\infty }c_{k}z^{k} +\sum _{k=1}^{\infty }c_{-k}{\bar{z}}^{k}, \quad z\in {\mathbb {D}}. \end{aligned}$$

(1.9)

It is obvious that v(z) is a harmonic mapping, i.e., $\triangle v=0$. We observe that u(z) of (1.7) and v(z) have same coefficient sequence $\{c_{k}\}_{-\infty }^{\infty }$. Actually, if $\alpha =0$, then $u(z)=v(z)$.

Observe that the kernel $K_{\alpha }$ in (1.4) is real. We call u of (1.3) or (1.7) as real kernel $\alpha$-harmonic mappings. Furthermore, suppose u(z) and v(z) have the expansions of (1.7) and (1.9), respectively. We call v(z) as the corresponding harmonic mapping of u(z). Conversely, we call u(z) as the corresponding real kernel $\alpha$-harmonic mapping of v(z).

If we take $\alpha =2(p-1)$, then a real kernel $\alpha$-harmonic mapping u is polyharmonic (or p-harmonic), where $p\in \{1,2,\ldots \}$ (cf. [1, 2, 5, 6, 11, 13, 15, 27]). In particular, if $\alpha =0$, then u is harmonic (cf. [10, 18,19,20]). Thus, the real kernel $\alpha$-harmonic mapping is a kind of generalization of classical harmonic mapping. Furthermore, by Olofsson [25], we know that it is related to standard weighted harmonic mappings. For the related discussion on standard weighted harmonic mappings, see [8, 16, 17, 23].

For the real kernel $\alpha$-harmonic mappings, the Schwarz–Pick type estimates and coefficient estimates are obtained in [7]; the starlikeness, convexity and Landau type theorem are studied in [22]; the sharp Heinz type inequality is established and the extremal functions of Schwartz type lemma are explored in [21]; the Lipschitz continuity with respect to the distance ratio metric is proved in [14]. In [12], using the properties of the real kernel $\alpha$-harmonic mappings, the authors established some Schwarz type lemmas for mappings satisfying a class of inhomogeneous biharmonic Dirichlet problem.

In this paper, we continue to study the properties of the real kernel $\alpha$-harmonic mappings. The main idea of this paper is that by establishing the relationship between harmonic mapping and the corresponding real kernel $\alpha$-harmonic mapping, we use the harmonic mapping to characterize the corresponding real kernel $\alpha$-harmonic mapping. In Sect. 2, for a nonnegative even number $\alpha$, we get an explicit representation theorem which determines the relation between the real kernel $\alpha$-harmonic mapping and the corresponding harmonic mapping. As its application, in Sect. 3, we show that the Lipschitz continuity of a real kernel $\alpha$-harmonic mapping is determined by the corresponding harmonic mapping. In Sect. 4, for a subclass of the real kernel $\alpha$-harmonic mappings, we discuss its univalency and explore its Radó–Kneser–Choquet type theorem. In Sect. 5, we explore the influence of parameters $\alpha$ on the image area of the real kernel $\alpha$-harmonic mappings.

2 Representation Theorem

Theorem 2.1

Let $v(z)=h(z)+\overline{g(z)}=\sum _{k=0}^{\infty }c_{k}z^{k} +\sum _{k=1}^{\infty }c_{-k}{\bar{z}}^{k}$ be a harmonic mapping defined on the unit disk ${\mathbb {D}}$. If $\frac{\alpha }{2}=p-1$ is a nonnegative integer, then the corresponding real kernel $\alpha$-harmonic mapping of v(z) can be represented by

$$\begin{aligned} u(z)=\sum _{n=0}^{p-1}|z|^{2n}\frac{(1-p)_{n}}{n!} \left( I_{n}+\overline{J_{n}}\right) , \end{aligned}$$

(2.1)

where $I_{n}$ and $J_{n}$ satisfy the recurrence formulas

$$\begin{aligned} I_{n}&=I_{n-1} -p\frac{\int _{0}^{z}z^{n-1}I_{n-1}dz}{z^{n}}, \end{aligned}$$

(2.2)

$$\begin{aligned} J_{n}&=J_{n-1}-p\frac{\int _{0}^{z}z^{n-1}J_{n-1}dz}{z^{n}}\quad n=1,2,\ldots , p-1, \end{aligned}$$

(2.3)

$I_{0}=h(z)$, and $J_{0}=g(z)$.

Proof

Let $H(z)=\sum _{k=0}^{\infty }c_{k}F(-\frac{\alpha }{2}, k-\frac{\alpha }{2}; k+1; |z|^{2})z^{k}$ and $G(z)=\sum _{k=1}^{\infty }\overline{c_{-k}}F(-\frac{\alpha }{2}, k-\frac{\alpha }{2}; k+1; |z|^{2})z^{k}$. Then by the assumption and (1.7), we have

$$\begin{aligned} u(z)=H(z)+\overline{G(z)}. \end{aligned}$$

(2.4)

When $\frac{\alpha }{2}=p-1$, rewrite H(z) as

$$\begin{aligned} H(z)&=\sum _{k=0}^{\infty }c_{k}F(1-p,k+1-p; k+1; |z|^{2})z^{k}\nonumber \\&=\sum _{k=0}^{\infty }c_{k}z^{k}\left( \sum _{n=0}^{\infty } \frac{(1-p)_{n}(k+1-p)_{n}}{(k+1)_{n}}\frac{|z|^{2n}}{n!}\right) \nonumber \\&=\sum _{k=0}^{\infty }c_{k}z^{k}\left( \sum _{n=0}^{p-1} \frac{(1-p)_{n}(k+1-p)_{n}}{(k+1)_{n}}\frac{|z|^{2n}}{n!}\right) \nonumber \\&=\sum _{n=0}^{p-1}|z|^{2n}\frac{(1-p)_{n}}{n!}I_{n}, \end{aligned}$$

