††footnotetext:  File: main.tex, printed: 2024-8-28, 3.07

# Multidimensional Bohr radii for vector-valued holomorphic functions

Vasudevarao Allu  Vasudevarao Allu, School of Basic Sciences, Indian Institute
of Technology Bhubaneswar, Bhubaneswar-752050, Odisha, India.
<EMAIL_ADDRESS>,  Himadri Halder  Himadri Halder, Department of
Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai,
Maharashtra-400076, India<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Subhadip Pal  Subhadip Pal, School of Basic
Sciences, Indian Institute of Technology Bhubaneswar, Bhubaneswar-752050,
Odisha, India<EMAIL_ADDRESS>
###### Abstract.

The main aim of this paper is to answer certain open questions related to the
exact values of multidimensional Bohr radii by using the concept of arithmetic
Bohr radius for vector-valued holomorphic functions defined in complete
Reinhardt domains in $\mathbb{C}^{n}$. More precisely, we study the asymptotic
estimates of the arithmetic Bohr radius for holomorphic functions in the unit
ball of $\ell^{n}_{q}$ $(1\leq q\leq\infty)$ spaces with values in arbitrary
complex Banach spaces. Many of our results generalize the results obtained by
Defant, Maestre, and Prengel [Q. J. Math. 59, (2008), pp. 189–205].

###### Key words and phrases:

Bohr radius, Arithmetic Bohr radius, Complete Reinhardt domain, Complex Banach
spaces

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

Primary 32A05, 32A10, 46B45; Secondary 46G20

##  1\. Introduction

The Bohr radius problem plays a crucial role in connecting two major areas of
research in pure mathematics, namely complex analysis and functional analysis.
The situation in multidimensional settings of Bohr radius problem is much more
complicated. Due to the lack of certain essential tools with regard to the
behavior of the multidimensional Bohr radius, several research problems with
regard to Bohr radius are still open. The more interesting aspect of the Bohr
radius problem is when we consider this in multidimensional settings for
holomorphic functions with values in complex Banach spaces.

For the last two decades, a lot of attention has been paid to the
generalizations of multidimensional Bohr radius problem in various settings
(see [1, 3, 6, 10, 12, 15, 20]). One such generalization is studying Bohr’s
theorem for the power series of holomorphic functions defined in complete
Reinhardt domains in $\mathbb{C}^{n}$. A complete Reinhardt domain $\Omega$ in
$\mathbb{C}^{n}$ is a domain in $\mathbb{C}^{n}$ such that if
$z=(z_{1},\ldots,z_{n})\in\Omega$, then $(\lambda
e^{i\theta_{1}}z_{1},\ldots,\lambda e^{i\theta_{n}}z_{n})\in\Omega$ for all
$\lambda\in\overline{\mathbb{D}}$ and all $\theta_{i}\in\mathbb{R}$,
$i=1,\ldots,n$. For a complex Banach space $X$, let $\mathcal{F}(\Omega,X)$ be
the space of all holomorphic mappings $f$ in $\Omega$ into $X$ and
$\mathcal{P}^{k}(\Omega,X)$ be the space of all $k$-homogeneous vector-valued
polynomials, where $k\in\mathbb{N}$. Let $H_{\infty}(\Omega,X)$ be the set of
all bounded holomorphic mappings on $\Omega$ into a complex Banach space $X$.
In this article, we mostly focus on the aforesaid spaces with
$\Omega=B_{\ell^{n}_{q}}$, where $q\in[1,\infty]$. For $q\in[1,\infty)$,

$B_{\ell^{n}_{q}}=\\{z\in\mathbb{C}^{n}:\left\lVert
z\right\rVert_{q}=(\sum_{i=1}^{n}|z_{i}|^{q})^{1/q}<1\\}$

and $B_{\ell^{n}_{\infty}}=\\{z\in\mathbb{C}^{n}:\left\lVert
z\right\rVert_{\infty}=\sup_{1\leq i\leq n}|z_{i}|<1\\}$. For any given
$p\in[1,\infty)$, $\lambda\geq 1$, and Reinhardt domain $\Omega$, the
$\lambda_{p}$-Bohr radius $r^{n}_{p}(\Omega,X,\lambda)$ of $\Omega$ with
respect to $\mathcal{F}(\Omega,X)$ is defined as the supremum of all
$r\in[0,1]$ such that

$\sup_{z\in r\Omega}\sum_{\alpha\in\mathbb{N}^{n}_{0}}\left\lVert
x_{\alpha}(f)z^{\alpha}\right\rVert^{p}\leq\lambda\left\lVert
f\right\rVert^{p}_{\Omega}$

for all $f\in\mathcal{F}(\Omega,X)$ with
$f(z)=\sum_{\alpha\in\mathbb{N}^{n}_{0}}x_{\alpha}(f)z^{\alpha}$ and
$\left\lVert f\right\rVert_{\Omega}=\sup\\{\|f(z)\|:z\in\Omega\\}$ and
$\mathbb{N}_{0}=\mathbb{N}\cup\\{0\\}$. We write
$r^{n}_{p}(\Omega,X,\lambda)=r_{p}(\Omega,X,\lambda)$ for $n=1$. The
remarkable result of Bohr [8] states that
$r_{1}(\mathbb{D},\mathbb{C},1)=1/3$. Much of the recent interest in the
Bohr’s theorem stems from the work of Dixon [16] when the author used it to
disprove a long-standing open problem on the characterization of the Banach
algebra satisfying the von Neumann inequality. Since then, a lot of research
has been done to generalize Bohr’s theorem in multidimensional and more other
abstract settings (see [2]). A major part of this research concentrates on
studying Bohr’s theorem in several variables for vector-valued functions and
this led to introduce the quantity $r^{n}_{p}(\Omega,X,\lambda)$. However, to
determine the exact value of $r^{n}_{p}(\Omega,X,\lambda)$ is very much
challenging and it seems there has been not much lack of progress on this
problem even for $X=\mathbb{C}$. For $\lambda\in[1,\sqrt{2}]$, Bombieri [9]
has shown that

$r_{1}(\mathbb{D},\mathbb{C},\lambda)=\frac{1}{3\lambda-2\sqrt{2(\lambda^{2}-1)}}$

and for $\lambda$ very close to $\infty$, Bombieri and Bourgain [10] have
obtained the following estimate

$r_{1}(\mathbb{D},\mathbb{C},\lambda)\sim\frac{\sqrt{\lambda^{2}-1}}{\lambda}.$

In $1989$, Dineen and Timoney [17] extensively studied the constant
$r^{n}_{1}(B_{\ell^{n}_{\infty}},\mathbb{C},1)$ and their result has been
clarified in [6]. In $1997$, Boas and Khavinson [6] obtained the following
estimate for each $n\in\mathbb{N}$ with $n\geq 2$,

(1.1)
$\frac{1}{3\sqrt{n}}<r^{n}_{1}(B_{\ell^{n}_{\infty}},\mathbb{C},1)<2\sqrt{\frac{\log
n}{n}}.$

The exact value of $r^{n}_{1}(B_{\ell^{n}_{\infty}},\mathbb{C},1)$ is still an
open problem and it has drawn the attention of many mathematicians to work
further on Bohr radius problem. Motivated by the above estimate of Boas and
Khavinson, in $2000$, Aizenberg [1] studied the constant
$r^{n}_{1}(B_{\ell^{n}_{q}},\mathbb{C},1)$ with $q=1$ and proved that

(1.2)
$\frac{1}{3e^{1/3}}<r^{n}_{1}(B_{\ell^{n}_{1}},\mathbb{C},1)\leq\frac{1}{3}.$

In the same year, Boas [7] generalized (1.1) and (1.2) to
$r^{n}_{1}(B_{\ell^{n}_{q}},\mathbb{C},1)$ for $1<q<\infty$. For $n>1$, Boas
[7] has proved that, if $1\leq q<2$, then

(1.3) $\frac{1}{3\sqrt[3]{e}}\left(\frac{1}{n}\right)^{1-\frac{1}{q}}\leq
r^{n}_{1}(B_{\ell^{n}_{q}},\mathbb{C},1)<3\left(\frac{\log\,n}{n}\right)^{1-\frac{1}{q}}$

and if $2\leq q\leq\infty$, then

(1.4) $\frac{1}{3}\sqrt{\frac{1}{n}}\leq
r^{n}_{1}(B_{\ell^{n}_{q}},\mathbb{C},1)<2\sqrt{\frac{\log\,n}{n}}.$

