# Harmonic function for which the second dilatation is α-spiral

## Abstract

Let $f=h+\overline{g}$ be a harmonic function in the unit disc $\mathbb{D}$. We will give some properties of f under the condition the second dilatation is α-spiral.

MSC:30C45, 30C55.

## 1 Introduction

A planar harmonic mapping in the unit disc $\mathbb{D}=\left\{z\in \mathbb{C}||z|<1\right\}$ is a complex-valued harmonic function f which maps $\mathbb{D}$ onto some planar domain $f\left(\mathbb{D}\right)$. Since $\mathbb{D}$ is simply connected, the mapping f has a canonical decomposition $f=h+\overline{g}$, where h and g are analytic in $\mathbb{D}$. As usual, we call h the analytic part of f and g the co-analytic part of f. An elegant and complete account of the theory of planar harmonic mapping is given in Duren’s monograph [1].

Lewy [2] proved in 1936 that the harmonic function f is locally univalent in a simply connected domain ${\mathbb{D}}_{1}$ if and only if its Jacobian

${J}_{f}\left(z\right)=|{h}^{\prime }\left(z\right){|}^{2}-|{g}^{\prime }\left(z\right){|}^{2}>0$

is different from zero in ${\mathbb{D}}_{1}$. In view of this result, locally univalent harmonic mappings in the unit disc are either sense-reversing if

$|{g}^{\prime }\left(z\right)|>|{h}^{\prime }\left(z\right)|$

in ${\mathbb{D}}_{1}$ or sense-preserving if

$|{g}^{\prime }\left(z\right)|<|{h}^{\prime }\left(z\right)|$

in ${\mathbb{D}}_{1}$. Throughout this paper, we will restrict ourselves to the study of sense-preserving harmonic mappings. However, since f is sense-preserving if and only if $\overline{f}$ is sense-reserving, all the results obtained in this article regarding sense-preserving harmonic mappings can be adapted to sense-reversing ones. Note that $f=h+\overline{g}$ is sense-preserving in $\mathbb{D}$ if and only if ${h}^{\prime }\left(z\right)$ does not vanish in the unit disc and the second-complex dilatation $w\left(z\right)=\frac{{g}^{\prime }\left(z\right)}{{h}^{\prime }\left(z\right)}$ has the property $|w\left(z\right)|<1$ in $\mathbb{D}$; therefore, we can take $h\left(z\right)=z+{a}_{2}{z}^{2}+\cdots$ , $g\left(z\right)={b}_{1}z+{b}_{2}{z}^{2}+\cdots$ . Thus, the class of all harmonic mappings being sense-preserving in the unit disc can be defined by

Let Ω be the family of functions $\varphi \left(z\right)$ which are regular in $\mathbb{D}$ and satisfy the conditions $\varphi \left(0\right)=0$, $|\varphi \left(z\right)|<1$ for all $z\in \mathbb{D}$. Denote by P the family of functions $p\left(z\right)=1+{p}_{1}z+{p}_{2}{z}^{2}+\cdots$ which are regular in $\mathbb{D}$ such that

$p\left(z\right)=\frac{1+\varphi \left(z\right)}{1-\varphi \left(z\right)}$
(1.1)

for some function $\varphi \left(z\right)\in \mathrm{\Omega }$ for all $z\in \mathbb{D}$.

Next, let ${S}^{\ast }$ denote the family of functions $s\left(z\right)=z+{c}_{2}{z}^{2}+{c}_{3}{z}^{3}+\cdots$ which are regular in $\mathbb{D}$ such that

$z\frac{{s}^{\prime }\left(z\right)}{s\left(z\right)}=p\left(z\right)$
(1.2)

for some $p\left(z\right)\in P$ for all $z\in \mathbb{D}$.

Let ${s}_{1}\left(z\right)=z+{\alpha }_{2}{z}^{2}+{\alpha }_{3}{z}_{3}+\cdots$ and ${s}_{2}\left(z\right)=z+{\beta }_{2}{z}^{2}+{\beta }_{3}{z}^{3}+\cdots$ be analytic functions in $\mathbb{D}$. If there exists $\varphi \left(z\right)\in \mathrm{\Omega }$ such that ${s}_{1}\left(z\right)={s}_{2}\left(\varphi \left(z\right)\right)$ for all $z\in \mathbb{D}$, then we say that ${s}_{1}\left(z\right)$ is subordinate to ${s}_{2}\left(z\right)$ and we write ${s}_{1}\left(z\right)\prec {s}_{2}\left(z\right)$, then ${s}_{1}\left(\mathbb{D}\right)\subset {s}_{2}\left(\mathbb{D}\right)$.