(2.5)

where

$$\begin{aligned} I_{n}=\sum _{k=0}^{\infty }\frac{(k+1-p)_{n}}{(k+1)_{n}}c_{k}z^{k}. \end{aligned}$$

Because

$$\begin{aligned} \frac{(k+1-p)_{n}}{(k+1)_{n}}&=\frac{(k+1-p)_{n-1}(k+n-p)}{(k+1)_{n-1}(k+n)}\\&=\frac{(k+1-p)_{n-1}}{(k+1)_{n-1}}-\frac{p}{k+n}\frac{(k+1-p)_{n-1}}{(k+1)_{n-1}}, \end{aligned}$$

we can get

$$\begin{aligned} I_{n}&=\sum _{k=0}^{\infty }\frac{(k+1-p)_{n-1}}{(k+1)_{n-1}}c_{k}z^{k}-\sum _{k=0}^{\infty } \frac{p}{k+n}\frac{(k+1-p)_{n-1}}{(k+1)_{n-1}}c_{k}z^{k}\\&=I_{n-1}-p\frac{\int _{0}^{z}z^{n-1}I_{n-1}dz}{z^{n}}. \end{aligned}$$

This is (2.2).

Similarly, we can get

$$\begin{aligned} G(z)=\sum _{n=0}^{p-1}|z|^{2n}\frac{(1-p)_{n}}{n!}J_{n}, \end{aligned}$$

(2.6)

where $J_{n}$ is defined as in (2.3). Therefore, Eq. (2.1) follows from Eqs. (2.4)–(2.6). $\square$

Example 2.1

From the recurrence formula (2.1), we have the following:

(i) When $\alpha =0$, i.e. $p=1$,

$$\begin{aligned}u(z)=v(z); \end{aligned}$$

(ii) When $\alpha =2$, i.e. $p=2$,

$$\begin{aligned} u(z)&=h+{\bar{g}}-|z|^{2}\left( h-2\frac{\int _{0}^{z}h(z)dz}{z} +\overline{g-2\frac{\int _{0}^{z}g(z)dz}{z}}\right) \nonumber \\&=\sum _{k=0}^{\infty }c_{k}z^{k}+\sum _{k=1}^{\infty }c_{-k} {\bar{z}}^{k}-|z|^{2}\left( \sum _{k=0}^{\infty }c_{k}\frac{k-1}{k+1}z^{k} +\sum _{k=1}^{\infty }c_{-k}\frac{k-1}{k+1}{\bar{z}}^{k}\right) ; \end{aligned}$$

(2.7)

(iii) When $\alpha =4$, i.e. $p=3$,

$$\begin{aligned} u(z)&=h+{\bar{g}}-2|z|^{2}\left( h-3\frac{\int _{0}^{z}h(z)dz}{z} +\overline{g-3\frac{\int _{0}^{z}g(z)dz}{z}}\right) \\&\quad +|z|^{4}\left( h-3\frac{\int _{0}^{z}h(z)dz}{z} -3\frac{\int _{0}^{z}zh(z)dz}{z^{2}}+9\frac{\int _{0}^{z} \int _{0}^{z}h(z)dzdz}{z^{2}}\right. \\&\quad \left. +\overline{g-3\frac{\int _{0}^{z}g(z)dz}{z} -3\frac{\int _{0}^{z}zg(z)dz}{z^{2}}+9\frac{\int _{0}^{z} \int _{0}^{z}g(z)dzdz}{z^{2}}}\right) \\&=\sum _{k=0}^{\infty }c_{k}z^{k}+\sum _{k=1}^{\infty } c_{-k}{\bar{z}}^{k}-2|z|^{2}\left( \sum _{k=0}^{\infty }c_{k} \frac{k-2}{k+1}z^{k}+\sum _{k=1}^{\infty }c_{-k} \frac{k-2}{k+1}{\bar{z}}^{k}\right) \\&\quad +|z|^{4}\left( \sum _{k=0}^{\infty }c_{k} \frac{(k-1)(k-2)}{(k+1)(k+2)}z^{k} +\sum _{k=1}^{\infty }c_{-k} \frac{(k-1)(k-2)}{(k+1)(k+2)}{\bar{z}}^{k}\right) . \end{aligned}$$

3 Lipschitz Continuity

Theorem 3.1

Let u(z) be the corresponding real kernel $\alpha$-harmonic mapping of $v(z)=h+{\bar{g}}$ on the unit disk ${\mathbb {D}}$. If v(z) is Lipschitz continuous on the unit disk ${\mathbb {D}}$ and $\frac{\alpha }{2}=p-1$ is a nonnegative integer, then u is Lipschitz continuous on the unit disk ${\mathbb {D}}$ as well.

Proof

By the assumption and (2.1), it is sufficient to prove that $I_{n}$ and $J_{n}$ are Lipschitz continuous on the unit disk ${\mathbb {D}}$ for $n=0, 1,2,\ldots ,p-1$. In the following, we just prove the Lipschitz continuity of $I_{n}$. The case of $J_{n}$ is similar.

