††footnotetext:  File: Ahamed-Allu-P3-06-04-2021.tex, printed: 2024-8-28,
19.08

# Bohr-Rogosinski and improved Bohr type inequalities for certain fully
starlike harmonic mappings

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

The classical Bohr inequality states that if $f$ is an analytic function with
the power series representation $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ in the
unit disk $\mathbb{D}:=\\{z\in\mathbb{C}:|z|<1\\}$ such that $|f(z)|\leq 1$
for all $z\in\mathbb{D}$, then

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

and the constant $1/3$ cannot be improved. The constant $r_{0}=1/3$ is known
as Bohr radius and the inequality $\sum_{n=0}^{\infty}|a_{n}|r^{n}\leq 1$ is
known as Bohr inequality. Let $\mathcal{H}$ be the class of complex-valued
harmonic mappings $f=h+\bar{g}$ defined in the unit disk $\mathbb{D}$, where
$h$ and $g$ are analytic functions in $\mathbb{D}$ with the normalization
$h(0)=0=h^{\prime}(0)-1$ and $g(0)=0$. Let
$\mathcal{H}_{0}=\\{f=h+\bar{g}\in\mathcal{H}:g^{\prime}(0)=0\\}.$ Let
$\mathcal{P}^{0}_{\mathcal{H}}(M):=\\{f=h+\overline{g}\in\mathcal{H}_{0}:(zh^{\prime\prime}(z))>-M+|zg^{\prime\prime}(z)|,\;z\in\mathbb{D},\;M>0\\}$.
Functions in the class $\mathcal{P}^{0}_{\mathcal{H}}(M)$ are called fully
starlike univalent functions for $0<M<1/\log 4$. In this paper, we obtain the
sharp Bohr-Rogosinski type inequality and improved Bohr inequality and the
corresponding Bohr radius for the class $\mathcal{P}_{\mathcal{H}}^{0}(M)$.

###### Key words and phrases:

Analytic, univalent, harmonic functions; starlike, convex, close-to-convex
functions; coefficient estimate, growth theorem, Bohr radius.

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

Primary 30C45, 30C50, 30C80

##  1\. Introduction

Let $f$ be an analytic function with the power series representation
$f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ in the unit disk
$\mathbb{D}:=\\{z\in\mathbb{C}:|z|<1\\}$ such that $|f(z)|\leq 1$ for all
$z\in\mathbb{D}$. The Bohr inequality is given by

(1.1) $M_{f}(r):=\sum_{n=0}^{\infty}|a_{n}|r^{n}\leq 1,\;\;\text{for
all}\;\;|z|=r\leq\frac{1}{3}$

and the constant $1/3$ cannot be improved. The inequality (1.1) was
established by Bohr [12] in $1914$. The constant $r_{0}=1/3$ is known as Bohr
radius, while the inequality $\sum_{n=0}^{\infty}|a_{n}|r^{n}\leq 1$ is known
as Bohr inequality. Bohr actually obtained the inequality (1.1) for $r\leq
1/6$ and subsequently later, Weiner, Riesz and Schur have independently
obtained the inequality (1.1) for the radius $1/3$, which is sharp. Moreover,
for

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

it follows easily that $M_{\phi_{b}}(r)>1$ if, and only if, $r>1/(1+2b)$.
Consequently, the radius $1/3$ is optimal when $b\rightarrow 1$.

For $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ in $\mathbb{D}$, the Bohr inequality
can be written in terms of the Euclidean distance as follows

(1.2)
$d\left(\sum_{n=0}^{\infty}|a_{n}z^{n}|,|a_{0}|\right)=\sum_{n=1}^{\infty}|a_{n}z^{n}|\leq
1-|f(0)|=d(f(0),\partial\mathbb{D}),$

where $d$ is the Euclidean distance and $\partial\mathbb{D}$ is the boundary
of the unit disk $\mathbb{D}.$ More generally, a class $\mathcal{F}$ of
analytic (or harmonic) functions $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ mapping
$\mathbb{D}$ into a domain $\Omega$ is said to satisfy a Bohr phenomenon if an
inequality of type (1.2) hols uniformly in $|z|\leq\rho_{0}$, where
$0<\rho_{0}\leq 1$ for functions in the class $\mathcal{F}$. In $2004$,
Beneteau _et al._ [10] first introduced the notion of the Bohr phenonomenon
for a Banach space of analytic functions in the unit disk $\mathbb{D}$.
Aizenberg _et al._ [5] have presented an abstract apporach to the the problem
of finding Bohr radius and proved that Bohr’s phenomenon occurs in a general
condition.

In $1995$, Dixon [13] showed via methods related to Bohr’s inequality for
power series [12], that there are Banach algebras $\mathcal{A}$ which are not
isometrically operator algebras and yet satisfy a special case of von Neumann
inequality: $||p(a)||\leq\sup\\{|p(z)|:z\in\mathbb{D}\\}$ for all
$a\in\mathcal{A}$ and polynomial $p(z)$ with $p(0)=0$. In the recent years,
the generalization of Bohr’s inequality for various classes of analytic
functions becomes an active research area (see [22, 23, 25]). The Bohr radius,
for the holomorphic functions has been studied by Aizenberg et al. [6], Aytuna
and Djakov [11]. The Bohr phenomenon for the class of starlike log-harmonic
mappings has been studied by Ali et al. [7], and Ahamed and Allu [3]. In
$2018$, Kayumov and Ponnusamy [22] introduced the notion of $p$-Bohr radius
for harmonic functions, and established result obtaining $p$-Bohr radius for
the class of odd analytic functions. The improved Bohr inequality for locally
univalent harmonic mappings has been sudied by Evdoridis _et al._ [15]. In
$2021$, Huang _et al._ [18] determined Bohr inequality for the class of
harmonic mappings $f=h+\overline{g}$ in the unit disk $\mathbb{D}$, where both
$h(z)=\sum_{n=0}^{\infty}a_{pn+m}z^{pn+m}$ and
$g(z)=\sum_{n=0}^{\infty}b_{pn+m}z^{pn+m}$ are analytic and bounded in
$\mathbb{D}$ and also investigated Bohr-type inequalities of harmonic mappings
with a multiple zero at the origin. For more exciting aspects of Bohr
phenomenon, we refer to [1, 2, 4, 8, 9, 15, 16, 17, 19, 23, 24, 26] and the
references therein.

In addition to the Bohr inequality, the concept of the Rogosinski inequality
is also used in [29], which is defined as follows: Let
$f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ be analytic in $\mathbb{D}$ and its
corresponding partial sum of $f$ is defined by
$S_{N}(z):=\sum_{n=0}^{N-1}a_{n}z^{n}$. Then, for every $N\geq 1$, we have
$|\sum_{n=0}^{N-1}a_{n}z^{n}|<1$ in the disk $|z|<1/2$ and the radius $1/2$ is
sharp. Motivated by Rogosinski radius for bounded analytic functions in
$\mathbb{D}$, Kayumov and Ponnusamy [21] have introduced Bohr-Rogosinski
radius and considered the Bohr-Rogosinski sum $R_{N}^{f}(z)$ which is defined
by

$R_{N}^{f}(z):=|f(z)|+\sum_{n=N}^{\infty}|a_{n}||z|^{n}.$

It is easy to see that
$|S_{N}(z)|=\big{|}f(z)-\sum_{n=N}^{\infty}a_{n}z^{n}\big{|}\leq|R_{N}^{f}(z)|$.
Thus, the validity of Bohr-type radius for $R_{N}^{f}(z)$, which is related to
the classical Bohr sum (Majorant series) in which $f(0)$ is replaced by
$f(z)$, gives Rogosinski radius in the case of bounded analytic functions in
$\mathbb{D}$. Kayumov and Ponnusamy [21] have obtained the following
interesting result.