Now, we consider the following class of harmonic mappings in the plane:

$\begin{array}{rcl}{S}_{\mathrm{HPST}}^{\ast }\left(\alpha \right)& =& \left\{f=h\left(z\right)+\overline{g\left(z\right)}|f\in {S}_{H},h\left(z\right)\in {S}^{\ast },\\ Re\left({e}^{i\alpha }w\left(z\right)\right)=Re\left({e}^{i\alpha }\frac{{g}^{\prime }\left(z\right)}{{h}^{\prime }\left(z\right)}\right)>0,|\alpha |<\frac{\pi }{2}\right\}.\end{array}$
(1.3)

In the present paper, we will investigate the class ${S}_{\mathrm{HPST}}^{\ast }\left(\alpha \right)$.

We will need the following lemma and theorem in the sequel.

Theorem 1.1 ([3, 4])

Let $h\left(z\right)$ be an element of ${S}^{\ast }$, then

$\frac{r}{{\left(1+r\right)}^{2}}\le |h\left(z\right)|\le \frac{r}{{\left(1-r\right)}^{2}},$

for all $|z|=r<1$.

$\frac{1-r}{{\left(1+r\right)}^{3}}\le |{h}^{\prime }\left(z\right)|\le \frac{1+r}{{\left(1-r\right)}^{3}}.$

These inequalities are sharp because the extremal function is $h\left(z\right)=\frac{z}{{\left(1-z\right)}^{2}}$.

Lemma 1.2 ([2, 5])

Let $h\left(z\right)$ and $g\left(z\right)$ be regular in $\mathbb{D}$, $h\left(z\right)$ map $|z|<1$ onto a many-sheeted starlike region, $Re\left({e}^{i\alpha }\frac{{g}^{\prime }\left(z\right)}{{h}^{\prime }\left(z\right)}\right)>0$, $|\alpha |<\frac{\pi }{2}$ for $|z|<1$. $h\left(0\right)=g\left(0\right)=0$. Then $Re\left({e}^{i\alpha }\frac{g\left(z\right)}{h\left(z\right)}\right)>0$ for $|z|<1$.

## 2 Main results

Lemma 2.1 Let $f=h\left(z\right)+\overline{g\left(z\right)}$ be an element of ${S}_{\mathrm{HPST}}^{\ast }\left(\alpha \right)$ then

$\frac{|{b}_{1}|-r}{1-|{b}_{1}|r}\le |\frac{{g}^{\prime }\left(z\right)}{{h}^{\prime }\left(z\right)}|\le \frac{|{b}_{1}|+r}{1+|{b}_{1}|r}$
(2.1)

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

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

where $b={e}^{i\alpha }{b}_{1}$.

Proof

Since

then the function

$\varphi \left(z\right)=\frac{W\left(z\right)-W\left(0\right)}{1-\overline{W\left(0\right)}W\left(z\right)}=\frac{W\left(0\right)-b}{1-\overline{b}W\left(0\right)}=\frac{b-b}{1-{b}^{2}}=0$

satisfies the condition of the Schwarz lemma. Using the definition of subordination, we have

$W\left(z\right)={e}^{i\alpha }w\left(z\right)={e}^{i\alpha }\frac{{g}^{\prime }\left(z\right)}{{h}^{\prime }\left(z\right)}=\frac{b+\varphi \left(z\right)}{1+\overline{b}\varphi \left(z\right)}\phantom{\rule{1em}{0ex}}⇔\phantom{\rule{1em}{0ex}}{e}^{i\alpha }\frac{{g}^{\prime }\left(z\right)}{{h}^{\prime }\left(z\right)}\prec \frac{b+z}{1+\overline{b}z}.$

On the other hand, the transformation $\left(\frac{b+z}{1+\overline{b}z}\right)$ maps $|z|<1$ onto the disc with the center

$C\left(r\right)=\left(\frac{{\alpha }_{1}\left(1-{r}^{2}\right)}{1-{|{b}_{1}|}^{2}{r}^{2}},\frac{{\alpha }_{2}\left(1-{r}^{2}\right)}{1-{|{b}_{1}|}^{2}{r}^{2}}\right),\phantom{\rule{1em}{0ex}}b={\alpha }_{1}+i{\alpha }_{2}$

$\rho \left(r\right)=\frac{\left(1-{|{b}_{1}|}^{2}\right)r}{1-{|{b}_{1}|}^{2}{r}^{2}}.$

Therefore, we can write

$|{e}^{i\alpha }\frac{{g}^{\prime }\left(z\right)}{{h}^{\prime }\left(z\right)}-\frac{{b}_{1}\left(1-{r}^{2}\right)}{1-{|{b}_{1}|}^{2}{r}^{2}}|\le \frac{\left(1-{|{b}_{1}|}^{2}\right)r}{1-{|{b}_{1}|}^{2}{r}^{2}}$
(2.2)

which gives (2.1). □

Corollary 2.2 Let $f\in {S}_{\mathrm{HPST}}^{\ast }\left(\alpha \right)$, then

(2.3)
(2.4)

for all $|z|=r<1$.

