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# Improved Bohr radius for the class of starlike log-harmonic mappings

Molla Basir Ahamed  Molla Basir Ahamed, School of Basic Science, Indian
Institute of Technology Bhubaneswar, Bhubaneswar-752050, Odisha, India.
<EMAIL_ADDRESS>and  Vasudevarao Allu  Vasudevarao Allu, School of Basic
Science, Indian Institute of Technology Bhubaneswar, Bhubaneswar-752050,
Odisha, India<EMAIL_ADDRESS>
###### Abstract.

Let $\mathcal{H}(\mathbb{D})$ be the linear space of analytic functions on the
unit disk $\mathbb{D}=\\{z\in\mathbb{C}:|z|<1\\}$ and let
$\mathcal{B}=\\{w\in\mathcal{H}(\mathbb{D}:|w(z)|<1)\\}$. The classical Bohr’s
inequality states that if a power series $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$
converges in $\mathbb{D}$ and $|f(z)|<1$ for $z\in\mathbb{D}$, then

$\sum_{n=0}^{\infty}|a_{n}|r^{n}\leq 1\;\;\mbox{for}\;\;r\leq\frac{1}{3}$

and the constant $1/3$ is the best possible. The constant $1/3$ is known as
Bohr radius. A function $f:\mathbb{D}\rightarrow\mathbb{C}$ is said to be log-
harmonic if there is a $w\in\mathcal{B}$ such that $f$ is a non-constant
solution of the non-linear elliptic partial differential equation

$\bar{f}_{\bar{z}}(z)/\bar{f}(z)=w(z)f_{z}(z)/f(z).$

The class of log-harmonic mappings is denoted by $\mathcal{S}_{LH}$. The set
of all starlike log-harmonic mapping is defined by

$\mathcal{ST}_{LH}=\bigg{\\{}f\in\mathcal{S}_{LH}:\frac{\partial}{\partial\theta}{\rm
Arg}(f(e^{i\theta}))={\rm
Re}\left(\frac{zf_{z}-\bar{z}f_{\bar{z}}}{f}\right)>0\;\;\mbox{in}\;\;\mathbb{D}\bigg{\\}}.$

In this paper, we study several improved Bohr radius for the class
$\mathcal{ST}^{0}_{LH}$, a subclass of $\mathcal{ST}_{LH}$, consisting of
functions $f\in\mathcal{ST}_{LH}$ which map the unit disk $\mathbb{D}$ onto a
starlike domain (with respect to the origin).

###### Key words and phrases:

Analytic functions, harmonic functions, log-harmonic mappings, subordination,
improved Bohr radius

###### AMS Subject Classification: Mathematics Subject Classification:

Primary 30C45, 30C50, 30C80

##  1\. Introduction and preliminaries

A complex-valued function $f$ in $\mathbb{D}$ is said to be harmonic if it
satisfies the Laplace equation $\Delta f=4f_{z\bar{z}}=0$ in $\mathbb{D}$.
Every harmonic function $f$ in $\mathbb{D}$ has the unique canocical form
$f=h+\bar{g}$, where $h$ and $g$ are analytic in $\mathbb{D}$ with $g(0)=0$.
Every analyitc function is a harmonic function. Let $\mathcal{H}$ be the class
of all complex-valued harmonic functions $f=h+\bar{g}$ defined on
$\mathbb{D}$, where $h$ and $g$ are analytic in $\mathbb{D}$ with the
normalization $h(0)=h^{\prime}(0)-1=0$ and $g(0)=0.$ Here $h$ is called
analytic part and $g$ is called co-analytic part of $f$.

Harmonic mappings play the natural role in parameterizing minimal surfaces in
the context of differential geometry. Planner harmonic mappings have
application not only in the differential geometry but also in various field of
engineering, physics, operations research and other intriguing aspects of
applied mathematics. The theory of harmonic functions has been used to study
and solve fluid flow problems [9]. The theory of univalent harmonic functions
having prominent geometric properties like starlikeness, convexity and close-
to-convexity appear naturally while dealing with planner fluid dynamical
problems. For instance, the fluid flow problem on a convex domain satisfying
an interesting geometric property has been extensively studied by Aleman and
Constantin [9]. With the help of geometric properties of harmonic mappings,
Constantin and Martin [22] have obtained a complete solution of classifying
all two dimensional fluid flows.

Let $\mathcal{H}(\mathbb{D})$ be the class of analytic functions
$f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ in the unit disk $\mathbb{D}$. The origin
of the Bohr phenomenon lies in the seminal work by Harald Bohr [21], which
include the following result: Let $f\in\mathcal{H}(\mathbb{D})$ satisfies
$|f(z)|<1$ for all $z\in\mathbb{D}$, then
$\sum_{n=0}^{\infty}|a_{n}||z|^{n}\leq 1$ for all $z\in\mathbb{D}$ with
$|z|=r\leq 1/3$ and the constant $1/3$ is the best possible. For
$f\in\mathcal{H}(\mathbb{D})$, the majorant series is denoted by $M_{f}(r)$
and is defined by $M_{f}(r)=\sum_{n=0}^{\infty}|a_{n}||z|^{n}$. Bohr actually
obtained the inequality $M_{f}(r)\leq 1$ for $|z|\leq 1/6$, but subsequently
later, M. Riesz, I. Schur and F. Weiner, independently established this
inequality for $|z|\leq 1/3$ and the constant $1/3$ cannot be improved [39].
The constant $r_{0}=1/3$ is called the Bohr radius and the inequality
$M_{f}(r)\leq 1$ is called Bohr inequality for bounded analytic functions in
the unit disk $\mathbb{D}.$ Moreover, for the function $\phi_{a}$ defined by

$\phi_{a}(z)=\frac{a-z}{1-az},\;\;a\in[0,1)$

it follows that $M_{\phi_{a}}(r)>1$ if, and only if, $r>1/(1+2a)$, for which
$a\rightarrow 1$ shows that $1/3$ is optimal.

Using the Euclidian distance $d$, the Bohr inequality for
$f\in\mathcal{H}(\mathbb{D})$ can be written as

(1.1)
$d\left(\sum_{n=0}^{\infty}|a_{n}z^{n}|,|a_{0}|\right)=\sum_{n=1}^{\infty}|a_{n}z^{n}|\leq
1-|f(0)|=d(f(0),\partial\mathbb{D}),$

where $\partial\mathbb{D}$ is the boundary of the unit disk $\mathbb{D}$.

Let $\mathcal{M}$ be a class of analytic functions
$f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ which map the unit disk $\mathbb{D}$ into
a domain $\Omega\subset\mathbb{C}$. We say the class $\mathcal{M}$ satisfies
Bohr phenomenon if there exists $r^{*}$ such that (1.1) holds for $|z|=r\leq
r^{*}$. The largest such $r^{*}$ is called the Bohr radius for the class
$\mathcal{M}$.

In the recent years, studying Bohr inequalities become an interesting topic of
research for the functions of one as well as several complex variables. The
notion of Bohr inequality has been generalized to several complex variables
(see [5, 8, 20, 37]), to planner harmonic mappings (see [26, 30, 31]) to
polynomials (see [27]), to the solutions of elliptic partial differential
equations (see [2, 12]), to elliptic equations (see [6]), to vector valued
functions and operaotor valued functions (see [17, 18]), to analytic functions
in norm linear spaces (see [14]) and in a more abstract setting (see [7]). In
$1977$, Boas and Khavinshon [20] extended the Bohr inequality to several
complex varibales by finding multidimensional Bohr radius. Bohr’s theorem
attracted a greater interest after it was used by Dixon [24] in $1995$ to
characterize Banach algebras that satisfy von Neumann inequality. The
generalization of Bohr’s theorem become now-a-days an active topic of
research. In $2001$, Aizenberg _et al._ [8], and in $2013$ Aytuna and Djakov
[15] studied the Bohr property of holomorphic functions while Paulsen _et al._
[39] extended the Bohr inequality to Banach algebra. The relevance between
Banach theory and Bohr’s theorem was explored in [19, 23, 24].