Observe that $I_{0}=h(z)$ is holomorphic on ${\mathbb {D}}$. Then by the recurrence formula (2.2), it is easy to see that all $I_{n}$ are holomorphic on ${\mathbb {D}}$. It follows that all $I^{\prime }_{n}$ are holomorphic on ${\mathbb {D}}$ too, where

$$\begin{aligned} I^{\prime }_{n}=I^{\prime }_{n-1}-p\frac{z^{n}I_{n-1}-n\int _{0}^{z}z^{n-1}I_{n-1}dz}{z^{n+1}}, \quad n=1,2,\ldots ,p-1. \end{aligned}$$

(3.1)

Taking account of the maximum modulus principle of holomorphic functions, from Eqs. (2.2) and (3.1), we get

$$\begin{aligned}&\sup _{z\in {\mathbb {D}}}|I_{n}|\le \sup _{z\in {\mathbb {D}}}|I_{n-1}|+p \sup _{z\in {\mathbb {D}}}|I_{n-1}|=(p+1) \sup _{z\in {\mathbb {D}}}|I_{n-1}| \end{aligned}$$

and

$$\begin{aligned}&\sup _{z\in {\mathbb {D}}}|I^{\prime }_{n}|\le \sup _{z\in {\mathbb {D}}}|I^{\prime }_{n-1}|+p \sup _{z\in {\mathbb {D}}}|I_{n-1}|+np\sup _{z\in {\mathbb {D}}}|I_{n-1}|\\&=\sup _{z\in {\mathbb {D}}}|I^{\prime }_{n-1}|+(n+1)p\sup _{z\in {\mathbb {D}}}|I_{n-1}|, \end{aligned}$$

respectively. It follows that

$$\begin{aligned}&\sup _{z\in {\mathbb {D}}}|I_{n}|\le (p+1)^{n} \sup _{z\in {\mathbb {D}}}|I_{0}| \end{aligned}$$

and

$$\begin{aligned} \sup _{z\in {\mathbb {D}}}|I^{\prime }_{n}|&\le \sup _{z\in {\mathbb {D}}}|I^{\prime }_{n-1}| +(n+1)p\sup _{z\in {\mathbb {D}}}|I_{n-1}|\nonumber \\&\le \sup _{z\in {\mathbb {D}}}|I^{\prime }_{n-2}| +np\sup _{z\in {\mathbb {D}}}|I_{n-2}|+(n+1)p \sup _{z\in {\mathbb {D}}}|I_{n-1}|\nonumber \\&\le \cdots \nonumber \\&\le \sup _{z\in {\mathbb {D}}}|I^{\prime }_{0}| +p\sum _{i=1}^{n}(i+1)\sup _{z\in {\mathbb {D}}}|I_{i-1}|\nonumber \\&\le \sup _{z\in {\mathbb {D}}}|I^{\prime }_{0}| +p\sum _{i=1}^{n}(i+1)(p+1)^{i-1}\sup _{z\in {\mathbb {D}}}|I_{0}|. \end{aligned}$$

(3.2)

Because $v=h+{\bar{g}}$ is Lipschitz, there exists a constant M such that

$$\begin{aligned} |h^{\prime }|=|I^{\prime }_{0}|\le M \end{aligned}$$

(3.3)

for $z\in {\mathbb {D}}$. It follows that

$$\begin{aligned} \sup _{z\in {\mathbb {D}}}|I_{0}|=\sup _{z\in {\mathbb {D}}}|h|\le M. \end{aligned}$$

(3.4)

Therefore, by inequalities (3.2)–(3.4), we get that there exists a constant $C=C(M, p, n)$, such that

$$\begin{aligned}&\sup _{z\in {\mathbb {D}}}|I^{\prime }_{n}|\le \left( 1+p\sum _{i=1}^{n}(i+1)(p+1)^{i-1}\right) M=:C \end{aligned}$$

for $n=1,2,\ldots ,p-1$. It means that $I_{n}$ is Lipschitz continuous on ${\mathbb {D}}$. $\square$

4 Univalency of a Subclass of Real Kernel $\alpha$-Harmonic Mappings

In the rest of this paper, we use the following notations. Let $\alpha >-1$, $z=re^{i\theta }$, and

$$\begin{aligned} t&=|z|^{2}=r^{2},\\ F&=F_{k}=F\left( -\frac{\alpha }{2},k-\frac{\alpha }{2};k+1;t\right) ,\quad k=1,2,\ldots ,\\ F_{t}&=F_{k,t}=F_{k,t}\left( -\frac{\alpha }{2},k -\frac{\alpha }{2};k+1;t\right) =\frac{dF_{k}}{dt}=\frac{dF}{dt}. \end{aligned}$$

Furthermore, let

$$\begin{aligned} F_{k}(1)=\lim _{t\rightarrow 1^{-}}F\left( -\frac{\alpha }{2},k-\frac{\alpha }{2};k+1;t\right) . \end{aligned}$$

Then by (1.6), we have

$$\begin{aligned} F_{k}(1)=\frac{\Gamma (k+1)\Gamma (1+\alpha )}{\Gamma \left( k+1+\frac{\alpha }{2}\right) \Gamma \left( 1+\frac{\alpha }{2}\right) }. \end{aligned}$$

(4.1)

Lemma 4.1

Let $r_{n}$ and $s_{n}$ $(n=0,1,2,\ldots )$ be real numbers, and let the power series

$$\begin{aligned} R(x)=\sum _{n=0}^{\infty }r_{n}x^{n} \quad \text{ and }\quad S(x)=\sum _{n=0}^{\infty }s_{n}x^{n} \end{aligned}$$

be convergent for $|x|<r$, $(r>0)$ with $s_{n}>0$ for all n. If the non-constant sequence $\{r_{n}/s_{n}\}$ is increasing (decreasing) for all n, then the function $x\mapsto R(x)/S(x)$ is strictly increasing (resp. decreasing) on (0, r).

