# A Graph-based Conflict-free Cooperation Method
for Intelligent Electric Vehicles at Unsignalized Intersections

Chaoyi Chen1, Qing Xu1, Mengchi Cai1, Jiawei Wang1, Biao Xu2,
Xiangbin Wu3, Jianqiang Wang1, Keqiang Li*1 and Chunyu Qi4 *This work was
supported by the National Key Research and Development Program of China under
Grant 2019YFB1600804, the National Natural Science Foundation of China under
Grant 52072212, and Intel Collaborative Research Institute Intelligent and
Automated Connected Vehicles.1Chaoyi Chen, Qing Xu, Mengchi Cai, Jianqiang
Wang and Keqiang Li are with School of Vehicle and Mobility, Tsinghua
University, Beijing 100084, P.R.China2Biao Xu is with College of Mechanical
and Vehicle Engineering, Hunan University3Xiangbin Wu is with Intel Lab
China4Chunyu Qi is with Beijing BITNEI Corp., LtdCorresponding author: Keqiang
Li, Email<EMAIL_ADDRESS>
###### Abstract

Electric, intelligent, and network are the most important future development
directions of automobiles. Intelligent electric vehicles have shown great
potentials to improve traffic mobility and reduce emissions, especially at
unsignalized intersections. Previous research has shown that vehicle passing
order is the key factor in traffic mobility improvement. In this paper, we
propose a graph-based cooperation method to formalize the conflict-free
scheduling problem at unsignalized intersections. Firstly, conflict directed
graphs and coexisting undirected graphs are built to describe the conflict
relationship of the vehicles. Then, two graph-based methods are introduced to
solve the vehicle passing order. One method is an optimized depth-first
spanning tree method which aims to find the local optimal passing order for
each vehicle. The other method is a maximum matching algorithm that solves the
global optimal problem. The computational complexity of both methods is also
derived. Numerical simulation results demonstrate the effectiveness of the
proposed algorithms.

##  I INTRODUCTION

Intersections are the most common merging points in urban traffic scenarios
[1]. According to Federal Highway Administration, each year more than 2.8
million intersection-related crashes occur in the U.S., which hold 44 percent
of all the crashes [2]. Through V2X technology, the centralized controller at
the intersection coordinates the intelligent electric vehicles to pass the
intersection, which guarantees conflict-free vehicle cooperation. On the other
hand, electric vehicles have a natural sort of advantage in vehicles’
automated control since their power-train is highly electrified [3]. Thus,
electric vehicles have great potentials in improving traffic safety and
mobility at intersections [4, 5].

Considering the fully-autonomous scenario, the traffic light becomes
unnecessary when all the vehicles share their driving information in real
time. In perspective of single vehicle control, multiple methods have been
proposed to optimize the vehicle speed trajectory, e.g., fuzzy logic [6],
Model Predictive Control (MPC) [7] or optimal control [8].

In terms of a large scale of vehicles, researchers have pointed out that the
vehicles’ passing order is the key factor that influences the traffic mobility
[9]. As vehicles approaching the intersection, the optimal passing order
changes dynamically and constantly. One straightforward solution is First-in-
First-out (FIFO) strategy [8], which means firstly-entered vehicles are
scheduled to leave the intersection first. However, the FIFO passing order is
not likely to be the optimal solution in most cases. To tackle this problem,
various methods have been proposed to find the optimal passing order, e.g.,
spanning tree [9], Monte Carlo tree search [10] and dynamic programming [11].
Since optimal passing order is a discrete rather than a continuous problem,
geometry topology is also introduced in system modeling [12]. However, the
above-mentioned research mainly focus on specific methods to obtain the
passing order and only a few of them analyze the optimality and computational
complexity. Some research have focus on the reduction of the optimal passing
order problem [13, 14], but they mainly focus on the feasibility of a
conflict-free solution. In practice, however, the traffic efficiency is
another critical topic that has not been fully considered in intersection
coordination.