It is easy to see in (1.3) and (1.4) that the upper bounds contain a
logarithmic factor which is not present in the lower bounds. Therefore, Boas
observed that (1.3) and (1.4) do not give sharp decay rate of Bohr radius with
the dimension, and hence he conjectured that this logarithmic factor should
not be really present. In $2006$, Defant et al. [14] disproved this
conjectured by obtaining a logarithmic lower bound which is almost correct
asymptotic estimates for the Bohr radius
$r^{n}_{1}(B_{\ell^{n}_{q}},\mathbb{C},1)$ with $1\leq q\leq\infty$. In
particular, Defant et al. have proved that, if $1\leq q\leq\infty$ then there
is a constant $c>0$ such that

(1.5)
$\frac{1}{c}\left(\frac{\log\,n/\log\,\log\,n}{n}\right)^{1-\frac{1}{\min(q,2)}}\leq
r^{n}_{1}(B_{\ell^{n}_{q}},\mathbb{C},1)\,\,\,\,\,\mbox{for all}\,\,\,n>1.$

In $2017$, Blasco [5] extensively studied the constant
$r^{n}_{p}(\Omega,X,\lambda)$ for $\Omega=\mathbb{D},n=1,\lambda=1$ which is
usually known as $p$-Bohr radius $r_{p}(\mathbb{D},X,1)$ for arbitrary Banach
space $X$. It is worth noting that the precise value of
$r_{p}(\mathbb{D},X,1)$ is very difficult to compute even for $X=\mathbb{C}$.
Blasco [5] has obtained a lower estimate of $r_{p}(\mathbb{D},\mathbb{C},1)$
and showed that $r_{p}(\mathbb{D},\mathbb{C},1)\geq p/(p+2)$ which is sharp
only for $p=1$.

In view of the above discussions together with the estimates (1.3), (1.4), and
(1.5), we observe that it is very challenging to determine the exact value of
$r^{n}_{p}(B_{\ell^{n}_{q}},X,1)$ even for $X=\mathbb{C}$. This led us to
propose the following challenging problem in more general settings.

######  Problem 1.6.

Let $X$ be an arbitrary complex Banach space. For every $p\in[1,\infty)$,
$q\in[1,\infty]$, $\lambda\geq 1$, and $n>1$, what is the exact value of
$r^{n}_{p}(B_{\ell^{n}_{q}},X,\lambda)?$

One of our main contributions of this paper is to answer the aforesaid problem
affirmatively for arbitrary complex Banach spaces in terms of arithmetic Bohr
radius. The another aspect of the multidimensional Bohr radius is the
arithmetic Bohr radius which is more than a curious variant of the classical
radius. Let $\Omega\subset\mathbb{C}^{n}$ be a complete Reinhardt domain. For
each $1\leq p<\infty$ and $\lambda\geq 1$, the $\lambda_{p}$-arithmetic Bohr
radius of $\Omega$ with respect to $\mathcal{F}(\Omega,X)$ is defined as

$A_{p}(\mathcal{F}(\Omega,X),\lambda):=\sup\left\\{\frac{1}{n}\sum_{i=1}^{n}r_{i}\,|\,r\in\mathbb{R}^{n}_{\geq
0},\,\mbox{for
all}\,f\in\mathcal{F}(\Omega,X):\sum_{\alpha\in\mathbb{N}^{n}_{0}}\left\lVert
x_{\alpha}(f)\right\rVert^{p}r^{p\alpha}\leq\lambda\left\lVert
f\right\rVert^{p}_{\Omega}\right\\},$

where $\mathbb{R}^{n}_{\geq
0}=\\{r=(r_{1},.\,.\,.\,,r_{n})\in\mathbb{R}^{n}:r_{i}\geq 0,1\leq i\leq
n\\}.$ We write $A_{p}(\Omega,X,\lambda)$ for
$A_{p}(H_{\infty}(\Omega,X),\lambda)$, $A_{p}(\Omega,X)$ for
$A_{p}(\Omega,X,1)$ and $A^{k}_{p}(\Omega,X,\lambda)$ for
$A_{p}(\mathcal{P}^{k}(\Omega,X),\lambda)$. For $X=\mathbb{C}$ and $p=1$,
Defant et al. [15] have first considered the constant
$A_{1}(\mathcal{F}(\Omega,\mathbb{C}),\lambda)$ which is known as usual
arithmetic Bohr radius. Recently, Kumar [18] has also studied the constant
$A_{1}(\mathcal{F}(\Omega,\mathbb{C}),\lambda)$. The arithmetic Bohr radius
$A_{1}(\mathcal{F}(\Omega,\mathbb{C}),\lambda)$ forms a useful tool to
describe the domain of existence of the monomial expansion of complex-valued
holomorphic functions in a complete Reinhardt domain (cf. [19]). This led us
to introduce the notion of $\lambda_{p}$-arithmetic Bohr radius
$A_{p}(\mathcal{F}(\Omega,X),\lambda)$ for arbitrary complex Banach spaces
which plays a crucial role to estimate sequences in the set of monomial
convergence of vector-valued holomorphic functions in a Reinhardt domain
$\Omega$ in $\mathbb{C}^{n}$. The main purpose of this paper is to investigate
the constant $A_{p}(\mathcal{F}(\Omega,X),\lambda)$ by establishing relation
between $\lambda_{p}$\- Bohr radius and $\lambda_{p}$-arithmetic Bohr radius.
Our results mainly generalize the results obtained in [15].

We now concentrate another multidimensional generalization of the Bohr radius
$R_{p,q}(X)$ which has been introduced for the first time by Blasco [4] in
order to study the Bohr inequality for Banach space valued holomorphic
functions in the unit disk $\mathbb{D}$. In this paper, we introduce a new
notion of $R_{p,q}(X)$ in more general multidimensional settings. Let
$\Omega\subset\mathbb{C}^{n}$ be a complete Reinhardt domain. For each $1\leq
p,q<\infty$ and $\lambda\geq 1$, we define $\lambda_{p,q}$-Bohr radius,
denoted by $R^{n}_{p,q}(\mathcal{F}(\Omega,X),\lambda)$, of $\Omega$ with
respect to $\mathcal{F}(\Omega,X)$ to be the supremum of all $r\in[0,1]$ such
that

(1.7) $\sup_{z\in r\Omega}\left(\left\lVert
x_{0}\right\rVert^{p}+\left(\sum_{m=1}^{\infty}\sum_{|\alpha|=m}\left\lVert
x_{\alpha}(f)z^{\alpha}\right\rVert\right)^{q}\right)\leq\lambda\left\lVert
f\right\rVert_{\Omega}$

for all $f\in\mathcal{F}(\Omega,X)$ with
$f(z)=\sum_{k=0}^{\infty}\sum_{|\alpha|=k}x_{\alpha}(f)z^{\alpha}$. We use the
convention $R^{n}_{p,q}(\Omega,X,\lambda)$ instead of
$R^{n}_{p,q}(\mathcal{F}(\Omega,X),\lambda)$ and
$R^{n}_{p,q}(\Omega,X,\lambda)=R_{p,q}(\Omega,X,\lambda)$ for $n=1$. Blasco
[4], has extensively studied the Bohr radius $R_{p,q}(\mathbb{D},X,1)$ for
complex Banach spaces $X$. In the same paper, it has been established [4,
Theorem 1.2] that the classical Bohr radius $R(\mathbb{D},\ell^{n}_{p},1)=0$
for $1\leq p<\infty$. To make this modified Bohr radius more significant, we
introduce the constant $\lambda$ in (1.7). Determining the exact value of
$\lambda_{p,q}$-Bohr radius is difficult in-general for different complex
Banach spaces. Recently, Das [11] has obtained the exact value of
$R_{p,q}(\mathbb{D},\mathbb{C},1)$ for $p,q\in[1,\infty)$ and asymptotic
behavior of $R^{n}_{p,q}(B_{\ell^{n}_{\infty}},X,1)$ for infinite dimensional
Banach spaces. In this situation, our aim is to study $\lambda_{p,q}$-Bohr
radius $R_{p,q}(\mathbb{D},X,\lambda)$ in terms of the another notion of
arithmetic Bohr radius namely, $\lambda_{p,q}$-arithmetic Bohr radius
$A_{p,q}(\mathcal{F}(\Omega,X),\lambda)$ of $\Omega$ with respect to
$\mathcal{F}(\Omega,X)$, which is defined by

