Several properties of a class of generalized harmonic mappings

Bo-Yong Long Qi-Han Wang

School of Mathematical Sciences, Anhui University, Hefei 230601, China

^†^†footnotetext: Supported by NSFC (No.12271001) and Natural Science Foundation of Anhui Province (2308085MA03), China. E-mail: [email protected] [email protected]

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.
Keywords: Weighted Laplacian operator; univalency; Polyharmonic mappings; Lipschitz continuity; Gauss hypergeometric function
2010 Mathematics Subject Classification: Primary 30C45 Secondary 33C05, 30B10

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

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

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

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

The corresponding partial differential equation is

\displaystyle T_{\alpha}(u)=0\quad\mbox{in}\,\mathbb{D}.

(1.1)

The associated Dirichlet boundary value problem is

\;\left\{\begin{array}[]{rr}T_{\alpha}(u)=0&\quad\mbox{in}\,\mathbb{D},\\ u=u^{*}&\quad\mbox{on}\,\mathbb{T}.\end{array}\right.

(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

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

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

u(z)=\frac{1}{2\pi}\int_{0}^{2\pi}K_{\alpha}(ze^{-i\tau)})u^{*}(e^{i\tau})d\tau,\quad\mbox{for}\,\,z\in\mathbb{D},

(1.3)

where

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

(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 $\alpha\leq-1$ , $u\in\mathcal{C}^{2}(\mathbb{D})$ 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 $c\neq 0,\,-1,\,-2,...$ , the Gauss hypergeometric function is defined by the series

F(a,b;c;x)=\sum_{n=0}^{\infty}\frac{(a)_{n}(b)_{n}}{(c)_{n}}\frac{x^{n}}{n!}

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

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

(1.5)

Furthermore, it holds that (cf.[3] )

\lim_{x\rightarrow 1}F(a,b;c;x)=\frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}

(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 $c>0$ , $a\leq c$ , $b\leq c$ and $ab\leq 0$ $(ab\geq 0)$ . 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

u(z)=\sum_{k=0}^{\infty}c_{k}F(-\frac{\alpha}{2},k-\frac{\alpha}{2};k+1;|z|^{2})z^{k}+\sum_{k=1}^{\infty}c_{-k}F(-\frac{\alpha}{2},k-\frac{\alpha}{2};k+1;|z|^{2})\bar{z}^{k},\quad z\in\mathbb{D},

(1.7)

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

\lim_{|k|\rightarrow\infty}\sup|c_{k}|^{\frac{1}{|k|}}\leq 1.

(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

v(z)=\sum_{k=0}^{\infty}c_{k}z^{k}+\sum_{k=1}^{\infty}c_{-k}\bar{z}^{k},\quad z\in\mathbb{D}.

(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 $p\in\{1,2,...\}$ (cf.[1, 6, 13, 5, 27, 11, 2, 15]). 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 [25], we know that it is related to standard weighted harmonic mappings. For the related discussion on standard weighted harmonic mappings, see [8, 17, 16, 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 section 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 section 3, we show that the Lipschitz continuity of a real kernel $\alpha-$ harmonic mapping is determined by the corresponding harmonic mapping. In section 4, for a subclass of the real kernel $\alpha-$ harmonic mappings, we discuss its univalency and explore its Radó-Kneser-Choquet type theorem. In section 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

\displaystyle u(z)=\sum_{n=0}^{p-1}|z|^{2n}\frac{(1-p)_{n}}{n!}(I_{n}+\overline{J_{n}}),

(2.1)

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

		$\displaystyle I_{n}=I_{n-1}-p\frac{\int_{0}^{z}z^{n-1}I_{n-1}dz}{z^{n}},$		(2.2)
		$\displaystyle J_{n}=J_{n-1}-p\frac{\int_{0}^{z}z^{n-1}J_{n-1}dz}{z^{n}}\qquad n=1,2,...,p-1,$		(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

\displaystyle u(z)=H(z)+\overline{G(z)}.