Generally speaking, the majority of existing works focus on obtaining the
passing order through a specific method without considering the optimality and
computational complexity. Some research focuses on the feasibility of a
conflict-free solution, but traffic efficiency is neglected. Therefore, a
concise and rigid solution of optimal passing order which considers both
traffic safety and mobility has yet not been developed. In this paper, we
propose a graph-based conflict-free cooperation method. Based on that, two
novel methods are introduced to obtain the local or global optimal vehicle
passing order.

Specifically, our contributions are:

  1. 1.

An optimized depth-first spanning tree method is proposed based on [15]. After
separating different conflict types by Conflict Directed Graph (CDG), the
overall depth of the spanning tree, i.e., evacuation time of the vehicles, is
further reduced. The method has low computational complexity and the passing
order solution it generates is proved to be local optimal.

  2. 2.

From graphic analysis, Coexisting Undirected Graph (CUG) is introduced to
describe vehicles which could pass the intersection simultaneously. A maximum
matching method is further applied to obtain the spanning tree, which is
proved to be the global optimal passing order solution.

The rest of this paper is organized as follows. Section II introduces the
scenario setup and conflict analysis. Section III presents the optimized
depth-first spanning tree method and the maximum matching method. Simulation
results are in Section IV and Section V concludes the paper.

##  II SCENARIO SETUP

In this paper, we consider a three-lane intersection as shown in Fig. 1, where
the incoming vehicles are separated into different lanes referring to their
destination. Similar to existing research[10, 8, 15], lane change behavior is
not permitted to guarantee vehicle safety and improve traffic efficiency
inside the control zone $R_{c}$, as shown as the blue dashed line in Fig. 1.

Figure 1: Illustration of the traffic scenario.

A typical intersection has four conflict modes, including crossing,
converging, diverging, and no conflict [15]. 16 red circles in Fig. 1
represent crossing conflict points, where the vehicles have potentials to
collide as shown in Fig. 2(a). Then, 12 orange squares represent diverging
conflict points, where two vehicles come from the same lane as shown in Fig.
2(b). Converging conflict means the vehicles are going into the same lane,
which doesn’t exist in our scenario.

(a) Crossing Conflict (b) Diverging Conflict

Figure 2: Different conflict modes.

The incoming vehicles are indexed from $1$ to $N$ according to their arriving
sequence at the control zone $R_{c}$. We define different conflict sets to
describe the conflict relationship of the vehicles. For each vehicle
$i\left(i\leq N,i\in\mathbb{N}^{+}\right)$, its crossing set is defined as
$\mathcal{C}_{i}$, and the diverging set as $\mathcal{D}_{i}$. Note that
because the conflict sets are determined when the vehicle reaches the control
zone, vehicle indexes in the conflict sets are smaller than that of the ones
on the control zone border.

###### Example 1

Considering an example case as shown in Fig. 3(a). Vehicle $1$ and $2$ are
coming from the east direction on the middle lane and left lane respectively.
Vehicle $3$ and $4$ are approaching on the middle lane from south and west.
Vehicle $5$ and $6$ are both on the middle lane from the north.

In this example, $\mathcal{C}_{1}=\varnothing$, $\mathcal{D}_{1}=\varnothing$,
$\mathcal{C}_{2}=\varnothing$, $\mathcal{D}_{2}=\varnothing$,
$\mathcal{C}_{3}=\\{1,2\\}$, $\mathcal{D}_{3}=\varnothing$,
$\mathcal{C}_{4}=\\{2,3\\}$, $\mathcal{D}_{4}=\varnothing$,
$\mathcal{C}_{5}=\\{1,4\\}$, $\mathcal{D}_{5}=\varnothing$,
$\mathcal{C}_{6}=\\{1,4\\}$, $\mathcal{D}_{6}=\\{5\\}$.

(a) Scenario Case (b) Conflict Directed Graph (c) Coexisting Undirected Graph

Figure 3: A typical intersection scenario and conflict analysis.