$\displaystyle A_{p,q}(\mathcal{F}(\Omega,X),\lambda):=$
$\displaystyle\sup\left\\{\frac{1}{n}\sum_{i=1}^{n}r_{i}\,|\,r\in\mathbb{R}^{n}_{\geq
0},\,\mbox{for all}\,f\in\mathcal{F}(\Omega,X):\left\lVert
x_{0}\right\rVert^{p}+\left(\sum_{m=1}^{\infty}\sum_{|\alpha|=m}\left\lVert
x_{\alpha}(f)\right\rVert r^{\alpha}\right)^{q}\leq\lambda\left\lVert
f\right\rVert_{\Omega}\right\\},$

where $\mathbb{R}^{n}_{\geq
0}=\\{r=(r_{1},.\,.\,.,r_{n})\in\mathbb{R}^{n}:r_{i}\geq 0,1\leq i\leq n\\}$
and $\left\lVert f\right\rVert_{\Omega}=\sup\\{\|f(z)\|:z\in\Omega\\}$.
The organization of this paper as follows. In Theorem 2.1, we provide the
estimate of $\lambda_{p}$-arithmetic Bohr radius of bounded holomorphic
functions in terms of $\lambda_{p}$-arithmetic Bohr radius of $k$-homogeneous
polynomials in complete Reinhardt domains. Theorem 2.2 gives the asymptotic
estimates of the $\lambda_{p}$-arithmetic Bohr radius for the $n$-dimensional
$\ell^{n}_{q}$-balls. Next, Theorem 2.5 relates the classical
$\lambda_{p}$-Bohr radius and $\lambda_{p}$-arithmetic Bohr radius in unit
ball of $\ell^{n}_{q}$ spaces. After giving an important estimate of
arithmetic Bohr radius of homogeneous polynomials in Proposition 2.5, we give
the a positive answer of Problem 1.6 in Theorem 2.4 with regard to the exact
value of $\lambda_{p}$-Bohr radii in unit ball of $\ell^{n}_{q}$ spaces. We
establish the connection between $\lambda_{p,q}$-arithmetic Bohr radii of
different vector-valued holomorphic functions spaces and the bounded
holomorphic function spaces in Theorem 2.5. Finally, we investigate the exact
value of $\lambda_{p,q}$-arithmetic Bohr radius of unit ball in $\ell^{n}_{1}$
space through the proof of Theorem 2.6.

##  2\. Main Results

Our first result provides an estimate of $\lambda_{p}$-arithmetic Bohr radius
of bounded holomorphic functions in terms of $\lambda_{p}$-arithmetic Bohr
radius of $k$-homogeneous polynomials in complete Reinhardt domains.

######  Theorem 2.1.

Let $\Omega$ be a complete Reinhardt domain in $\mathbb{C}^{n}$, $\lambda\geq
1$, $1\leq p<\infty$, and $C=\max\\{1/3,((\lambda-1)/(2\lambda-1))^{1/p}\\}$.
Then we have

$C\,A_{p}\left(\bigcup_{k=1}^{\infty}\mathcal{P}^{k}(\Omega),\mathbb{C},\lambda\right)\leq
A_{p}\left(H_{\infty}(\Omega),\lambda\right)\leq
A_{p}\left(\bigcup_{k=1}^{\infty}\mathcal{P}^{k}(\Omega),\mathbb{C},\lambda\right).$

It is worth to mention that the result of Defant et. al. [15, Proposition 2.2]
follows from Theorem 2.1 for $p=1$. In the following result Theorem 2.2, we
are going to give the asymptotic estimates of the $\lambda_{p}$-arithmetic
Bohr radius for the $n$-dimensional $\ell^{n}_{p}$-balls. The main ingredients
of our proof of Theorem 2.3 are the following facts which we record as Lemma
2.3 , Lemma 2.4 and Proposition 2.5. To make the statement of these results
brief, let us recall the following convention from [15]. For bounded Reinhardt
domains $\Omega_{1},\Omega_{2}\subset\mathbb{C}^{n}$, let

$S(\Omega_{1},\Omega_{2}):=\inf\left\\{s>0:\Omega_{1}\subset
s\Omega_{2}\right\\}.$

By a Banach sequence space $X$, we mean a complex Banach space
$X\subset\mathbb{C}^{\mathbb{N}}$ such that $\ell_{1}\subset
X\subset\ell_{\infty}$. If $\Omega$ is a bounded Reinhardt domain in
$\mathbb{C}^{n}$ and $X$ and $Y$ are Banach sequence spaces, we write

(2.1) $S(\Omega,B_{X_{n}})=\sup_{z\in\Omega}\left\lVert
z\right\rVert_{X}\quad\mbox{and }\quad
S(B_{X_{n}},B_{Y_{n}})=\left\lVert\mbox{id}:X_{n}\rightarrow
Y_{n}\right\rVert,$

where $X_{n}$(resp. $Y_{n}$) is the space spanned by first $n$ canonical basis
vectors $e_{n}$ in $X$(resp. $Y$). Throughout the paper, $X$ will stand for a
complex Banach space. The next result compares the $\lambda_{p}$-arithmetic
Bohr radii for two bounded Reinhardt domains in $\mathbb{C}^{n}$.

######  Lemma 2.2.

Let $\Omega_{1},\Omega_{2}$ be bounded Reinhardt domains in $\mathbb{C}^{n}$
and $1\leq p<\infty$. Then for $\lambda\geq 1$, we have

$A_{p}(\Omega_{1},X,\lambda)\leq
S(\Omega_{1},\Omega_{2})\,A_{p}(\Omega_{2},X,\lambda).$

######  Remark 2.1.

  1. (a)

For a bounded Reinhardt domain $\Omega$ in $\mathbb{C}^{n}$, it is easy to
observe that $S(\Omega,t\Omega)=1/t$ and $S(t\Omega,\Omega)=t$ for all $t>0.$
Therefore, an immediate consequence from Lemma 2.2 we have that for any $1\leq
p<\infty$,

$A_{p}(t\Omega_{1},X,\lambda)=tA_{p}(\Omega_{1},X,\lambda),\quad\mbox{for
all}\quad t>0.$

  2. (b)

$A_{p}(\cdot,X,\lambda)$ is increasing, that is,
$A_{p}(\Omega_{1},X,\lambda)\leq A_{p}(\Omega_{2},X,\lambda)$ whenever
$\Omega_{1}\subset\Omega_{2}.$

  3. (c)

It is obvious that $A_{p}(\Omega,X,\lambda)\leq A^{k}_{p}(\Omega,X,\lambda)$.

  4. (d)

We note that Lemma 2.2 generalizes the result of Defant et. al. [15, Lemma
4.2].

Using the above notation, the following lemma gives a connection between the
$\lambda_{p}$-arithmetic Bohr radius $A_{p}(\Omega,X,\lambda)$ for bounded
Reinhardt domain $\Omega\subset\mathbb{C}^{n}$ and the $p$-Bohr radius
$r^{n}_{p}(\Omega,X,1)$ of a complex Banach space $X$, which helps us to
obtain the lower bound in Theorem 2.2.

######  Lemma 2.3.

Let $\Omega\subset\mathbb{C}^{n}$ be a bounded Reinhardt domain and $1\leq
p<\infty$. Then for $\lambda\geq 1$ we have,

$A_{p}(\Omega,X,\lambda)\geq\frac{S(\Omega,B_{\ell^{n}_{1}})}{n}r^{n}_{p}(\Omega,X,1).$

We need the following estimate for $k$-homogeneous polynomials $(k\geq 2)$ in
$\mathbb{C}^{n}$ which has been proved by Boas [7].

