Bohr radius for subordination and $K$-quasiconformal harmonic mappings

TL;DR

This paper investigates the Bohr radius for $K$-quasiconformal harmonic mappings in the unit disk, focusing on cases where the analytic part is subordinate to convex or univalent functions, providing bounds depending on geometric properties.

Contribution

It establishes new bounds for the Bohr radius for $K$-quasiconformal harmonic mappings with subordinate analytic parts, depending on geometric properties of the image domain.

Findings

Derived bounds for the Bohr radius depending on $ ext{dist}( ext{0}, ext{boundary})$

Provided estimates for the maximal radius $r^*$ based on geometric properties and $K$

Extended results for cases with $b_1=0$ and as $K o \infty$

Abstract

The present article concerns the Bohr radius for $K$-quasiconformal sense-preserving harmonic mappings $f=h+\overline{g}$ in the unit disk $\mathbb{D}$ for which the analytic part $h$ is subordinated to some analytic function $\varphi$, and the purpose is to look into two cases: when $\varphi$ is convex, or a general univalent function in $\ID$. The results state that if $h(z) =\sum_{n=0}^{\infty}a_n z^n$ and $g(z)=\sum_{n=1}^{\infty}b_n z^n$, then $$\sum_{n=1}^{\infty}(|a_n|+|b_n|)r^n\leq \dist (\varphi(0),\partial\varphi(\ID)) ~\mbox{ for $r\leq r^*$} $$ and give estimates for the largest possible $r^*$ depending only on the geometric property of $\varphi (\ID)$ and the parameter $K$. Improved versions of the theorems are given for the case when $b_1 = 0$ and corollaries are drawn for the case when $K\rightarrow \infty$.

Equations

$$\sum_{n=1}^{\infty}(|a_{n}|+|b_{n}|)r^{n}\leq{\operatorname{dist}}(\varphi(0),\partial\varphi(\mathbb{D}))~{}\mbox{ for $r\leq r^{*}$}$$

$$B_f(r):=\sum _{n=0}^{\infty }|a_n|r^n\le \Vert f{\Vert }_{\infty }\text{mbox}forall|z|=r\le \frac{1}{3}$$

$$\sum _{k=0}^{\infty }|a_k|r^k+\frac{16}{9}\left(\frac{S_r}{\pi }\right)\le 1\text{mbox}forr\le \frac{1}{3},$$

$$|a_0{|}^2+\sum _{k=1}^{\infty }|a_k|r^k+\frac{9}{8}\left(\frac{S_r}{\pi }\right)\le 1\text{mbox}forr\le \frac{1}{2},$$

$$h(z)=\sum _{n=0}^{\infty }{a}_n{z}^n\text{mbox}andg(z)=\sum _{n=1}^{\infty }{b}_n{z}^n,z\in \mathbb{D}.$$

$$\left|\frac{g^{\prime }(z)}{h^{\prime }(z)}\right|\le k<1,$$

$$\sum _{n=1}^{\infty }(|a_n|+|b_n|)r^n\le \max \{||h|{|}_{\infty },||g|{|}_{\infty }\}\text{mbox}forr\le \sqrt{\frac{7}{32}}.$$

$$g(z)=\varphi(\omega(z))~{}\mbox{ for $|z|<1$}$$

$$S(\phi )=\left\{g:g\prec \phi \right\}.$$

$$\sum_{n=1}^{\infty}\left|b_{n}z^{n}\right|\leq{\operatorname{dist}}(\varphi(0),\partial\Omega)~{}\mbox{for $|z|<r_{\varphi}$,}$$

$$\frac{1}{4}|{\phi }^{\prime }(0)|\le \mathrm{dist}(\phi (0),\partial \Omega )\le |{\phi }^{\prime }(0)|.$$

$$|b_n|\le n|{\phi }^{\prime }(0)|\le 4n\mathrm{dist}(\phi (0),\partial \Omega ).$$

$$\frac{1}{2}|{\phi }^{\prime }(0)|\le \mathrm{dist}(\phi (0),\partial \Omega )\le |{\phi }^{\prime }(0)|.$$

$$|b_n|\le |{\phi }^{\prime }(0)|\le 2\mathrm{dist}(\phi (0),\partial \Omega ).$$

$$\frac{1}{4}\le \mathrm{dist}(0,\partial \phi (\mathbb{D}))\le 1$$

$$\frac{1}{2}\le \mathrm{dist}(0,\partial \phi (\mathbb{D}))\le 1.$$

$$\sum _{n=1}^{\infty }|b_n{|}^2{r}^n\le k^2\sum _{n=1}^{\infty }|a_n{|}^2{r}^n\text{mbox}for|z|=r<1.$$

