\section{\bf Introduction}
Let $X$ be a complex Banach space. For domain $G\subset X$, let $H(G,D)$ be the set of holomorphic mappings from $G$ into $D$. Every $f\in H(G,D)$ can be expanded, in a neighbourhood of any given $x_0 \in G$, into the series
\begin{equation}\label{eqf}
	f(x)=\sum_{m=0}^{\infty}\cfrac{1}{m!}\,D^m f(x_0)\big(\underbrace{(x-x_0),(x-x_0),\dots,(x-x_0)}_{m-times}\big)
\end{equation}
where $D^mf(x_0)$, $m\in \mathbb{N}$, denote the $m$-th Fr\'{e}chet derivative of $f$ at $x_0$, which is a bounded symmetric $m$-linear mapping from $\prod_{i=1}^m X$ to $\mathbb{C}$. For convenience, in this article, further we write $D^m f(x_0)\big((x-x_0)^m\big)$ instead of 
$$
D^m f(x_0)\big(\underbrace{(x-x_0),(x-x_0),\dots,(x-x_0)}_{m-times}\big).
$$ 
For more information on this topic, we refer \cite{GK} to the reader.

\medskip

A domain $G\subset X$ is said to be {\em balanced}, if $zG\subset G$ for all $z\in \overline{\mathbb{D}}$, where $\mathbb{D}:=\{z\in \mathbb{C}: |z|<1\}$.
Given a balanced domain $G$, we denote higher dimensional Bohr radius by $K^G_X(D)$ the largest non-negative number $r$ with the property that if the power series
\begin{equation*}\label{eqbi}
	\sum_{m=1}^{\infty}\bigg|\cfrac{1}{m!}\,D^m f(0)\big(x^m\big)\bigg|\le d(f(0),\partial D)
\end{equation*}
holds for all $x\in r.G$ and for all holomorphic functions $f \in H(G,D)$ with an expansion \eqref{eqf} about origin. Here $d(f(0),\partial D)$ denotes the Euclidean distance between $f(0)$ and the boundary $\partial D$ of the domain $D$. In 1914,
Bohr gave a remarkable result \cite{Bohr14} (due to the work of Riesz, Schur, Wiener) states that 
$K^\mathbb{C}_\mathbb{D}(\mathbb{D})=1/3
$. An application of this result in the Banach algebras given by Dixon in \cite{D95}. After this, Bohr's result have been very popular to many mathematician. To see the progress on this topic, we refer survey article \cite{Abu-M16} and for recent results related to the Bohr phenomenon with complex variables, reader can refer to \cite{AAL20,BD19,Kaypon18,LLP2020,LPW2020,LP2019}.

\medskip

In recent years, a lot of attention has been paid to the multidimensional generalizations of Bohr’s theorem which was first given by Boas and Khavinson \cite{BK97} in 1997.
The result $K_{\mathbb{C}^n}^G(\mathbb{D})\ge 1/3$ was proved by Aizenberg \cite{A00} for any balanced domain $G$ in $\mathbb{C}^n$, in the same paper he gave the optimal result $K^G_{\mathbb{C}^n}(\mathbb{D})= 1/3$ by assuming $G$ is convex. In 2009, Hamada et al. \cite[Corollary 3.2]{HHK} discussed a more general result $K^G_X(\mathbb{D})=1/3$ for any bounded balanced domain $G$ subset of complex Banach space $X$. Also, by taking a restriction on $f\in H(G,\mathbb{D})$ such that $f(0)=0$ and $G\subset \mathbb{C}^n$, Liu and Ponnusamy \cite{LP21} improved the quantity $K^G_{\mathbb{C}^n}(\mathbb{D})=1/\sqrt{2}$ and obtain a sharp radius $K^G_{\mathbb{C}^n}(\mathbb{D})=1/\sqrt{2}$ if $G$ is convex. Moreover, Bhowmik and Das \cite{BD21} calculated the following quantities
$$
\inf\{K^G_X(\Omega):\Omega\subset \mathbb{C} \mbox{ is simply connected}\}=3-2\sqrt{2}
$$
and 
$$
\inf\{K^G_X(\Omega):\Omega\subset \mathbb{C} \mbox{ is convex}\}=\frac{1}{3}.
$$
In higher dimension, there are many significant and attractive results (see for instance \cite{A00,A07}).