Lemma 4.1 is basically due to [4] (see also [28]) and in this form with a general setting was stated in [26] along with many applications which were later adopted by a number of researchers.

Lemma 4.2

[22] Let $\frac{\alpha }{2}\in (0,1]$. Then it holds that

$$\begin{aligned}&(1)\quad \frac{F_{k}}{F_{1}}\le 1\,\text{ for } \,k=1,2,3,\ldots \,\text{ and }\,t\in [0,1);\\&(2)\quad \frac{|F_{k,t}|}{F_{1}}<\frac{\left( k-\frac{\alpha }{2} \right) \Gamma (k+1)\Gamma \left( 2+\frac{\alpha }{2}\right) }{2\Gamma \left( k+1+\frac{\alpha }{2}\right) }\, \text{ for }\,k=1, 2,3,\ldots \,\text{ and }\,t\in (0,1). \end{aligned}$$

Theorem 4.3

If $\alpha \in (0,2]$, $c_{-k}\in (-N, N)$, where

$$\begin{aligned} N=\frac{\alpha }{2\left( \frac{\left( k-\frac{\alpha }{2}\right) \Gamma (k+1)\Gamma \left( 2+\frac{\alpha }{2}\right) }{\Gamma \left( k+1+\frac{\alpha }{2}\right) }+k\right) }, \end{aligned}$$

(4.2)

then the real kernel $\alpha$-harmonic mapping

$$\begin{aligned} u(z)=F_{1}z+c_{-k}F_{k}{\overline{z}}^{k}, \quad k=1,2,3\ldots , \end{aligned}$$

(4.3)

is sense-preserving univalent in ${\mathbb {D}}$.

Proof

We divide the proof into two steps.

First step: Formula (4.3) implies that

$$\begin{aligned} u_{z}=F_{1}+F_{1,t}t+c_{-k}F_{k,t}{\bar{z}}^{k+1}, \quad u_{{\bar{z}}}=F_{1,t}z^{2}+c_{-k}\left( F_{k,t}z{\bar{z}}^{k}+kF_{k}{\bar{z}}^{k-1}\right) . \end{aligned}$$

It follows that

$$\begin{aligned} |u_{z}|-|u_{{\bar{z}}}|&\ge F_{1}-|F_{1,t}t|-|c_{-k}F_{k,t}{\bar{z}}^{k+1}|\\&\quad -\left| F_{1,t}z^{2}\right| -\left| c_{-k}F_{k,t}z{\bar{z}}^{k}\right| -k\left| c_{-k}F_{k}{\bar{z}}^{k-1}\right| \\&\quad>F_{1}-|F_{1,t}|-|c_{-k}||F_{k,t}|-|F_{1,t}| -|c_{-k}||F_{k,t}|-k|c_{-k}||F_{k}|\\&=F_{1}\left[ 1-\frac{2|F_{1,t}|}{F_{1}}-|c_{-k}| \left( \frac{2|F_{k,t}|}{F_{1}}+k\frac{|F_{k}|}{F_{1}}\right) \right] \\&\quad>F_{1}\left[ 1-\left( 1-\frac{\alpha }{2}\right) -|c_{-k}| \left( \frac{\left( k-\frac{\alpha }{2}\right) \Gamma (k+1)\Gamma \left( 2+\frac{\alpha }{2}\right) }{\Gamma \left( k+1+\frac{\alpha }{2}\right) }+k\right) \right] >0 \end{aligned}$$

for $c_{-k}\in (-N, N)$. The third inequality of the above holds because of Lemma 4.2. Therefore, u(z) is sense-preserving.

Second step: Let $c_{-k}=|c_{-k}|e^{i\beta }$. By assumption, we have $\beta =0$ or $\pi$. Let $z=re^{i\theta }$ and $u(z)=Re^{i\varphi }$. Rewrite u(z) of (4.3) as

$$\begin{aligned} u(z)&=F_{1}re^{i\theta }+|c_{-k}|F_{k}r^{k}e^{i(\beta -k\theta )}\nonumber \\&=F_{1}r\cos \theta +|c_{-k}|F_{k}r^{k}\cos (\beta -k\theta )+i(F_{1}r\sin \theta +|c_{-k}|F_{k}r^{k}\sin (\beta -k\theta )). \end{aligned}$$

(4.4)

Then

$$\begin{aligned} \tan \varphi =\frac{F_{1}r\sin \theta +|c_{-k}|F_{k}r^{k}\sin (\beta -k\theta )}{F_{1}r\cos \theta +|c_{-k}|F_{k}r^{k}\cos (\beta -k\theta )}, \end{aligned}$$