(2.4)

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

		$\displaystyle H(z)=\sum_{k=0}^{\infty}c_{k}F(1-p,k+1-p;k+1;\|z\|^{2})z^{k}$
		$\displaystyle\qquad\,=\sum_{k=0}^{\infty}c_{k}z^{k}\left(\sum_{n=0}^{\infty}\frac{(1-p)_{n}(k+1-p)_{n}}{(k+1)_{n}}\frac{\|z\|^{2n}}{n!}\right)$
		$\displaystyle\qquad\,=\sum_{k=0}^{\infty}c_{k}z^{k}\left(\sum_{n=0}^{p-1}\frac{(1-p)_{n}(k+1-p)_{n}}{(k+1)_{n}}\frac{\|z\|^{2n}}{n!}\right)$
		$\displaystyle\qquad\,=\sum_{n=0}^{p-1}\|z\|^{2n}\frac{(1-p)_{n}}{n!}I_{n},$		(2.5)

where

I_{n}=\sum_{k=0}^{\infty}\frac{(k+1-p)_{n}}{(k+1)_{n}}c_{k}z^{k}.

Because

	$\displaystyle\frac{(k+1-p)_{n}}{(k+1)_{n}}=\frac{(k+1-p)_{n-1}(k+n-p)}{(k+1)_{n-1}(k+n)}$
	$\displaystyle\qquad\qquad\quad\,\,\,=\frac{(k+1-p)_{n-1}}{(k+1)_{n-1}}-\frac{p}{k+n}\frac{(k+1-p)_{n-1}}{(k+1)_{n-1}},$

we can get

	$\displaystyle I_{n}=\sum_{k=0}^{\infty}\frac{(k+1-p)_{n-1}}{(k+1)_{n-1}}c_{k}z^{k}-\sum_{k=0}^{\infty}\frac{p}{k+n}\frac{(k+1-p)_{n-1}}{(k+1)_{n-1}}c_{k}z^{k}$
	$\displaystyle\quad=I_{n-1}-p\frac{\int_{0}^{z}z^{n-1}I_{n-1}dz}{z^{n}}.$

This is (2.2).

Similarly, we can get

\displaystyle G(z)=\sum_{n=0}^{p-1}|z|^{2n}\frac{(1-p)_{n}}{n!}J_{n},

(2.6)

where $J_{n}$ is defined as in (2.3). Therefore, equation (2.1) follows from equations (2.4)-(2.6). ∎

Example 2.1.

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

(i) When $\alpha=0$ , i.e. $p=1$ ,

\displaystyle u(z)=v(z);

(ii) When $\alpha=2$ , i.e. $p=2$ ,

		$\displaystyle u(z)=h+\bar{g}-\|z\|^{2}\left(h-2\frac{\int_{0}^{z}h(z)dz}{z}+\overline{g-2\frac{\int_{0}^{z}g(z)dz}{z}}\right)$
		$\displaystyle\qquad=\sum_{k=0}^{\infty}c_{k}z^{k}+\sum_{k=1}^{\infty}c_{-k}\bar{z}^{k}-\|z\|^{2}\left(\sum_{k=0}^{\infty}c_{k}\frac{k-1}{k+1}z^{k}+\sum_{k=1}^{\infty}c_{-k}\frac{k-1}{k+1}\bar{z}^{k}\right);$		(2.7)

(iii) When $\alpha=4$ , i.e. $p=3$ ,

	$\displaystyle u(z)=h+\bar{g}-2\|z\|^{2}\left(h-3\frac{\int_{0}^{z}h(z)dz}{z}+\overline{g-3\frac{\int_{0}^{z}g(z)dz}{z}}\right)$
	$\displaystyle\qquad\quad+\|z\|^{4}\left(h-3\frac{\int_{0}^{z}h(z)dz}{z}-3\frac{\int_{0}^{z}zh(z)dz}{z^{2}}+9\frac{\int_{0}^{z}\int_{0}^{z}h(z)dzdz}{z^{2}}\right.$
	$\displaystyle\qquad\quad\left.+\overline{g-3\frac{\int_{0}^{z}g(z)dz}{z}-3\frac{\int_{0}^{z}zg(z)dz}{z^{2}}+9\frac{\int_{0}^{z}\int_{0}^{z}g(z)dzdz}{z^{2}}}\right)$
	$\displaystyle\qquad=\sum_{k=0}^{\infty}c_{k}z^{k}+\sum_{k=1}^{\infty}c_{-k}\bar{z}^{k}-2\|z\|^{2}\left(\sum_{k=0}^{\infty}c_{k}\frac{k-2}{k+1}z^{k}+\sum_{k=1}^{\infty}c_{-k}\frac{k-2}{k+1}\bar{z}^{k}\right)$
	$\displaystyle\quad\qquad+\|z\|^{4}\left(\sum_{k=0}^{\infty}c_{k}\frac{(k-1)(k-2)}{(k+1)(k+2)}z^{k}+\sum_{k=1}^{\infty}c_{-k}\frac{(k-1)(k-2)}{(k+1)(k+2)}\bar{z}^{k}\right).$

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 $n=0,1,2,...,p-1$ . 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

\displaystyle 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}},\qquad n=1,2,...,p-1.