######  Theorem 1.3.

[21] Let $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ be analytic in $\mathbb{D}$ and
$|f(z)|\leq 1$. Then

(1.4) $|f(z)|+\sum_{n=N}^{\infty}|a_{n}||z|^{n}\leq 1$

for $|z|=r\leq R_{N}$, where $R_{N}$ is the positive root of the equation
$\psi_{N}(r)=0$, $\psi_{N}(r)=2(1+r)r^{N}-(1-r)^{2}$. The radius $R_{N}$ is
the best possible. Moreover,

(1.5) $|f(z)|^{2}+\sum_{n=N}^{\infty}(|a_{n}|+|b_{n}|)|z|^{n}\leq 1$

for $|z|=r\leq R^{\prime}_{N}$, where $R^{\prime}_{N}$ is the positive root of
the equation $(1+r)r^{N}-(1-r)^{2}$. The radius $R^{\prime}_{N}$ is the best
possible.

In $2020$, Ponnusamy et al. [28] established the following refined Bohr
inequality by applying a refined version of the coefficient inequalities.

######  Theorem 1.6.

[28] Let $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ be analytic in $\mathbb{D}$ and
$|f(z)|\leq 1$. Then

$\sum_{n=0}^{\infty}|a_{n}|r^{n}+\left(\frac{1}{1+|a_{0}|}+\frac{r}{1-r}\right)\sum_{n=1}^{\infty}|a_{n}|^{2}r^{2n}\leq
1$

for $|z|=r\leq 1/(2+|a_{0}|)$ and the numbers $1/(1+|a_{0}|)$ and
$1/(2+|a_{0}|)$ cannot be improved. Moreover,

$|a_{0}|^{2}+\sum_{n=1}^{\infty}|a_{n}|r^{n}+\left(\frac{1}{1+|a_{0}|}+\frac{r}{1-r}\right)\sum_{n=1}^{\infty}|a_{n}|^{2}r^{2n}\leq
1$

for $|z|=r\leq 1/2$ and the numbers $1/(1+|a_{0}|)$ and $1/2$ cannot be
improved.

A harmonic mapping in the unit disk $\mathbb{D}$ is a complex-valued function
$f=u+iv$ of $z=x+iy$ in $\mathbb{D}$, which satisfies the Laplace equation
$\Delta f=4f_{z\bar{z}}=0$, where $f_{z}=(f_{x}-if_{y})/2$ and
$f_{\bar{z}}=(f_{x}+if_{y})/2$ and $u$ and $v$ are real-valued harmonic
functions in $\mathbb{D}$. It follows that the function $f$ admits the
canonical representation $f=h+\bar{g}$, where $h$ and $g$ are analytic in
$\mathbb{D}$. Let $\mathcal{H}$ be the class of complex-valued harmonic
mappings $f=h+\bar{g}$ defined in the unit disk $\mathbb{D}$, where $h$ and
$g$ are analytic functions in $\mathbb{D}$ with the normalization
$h(0)=0=h^{\prime}(0)-1$ and $g(0)=0$. Let
$\mathcal{H}_{0}=\\{f=h+\bar{g}\in\mathcal{H}:g^{\prime}(0)=0\\}.$

A domain $\Omega\subseteq\mathbb{C}$ is called starlike with respect to a
point $z_{0}\in\Omega$ if the line segment joining $z_{0}$ to any other point
in $\Omega$ must lies in $\Omega$. In particular, if $z_{0}=0$, then $\Omega$
is called starlike. A complex-valued harmonic mapping $f$ is said to be
starlike if $f(\mathbb{D})$ is starlike. It is well-known that starlikeness is
a heriditary property. That is, if an anlytic function maps the unit disk
$\mathbb{D}$ univalently into a starlike domain, then it also maps the
concentric circle onto a starlike domain. In general, starlike harmonic
functions do not have this property. The failure of this hereditary property
of starlike harmonic mappings led to the introduction of fully starlike
harmonic mappings.

It is well-known [27] that a sense-preserving harmonic function
$f=h+\overline{g}$ is fully starlike in $\mathbb{D}$ if the analytic function
$h+\epsilon g$ is starlike in $\mathbb{D}$ for each $\epsilon$ with
$|\epsilon|=1$.

Let

$\mathcal{P}^{0}_{\mathcal{H}}(M)=\\{f=h+\overline{g}\in\mathcal{H}_{0}:(zh^{\prime\prime}(z))>-M+|zg^{\prime\prime}(z)|,\;z\in\mathbb{D}\;\mbox{and
}\;M>0\\},$

where

(1.7)
$f(z)=h(z)+\overline{g(z)}=z+\sum_{n=2}^{\infty}a_{n}z^{n}+\overline{\sum_{n=2}^{\infty}b_{n}z^{n}}.$

It is well-known that [17] functions in $\mathcal{P}^{0}_{\mathcal{H}}(M)$ are
starlike for $0<M<1/\log 4$. In particular, functions in
$\mathcal{P}^{0}_{\mathcal{H}}(M)$ are fully starlike for $0<M<1/\log 4$ (see
[17]). The growth theorem and sharp coefficient bounds for the functions in
$\mathcal{P}^{0}_{\mathcal{H}}(M)$ have been studied in [17]. The following
results on the coefficient bounds and growth estimation play vital role in
proving our main results.

######  Lemma 1.8.

[17] Let $f=h+\overline{g}\in\mathcal{P}^{0}_{\mathcal{H}}(M)$ for some $M>0$
and be of the form (1.7). Then for $n\geq 2,$

  1. (i)

$\displaystyle|a_{n}|+|b_{n}|\leq\frac{2M}{n(n-1)};$

  2. (ii)

$\displaystyle||a_{n}|-|b_{n}||\leq\frac{2M}{n(n-1)};$

  3. (iii)

$\displaystyle|a_{n}|\leq\frac{2M}{n(n-1)}.$

The inequalities are sharp with extremal function $f_{M}$ given by
$f_{M}(z)=z+2M\sum\limits_{n=2}^{\infty}\dfrac{z^{n}}{n(n-1)}.$

######  Lemma 1.9.

[17] Let $f\in\mathcal{P}^{0}_{\mathcal{H}}(M)$. Then

(1.10)
$|z|+2M\sum\limits_{n=2}^{\infty}\dfrac{(-1)^{n-1}|z|^{n}}{n(n-1)}\leq|f(z)|\leq|z|+2M\sum\limits_{n=2}^{\infty}\dfrac{|z|^{n}}{n(n-1)}.$

Both inequalities are sharp for the function $f_{M}$ given by
$f_{M}(z)=z+2M\sum\limits_{n=2}^{\infty}\dfrac{z^{n}}{n(n-1)}.$

In $2017$, Kayumov and Ponnusamy [21] proved the following improved version of
Bohr’s inequality.

######  Theorem 1.11.

[21] Let $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ be analytic in $\mathbb{D}$,
$|f(z)|\leq 1$ and $S_{r}$ denotes the area of the image of the subdisk
$|z|<r$ under mapping $f$. Then

(1.12)
$B_{1}(r):=\sum_{n=0}^{\infty}|a_{n}|r^{n}+\frac{16}{9}\left(\frac{S_{r}}{\pi}\right)\leq
1\quad\mbox{for}\quad r\leq\frac{1}{3}$

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

(1.13)
$B_{2}(r):=|a_{0}|^{2}+\sum_{n=1}^{\infty}|a_{n}|r^{n}+\frac{9}{8}\left(\frac{S_{r}}{\pi}\right)\leq
1\quad\mbox{for}\quad r\leq\frac{1}{2}$

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

The main aim of this paper is to obtain the sharp Bohr-Rogosinski inequality,
improved Bohr inequality and Refined Bohr inequality and Bohr radius for the
class $\mathcal{P}^{0}_{\mathcal{H}}(M)$. The organization of this paper as
follows: In Section 2, we prove the sharp Bohr-Rogosinski, improved Bohr
radius and Bohr radius for the class $\mathcal{P}^{0}_{\mathcal{H}}(M)$. In
Section 3, we give the proof of all the main results of this paper.