Proof

Using Lemma 1.2 and Lemma 2.1, then we can write

(2.5)
(2.6)

If we use Theorem 1.1 in the inequalities (2.5) and (2.6), we get (2.3) and (2.4). □

Corollary 2.3 Let $f=h\left(z\right)+\overline{g\left(z\right)}$ be an element of ${S}_{HPTS}^{\ast }\left(\alpha \right)$, then

$\frac{\left(1-|{b}_{1}{|}^{2}\right){\left(1-r\right)}^{3}}{{\left(1+r\right)}^{5}{\left(1+|{b}_{1}|r\right)}^{2}}\le {J}_{f\left(z\right)}\le \frac{\left(1-|{b}_{1}{|}^{2}\right){\left(1+r\right)}^{3}}{{\left(1-r\right)}^{5}{\left(1+|{b}_{1}|r\right)}^{2}}$
(2.7)

for all $|z|=r<1$.

Proof

Since

${J}_{f\left(z\right)}=|{h}^{\prime }\left(z\right){|}^{2}-|{g}^{\prime }\left(z\right){|}^{2}=|{h}^{\prime }\left(z\right){|}^{2}\left(1-|w\left(z\right){|}^{2}\right),$
(2.8)

using Lemma 2.1 and Theorem 1.1 in the equality (2.8) and after simple calculations, we get (2.7). □

Corollary 2.4 If $f=h\left(z\right)+\overline{g\left(z\right)}$ is an element of ${S}_{HPTS}^{\ast }\left(\alpha \right)$, then

(2.9)

where $a=|{b}_{1}|$ for all $|z|=r<1$.

Proof

Using Corollary 2.2 and Theorem 1.1, we obtain

$\left(|{h}^{\prime }\left(z\right)|-|{g}^{\prime }\left(z\right)|\right)\ge \frac{\left(1-{r}^{4}\right)\left(1+|{b}_{1}|r\right)-\left(1+{r}^{4}\right)\left(|{b}_{1}|+r\right)}{{\left(1-r\right)}^{3}{\left(1+r\right)}^{3}\left(1+|{b}_{1}|r\right)},$

and

$\left(|{h}^{\prime }\left(z\right)|+|{g}^{\prime }\left(z\right)|\right)\le \frac{{\left(1+r\right)}^{2}\left(1+|{b}_{1}|\right)}{{\left(1-r\right)}^{3}\left(1+|{b}_{1}|r\right)}.$

Therefore, we have

(2.10)

Integrating the last inequality (2.10), we get (2.9). □

Theorem 2.5 Let $f=h\left(z\right)+\overline{g\left(z\right)}$ be an element of ${S}_{HPTS}^{\ast }\left(\alpha \right)$, then

$\sum _{k=1}^{n}|{A}_{k}{|}^{2}\le {|t+1|}^{2}+\sum _{k=1}^{n}|{B}_{k}{|}^{2}$
(2.11)

where ${A}_{k}=\left(k+1\right)\left(\frac{{b}_{k+1}}{{b}_{1}}-{a}_{k+1}\right)$; ${B}_{k}=\left(k+1\right)\left(\frac{{b}_{k+1}}{{b}_{1}}+t{a}_{k+1}\right)$; ${a}_{k}$ and ${b}_{k}$ are the coefficients of the functions $h\left(z\right)$ and $g\left(z\right)$; $k=1,2,3,\dots ,n$; $t=2s-1$; $s={e}^{-i\alpha }cos\alpha$.

Proof

Since

$g\left(z\right)={b}_{1}z+{b}_{2}{z}^{2}+{b}_{3}{z}^{3}+\cdots \phantom{\rule{1em}{0ex}}⇒\phantom{\rule{1em}{0ex}}{g}^{\prime }\left(z\right)={b}_{1}+2{b}_{2}z+3{b}_{3}{z}^{2}+\cdots .$

We denote by $G\left(z\right)=\frac{1}{{b}_{1}}g\left(z\right)$

then we have

$\left\{\begin{array}{c}\frac{1}{cos\alpha }\left({e}^{i\alpha }\frac{\frac{1}{{b}_{1}}{g}^{\prime }\left(z\right)}{{h}^{\prime }\left(z\right)}-isin\alpha \right)=p\left(z\right)\phantom{\rule{1em}{0ex}}⇔\phantom{\rule{1em}{0ex}}{e}^{i\alpha }\frac{\frac{1}{{b}_{1}}{g}^{\prime }\left(z\right)}{{h}^{\prime }\left(z\right)}=cos\alpha p\left(z\right)+isin\alpha ,\hfill \\ \phantom{\rule{1em}{0ex}}⇔\phantom{\rule{1em}{0ex}}\frac{\frac{1}{{b}_{1}}{g}^{\prime }\left(z\right)}{{h}^{\prime }\left(z\right)}=1+{e}^{-i\alpha }cos\alpha \left(p\left(z\right)-1\right).\hfill \end{array}$
(2.12)