(4.5)

where $\varphi$ is the argument of u(z). It follows that

$$\begin{aligned} \frac{d}{d\theta }(\tan \varphi )&=\frac{d}{d\theta }\left( \frac{F_{1}r\sin \theta +|c_{-k}|F_{k}r^{k}\sin (\beta -k\theta )}{F_{1}r\cos \theta +|c_{-k}|F_{k}r^{k}\cos (\beta -k\theta )}\right) \nonumber \\&=\frac{F_{1}^{2}-|c_{-k}|^{2}F_{k}^{2}r^{2(k-1)}k-(k-1)|c_{-k}|F_{1} F_{k}r^{k-1}\cos (\beta -(k+1)\theta )}{\left[ F_{1}\cos \theta +|c_{-k}|F_{k}r^{k-1}\cos (\beta -k\theta )\right] ^{2}}\nonumber \\&\ge \frac{F_{1}^{2}-|c_{-k}|^{2}F_{k}^{2}r^{2(k-1)} k-(k-1)|c_{-k}|F_{1}F_{k}r^{k-1}}{\left[ F_{1}\cos \theta +|c_{-k}|F_{k}r^{k-1}\cos (\beta -k\theta )\right] ^{2}}\nonumber \\&=\frac{\left( F_{1}+|c_{-k}|F_{k}r^{k-1}\right) \left( F_{1}-|c_{-k}|kF_{k}r^{ k-1}\right) }{\left[ F_{1}\cos \theta +|c_{-k}|F_{k}r^{k-1}\cos (\beta -k\theta )\right] ^{2}} >0 \end{aligned}$$

(4.6)

for $|c_{-k}|<\frac{1}{k}$. The last inequality of the above holds because of Lemma 4.2(1). That is to say, $\tan \varphi$ is strictly increasing with respect to $\theta$. So is $\varphi$, too.

In the following we divide into two cases to discuss.

Case 1 $\beta =0$. It follows from (4.5) that

$$\begin{aligned} \cot \varphi =\frac{\cos \theta +|c_{-k}|\frac{F_{k}}{F_{1}}r^{k-1}\cos k\theta }{\sin \theta -|c_{-k}|\frac{F_{k}}{F_{1}}r^{k-1}\sin k\theta }. \end{aligned}$$

(4.7)

Let’s take a close look at the changes in the value of the function $\cot \varphi$. Firstly, as is well-known, it is easy to verify by mathematical induction that

$$\begin{aligned} \left| \frac{\sin k\theta }{\sin \theta }\right| \le k \end{aligned}$$

(4.8)

for $k=1,2,\ldots$ and $\theta \in [0,2\pi )$. If $|c_{-k}|<\frac{1}{k}$, $\alpha \in (0,2]$ and $\sin \theta \ne 0$, then Lemma 4.2(1) and inequality (4.8) imply that $|\sin \theta |>|c_{-k}|\frac{F_{k}}{F_{1}}r^{k-1}|\sin k\theta |$. So, $\sin \theta \ne 0$ implies $\sin \theta -|c_{-k}|\frac{F_{k}}{F_{1}}r^{k-1}\sin k\theta \ne 0$. In another words, the zero of the denominator of the right side of equation (4.7) comes only from the zero of $\sin \theta$. Secondly, $\sin \theta$ only have two zeros in the intervals $[0, 2\pi )$. That is $\theta =0$ and $\pi$. By (4.7), we have that if $\theta =0^{+}$, then $\cot \varphi =+\infty$; if $\theta =\pi ^{-}$, then $\cot \varphi =-\infty$; if $\theta =\pi ^{+}$, then $\cot \varphi =+\infty$; if $\theta =2\pi ^{-}$, then $\cot \varphi =-\infty$. Therefore, considering the continuity and monotonicity of $\cot \varphi$, we can get that the $u(re^{i\theta })$ maps every circle $|z|=r<1$ in a one-to-one manner onto a closed Jordan curve.

Case 2 $\beta =\pi$. Considering (4.5), we have

$$\begin{aligned} \cot \varphi =\frac{\cos \theta -|c_{-k}|\frac{F_{k}}{F_{1}}r^{k-1}\cos k\theta }{\sin \theta +|c_{-k}|\frac{F_{k}}{F_{1}}r^{k-1}\sin k\theta }. \end{aligned}$$

Follow the discussion of Case 1. We omit the further details.

It is easy to see that $N<\frac{1}{k}$, where N defined by (4.2). Therefore, considering the above two steps of the proof, by degree principle [9], we can get that u(z) is univalent in ${\mathbb {D}}$. $\square$

The following is the well known Radó–Kneser–Choquet theorem, which can be seen in the page 29 of [10].

Theorem 4.4

If $\Omega \in {\mathbb {C}}$ is a bounded convex domain whose boundary is a Jordan curve $\gamma$ and f is a homeomorphism of the unit circle ${\mathbb {T}}$ onto $\gamma$, then its harmonic extension

$$\begin{aligned} u(z)=\frac{1}{2\pi }\int _{0}^{2\pi } \frac{1-|z|^{2}}{|e^{it}-z|^{2}}f(e^{it})dt \end{aligned}$$

is univalent in ${\mathbb {D}}$ and defines a harmonic mapping of ${\mathbb {D}}$ onto $\Omega$.

Next, we want to explore the Radó–Kneser–Choquet type theorem for real kernel $\alpha$-harmonic mappings. We need the following Proposition at first.