##  2\. Main results

Before stating the main results of this paper, we recall the definiton of
dilogarithm. The dilogarithm function ${\rm Li}_{2}(z)$ is defined by the
power series

${\rm
Li}_{2}(z)=\sum_{n=1}^{\infty}\frac{z^{n}}{n^{2}}\;\;\mbox{for}\;\;|z|<1.$

The definition and the name, of course, come from the analogy with the Taylor
series of the ordinary logarithm around $1$,

$-\log(1-z)=\sum_{n=1}^{\infty}\frac{z^{n}}{n}\;\;\mbox{for}\;\;|z|<1,$

which leads similarly to the defintion of the polylogarithm

${\rm
Li}_{m}(z)=\sum_{n=1}^{\infty}\frac{z^{n}}{n^{m}}\;\;\mbox{for}\;\;|z|<1,\;\;m=1,2,3,\ldots.$

In addition, the relation

$\frac{d}{dz}\left({\rm Li}_{m}(z)\right)=\frac{1}{z}{\rm
Li}_{m-1}(z)\;\;\mbox{for}\;\;m\geq 2$

is obvious and leads by induction to the extension of the domain of definition
of ${\rm Li}_{m}$ to the cut plane $\mathbb{C}\setminus[1,\infty)$. In
particular, the analytic continuation of the dilogarithm is given by

${\rm
Li}_{2}(z)=-\int_{0}^{z}\log(1-u)\frac{du}{u}\;\;\mbox{for}\;\;z\in\mathbb{C}\setminus[1,\infty).$

Using Lemma 1.8 and Lemma 1.9, for the functions in the class
$\mathcal{P}^{0}_{\mathcal{H}}(M)$, we prove the following sharp Bohr
inequality which is an harmonic analogue of Theorem 1.3.

######  Theorem 2.1.

Let $f\in\mathcal{P}^{0}_{\mathcal{H}}(M)$ be given (1.7) with $0<M<1/(2(\ln
4-1))$. Then

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

for $|z|=r\leq r_{{}_{M}}$, where $r_{{}_{M}}$ is the unique root of the
equation

(2.2) $2r+4M\left(r+(1-r)\log(1-r)\right)-1-2M(1-2\log 2)=0$

in $(0,1)$. The radius $r_{{}_{M}}$ is the best possible.

######  Remark 2.1.

For particular values of $M$ in Theorem 2.1, we have

$0<M<{1}/{(2(2\ln 2-1))}\approx 1.294349.$

A simple computation gives the Bohr radii $r_{0.1}\approx 0.438485$,
$r_{0.2}\approx 0.387786$, $r_{0.3}\approx 0.343722$, $r_{0.4}\approx
0.304054$, $r_{0.5}\approx 0.267404$, $r_{0.6}\approx 0.23283$,
$r_{0.7}\approx 0.19963$, $r_{0.8}\approx 0.167229$, $r_{0.9}\approx
0.135111$, $r_{1.0}\approx 0.102764$, $r_{1.25}\approx 0.0167782$.

Figure 1. The roots of (2.2) for different values of $M$ when
$0<M<{1}/{(2(2\ln 2-1))}$.

Figure 1 illustrates the roots of (2.2) for different values of $M$ when
$0<M<{1}/{(2(2\ln 2-1))}$. Let $S_{r}$ denote the area of the image of of the
subdisk $\mathbb{D}_{r}:=\\{z\in\mathbb{C}:|z|<r\\}$ under the mappings in the
class $\mathcal{P}^{0}_{\mathcal{H}}(M)$. We now prove the sharp Bohr
inequality which is a harmonic analogue of Theorem 1.11.

######  Theorem 2.3.

Let $f\in\mathcal{P}^{0}_{\mathcal{H}}(M)$ be given (1.7) with $0<M<1/(2(\ln
4-1))$. Then

$|f(z)|+\sum_{n=2}^{\infty}(|a_{n}|+|b_{n}|)|z|^{n}+\frac{S_{r}}{\pi}\leq
d\left(f(0),\partial f(\mathbb{D})\right)$

for $|z|=r\leq r^{*}_{{}_{M}}$, where $r^{*}_{{}_{M}}$ is the unique root of
the equation

(2.4) $\displaystyle r+4M\left(r+(1-r)\log(1-r)\right)+4M^{2}\left(r^{2}{\rm
Li}_{2}(r^{2})-r^{2}-\left(1-r^{2}\right)\log\left(1-r^{2}\right)\right)$
$\displaystyle\quad\quad-1-2M(1-2\log 2)=0$

in $(0,1)$. The radius $r^{*}_{{}_{M}}$ is the best possible.

######  Remark 2.2.

For particular values of $M$ in Theorem 2.3, a simple computation yields the
Bohr radii $r^{*}_{0.1}\approx 0.546723$, $r^{*}_{0.2}\approx 0.487374$,
$r^{*}_{0.3}\approx 0.435926$, $r^{*}_{0.4}\approx 0.389886$,
$r^{*}_{0.5}\approx 0.0.347564$, $r^{*}_{0.6}\approx 0.307711$,
$r^{*}_{0.7}\approx 0.269313$, $r^{*}_{0.8}\approx 0.231445$,
$r^{*}_{0.9}\approx 0.193148,$ $r^{*}_{1.0}\approx 0.153247$,
$r^{*}_{1.25}\approx 0.0371406$.

Figure 2. The roots of (2.4) for different values of $M$ when
$0<M<{1}/{(2(2\ln 2-1))}$.

Figure 2 illustrates the roots of (2.4) for different values of $M$ when
$0<M<{1}/{(2(2\ln 2-1))}$. The Jacobian of a complex-valued functions of the
form $f=h+\bar{g}$ is defined by
$J_{f}(z)=|h^{\prime}(z)|^{2}-|g^{\prime}(z)|^{2}$. It is known that $f$ is
sense-preserving if $J_{f}(z)>0$ and $f$ is sense-reversing if $J_{f}(z)<0$ in
$\mathbb{D}$. Using Lemma 1.8 and Lemma 1.9, we prove the following result for
the functions in the class $\mathcal{P}^{0}_{\mathcal{H}}(M)$.

######  Theorem 2.5.

Let $f\in\mathcal{P}^{0}_{\mathcal{H}}(M)$ be given (1.7) with $0<M<1/(2(\ln
4-1))$. Then

$\sum_{n=2}^{\infty}(|a_{n}|+|b_{n}|)|z|^{n}+\sqrt{|J_{f}(z)|}|z|\leq
d\left(f(0),\partial f(\mathbb{D})\right)$

for $|z|=r\leq r^{**}_{{}_{M}}$, where $r^{**}_{{}_{M}}$ is the unique root of
the equation

(2.6) $\displaystyle
2r+2M\left(r+(1-r)\log(1-r)\right)-2Mr\log(1-r)-1-2M(1-2\log 2)=0$

in $(0,1)$. The radius $r^{**}_{{}_{M}}$ is the best possible.

Figure 3. The roots of (2.6) for different values of $M$ when
$0<M<{1}/{(2(2\ln 2-1))}$.

######  Remark 2.3.

For particular values of $M$ in Theorem 2.5, a simple computation yields the
Bohr radii $r^{**}_{0.1}\approx 0.426832$, $r^{**}_{0.2}\approx 0.372123$,
$r^{**}_{0.3}\approx 0.327085$, $r^{**}_{0.4}\approx 0.287924$,
$r^{**}_{0.5}\approx 0.252589$, $r^{**}_{0.6}\approx 0.2198$,
$r^{**}_{0.7}\approx 0.18866$, $r^{**}_{0.8}\approx 0.158469$,
$r^{**}_{0.9}\approx 0.128614$, $r^{**}_{1.0}\approx 0.0984794$,
$r^{**}_{1.25}\approx 0.0166108$.

Figure 3 illustrates the roots of (2.6) for different values of $M$ when
$0<M<{1}/{(2(2\ln 2-1))}$. We prove the following refine sharp bohr inequality
for functions in the class $\mathcal{P}^{0}_{\mathcal{H}}(M)$ which is a
harmonic analogue version of Theorem 1.6 .