$$\phi (z)=\frac{\alpha z}{(1+z)(1+\lambda z)},z\in \mathbb{D}.$$

$$\frac{\alpha }{\phi (z)}=\frac{1}{z}+\lambda z+(1+\lambda )$$

$$\frac{U^2}{(1+\lambda {)}^2}+\frac{V^2}{(1-\lambda {)}^2}=1.$$

$$\phi (z)=\frac{z}{1-2\lambda z+z^2}=\frac{1}{\frac{1}{z}+z-2\lambda },z\in \mathbb{D}.$$

$$\mathrm{dist}(\phi (0),\partial \phi (\mathbb{D}))=1/(2(1+\lambda )).$$

$$\phi (z)=a\left(\frac{1+z}{1-z}\right)+2(\sqrt{c^2+a^2}-a)\frac{z}{1-z^2},z\in \mathbb{D}.$$

$$\mathrm{dist}(\phi (0),\partial \phi (\mathbb{D}))=\mathrm{dist}(a,\partial \phi (\mathbb{D}))=\sqrt{c^2+a^2}.$$

$${\phi }_{a,n}(z)=\frac{a(1+z)}{1-z}+\frac{2nz}{1-z^2}$$

$$\mathrm{dist}(\phi (0),\partial \phi (\mathbb{D}))=n+a.$$

$$\phi (z)=\frac{z-\lambda z^2}{(1-z{)}^2}.$$

$$\phi (z)=\frac{1}{4(1-\lambda )}({\zeta }^2-1),$$

$$x<{\lambda }^2-\frac{y^2}{4{\lambda }^2}.$$

$$\mathrm{dist}(\phi (0),\partial \phi (\mathbb{D}))=-\frac{1-{\lambda }^2}{4(1-\lambda )}=-\frac{1+\lambda }{4}.$$

Full text

[TABLE]

and give estimates for the largest possible $r^{*}$ depending only on the geometric property of $\varphi(\mathbb{D})$ and the parameter $K$. Improved versions of the theorems are given for the case when $b_{1}=0$ and corollaries are drawn for the case when $K\rightarrow\infty$.

File: 1905.10334.tex, printed: 19-3-2024, 17.22

1. Introduction and Preliminaries

A classical theorem of Bohr states that [7], if $f$ is a bounded analytic function on the open unit disk $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$, with power series of the form $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$, then

[TABLE]

and the constant $1/3$, often called the Bohr radius, cannot be improved. This inequality known as Bohr’s inequality, was originally obtained in 1914 by H. Bohr for $0\leq r\leq 1/6$. The fact that the inequality is actually true for $0\leq r\leq 1/3$ and that $1/3$ is the best possible constant was obtained independently by M. Riesz, I. Schur and F. Wiener. Bohr’s and Wiener’s proofs can be found in [7]. Several improved versions of the Bohr inequality are established.

For example, in a related development, Kayumov and Ponnusamy [15] gave several improved versions of Bohr’s inequality. Some of them may now be recalled.

Theorem A**.**

Suppose that $f(z)=\sum_{k=0}^{\infty}a_{k}z^{k}$ is analytic in $\mathbb{D}$, $|f(z)|\leq 1$ in $\mathbb{D}$ and $S_{r}$ denotes the area of the image of the subdisk $|z|<r$ under the mapping $f$. Then

[TABLE]

and the constants $1/3$ and $16/9$ cannot be improved. Moreover,

[TABLE]

and the constants $1/2$ and $9/8$ cannot be improved.

Theorem B**.**

Suppose that $f(z)=\sum_{k=0}^{\infty}a_{k}z^{k}$ is analytic in $\mathbb{D}$ and $|f(z)|\leq 1$ in $\mathbb{D}$. Then we have

1. (1)

$\displaystyle|a_{0}|+\sum_{k=1}^{\infty}\left(|a_{k}|+\frac{1}{2}|a_{k}|^{2}\right)r^{k}\leq 1~{}\mbox{ for }~{}r\leq\frac{1}{3},$* and the constants $1/3$ and $1/2$ cannot be improved.* 2. (2)

$\displaystyle\sum_{k=0}^{\infty}|a_{k}|r^{k}+|f(z)-a_{0}|^{2}\leq 1~{}\mbox{ for }~{}r\leq\frac{1}{3},$* and the constant $1/3$ cannot be improved.* 3. (3)