(3.1)

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

\displaystyle\sup_{z\in\mathbb{D}}|I_{n}|\leq\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}|

and

\displaystyle\sup_{z\in\mathbb{D}}|I^{\prime}_{n}|\leq\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}|,

respectively. It follows that

\displaystyle\sup_{z\in\mathbb{D}}|I_{n}|\leq(p+1)^{n}\sup_{z\in\mathbb{D}}|I_{0}|

and

	$\displaystyle\sup_{z\in\mathbb{D}}\|I^{\prime}_{n}\|\leq\sup_{z\in\mathbb{D}}\|I^{\prime}_{n-1}\|+(n+1)p\sup_{z\in\mathbb{D}}\|I_{n-1}\|$
	$\displaystyle\quad\qquad\leq\sup_{z\in\mathbb{D}}\|I^{\prime}_{n-2}\|+np\sup_{z\in\mathbb{D}}\|I_{n-2}\|+(n+1)p\sup_{z\in\mathbb{D}}\|I_{n-1}\|$
	$\displaystyle\quad\qquad\leq...$
	$\displaystyle\quad\qquad\leq\sup_{z\in\mathbb{D}}\|I^{\prime}_{0}\|+p\sum_{i=1}^{n}(i+1)\sup_{z\in\mathbb{D}}\|I_{i-1}\|$
	$\displaystyle\quad\qquad\leq\sup_{z\in\mathbb{D}}\|I^{\prime}_{0}\|+p\sum_{i=1}^{n}(i+1)(p+1)^{i-1}\sup_{z\in\mathbb{D}}\|I_{0}\|.$		(3.2)

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

\displaystyle|h^{\prime}|=|I^{\prime}_{0}|\leq M

(3.3)

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

\displaystyle\sup_{z\in\mathbb{D}}|I_{0}|=\sup_{z\in\mathbb{D}}|h|\leq M.

(3.4)

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

\displaystyle\sup_{z\in\mathbb{D}}|I^{\prime}_{n}|\leq(1+p\sum_{i=1}^{n}(i+1)(p+1)^{i-1})M=:C

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

∎

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

	$\displaystyle t=\|z\|^{2}=r^{2},$
	$\displaystyle F=F_{k}=F(-\frac{\alpha}{2},k-\frac{\alpha}{2};k+1;t),\quad k=1,2,....,$
	$\displaystyle F_{t}=F_{k,t}=F_{k,t}(-\frac{\alpha}{2},k-\frac{\alpha}{2};k+1;t)=\frac{dF_{k}}{dt}=\frac{dF}{dt}.$

Furthermore, let

\displaystyle F_{k}(1)=\lim_{t\rightarrow 1^{-}}F(-\frac{\alpha}{2},k-\frac{\alpha}{2};k+1;t).

Then by (1.6), we have

\displaystyle F_{k}(1)=\frac{\Gamma(k+1)\Gamma(1+\alpha)}{\Gamma(k+1+\frac{\alpha}{2})\Gamma(1+\frac{\alpha}{2})}.

(4.1)

Lemma 4.1.

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

R(x)=\sum_{n=0}^{\infty}r_{n}x^{n}\quad\mbox{and}\quad S(x)=\sum_{n=0}^{\infty}s_{n}x^{n}

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

	$\displaystyle(1)\quad\frac{F_{k}}{F_{1}}\leq 1\,\,\mbox{for}\,\,k=1,2,3,...\,\,\mbox{and}\,\,t\in[0,1);$
	$\displaystyle(2)\quad\frac{\|F_{k,t}\|}{F_{1}}<\frac{(k-\frac{\alpha}{2})\Gamma(k+1)\Gamma(2+\frac{\alpha}{2})}{2\Gamma(k+1+\frac{\alpha}{2})}\,\,\mbox{for}\,\,k=1,2,3,...\mbox{and}\,\,t\in(0,1).$

Theorem 4.3.

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

\displaystyle N=\frac{\alpha}{2\left(\frac{(k-\frac{\alpha}{2})\Gamma(k+1)\Gamma(2+\frac{\alpha}{2})}{\Gamma(k+1+\frac{\alpha}{2})}+k\right)},

(4.2)

then the real kernel $\alpha-$ harmonic mapping

\displaystyle u(z)=F_{1}z+c_{-k}F_{k}\overline{z}^{k},\quad k=1,2,3...,

(4.3)

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

Proof.