Recently, Ali and Ng [11] have extended the classical Bohr inequality in the
Poincare disk model of hyperbolic plane. Kayumove and Ponnusamy [31] have
determined the Bohr radius for the class of analytic functions
$f(z)=z^{m}\sum_{k=0}^{\infty}a_{kp}z^{kp}$, $p\geq m\geq 0$ with $|f(z)|\leq
1$. In $2018$, Kayumov _et al._ [30] introduced the idea of $p$\- Bohr radius
for harmonic functions and obtained the $p$\- Bohr radius for the class of odd
harmonic functions. Kayumov _et al._ [30] have obtained the Bohr radius for
the class of analytic Bloch functions and harmonic functions. Alkhaleefah _et
al._ [13] have studied the Bohr radius for the class of quasi-subordinate
functions which in particular gives the classical Bohr radius. Number of
improved versions of the classical Bohr inequality have been proved in [32].

We now define Bohr radius in subordination and bounded harmonic classes. Let
$f$ and $g$ be two analytic functions in the unit disk $\mathbb{D}$. We say
that $g$ is subordinate to $f$ if there exists an anlytic function
$\phi:\mathbb{D}\rightarrow\mathbb{D}$ with $\phi(0)=0$ so that $g=f\circ\phi$
and it is denoted by $f\prec g$. If $g$ is univalent and $f(0)=g(0)$ then
$f(\mathbb{D})\subset g(\mathbb{D})$. We denote the class of all functions
subordinate to a fixed function $f$ by $\mathcal{S}(f)$ and
$f(\mathbb{D})=\Omega$. The class $\mathcal{S}(f)$ is said to have Bohr’s
phenomenon if for any $g(z)=\sum_{n=0}^{\infty}b_{n}z^{n}\in\mathcal{S}(f)$
and $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ there is a $r_{0}$ in $(0,1]$ such
that

(1.2) $\sum_{n=0}^{\infty}|b_{n}z^{n}|\leq
d(f(0),\partial\Omega)\;\;\mbox{for}\;\;|z|<r_{0}.$

In $2010$, it was established by Abu-Muhanna [3, Theorem] that the class
$\mathcal{S}(f)$ has Bohr phenomenon when $f$ is univalent in $\mathbb{D}$. In
particular, the following interesting result was obtained.

######  Theorem 1.1.

[3] If $g(z)=\sum_{n=0}^{\infty}b_{n}z^{n}\in\mathcal{S}(f)$ and
$f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ is univalent, then

(1.3) $\sum_{n=1}^{\infty}|b_{n}z^{n}|\leq
d(f(0),\partial\Omega)\;\;\mbox{for}\;\;|z|\leq r_{0}=3-\sqrt{8}=0.17157.$

Here $r_{0}$ is sharp for the Koebe function $f_{K}(z)=z/(1-z)^{2}.$

In [3], Abu-Muhanna has proved the following lemma to find the lower bound of
the distance $d(f(0),\partial\Omega)$.

######  Lemma 1.4.

[3] Let $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ be an analytic univalent function
from $\mathbb{D}$ onto a simply connected domain $\Omega$. Then

(1.5) $\frac{1}{4}|f^{\prime}(0)|\leq
d(f(0),\partial\Omega)\leq|f^{\prime}(0)|.$

Next we discuss improved Bohr radius for starlike log-harmonic mappings. A
function $f:\mathbb{D}\rightarrow\mathbb{C}$ is said to be log-harmonic if
there is a $w\in\mathcal{B}$ such that $f$ is a non-constant solution of the
non-linear elliptic partial differential equation

(1.6) $\bar{f}_{\bar{z}}(z)/\bar{f}(z)=w(z)f_{z}(z)/f(z),$

where the second dilation function $w$ is such that $|w(z)|<1$ for all
$z\in\mathbb{D}$. The Jacobian

$J_{f}=|f_{z}|^{2}-|f_{\bar{z}}|^{2}=|f_{z}|^{2}(1-|w(z)|^{2})$

is positive, and therefore all the non-constant log-harmonic mappings are
sense-preserving and open in $\mathbb{D}$.

In $2013$, Li et al. [34] proved a necessary and sufficient condition for a
function to be log-$p$-harmonic and also studied local log-$p$-harmonic
mappings. Mao et al. [38] have established Schwarz’ lemma for log-harmonic
mappings, through which they proved two versions of the Landau’s theorem for
these functions. In $2018$, Liu and Ponnusamy [35] obtained the coefficient
estimates and hence studied Bohr radius for log-harmonic mappings. Inner
mapping radius by constructing a family of $1$-slit log-harmonic mappings have
been established in [35]. Several interesting properties have been established
in [35] of log-harmonic mappings. In $2019$, Liu and Ponnusamy [36] obtained
the precise ranges of log-harmonic Koebe mapping, log-harmonic right half-
plane mapping and log-harmonic two-slits mappings. Further, the coefficient
estimates for univalent log-harmonic starlike mappings has been established in
[36].

Let $h_{0}$ and $g_{0}$ be two functions defined by

(1.7)
$h_{0}(z)=\frac{1}{1-z}\exp\left(\frac{2z}{1-z}\right)=\exp\left(\sum_{n=1}^{\infty}\left(2+\frac{1}{n}\right)z^{n}\right)$
(1.8)
$g_{0}(z)=(1-z)\exp\left(\frac{2z}{1-z}\right)=\exp\left(\sum_{n=1}^{\infty}\left(2-\frac{1}{n}\right)z^{n}\right).$

Then the function $f_{0}$ defined by

(1.9)
$f_{0}(z)=zh_{0}(z)\overline{g_{0}(z)}=\frac{z(1-\bar{z})}{1-z}\exp\left({\rm
Re}\left(\frac{4z}{1-z}\right)\right)\;\;\mbox{for}\;\;z\in\mathbb{D}$

is the log-harmonic Koebe function.

In 2011, Duman [25] obtained the upper bound for $|h(z)|$ and $|g(z)|$. In
$2016$, Ali et al. [10, Theorem 2] established the sharp lower bounds and
exhibited the corresponding extremal functions $h_{0}$, $g_{0}$ and $f_{0}$.
Ali et al. [10] extended the Bohr phenomenon to the context of starlike
univalent log-harmonic mappings of the form

(1.10) $f(z)=zh(z)\overline{g(z)}\;\;\text{in}\;\;\mathcal{ST}^{0}_{LH},$

and proved the following interesting result.

######  Theorem 1.2.

[10] Let $f$ be a function given by (1.10). Also, let $H(z)=zh(z)$ and
$G(z)=zg(z)$. Then

$\begin{cases}\displaystyle\frac{1}{2e}\leq d(0,\partial H(\mathbb{D}))\leq
1\vspace{3.5mm}\\\ \displaystyle\frac{2}{e}\leq d(0,\partial
G(\mathbb{D}))\leq 1\vspace{3.5mm}\\\ \displaystyle\frac{1}{e^{2}}\leq
d(0,\partial f(\mathbb{D}))\leq 1.\end{cases}$

Equalities occur if, and only if, $h$, $g$ and $f$ are suitable rotation of
$h_{0}$, $g_{0}$ and $f_{0}$.