$\displaystyle|f(z)|^{2}+\sum_{k=1}^{\infty}|a_{k}|^{2}r^{2k}\leq 1~{}\mbox{ for }~{}r\leq\sqrt{\frac{11}{27}},$* and the constant $11/27$ cannot be improved.*

For the last two decades, Bohr’s inequality has been revived and improved in many ways due to the discovery of generalizations to domains in $\mathbb{C}^{n}$ and to more abstract settings. For background information about this result and further work related to Bohr’s phenomenon, we refer to the recent surveys by Abu-Muhanna et al. [3], Bénéteau et al. [6], Ismagilov et al. [10], Kayumova et al. [12] and the references therein. Some of the recent results from [4, 13, 14, 15] are included in the latest two surveys. More generally, harmonic version of Bohr’s inequality was discussed by Kayumov et al. in [16] which will also be recalled below. For certain other results on harmonic Bohr’s inequality, we refer to [16, 9]. We refer to [18] for Bohr’s inequality for the class of harmonic $\nu$-Bloch-type mappings as a generalization of harmonic $\nu$-Bloch mappings and to [5] for the class of quasi-subordinations.

A harmonic mapping $f$ defined on $\mathbb{D}$ is a complex-valued function $f=u+iv$, where $u$ and $v$ are real-valued harmonic functions of $\mathbb{D}$. It follows that $f$ admits the representation $f=h+\overline{g}$, where $h$ and $g$ are analytic in $\mathbb{D}$ known as the analytic and co-analytic parts of $f$, respectively. We follow the convention that $g(0)=0$ so that the representation $f=h+\overline{g}$ is unique and is called the canonical representation of $f$ and thus $h$ and $g$ admit power series expansions of the form

[TABLE]

A locally univalent harmonic function $f$ in $\mathbb{D}$ is said to be sense-preserving if the Jacobian $J_{f}(z)$ of $f$ given by $J_{f}(z)=|h^{\prime}(z)|^{2}-|g^{\prime}(z)|^{2}$, is positive in $\mathbb{D}$; or equivalently, its dilatation $\omega=g^{\prime}/h^{\prime}$ is an analytic function in $\mathbb{D}$ which maps $\mathbb{D}$ into itself (See [17] or [8]).

If a locally univalent and sense-preserving harmonic mapping $f=h+\overline{g}$ satisfies the condition

[TABLE]

then $f$ is called $K$-quasiconformal harmonic mapping on $\mathbb{D}$, where $K=\frac{1+k}{1-k}\geq 1$ (cf. [19, 11], and also [21] for some recent investigation on harmonic $K$-quasiconformal self-mapping of $\mathbb{D}$). Obviously $k\rightarrow 1$ corresponds to the case $K\rightarrow\infty$. Harmonic extension of the classical Bohr theorem was established in [16]. For example, they proved the following results.

Theorem C**.**

Suppose that $f(z)=h(z)+\overline{g(z)}=\sum_{n=0}^{\infty}a_{n}z^{n}+\overline{\sum_{n=1}^{\infty}b_{n}z^{n}}$ is a sense-preserving $K$–quasiconformal harmonic mapping of the disk $\mathbb{D}$, where $h$ is a bounded function in $\mathbb{D}$. Then we have

1. (1)

$\displaystyle\sum_{n=0}^{\infty}|a_{n}|r^{n}+\sum_{n=1}^{\infty}|b_{n}|r^{n}\leq||h||_{\infty}~{}\mbox{ for }~{}r\leq\frac{K+1}{5K+1}.$* The constant $(K+1)/(5K+1)$ is sharp.* 2. (2)

$\displaystyle|a_{0}|^{2}+\sum_{n=1}^{\infty}(|a_{n}|+|b_{n}|)r^{n}\leq||h||_{\infty}~{}\mbox{ for }~{}r\leq\frac{K+1}{3K+1}.$* The constant $(K+1)/(3K+1)$ is sharp.*

Theorem D**.**

Suppose that either $f=h+g$ or $f=h+\overline{g}$, where $h(z)=\sum_{n=1}^{\infty}a_{n}z^{n}$ and $g(z)=\sum_{n=1}^{\infty}b_{n}z^{n}$ are bounded analytic functions in $\mathbb{D}$. Then

[TABLE]

This number $\sqrt{7/32}$ is sharp.

The purpose of this article is to determine the Bohr radius for the class of $K$-quasiconformal sense-preserving harmonic mappings $f=h+\overline{g}$, where $h$ is subordinate to $\varphi$, where $\varphi$ is either a general function in the convex family or in the univalent family.