We divide the proof into two steps.

First step : Formula (4.3) implies that

\displaystyle 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}(F_{k,t}z\bar{z}^{k}+kF_{k}\bar{z}^{k-1}).

It follows that

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

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

	$\displaystyle u(z)=F_{1}re^{i\theta}+\|c_{-k}\|F_{k}r^{k}e^{i(\beta-k\theta)}$
	$\displaystyle\qquad=F_{1}r\cos\theta+\|c_{-k}\|F_{k}r^{k}\cos(\beta-k\theta)+i(F_{1}r\sin\theta+\|c_{-k}\|F_{k}r^{k}\sin(\beta-k\theta)).$		(4.4)

Then

\displaystyle\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)},

(4.5)

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

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

\displaystyle\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}.

(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

\displaystyle\left|\frac{\sin k\theta}{\sin\theta}\right|\leq k

(4.8)

for $k=1,2,...$ and $\theta\in[0,2\pi)$ . If $|c_{-k}|<\frac{1}{k}$ , $\alpha\in(0,2]$ and $\sin\theta\neq 0$ , 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, $\sin\theta\neq 0$ implies $\sin\theta-|c_{-k}|\frac{F_{k}}{F_{1}}r^{k-1}\sin k\theta\neq 0$ . 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

\displaystyle\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}.

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 $\mathbb{D}$ .

∎

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

\displaystyle u(z)=\frac{1}{2\pi}\int_{0}^{2\pi}\frac{1-|z|^{2}}{|e^{it}-z|^{2}}f(e^{it})dt

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

\displaystyle f(e^{i\theta})=F_{1}(1)e^{i\theta}+c_{-k}F_{k}(1)e^{-ik\theta},\quad k=1,2,3,....

(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

\displaystyle M=\frac{\Gamma(k+1+\frac{\alpha}{2})}{k^{2}\Gamma(k+1)\Gamma(2+\frac{\alpha}{2})}.

(4.10)

Proof.

Direct computation leads to

\displaystyle\frac{d}{d\theta}(f(e^{i\theta}))=-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).

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

	$\displaystyle\frac{d}{d\theta}(\tan\psi(\theta))$	$\displaystyle=\frac{(F_{1}(1))^{2}-k^{3}(c_{-k}F_{k}(1))^{2}+k(k-1)c_{-k}F_{1}(1)F_{k}(1)\cos((k+1)\theta)}{(F_{1}(1)\sin\theta+kc_{-k}F_{k}(1)\sin k\theta)^{2}}$
		$\displaystyle\geq\frac{(F_{1}(1))^{2}-k^{3}(c_{-k}F_{k}(1))^{2}-k(k-1)\|c_{-k}\|F_{1}(1)F_{k}(1))}{(F_{1}(1)\sin\theta+kc_{-k}F_{k}(1)\sin k\theta)^{2}}$
		$\displaystyle=\frac{(F_{1}(1)+k\|c_{-k}\|F_{k}(1))(F_{1}(1)-k^{2}\|c_{-k}\|F_{k}(1))}{(F_{1}(1)\sin\theta+kc_{-k}F_{k}(1)\sin k\theta)^{2}}$

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

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 $k=1,2,3,...$ , $\alpha\in(0,2]$ , $c_{-k}\in(-L,L)$ , 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

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

Furthermore, let $u^{*}(e^{i\theta})=f(e^{i\theta})$ . Then (1.3), or (2.1), implies that the corresponding real kernel $\alpha$ -harmonic mapping is

\displaystyle 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}.

(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 $k=3,4,5,...$ , 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 $S_{u}$ denote the area of the Riemann surface of $u$ . 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 $|c_{k}|\geq|c_{-k}|$ for $k=1,2,...$ , then

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

(2) $S_{u}(\alpha)$ is strictly decreasing with respect to $\alpha\in(-1,0.8)$ , where $S_{u}(\alpha)=S_{u}$ .

Proof.