######  Lemma 2.4.

[7] Let $1\leq p\leq\infty$ and $n$ and $k$ are positive integers greater than
$1$. Then there exists a homogeneous polynomial of degree $k$ in
$\mathbb{C}^{n}$ and signs $\epsilon_{\alpha}\in\\{-1,1\\}$ such that

$\bigg{\|}\sum_{|\alpha|=k}\epsilon_{\alpha}\frac{k!}{\alpha!}z^{\alpha}\bigg{\|}_{B_{\ell^{n}_{p}}}\leq\sqrt{32k\log(6k)}\,\times\,\left\\{\begin{array}[]{ll}n^{\frac{1}{2}}(k!)^{1-\frac{1}{p}}&\mbox{
if \,$p\leq 2$},\\\\[2.84526pt]
n^{\frac{1}{2}+\left(\frac{1}{2}-\frac{1}{p}\right)k}(k!)^{\frac{1}{2}}&\mbox{
if \,$p\geq 2$}.\end{array}\right.$

Now we obtain an upper bound for $\lambda_{p}$-arithmetic Bohr radius of
$k$-homogeneous polynomials in $\mathbb{C}^{n}$ which later gives an upper
bound for $\lambda_{p}$-arithmetic Bohr radius of $B_{\ell^{n}_{q}}\,\,(1\leq
q\leq\infty)$ with respect to $H_{\infty}(B_{\ell^{n}_{q}},X).$

######  Proposition 2.5.

Let $1\leq p<\infty$ and $\lambda\geq 1$. Then for all $1\leq q\leq\infty$ and
$k,n\geq 2$ we have

$A^{k}_{p}(B_{\ell^{n}_{q}},X,\lambda)\leq d\sqrt[pk]{\lambda}\sqrt[2k]{k\log
k(k!)^{2(1-(1/\min\\{2,q\\}))}n}\,\left(\frac{1}{n}\right)^{\frac{1}{p}+\frac{1}{\max\\{2,q\\}}-\frac{1}{2}},$

where $d$ is a uniform constant.

Following the techniques from [15, 13] and all above preparation, we now wish
to state one of our main results with regard to the asymptotic estimates of
the $\lambda_{p}$-arithmetic Bohr radius for the $n$-dimensional
$\ell^{n}_{q}$-balls.

######  Theorem 2.2.

Let $1\leq p<\infty$ and $\lambda\geq 1$. Then there exists a constant $d>0$
such that for all $1\leq q\leq\infty$ and $n>1$, we have

(2.6) $\frac{r^{n}_{p}(B_{\ell^{n}_{q}},X,1)}{n^{1/q}}\leq
A_{p}(B_{\ell^{n}_{q}},X,\lambda)\leq d\,\frac{\left(\log
n\right)^{1-\left(1/\min\\{q,2\\}\right)}}{n^{\frac{1}{p}+\frac{1}{\max\\{2,q\\}}-\frac{1}{2}}}.$

######  Remark 2.2.

In Theorem 2.2, for $p=1$ we obtain the upper estimate of [15, Theorem 4.1],
but we feel that the lower estimate of (2.6) can be improved for arbitrary
complex Banach spaces.

Next, we present a relationship between arithmetic Bohr radius and geometric
property of complex Banach spaces. Recently, Das [11] has studied the $p$-Bohr
radius $r_{p}(B_{\ell^{n}_{\infty}},X,1)$ of a complex Banach space $X$. In
the same paper [11, Theorem 3], the author has established the asymptotic
estimates of $r_{p}(B_{\ell^{n}_{\infty}},X,1)$ for any $p$-uniformly
$PL$-convex $(p\geq 2)$ complex Banach spaces $X$. A complex Banach space $X$
is said to be $p$-uniformly $PL$-convex $(2\leq p<\infty)$ if there exists a
constant $\delta>0$ such that

(2.7) $\left\lVert x\right\rVert^{p}+\delta\left\lVert
y\right\rVert^{p}\leq\frac{1}{2\pi}\int_{0}^{2\pi}\left\lVert
x+e^{i\theta}y\right\rVert^{p}\,d\theta$

for all $x,y\in X$. We denote $I_{p}(X)$ be the supremum of all such
$\delta>0$ satisfying (2.7).

In view of [11, Theorem 3], Lemma 2.3 and the fact that
$S(B_{\ell^{n}_{\infty}},B_{\ell^{n}_{1}})=1/n$, we have the following
consequence.

######  Corollary 2.8.

Let $B_{\ell^{n}_{\infty}}$ be the unit polydisk in $\mathbb{C}^{n}$ and
$2\leq p<\infty$. Then for all $\lambda\geq 1$ and $n>1$, we have

$A_{p}(B_{\ell^{n}_{\infty}},X,\lambda)\geq\frac{1}{n^{2}}\left(\frac{I_{p}(X)}{2^{p}+I_{p}(X)}\right)^{2/p},$

where $X$ is any $p$-uniformly PL-convexity complex Banach space $X$ with
dim$(X)\geq 2$.

Now we are in position to give an interesting connection between the classical
$\lambda_{p}$-Bohr radius and $\lambda_{p}$-arithmetic Bohr radius in unit
ball of $\ell^{n}_{q}$ spaces.

######  Theorem 2.3.

Let $1\leq\lambda<\infty$ and $1\leq p<\infty$. Then for every $n$ and $1\leq
q<\infty$ we have

$\frac{r_{p}(\mathbb{D},X,\lambda)}{n}\leq
A_{p}(B_{\ell^{n}_{q}},X,\lambda)\leq\left(\frac{r_{p}(\mathbb{D},X,\lambda)}{n}\right)^{1/q}.$

As we promised before, we obtain an affirmative answer of our proposed Problem
1.6 in the following result.

######  Theorem 2.4.

Let $X$ be a complex Banach space and $1\leq p<\infty$ and $1\leq
q\leq\infty$. Then for $\lambda\geq 1$, and for all $n\in\mathbb{N}$, we have

$A_{p}(B_{\ell^{n}_{q}},X,\lambda)=\frac{r^{n}_{p}(B_{\ell^{n}_{q}},X,\lambda)}{n^{1/q}}.$

From Theorem 2.3 and Theorem 2.4, we obtain the following important result.

######  Corollary 2.9.

Let $1\leq p<\infty$ and $\lambda\geq 1$. Then for every $1\leq q<\infty$ and
$n\in\mathbb{N}$, we have

$\frac{r_{p}(\mathbb{D},X,\lambda)}{n^{1-(1/q)}}\leq
r^{n}_{p}(B_{\ell^{n}_{q}},X,\lambda)\leq r_{p}(\mathbb{D},X,\lambda).$

In view of Corollary 2.9, we obtain a surprising consequence that the
$\lambda_{p}$-Bohr radius of unit ball in $\ell^{n}_{1}$ space coincides with
the classical $\lambda_{p}$-Bohr radius in unit disk in complex plane. We
state the consequence as our next corollary which reduces the result of Kumar
[18, Corollary 2.6] for $p=1$.

######  Corollary 2.10.

For every $\lambda\in[1,\infty)$ and $n\in\mathbb{N}$, we have
$r^{n}_{p}(B_{\ell^{n}_{1}},X,\lambda)=r_{p}(\mathbb{D},X,\lambda).$

We want to recall that we have provided the notion of
$\lambda_{p,q}$-arithmetic Bohr radius in Section 1. In the following main
result, we establish that the $\lambda_{p,q}$-arithmetic Bohr radii of
different vector-valued holomorphic function spaces $\mathcal{O}$ corresponds
to $\lambda_{p,q}$-arithmetic Bohr radius of bounded holomorphic function
space if $\mathcal{O}$ contains all vector-valued polynomials.

######  Theorem 2.5.

Let $\Omega$ be a complete Reinhardt domain in $\mathbb{C}^{n}$ and
$\lambda\geq 1$. If $\mathcal{O}(\Omega,X)$ is a space of all holomorphic
mappings on $\Omega$ into a complex Banach space $X$ containing all vector-
valued polynomials then for every $1\leq p,q,<\infty$, we have

$A_{p,q}(\mathcal{O}(\Omega,X),\lambda)=A_{p,q}(H_{\infty}(\Omega,X),\lambda).$

An analogous version of Theorem 2.3 can also be shown in the setting of
$\lambda_{p,q}$-arithmetic Bohr radius which later provides us an exact value
of arithmetic Bohr radius.

######  Theorem 2.6.

Let $1\leq q<\infty$ and $\lambda\geq 1$. Then for every $1\leq
p_{1},q_{1}<\infty$ and $n$ we have

$\frac{R_{p_{1},q_{1}}(\mathbb{D},X,\lambda)}{n}\leq
A_{p_{1},q_{1}}(B_{\ell^{n}_{q}},X,\lambda)\leq\left(\frac{R_{p_{1},q_{1}}(\mathbb{D},X,\lambda)}{n}\right)^{1/q}.$

In 2010, O. Blasco [4, Proposition 1.4], established the exact value of Bohr
radius $R_{p,1}(\mathbb{D},\mathbb{C})$ as $p/(p+2)$ for $1\leq p\leq 2$ and
$\lambda=1$. In view of of Blasco’s result, we record the following easy
consequence of Theorem 2.6 as the exact value of arithmetic Bohr radius of the
unit ball in $\ell^{n}_{1}$.