In $1989$, Abdulhadi and Hengartner [2] established the sharp coefficient
bounds for the function in the class $\mathcal{ST}^{0}_{LH}$.

######  Theorem 1.3.

[2] Let $f$ be a function given by (1.10). Then

$|a_{n}|\leq 2+\frac{1}{n}\;\;\text{and}\;\;|b_{n}|\leq
2-\frac{1}{n}\;\;\text{for all}\;n\geq 1.$

Equalities hold for rotation of the function $f_{0}$.

In $2016$, Ali _et al._ [10] obtained Bohr radius for log-harmonic mappings of
the class $\mathcal{ST}^{0}_{LH}$.

######  Theorem 1.4.

[10] Let $f(z)=zh(z)\overline{g(z)}\in\mathcal{ST}^{0}_{LH}$ and $H(z)=zh(z)$
and $G(z)=zg(z)$. Then

  1. (i)

the inequality

$M_{h}(r):=|z|\exp\left(\sum_{n=1}^{\infty}|a_{n}||z|^{n}\right)\leq
d(0,\partial H(\mathbb{D}))$

holds for $|z|\leq r_{H}\approx 0.1222$, where $r_{H}$ is the unique root in
$(0,1)$ of

$\frac{r}{1-r}\exp\left(\frac{2r}{1-r}\right)=\frac{1}{2e}.$

  2. (ii)

the inequality

$M_{g}(r):=|z|\exp\left(\sum_{n=1}^{\infty}|b_{n}||z|^{n}\right)\leq
d(0,\partial G(\mathbb{D}))$

holds for $|z|\leq r_{G}\approx 0.3659$, where $r_{G}$ is the unique root in
$(0,1)$ of

$r(1-r)\exp\left(\frac{2r}{1-r}\right)=\frac{2}{e}.$

Both the radii are sharp and are attained by appropriate rotation of the
functions $H_{0}(z)=zh_{0}(z)$ and $G_{0}(z)=zg_{0}(z)$.

######  Theorem 1.5.

[10] Let $f$ be a function given by (1.10). Then for any real $t$, the
inequality

$|z|\exp\left(\sum_{n=1}^{\infty}|a_{n}+e^{it}b_{n}||z|^{n}\right)\leq
d(0,\partial f(\mathbb{D}))$

holds for $|z|\leq r_{f}\approx 0.09078$, where $r_{f}$ is the unique root in
$(0,1)$ of

$r\exp\left(\frac{4r}{1-r}\right)=\frac{1}{e^{2}}.$

The bound is sharp and is attained by suitable rotation of the log-harmonic
Koebe function $f_{0}$.

Our another interest in this paper is to study Bohr radius for the class of
analytic functions $f$ which map unit disk $\mathbb{D}$ into a concave-wedge
domain. The concave-wedge domain is defined (see [4]) by

$W_{\alpha}=\bigg{\\{}w\in\mathbb{C}:|\arg
w|<\frac{\alpha\pi}{2},\;1\leq\alpha\leq 2\bigg{\\}}.$

It is known that the conformal mapping from $\mathbb{D}$ onto $W_{\alpha}$ is
given by

(1.11)
$F_{\alpha,t}(z)=t\left(\frac{1+z}{1-z}\right)^{\alpha}=t\left(1+\sum_{n=1}^{\infty}A_{n}z^{n}\right)\;\;\mbox{for}\;1\leq\alpha\leq
2\;\mbox{and}\;\;t>0.$

It is easy to see that when $\alpha=1$, the domain turns out to be a convex
half-plane and when $\alpha=2$ it gives a slit domain. Let $S_{W_{\alpha}}$ be
the class of analytic functions $f$ which maps the unit disk $\mathbb{D}$ into
the wedge domain $W_{\alpha}$.

In $2014$, Abu-Muhana _et al._ [4] proved the following interesting result for
functions in the class $S_{W_{\alpha}}$.

######  Theorem 1.6.

[4] Let $\alpha\in[1,2]$. If
$f(z)=a_{0}+\sum_{n=1}^{\infty}a_{n}z^{n}\in\mathcal{S}_{W_{\alpha}}$ with
$a_{0}>0$, then the inequlaity

$\sum_{n=1}^{\infty}|a_{n}||z|^{n}\leq d(a_{0},\partial W_{\alpha})$

holds for $|z|\leq r_{\alpha}=(2^{1/\alpha}-1)/(2^{1/\alpha}+1)$. The function
$f=F_{\alpha,a_{0}}$ in (1.11) shows that $r_{\alpha}$ is sharp.

The following lemma is useful to prove one of our main results for functions
in the Class $S_{W_{\alpha}}$.

######  Lemma 1.12.

[4] Let $F_{\alpha,t}$ be given by (1.11), where $\alpha\in[1,2]$. Then
$A_{n}>0$ for all $n\geq 1$.

##  2\. Mian results

###  2.1. Bohr radius in subordination and bounded harmonic classes

It is natural to investigate the improved version of the Theorem 1.1. We prove
the following improved sharp Bohr radius for the class $\mathcal{S}(f)$.

######  Theorem 2.1.

Let $\beta\in[0,1/4)$. If
$g(z)=\sum_{n=0}^{\infty}b_{n}z^{n}\in\mathcal{S}(f)$ and
$f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ is univalent, then

(2.1) $\beta|f^{\prime}(0)|+\sum_{n=0}^{\infty}|b_{n}z^{n}|\leq
d(f(0),\partial\Omega)\;\;\mbox{for}\;\;|z|\leq
r_{\beta}=\frac{3-4\beta-\sqrt{8}\sqrt{1-2\beta}}{1-4\beta}.$

The radius $r_{\beta}$ is sharp for the Koebe function $f_{K}(z)=z/(1-z)^{2}.$

######  Remark 2.1.

In particular, when $\beta=0$, the radius $r_{\beta}$ which has been proved in
Theorem 2.1 coincides exactly with $r_{0}=3-\sqrt{8}=0.17159$ in Theorem 1.1.
Further, in particular, we obtain $r_{\beta}=5-2\sqrt{6}\approx 0.10102$ for
$\beta=1/8$, $r_{\beta}=9-4\sqrt{5}\approx 0.05572$ for $\beta=3/16$ and
$r_{\beta}=17-12\sqrt{2}\approx 0.02943$ for $\beta=7/32$. In fact, we see
that $\lim_{\beta\rightarrow({1}/{4})^{-}}r_{\beta}=0.$

###  2.2. Improved Bohr radius for starlike log-harmonic mappings

It is known that if $f$ is a non-vanishing log-harmonic mapping then $f$ can
be written as $f(z)=h(z)\overline{g(z)}$ where $h$ and $g$ are analytic
functions in $\mathbb{D}$. On the other hand, if $f$ vanishes at $z=0$ but is
not identically zero, then $f$ admits the following representation

(2.2) $f(z)=z^{m}|z|^{2\beta m}h(z)\overline{g(z)}$

where $m$ is a non-negative integer and ${\rm Re}\;\beta>-1/2$, and $h$, $g$
are analytic functions in $\mathbb{D}$ with $g(0)=1$ and $h(0)\neq 1$ (see
[1]). The exponent $\beta$ in (2.2) depends only on $w(0)$ and it can be
expresses as

$\beta=\overline{w(0)}\frac{1+w(0)}{1-|w(0)|^{2}}.$

Note that $f(0)\neq 0$ if, and only if, $m=0$, and that a univalent log-
harmonic mapping on $\mathbb{D}$ vanishes at the origin if, and only if,
$m=1$. Univalent log-harmonic mappings have been studied extensively by many
researchers (see [10, 25, 29]). The class of log-harmonic mappings is denoted
by $\mathcal{S}_{LH}$. Let $z|z|^{2\beta}h(z)\overline{g(z)}$ be a log-
harmonic univalent function. We say that $f$ is a starlike log-harmonic
mapping if

(2.3) $\frac{\partial}{\partial\theta}{\rm Arg}(f(e^{i\theta}))={\rm
Re}\left(\frac{zf_{z}-\bar{z}f_{\bar{z}}}{f}\right)>0\;\;\mbox{in}\;\;\mathbb{D}$

and we denote the set of all strlike log-harmonic functions by
$\mathcal{ST}_{LH}$. Let $\mathcal{ST}^{0}_{LH}$ be a subclass of
$\mathcal{ST}_{LH}$, consisiting of functions $f\in\mathcal{ST}_{LH}$ which
map the unit disk $\mathbb{D}$ onto a starlike domain (with repsect to the
origin).