By (1.7), direct computation leads to

	$\displaystyle u_{z}$	$\displaystyle=\sum_{k=1}^{\infty}c_{k}[F_{t}\bar{z}z^{k}+kFz^{k-1}]+\sum_{k=1}^{\infty}c_{-k}F_{t}\bar{z}^{k+1}$
		$\displaystyle=\sum_{k=1}^{\infty}c_{k}[F_{t}r^{k+1}+kFr^{k-1}]e^{i(k-1)\theta}+\sum_{k=1}^{\infty}c_{-k}F_{t}r^{k+1}e^{-i(k+1)\theta}$

and

	$\displaystyle u_{\bar{z}}$	$\displaystyle=\sum_{k=1}^{\infty}c_{k}F_{t}z^{k+1}+\sum_{k=1}^{\infty}c_{-k}[F_{t}z\bar{z}^{k}+kF\bar{z}^{k-1}]$
		$\displaystyle=\sum_{k=1}^{\infty}c_{-k}[F_{t}r^{k+1}+kFr^{k-1}]e^{-i(k-1)\theta}+\sum_{k=1}^{\infty}c_{k}F_{t}r^{k+1}e^{i(k+1)\theta}.$

So,

$\displaystyle S_{u}$	$\displaystyle=\int_{0}^{2\pi}\int_{0}^{1}J_{u}(z)rdrd\theta$
	$\displaystyle=2\pi\int_{0}^{1}\sum_{k=1}^{\infty}\left[\|c_{k}(F_{t}r^{k+1}+kFr^{k-1})\|^{2}+\|c_{-k}F_{t}r^{k+1}\|^{2}-\|c_{-k}(F_{t}r^{k+1}+kFr^{k-1})\|^{2}-\|c_{k}F_{t}r^{k+1}\|^{2}\right]rdr$
	$\displaystyle=2\pi\int_{0}^{1}\left[\sum_{k=1}^{\infty}(\|c_{k}\|^{2}-\|c_{-k}\|^{2})(k^{2}F^{2}r^{2k-1}+2kFF_{t}r^{2k+1})\right]dr$
	$\displaystyle=2\pi\sum_{k=1}^{\infty}\left[(\|c_{k}\|^{2}-\|c_{-k}\|^{2})k\int_{0}^{1}(kF^{2}r^{2k-1}+2FF_{t}r^{2k+1})dr\right]$
	$\displaystyle=\pi\sum_{k=1}^{\infty}\left[(\|c_{k}\|^{2}-\|c_{-k}\|^{2})k\int_{0}^{1}d(F^{2}r^{2k})\right]$
	$\displaystyle=\pi\frac{\Gamma^{2}(1+\alpha)}{\Gamma^{2}(1+\frac{\alpha}{2})}\sum_{k=1}^{\infty}\left[k(\|c_{k}\|^{2}-\|c_{-k}\|^{2})\frac{\Gamma^{2}(k+1)}{\Gamma^{2}(k+1+\frac{\alpha}{2})}\right].$	(5.1)

The last equality holds because of (1.6).

Similarly, we have

\displaystyle S_{v}=\pi\sum_{k=1}^{\infty}k(|c_{k}|^{2}-|c_{-k}|^{2}).

(5.2)

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

Let

\displaystyle f(x)=\frac{\Gamma(1+\alpha)\Gamma(x+1)}{\Gamma(1+\frac{\alpha}{2})\Gamma(x+1+\frac{\alpha}{2})}.

Then we have

\displaystyle(\log f(x))^{\prime}

\displaystyle=\psi(x+1)-\psi(x+1+\frac{\alpha}{2}).

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

\displaystyle f(\alpha/2)=1.

Therefore, for $k=1,2,...$ , 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])

		$\displaystyle\psi(1+x)=\frac{1}{x}+\psi(x),$		(5.3)
		$\displaystyle\psi(x)=-\gamma+\sum^{\infty}_{n=0}\left(\frac{1}{n+1}-\frac{1}{n+x}\right),$		(5.4)

and

\displaystyle\psi^{\prime}(x)=\sum^{\infty}_{n=0}\frac{1}{(x+n)^{2}}

(5.5)

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

Let

h(\alpha)=\psi(1+\alpha)-\psi(1+\frac{\alpha}{2})-\frac{1}{2+\alpha}.