\medskip

Before proceeding further we set some notations.  
Let $\mathbb{H}:=\{z\in \mathbb{C}:{\rm Re}\,z>0\}$ be the right half-plane, $
T:=\{z\in \mathbb{C}:|\arg z|<\pi\}
$ be the slit region, $\mathbb{D}_0:=\{z\in \mathbb{C}:0<|z|<1\}$ be the punctured unit disk, and $\overline{\mathbb{D}}^c:=\{z\in \mathbb{C}:|z|>1\}$ be the exterior of the closed unit disk $\overline{\mathbb{D}}$. The spherical distance between two complex numbers of extended complex plan is given by
$$
\lambda (z,w)=
\begin{cases}\vspace{0.1cm}
	\frac{|z-w|}{\sqrt{1+|z|^2}\sqrt{1+|w|^2}},& \text{if $z,w\in \mathbb{C}$,}\\\vspace{0.2cm}
	\frac{1}{\sqrt{1+|z|^2}}, & \text{if $w=\infty$}.
\end{cases}
$$ 
In \cite{AbuAli}, Abu-Muhanna and Ali observed that if $f(z)=\sum_{n=0}^{\infty}a_nz^n \in H(\mathbb{D},\overline{\mathbb{D}}^c)$ then 
\begin{equation}\label{EUD}
	\lambda\bigg(\sum_{n=0}^{\infty}|a_nz^n|, |a_0|\bigg)\le \lambda(f(0),\partial \overline{\mathbb{D}}^c)
\end{equation}
for all $|z|\le 1/3$, here the bound $1/3$ is sharp. 
In 2013, Abu-Muhanna and Ali in \cite{Abu4} obtained $K_\mathbb{C}^\mathbb{D}(\mathbb{H})=1/3$ and $K_\mathbb{C}^\mathbb{D}(T)=(\sqrt{2}-1)/(\sqrt{2}+1)$. Further, Abu-Muhanna et al. \cite{AAN17} determined $K_\mathbb{C}^\mathbb{D}(\mathbb{D}_0)=1/3$.  

\medskip

The goal of this paper is establish sharp bound of $K^G_X(\mathbb{H})$, $K^G_X(\mathbb{D}_0)$, and
$K^G_X(T)$, where $G$ is a balanced domain in a Banach space $X$. Moreover, we obtain a radius under which the inequality \eqref{EUD} holds for functions of the class $H(G,\overline{\mathbb{D}}^c)$.

\medskip
\section{\bf Preliminaries and Main Results}
In this section, we will use the concept of  subordination to prove our results. Therefore, it is worth to recall here the definition of subordination. Suppose $f$ and $g$ are analytic functions in $\mathbb{D}$. We say that $g$ is {\em subordinate} to $f$, or $g\prec f$, if there is an analytic function 
$w:\mathbb{D}\rightarrow \mathbb{D}$ with $w(0)=0$ such that $g=f\circ w$. 
To know more about subordination, reader can refer to \cite{Duren83,MM-Book}.
The Bohr phenomenon for the class of subordinations studied in \cite{Abu, AA,BD18,KS21}.

\medskip
 
\subsection{Half-plane and Convex domains}
Let $\mathcal{P}$ denote the class of all analytic functions $p$ with $p(0)=1$ having positive real part in $\mathbb{D}$, with the form
\begin{equation}\label{p}
	p(z)=1+c_1z+c_2z^2+\cdots.
\end{equation}
A member of $\mathcal{P}$ is called a {\em Carath\'{e}odory function}.
In order to establish our first main result, we need the following fundamental result for Carath\'{e}odory functions.

\begin{lemma} \label{lemmap}
	For every $p\in \mathcal{P}$ of the form \eqref{p}, we have $|c_n|\le 2$ for $n\ge 1$.
\end{lemma}

\medskip

We can now state and prove our first result on multidimensional Bohr's phenomenon for functions  in the class $H(G,\mathbb{H})$.