Our main aim is to study Bohr radius for the class of sense-preserving
satrlike log-harmonic mappings in $\mathbb{D}$ of the form
$f(z)=zh(z)\overline{g(z)}$ with

$h(z)=\exp\left(\sum_{n=1}^{\infty}a_{k}z^{k}\right)\;\;\mbox{and}\;\;g(z)=\exp\left(\sum_{n=1}^{\infty}b_{n}z^{n}\right),$

where $h(z)$ and $g(z)$ may be called as analytic and co-analytic factors of
the function $f(z).$

We prove the following improved Bohr radius for functions in the class
$\mathcal{ST}^{0}_{LH}.$

######  Theorem 2.2.

Let $f$ be a function given by (1.10). Then for any real $t$, the inequality

$|z|\exp\left(\sum_{n=1}^{\infty}\bigg{|}a_{n}+e^{it}b_{n}+\frac{n}{4n^{2}-1}a_{n}b_{n}\bigg{|}|z|^{n}\right)\leq
d(0,\partial f(\mathbb{D}))$

holds for $|z|\leq r_{f}\approx 0.08528$, where $r_{f}$ is the unique root in
$(0,1)$ of

(2.4)
$\frac{r}{1-r}\exp\left(\frac{4r}{1-r}\right)=\frac{1}{e^{2}}\;\;\mbox{in}\;\;(0,1).$

The radius $r_{f}$ is sharp and is attained by a suitable rotation of the log-
harmonic Koebe function $f_{0}$ given by (1.9).

Figure 1. The radius $r_{f}\approx 0.08528$ is a root of (2.4) in $(0,1)$.

Figure 2. Image of unit disk $\mathbb{D}$ under the Koebe function
$f(z)=\frac{z}{(1-z)^{2}}$ and log-harmonic Koebe function
$f_{0}(z)=\frac{z(1-\bar{z})}{1-z}\exp\left({\rm
Re}\left(\frac{4z}{1-z}\right)\right)$.

Figure 3. Image of unit disk $\mathbb{D}$ under the map
$h_{0}(z)=\frac{1}{1-z}\exp\left(\frac{2z}{1-z}\right)$ and
$g_{0}(z)=(1-z)\exp\left(\frac{2z}{1-z}\right)$.

Next we prove the sharp Bohr radius for the class $\mathcal{ST}^{0}_{LH}$ in
view of additional terms $|a_{n}|^{2}$ and $|b_{n}|^{2}$ in the series
expansion of $h$ and $g$ respectively.

######  Theorem 2.3.

Let $f$ be a function given by (1.10) and $H(z)=zh(z)$ and $G(z)=zg(z)$. Then

  1. (i)

the inequality

$|z|\exp\left(\sum_{n=1}^{\infty}\left(|a_{n}|+\frac{n}{(2n+1)^{2}}|a_{n}|^{2}\right)|z|^{n}\right)\leq
d(0,\partial H(\mathbb{D}))$

holds for $|z|\leq r_{H}\approx 0.09735$, where $r_{H}$ is the unique root of

(2.5)
$\frac{r}{(1-r)^{2}}\exp\left(\frac{2r}{1-r}\right)=\frac{1}{2e}\;\;\mbox{in}\;\;(0,1).$

Figure 4. The radii $r_{H}\approx 0.09735$ and $r_{G}\approx 0.30539$ are
roots of (2.5) and (2.6) respectively in $(0,1)$.

  2. (ii)

the inequality

$|z|\exp\left(\sum_{n=1}^{\infty}\left(|b_{n}|+\frac{n}{(2n-1)^{2}}|b_{n}|^{2}\right)|z|^{n}\right)\leq
d(0,\partial G(\mathbb{D}))$

holds for $|z|\leq r_{G}\approx 0.30539$, where $r_{G}$ is the unique root of

(2.6) $r\exp\left(\frac{2r}{1-r}\right)=\frac{2}{e}\;\;\mbox{in}\;\;(0,1).$

Both the radii are sharp and are attained by appropriate rotation of
$H_{0}(z)=zh_{0}(z)$ and $G_{0}(z)=zg_{0}(z)$.

We prove the next improved sharp Bohr radius for the class
$\mathcal{ST}^{0}_{LH}$ adding $|H(z)|$ and $|G(z)|$ with $M_{h}(r)$ and
$M_{g}(r)$ respecively.

######  Theorem 2.4.

Let $f$ be a function given by (1.10) and $H(z)=zh(z)$ and $G(z)=zg(z)$. Then

  1. (i)

the inequality

$|H(z)|+|z|\exp\left(\sum_{n=1}^{\infty}|a_{n}||z|^{n}\right)\leq d(0,\partial
H(\mathbb{D}))$

holds for $|z|\leq r_{H}\approx 0.1073$, where $r_{H}$ is the unqiue root of

(2.7)
$r\left(\frac{2r}{1-r}-\log(1-r)+\frac{1}{1-r}\exp\left(\frac{2r}{1-r}\right)\right)=\frac{1}{2e}\;\;\mbox{in}\;\;(0,1).$

  2. (ii)

the inequality

$|G(z)|+|z|\exp\left(\sum_{n=1}^{\infty}|b_{n}||z|^{n}\right)\leq d(0,\partial
G(\mathbb{D}))$

holds for $|z|\leq r_{G}\approx 0.3063$, where $r_{G}$ is the unique root of

(2.8)
$r\left(\frac{2r}{1-r}+\log(1-r)+(1-r)\exp\left(\frac{2r}{1-r}\right)\right)=\frac{2}{e}\;\;\mbox{in}\;\;(0,1).$

Both the radii are sharp and are attained by appropriate rotation of
$H_{0}(z)=zh_{0}(z)$ and $G_{0}(z)=zg_{0}(z)$.

Figure 5. The radii $r_{H}\approx 0.1073$ and $r_{G}\approx 0.3063$ are roots
of (2.7) and (2.8) respectively in $(0,1)$.

For any positive integer $m$, considering $|h(z)|^{m}$ and $|g(z)|^{m}$, next
we prove the improved sharp Bohr radius for the class $\mathcal{ST}^{0}_{LH}$.

######  Theorem 2.5.

Let $f$ be a function given by (1.10) and $H(z)=zh(z)$ and $G(z)=zg(z)$.