##  III METHODOLOGY

In [15], Xu et al. propose a method of projecting vehicles from different
lanes into a virtual lane, which forms a virtual platoon. In this way, the
vehicles from different lanes can drive through the intersection as if they
were in the same lane to improve driving smoothness. A Depth-first Spanning
Tree (DFST) method is then introduced to find the vehicle passing order. In
Section III-A, we optimize the DFST method to obtain a local optimal passing
order solution, which we name as Optimized Depth-first Spanning Tree (OPT-
DFST) method. Then in Section III-B, we introduce the maximum matching (MM)
method to find the global optimal passing order solution.

###  III-A Optimized Depth-first Spanning Tree Method

In the virtual platoon, there exists a virtual leading vehicle $0$ with the
constant moving velocity in front of vehicles that are closest to the
intersection. In this way, the leading vehicles in the real world maintain a
typical platooning behavior. Accordingly, if the vehicle is the closest to the
intersection on each lane, vehicle $0$ is added into its diverging set, e.g.,
in Example 1, $\mathcal{D}_{1}=\\{0\\}$, $\mathcal{D}_{2}=\\{0\\}$,
$\mathcal{D}_{3}=\\{0\\}$, $\mathcal{D}_{4}=\\{0\\}$,
$\mathcal{D}_{5}=\\{0\\}$.

Based on the conflict sets analysis in Section II, we further define a
Conflict Directed Graph (CDG) $\mathcal{G}_{N+1}$ to represent the conflict
relationship.

######  Definition 1 (Conflict Directed Graph)

The conflict directed graph is denoted as
$\mathcal{G}_{N+1}=\left\\{\mathcal{V}_{N+1},\mathcal{E}_{N+1}\right\\}$. If
there are $N$ vehicles in the control zone, we have vertex set
$\mathcal{V}_{N+1}=\\{0,1,2,\dots,N\\}$ (vertex $0$ stands for the virtual
leading vehicle). Unidirectional edge set is defined as
$\mathcal{E}_{N+1}^{u}=\\{(i,j)|i<j,i\in\mathcal{D}_{j}\\}$, and bidirectional
edge set is defined as
$\mathcal{E}_{N+1}^{b}=\\{(i,j)|i<j,i\in\mathcal{C}_{j}\\}$. Edge set is the
union of these two sets as
$\mathcal{E}_{N+1}=\mathcal{E}_{N+1}^{u}\cup\mathcal{E}_{N+1}^{b}$.

The CDG of the case scenario is drawn in Fig. 3(b). The vertices in CDG
represent the vehicles in the control zone. The black unidirectional edges
represent the diverging conflicts. If there exist a unidirectional edge
$(i,j)$, it means that vehicle $j$ cannot reach the intersection earlier than
vehicle $i$. The red bidirectional edges stand for the crossing conflict,
which means these vehicles’ arriving sequences can be exchanged. Note that in
DFST method, the crossing conflict $\mathcal{C}_{i}$ and diverging conflict
$\mathcal{D}_{i}$ are treated as one union conflict type. In the following
paragraph, we will interpret the separation of these different conflict sets
helps to improve the original DFST algorithm performance.

It is straightforward that CDG describes all the conflict relationships of
vehicles. Previous research has proved that CDG in DFST method has a depth-
first spanning tree, as shown in Lemma 1. Since this conclusion is drawn with
only unidirectional edges in CDG, it still stands for our OPT-DFST method.

######  Lemma 1 ([15])

A conflict directed graph has a depth-first spanning tree with the root node
vertex 0.

It is evident that for a group of vehicles in the control zone, a passing
order solution is related to the depth-first spanning tree addressed in Lemma
1. The CDG describes all the conflict relationships of the vehicles and the
edges specifically describe the conflict type. If the depth-first spanning
tree is built according to CDG, a feasible passing order solution can be
built. We propose the OPT-DFST method in Algorithm 1 and 2.