The paper is organized as follows. In Section 2, we present main definitions and necessary lemmas that are required to state and prove our main results. Section 3 begins with examples containing test functions for which our main results could be used to derive several new theorems and corollaries, and then we state and prove our main theorems and several of their consequences. More precisely Theorems 1 and 2 generalize Theorem A(1) whereas Theorems 3 and 4 essentially deal with the case when the subordinating function is univalent instead of convex. The article concludes with a conjecture.

2. Necessary Lemmas

We need to recall some basic notions and results on subordination.

Definition 1**.**

Let $\varphi$ and $g$ be analytic in $\mathbb{D}$ with $\varphi(0)=g(0)$. Then we say that $g$ is subordinate to $\varphi$ (written by $g\prec\varphi$ or $g(z)\prec\varphi(z)$) if

[TABLE]

for some analytic function $\omega$ on $\mathbb{D}$ with $|\omega(z)|\leq|z|$ for $z\in\mathbb{D}$. When $\varphi$ is univalent, $g\prec\varphi$ precisely when $\varphi(0)=g(0)$ and $g(\mathbb{D})\subset\varphi(\mathbb{D})$.

For basic details and results on subordination classes, see for example [8, Chapter 6] or [20, p. 35]. Let $\mathcal{S}$ denote the class of all univalent analytic mappings $\varphi$ on $\mathbb{D}$ normalized by $\varphi(0)=0$ and $\varphi^{\prime}(0)=1$. Denote by $\mathcal{S}^{*}$ and $\mathcal{C}$ the subclass of $\mathcal{S}$ of mappings that map $\mathbb{D}$ onto starlike and convex domains, respectively. See [8] for details on these classes and many other related subclasses of $\mathcal{S}$. If $\varphi$ is univalent, then the following coefficient inequalities are well-known.

Theorem E**.**

(L. de Branges’ Theorem) Suppose that $g\prec\varphi$ and $g(z)=\sum_{n=1}^{\infty}b_{n}z^{n}$. If $\varphi\in\mathcal{S}$, then $|b_{n}|\leq n$ for $n\geq 2$.

Because $\varphi\in\mathcal{C}$ if and only if $z\varphi^{\prime}\in\mathcal{S^{*}}$, and $\mathcal{S^{*}}\subset\mathcal{S}$, Theorem E, in particular gives the following:

1. (1)

if $\varphi\in\mathcal{C}$, then $|b_{n}|\leq 1$ for $n\geq 2$; 2. (2)

if $\varphi\in\mathcal{S^{*}}$, then $|b_{n}|\leq n$ for $n\geq 2$.

Throughout this paper, we denote the class of all analytic functions $g$ in $\mathbb{D}$ subordinate to a fixed univalent function $\varphi$ in $\mathbb{D}$ by

[TABLE]

We say that the family $S(\varphi)$ has Bohr’s phenomenon if for any $g\in S(\varphi)$ and $g(z)=\sum_{n=0}^{\infty}b_{n}z^{n}\prec\varphi(z)$ there is an $r_{\varphi}$, $0<r_{\varphi}\leq 1$, such that (see [1, 3])

[TABLE]

where ${\operatorname{dist}}(\varphi(0),\partial\Omega)$ denotes the distance between $\varphi(0)$ and the boundary $\partial\Omega$ of $\Omega=\varphi(\mathbb{D})$.

We observe that if $\varphi(z)=(\alpha-z)/(1-\overline{\alpha}z)$ with $|\alpha|<1$, then $\Omega=\varphi(\mathbb{D})=\mathbb{D}$, $\varphi(0)=\alpha$ and ${\operatorname{dist}}(\varphi(0),\partial\Omega)=1-|\alpha|=1-|b_{0}|$ so that (2.1) (and hence (1.1)) holds with $r_{\varphi}=1/3$.

We can easily to obtain the following two lemmas from [8, p. 195-196] (see also [3, 20]).

Lemma A**.**

Let $\varphi$ be an analytic univalent map from $\mathbb{D}$ onto a simply connect domain $\Omega=\varphi(\mathbb{D})$. Then

[TABLE]

If $g(z)=\sum_{n=0}^{\infty}b_{n}z^{n}\prec\varphi(z)$, then

[TABLE]

Lemma B**.**

Let $\varphi$ be an analytic univalent map from $\mathbb{D}$ onto a convex domain $\Omega=\varphi(\mathbb{D})$. Then

[TABLE]

If $g(z)=\sum_{n=0}^{\infty}b_{n}z^{n}\prec\varphi(z)$, then

[TABLE]

Particularly, the well-known Growth Theorem implies that if $\varphi\in\mathcal{S}$ then

[TABLE]

and if $\varphi\in\mathcal{C}$ then

[TABLE]

See [8, Theorems 2.6 and 2.15] or [20, p. 22]. Note that (2.3) and (2.4) follow from Lemmas A and B, respectively.