  1. (i)

If $|h(z)|\leq 1$, then for any $m\in\mathbb{N}$, the inequality

$|z|\exp\left(|h(z)|^{m}+\sum_{n=1}^{\infty}|a_{n}||z|^{n}\right)\leq
d(0,\partial H(\mathbb{D}))$

holds for $|z|\leq r_{H}\approx 0.0566$, where $r_{H}$ is the unique root of

(2.9)
$\frac{re}{1-r}\exp\left(\frac{2r}{1-r}\right)=\frac{1}{2e}\;\;\mbox{in}\;\;(0,1).$

  2. (ii)

If $|g(z)|\leq 1$, then for any $m\in\mathbb{N}$, the inequality

$|z|\exp\left(|g(z)|^{m}+\sum_{n=1}^{\infty}|b_{n}||z|^{n}\right)\leq
d(0,\partial G(\mathbb{D}))$

holds for $|z|\leq r_{G}\approx 0.1764$, where $r_{G}$ is the unique root of

(2.10)
$re(1-r)\exp\left(\frac{2r}{1-r}\right)=\frac{2}{e}\;\;\mbox{in}\;\;(0,1).$

Both the radii are sharp and are attained by a suitable rotation of
$H_{0}(z)=zh_{0}(z)$ and $G_{0}(z)=zg_{0}(z)$.

Figure 6. The radii $r_{H}\approx 0.0566$ and $r_{G}\approx 0.1764$ are roots
of (2.9) and (2.10) respectively in $(0,1)$.

######  Remark 2.2.

It is worth to notice in Theorem 2.5 that the Bohr radii $r_{H}$ and $r_{G}$
are independent of the choice of the positive integer $m$.

We prove the improved sharp Bohr radius adding $|h(z)+g(z)|$ with the series
$\sum_{n=1}^{\infty}|a_{n}+e^{it}b_{n}||z|^{n}$ for the class
$\mathcal{ST}^{0}_{LH}$.

######  Theorem 2.6.

Let $f$ be a function given by (1.10) with $|h(z)|+|g(z)|\leq 1$. Then for any
real $t$, the inequality

$|z|\exp\left(|h(z)+g(z)|+\sum_{n=1}^{\infty}|a_{n}+e^{it}b_{n}||z|^{n}\right)\leq
d(0,\partial f(\mathbb{D}))$

holds for $|z|\leq r_{f}\approx 0.04181$, where $r_{f}$ is the unique root of

(2.11)
$er\exp\left(\frac{4r}{1-r}\right)=\frac{1}{e^{2}}\;\;\mbox{in}\;\;(0,1).$

The Bohr radius $r_{f}$ is sharp and is attained by suitable rotation of the
log-harmonic Koebe function $f_{0}$.

Next we prove the improved sharp Bohr radius for the class
$\mathcal{ST}^{0}_{LH}$ adding $|f(z)|$.

######  Theorem 2.7.

Let $f$ be a function given by (1.10) with $|h(z)|\leq 1$ and $|g(z)|\leq 1$.
Then for any real $t$, the inequality

$|f(z)|+|z|\exp\left(\sum_{n=1}^{\infty}|a_{n}+e^{it}b_{n}||z|^{n}\right)\leq
d(0,\partial f(\mathbb{D}))$

holds for $|z|\leq r_{f}\approx 0.0592$, where $r_{f}$ is the unique root of

(2.12)
$r\left(1+\exp\left(\frac{4r}{1-r}\right)\right)=\frac{1}{e^{2}}\;\;\mbox{in}\;\;(0,1).$

The Bohr radius $r_{f}$ is sharp for a suitable rotation of the log-harmonic
Koebe function $f_{0}$.

Figure 7. The radius $r_{f}\approx 0.0592$ is the root of (2.12) in $(0,1)$.

###  2.3. Bohr radius for concave-wedge domain

We prove the following result which is an improved version of Theorem 1.6.

######  Theorem 2.8.

Let $\alpha\in[1,2]$ and $\beta\in[0,2)$. If
$f(z)=a_{0}+\sum_{n=1}^{\infty}a_{n}z^{n}\in S_{W_{\alpha}}$ with $a_{0}>0$,
then the inequlaity

$\frac{2\beta a_{0}}{\alpha\pi}|\arg
f(z)|+\sum_{n=1}^{\infty}|a_{n}||z|^{n}\leq d(a_{0},\partial W_{\alpha})$

holds for $|z|\leq
r_{\alpha,\beta}=((2-\beta)^{1/\alpha}-1)/((2-\beta)^{1/\alpha}+1)$. The
function $f=F_{\alpha,a_{0}}$ in (1.11) shows that $r_{\alpha,\beta}$ is
sharp.

######  Remark 2.3.

Since $W_{\alpha}$ turns out to be a convex half-plane when $\alpha=1$, it is
evident that, for $\alpha=1$ and $\beta=0$, the radius $r_{\alpha,\beta}$
coincides exactly with the Bohr radius $1/3$.

Figure 8. Image of unit disk $\mathbb{D}$ under the maps $F_{1,1}(z)$,
$F_{1.5,20}(z)$ and $F_{2,3}(z)$ repsectively.

##  3\. Proof of the main results

###### Proof of Theorem 2.1.

Since $g(z)=\sum_{n=0}^{\infty}b_{n}z^{n}\in S(f)$ and
$f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ are univalent, by the famous well-known
de Brandge’s theorem [28, p.17], we have

(3.1) $|b_{n}|\leq n|f^{\prime}(0)|.$

Therefore, from (1.5) and (3.1), it is easy to see that

(3.2) $|b_{n}|\leq 4nd(f(0),\partial\Omega).$

By a simple computation using (3.2), we obtain

$\displaystyle\beta|f^{\prime}(0)|+\sum_{n=1}^{\infty}|b_{n}z^{n}|$
$\displaystyle=\beta|f^{\prime}(0)|+\sum_{n=1}^{\infty}|b_{n}||z|^{n}$
$\displaystyle\leq 4\beta
d(f(0),\partial\Omega)+4d(f(0),\partial\Omega)\sum_{n=1}^{\infty}nr^{n}$
$\displaystyle=4d(f(0),\partial\Omega)\left(\beta+\frac{r}{(1-r)^{2}}\right)$
$\displaystyle\leq d(f(0),\partial\Omega)$

if, and only if,

$4\left(\beta+\frac{r}{(1-r)^{2}}\right)\leq 1.$

Therefore, (2.1) holds for

$|z|\leq r_{0}=\frac{3-4\beta-\sqrt{8}\sqrt{1-2\beta}}{1-4\beta}.$

Since

$f_{K}(z)=\frac{z}{(1-z)^{2}}=\sum_{n=1}^{\infty}nz^{n}\;\;\text{and}\;\;d(f(0).\partial\Omega)=\frac{1}{4},$

a simple computation shows that

$\displaystyle\beta|f_{K}^{\prime}(0)|+\sum_{n=1}^{\infty}|b_{n}z^{n}|$
$\displaystyle=\beta|f_{K}^{\prime}(0)|+\sum_{n=1}^{\infty}|b_{n}|r_{0}^{n}$
$\displaystyle=4\beta
d(f_{K}(0),\partial\Omega)+4d(f_{K}(0),\partial\Omega)\sum_{n=1}^{\infty}nr_{0}^{n}$
$\displaystyle=4d(f_{K}(0),\partial\Omega)\left(\beta+\frac{r_{o}}{(1-r_{0})^{2}}\right)$
$\displaystyle=d(f(0),\partial\Omega).$

This shows that the radius $r_{0}$ is the best possible. This completes the
proof. ∎

###### Proof of Theorem 2.2.

Let $f(z)=zh(z)\overline{g(z)}\in\mathcal{ST}^{0}_{LH}$. Then in view of
Theorem 1.3, we have the following coefficient bounds

$|a_{n}|\leq 2+\frac{1}{n}\;\;\text{and}\;\;|b_{n}|\leq
2-\frac{1}{n}\;\;\text{for all}\;n\geq 1.$