######  Theorem 2.7.

Let $f\in\mathcal{P}^{0}_{\mathcal{H}}(M)$ be given (1.7) with $0<M<1/(2(\ln
4-1))$. Then

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

for $|z|=r\leq r^{\prime}_{{}_{M}}$, where $r^{\prime}_{{}_{M}}$ is the unique
root of the equation

(2.8)
$\displaystyle\left(r+2M\left(r+(1-r)\log(1-r)\right)\right)^{2}+2M\left(r+(1-r)\log(1-r)\right)$
$\displaystyle\quad\quad-2Mr\log(1-r)-1-2M(1-2\log 2)=0$

in $(0,1)$. The radius $r^{\prime}_{{}_{M}}$ is the best possible.

Figure 4. The roots of (2.8) for different values of $M$ when
$0<M<{1}/{(2(2\ln 2-1))}$.

######  Remark 2.4.

For particular values of $M$ in Theorem 2.7, a simple computation gives the
Bohr radii as follow: $r^{\prime}_{0.1}\approx 0.802472$,
$r^{\prime}_{0.2}\approx 0.696255$, $r^{\prime}_{0.3}\approx 0.619712$,
$r^{\prime}_{0.4}\approx 0.558217$, $r^{\prime}_{0.5}\approx 0.505494$,
$r^{\prime}_{0.6}\approx 0.458107$, $r^{\prime}_{0.7}\approx 0.413815$,
$r^{\prime}_{0.8}\approx 0.370862$, $r^{\prime}_{0.9}\approx 0.327549$,
$r^{\prime}_{1.0}\approx 0.281757$, $r^{\prime}_{1.25}\approx 0.125838$.

Figure 4 illustrates the roots of (2.8) for different values of $M$ when
$0<M<{1}/{(2(2\ln 2-1))}$

##  3\. Proof of the main results

For $f\in\mathcal{P}^{0}_{\mathcal{H}}(M)$, we have

(3.1)
$|f(z)|\geq|z|+2M\sum\limits_{n=2}^{\infty}\dfrac{(-1)^{n-1}}{n(n-1)}|z|^{n}\quad\mbox{for
}\quad|z|<1.$

Then the Euclidean distance between $f(0)$ and the boundary of $f(\mathbb{D})$
is

(3.2) $d(f(0),\partial f(\mathbb{D}))=\liminf\limits_{|z|\rightarrow
1}|f(z)-f(0)|.$

Therefore, a simple computation using (1.10) and (3.2) shows that

(3.3) $d(f(0),\partial f(\mathbb{D}))\geq
1+2M\sum\limits_{n=2}^{\infty}\dfrac{(-1)^{n-1}}{n(n-1)}=1+2M(1-\ln 4).$

Proof of Theorem 2.1. Let $f\in\mathcal{P}^{0}_{\mathcal{H}}(M)$ be given by
(1.7). Using Lemma 1.8 and Lemma 1.9, for $|z|=r$, we obtain

(3.4) $\displaystyle|z|+|f(z)|+\sum_{n=2}^{\infty}(|a_{n}|+|b_{n}|)|z|^{n}$
$\displaystyle\leq
2r+\sum_{n=2}^{\infty}\frac{2Mr^{n}}{n(n-1)}+\sum_{n=2}^{\infty}\frac{2Mr^{n}}{n(n-1)}$
$\displaystyle=2r+4M\sum_{n=2}^{\infty}\frac{r^{n}}{n(n-1)}.$

A simple computation shows that

(3.5)
$\displaystyle\sum_{n=2}^{\infty}\frac{r^{n}}{n(n-1)}=r\sum_{n=1}^{\infty}\frac{r^{n}}{n}+\sum_{n=2}^{\infty}-\frac{r^{n}}{n}=r+(1-r)\log(1-r)$

and

$\sum_{n=2}^{\infty}\frac{(-1)^{n-1}}{n(n-1)}=1-2\log 2\approx-0.386294.$

Therefore, using (3.5) in (3.4), we obtain

$\displaystyle|z|+|f(z)|+\sum_{n=2}^{\infty}(|a_{n}|+|b_{n}|)r^{n}\leq
2r+4M\left(r+(1-r)\log(1-r)\right).$

It is easy to see that

$2r+4M\left(r+(1-r)\log(1-r)\right)\leq 1+2M(1-2\ln 2)$

for $r\leq r_{{}_{M}}$, where $r_{{}_{M}}$ is a root of $H_{1}(r)=0$ in
$(0,1)$, where $H_{1}:[0,1)\rightarrow\mathbb{R}$ is defined by

$\displaystyle H_{1}(r):$
$\displaystyle=2r+4M\left(r+(1-r)\log(1-r)\right)-1-2M(1-2\ln 2)$
$\displaystyle=2r+4M\sum_{n=2}^{\infty}\frac{r^{n}}{n(n-1)}-1-2M\sum_{n=2}^{\infty}\frac{(-1)^{n-1}}{n(n-1)}.$

The existence of the root $r_{{}_{M}}$ is guaranteed by the Intermediate value
theorem since $H_{1}$ is a continuous function on $[0,1]$, differentiable on
$(0,1)$ and $H_{1}(0)=-1-2M(1-2\ln 2)<0$ for $0<M<1/(2(\ln 4-1))$ and
$\displaystyle\lim_{r\rightarrow 1}H_{1}(r)=+\infty.$ To show that
$r_{{}_{M}}$ is the unique root of $H_{1}(r)=0$, it is enough to show that
$H_{1}$ is strictly monotone function on $(0,1)$. A simple computation shows
that

$\frac{d}{dr}\left(H_{1}(r)\right)=2+4M\sum_{n=2}^{\infty}\frac{r^{n-1}}{n-1}>0$

for $r\in(0,1)$, which shows that $H_{1}$ is strictly increasing in $(0,1)$.
Hence, the root $r_{{}_{M}}$ is the unique root of $H_{1}$ in $(0,1)$ and
hence, we have

(3.6)
$2r_{{}_{M}}+4M\left(r_{{}_{M}}+(1-r_{{}_{M}})\log(1-r_{{}_{M}})\right)=1+2M(1-2\ln
2).$

To show that $r_{{}_{M}}$ is the best possible, we consider the function
$f=f_{M}$ is given by

(3.7) $f_{M}(z)=z+\sum_{n=2}^{\infty}\frac{2Mz^{n}}{n(n-1)}.$

It is easy to see that $f_{M}\in\mathcal{P}^{0}_{\mathcal{H}}(M)$ and for
$f=f_{M}$, we have

(3.8) $d(f_{M}(0),\partial f_{M}(\mathbb{D}))=1+2M(1-2\ln 2).$

For the function $f=f_{M}$ and $z=r$, a simple computation shows that

(3.9) $\displaystyle
r+|f_{M}(r)|+\sum_{n=2}^{\infty}(|a_{n}|+|b_{n}|)r^{n}=2r+\sum_{n=2}^{\infty}\frac{4Mr^{n}}{n(n-1)}.$

Therefore, a simple computation using (3.7), (3.8) and (3.9) for $f=f_{M}$ and
$r>r_{{}_{M}}$ shows that