Then (5.5) implies that

	$\displaystyle h^{\prime}(\alpha)=\psi^{\prime}(1+\alpha)-\frac{1}{2}\psi^{\prime}(1+\frac{\alpha}{2})+\frac{1}{(2+\alpha)^{2}}$
	$\displaystyle\qquad\,=\sum^{\infty}_{n=0}\left(\frac{1}{(1+\alpha+n)^{2}}-\frac{1}{2(1+\frac{\alpha}{2}+n)^{2}}\right)+\frac{1}{(2+\alpha)^{2}}$
	$\displaystyle\qquad\,=\sum^{\infty}_{n=0}\frac{(n+1)^{2}-\frac{\alpha^{2}}{2}}{2(1+\frac{\alpha}{2}+n)^{2}(1+\alpha+n)^{2}}+\frac{1}{(2+\alpha)^{2}}$
	$\displaystyle\qquad\,>0$

for $\alpha\in(-1,1)$ . Furthermore, using (5.4), direct numerical computation shows

\displaystyle h(0.8)=-0.0108<0.

Thus, $h(\alpha)<0$ for $\alpha\in(-1,0.8)$ . Let

g(\alpha)=\frac{\Gamma(1+\alpha)}{\Gamma(1+\frac{\alpha}{2})\Gamma(k+1+\frac{\alpha}{2})},\quad k=1,2,....

Then it follows that

	$\displaystyle\frac{d\log g(\alpha)}{d\alpha}=\psi(1+\alpha)-\frac{1}{2}\psi(1+\frac{\alpha}{2})-\frac{1}{2}\psi(k+1+\frac{\alpha}{2})$
	$\displaystyle\qquad\qquad\,<\psi(1+\alpha)-\frac{1}{2}\psi(1+\frac{\alpha}{2})-\frac{1}{2}\psi(2+\frac{\alpha}{2})$
	$\displaystyle\qquad\qquad\,=h(\alpha).$

That is to say $g(\alpha)$ is strictly decreasing on $(-1,0.8)$ . Therefore, (5.1) implies that $S_{u}(\alpha)$ is strictly decreasing with respect to $\alpha\in(-1,0.8)$ .

∎

Acknowledgements The authors heartily thank the anonymous reviewers for their careful review and for their effective suggestions.

Declarations

Conflict of interests The authors declare that they have no conflict of interest.

Data availability statement My manuscript has no associated date.