######  Corollary 2.11.

For every $1\leq p\leq 2$ and $n\in\mathbb{N}$, we have
$A_{p,1}(B_{\ell^{n}_{1}},\mathbb{C},1)=p/(n(p+2)).$

##  3\. Proof of Main Results

Proof of Theorem 2.1. The right-hand inequality is clear from the fact that

$\bigcup_{k=1}^{\infty}\mathcal{P}^{k}(\Omega)\subset H_{\infty}(\Omega).$

Choose $r\in\mathbb{R}^{n}_{\geq 0}$ such that for all $k\geq 1$ and for all
$k$-homogeneous polynomial $h\in\mathcal{P}^{k}(\mathbb{C}^{n})$, we have

(3.1) $\sum_{|\alpha|=k}|c_{\alpha}(h)r^{\alpha}|^{p}\leq\lambda\left\lVert
h\right\rVert^{p}_{\Omega}.$

We wish to show that

(3.2)
$\max\left\\{\frac{1}{3},\left(\frac{\lambda-1}{2\lambda-1}\right)^{1/p}\right\\}\frac{1}{n}\sum_{i=1}^{n}r_{i}\leq
A_{p}(H_{\infty}(\Omega)_{,}\,\lambda).$

Fix $f\in H_{\infty}(\Omega)$. Without loss of generality, we assume that
$\left\lVert f\right\rVert_{\infty}=1$ and consider its monomial series
expansion as $f(z)=\sum_{\alpha\in\mathbb{N}^{n}_{0}}c_{\alpha}(f)z^{\alpha}$.
Given $z\in\Omega$, we define the function $h:\mathbb{D}\rightarrow\mathbb{C}$
by

$h(\zeta)=f(\zeta
z)=\sum_{\alpha\in\mathbb{N}^{n}_{0}}c_{\alpha}(f)\zeta^{|\alpha|}z^{\alpha}=\sum_{k=0}^{\infty}\left(\sum_{|\alpha|=k}c_{\alpha}(f)z^{\alpha}\right)\zeta^{k}.$

Let $\theta\in\mathbb{R}$ be such that $e^{i\theta}c_{0}(f)=|c_{0}(f)|$ and
consider the function $F(\zeta)=1-e^{i\theta}h(\zeta)$, $\eta\in\mathbb{D}$.
It is worth to notice that $\zeta z\in\Omega$ for every $\zeta\in\mathbb{D}$,
thus we have $\left\lVert h\right\rVert_{\infty}\leq 1$ on $\mathbb{D}$ and

$\mbox{Re}\,F(\zeta)=\mbox{Re}\left(1-e^{i\theta}h(\zeta)\right)\geq 0.$

Suppose $F(\zeta)=\sum_{k=0}^{\infty}a_{k}\zeta^{k}$, then

$\sum_{k=0}^{\infty}a_{k}\zeta^{k}=1-\sum_{k=0}^{\infty}\left(e^{i\theta}\sum_{|\alpha|=k}c_{\alpha}(f)z^{\alpha}\right)\zeta^{k}=1-e^{i\theta}c_{0}(f)+\sum_{k=1}^{\infty}\left(e^{i\theta}\sum_{|\alpha|=k}c_{\alpha}(f)z^{\alpha}\right)\zeta^{k}.$

It follows that $a_{0}=1-|c_{0}(f)|$ and
$a_{k}=e^{i\theta}\sum_{|\alpha|=k}c_{\alpha}(f)z^{\alpha}$, for
$k\in\mathbb{N}.$
In view of Carathéodory’s theorem [3], we obtain that

(3.3) $\left|\sum_{|\alpha|=k}c_{\alpha}(f)z^{\alpha}\right|\leq
2(1-|c_{0}(f)|).$

From (3.1) and (3.3), we have

$\displaystyle\sum_{\alpha\in\mathbb{N}^{n}_{0}}\left|c_{\alpha}(f)\left(\frac{r}{3}\right)^{\alpha}\right|^{p}$
$\displaystyle=|c_{0}(f)|^{p}+\sum_{k=1}^{\infty}\sum_{|\alpha|=k}\left|c_{\alpha}(f)\left(\frac{r}{3}\right)^{\alpha}\right|^{p}$
$\displaystyle=|c_{0}(f)|^{p}+\sum_{k=1}^{\infty}\frac{1}{3^{pk}}\sum_{|\alpha|=k}|c_{\alpha}(f)|^{p}r^{p\alpha}$
$\displaystyle\leq|c_{0}(f)|^{p}+\sum_{k=1}^{\infty}\frac{1}{3^{pk}}\,\lambda\sup_{z\in\Omega}\left|\sum_{|\alpha|=k}c_{\alpha}(f)z^{\alpha}\right|^{p}$
$\displaystyle\leq|c_{0}(f)|^{p}+\lambda\,\frac{2^{p}}{3^{p}-1}\left(1-|c_{0}(f)|\right)^{p}\leq\lambda=\lambda\left\lVert
f\right\rVert_{\Omega}.$

This gives one of the estimates in (3.2). To obtain the other estimate, we set
$s=({(2\lambda-1)}/{(\lambda-1)})^{1/p}.$ By going similar lines of argument
as above and using the well-known Cauchy’s inequalities, we obtain

$\displaystyle\sum_{\alpha\in\mathbb{N}^{n}_{0}}\left|c_{\alpha}(f)\left(\frac{r}{s}\right)^{\alpha}\right|^{p}$
$\displaystyle=|c_{0}(f)|^{p}+\sum_{k=1}^{\infty}\frac{1}{s^{pk}}\sum_{|\alpha|=k}|c_{\alpha}(f)|^{p}r^{p\alpha}$
$\displaystyle\leq|c_{0}(f)|^{p}+\lambda\sum_{k=1}^{\infty}\frac{1}{s^{pk}}\left\lVert
P_{k}(f)\right\rVert^{p}_{\Omega}$
$\displaystyle\leq|c_{0}(f)|^{p}+\lambda\left\lVert
f\right\rVert^{p}_{\Omega}\sum_{k=1}^{\infty}\frac{1}{s^{pk}}\leq\left(1+\frac{\lambda}{s^{p}-1}\right)\left\lVert
f\right\rVert^{p}_{\Omega}=\lambda\left\lVert f\right\rVert^{p}_{\Omega}.$

Therefore, $(1/sn)\sum_{i=1}^{n}r_{i}\leq
A_{p}(H_{\infty}(\Omega),\mathbb{C},\lambda).$ This completes the proof. ∎

Proof of Lemma 2.2. Let $r=(r_{1},\cdots,r_{n})\in\mathbb{R}^{n}_{\geq 0}$ be
such that

$\sum_{\alpha\in\mathbb{N}^{n}_{0}}\left\lVert
x_{\alpha}(f)r^{\alpha}\right\rVert^{p}\leq\lambda\left\lVert
f\right\rVert^{p}_{\Omega_{1}}$

for all $f(z)=\sum_{\alpha\in\mathbb{N}^{n}_{0}}x_{\alpha}(f)z^{\alpha}\in
H_{\infty}(\Omega_{1},X)$. Set $t:=S(\Omega_{1},\Omega_{2})$. Then for all
$\epsilon>0$, we have

$\sum_{\alpha\in\mathbb{N}^{n}_{0}}\left\lVert
x_{\alpha}(f)\right\rVert^{p}\left(\frac{r}{t+\epsilon}\right)^{p\alpha}\leq\lambda\bigg{\|}\sum_{\alpha\in\mathbb{N}^{n}_{0}}x_{\alpha}(f)\left(\frac{z}{t+\epsilon}\right)^{\alpha}\bigg{\|}^{p}_{\Omega_{1}}\leq\lambda\left\lVert
f\right\rVert^{p}_{\Omega_{2}}$

for all $f\in H_{\infty}(\Omega_{2},X).$ By letting $\epsilon\rightarrow 0$,
we obtain