$\displaystyle|z|+|f_{M}(z)|+\sum_{n=2}^{\infty}(|a_{n}|+|b_{n}|)|z|^{n}$
$\displaystyle=2r+\sum_{n=2}^{\infty}\frac{2Mr^{n}}{n(n-1)}+\sum_{n=2}^{\infty}\frac{2Mr^{n}}{n(n-1)}$
$\displaystyle>2r_{{}_{M}}+\sum_{n=2}^{\infty}\frac{2Mr^{n}_{{}_{M}}}{n(n-1)}+\sum_{n=2}^{\infty}\frac{2Mr^{n}_{{}_{M}}}{n(n-1)}$
$\displaystyle=2r+4M\sum_{n=2}^{\infty}\frac{r^{n}_{{}_{M}}}{n(n-1)}$
$\displaystyle=2r_{{}_{M}}+4M\left(r_{{}_{M}}+(1-r_{{}_{M}})\log(1-r_{{}_{M}})\right)$
$\displaystyle=1+2M(1-2\ln 2)$ $\displaystyle=d(f_{M}(0),\partial
f_{M}(\mathbb{D}))$

and hence $r_{{}_{M}}$ is the best possible. This completes the proof. ∎

Proof of Theorem 2.3. Let $f\in\mathcal{P}^{0}_{\mathcal{H}}(M)$ be given by
(1.7). It is well-known that (see [14, p.113]) the area of the disk
$\mathbb{D}_{r}:=\\{z\in\mathbb{C}:|z|<r\\}$ under the harmonic map
$f=h+\bar{g}$ is

(3.10) $\displaystyle
S_{r}=\iint\limits_{\mathbb{D}_{r}}J_{f}(z)dxdy=\iint\limits_{\mathbb{D}_{r}}\left(|{h^{\prime}(z)}|^{2}-|{g^{\prime}(z)}|^{2}\right)dxdy.$

It is easy to see that

(3.11) $\displaystyle\iint\limits_{\mathbb{D}_{r}}|h^{\prime}(z)|^{2}dxdy$
$\displaystyle=\int_{0}^{r}\int_{0}^{2\pi}|h^{\prime}(\rho
e^{i\theta})|^{2}\rho d\theta d\rho$
$\displaystyle=\int_{0}^{r}\int_{0}^{2\pi}\rho\left(\sum_{n=1}^{\infty}n\;a_{n}\rho^{n-1}e^{i(n-1)\theta}\right)\left(\sum_{n=1}^{\infty}n\;\bar{a}_{n}\rho^{n-1}e^{-i(n-1)\theta}\right)d\theta
d\rho$ $\displaystyle=\int_{0}^{r}\left(\sum_{n=1}^{\infty}2\pi
n^{2}|a_{n}|^{2}\rho^{2n-1}\right)d\rho$
$\displaystyle=\sum_{n=1}^{\infty}2\pi n^{2}|a_{n}|^{2}\frac{r^{2n}}{2n}$
$\displaystyle=\pi\sum_{n=1}^{\infty}n|a_{n}|^{2}r^{2n}.$

Similarly, for $g(z)=\sum_{n=2}^{\infty}b_{n}z^{n}$, we obtain

(3.12) $\displaystyle\iint\limits_{\mathbb{D}_{r}}$
$\displaystyle|g^{\prime}(z)|^{2}dxdy=\pi\sum_{n=2}^{\infty}n|b_{n}|^{2}r^{2n}.$

In view of Lemma 1.8, (3.10), (3.11) and (3.12), we obtain

(3.13) $\displaystyle\frac{S_{r}}{\pi}$
$\displaystyle=\frac{1}{\pi}\iint\limits_{\mathbb{D}_{r}}\left(|h^{\prime}(z)|^{2}-|g^{\prime}(z)|^{2}\right)dxdy$
$\displaystyle=r^{2}+\sum_{n=2}^{\infty}n\left(|a_{n}|^{2}-|b_{n}|^{2}\right)r^{2n}$
$\displaystyle=r^{2}+\sum_{n=2}^{\infty}n(|a_{n}|+|b_{n}|)(|a_{n}|-|b_{n}|)r^{2n}$
$\displaystyle=r^{2}+\sum_{n=2}^{\infty}\frac{4M^{2}r^{2n}}{n\left(n-1\right)^{2}}.$

Therefore, in view of Lemma 1.8, Lemma 1.9 and (3.12), for $|z|=r$ yields

(3.14)
$\displaystyle|f(z)|+\sum_{n=2}^{\infty}\left(|a_{n}|+|b_{n}|\right)|z|^{n}+\frac{S_{r}}{\pi}$
$\displaystyle\leq
r+4M\sum_{n=2}^{\infty}\frac{r^{n}}{n(n-1)}+\left(r^{2}+\sum_{n=2}^{\infty}\frac{4M^{2}r^{2n}}{n(n-1)^{2}}\right).$

A simple computation shows that

(3.15) $\displaystyle\sum_{n=2}^{\infty}\frac{r^{2n}}{n(n-1)^{2}}$
$\displaystyle=\sum_{n=2}^{\infty}\frac{r^{2n}}{n}+\sum_{n=2}^{\infty}\frac{r^{2n}}{(n-1)^{2}}+\sum_{n=2}^{\infty}-\frac{r^{2n}}{n-1}$
$\displaystyle=-r^{2}+\sum_{n=1}^{\infty}\frac{r^{2n}}{n}+r^{2}\sum_{n=2}^{\infty}\frac{r^{2n-2}}{(n-1)^{2}}+r^{2}\sum_{n=2}^{\infty}-\frac{r^{2n-2}}{n-1}$
$\displaystyle=-r^{2}-\log(1-r^{2})+r^{2}\sum_{n=1}^{\infty}\frac{r^{2n}}{n^{2}}-r^{2}\sum_{n=1}^{\infty}\frac{r^{2n}}{n}$
$\displaystyle=r^{2}{\rm
Li}_{2}\left(r^{2}\right)-r^{2}-\left(1-r^{2}\right)\log\left(1-r^{2}\right).$

Therefore, using (3.5) and (3.15), from (3.14), we obtain

$\displaystyle|f(z)|+\sum_{n=2}^{\infty}\left(|a_{n}|+|b_{n}|\right)|z|^{n}+\frac{S_{r}}{\pi}$
$\displaystyle\leq
r^{2}+r+4M\left(r+(1-r)\log(1-r)\right)+4M^{2}\left(r^{2}{\rm
Li}_{2}\left(r^{2}\right)-r^{2}-\left(1-r^{2}\right)\log\left(1-r^{2}\right)\right).$

It is easy to see that

$\displaystyle r^{2}+r+4M\left(r+(1-r)\log(1-r)\right)+4M^{2}\left(r^{2}{\rm
Li}_{2}\left(r^{2}\right)-r^{2}-\left(1-r^{2}\right)\log\left(1-r^{2}\right)\right)$
$\displaystyle\leq 1+2M\left(1-2\log 2\right)$

for $r\leq r^{*}_{{}_{M}}$, where $r^{*}_{{}_{M}}$ is a root of $H_{2}(r)=0$,
where $H_{2}:[0,1]\rightarrow\mathbb{R}$ is defined by

$\displaystyle H_{2}(r):$
$\displaystyle=r^{2}+r+4M\left(r+(1-r)\log(1-r)\right)-1-2M\left(1-2\log
2\right)$ $\displaystyle\quad\quad+4M^{2}\left(r^{2}{\rm
Li}_{2}\left(r^{2}\right)-r^{2}-\left(1-r^{2}\right)\log\left(1-r^{2}\right)\right)$
$\displaystyle=r^{2}+r+4M\sum_{n=2}^{\infty}\frac{r^{n}}{n(n-1)}+\sum_{n=2}^{\infty}\frac{4M^{2}r^{2n}}{n(n-1)^{2}}-1-2M\sum_{n=2}^{\infty}\frac{(-1)^{n-1}}{n(n-1)}.$

A simple computation shows that $H_{2}(0)=-1-2M\left(1-2\log 2\right)<0$ for
$0<M<1/(2(\ln 4-1))$ and