Since $p\left(z\right)$ is in P, there is a function $\varphi \left(z\right)$ satisfying the conditions of the Schwarz lemma such that

$p\left(z\right)=\frac{1+\varphi \left(z\right)}{1-\varphi \left(z\right)}\phantom{\rule{1em}{0ex}}⇔\phantom{\rule{1em}{0ex}}p\left(z\right)-1=\frac{2\varphi \left(z\right)}{1-\varphi \left(z\right)}.$
(2.13)

Using this equation in (2.12) and after the following calculations given above

$\frac{\frac{1}{{b}_{1}}{g}^{\prime }\left(z\right)}{{h}^{\prime }\left(z\right)}=1+{e}^{-i\alpha }cos\alpha \left(p\left(z\right)-1\right)=1+s\left(\frac{2\varphi \left(z\right)}{1-\varphi \left(z\right)}\right)\phantom{\rule{1em}{0ex}}⇒,$

we get the following equality:

$\frac{1}{{b}_{1}}{g}^{\prime }\left(z\right)-{h}^{\prime }\left(z\right)=\left(t{h}^{\prime }\left(z\right)+\frac{1}{{b}_{1}}{g}^{\prime }\left(z\right)\right).$
(2.14)

If $\varphi \left(z\right)={c}_{1}z+{c}_{2}{z}^{2}+{c}_{3}{z}^{3}+\cdots$ , we have

$\sum _{k=1}^{n}{A}_{k}{z}^{k}+\sum _{k=n+1}^{\mathrm{\infty }}{D}_{k}{z}^{k}=\left[\left(1+t\right)+\sum _{k=1}^{n}{B}_{k}{z}^{k}\right]\varphi \left(z\right),$
(2.15)

where

$\sum _{k=n+1}^{\mathrm{\infty }}{D}_{k}{z}^{k}=\sum _{k=n+1}^{\mathrm{\infty }}{A}_{k}{z}^{k}-\left({c}_{1}{B}_{n}{z}^{n+1}+{c}_{1}{B}_{n+1}{z}^{n+2}+\cdots \right).$

Therefore, the equality (2.15) can be considered in the following form:

$F\left(z\right)=G\left(z\right)\varphi \left(z\right).$
(2.16)

Using the Clunie method [6], then we can write

$\frac{1}{2\pi }{\int }_{0}^{2\pi }{|F\left(r{e}^{i\theta }\right)|}^{2}d\theta \le \frac{1}{2\pi }{\int }_{0}^{2\pi }{|G\left(r{e}^{i\theta }\right)|}^{2}\phantom{\rule{0.2em}{0ex}}d\theta ,$

which gives

$\sum _{k=1}^{n}{|{A}_{k}|}^{2}{r}^{2k}+\sum _{k=n+1}^{\mathrm{\infty }}{|{D}_{k}|}^{2}{r}^{2k}\le \left({|t+1|}^{2}+\sum _{k=1}^{n}{|{B}_{k}|}^{2}{r}^{2k}\right).$
(2.17)

Eventually, we will let $r\to {1}^{-}$, then we have

$\sum _{k=1}^{n}{|{A}_{k}|}^{2}\le {|t+1|}^{2}+\sum _{k=1}^{n}{|{B}_{k}|}^{2}.$

□

## References

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2. Lewy H: On the non-vanishing of the Jacobian in certain in one-to-one mappings. Bull. Am. Math. Soc. 1936, 42: 689–692. 10.1090/S0002-9904-1936-06397-4

3. Goodman AW I. In Univalent Functions. Mariner Publishing Company, Tampa; 1983.

4. Goodman AW II. In Univalent Functions. Mariner Publishing Company, Tampa; 1983.

5. Bernardi SD: Convex and starlike univalent functions. Trans. Am. Math. Soc. 1969, 135: 429–446.

6. Clunie J: On meromorphic Schlicht functions. J. Lond. Math. Soc. 1959, 34: 215–216. 10.1112/jlms/s1-34.2.215

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Correspondence to Melike Aydog̃an.

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The authors declare that they have no competing interests.

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All authors contributed equally and significantly to writing this paper. All authors read and approved the final manuscript.

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Aydog̃an, M., Duman, E.Y., Polatog̃lu, Y. et al. Harmonic function for which the second dilatation is α-spiral. J Inequal Appl 2012, 262 (2012). https://doi.org/10.1186/1029-242X-2012-262