The following lemma plays an important role in the proof of our results.

Lemma C**.**

(see [16, Lemma 2.1])* Suppose that $h(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ and $g(z)=\sum_{n=0}^{\infty}b_{n}z^{n}$ are two analytic functions in the unit disk $\mathbb{D}$ such that $|g^{\prime}(z)|\leq k|h^{\prime}(z)|$ in $\mathbb{D}$ and for some $k\in[0,1]$. Then*

[TABLE]

3. Main results and their proofs

Before we state and prove our main theorems, it is worth pointing out that our approach provides many results by different choices of $\varphi$ in the main theorems. To demonstrate this, we first present a set of test functions for which our results are applicable.

Examples 1**.**

1. (a)

For $\lambda\in(0,1]$ and $\alpha\in{\mathbb{C}}\backslash\{0\}$, consider

[TABLE]

Then it is easy to see that ${\operatorname{dist}}(\varphi(0),\partial\varphi(\mathbb{D}))=|\alpha|/(2(1+\lambda))$, because

[TABLE]

and for $\lambda\in(0,1)$, $w=\frac{1}{z}+\lambda z$ maps $\mathbb{D}$ onto the exterior of the ellipse bounded by

[TABLE]

Also, we see that $\varphi$ is univalent in $\mathbb{D}$. 2. (b)

For $\lambda\in[0,1)$, consider the univalent function

[TABLE]

Then it can be easily seen that $\varphi(z)$ maps $\mathbb{D}$ onto the complement of segment $(-1/(2(1+\lambda)),1/(2(1-\lambda)))$ and thus,

[TABLE] 3. (c)

For $a\in{\mathbb{R}}\backslash\{0\}$ and $c>0$, consider

[TABLE]

Using the range of the function $\psi(z)=2cz/(1-z^{2})$, it can be easily shown that $\varphi(z)$ maps $\mathbb{D}$ onto the complex plane with slits along half-lines ${\rm Re}\,w=0$ and $|{\rm Im}\,w|\geq c:=c(a)$ such that $\varphi(0)=a$ and

[TABLE]

Obviously, $\varphi$ is univalent and starlike in $\mathbb{D}$.

In particular, for $a>0$ or $-n/2<a<0$, set $\sqrt{c^{2}+a^{2}}\,-a=n$, where $n>0$. Then we obtain that the function

[TABLE]

maps $\mathbb{D}$ onto the complex plane with slits along half-lines ${\rm Re}\,w=0$ and $|{\rm Im}\,w|\geq c:=c(a,n)=\sqrt{n(n+2a)}$ such that $\varphi(0)=a$ and

[TABLE] 4. (d)

For $\lambda\in{\mathbb{C}}$, consider the function

[TABLE]

We see that it is univalent in $\mathbb{D}$ if and only if $|2\lambda-1|\leq 1$. The function $\varphi(z)$ in general is not starlike, for example, for $\lambda=1/2$, this function is known to be close-to-convex (univalent) but is not starlike in $\mathbb{D}$.

In particular, for $\lambda\in[0,1]$, we may write $\varphi$ as

[TABLE]

where $\zeta=\psi(z)=\frac{1+(1-2\lambda)z}{1-z}$ and $w=\Psi(\zeta)=\zeta^{2}.$ It follows that $\zeta=\psi(z)$ maps the disk $\mathbb{D}$ onto the half-plane ${\rm Re}\,w>\lambda$ and $w=\Psi(\zeta)$ maps the half-plane ${\rm Re}\,w>\lambda$ onto the parabolic region

[TABLE]

Consequently, for $\lambda\in[0,1]$, $\varphi$ maps $\mathbb{D}$ onto a parabolic region such that

[TABLE] 5. (e)

For $|\lambda|<1$, consider

[TABLE]

where $\xi=e^{-i\pi{\rm Im}\,\lambda}$. Then it is a simple exercise to show that $\varphi(z)$ maps $\mathbb{D}$ onto the strip $\Omega=\{w:\,|{\rm Im}\,w|<1\}$ with $\varphi(0)=\lambda$ and $\varphi^{\prime}(0)=\frac{2}{\pi}(1+\xi)$ such that

[TABLE]

Note that $\varphi$ is convex. 6. (f)

For ${\rm Re}\,\lambda>0$, consider

[TABLE]

Then we see that $\varphi(z)$ maps $\mathbb{D}$ onto the right half-plane ${\rm Re}\,w>0$ such that $\varphi(0)=\lambda$ and

[TABLE]

Clearly, $\varphi$ is convex. 7. (g)

For $\alpha\in[1,2]$, the function

[TABLE]

is univalent in $\mathbb{D}$ and

[TABLE] 8. (h)

For $\alpha\in[0,1)$, the function

[TABLE]

is univalent (and is in fact starlike of order $\alpha$) in $\mathbb{D}$ and

[TABLE]

The following result is a generalization of [16, Theorems 1.1 and 1.3] (see also Theorem A) for appropriate choices of $\varphi$.