$\displaystyle H_{2}(1)$
$\displaystyle=2+4M\sum_{n=2}^{\infty}\frac{1}{n(n-1)}+4M^{2}\sum_{n=2}^{\infty}\frac{1}{n(n-1)^{2}}-2M(1-2\log
2)$
$\displaystyle=2+4M\sum_{n=2}^{\infty}\left(\frac{1}{n-1}-\frac{1}{n}\right)+4M^{2}\left(\sum_{n=2}^{\infty}\frac{1}{(n-1)^{2}}+\sum_{n=2}^{\infty}\left(\frac{1}{n}-\frac{1}{n-1}\right)\right)$
$\displaystyle\quad\quad-2M(1-2\log 2)$
$\displaystyle=2+4M\left(\left(1-\frac{1}{2}\right)+\left(\frac{1}{2}-\frac{1}{3}\right)+\left(\frac{1}{3}-\frac{1}{4}\right)+\cdots\right)-2M(1-2\log
2)$
$\displaystyle\quad\quad+4M^{2}\left(\left(\frac{1}{2}-1\right)+\left(\frac{1}{3}-\frac{1}{2}\right)+\left(\frac{1}{4}\right)-\frac{1}{3}+\cdots+\sum_{n=1}^{\infty}\frac{1}{n^{2}}\right)$
$\displaystyle=2+2M+4M^{2}\left(\frac{\pi^{2}}{6}-1\right)+4M\log 2$
$\displaystyle>0.$

It is easy to see that $H_{2}$ is strictly increasing in $(0,1)$ because

$\displaystyle\frac{d}{dr}\left(H_{2}(r)\right)=2r+1+4M\sum_{n=2}^{\infty}\frac{r^{n-1}}{n-1}+8M^{2}\sum_{n=2}^{\infty}\frac{r^{2n}}{(n-1)^{2}}>0$

for $r\in(0,1)$. By the similar argument that is being used in the proof of
Theorem 2.1, we can show that $r^{*}_{{}_{M}}$ is the unique root of $H_{2}$
in $(0,1)$ and hence, we have

(3.16)
$\displaystyle\left(r^{*}_{{}_{M}}\right)^{2}+r^{*}_{{}_{M}}+4M\left(r^{*}_{{}_{M}}+(1-r^{*}_{{}_{M}})\log(1-r)\right)$
$\displaystyle\quad+4M^{2}\left(\left(r^{*}_{{}_{M}}\right)^{2}{\rm
Li}_{2}\left(\left(r^{*}_{{}_{M}}\right)^{2}\right)-\left(r^{*}_{{}_{M}}\right)^{2}-\left(1-\left(r^{*}_{{}_{M}}\right)^{2}\right)\log\left(1-\left(r^{*}_{{}_{M}}\right)^{2}\right)\right)$
$\displaystyle=1+2M\left(1-2\log 2\right).$

To show that $r^{*}_{{}_{M}}$ is the best possible radius, we consider
$f=f_{M}$ given by (3.7). A simple computation, using (3.8), (3.14) and (3.16)
for $f=f_{M}$ and $r^{*}_{{}_{M}}<r$ shows that

$\displaystyle|f(z)|+\sum_{n=2}^{\infty}\left(|a_{n}|+|b_{n}|\right)|z|^{n}+\frac{S_{r}}{\pi}$
$\displaystyle=r^{2}+r+4M\sum_{n=2}^{\infty}\frac{r^{n}}{n(n-1)}+\left(r^{2}+\sum_{n=2}^{\infty}\frac{4M^{2}r^{2n}}{n(n-1)^{2}}\right)$
$\displaystyle>\left(r^{*}_{{}_{M}}\right)^{2}+r^{*}_{{}_{M}}+4M\sum_{n=2}^{\infty}\frac{\left(r^{*}_{{}_{M}}\right)^{n}}{n(n-1)}+\left(\left(r^{*}_{{}_{M}}\right)^{2}+\sum_{n=2}^{\infty}\frac{4M^{2}\left(r^{*}_{{}_{M}}\right)^{2n}}{n(n-1)^{2}}\right)$
$\displaystyle=\left(r^{*}_{{}_{M}}\right)^{2}+r^{*}_{{}_{M}}+4M\left(r^{*}_{{}_{M}}+(1-r^{*}_{{}_{M}})\log(1-r)\right)$
$\displaystyle\quad+4M^{2}\left(\left(r^{*}_{{}_{M}}\right)^{2}{\rm
Li}_{2}\left(\left(r^{*}_{{}_{M}}\right)^{2}\right)-\left(r^{*}_{{}_{M}}\right)^{2}-\left(1-\left(r^{*}_{{}_{M}}\right)^{2}\right)\log\left(1-\left(r^{*}_{{}_{M}}\right)^{2}\right)\right)$
$\displaystyle=1+2M\left(1-2\log 2\right)$ $\displaystyle=d(f_{M}(0),\partial
f_{M}(\mathbb{D}))$

and hence $r^{*}_{{}_{M}}$ is the best possible radius. This completes the
proof. ∎

Proof of Theorem 2.5. Let $f\in\mathcal{P}^{0}_{\mathcal{H}}(M)$ be given by
(1.7). The Jacobian of a complex-valued function
$f\in\mathcal{P}^{0}_{\mathcal{H}}(M)$ has the following property

(3.17)
$\displaystyle|J_{f}(z)|=|h^{\prime}(z)|^{2}-|g^{\prime}(z)|^{2}\leq|h^{\prime}(z)|^{2}\leq\left(1+2M\sum_{n=1}^{\infty}\frac{r^{n}}{n}\right)^{2}.$

In view of Lemma 1.8, Lemma 1.9 and (3.17), for $|z|=r$, we obtain

(3.18)
$\displaystyle\sum_{n=2}^{\infty}\left(|a_{n}|+|b_{n}|\right)|z|^{n}+\sqrt{|J_{f}(z)|}|z|$
$\displaystyle\leq
r+\sum_{n=2}^{\infty}\frac{2Mr^{n}}{n(n-1)}+\left(1+2M\sum_{n=1}^{\infty}\frac{r^{n}}{n}\right)r.$

A simple computation shows that

(3.19)
$\begin{cases}\displaystyle\sum_{n=2}^{\infty}\frac{r^{n}}{n(n-1)}=r+(1-r)\log(1-r),\vspace{1.5mm}\\\
\displaystyle\sum_{n=1}^{\infty}\frac{r^{n}}{n}=-\log(1-r).\end{cases}$

Therefore, using (3.19) in (3.18), we obtain

$\displaystyle\sum_{n=2}^{\infty}\left(|a_{n}|+|b_{n}|\right)|z|^{n}+\sqrt{|J_{f}(z)}|z|\leq
2r+2M\left(r+(1-2r)\log(1-r)\right).$

It is easy to see that

$\displaystyle 2r+2M\left(r+(1-2r)\log(1-r)\right)\leq 1+2M(1-2\log 2)$

for $r\leq r^{**}_{{}_{M}}$, where $r^{**}_{{}_{M}}$ is a root of
$H_{3}(r)=0$, where $H_{3}:[0,1]\rightarrow\mathbb{R}$ is defined by

$\displaystyle H_{3}(r):$
$\displaystyle=2r+2M\left(r+(1-2r)\log(1-r)\right)-1-2M(1-2\log 2)$
$\displaystyle=r+\sum_{n=2}^{\infty}\frac{2Mr^{n}}{n(n-1)}+\left(1+2M\sum_{n=1}^{\infty}\frac{r^{n}}{n}\right)r-1-2M\sum_{n=2}^{\infty}\frac{(-1)^{n-1}}{n(n-1)}.$

With the help of the similar argument that is being used in the proof of
Theorem 2.1 and Theorem 2.3, we can show that $r^{**}_{{}_{M}}$ is the unique
root of $H_{3}(r)$ in $(0,1)$. Thus, we have

(3.20)
$2r^{**}_{{}_{M}}+2M\left(r^{**}_{{}_{M}}+(1-2r^{**}_{{}_{M}})\log(1-r)\right)=1+2M(1-2\log
2).$

To show that $r^{**}_{{}_{M}}$ is the best possible, we consider $f=f_{M}$
given by (3.7). A simple computation, using (3.8), (3.18) and (3.20) for
$f=f_{M}$ and $r>r^{**}_{{}_{M}}$ shows that