Proposition 4.5

Suppose $\alpha >-1$ and $c_{-k}\in {\mathbb {R}}$. Let

$$\begin{aligned} f(e^{i\theta })=F_{1}(1)e^{i\theta }+c_{-k}F_{k}(1)e^{-ik\theta },\quad k=1,2,3,\ldots . \end{aligned}$$

(4.9)

Then f maps the unit circle ${\mathbb {T}}$ onto a convex Jordan curve if and only if $c_{-k}\in (-M, M)$, where

$$\begin{aligned} M=\frac{\Gamma \left( k+1+\frac{\alpha }{2}\right) }{k^{2} \Gamma (k+1)\Gamma \left( 2+\frac{\alpha }{2}\right) }. \end{aligned}$$

(4.10)

Proof

Direct computation leads to

$$\begin{aligned} \frac{d}{d\theta }\left( f(e^{i\theta })\right) =-F_{1}(1)\sin \theta -kc_{-k}F_{k}(1)\sin k\theta +i(F_{1}(1)\cos \theta -kc_{-k}F_{k}(1)\cos k\theta ). \end{aligned}$$

Let $\psi =\psi (\theta )=\arg \{\frac{d}{d\theta }f(e^{i\theta })\}$. Then we have

$$\begin{aligned} \frac{d}{d\theta }(\tan \psi (\theta ))&=\frac{(F_{1}(1))^{2} -k^{3}(c_{-k}F_{k}(1))^{2}+k(k-1)c_{-k}F_{1}(1)F_{k}(1)\ \cos ((k+1)\theta )}{(F_{1}(1)\sin \theta +kc_{-k}F_{k}(1)\sin k\theta )^{2}}\\&\ge \frac{(F_{1}(1))^{2}-k^{3}(c_{-k}F_{k}(1))^{2} -k(k-1)|c_{-k}|F_{1}(1)F_{k}(1))}{(F_{1}(1)\sin \theta +kc_{-k}F_{k}(1)\sin k\theta )^{2}}\\&=\frac{(F_{1}(1)+k|c_{-k}|F_{k}(1))(F_{1}(1) -k^{2}|c_{-k}|F_{k}(1))}{(F_{1}(1)\sin \theta +kc_{-k}F_{k}(1)\sin k\theta )^{2}} \end{aligned}$$

Hence, $\frac{d}{d\theta }(\tan \psi (\theta ))\ge 0$ if and only if $|c_{-k}|\le \frac{F_{1}(1)}{k^{2}F_{k}(1)}= \frac{\Gamma (k+1+\frac{\alpha }{2})}{k^{2}\Gamma (k+1)\Gamma (2+\frac{\alpha }{2})}$. $\square$

Now let $f(e^{i\theta })$ be defined as in (4.9) with $\alpha \in (0,2]$, $c_{-k}\in (-L, L)$, where $L=\min \{M, N\}$. Observe that $\lim _{r\rightarrow 1}u(z):=u^{*}(e^{i\theta })=f(e^{i\theta })$, where u(z) are defined by (4.3). Similar to the second step of the proof of Theorem 4.3, we can verify that $f(e^{i\theta })$ maps unit circle ${\mathbb {T}}$ onto a closed Jordan curve in a one-to-one manner, too. Therefore, considering Theorem 3.3 of [24] and Theorem 4.3 of the above, we actually get a Radó–Kneser–Choquet type theorem as follows:

Proposition 4.6

Let $u^{*}(e^{i\theta })=f(e^{i\theta })$ be defined by (4.9) with $k=1,2,3,\ldots$, $\alpha \in (0,2]$, $c_{-k}\in (-L, L)$, where $L=\min \{M, N\}$, N and M are defined by (4.2) and (4.10), respectively. Then $u^{*}(e^{i\theta })$ is a homeomorphism of the unit circle ${\mathbb {T}}$ onto a convex Jordan curve $\gamma$ which is a boundary of a bounded convex domain $\Omega \subset {\mathbb {C}}$. Furthermore, u(z) defined by (1.3) defines a univalent real kernel $\alpha$-harmonic mapping of ${\mathbb {D}}$ onto $\Omega$.

Let us have a look at some special cases of Theorem 4.3 or Proposition 4.6.

Example 4.1

Let $\alpha =2$. Then $M=\frac{k+1}{2k^{2}}$ and $N=\frac{k+1}{k^{2}+3k-2}$. Formula (4.9) deduces to

$$\begin{aligned} f(e^{i\theta })=e^{i\theta }+\frac{2}{k+1}c_{-k}e^{-ik\theta }. \end{aligned}$$

Furthermore, let $u^{*}(e^{i\theta })=f(e^{i\theta })$. Then (1.3), or (2.7), implies that the corresponding real kernel $\alpha$-harmonic mapping is

$$\begin{aligned} u(z)=F_{1}z+c_{-k}F_{k}{\bar{z}}^{k}=z+c_{-k}\left( 1 -\frac{k-1}{k+1}|z|^{2}\right) {\bar{z}}^{k}. \end{aligned}$$

(4.11)

Actually, it is biharmonic.

• (1) If $k=1$ or $k=2$, then $L=M=N$. If $c_{-k}\in (-L, L)$, then Proposition 4.6 says that the u(z) given by (4.11) is univalent, and $u({\mathbb {D}})=\Omega$ is a convex domain.

• (2) If $k=3,4,5,\ldots$, then a direct computation leads to $N>M$. Taking $c_{-k}\in (M, N)$, Theorem 4.3 and Proposition 4.5 imply that the above u(z) is still univalent, but $u({\mathbb {D}})=\Omega$ is not a convex domain.

5 Area $S_{u}$

Let $S_{u}(\alpha )$ denote the area of the Riemann surface of real kernel $\alpha$-harmonic mapping u. Then we have the following results.