On the other hand, from Theorem 1.2, we have

$d(0,\partial f(\mathbb{D}))\geq\frac{1}{e^{2}}.$

Therefore, a simple computation shows that

$\displaystyle
r\exp\left(\sum_{n=1}^{\infty}|a_{n}|r^{n}+\sum_{n=1}^{\infty}|b_{n}|r^{n}+\sum_{n=1}^{\infty}\frac{n}{4n^{2}-1}|a_{n}||b_{n}|r^{n}\right)$
$\displaystyle\leq
r\exp\left(\sum_{n=1}^{\infty}\left(2+\frac{1}{n}\right)r^{n}+\sum_{n=1}^{\infty}\left(2-\frac{1}{n}\right)r^{n}+\sum_{n=1}^{\infty}\frac{n}{4n^{2}-1}\left(2+\frac{1}{n}\right)\left(2-\frac{1}{n}\right)r^{n}\right)$
$\displaystyle=r\exp\left(4\sum_{n=1}^{\infty}r^{n}+\sum_{n=1}^{\infty}\frac{r^{n}}{n}\right)$
$\displaystyle=r\exp\left(\frac{4r}{1-r}-\log(1-r)\right)$ $\displaystyle\leq
d(0,\partial f(\mathbb{D}))$

if, and only if,

$r\exp\left(\frac{4r}{1-r}-\log(1-r)\right)\leq\frac{1}{e^{2}}$

which is equivalent to

$\frac{r}{1-r}\exp\left(\frac{4r}{1-r}\right)\leq\frac{1}{e^{2}}.$

The Bohr radius $r_{f}$ is the unique root of the equation

$\frac{r}{1-r}\exp\left(\frac{4r}{1-r}\right)=\frac{1}{e^{2}}$

in $(0,1),$ a simple com putation shows that $r_{f}\approx 0.08528$.

In order to show the sharpness of $r_{f}$, let $h_{0}$, $g_{0}$ and $f_{0}$ be
given by (1.7), (1.8) and (1.9) repsectively. For these functions, it is easy
to see that

(3.3) $|a_{n}|=2+\frac{1}{n},\;\;|b_{n}|=2-\frac{1}{n}\;\;\text{for
all}\;n\in\mathbb{N}\;\;\text{and}\;\;d(0,\partial
f_{0}(\mathbb{D}))=\frac{1}{e^{2}}.$

A simple computation using (3.3) shows that

$\displaystyle
r_{f}\exp\left(\sum_{n=1}^{\infty}|a_{n}|r_{f}^{n}+\sum_{n=1}^{\infty}|b_{n}|r_{f}^{n}+\sum_{n=1}^{\infty}\frac{n}{4n^{2}-1}|a_{n}||b_{n}|r_{f}^{n}\right)$
$\displaystyle=r_{f}\exp\left(\sum_{n=1}^{\infty}\left(2+\frac{1}{n}\right)r_{f}^{n}+\sum_{n=1}^{\infty}\left(2-\frac{1}{n}\right)r_{f}^{n}+\sum_{n=1}^{\infty}\frac{n}{4n^{2}-1}\left(2+\frac{1}{n}\right)\left(2-\frac{1}{n}\right)r_{f}^{n}\right)$
$\displaystyle=r_{f}\exp\left(4\sum_{n=1}^{\infty}r_{f}^{n}+\sum_{n=1}^{\infty}\frac{r_{f}^{n}}{n}\right)$
$\displaystyle=r_{f}\exp\left(\frac{4r_{f}}{1-r_{f}}-\log(1-r_{f})\right)$
$\displaystyle=\frac{1}{e^{2}}$ $\displaystyle=d(0,\partial
f_{0}(\mathbb{D})).$

Therefore, the radius $r_{f}$ is the best possible. ∎

###### Proof of Theorem 2.3.

Let $f(z)=zh(z)\overline{g(z)}\in\mathcal{ST}^{0}_{LH}$, $H(z)=zh(z)$ and
$G(z)=zg(z).$ (i) In view of Theorem 1.2 and Theorem 1.3, we have the
following sharp coefficient estimates

$|a_{n}|\leq 2+\frac{1}{n}\;\;\text{for all}\;\;n\geq
1\;\;\text{and}\;\;d(0,\partial H(\mathbb{D}))\geq\frac{1}{2e}$

and equality holds for the function $h_{0}(z)$ and $H(z)=zh_{0}(z)$
respectively. A simple computation shows that

$\displaystyle|z|\exp\left(\sum_{n=1}^{\infty}\left(|a_{n}|+\frac{n}{(2n+1)^{2}}|a_{n}|^{2}\right)|z|^{n}\right)$
$\displaystyle\leq
r\exp\left(2\sum_{n=1}^{\infty}r^{n}+2\sum_{n=1}^{\infty}\frac{r^{n}}{n}\right)$
$\displaystyle=r\exp\left(\frac{2r}{1-r}-2\log(1-r)\right)$
$\displaystyle=\frac{r}{(1-r)^{2}}\exp\left(\frac{2r}{1-r}\right)$
$\displaystyle\leq d(0,\partial H(\mathbb{D}))$

if, and only if,

$\frac{r}{(1-r)^{2}}\exp\left(\frac{2r}{1-r}\right)\leq\frac{1}{2e}.$

Therefore, the Bohr radius $r_{H}$ is the unique root of the equation

$\frac{r}{(1-r)^{2}}\exp\left(\frac{2r}{1-r}\right)=\frac{1}{2e}$

in $(0,1)$, a computation shows that $r_{H}\approx 0.09735.$

To show the sharpness of the radius $r_{H}$, let $H_{0}(z)=zh_{0}(z)$, where
$h_{0}$ is given by (1.7). It is easy to see that

$\displaystyle|z|\exp\left(\sum_{n=1}^{\infty}\left(|a_{n}|+\frac{n}{(2n+1)^{2}}|a_{n}|^{2}\right)|z|^{n}\right)$
$\displaystyle=r_{H}\exp\left(2\sum_{n=1}^{\infty}r_{H}^{n}+2\sum_{n=1}^{\infty}\frac{r_{H}^{n}}{n}\right)$
$\displaystyle=r\exp\left(\frac{2r}{1-r_{H}}-2\log(1-r_{H})\right)$
$\displaystyle=\frac{r_{H}}{(1-r_{H})^{2}}\exp\left(\frac{2r_{H}}{1-r_{H}}\right)$
$\displaystyle\vspace{2mm}=\frac{1}{2e}$
$\displaystyle\vspace{2mm}=d(0,\partial H_{0}(\mathbb{D})).$

Therefore, the radius $r_{H}$ is the best possible.

(ii) In view of Theorem 1.2 and Theorem 1.3, we have

$|b_{n}|\leq 2-\frac{1}{n}\;\;\text{for all}\;\;n\geq
1\;\;\text{and}\;\;d(0,\partial G(\mathbb{D}))\geq\frac{2}{e}.$

Both the equalities hold for the function $g_{0}(z)$ and $G_{0}(z)=zg_{0}(z)$
respectively.