\medskip

\begin{theorem}\label{ThmH}
Let $f\in H(G,\mathbb{H})$ be of the form \eqref{eqf} in a neighbourhood of the origin. Then 
\begin{equation*}
\sum_{m=1}^{\infty}\bigg|	\cfrac{1}{m!}D^m f(0)\big(x^m\big)\bigg|\le d(f(0),\partial \mathbb{H})
\end{equation*}
for $x\in (1/3)G$. Moreover, the constant $1/3$ is best possible.
\end{theorem}

\begin{proof}
Note that $|f(0)|>0$, for $f\in H(G,\mathbb{H})$. Assume any fixed $y\in G$ and let 
\begin{equation*}
F(z):=f(zy),\quad z\in \mathbb{D}.
\end{equation*}
Then $F:\mathbb{D}\to \mathbb{H}$ is holomorphic and 
\begin{equation}\label{eqHF}
F(z)=f(0)+\sum_{m=1}^{\infty}\cfrac{1}{m!}\,D^m f(0)\big(y^m\big)z^m.
\end{equation}
Consider a function
\begin{equation*}
g(z)=\cfrac{F(z)-i{\rm Im}(f(0))}{{\rm Re}(f(0))}=1+\cfrac{1}{{\rm Re}(f(0))}\sum_{m=1}^{\infty}\cfrac{1}{m!}\,D^m f(0)\big(y^m\big)z^m.
\end{equation*}
It easily follow that $g\in \mathcal{P}$. We now apply Lemma \ref{lemmap} on the  Carath\'{e}odory function $g$ and obtain
\begin{equation*}
\bigg|	\cfrac{\,D^m f(0)\big(y^m\big)}{m!}\bigg| \le 2 {\rm Re}(f(0))
\end{equation*}
for $m\ge 1$. Thus we have
\begin{equation*}
\sum_{m=1}^{\infty}\bigg|	\cfrac{\,D^m f(0)\big(x^m\big)}{m!}\bigg|=\sum_{m=1}^{\infty}\bigg|	\cfrac{\,D^m f(0)\big(y^m\big)}{m!}\bigg| \bigg(\frac{1}{3}\bigg)^m\le  {\rm Re}(f(0))=d(f(0),\partial \mathbb{H})
\end{equation*}
for $x\in (1/3)G.$

\medskip

Now let us prove that $1/3$ is optimal number using the technique given in \cite{HHK}. For any $1/3<r_0<1$, there exist a $0<c<1$ and $v\in \partial G$ such that $cr_0>1/3$ and $c\sup_{x\in \partial G}||x||<||v||$. Next, we consider a function $f_0$ on $G$ by 
$$
f_0(x)=L\bigg(\cfrac{c\psi_v(x)}{||v||}\bigg)
$$
where $L(z)=(1+z)/(1-z),\,z\in \mathbb{D}$, $\psi_v$ is a bounded linear functional on $X$ with $\psi_v(v)=||v||$ and $||\psi_v||=1$. It is easy to check that $c\psi_v(x)/||v||\in\mathbb{D}$ and $f_0\in H(G,\mathbb{H})$. Choose $x=r_0v$, we get
\begin{equation*}
f_0(r_0v)=\cfrac{1+cr_0}{1-cr_0}=1+2\sum_{n=1}^{\infty}{cr_0}^n=f_0(0)+\sum_{m=1}^{\infty}\bigg|	\cfrac{\,D^m f_0(0)\big(x^m\big)}{m!}\bigg| >2=1+d(f_0(0),\partial\mathbb{H}).
\end{equation*}
The proof of theorem is complete.
\end{proof}

\medskip

It is now appropriate to remark that when $X=\mathbb{C}$ and $G=\mathbb{D}$, Theorem \ref{ThmH} coincides with \cite[Theorem 2.1]{Abu4}.
Furthermore, the next result, for holomorphic functions defined on a bounded balanced domain which takes values in a convex domain, can be derived from Theorem \ref{ThmH}:

\medskip

\begin{corollary}
Let $f\in H(G,C)$, where $C$ is a convex domain, be of the form \eqref{eqf} in a neighbourhood of the origin. Then 
\begin{equation*}
	\sum_{m=1}^{\infty}\bigg|	\cfrac{\,D^m f(0)\big(x^m\big)}{m!}\bigg|\le d(f(0),\partial C)
\end{equation*}
for $x\in (1/3)G$. This result is sharp.
\end{corollary}

\begin{proof}
Let $u\in \partial C$ be nearest to $f(0)$. Further let $T_u$ be the tangent line at $u$, and $H_u$ the half-plane containing $C$. Then $f\in H(G,H_u)$. Now choose $t$ real so that $(H_u-u)e^{it}=\mathbb{H}$, the right half-plane and let $K=(C-u)e^{it}\subset\mathbb{H}$. Hence it yields that $g(z)=(f(z)-u)e^{it}\in H(G,K)\subset H(G, \mathbb{H})$ and $g(0)=(f(0)-u)e^{it}=|f(0)-u|=d(f(0),\partial C)$. Then we can apply Theorem \ref{ThmH} on $g$ and it follows the desired result as well as the sharpness.
\end{proof}

\subsection{Punctured unit disk} Our next objective is to give multidimensional analogue of \cite[Theorem 2.1]{AAN17} for the class $H(G,\mathbb{D}_0)$.

\medskip

\begin{theorem}
Let $f\in H(G,\mathbb{D}_0)$ be of the form \eqref{eqf} in a neighbourhood of the origin. Then 
\begin{equation}\label{eqD0}
\sum_{m=1}^{\infty}\bigg|	\cfrac{1}{m!}D^m f(0)\big(x^m\big)\bigg|\le d(f(0),\partial \mathbb{D}_0)=1
\end{equation}
for $x\in (1/3)G$. This result is sharp.
\end{theorem}


\begin{proof}
Because of the inclusion $H(G,\mathbb{D}_0)\subset H(G,\mathbb{D})$, the required inequality \eqref{eqD0} for $x\in (1/3)G$ directly follows from \cite[Corollary 3.2]{HHK}. To prove sharpness part, we apply the same technique as in the proof of Theorem \ref{ThmH}. We consider for any $r_0\in (1/3,1)$ there exist $0<c<1$ and $v\in \partial G$ such that $cr_0>1/3$ and $c\sup_{x\in \partial G}||x||<||v||$. Next, we define
\begin{equation*}
H_t(z)=\exp \bigg(-t \cfrac{1+z}{1-z}\bigg)=\cfrac{1}{e^t}+\cfrac{1}{e^t}\sum_{n=1}^{\infty}\bigg[\sum_{m=1}^{n}\cfrac{(-2t)^m}{m!}{n-1 \choose m-1}\bigg]z^n,\quad t>0 
\end{equation*}
which is in $H(\mathbb{D},\mathbb{D}_0)$. Now consider a function $f_1\in H(G,\mathbb{D}_0)$ defined by
$$
f_1(x)=H_t\bigg(\cfrac{c\psi_v(x)}{||v||}\bigg),
$$
where $\psi_v$ is a bounded linear functional on $X$ with $\psi_v(v)=||v||$ and $||\psi_v||=1$. Thus, for $x=r_0v$, we get
$$
f_1(r_0v)=H_t(cr_0)=\sum_{m=1}^{\infty}\bigg|	\cfrac{\,D^m f_1(0)\big(x^m\big)}{m!}\bigg|>1
$$
by using the argument given in \cite[Theorem 2.1]{AAN17}. This concludes the proof.
\end{proof}

\subsection{Slit mappings}
The following well-known result of de Branges \cite{DeB1} on subordination is useful to prove our next result.

\medskip

\begin{lemma}\label{lemmasub}
If $f(z)=\sum_{n=0}^{\infty}a_nz^n$ and $g(z)=\sum_{n=0}^{\infty}b_nz^n$ are two analytic functions in $\mathbb{D}$ such that $f(0)=g(0)$, $g$ is univalent (one-to-one) and $f\prec g$, then $|a_n|\le n|b_1|$.
\end{lemma}

\medskip

We now state and prove our next result which provides the sharp Bohr inequality for functions belonging to the class $H(G,T)$.