$\frac{1}{nt}\sum_{i=1}^{n}r_{i}\leq A_{p}(\Omega_{2},X,\lambda).$

Therefore, it follows that $A_{p}(\Omega_{1},X,\lambda)\leq
S(\Omega_{1},\Omega_{2})\,A_{p}(\Omega_{2},X,\lambda).$ ∎

Proof of Lemma 2.3. It is sufficient to give a proof for the case $\lambda=1$.
That is, for arithmetic Bohr radius $A_{p}(\Omega,X)$, as
$A_{p}(\Omega,X,\lambda)$ is increasing in $\lambda$. In view of (2.1), we
have

$S(\Omega,B_{\ell^{n}_{1}})=\sup_{z\in\Omega}\left\lVert
z\right\rVert_{\ell^{n}_{1}}.$

Therefore, for given $0<\epsilon<r^{n}_{p}(\Omega,X,1)$, we can find an
element $\tilde{z}\in\Omega$ such that

$\left\lVert\tilde{z}\right\rVert_{\ell^{n}_{1}}\geq
S(\Omega,B_{\ell^{n}_{1}})-\epsilon.$

Let $s:=r^{n}_{p}(\Omega,X,1)-\epsilon$, $w:=s\tilde{z}$, and
$r:=s|\tilde{z}|=|w|.$ Since $w\in s\Omega$ and $s<r^{n}_{p}(\Omega,X,1)$, for
$f=\sum_{\alpha\in\mathbb{N}^{n}_{0}}x_{\alpha}z^{\alpha}\in
H_{\infty}(\Omega,X)$, we have

$\sum_{\alpha\in\mathbb{N}^{n}_{0}}\left\lVert
x_{\alpha}\right\rVert^{p}r^{p\alpha}=\sum_{\alpha\in\mathbb{N}^{n}_{0}}\left\lVert
x_{\alpha}w^{\alpha}\right\rVert^{p}\leq\bigg{\|}\sum_{\alpha\in\mathbb{N}^{n}_{0}}x_{\alpha}z^{\alpha}\bigg{\|}^{p}_{s\Omega}\leq\left\lVert
f\right\rVert^{p}_{\Omega}.$

Therefore, we obtain

$A_{p}(\Omega,X)\geq\frac{1}{n}\sum_{i=1}^{n}r_{i}=\frac{r^{n}_{p}(\Omega,X,1)-\epsilon}{n}\left\lVert\tilde{z}\right\rVert_{\ell^{n}_{1}}\geq\frac{r^{n}_{p}(\Omega,X,1)-\epsilon}{n}\left(S(\Omega,B_{\ell^{n}_{1}})-\epsilon\right)$

holds for all $\epsilon>0.$ This completes the proof. ∎

Proof of Proposition 2.5. Let $r=(r_{1},.\,.\,.,r_{n})\in\mathbb{R}^{n}_{\geq
0}$ be such that

$\sum_{|\alpha|=k}\left\lVert
x_{\alpha}(f)\right\rVert^{p}r^{p\alpha}\leq\lambda\left\lVert
f\right\rVert^{p}_{B_{\ell^{n}_{q}}}$

for all $f\in\mathcal{P}^{k}(\mathbb{C}^{n},X)$. From Hölder’s inequality, we
have $\left\lVert r\right\rVert^{p}_{\ell^{n}_{1}}\leq n^{p-1}\left\lVert
r\right\rVert^{p}_{\ell^{n}_{p}}$. In view of Lemma 2.4, there exist signs
$\epsilon_{\alpha}\in\\{-1,1\\}$ such that

$\left(\frac{1}{n}\sum_{i=1}^{n}r_{i}\right)^{pk}\leq\frac{1}{n^{k}}\sum_{|\alpha|=k}\frac{k!}{\alpha!}r^{p\alpha}\leq\frac{\lambda}{n^{k}}\bigg{\|}\sum_{|\alpha|=k}\epsilon_{\alpha}\frac{k!}{\alpha!}z^{\alpha}\bigg{\|}^{p}_{B_{\ell^{n}_{p}}}\leq\frac{\lambda}{n^{k}}R_{k},$

where $R_{k}$ is the expression in the right-hand side of Lemma 2.4. Taking
the $pk$-th root of both sides, we obtain

$A^{k}(B_{\ell^{n}_{q}},X,\lambda)\leq\frac{\sqrt[pk]{\lambda}\sqrt[k]{R_{k}}}{n^{1/p}}.$

This completes the proof. ∎

Proof of Theorem 2.2. In view of Lemma 2.3 and using the fact that

$S(B_{\ell^{n}_{q}},B_{\ell^{n}_{1}})=n^{1-(1/q)}\quad\,\mbox{for
all}\,\,\,1\leq q\leq\infty,$

we obtain the lower bound of (2.6). For the upper bound of
$A_{p}(B_{\ell^{n}_{q}},X,\lambda)$, by Proposition 2.5, it is enough to show
that there exists a constant $d>0$ such that for all $n>1$ we are able to find
a $k\,(\geq 2)$ with

(3.4) $\sqrt[pk]{\lambda}\sqrt[2k]{k\log
k(k!)^{2\left(1-\left(1/\min\\{2,q\\}\right)\right)}n}\leq d\,\left(\log
n\right)^{1-\left(1/\min\\{2,q\\}\right)}.$

It is worth to note that the quantity $\sqrt[2k]{k\log k}$ is bounded in $k$
and also $\sqrt[k]{k!}\leq k/\sqrt{2}.$ So we can estimate the left-hand side
of (3.4) upto a constant independent of $k$ and $n$ by

(3.5) $\sqrt[pk]{\lambda}\,k^{1-\left(1/\min\\{2,q\\}\right)}\sqrt[2k]{n}.$

Let us choose $k=2$ for $n=2,.\,.\,.,8$ and $k=[\log n]$ (the greatest integer
less than or equal to $\log n$) for $n\geq 9$. Then, the above expression
(3.5) is bounded above by

$\lambda^{\frac{2}{p\log n}}(\log n)^{1-\frac{1}{\min\\{2,q\\}}}e,$

which gives the desired bound. This completes the proof. ∎

Proof of Theorem 2.3. Let $r=r_{p}(\mathbb{D},X,\lambda)$ and
$f(z)=\sum_{\alpha\in\mathbb{N}^{n}_{0}}c_{\alpha}(f)z^{\alpha}\in
H_{\infty}(B_{\ell^{n}_{q}},X)$. Consider the function $g(z)=f(\xi z)$, where
$\xi=(1,0,.\,.\,.,0)\in B_{\ell^{n}_{q}}$ and $z\in\mathbb{D}$. Clearly, $g\in
H_{\infty}(\mathbb{D},X)$ with $\left\lVert
g\right\rVert_{\mathbb{D}}=\left\lVert f\right\rVert_{B_{\ell^{n}_{q}}}$.
Therefore, we have

$\sum_{\alpha\in\mathbb{N}^{n}_{0}}\left\lVert
x_{\alpha}(f)\right\rVert^{p}(r,0,\,.\,.\,.,0)^{p\alpha}=\sum_{k=0}^{\infty}\left\lVert
x_{k\xi}(f)\right\rVert^{p}r^{pk}=\sum_{k=0}^{\infty}\left\lVert
x_{k}(g)\right\rVert^{p}r^{pk}\leq\lambda\left\lVert
g\right\rVert_{\mathbb{D}}=\lambda\left\lVert
f\right\rVert_{B_{\ell^{n}_{q}}}.$

This shows that $(r/n)\leq A_{p}(B_{\ell^{n}_{q}},X,\lambda)$, and hence we
obtain $(r_{p}(\mathbb{D},X,\lambda)/n)\leq
A_{p}(B_{\ell^{n}_{q}},X,\lambda).$ Conversely, we prove that

$A_{p}(B_{\ell^{n}_{q}},X,\lambda)\leq\left(\frac{r_{p}(\mathbb{D},X,\lambda)}{n}\right)^{1/q}.$

Suppose $r\in\mathbb{R}^{n}_{\geq 0}$ such that for all $h\in
H_{\infty}(B_{\ell^{n}_{q}},X)$,

$\sum_{\alpha\in\mathbb{N}^{n}_{0}}\left\lVert
h\right\rVert^{p}r^{p\alpha}\leq\left\lVert
h\right\rVert^{p}_{B_{\ell^{n}_{q}}}.$