Theorem 1**.**

Suppose that $f(z)=h(z)+\overline{g(z)}=\sum_{n=0}^{\infty}a_{n}z^{n}+\overline{\sum_{n=1}^{\infty}b_{n}z^{n}}$ is a $K$-quasiconformal sense-preserving harmonic mapping in $\mathbb{D}$ and $h\prec\varphi$, where $\varphi$ is univalent and convex in $\mathbb{D}$. Then

[TABLE]

The result is sharp.

Proof.

By assumption $h\prec\varphi$ and $\varphi(\mathbb{D})$ is a convex domain. Then, by Lemma B, we have

[TABLE]

Consequently,

[TABLE]

Because $f=h+\overline{g}$ is a $K$-quasiconformal sense-preserving harmonic mapping so that $|g^{\prime}(z)|\leq k|h^{\prime}(z)|$ in $\mathbb{D}$, where $0\leq k<1$, by Lemma C and the Cauchy-Schwarz inequality, it follows that

[TABLE]

Thus, we have

[TABLE]

which is less than or equal to ${\operatorname{dist}}(\varphi(0),\partial\varphi(\mathbb{D}))$ for $r\leq\frac{1}{3+2k}$. Substituting $k=\frac{K-1}{K+1}$ gives the desired result.

In order to prove the sharpness, we consider

[TABLE]

and $g^{\prime}(z)=k\lambda h^{\prime}(z)$, where $\lambda\in\mathbb{D}$. Then it is easy to see that

[TABLE]

and

[TABLE]

So it is a simple exercise to yield

[TABLE]

which is bigger than or equal to $1/2$ if and only if

[TABLE]

This shows that the number $\frac{K+1}{5K+1}$ cannot be improved since $|\lambda|$ could be chosen so close to $1$ from left. This completes the proof. ∎

Also, it is interesting to note that when $k=0$ (or, equivalently, $K=1$) one retrieves Aizenberg’s [2] result, according to which for convex functions $\varphi$, the Bohr inequality (1.1) holds with $1/3$ as its Bohr radius. Because of its independent interest, it might be worth stating the following two corollaries.

Corollary 1**.**

Suppose that $f(z)=h(z)+\overline{g(z)}=\sum_{n=0}^{\infty}a_{n}z^{n}+\overline{\sum_{n=1}^{\infty}b_{n}z^{n}}$ is a sense-preserving harmonic mapping in $\mathbb{D}$ and $h\prec\varphi$, where $\varphi(z)$ is univalent and convex in $\mathbb{D}$. Then

[TABLE]

for $|z|=r\leq 1/5$. The number $1/5$ is sharp.

Proof.

Allow $k=1$ in the proof of Theorem 1. Indeed, since $f(z)$ is locally univalent and sense-preserving in $\mathbb{D}$, we have $|g^{\prime}(z)|<|h^{\prime}(z)|$ in $\mathbb{D}$ and thus, we can allow $K\rightarrow\infty$ to obtain the desired conclusion. ∎

If we choose $\varphi(z)=(\alpha-z)/(1-\overline{\alpha}z)$ with $|\alpha|<1$, then $\varphi(0)=\alpha$ and ${\operatorname{dist}}(\varphi(0),\partial\Omega)=1-|\alpha|$ and this clearly give the following corollary (see also [16] or Theorem C with $K\rightarrow\infty$).

Corollary 2**.**

Suppose that $f(z)=h(z)+\overline{g(z)}=\sum_{n=0}^{\infty}a_{n}z^{n}+\overline{\sum_{n=1}^{\infty}b_{n}z^{n}}$ is a sense-preserving harmonic mapping of the disk $\mathbb{D}$, where $|h(z)|<1$ in $\mathbb{D}$. Then the following sharp inequality hold:

[TABLE]

Theorem 2**.**

Suppose that $f(z)=h(z)+\overline{g(z)}=\sum_{n=0}^{\infty}a_{n}z^{n}+\overline{\sum_{n=2}^{\infty}b_{n}z^{n}}$ is a $K$-quasiconformal sense-preserving harmonic mapping in $\mathbb{D}$ and $h\prec\varphi$, where $\varphi$ is univalent and convex in $\mathbb{D}$. Then

[TABLE]

for $|z|=r\leq r_{c}(k)$, where $r_{c}(k)$ is the positive root of the equation

[TABLE]

and $k=(K-1)/(K+1)$. The number $r_{c}(k)$ cannot be replaced by the number greater than $R:=R(k)$, where $R$ is the positive root of the equation

[TABLE]

Proof.

