An investigation on a new class of harmonic mappings

Abstract

In the present paper, we give an extension of the idea which was introduced by Sakaguchi (J. Math. Soc. Jpn. 11:72-75, 1959), and we give some applications of this extended idea for the investigation of the class of harmonic mappings.

MSC:30C45, 30C55.

1 Introduction

Let $\mathbb{D}=\{z\mid |z|<1\}$ be the open unit disc in the complex plane ℂ. A complex valued harmonic function $f:\mathbb{D}\to \mathbb{C}$ has the representation

$$f=h(z)+\overline{g(z)},$$

(1.1)

where $h(z)$ and $g(z)$ are analytic in and have the following power series expansions:

$$h(z)=\sum _{n=0}^{\mathrm{\infty }}{a}_n{z}^n,g(z)=\sum _{n=0}^{\mathrm{\infty }}{b}_n{z}^n,z\in \mathbb{D},$$

where $a_n,b_n\in \mathbb{C}$, $n=0,1,2,\dots$ . Choose $g(0)=0$ (i.e., $b_0=0$), so the representation (1.1) is unique in and is called the canonical representation of f in . It is convenient to make further normalization (without loss of generality) $h(0)=0$ (i.e., $a_0=0$) and $h^{\prime }(0)=1$ (i.e., $a_1=1$). The family of such functions f is denoted by ${\mathcal{S}}_{\mathcal{H}}$ [1]. It is known that if f is a sense-preserving harmonic mapping of onto some other region, then by Lewy’s theorem its Jacobian is strictly positive, i.e.,

$$J_{f(z)}={|h^{\prime }(z)|}^2-{|g^{\prime }(z)|}^2>0.$$

Equivalently, the inequality $|g^{\prime }(z)|<|h^{\prime }(z)|$ holds for all $z\in \mathbb{D}$. The family of all functions $f\in {\mathcal{S}}_{\mathcal{H}}$ with the additional property that $g^{\prime }(0)=0$ (i.e., $b_1=0$) is denoted by ${\mathcal{S}}_{\mathcal{H}}^0$ [1]. Observe that the classical family of univalent functions consists of all functions $f\in {\mathcal{S}}_{\mathcal{H}}^0$ such that $g(z)=0$ for all $z\in \mathbb{D}$. Thus, it is clear that $\mathcal{S}\subset {\mathcal{S}}_{\mathcal{H}}^0\subset {\mathcal{S}}_{\mathcal{H}}$.

Let Ω be the family of functions $\varphi (z)$ regular in the open unit disc and satisfy the conditions $\varphi (0)=0$, $|\varphi (z)|<1$ for all $z\in \mathbb{D}$.

Denote by the family of functions $p(z)=1+p_{1}z+p_2{z}^2+\cdots$ regular in such that $p(z)$ in if and only if

$$p(z)=\frac{1+\varphi (z)}{1-\varphi (z)}$$

(1.2)

for some $\varphi (z)\in \mathrm{\Omega }$ and every $z\in \mathbb{D}$.

If $s(z)=z+c_2{z}^2+\cdots$ is regular in the open unit disc and satisfies the condition

$$Re\left(e^{i\alpha }z\frac{s^{\prime }(z)}{s(z)}\right)>0,z\in \mathbb{D}$$

(1.3)

for some real α, $|\alpha |<\pi /2$, then $s(z)$ is said to be an α-spirallike function in [2, 3]. Such functions are known to be univalent [4]. The class of such functions is denoted by ${\mathcal{S}}_{\alpha }^{\ast }$.

Let $F_1(z)=z+{\alpha }_2{z}^2+{\alpha }_3{z}^3+\cdots$ and $F_2(z)=z+{\beta }_2{z}^2+{\beta }_3{z}^3+\cdots$ be analytic functions in . If there exists a function $\varphi (z)\in \mathrm{\Omega }$ such that $F_1(z)=F_2(\varphi (z))$ for every $z\in \mathbb{D}$, then we say that $F_1(z)$ is subordinate to $F_2(z)$ and we write $F_1\prec F_2$. We also note that if $F_1\prec F_2$, then $F_1(\mathbb{D})\subset F_2(\mathbb{D})$ [2, 3]. We also note that $w(z)=g^{\prime }(z)/h^{\prime }(z)$ is the analytic second dilatation of f and $|w(z)|<1$ for every $z\in \mathbb{D}$.

In this paper we investigate the class of harmonic mappings defined by

$${\mathcal{S}}_{\mathcal{H}(\alpha )}=\{f=h(z)+\overline{g(z)}|w(z)\prec \frac{1+z}{1-z},h(z)\in {\mathcal{S}}_{\alpha }^{\ast }\}.$$

For this aim we need the following theorems and lemmas.