References

[1] Z. Abdulhadi and Y. Abu Muhanna. Landau’s theorem for biharmonic mappings. J. Math. Anal. Appl., 338(1):705–709, 2008.
[2] K. F. Amozova, E. G. Ganenkova, and S. Ponnusamy. Criteria of univalence and fully $\alpha$ -accessibility for $p$ -harmonic and $p$ -analytic functions. Complex Var. Elliptic Equ., 62(8): 1165–1183, 2017.
[3] G. E. Andrews, R. Askey, and R. Roy. Special functions, volume 71 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1999.
[4] M. Biernacki and J. Krzyż On the monotonicity of certain functionals in the theory of analytic functions. Ann. Univ. M. Curie-Skłodowska, 9: 134-145, 1955.
[5] J.-L. Chen, A. Rasila, and X.-T. Wang. Coefficient estimates and radii problems for certain classes of polyharmonic mappings. Complex Var. Elliptic Equ., 60(3):354–371, 2015.
[6] S.-L. Chen, S. Ponnusamy, and X.-T. Wang. Bloch constant and Landau’s theorem for planar $p$ -harmonic mappings. J. Math. Anal. Appl., 373(1):102–110, 2011.
[7] S.-L. Chen and M. Vuorinen. Some properties of a class of elliptic partial differential operators. J. Math. Anal. Appl., 431(2):1124–1137, 2015.
[8] X.-D. Chen and D. Kalaj. A representation theorem for standard weighted harmonic mappings with an integer exponent and its applications. J. Math. Anal. Appl., 444(2):1233–1241, 2016.
[9] M. Cristea. A generalization of the argument principle. Complex Variables Theory Appl., 42(4):333–345, 2000.
[10] P. Duren. Harmonic mappings in the plane, volume 156 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2004.
[11] L. El Hajj. On the univalence of polyharmonic mappings. J. Math. Anal. Appl., 452(2):871–882, 2017.
[12] A. Khalfallah, F. Haggui, and M. Mhamdi. Generalized harmonic functions and Schwarz lemma for biharmonic mappings. Monatsh. Math., 196(4): 823–849, 2021.
[13] P.-J. Li, S. A. Khuri, and X.-T. Wang. On certain geometric properties of polyharmonic mappings. J. Math. Anal. Appl., 434(2):1462–1473, 2016.
[14] P.-J. Li and S. Ponnusamy. Lipschitz continuity of quasiconformal mappings and of the solutions to second order elliptic PDE with respect to the distance ratio metric. Complex Anal. Oper. Theory, 12(8):1991–2001, 2018.
[15] P.-J. Li, S. Ponnusamy, and X.-T. Wang. Some properties of planar $p$ -harmonic and $\log$ - $p$ -harmonic mappings. Bull. Malays. Math. Sci. Soc., 36(3): 595–609, 2013.
[16] P.-J. Li and X.-T. Wang. Lipschitz continuity of $\alpha$ -harmonic functions. Hokkaido Math. J., 48(1):85–97, 2019.
[17] P.-J. Li, X.-T. Wang, and Q.-H. Xiao. Several properties of $\alpha$ -harmonic functions in the unit disk. Monatsh. Math., 184(4):627–640, 2017.
[18] M.-S. Liu and H.-H. Chen. The Landau-Bloch type theorems for planar harmonic mappings with bounded dilation. J. Math. Anal. Appl., 468(2):1066–1081, 2018.
[19] Z.-H. Liu, Y.-P. Jiang, and Y. Sun. Convolutions of harmonic half-plane mappings with harmonic vertical strip mappings. Filomat, 31(7):1843–1856, 2017.
[20] B.-Y. Long, T. Sugawa, and Q.-H. Wang. Completely monotone sequences and harmonic mappings. Ann. Fenn. Math., 47(1): 237–250, 2022.
[21] B.-Y. Long and Q.-H. Wang. Some coefficient estimates on real kernel $\alpha-$ harmonic mappings. Proc. Amer. Math. Soc., 150(4): 1529–1540, 2022.
[22] B.-Y. Long and Q.-H. Wang. Starlikeness, convexity and Landau type theorem of the real kernel $\alpha-$ harmonic mappings. Filomat, 35(8): 2629–2644, 2021.
[23] B.-Y Long and Q.-H Wang. Some geometric properties of complex-valued kernel $\alpha$ -harmonic mappings. Bull. Malays. Math. Sci. Soc., 44(4):2381–2399, 2021.
[24] A. Olofsson. Differential operators for a scale of Poisson type kernels in the unit disc. J. Anal. Math., 123: 227–249, 2014.
[25] A. Olofsson and J. Wittsten. Poisson integrals for standard weighted Laplacians in the unit disc. J. Math. Soc. Japan, 65(2):447–486, 2013.
[26] S. Ponnusamy and M. Vuorinen. Asymptotic expansions and inequalities for hypergeometric functions. Mathematika 44(2): 278–301, 1997.
[27] J.-J. Qiao. Univalent harmonic and biharmonic mappings with integer coefficients in complex quadratic fields. Bull. Malays. Math. Sci. Soc., 39(4):1637–1646, 2016.
[28] Z.-H. Yang, Y.-M. Chu, and M.-K. Wang. Monotonicity criterion for the quotient of power series with applications. J. Math. Anal. Appl., 428(1):587–604, 2015.

		$\displaystyle H(z)=\sum_{k=0}^{\infty}c_{k}F(1-p,k+1-p;k+1;\|z\|^{2})z^{k}$
		$\displaystyle\qquad\,=\sum_{k=0}^{\infty}c_{k}z^{k}\left(\sum_{n=0}^{\infty}\frac{(1-p)_{n}(k+1-p)_{n}}{(k+1)_{n}}\frac{\|z\|^{2n}}{n!}\right)$
		$\displaystyle\qquad\,=\sum_{k=0}^{\infty}c_{k}z^{k}\left(\sum_{n=0}^{p-1}\frac{(1-p)_{n}(k+1-p)_{n}}{(k+1)_{n}}\frac{\|z\|^{2n}}{n!}\right)$
		$\displaystyle\qquad\,=\sum_{n=0}^{p-1}\|z\|^{2n}\frac{(1-p)_{n}}{n!}I_{n},$		(2.5)