$\displaystyle\sum_{n=2}^{\infty}\left(|a_{n}|+|b_{n}|\right)|z|^{n}+\sqrt{|J_{f_{M}}(z)}|z|$
$\displaystyle=r+\sum_{n=2}^{\infty}\frac{2Mr^{n}}{n(n-1)}+\left(1+2M\sum_{n=1}^{\infty}\frac{r^{n}}{n}\right)r$
$\displaystyle>r^{**}_{{}_{M}}+\sum_{n=2}^{\infty}\frac{2M\left(r^{**}_{{}_{M}}\right)^{n}}{n(n-1)}+\left(1+2M\sum_{n=1}^{\infty}\frac{\left(r^{**}_{{}_{M}}\right)^{n}}{n}\right)r$
$\displaystyle=2r^{**}_{{}_{M}}+2M\left(r^{**}_{{}_{M}}+(1-2r^{**}_{{}_{M}})\log(1-r)\right)$
$\displaystyle=1+2M(1-2\log 2)$ $\displaystyle=d(f_{M}(0),\partial
f_{M}(\mathbb{D}))$

and hence $r^{**}_{{}_{M}}$ is the best possible radius. This completes the
proof. ∎

Proof of Theorem 2.7. Let $f\in\mathcal{P}^{0}_{\mathcal{H}}(M)$ be given by
(1.7). In view of Lemma 1.8, Lemma 1.9, for $|z|=r$, we obtain

(3.21)
$\displaystyle|f(z)|^{2}+\sum_{n=2}^{\infty}(|a_{n}|+|b_{n}|)|z|^{n}+\frac{r}{1-r}\sum_{n=2}^{\infty}(|a_{n}|+|b_{n}|)^{2}|z|^{2n}$
$\displaystyle\leq\left(r+2M\sum_{n=2}^{\infty}\frac{r^{n}}{n(n-1)}\right)^{2}+2M\sum_{n=2}^{\infty}\frac{r^{n}}{n(n-1)}+\frac{4M^{2}r}{1-r}\sum_{n=2}^{\infty}\frac{r^{2n}}{n^{2}(n-1)^{2}}$
$\displaystyle=\left(r+2M\sum_{n=2}^{\infty}\frac{r^{n}}{n(n-1)}\right)^{2}+2M\sum_{n=2}^{\infty}\frac{r^{n}}{n(n-1)}$
$\displaystyle\quad\quad+\frac{4M^{2}r}{1-r}\left(\displaystyle\sum_{n=2}^{\infty}\frac{r^{2n}}{n^{2}}+\displaystyle\sum_{n=2}^{\infty}\frac{r^{2n}}{(n-1)^{2}}+\displaystyle\sum_{n=2}^{\infty}-\frac{2r^{2n}}{n-1}+\displaystyle\sum_{n=2}^{\infty}\frac{2r^{2n}}{n}\right).$

A simple computation shows that

(3.22)
$\begin{cases}\displaystyle\sum_{n=2}^{\infty}\frac{r^{2n}}{n^{2}}=-r^{2}+{\rm
Li}_{2}\left(r^{2}\right),\vspace{1.5mm}\\\
\displaystyle\sum_{n=2}^{\infty}\frac{r^{2n}}{(n-1)^{2}}=r^{2}{\rm
Li}_{2}\left(r^{2}\right),\vspace{1.5mm}\\\
\displaystyle\sum_{n=2}^{\infty}-\frac{2r^{2n}}{n-1}=2r^{2}\log\left(1-r^{2}\right),\vspace{1.5mm}\\\
\displaystyle\sum_{n=2}^{\infty}\frac{2r^{2n}}{n}=-2r^{2}-2\log\left(1-r^{2}\right).\end{cases}$

Therefore, using (3.5), (3.21) and (3.22), we obtain

$\displaystyle|f(z)|^{2}+\sum_{n=2}^{\infty}(|a_{n}|+|b_{n}|)|z|^{n}+\frac{r}{1-r}\sum_{n=2}^{\infty}(|a_{n}|+|b_{n}|)^{2}|z|^{2n}$
$\displaystyle\leq\left(r+2M(r+(1-r)\log(1-r))\right)^{2}+2M\left(r+(1-r)\log(1-r)\right)$
$\displaystyle\quad\quad+\frac{4M^{2}r}{1-r}\bigg{(}-3r^{2}-2\left(1-r^{2}\right)\log\left(1-r^{2}\right)+\left(1+r^{2}\right){\rm
Li}_{2}\left(r^{2}\right)\bigg{)}.$

It is easy to see that

$\displaystyle\left(r+2M(r+(1-r)\log(1-r))\right)^{2}+2M\left(r+(1-r)\log(1-r)\right)$
$\displaystyle\quad\quad+\frac{4M^{2}r}{1-r}\bigg{(}-3r^{2}-2\left(1-r^{2}\right)\log\left(1-r^{2}\right)+\left(1+r^{2}\right){\rm
Li}_{2}\left(r^{2}\right)\bigg{)}\leq 1+2(1-2\log 2)$

for $r\leq r^{\prime}_{{}_{M}}$, where $r^{\prime}_{{}_{M}}$ is a root of
$H_{4}(r)=0$, where $H_{4}:[0,1]\rightarrow\mathbb{R}$ is defined by

$\displaystyle H_{4}(r):$
$\displaystyle=\left(r+2M(r+(1-r)\log(1-r))\right)^{2}+2M\left(r+(1-r)\log(1-r)\right)$
$\displaystyle\quad\quad+\frac{4M^{2}r}{1-r}\bigg{(}-3r^{2}-2\left(1-r^{2}\right)\log\left(1-r^{2}\right)+\left(1+r^{2}\right){\rm
Li}_{2}\left(r^{2}\right)\bigg{)}$ $\displaystyle\quad\quad-1-2(1-2\log 2)$
$\displaystyle=\left(r+2M\sum_{n=2}^{\infty}\frac{r^{n}}{n(n-1)}\right)^{2}+2M\sum_{n=2}^{\infty}\frac{r^{n}}{n(n-1)}+\frac{4M^{2}r}{1-r}\sum_{n=2}^{\infty}\frac{r^{2n}}{n^{2}(n-1)^{2}}$
$\displaystyle\quad\quad-1-2M\sum_{n=2}^{\infty}\frac{(-1)^{n-1}}{n(n-1)}.$

By the similar argument that being used in the proof of the preceeding
results, a simple computation shows that $r^{\prime}_{{}_{M}}$ is the unique
root of $H_{4}(r)$ in $(0,1)$, and hence we have

(3.23)
$\displaystyle\left(r^{\prime}_{{}_{M}}+2M(r^{\prime}_{{}_{M}}+(1-r^{\prime}_{{}_{M}})\log(1-r^{\prime}_{{}_{M}}))\right)^{2}+2M\left(r^{\prime}_{{}_{M}}+(1-r^{\prime}_{{}_{M}})\log(1-r^{\prime}_{{}_{M}})\right)$
$\displaystyle\quad\quad+\frac{4M^{2}r^{\prime}_{{}_{M}}}{1-r^{\prime}_{{}_{M}}}\bigg{(}-3\left(r^{\prime}_{{}_{M}}\right)^{2}-2\left(1-\left(r^{\prime}_{{}_{M}}\right)^{2}\right)\log\left(1-\left(r^{\prime}_{{}_{M}}\right)^{2}\right)$
$\displaystyle\quad\quad+\left(1+\left(r^{\prime}_{{}_{M}}\right)^{2}\right){\rm
Li}_{2}\left(\left(r^{\prime}_{{}_{M}}\right)^{2}\right)\bigg{)}$
$\displaystyle=1+2(1-2\log 2).$

To show that $r^{\prime}_{{}_{M}}$ is the best possible radius, we consider
$f=f_{M}$ given by (3.7). Then a simple computation, using (3.8), (3.13) and
(3.23) for $f=f_{M}$ and $r>r^{\prime}_{{}_{M}}$ shows that