\medskip

\begin{theorem}
Let $f\in H(G,T)$ of the form \eqref{eqf} in a neighbourhood of the origin with $f(0)>0$. Then
\begin{equation}\label{eqT}
\sum_{m=1}^{\infty}\bigg|	\cfrac{\,D^m f(0)\big(x^m\big)}{m!}\bigg|\le d(f(0),\partial T)=f(0)
\end{equation}
for $x\in (3-2\sqrt{2})G$. This result is sharp.
\end{theorem}


\begin{proof}
Suppose that $f\in H(G,T)$. Now, we construct $F$ as in the proof of Theorem \ref{ThmH} which is in $H(\mathbb{D},T)$ having series expansion \eqref{eqHF}. Since $F\in H(\mathbb{D},T)$, we may write the given condition as
$$
F(z)\prec g(z)
$$
where
$$
g(z)=f(0)\bigg(\cfrac{1+z}{1-z}\bigg)^2=f(0)+4f(0)\sum_{n=1}^{\infty}nz^n.
$$
Here $g$ is univalent mapping in $\mathbb{D}$ such that $F(\mathbb{D})\subset g(\mathbb{D})$ and $F(0)=g(0)$. According to Lemma \ref{lemmasub}, we have
\begin{equation*}
\bigg|	\cfrac{\,D^m f(0)\big(y^m\big)}{m!}\bigg| \le 4f(0)m,\quad m\ge 1.
\end{equation*}
This gives, for $z\in \mathbb{D}$ and $y\in G$, that
\begin{equation*}
\sum_{m=1}^{\infty}\bigg|	\cfrac{\,D^m f(0)\big(y^m\big)}{m!}\bigg| |z|^m\le  4f(0)\sum_{m=1}^{\infty}m |z|^m=4f(0)\cfrac{|z|}{(1-|z|)^2}
\end{equation*}
which is less than or equal to $f(0)$ for all $|z|\le 3-2\sqrt{2}\approx0.17157$. Hence the inequality \eqref{eqT} is hold in $(3-2\sqrt{2})G$.

\medskip

To prove that the result is sharp, we adopt the approach of Theorem \ref{ThmH} and consider the following situation: for any $r_0\in (3-2\sqrt{2},1)$, there exist a $c\in (0,1)$ such that $cr_0>3-2\sqrt{2}$. Also there exist a $v\in \partial G$ such that
$$
c\sup \{||x||:x\in \partial G\}<||v||.
$$
 Next, we consider a function $f_2$ on $G$ defined by
 $$
 f_2(x)=U\bigg(\cfrac{c\psi_v(x)}{||v||}\bigg),
 $$
 where
 $$
 U(z)=\bigg(\cfrac{1+z}{1-z}\bigg)^2,\quad z\in \mathbb{D}
 $$
and, $\psi_v$ is a bounded linear functional on $X$ with $\psi_v(v)=||v||$ and $||\psi_v||=1$.
It is easy to check that $c\psi_v(x)/||v||\in\mathbb{D}$ and $f_2\in H(G,T)$. Then, for $x=r_0v$, we have
\begin{equation*}
f_2(x)	=f_2(0)+\sum_{m=1}^{\infty}\bigg|	\cfrac{\,D^m f_2(0)\big(y^m\big)}{m!}\bigg| |z|^m=\bigg(\cfrac{1+cr_0}{1-cr_0}\bigg)^2=1+4\sum_{n=1}^{\infty}n(cr_0)^n>2.
 \end{equation*}
This shows the sharpness.
\end{proof}

\subsection{Exterior of the closed unit disk} The following lemma by Abu-Muhanna and Ali \cite{AbuAli} will be required to establish our next result.
 
\begin{lemma}\label{lemma2.7}
Let $f\in H(\mathbb{D},\overline{\mathbb{D}}^c)$. Then $f\prec W$, where
\begin{equation*}
W(z)=\exp\bigg(\cfrac{1+\phi(z)}{1-\phi(z)}\bigg),
\end{equation*}
with 
\begin{equation*}
\phi(z)=\cfrac{z+b}{1+\overline{z}b} \mbox{ and }  b=\cfrac{\log f(0)-1}{\log f(0)+1}.
\end{equation*}
\end{lemma}