A simple computation shows that

$\displaystyle|z|\exp\left(\sum_{n=1}^{\infty}\left(|b_{n}|+\frac{n}{(2n-1)^{2}}|b_{n}|^{2}\right)|z|^{n}\right)$
$\displaystyle\leq r\exp\left(2\sum_{n=1}^{\infty}r^{n}\right)$
$\displaystyle=r\exp\left(\frac{2r}{1-r}\right)$ $\displaystyle\leq
d(0,\partial G(\mathbb{D}))$

if, and only if,

$r\exp\left(\frac{2r}{1-r}\right)\leq\frac{2}{e}.$

Therefore, the Bohr radius $r_{G}$ is the unique root of the equation

$r\exp\left(\frac{2r}{1-r}\right)=\frac{2}{e}$

in $(0,1)$ which can be computed as $r_{G}\approx 0.30539.$

To show the sharpness of the radius $r_{G}$, let $G_{0}(z)=zg_{0}(z)$, where
$g_{0}$ is given by (1.7). It is easy to see that

$\displaystyle|z|\exp\left(\sum_{n=1}^{\infty}\left(|b_{n}|+\frac{n}{(2n-1)^{2}}|b_{n}|^{2}\right)|z|^{n}\right)$
$\displaystyle=r_{G}\exp\left(2\sum_{n=1}^{\infty}r_{G}^{n}\right)$
$\displaystyle=r_{G}\exp\left(\frac{2r_{G}}{1-r_{G}}\right)$
$\displaystyle=\frac{2}{e}$ $\displaystyle=d(0,\partial G_{0}(\mathbb{D})).$

Therefore, the radius $r_{G}$ is the best possible. ∎

###### Proof of Theorem 2.5.

Let $f(z)=zh(z)\overline{g(z)}\in\mathcal{ST}^{0}_{LH}$, $H(z)=zh(z)$ and
$G(z)=zg(z)$.

(i) In view of Theorem 1.2 and Theorem 1.3 and using the fact that $|h(z)|\leq
1$, for $m\in\mathbb{N}$, we obtain

$\displaystyle|z|\exp\left(|h(z)|^{m}+\sum_{n=1}^{\infty}|a_{n}||z|^{n}\right)$
$\displaystyle=|z|\exp(|h(z)|^{m})\exp\left(\sum_{n=1}^{\infty}|a_{n}||z|^{n}\right)$
$\displaystyle\leq re\exp\left(\frac{2r}{1-r}-2\log(1-r)\right)$
$\displaystyle=\frac{re}{1-r}\exp\left(\frac{2r}{1-r}\right)$
$\displaystyle\leq d(0,\partial H(\mathbb{D}))$

if, and only if,

$\frac{re}{1-r}\exp\left(\frac{2r}{1-r}\right)\leq\frac{1}{2e}.$

Therefore, the Bohr radius $r_{H}$ is the solution of

$\frac{er}{1-r}\exp\left(\frac{2r}{1-r}\right)=\frac{1}{2e}.$

It is easy to see that $r_{H}\approx 0.0566$. The radius $r_{H}$ is best
possible and it can be shown by using the function $H_{0}(z)=zh_{0}(z)$.

(ii) Since $|g(z)|\leq 1$, in view of Theorem 1.2 and Theorem 1.3, by a simple
computation, we obtain

$\displaystyle|z|\exp\left(|g(z)|^{m}+\sum_{n=1}^{\infty}|b_{n}||z|^{n}\right)=$
$\displaystyle|z|\exp\left(|g(z)|^{m}\right)\exp\left(\sum_{n=1}^{\infty}|b_{n}||z|^{n}\right)$
$\displaystyle\leq
er\exp\left(\sum_{n=1}^{\infty}\left(2-\frac{1}{n}\right)r^{n}\right)$
$\displaystyle=re\exp\left(\frac{2r}{1-r}+\log(1-r)\right)$
$\displaystyle=re(1-r)\exp\left(\frac{2r}{1-r}\right)$ $\displaystyle\leq
d(0,\partial G(\mathbb{D}))$

if, and only if,

$re(1-r)\exp\left(\frac{2r}{1-r}\right)\leq\frac{2}{e}.$

Therefore, the Bohr radius $r_{G}$ is the unieque root of the equation

$re(1-r)\exp\left(\frac{2r}{1-r}\right)=\frac{2}{e}.$

A simple computation shows $r_{G}\approx 0.1764$. The radius $r_{G}$ is the
best possible which can be shown by considering the function
$G_{0}(z)=zg_{0}(z)$, where $g_{0}(z)$ is defined in (1.8). ∎

###### Proof of Theorem 2.6.

In view of Theorem 1.3, we have the following sharp coefficient bounds

$|a_{n}|\leq 2+\frac{1}{n}\;\;\text{and}\;\;|b_{n}|\leq
2-\frac{1}{n}\;\;\text{for all}\;n\geq 1,$

which are attained by the function $h_{0}$ and $g_{0}$ respectively defined in
(1.7) and (1.8). On the other hand, by Theorem 1.2, we have the sharp distance

$d(0,\partial f(\mathbb{D}))\geq\frac{1}{e^{2}}$

which is attained by the function $f_{0}$ defined in (1.9).

Since $|h(z)|+|g(z)|\leq 1$, we obtain

$\displaystyle|z|\exp\left(|h(z)+g(z)|+\sum_{n=1}^{\infty}|a_{n}||z|^{n}+\sum_{n=1}^{\infty}|b_{n}||z|^{n}\right)$
$\displaystyle=r\exp\left(|h(z)|+|g(z)|\right)\exp\left(\sum_{n=1}^{\infty}|a_{n}|r^{n}+\sum_{n=1}^{\infty}|b_{n}|r^{n}\right)$
$\displaystyle\leq er\exp\left(4\sum_{n=1}^{\infty}|a_{n}|r^{n}\right)$
$\displaystyle=er\exp\left(\frac{4r}{1-r}\right)$ $\displaystyle\leq
d(0,\partial f(\mathbb{D}))$

if, and only if,

$er\exp\left(\frac{4r}{1-r}\right)\leq\frac{1}{e^{2}}.$

Therefore, the Bohr radius $r_{f}$ is the unique root of the equation

$er\exp\left(\frac{4r}{1-r}\right)=\frac{1}{e^{2}}$

in $(0,1)$ which yields $r_{f}\approx 0.04181$. The sharpness of the radius
$r_{f}$ can be shown by considering a suitable rotation of the log-harmonic
Koebe function $f_{0}$. This completes the proof. ∎

###### Proof of Theorem 2.7.

Since $|h(z)|\leq 1$ and $|g(z)|\leq 1$, in view of Theorem 1.2 and Theorem
1.3, we obtain

$\displaystyle|f(z)|+|z|\exp\left(\sum_{n=1}^{\infty}|a_{n}||z|^{n}+\sum_{n=1}^{\infty}|b_{n}||z|^{n}\right)$
$\displaystyle\leq
r|h(z)||g(z)|+r\exp\left(\sum_{n=1}^{\infty}|a_{n}|r^{n}+\sum_{n=1}^{\infty}|b_{n}|r^{n}\right)$
$\displaystyle\leq r+r\exp\left(4\sum_{n=1}^{\infty}\frac{r^{n}}{r}\right)$
$\displaystyle=r+r\exp\left(\frac{4r}{1-r}\right)$ $\displaystyle\leq
d(0,\partial f(\mathbb{D}))$

if, and only if,

$r\left(1+\exp\left(\frac{4r}{1-r}\right)\right)\leq\frac{1}{e^{2}}.$

Thus, the Bohr radius $r_{f}$ is the unique root of the equation

$r\left(1+\exp\left(\frac{4r}{1-r}\right)\right)=\frac{1}{e^{2}}$

which gives $r_{f}\approx 0.0592$. The radius $r_{f}$ is sharp and can be
shown by considering a suitable rotation of the log-harmonic Koebe function
$f_{0}.$ This completes the proof. ∎

###### Proof of Theorem 2.4.

By Theorem 1.2, we have

(3.4) $d(0,\partial
H(\mathbb{D}))\geq\frac{1}{2e}\;\;\text{and}\;\;d(0,\partial
G(\mathbb{D}))\geq\frac{2}{e}$

and the sharp coefficient bounds

$|a_{n}|\leq 2+\frac{1}{n}\;\;\text{and}\;\;|b_{n}|\leq
2-\frac{1}{n}\;\;\text{for all}\;n\geq 1.$

All the inequlities are attained by the extremal functions
$H_{0}(z)=zh_{0}(z)$ and $G_{0}(z)=zg_{0}(z)$, where $h_{0}$ and $g_{0}$ are
defined respectively in (1.7) and (1.8).