We want to prove that

$\frac{\sum_{j=1}^{n}r_{j}}{n}\leq\left(\frac{r_{p}(\mathbb{D},X,\lambda)}{n}\right)^{1/q}.$

Let $f:\mathbb{D}\rightarrow X$ be a bounded holomorphic map and we consider
the function $s:B_{\ell^{n}_{q}}\rightarrow\mathbb{D}$ defined by

$s(z)=z^{q}_{1}+\cdots+z^{q}_{n},\quad z\in B_{\ell^{n}_{q}}.$

Now if we set $h=f\circ s$, then we have $\left\lVert
h\right\rVert_{B_{\ell^{n}_{q}}}=\left\lVert f\right\rVert_{\mathbb{D}}.$
Moreover, for each $z\in B_{\ell^{n}_{q}}$,

$h(z)=\sum_{k=0}^{\infty}x_{k}(f)s(z)^{k}=\sum_{k=0}^{\infty}x_{k}(f)\sum_{|\alpha|=k}\frac{k!}{\alpha!}z^{q\alpha}=\sum_{\alpha\in\mathbb{N}^{n}_{0}}x_{\alpha}(h)z^{q\alpha},$

where

$x_{\alpha}(h)=\frac{k!}{\alpha!}x_{k}(f)$

whenever $|\alpha|=k$. Then for all $z\in\mathbb{C}^{n}$, we have

$\displaystyle\sum_{\alpha\in\mathbb{N}^{n}_{0}}\left\lVert
x_{\alpha}(h)\right\rVert^{p}|z|^{pq\alpha}$
$\displaystyle=\sum_{k=0}^{\infty}\left\lVert
x_{k}(f)\right\rVert^{p}\sum_{|\alpha|=k}\left(\frac{k!}{\alpha!}\right)^{p}|z|^{pq\alpha}$
$\displaystyle\geq\sum_{k=0}^{\infty}\left\lVert
x_{k}(f)\right\rVert^{p}\sum_{|\alpha|=k}\frac{k!}{\alpha!}|z|^{pq\alpha}=\sum_{k=0}^{\infty}\left\lVert
x_{k}(f)\right\rVert^{p}\left\lVert z\right\rVert^{pqk}_{q}.$

An observation shows that

$\sum_{k=0}^{\infty}\left\lVert x_{k}(f)\right\rVert^{p}\left\lVert
r\right\rVert^{pqk}_{q}\leq\sum_{\alpha\in\mathbb{N}^{n}_{0}}\left\lVert
x_{\alpha}(h)r^{q\alpha}\right\rVert^{p}\leq\lambda\left\lVert
h\right\rVert^{p}_{B_{\ell^{n}_{q}}}=\lambda\left\lVert
f\right\rVert_{\mathbb{D}},$

which implies that $\left\lVert r\right\rVert^{q}_{q}\leq
r_{p}(\mathbb{D},X,\lambda).$ Again we have $\left\lVert
r\right\rVert^{q}_{1}\leq n^{q-1}\left\lVert r\right\rVert^{q}_{q}.$ Hence we
obtain $n^{1-q}\left\lVert r\right\rVert^{q}_{1}\leq
r_{p}(\mathbb{D},X,\lambda)$. This completes the proof. ∎

Proof of Theorem 2.4. The inequality
$A_{p}(B_{\ell^{n}_{q}},X,\lambda)\geq{r^{n}_{p}(B_{\ell^{n}_{q}},X,\lambda)}/{n^{1/q}}$
follows from Lemma 2.3. Therefore, we only need to show that

(3.6)
$A_{p}(B_{\ell^{n}_{q}},X,\lambda)\leq\frac{r^{n}_{p}(B_{\ell^{n}_{q}},X,\lambda)}{n^{1/q}}.$

Let $r=(r_{1},.\,.\,.,r_{n})\in\mathbb{R}^{n}_{\geq 0}$ be such that

$\sum_{\alpha\in\mathbb{N}^{n}_{0}}\left\lVert
x_{\alpha}(g)\right\rVert^{p}r^{p\alpha}\leq\lambda\left\lVert
g\right\rVert^{p}_{B_{\ell^{n}_{q}}}$

for all $g(z)=\sum_{\alpha\in\mathbb{N}^{n}_{0}}x_{\alpha}(g)z^{\alpha}\in
H_{\infty}(B_{\ell^{n}_{q}},X)$. To show (3.6), it suffices to prove that
$n^{(1/q)-1}\left\lVert r\right\rVert_{1}\leq
r_{p}(B_{\ell^{n}_{q}},X,\lambda).$ Fix $f\in H_{\infty}(B_{\ell^{n}_{q}},X)$,
then for $v\in n^{(1/q)-1}\left\lVert
r\right\rVert_{1}\overline{B_{\ell^{n}_{q}}}$, we have $\left\lVert
v\right\rVert_{1}\leq\left\lVert r\right\rVert_{1}$. Therefore,

$\sum_{\alpha\in\mathbb{N}^{n}_{0}}\left\lVert
x_{\alpha}(f)v^{\alpha}\right\rVert^{p}\leq\lambda\left\lVert
f\right\rVert^{p}_{B_{\ell^{n}_{q}}}$

holds for every $v\in n^{(1/q)-1}\left\lVert
r\right\rVert_{1}\overline{B_{\ell^{n}_{q}}}.$ Therefore, it follows that
$n^{(1/q)-1}A_{p}(B_{\ell^{n}_{q}},X,\lambda)\leq
r^{n}_{p}(B_{\ell^{n}_{q}},X,\lambda),$ which gives our conclusion. This
completes the proof. ∎

Proof of Theorem 2.5. We start with an observation that

$\mathcal{P}(\Omega,X)\subseteq\mathcal{O}(\Omega,X)\subseteq
H_{\infty}(\Omega,X).$

Then, we have the following inclusion

$A_{p,q}(H_{\infty}(\Omega,X),\lambda)\leq
A_{p,q}(\mathcal{O}(\Omega,X),\lambda)\leq
A_{p,q}(\mathcal{P}(\Omega,X),\lambda).$

It suffices to show that for any $\lambda\geq 1$,
$A_{p,q}(\mathcal{P}(\Omega,X),\lambda)\leq
A_{p,q}(H_{\infty}(\Omega,X),\lambda)$.
Assume $r\in\mathbb{R}^{n}_{\geq 0}$ such that for all
$P\in\mathcal{P}(\mathbb{C}^{n},X)$,

(3.7) $\left\lVert
x_{0}(P)\right\rVert^{p}+\left(\sum_{k=1}^{\infty}\left\lVert
x_{\alpha}(P)\right\rVert r^{\alpha}\right)^{q}\leq\lambda\left\lVert
P\right\rVert_{\Omega}.$

Fix $f\in H_{\infty}(\Omega,X)$ and $0<t<1$. Our aim is to show that

(3.8) $\left\lVert
x_{0}(f)\right\rVert^{p}+\left(\sum_{k=1}^{\infty}\left\lVert
x_{\alpha}(f)\right\rVert(tr)^{\alpha}\right)^{q}\leq\lambda\left\lVert
f\right\rVert_{\Omega}.$

For $k\geq 0$, consider the $k$-homogeneous polynomials
$\chi_{k}=\sum_{|\alpha|=k}x_{\alpha}(f)z^{\alpha}$, where $x_{\alpha}(f)$ is
the $\alpha$-th coefficient in the monomial series expansion
$\sum_{\alpha\in\mathbb{N}^{n}_{0}}x_{\alpha}(f)z^{\alpha}$ of $f$. Note that
$\Omega$ is balanced, so by Cauchy inequalities we have

(3.9)
$\left\lVert\chi_{k}\right\rVert_{t\Omega}=t^{k}\left\lVert\chi_{k}\right\rVert_{\Omega}\leq
t^{k}\left\lVert f\right\rVert_{\Omega}.$

Therefore, the series $\sum_{k=0}^{\infty}\chi_{k}$ converges absolutely and
uniformly to $f$ on $t\Omega$. Thus, given $\epsilon>0$, there exits
$n_{0}\in\mathbb{N}$ such that for all $n\geq n_{0}$,

$\bigg{\|}\sum_{k=0}^{n}\chi_{k}\bigg{\|}_{t\Omega}\leq\left\lVert
f\right\rVert_{t\Omega}+\epsilon\leq\left\lVert
f\right\rVert_{\Omega}+\epsilon.$