As $\varphi(z)$ is analytic and convex in $\mathbb{D}$, by (2.4) and Lemma B, we have

[TABLE]

so that

[TABLE]

Because $f=h+\overline{g}$ is locally univalent and $K$-quasiconformal sense-preserving harmonic mapping with $g^{\prime}(0)=0$, Schwarz’s lemma gives that $\omega=g^{\prime}/h^{\prime}$ is analytic in $\mathbb{D}$ and $|\omega(z)|\leq k|z|$ in $\mathbb{D}$. Thus, we have

[TABLE]

Integrate this inequality on the circle $|z|=r$, we obtain

[TABLE]

By using the Cauchy-Schwarz inequality, it follows that

[TABLE]

Consequently, by combining (3.3) with the last inequality, and (2.2), we find that

[TABLE]

where the last inequality holds if and only if

[TABLE]

The above inequality holds for $r\leq r_{c}(k)$, where $r_{c}(k)$ is the positive root of the equation (3.1).

Finally, we consider the functions

[TABLE]

Then we find that

[TABLE]

so that

[TABLE]

which is less than or equal to $1/2$ only in the case when $r\leq R$, where $R=R(k)$ is the positive root of the equation (3.2). ∎

Setting $k=0$ we see that Theorem 2 contains the classical Bohr theorem. The case $k=1$ leads to

Corollary 3**.**

Suppose that $f(z)=h(z)+\overline{g(z)}=\sum_{n=0}^{\infty}a_{n}z^{n}+\overline{\sum_{n=2}^{\infty}b_{n}z^{n}}$ is a sense-preserving harmonic mapping in $\mathbb{D}$ and $h\prec\varphi$, where $\varphi$ is univalent and convex in $\mathbb{D}$. Then

[TABLE]

for $|z|=r\leq r_{c}=0.294265\cdots$, where $r_{c}$ is the positive root of the equation

[TABLE]

The number $0.294265\cdots$ cannot be replaced by the number greater than $R=0.299823\cdots$, where $R$ is the positive root of the equation

[TABLE]

Remark 1**.**

Corollary 3 shows that the radius $r_{c}(k)$ obtained in Theorem 2 is close to the sharp value.

Theorem 3**.**

Suppose that $f(z)=h(z)+\overline{g(z)}=\sum_{n=0}^{\infty}a_{n}z^{n}+\overline{\sum_{n=1}^{\infty}b_{n}z^{n}}$ is a $K$-quasiconformal sense-preserving harmonic mapping in $\mathbb{D}$ and $h\prec\varphi$, where $\varphi$ is analytic and univalent in $\mathbb{D}$. Then

[TABLE]

for $|z|=r\leq r_{u}$, where $r_{u}=r_{u}(k)$ is the root of the equation

[TABLE]

in the interval $(0,1)$ and $k=(K-1)/(K+1)$.

Proof.

By the assumption and Lemma A, it follows that $|a_{n}|\leq 4n\,{\operatorname{dist}}(\varphi(0),\partial\varphi(\mathbb{D}))$ and thus,

[TABLE]

Moreover, because $|g^{\prime}(z)|\leq k|h^{\prime}(z)|$ in $\mathbb{D}$, as in the proof of Theorem 1, it follows from Cauchy-Schwarz inequality and Lemma C with $k=1$ that

[TABLE]

and thus, we have

[TABLE]

which is less than or equal to ${\operatorname{dist}}(\varphi(0),\partial\varphi(\mathbb{D}))$ if and only if

[TABLE]

This gives $|z|=r\leq r_{u}$, where $r_{u}=r_{u}(k)$ is as in the statement. ∎

Remark 2**.**

Theorem 3 for $k=0$ reduces to a result of Abu-Muhanna [1] with the sharp Bohr radius as $3-2\sqrt{2}=0.17157...$.