Theorem 1.1 [5]

Let $s(z)\in {\mathcal{S}}_{\alpha }^{\ast }$, then

$$rF(cos\alpha ,-r)\le |s(z)|\le rF(cos\alpha ,r)$$

(1.4)

and

$$\begin{array}{r}[(1-r)cos\alpha -(1+r)sin\alpha ]F(cos\alpha ,-r)\\ \le |s^{\prime }(z)|\le [(1+r)cos\alpha +(1-r)sin\alpha ]F(cos\alpha ,r),\end{array}$$

(1.5)

where

$$F(cos\alpha ,r)=\frac{1}{{(1+r)}^{cos\alpha (cos\alpha -1)}{(1-r)}^{cos\alpha (cos\alpha +1)}}$$

(1.6)

for all $|z|=r<1$ and $|\alpha |<\pi /2$.

Lemma 1.2 [6]

Let $\varphi (z)$ be regular in the unit disc with $\varphi (0)=0$, then if $|\varphi (z)|$ attains its maximum value on the circle $|z|=r$ at the point $z_1$, one has $z_1{\varphi }^{\prime }(z_1)=k\varphi (z_1)$ for some $k\ge 1$.

Lemma 1.3 [7]

If $s_1(z)$ and $s_2(z)$ are regular in , $s_1(0)=s_2(0)$, $s_2(z)$ maps onto a many-sheeted region which is starlike with respect to the origin, and $s_1^{\prime }(z)/s_2^{\prime }(z)\in \mathcal{P}$, then $s_1(z)/s_2(z)\in \mathcal{P}$.

Lemma 1.4 [8]

Let $f=h(z)+\overline{g(z)}$ be an element of ${\mathcal{S}}_{\mathcal{HPST}{(\alpha )}^{\ast }}$, then

$$\frac{|b_1|-r}{1-|b_1|r}\le \left|\frac{g^{\prime }(z)}{h^{\prime }(z)}\right|\le \frac{|b_1|+r}{1+|b_1|r}$$

(1.7)

for all $|z|=r<1$. This inequality is sharp because the extremal function is

$$e^{i\alpha }\frac{g^{\prime }(z)}{h^{\prime }(z)}=\frac{z+b}{1+\overline{b}z},$$

where $b=e^{i\alpha }{b}_1$.

2 Main results

Theorem 2.1 $s(z)\in {\mathcal{S}}_{\alpha }^{\ast }$ if and only if

$$z\frac{s^{\prime }(z)}{s(z)}-1\prec \frac{2cos\alpha e^{-i\alpha }z}{1-z}$$

(2.1)

for all $z\in \mathbb{D}$.

Proof Let $s(z)$ in ${\mathcal{S}}_{\alpha }^{\ast }$, then we have

$$e^{i\alpha }z\frac{s^{\prime }(z)}{s(z)}=cos\alpha \frac{1+\varphi (z)}{1-\varphi (z)}+isin\alpha$$

or

$$e^{i\alpha }z\frac{s^{\prime }(z)}{s(z)}=e^{i\alpha }\frac{1+\varphi (z)}{1-\varphi (z)}$$

for some $\varphi (z)\in \mathrm{\Omega }$ and all $z\in \mathbb{D}$. Thus

$$\begin{array}{rl}z\frac{s^{\prime }(z)}{s(z)}-1& =\frac{1+e^{-2i\alpha }\varphi (z)}{1-\varphi (z)}-1\\ =\frac{1+(cos2\alpha -isin2\alpha )\varphi (z)-1+\varphi (z)}{1-\varphi (z)}\\ =\frac{2cos\alpha e^{-i\alpha }\varphi (z)}{1-\varphi (z)}\end{array}$$

for some $\varphi (z)\in \mathrm{\Omega }$ and all $z\in \mathbb{D}$. Since $\varphi (z)\in \mathrm{\Omega }$, we have that (2.1) is true. The sufficient part of the proof can be seen by following the above steps in the opposite direction by considering the subordination principle. □

Theorem 2.2 Let $f=h(z)+\overline{g(z)}$ be an element of ${\mathcal{S}}_{\mathcal{H}(\alpha )}$. If $w(z)=\frac{g^{\prime }(z)}{h^{\prime }(z)}\in \mathcal{P}$, then $\frac{g(z)}{h(z)}\in \mathcal{P}$ for all $z\in \mathbb{D}$.

Proof A version of this theorem was proved by Sakaguchi for a univalent starlike function [7, 9]. Since $f=h(z)+\overline{g(z)}\in {\mathcal{S}}_{\mathcal{H}(\alpha )}$, then $h(z)$ and $g(z)$ are regular in and $h(0)=g(0)=0$. On the other hand, we have

$$w(z)=\frac{g^{\prime }(z)}{h^{\prime }(z)}\in \mathcal{P}\text{if and only if}\frac{g^{\prime }(z)}{h^{\prime }(z)}\prec \frac{1+z}{1+z}$$

(2.2)

for all $z\in \mathbb{D}$. Geometrically, this means that $\frac{g^{\prime }(z)}{h^{\prime }(z)}$ maps inside the open disc centered on the real axis with diameter end points $\frac{1-r}{1+r}$ and $\frac{1+r}{1-r}$. Now we define a function $\varphi (z)$ by

$$\frac{g(z)}{h(z)}=\frac{1+\varphi (z)}{1-\varphi (z)}(z\in \mathbb{D}).$$

(2.3)