Theorem 5.1

Let u be a sense-preserving real kernel $\alpha$-harmonic mapping that has the series expansion of the form (1.7) with $c_{0}=0$, continuous on $\overline{{\mathbb {D}}}$. Let v be the corresponding sense-preserving harmonic mapping that has the series expansion of the form (1.9), continuous on $\overline{{\mathbb {D}}}$. If $|c_{k}|\ge |c_{-k}|$ for $k=1,2,\ldots$, then

• (1) $S_{u}(\alpha )< S_{u}(0)$ for $\alpha \in (0, 2)$ and $S_{u}> S_{u}(0)$ for $\alpha \in (-1,0)$;

• (2) $S_{u}(\alpha )$ is strictly decreasing with respect to $\alpha \in (-1,\alpha _{0})$, where $\alpha _{0}$ is the unique solution of equation

  $$\begin{aligned} \psi (1+\alpha )-\psi \left( 1+\frac{\alpha }{2}\right) -\frac{1}{2+\alpha }=0 \end{aligned}$$

  in the interval (0.8, 1), $\psi$ is the digamma function. Furthermore, the constant $\alpha _{0}$ is sharp.

Proof

By (1.7), direct computation leads to

$$\begin{aligned} u_{z}&=\sum _{k=1}^{\infty }c_{k}\left[ F_{t}{\bar{z}}z^{k} +kFz^{k-1}\right] +\sum _{k=1}^{\infty }c_{-k}F_{t}{\bar{z}}^{k+1}\\&=\sum _{k=1}^{\infty }c_{k}\left[ F_{t}r^{k+1}+kFr^{k-1}\right] e^{i(k-1)\theta }+\sum _{k=1}^{\infty }c_{-k}F_{t}r^{k+1}e^{-i(k+1)\theta }\\ \end{aligned}$$

and

$$\begin{aligned} u_{{\bar{z}}}&=\sum _{k=1}^{\infty }c_{k}F_{t}z^{k+1} +\sum _{k=1}^{\infty }c_{-k}\left[ F_{t}z{\bar{z}}^{k}+kF{\bar{z}}^{k-1}\right] \\&=\sum _{k=1}^{\infty }c_{-k}\left[ F_{t}r^{k+1}+kFr^{k-1}\right] e^{-i(k-1)\theta }+\sum _{k=1}^{\infty }c_{k}F_{t}r^{k+1}e^{i(k+1)\theta }. \end{aligned}$$

So,

$$\begin{aligned} S_{u}(\alpha )&=\int _{0}^{2\pi }\int _{0}^{1}J_{u}(z)rdrd\theta \nonumber \\&=2\pi \int _{0}^{1}\sum _{k=1}^{\infty }\left[ \left| c_{k}\left( F_{t}r^{k+1} +kFr^{k-1}\right) \right| ^{2}+\left| c_{-k}F_{t}r^{k+1}\right| ^{2}\right. \nonumber \\&\left. \quad -\left| c_{-k}\left( F_{t}r^{k+1} +kFr^{k-1}\right) \right| ^{2}-\left| c_{k}F_{t}r^{k+1}\right| ^{2}\right] rdr\nonumber \\&=2\pi \int _{0}^{1}\left[ \sum _{k=1}^{\infty }\left( |c_{k}|^{2} -|c_{-k}|^{2}\right) \left( k^{2}F^{2}r^{2k-1}+2kFF_{t}r^{2k+1}\right) \right] dr\nonumber \\&=2\pi \sum _{k=1}^{\infty }\left[ \left( |c_{k}|^{2}-|c_{-k}|^{2} \right) k\int _{0}^{1}\left( kF^{2}r^{2k-1}+2FF_{t}r^{2k+1}\right) dr\right] \nonumber \\&=\pi \sum _{k=1}^{\infty }\left[ \left( |c_{k}|^{2}-|c_{-k}|^{2}\right) k\int _{0}^{1}d(F^{2}r^{2k})\right] \nonumber \\&=\pi \frac{\Gamma ^{2}(1+\alpha )}{\Gamma ^{2} \left( 1+\frac{\alpha }{2}\right) }\sum _{k=1}^{\infty }\left[ k\left( |c_{k}|^{2} -|c_{-k}|^{2}\right) \frac{\Gamma ^{2}(k+1)}{\Gamma ^{2}\left( k+1+\frac{\alpha }{2}\right) }\right] . \end{aligned}$$

(5.1)

The last equality holds because of (1.6).

Particularly, we have

$$\begin{aligned} S_{u}(0)=\pi \sum _{k=1}^{\infty }k\left( |c_{k}|^{2}-|c_{-k}|^{2}\right) . \end{aligned}$$

(5.2)

(1) Recall that the digamma function is defined as $\psi (x)=\Gamma ^{\prime }(x)/\Gamma (x)$. It is well known that (cf. [3]) $\psi (x)$ is strictly increasing on $(0, +\infty )$.

Let

$$\begin{aligned} f(x)=\frac{\Gamma (1+\alpha )\Gamma (x+1)}{\Gamma \left( 1+\frac{\alpha }{2}\right) \Gamma \left( x+1+\frac{\alpha }{2}\right) }. \end{aligned}$$

Then we have

$$\begin{aligned} (\log f(x))^{\prime }&=\psi (x+1)-\psi \left( x+1+\frac{\alpha }{2}\right) . \end{aligned}$$

It follows that $(\log f(x))^{\prime }<0$ provided $\alpha >0$, and $(\log f(x))^{\prime }>0$ provided $\alpha <0$. Observe that

$$\begin{aligned} f(\alpha /2)=1. \end{aligned}$$

Therefore, for $k=1,2,\ldots$, we have $f(k)<1$ if $\alpha \in (0,2)$ as well as $f(k)>1$ if $\alpha \in (-1,0)$. Taking account of (5.1) and (5.2), we can get Theorem 5.1(1).