$\displaystyle|f_{M}(z)|^{2}+\sum_{n=2}^{\infty}(|a_{n}|+|b_{n}|)|z|^{n}+\frac{r}{1-r}\sum_{n=2}^{\infty}(|a_{n}|+|b_{n}|)^{2}|z|^{2n}$
$\displaystyle=\left(r+2M\sum_{n=2}^{\infty}\frac{r^{n}}{n(n-1)}\right)^{2}+2M\sum_{n=2}^{\infty}\frac{r^{n}}{n(n-1)}+\frac{4M^{2}r}{1-r}\sum_{n=2}^{\infty}\frac{r^{2n}}{n^{2}(n-1)^{2}}$
$\displaystyle>\left(r^{\prime}_{{}_{M}}+2M\sum_{n=2}^{\infty}\frac{\left(r^{\prime}_{{}_{M}}\right)^{n}}{n(n-1)}\right)^{2}+2M\sum_{n=2}^{\infty}\frac{\left(r^{\prime}_{{}_{M}}\right)^{n}}{n(n-1)}+\frac{4M^{2}r^{\prime}_{{}_{M}}}{1-r^{\prime}_{{}_{M}}}\sum_{n=2}^{\infty}\frac{\left(r^{\prime}_{{}_{M}}\right)^{2n}}{n^{2}(n-1)^{2}}$
$\displaystyle=\left(r^{\prime}_{{}_{M}}+2M(r^{\prime}_{{}_{M}}+(1-r^{\prime}_{{}_{M}})\log(1-r^{\prime}_{{}_{M}}))\right)^{2}+2M\left(r^{\prime}_{{}_{M}}+(1-r^{\prime}_{{}_{M}})\log(1-r^{\prime}_{{}_{M}})\right)$
$\displaystyle\quad\quad+\frac{4M^{2}r^{\prime}_{{}_{M}}}{1-r^{\prime}_{{}_{M}}}\bigg{(}-3\left(r^{\prime}_{{}_{M}}\right)^{2}-2\left(1-\left(r^{\prime}_{{}_{M}}\right)^{2}\right)\log\left(1-\left(r^{\prime}_{{}_{M}}\right)^{2}\right)$
$\displaystyle\quad\quad+\left(1+\left(r^{\prime}_{{}_{M}}\right)^{2}\right){\rm
Li}_{2}\left(\left(r^{\prime}_{{}_{M}}\right)^{2}\right)\bigg{)}$
$\displaystyle=1+2(1-2\log 2)$ $\displaystyle=d(f_{M}(0),\partial
f_{M}(\mathbb{D}))$

and hence $r^{\prime}_{{}_{M}}$ is the best possible. 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-
CRG, India.

## References

  * [1] Y. Abu-Muhanna, Bohr’s phenomenon in subordination and bounded harmonic classes, Complex Var. Elliptic Equ. 55 (2010), 1071–1078.
  * [2] Y. Abu Muhanna, R. M. Ali, Z. C. Ng and S. F. M Hasni, Bohr radius for subordinating families of analytic functions and bounded harmonic mappings, J. Math. Anal. Appl. 420 (2014), 124–136.
  * [3] M. B. Ahamed and V. Allu, Improved Bohr radius for the class of starlike log-harmonic mappings, arXiv:2103.07507v1.
  * [4] L. Aizenberg, Multidimensional analogues of Bohr’s theorem on power series, Proc. Amer. Math. Soc. 128 (2000), 1147–1155.
  * [5] L. Aizenberg, A. Aytuna and P. Djakov, An abstract approach to Bohr’s phenomenon, Proc. Amer. Math. Soc. 128 (9) (2000), 2611–2619.
  * [6] L. Aizenberg, A. Aytuna and P. Djakov, Generalization of theorem on Bohr for bases in spaces of holomorphic functions of several complex variables, J. Math. Anal. Appl. 258 (2001), 429–447.
  * [7] R. M. Ali, Z. Abdulhadi and Z. C. Ng, The Bohr radius for starlike logharmonic mappings, Complex Var. Elliptic Equ. 61(1)(2016), 1–14.
  * [8] R. M. Ali, R. W. Barnard and A.Yu. Solynin, A note on Bohr’s phenomenon for power series, J. Math. Anal. Appl. 449 (2017), 154-167.
  * [9] S. A. Alkhaleefah, I.R. Kayumov and S. Ponnusamy, On the Bohr inequality with a fixed zero coefficient, Proc. Amer. Math. Soc. 147 (2019), 5263–5274.
  * [10] C. B${\rm\acute{E}}$n${\rm\acute{E}}$teau, A. Dahlner and D. Khavinson, Remarks on the Bohr phenomenon, Comput. Methods Funct. Theory 4 (2004), 1–19.
  * [11] A. Aytuna and P. Djakov, Bohr property of bases in the space of entire functions and its generalizations, Bull. London Math. Soc. 45 (2) (2013), 411–420
  * [12] H. Bohr, A theorem concerning power series, Proc. Lond. Math. Soc. s2-13 (1914), 1–5.
  * [13] P. G. Dixon, Banach algebras satisfying the non-unital von Neumann inequality, Bull. London Math. Soc. 27(4)(1995), 359–362.
  * [14] P. L. Duren, Harmonic mapping in the plan, Cambridge University Press, (2004)
  * [15] S. Evdoridis, S. Ponnusamy and A. Rasila, Improved Bohr’s inequality for locally univalent harmonic mappings, Indag. Math. (N.S.) 30 (1) (2019), 201–-213.
  * [16] N. Ghosh and V. Allu, Some basic properties of certain subclass of harmonic univalent functions, Complex Var. Elliptic Equ. 63 (2018), 1687–1703.
  * [17] N. Ghosh and V. Allu, On some subclasses of harmonic mappings, Bull. Aust. Math. Soc. 101 (2020), 130–140.
  * [18] Y. Huang, M-S. Liu and S. Ponnusamy, Bohr-Type Inequalities for Harmonic Mappings with a Multiple Zero at the Origin, Mediterr. J. Math. (2021) 18:75.
  * [19] I. R. Kayumov and S. Ponnusamy, On a powered Bohr inequality, Ann. Acad. Sci. Fenn. Ser. A, 44 (2019), 301–310.
  * [20] A. Ismagilov, I. R. Kayumov and S. Ponnusamy, Sharp Bohr type inequality, J. Math. Anal. Appl. 489 (2020), 124147.
  * [21] I. R. Kayumov and S. Ponnusamy, Bohr-Rogosinski radius for analytic functions, preprint, see https://arxiv.org/abs/1708.05585.
  * [22] I. R. Kayumov and S. Ponnusamy, Bohr’s inequalities for the analytic functions with lacunary series and harmonic functions, J. Math. Anal. Appl. 465 (2018), 857–871.
  * [23] I. R. Kayumov and S. Ponnusamy, Improved version of Bohr’s inequalities, C. R. Math. Acad. Sci. Paris 358 (5) (2020), 615–-620.
  * [24] I. R. Kayumov, S. Ponnusamy and N. Shakirov, Bohr radius for locally univalent harmonic mappings, Math. Nachr 291 (2018), 1757–-1768.
  * [25] G. Liu, Z. Liu and S. Ponnusamy, Refined Bohr inequality for bounded analytic functions, priprint, see https://arxiv.org/pdf/2006.08930.
  * [26] Z. Liu and S. Ponnusamy, Bohr radius for subordination and k-quasiconformal harmonic mappings, Bull. Malays. Math. Sci. Soc. 42 (2019) 2151–2168.
  * [27] S. Nagpal and V. Ravichandran, Fully starlike and fully convex harmonic mappings of order $\alpha$, Ann. Polon. Math. 108 (2013), 85–107.
  * [28] S. Ponnusamy, R. Viajayakumar and K-J. Wirths, New inequalities for the coefficients of unimodular bounded functions, Results Math (2020) 75 : 107.
  * [29] W. Rogosinski, Über Bildschranken bei Potenzreihen und ihren Abschnitten, Math. Z. 17 (1923), 260–276.