	$\displaystyle u(z)=h+\bar{g}-2\|z\|^{2}\left(h-3\frac{\int_{0}^{z}h(z)dz}{z}+\overline{g-3\frac{\int_{0}^{z}g(z)dz}{z}}\right)$
	$\displaystyle\qquad\quad+\|z\|^{4}\left(h-3\frac{\int_{0}^{z}h(z)dz}{z}-3\frac{\int_{0}^{z}zh(z)dz}{z^{2}}+9\frac{\int_{0}^{z}\int_{0}^{z}h(z)dzdz}{z^{2}}\right.$
	$\displaystyle\qquad\quad\left.+\overline{g-3\frac{\int_{0}^{z}g(z)dz}{z}-3\frac{\int_{0}^{z}zg(z)dz}{z^{2}}+9\frac{\int_{0}^{z}\int_{0}^{z}g(z)dzdz}{z^{2}}}\right)$
	$\displaystyle\qquad=\sum_{k=0}^{\infty}c_{k}z^{k}+\sum_{k=1}^{\infty}c_{-k}\bar{z}^{k}-2\|z\|^{2}\left(\sum_{k=0}^{\infty}c_{k}\frac{k-2}{k+1}z^{k}+\sum_{k=1}^{\infty}c_{-k}\frac{k-2}{k+1}\bar{z}^{k}\right)$
	$\displaystyle\quad\qquad+\|z\|^{4}\left(\sum_{k=0}^{\infty}c_{k}\frac{(k-1)(k-2)}{(k+1)(k+2)}z^{k}+\sum_{k=1}^{\infty}c_{-k}\frac{(k-1)(k-2)}{(k+1)(k+2)}\bar{z}^{k}\right).$

	$\displaystyle\sup_{z\in\mathbb{D}}\|I^{\prime}_{n}\|\leq\sup_{z\in\mathbb{D}}\|I^{\prime}_{n-1}\|+(n+1)p\sup_{z\in\mathbb{D}}\|I_{n-1}\|$
	$\displaystyle\quad\qquad\leq\sup_{z\in\mathbb{D}}\|I^{\prime}_{n-2}\|+np\sup_{z\in\mathbb{D}}\|I_{n-2}\|+(n+1)p\sup_{z\in\mathbb{D}}\|I_{n-1}\|$
	$\displaystyle\quad\qquad\leq...$
	$\displaystyle\quad\qquad\leq\sup_{z\in\mathbb{D}}\|I^{\prime}_{0}\|+p\sum_{i=1}^{n}(i+1)\sup_{z\in\mathbb{D}}\|I_{i-1}\|$
	$\displaystyle\quad\qquad\leq\sup_{z\in\mathbb{D}}\|I^{\prime}_{0}\|+p\sum_{i=1}^{n}(i+1)(p+1)^{i-1}\sup_{z\in\mathbb{D}}\|I_{0}\|.$		(3.2)

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

$\displaystyle\frac{d}{d\theta}(\tan\varphi)$	$\displaystyle=\frac{d}{d\theta}\left(\frac{F_{1}r\sin\theta+\|c_{-k}\|F_{k}r^{k}\sin(\beta-k\theta)}{F_{1}r\cos\theta+\|c_{-k}\|F_{k}r^{k}\cos(\beta-k\theta)}\right)$
	$\displaystyle=\frac{F_{1}^{2}-\|c_{-k}\|^{2}F_{k}^{2}r^{2(k-1)}k-(k-1)\|c_{-k}\|F_{1}F_{k}r^{k-1}\cos(\beta-(k+1)\theta)}{[F_{1}\cos\theta+\|c_{-k}\|F_{k}r^{k-1}\cos(\beta-k\theta)]^{2}}$
	$\displaystyle\geq\frac{F_{1}^{2}-\|c_{-k}\|^{2}F_{k}^{2}r^{2(k-1)}k-(k-1)\|c_{-k}\|F_{1}F_{k}r^{k-1}}{[F_{1}\cos\theta+\|c_{-k}\|F_{k}r^{k-1}\cos(\beta-k\theta)]^{2}}$
	$\displaystyle=\frac{(F_{1}+\|c_{-k}\|F_{k}r^{k-1})(F_{1}-\|c_{-k}\|kF_{k}r^{k-1})}{[F_{1}\cos\theta+\|c_{-k}\|F_{k}r^{k-1}\cos(\beta-k\theta)]^{2}}$
	$\displaystyle>0$	(4.6)

Several properties of a class of generalized harmonic mappings

1 Introduction

Lemma 1.1.

Theorem 1.2.

2 Representation theorem

Theorem 2.1.

Proof.

Example 2.1.

3 Lipschitz continuity

Theorem 3.1.

Proof.

4 Univalency of a subclass of real kernel α−\alpha-harmonic mappings

Lemma 4.1.

Lemma 4.2.

Theorem 4.3.

Proof.

Theorem 4.4.

Proposition 4.5.

Proof.

Proposition 4.6.

Example 4.1.

5 Area SuS_{u}

Theorem 5.1.

Proof.

References

4 Univalency of a subclass of real kernel $\alpha-$ harmonic mappings

5 Area $S_{u}$