We now consider the polynomials $\Gamma_{k}(z)=\chi_{k}(tz)$ for
$z\in\mathbb{C}^{n}$. Then,

$x_{\alpha}(\Gamma_{k})=\left\\{\begin{array}[]{ll}t^{k}x_{\alpha}(f)&\mbox{
if \,$|\alpha|=k$},\\\\[2.84526pt] 0&\mbox{ if \,$|\alpha|\neq
k$}.\end{array}\right.$

From (3.7) and (3.9), we obtain that

$\displaystyle\left\lVert
x_{0}(f)\right\rVert^{p}+\left(\sum_{k=1}^{\infty}\left\lVert
x_{\alpha}(f)\right\rVert(tr)^{\alpha}\right)^{q}$ $\displaystyle=\left\lVert
x_{0}(\Gamma_{0})\right\rVert^{p}+\left(\sum_{\alpha\in\mathbb{N}^{n}}\bigg{\|}x_{\alpha}\left(\sum_{k=1}^{n}\Gamma_{k}\right)r^{\alpha}\bigg{\|}\right)^{q}$
$\displaystyle\leq\lambda\bigg{\|}\sum_{k=0}^{n}\Gamma_{k}\bigg{\|}_{\Omega}=\lambda\bigg{\|}\sum_{k=0}^{n}\chi_{k}\bigg{\|}_{t\Omega}\leq\lambda\left(\left\lVert
f\right\rVert_{\Omega}+\epsilon\right),$

which holds for all $\epsilon>0$. Consequently, this proves (3.8). Hence for
every $r\in\mathbb{R}^{n}_{\geq 0}$ satisfying (3.7) and for all $f\in
H_{\infty}(\Omega,X)$ and $0<t<1$, we have shown that

$t\sum_{i=1}^{n}r_{i}\leq A_{p,q}(H_{\infty}(\Omega,X),\lambda),$

which proves our claim. This completes the proof. ∎

Proof of Theorem 2.6. First we show the left-hand inequality
$(R_{p_{1},q_{1}}(\mathbb{D},X,\lambda))/n\leq
A_{p_{1},q_{1}}(B_{\ell^{n}_{q}},X,\lambda).$ Assume
$r=R_{p_{1},q_{1}}(\mathbb{D},X,\lambda)$ and $f\in
H_{\infty}(B_{\ell^{n}_{q}},X)$. We define $g(z)=f(ze_{1})=f(z,0,.\,.\,.,0)$
for $z\in\mathbb{D}.$ Then $g\in H_{\infty}(\mathbb{D},X)$ and $\left\lVert
g\right\rVert_{\mathbb{D}}=\left\lVert f\right\rVert_{B_{\ell^{n}_{q}}}$.
Further,

$\displaystyle\left\lVert
x_{0}(f)\right\rVert^{p_{1}}+\left(\sum_{k=1}^{\infty}\left\lVert
x_{\alpha}(f)\right\rVert(r,0,.\,.\,.,0)^{\alpha}\right)^{q_{1}}$
$\displaystyle=\left\lVert
x_{0}(f)\right\rVert^{p_{1}}+\left(\sum_{k=1}^{\infty}\left\lVert
x_{ke_{1}}(f)\right\rVert r^{k}\right)^{q_{1}}$ $\displaystyle=\left\lVert
x_{0}(g)\right\rVert^{p_{1}}+\left(\sum_{k=1}^{\infty}\left\lVert
x_{k}(g)\right\rVert r^{k}\right)^{q_{1}}$
$\displaystyle\leq\lambda\left\lVert
g\right\rVert_{\mathbb{D}}=\lambda\left\lVert
f\right\rVert_{B_{\ell^{n}_{q}}}.$

Therefore, we have $r/n=A_{p_{1},q_{1}}(B_{\ell^{n}_{q}},X,\lambda)$, which
gives the desired inequality.
On the other hand, we want to prove that
$A_{p_{1},q_{1}}(B_{\ell^{n}_{q}},X,\lambda)\leq\left({R_{p_{1},q_{1}}(\mathbb{D},X,\lambda)}/{n}\right)^{1/q}.$
Let $r\in\mathbb{R}^{n}_{\geq 0}$ be such that

$\left\lVert
x_{0}(u)\right\rVert^{p_{1}}+\left(\sum_{k=1}^{\infty}\sum_{|\alpha|=k}\left\lVert
x_{\alpha}(u)\right\rVert r^{\alpha}\right)^{q_{1}}\leq\lambda\left\lVert
u\right\rVert_{B_{\ell^{n}_{q}}}$

for all $u\in H_{\infty}(B_{\ell^{n}_{q}},X)$. Now our aim is to show that

$\frac{1}{n}\left(\sum_{j=1}^{n}r_{j}\right)\leq\left(\frac{R_{p_{1},q_{1}}(\mathbb{D},X,\lambda)}{n}\right)^{1/q}.$

Fix $f\in H_{\infty}(\mathbb{D},X)$, and define the function

$v(z)=z^{q}_{1}+\cdots+z^{q}_{n},\quad z=(z_{1},.\,.\,.,z_{n})\in
B_{\ell^{n}_{q}}.$

Then for $u=f\circ v$, we have $\left\lVert
u\right\rVert_{B_{\ell^{n}_{q}}}=\left\lVert f\right\rVert_{\mathbb{D}}$. Also
for each $z\in B_{\ell^{n}_{q}}$, we obtain

$u(z)=\sum_{k=0}^{\infty}x_{k}(f)v(z)^{k}=\sum_{k=0}^{\infty}x_{k}(f)\sum_{|\alpha|=k}\frac{k!}{\alpha!}z^{q\alpha}=\sum_{\alpha\in\mathbb{N}^{n}_{0}}x_{\alpha}(u)z^{q\alpha},$

where $x_{\alpha}(u)=x_{k}(f)(k!/\alpha!)$ whenever $|\alpha|=k$. Then for all
$z\in\mathbb{C}^{n}$, we have

$\displaystyle\left\lVert
x_{0}(u)\right\rVert^{p_{1}}+\left(\sum_{k=1}^{\infty}\left\lVert
x_{\alpha}(u)\right\rVert|z|^{q\alpha}\right)^{q_{1}}$
$\displaystyle=\left\lVert
x_{0}(f)\right\rVert^{p_{1}}+\left(\sum_{k=1}^{\infty}\left\lVert
x_{k}(f)\right\rVert\sum_{|\alpha|=k}\frac{k!}{\alpha!}|z|^{q\alpha}\right)^{q_{1}}$
$\displaystyle=\left\lVert
x_{0}(f)\right\rVert^{p_{1}}+\left(\sum_{k=1}^{\infty}\left\lVert
x_{k}(f)\right\rVert\left\lVert z\right\rVert^{qk}_{q}\right)^{q_{1}},$

and hence

$\displaystyle\left\lVert
x_{0}(f)\right\rVert^{p_{1}}+\left(\sum_{k=1}^{\infty}\left\lVert
x_{k}(f)\right\rVert\left\lVert r\right\rVert^{qk}_{q}\right)^{q_{1}}$
$\displaystyle=\left\lVert
x_{0}(u)\right\rVert^{p_{1}}+\left(\sum_{k=1}^{\infty}\left\lVert
x_{\alpha}(u)\right\rVert r^{q\alpha}\right)^{q_{1}}$
$\displaystyle\leq\lambda\left\lVert
u\right\rVert_{B_{\ell^{n}_{q}}}=\lambda\left\lVert
f\right\rVert_{\mathbb{D}}.$

Therefore, it follows that $\left\lVert r\right\rVert^{q}_{q}\leq
R_{p_{1},q_{1}}(\mathbb{D},X,\lambda)$. Since we have $\left\lVert
r\right\rVert^{q}_{1}\leq n^{q-1}\left\lVert r\right\rVert^{q}_{q}$, finally
we obtain $n^{1-q}\left\lVert r\right\rVert^{q}_{1}\leq
R_{p_{1},q_{1}}(\mathbb{D},X,\lambda)$. This completes the proof. ∎

Acknowledgment: The research of the first named author is supported by SERB-
CRG, Govt. of India. The research of the second named author is supported by
Institute Post-Doctoral Fellowship of IIT Bombay, and the research of the
third named author is supported by DST-INSPIRE Fellowship (IF 190721), New
Delhi, India.

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