(i) Using Theorem 1.3 and (3.4), we obtain

$\displaystyle|H(z)|+|z|\exp\left(\sum_{n=1}^{\infty}|a_{n}||z|^{n}\right)$
$\displaystyle\leq|z|\left(\sum_{n=1}^{\infty}|a_{n}||z|^{n}\right)+|z|\exp\left(\sum_{n=1}^{\infty}|a_{n}||z|^{n}\right)$
$\displaystyle\leq
r\left(\sum_{n=1}^{\infty}\left(2+\frac{1}{n}\right)r^{n}\right)+r\exp\left(\sum_{n=1}^{\infty}\left(2+\frac{1}{n}\right)r^{n}\right)$
$\displaystyle=r\left(\frac{2r}{1-r}-\log(1-r)\right)+r\exp\left(\frac{2r}{1-r}-\log(1-r)\right)$
$\displaystyle\leq d(0,\partial H(\mathbb{D}))$

if, and only if,

$r\left(\frac{2r}{1-r}-\log(1-r)\right)+r\exp\left(\frac{2r}{1-r}-\log(1-r)\right)\leq\frac{1}{2e}.$

Therefore, the Bohr radius $r_{H}$ is the unique root of the equation

$r\left(\frac{2r}{1-r}-\log(1-r)\right)+r\exp\left(\frac{2r}{1-r}-\log(1-r)\right)=\frac{1}{2e}$

in $(0,1)$ which shows that $r_{H}\approx 0.1073$. In order to show the radius
$r_{H}$ is sharp, we consider the function $H_{0}(z)=zh_{0}(z)$, where $h_{0}$
is defined in (1.7). Therefore, a simple compuatation shows that

$\displaystyle|H_{0}(z)|+|z|\exp\left(\sum_{n=1}^{\infty}|a_{n}||z|^{n}\right)$
$\displaystyle=|z|\left(\sum_{n=1}^{\infty}|a_{n}||z|^{n}\right)+|z|\exp\left(\sum_{n=1}^{\infty}|a_{n}||z|^{n}\right)$
$\displaystyle=r_{H}\left(\sum_{n=1}^{\infty}\left(2+\frac{1}{n}\right)r_{H}^{n}\right)+r_{H}\exp\left(\sum_{n=1}^{\infty}\left(2+\frac{1}{n}\right)r_{H}^{n}\right)$
$\displaystyle=r_{H}\left(\frac{2r_{H}}{1-r_{H}}-\log(1-r_{H})\right)+r_{H}\exp\left(\frac{2r_{H}}{1-r_{H}}-\log(1-r_{H})\right)$
$\displaystyle=\frac{1}{2e}\vspace{2mm}$ $\displaystyle=d(0,\partial
H(\mathbb{D})).$

This shows that $r_{H}$ is the best possible.

(ii) Using Theorem 1.3 and (3.4), we obtain

$\displaystyle|G(z)|+|z|\exp\left(\sum_{n=1}^{\infty}|b_{n}||z|^{n}\right)$
$\displaystyle\leq|z|\left(\sum_{n=1}^{\infty}|b_{n}||z|^{n}\right)+|z|\exp\left(\sum_{n=1}^{\infty}|b_{n}||z|^{n}\right)$
$\displaystyle\leq
r\left(\sum_{n=1}^{\infty}\left(2-\frac{1}{n}\right)r^{n}\right)+r\exp\left(\sum_{n=1}^{\infty}\left(2+\frac{1}{n}\right)r^{n}\right)$
$\displaystyle=r\left(\frac{2r}{1-r}+\log(1-r)\right)+r\exp\left(\frac{2r}{1-r}+\log(1-r)\right)$
$\displaystyle\leq d(0,\partial H(\mathbb{D}))$

if, and only if,

$r\left(\frac{2r}{1-r}+\log(1-r)\right)+r\exp\left(\frac{2r}{1-r}+\log(1-r)\right)\leq\frac{2}{e}.$

Therefore, the Bohr radius $r_{G}$ is the unique root of the equation

$r\left(\frac{2r}{1-r}-\log(1-r)\right)+r\exp\left(\frac{2r}{1-r}-\log(1-r)\right)=\frac{2}{e}$

in $(0,1)$ which yeilds $r_{G}\approx 0.3063$. To show the radius $r_{G}$ is
best possible, we consider the function $G_{0}(z)=zg_{0}(z)$, where $g_{0}$ is
defined in (1.8).

Thus, it is easy to see that

$\displaystyle|G_{0}(z)|+|z|\exp\left(\sum_{n=1}^{\infty}|b_{n}||z|^{n}\right)$
$\displaystyle=|z|\left(\sum_{n=1}^{\infty}|b_{n}||z|^{n}\right)+|z|\exp\left(\sum_{n=1}^{\infty}|b_{n}||z|^{n}\right)$
$\displaystyle=r_{G}\left(\sum_{n=1}^{\infty}\left(2-\frac{1}{n}\right)r_{G}^{n}\right)+r_{G}\exp\left(\sum_{n=1}^{\infty}\left(2+\frac{1}{n}\right)r_{G}^{n}\right)$
$\displaystyle=r_{G}\left(\frac{2r_{G}}{1-r_{G}}+\log(1-r_{G})\right)+r_{G}\exp\left(\frac{2r_{G}}{1-r_{G}}+\log(1-r_{G})\right)$
$\displaystyle=\frac{2}{e}$ $\displaystyle=d(0,\partial G(\mathbb{D}))$

which shows that $r_{G}$ is sharp. ∎

###### Proof of Theorem 2.8.

Since $f\in S_{W_{\alpha}}$, we have $|\arg f(z)|\leq\pi\alpha/2$. Therefore,
in view of Lemma 1.4, it is easy to see that

$\displaystyle\frac{2\beta a_{0}}{\pi\alpha}|\arg
f(z)|+\sum_{n=1}^{\infty}|a_{n}z^{n}|$ $\displaystyle\leq\frac{2\beta
a_{0}}{\pi\alpha}\frac{\pi\alpha}{2}+a_{0}\sum_{n=1}^{\infty}A_{n}r^{n}$
$\displaystyle=a_{0}\left(\sum_{n=1}^{\infty}A_{n}r^{n}+\beta\right)$
$\displaystyle=a_{0}\left(\left(\frac{1+r}{1-r}\right)^{\alpha}-1+\beta\right)$
$\displaystyle=d(a_{0},\partial
W_{\alpha})\left(\left(\frac{1+r}{1-r}\right)^{\alpha}-1+\beta\right)$
$\displaystyle\leq d(a_{0},\partial W_{\alpha})$

if, and only if,

$\left(\frac{1+r}{1-r}\right)^{\alpha}-1+\beta\leq 1.$

Therefore, the Bohr radius is the unique root of the equation

$\left(\frac{1+r}{1-r}\right)^{\alpha}-1+\beta=1.$

A simple computation shows that
$r_{\alpha,\beta}=((2-\beta)^{1/\alpha}-1)/((2-\beta)^{1/\alpha}+1)$. By a
suitable rotation of the function $f=F_{\alpha,a_{0}}$ in (1.11), it can be
shown that the radius $r_{\alpha,\beta}$ is sharp. This completes the proof. ∎

Acknowledgment: The first author is supported by the Institute Post Doctoral
Fellowship of IIT Bhubaneswar, India, the second author is supported by SERB-
MATRICS, India.

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