Then $\varphi (z)$ is analytic in , and $\varphi (0)=0$. On the other hand,

$$w(z)=\frac{g^{\prime }(z)}{h^{\prime }(z)}=\frac{2z{\varphi }^{\prime }(z)}{1-\varphi (z)}\frac{1}{e^{i\alpha }(1+e^{-2i\alpha })\varphi (z)}+\frac{1+\varphi (z)}{1-\varphi (z)}(z\in \mathbb{D}).$$

(2.4)

Now, it is easy to realize that the subordination (2.2) is equivalent to $|\varphi (z)|<1$ in (2.3) for all $z\in \mathbb{D}$. Indeed, assume to the contrary that there exists $z_1\in \mathbb{D}$ such that $|\varphi (z_1)|=1$. Then by Jack’s lemma (Lemma 1.2), $z_1{\varphi }^{\prime }(z_1)=k\varphi (z_1)$, $k\ge 1$, for such $z_1$ we have

$$w(z_1)=\frac{g^{\prime }(z_1)}{h^{\prime }(z_1)}=\frac{2k\varphi (z_1)}{1-\varphi (z_1)}\frac{1}{e^{i\alpha }(1+e^{-2i\alpha })\varphi (z_1)}+\frac{1+\varphi (z_1)}{1-\varphi (z_1)}=w(\varphi (z_1))\notin w(\mathbb{D}),$$

since $|\varphi (z_1)|=1$ and $k\ge 1$. But this is a contradiction to the condition $w(z)=\frac{g^{\prime }(z)}{h^{\prime }(z)}\prec \frac{1+z}{1-z}$, and so the assumption is wrong, i.e., $|\varphi (z)|<1$ for all $z\in \mathbb{D}$. □

Remark 2.3 Theorem 2.2 is an extension of Lemma 1.3 to the harmonic mappings.

Corollary 2.4 Let $f=h(z)+\overline{g(z)}$ be an element of ${\mathcal{S}}_{\mathcal{H}(\alpha )}$, then

$$\frac{r(1-r)}{1+r}F(cos\alpha ,-r)\le |g(z)|\le \frac{r(1+r)}{1-r}F(cos\alpha ,r)$$

(2.5)

and

$$\begin{array}{r}[(1-r)cos\alpha -(1+r)sin\alpha ]\frac{1-r}{1+r}F(cos\alpha ,-r)\\ \le |g^{\prime }(z)|\le [(1+r)cos\alpha +(1-r)sin\alpha ]\frac{1+r}{1-r}F(cos\alpha ,r),\end{array}$$

(2.6)

where $F(cos\alpha ,r)$ is given by (1.6) for all $|z|=r<1$ and $|\alpha |<\pi /2$.

Proof The proof of this theorem is a simple consequence of Theorem 1.1 and Theorem 2.2 since

$$Rew(z)=Re\frac{g^{\prime }(z)}{h^{\prime }(z)}\Rightarrow \frac{g^{\prime }(z)}{h^{\prime }(z)}\prec \frac{1+z}{1-z}\Rightarrow \frac{1-r}{1+r}\le \left|\frac{g^{\prime }(z)}{h^{\prime }(z)}\right|\le \frac{1+r}{1-r},$$

then

$$|h^{\prime }(z)|\frac{1-r}{1+r}\le |g^{\prime }(z)|\le |h^{\prime }(z)|\frac{1+r}{1-r}$$

and

$$Re\frac{g(z)}{h(z)}>0\Rightarrow \frac{g(z)}{h(z)}\prec \frac{1+z}{1-z}\Rightarrow \frac{1-r}{1+r}\le \left|\frac{g(z)}{h(z)}\right|\le \frac{1+r}{1-r},$$

then

$$|h(z)|\frac{1-r}{1+r}\le |g(z)|\le |h(z)|\frac{1+r}{1-r}.$$

 □

Corollary 2.5 Let $f=h(z)+\overline{g(z)}$ be an element of ${\mathcal{S}}_{\mathcal{H}(\alpha )}$, then

$$\begin{array}{r}\frac{{[(1-r)cos\alpha -(1+r)sin\alpha ]}^2{(F(cos\alpha ,-r))}^2(1-r^2)(1-{|b_1|}^2)}{{(1+|b_1|r)}^2}\\ \le J_{f(z)}\le \frac{{[(1+r)cos\alpha +(1-r)sin\alpha ]}^2{(F(cos\alpha ,r))}^2(1-r^2)(1-{|b_1|}^2)}{{(1-|b_1|r)}^2},\end{array}$$

where $F(cos\alpha ,r)$ is given by (1.6) for all $|z|=r<1$, $|\alpha |<\pi /2$ and $J_{f(z)}$ is the Jacobian of f defined by $J_{f(z)}={|h^{\prime }(z)|}^2-{|g^{\prime }(z)|}^2$ for all $z\in \mathbb{D}$.

Proof The proof of this corollary is a simple consequence of Lemma 1.1 and Lemma 1.4. □