(2) As to digamma function $\psi (x)$, we have (cf. [3])

$$\begin{aligned} \psi (1+x)&=\frac{1}{x}+\psi (x), \end{aligned}$$

(5.3)

$$\begin{aligned} \psi (x)&=-\gamma +\sum ^{\infty }_{n=0}\left( \frac{1}{n+1}-\frac{1}{n+x}\right) , \end{aligned}$$

(5.4)

and

$$\begin{aligned}&\psi ^{\prime }(x)=\sum ^{\infty }_{n=0}\frac{1}{(x+n)^{2}} \end{aligned}$$

(5.5)

for any $x\in (0, +\infty )$, where $\gamma$ is the Euler–Mascheroni constant.

Let

$$\begin{aligned} h(\alpha )=\psi (1+\alpha )-\psi \left( 1+\frac{\alpha }{2}\right) -\frac{1}{2+\alpha }. \end{aligned}$$

Using (5.4), direct computation or numerical computation lead to

$$\begin{aligned} h(1)&=\psi (2)-\psi (\frac{3}{2})-\frac{1}{3}\\&=-\gamma +\sum ^{\infty }_{n=0}\left( \frac{1}{n+1}-\frac{1}{n+2}\right) - \left[ -\gamma +\sum ^{\infty }_{n=0}\left( \frac{1}{n+1} -\frac{1}{n+3/2}\right) \right] -\frac{1}{3}\\&=2\sum ^{\infty }_{N=1}\left( \frac{1}{2N+1} -\frac{1}{2N+2}\right) -\frac{1}{3}\\&=2\left( \log 2-\frac{1}{2}\right) -\frac{1}{3}>0 \end{aligned}$$

and

$$\begin{aligned} h(0.8)=-0.0108<0. \end{aligned}$$

Furthermore, (5.5) implies that

$$\begin{aligned} h^{\prime }(\alpha )&=\psi ^{\prime }(1+\alpha )-\frac{1}{2}\psi ^{\prime } (1+\frac{\alpha }{2})+\frac{1}{(2+\alpha )^{2}}\\&=\sum ^{\infty }_{n=0}\left( \frac{1}{(1+\alpha +n)^{2}} -\frac{1}{2\left( 1+\frac{\alpha }{2}+n\right) ^{2}}\right) +\frac{1}{(2+\alpha )^{2}}\\&=\sum ^{\infty }_{n=0}\frac{(n+1)^{2} -\frac{\alpha ^{2}}{2}}{2\left( 1+\frac{\alpha }{2}+n\right) ^{2} (1+\alpha +n)^{2}}+\frac{1}{(2+\alpha )^{2}}>0 \end{aligned}$$

for $\alpha \in (-1,1]$. Thus, there exists a unique $\alpha _{0}\in (0.8,1)$, such that $h(\alpha _{0})=0$ and $h(\alpha )<0$ for $\alpha \in (-1,\alpha _{0})$. Let

$$\begin{aligned} g(\alpha )=\frac{\Gamma (1+\alpha )}{\Gamma (1+\frac{\alpha }{2})\Gamma (k+1+\frac{\alpha }{2})}, \quad k=1,2,\ldots . \end{aligned}$$

Then it follows that

$$\begin{aligned} \frac{d \log g(\alpha )}{d \alpha }&=\psi (1+\alpha ) -\frac{1}{2}\psi \left( 1+\frac{\alpha }{2}\right) -\frac{1}{2}\psi \left( k+1+\frac{\alpha }{2}\right) \\&\quad <\psi (1+\alpha ) -\frac{1}{2}\psi \left( 1+\frac{\alpha }{2}\right) -\frac{1}{2}\psi \left( 2+\frac{\alpha }{2}\right) \\&= h(\alpha ). \end{aligned}$$

That is to say $g(\alpha )$ is strictly decreasing on $(-1,\alpha _{0})$. Therefore, (5.1) implies that $S_{u}(\alpha )$ is strictly decreasing with respect to $\alpha \in (-1,\alpha _{0})$.

Let

$$\begin{aligned} u(z)=F\left( -\frac{\alpha }{2}, 1-\frac{\alpha }{2}; 2; |z|^{2}\right) z. \end{aligned}$$

Obviously, it satisfies the conditions of Theorem 5.1. Thus

$$\begin{aligned} S_{u}(\alpha )=\pi \left( \frac{\Gamma (1+\alpha )}{\Gamma \left( 1+\frac{\alpha }{2}\right) \Gamma \left( 2+\frac{\alpha }{2}\right) }\right) ^{2}. \end{aligned}$$

It follows that $\frac{d \log S_{u}(\alpha )}{d \alpha }=h(\alpha ).$ Considering the positivity and negativity of function $h(\alpha )$, we have that $S_{u}(\alpha )$ is strictly decreasing with respect to $\alpha$ in $(-1,\alpha _{0})$ and strictly increasing in $(\alpha _{0},1)$. Therefore, the constant $\alpha _{0}$ is sharp. $\square$