Corollary 4**.**

Suppose that $f(z)=h(z)+\overline{g(z)}=\sum_{n=0}^{\infty}a_{n}z^{n}+\overline{\sum_{n=1}^{\infty}b_{n}z^{n}}$ is a sense-preserving harmonic mapping in $\mathbb{D}$ and $h\prec\varphi$, where $\varphi$ is analytic and univalent in $\mathbb{D}$. Then

[TABLE]

for $|z|=r\leq r_{u}=0.099064\cdots$, where $r_{u}$ is the root of the equation

[TABLE]

in the interval $(0,1)$.

Proof.

Allow $k\rightarrow 1$ in Theorem 3. ∎

Remark 3**.**

When $\varphi$ in Corollary 4 is univalent and $b_{1}=0$, then the result can be improved (see also Corollary 5).

Our next result is to improve Theorem 3 when $g^{\prime}(0)=b_{1}=0$.

Theorem 4**.**

Suppose that $f(z)=h(z)+\overline{g(z)}=\sum_{n=1}^{\infty}a_{n}z^{n}+\overline{\sum_{n=2}^{\infty}b_{n}z^{n}}$ is a $K$-quasiconformal sense-preserving harmonic mapping in $\mathbb{D}$ and $h\prec\varphi$, where $\varphi$ is univalent in $\mathbb{D}$. Then

[TABLE]

for $|z|=r\leq r_{s}$, where $r_{s}=r_{s}(k)$ is the positive real root of the equation

[TABLE]

in the interval $(0,1)$ and $k=(K-1)/(K+1)$. The number $r_{s}(k)$ cannot be replaced by the number greater than $R=R(k)$, where $R$ is the positive root of the equation

[TABLE]

Proof.

By assumption $\varphi$ is analytic and univalent in $\mathbb{D}$, $\varphi(0)=0$ and thus, by  (2.3) and Theorem E, we have

[TABLE]

so that

[TABLE]

As in the proof of Theorem 2, it follows that

[TABLE]

By using the classical Cauchy-Schwarz inequality, we deduce that

[TABLE]

Consequently, by combining (3.6) with the last inequality, we find that

[TABLE]

This gives $r\leq r_{s}$, where $r_{s}=r_{s}(k)$ is the positive real root of the equation (3.4) in the interval $(0,1)$.

Finally, we consider the functions

[TABLE]

So we find that

[TABLE]

and thus, we have

[TABLE]

which is less than or equal to $1/4$ only in the case when $r\leq R=R(k)$, where $R$ is the positive root of the equation (3.5). ∎

Corollary 5**.**

Suppose that $f(z)=h(z)+\overline{g(z)}=\sum_{n=1}^{\infty}a_{n}z^{n}+\overline{\sum_{n=2}^{\infty}b_{n}z^{n}}$ is a sense-preserving harmonic mapping in $\mathbb{D}$ and $h\prec\varphi$, where $\varphi$ is univalent in $\mathbb{D}$. Then

[TABLE]

for $|z|=r\leq r_{s}=0.155856\cdots$, where $r_{s}$ is the positive real root of the equation

[TABLE]

in the interval $(0,1)$. The number $0.1593\cdots$ cannot be replaced by the number greater than $R=0.161353\cdots$, where $R$ is the positive root of the equation

[TABLE]

In view of Corollaries 3 and 5 (see also Theorems 2 and 4 to propose general conjectures), it is natural to propose in particular the following two conjectures.

Conjecture 1**.**

Suppose that $f(z)=h(z)+\overline{g(z)}=\sum_{n=0}^{\infty}a_{n}z^{n}+\overline{\sum_{n=2}^{\infty}b_{n}z^{n}}$ is a sense-preserving harmonic mapping in $\mathbb{D}$ and $h\prec\varphi$.

1. (a)

If $\varphi$ is univalent and convex in $\mathbb{D}$, then

[TABLE]

for $|z|=r\leq r_{c}=0.299823\cdots$, where $r_{c}$ is the positive root of the equation (3.2). 2. (b)

If $\varphi$ is univalent in $\mathbb{D}$, then the inequality (3.7) holds for $|z|=r\leq r_{s}=0.161353\cdots$, where $r_{s}$ is the positive real root of the equation (3.5).

Acknowledgments

The research of the first author was supported by the Foundation of Guilin University of Technology under Grant No. GUTQDJJ2018080, the Natural Science Foundation of Guangxi under Grant No. 2018GXNSFAA050005. The research was financially supported by Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering (Changsha University of Science $\&$ Technology). The work of the second author is supported in part by Mathematical Research Impact Centric Support (MATRICS) grant, File No.: MTR/2017/000367, by the Science and Engineering Research Board (SERB), Department of Science and Technology (DST), Government of India.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.