\medskip
The following theorem extends a result, given by Abu-Muhanna and Ali \cite[Theorem 2.1]{AbuAli}, in higher dimension.
\begin{theorem}
Let $f\in H(G,\overline{\mathbb{D}}^c)$ of the form \eqref{eqf} in a neighbourhood of the origin. Then
\begin{equation}\label{eqDc}
\lambda \bigg(\sum_{m=1}^{\infty}\bigg|	\cfrac{\,D^m f(0)\big(x^m\big)}{m!}\bigg|, |f(0)|\bigg)\le \lambda(f(0),\partial \overline{\mathbb{D}}^c)
\end{equation}
for $x\in (1/3)G$. This result is sharp.
\end{theorem}
\begin{proof}
Let $f\in H(G,\overline{\mathbb{D}}^c)$.
Then we can construct a function $F$ as in the proof of Theorem \ref{ThmH} which will be in $H(\mathbb{D},\overline{\mathbb{D}}^c)$ and have the series expansion \eqref{eqHF}. Now, by using Lemma \ref{lemma2.7}, we have $F\prec W$.
It is easy to observe that, the $W(z)$ defined in Lemma \ref{lemma2.7} can be written as
$$
W(z)=F(0)\exp \bigg(\frac{\log |F(0)|^2z}{1-z}\bigg).
$$
Hence, from \cite[Equation 2.7]{AbuAli}, we obtain
$$
\sum_{m=0}^{\infty}\bigg|\cfrac{1}{m!}\,D^m f(0)\big(x^m\big)\bigg||z|^m \leq |F(0)|^\frac{1+|z|}{1-|z|}=|f(0)|^\frac{1+|z|}{1-|z|}.
$$
It leads to the fact that, for $|z|\leq 1/3$,
$$
\sum_{m=0}^{\infty}\bigg|\cfrac{1}{m!}\,D^m f(0)\big(x^m\big)\bigg||z|^m \leq |f(0)|^2.
$$ 
Then a simple computation shows that 
$$
\lambda \bigg(\sum_{m=0}^{\infty}\bigg|\cfrac{1}{m!}\,D^m f(0)\big(x^m\big)\bigg|,|f(0)|\bigg)\leq \lambda(|f(0)|,|f(0)|^2)\leq \lambda(|f(0)|,1)
$$
holds for $x\in (1/3)G$.
 
 \medskip
 
Finally, we prove that inequality \eqref{eqDc} is not holds true for $x\in r_0G$, where $r_0\in (1/3,1)$. As we know that there exist a $c\in (0,1)$ and $v\in \partial G$ such that $cr_0>1/3$ and
$$
c\sup \{||x||:x\in \partial G\}<||v||.
$$
Now, we consider a function $f_3$ on $G$ defined by
$$
f_3(x)=W\bigg(\cfrac{c\psi_v(x)}{||v||}\bigg),
$$
where $\psi_v$ is a bounded linear functional on $X$ with $\psi_v(v)=||v||$ and $||\psi_v||=1$.
Thus
$$
f_3(r_0v)=W(cr_0)=|f(0)|\exp \bigg(\frac{cr_0}{1-cr_0}\log |f(0)|^2\bigg)=|f(0)|^{(1+cr_0)/(1-cr_0)}.
$$
Also, a simple computation gives that  
$$
\frac{\lambda(|f(0)|,|f(0)|^{(1+cr_0)/(1-cr_0)})}{\lambda(|f(0)|,1)}= \frac{\sqrt{2}|f(0)|(|f(0)|^{(2cr_0)/(1-cr_0)}-1)}{(|f(0)|-1)\sqrt{1+|f(0)|^{2(1+cr_0)/(1-cr_0)}}}\to \frac{2cr_0}{1-cr_0}
$$
as $|f(0)|\to 1$. Since $2x/(1-x)>1$,
for $x>1/3$,
it leads to the fact that
$$
\lambda \bigg(\sum_{m=0}^{\infty}\bigg|\cfrac{1}{m!}\,D^m f_0(0)\big(x^m\big)\bigg||z|^m,|f(0)|\bigg)= \lambda(|f(0)|,|f(0)|^2)> \lambda(|f(0)|,1)
$$ 
as $|f(0)|\to 1$. This concludes the theorem.
\end{proof}


\bigskip
\noindent
{\bf Acknowledgement.} 
The authors would like to thank Professor Swadesh Kumar Sahoo for fruitful discussion on this topic.
The first author thanks IIT Bhubaneswar for providing Institute Post Doctoral Fellowship. The work of the second author is supported by the Institute Post Doctoral
Fellowship of NISER Bhubaneswar, India.

\medskip
\noindent
{\bf Conflict of Interests.} The authors declare that there is no conflict of interests 
regarding the publication of this paper.