Algorithm 1 Optimized Depth-first Spanning Tree Method

1:Conflict Directed Graph
$\mathcal{G}_{N+1}=\left\\{\mathcal{V}_{N+1},\mathcal{E}_{N+1}\right\\}$

2:Optimized Depth-first Spanning Tree
$\mathcal{G}_{N+1}^{\prime}=\left\\{\mathcal{V}_{N+1},\mathcal{E}_{N+1}^{\prime}\right\\}$

3:initialize: Set the depth of vertex $0$’s layer $d_{0}=0$

4:for $i=1,2,...,N$ do

5: $P=\\{m\in\mathcal{V}_{N+1}|(m,i)\in\mathcal{E}_{N+1}\text{ and }m<i\\}$

6: $k$ = FIND-OPT-PARENT($\mathcal{G}_{N+1}^{\prime}$,$P$)

7: Set vertex $k$ as the parent vertex of $i$ in the graph
$\mathcal{G}_{N+1}^{\prime}$, add a vertex $i$ and an edge $(k,i)$ to the
graph $\mathcal{G}_{N+1}^{\prime}$, and set the depth of the vertex $i$ to
$d_{k}+1$

8:end for

Algorithm 2 FIND-OPT-PARENT

1:$\mathcal{G}_{N+1}^{\prime}$,$P$

2:Optimal Parent Vertex $k$

3:for $j=1,2,...,|P|$ do

4: Find the largest depth $d_{\text{div}}$ of the diverging conflict parents
in $\mathcal{G}_{N+1}^{\prime}$

5: Find the union depths set of crossing conflict parents $D_{\text{swap}}$ in
$\mathcal{G}_{N+1}^{\prime}$

6:end for

7:Find $\min{d_{k}}$, s.t.$\\{k\in P|d_{k}+1>d_{\text{div}},\text{ and
}(d_{k}+1)\cap D_{\text{swap}}=\varnothing\\}$

8:return $k$

We design an iterative algorithm to generate a depth-first spanning tree.
Similar to the general graph theory, the depth of each node in the spanning
tree is calculated by its distance to the root node $0$. The vehicle’s
conflict-free attribute of the same depth has been proofed in Lemma 2.
Specifically, the depth of the nodes represent the passing order of the
vehicles, i.e., the vehicles in the same depth shall pass the intersection
simultaneously. It can be inferred that the depth of the spanning tree is
related to the traffic mobility at the intersection. _Evacuation time_ means
the overall time cost for all the vehicles to pass the intersection, i.e., the
leaving time of the last vehicle. It also means the largest depth of the
spanning tree nodes, denoted as $d_{\text{all}}$ in rest of the paper.

######  Lemma 2 ([15])

Consider a virtual platoon characterized by conflict directed graph
$\mathcal{G}_{N+1}$ with the spanning tree $\mathcal{G}_{N+1}^{\prime}$. The
trajectories of two vehicles with the same depth in
$\mathcal{G}_{N+1}^{\prime}$ have no conflict relationship with each other.

For each node $i$, we firstly find all the parent vertices $P$ of it in
$\mathcal{G}_{N+1}$. Note that both original DFST method and OPT-DFST method
are based on FIFO principle. Even though there are bidirectional edges in CDG,
the larger-index vehicles should not be considered in smaller-index vehicles’
passing order optimization process. Hence, only the smaller-index vehicles are
selected as the parent node candidates, as shown in Line 5 in Algorithm 1.

Then in Line 6, i.e., Algorithm 2, we treat conflict types differently. If the
parent node $j$ is in vehicle $i$’s diverging conflict set, i.e.,
$j\in\mathcal{D}_{i}$, it means $j$ is the vehicle in front of $j$, which also
means $i$ can not surpass $j$ (Overtaking is not permitted). In this case, we
find all the largest depth $d_{\text{div}}$ of all the parent vehicles,
because vehicle $i$’s target depth should not surpass any of these vehicles.
In other cases, the parent node $j$ is in vehicle $i$’s crossing conflict set
$\mathcal{C}_{i}$. It means vehicle $i$ and vehicle $j$ cannot arrive at the
intersection simultaneously. Either vehicle $i$ or vehicle $j$ can pass the
intersection first, so we find the union depth set of crossing conflict
parents $D_{\text{swap}}$. To find the optimal parent $k$, we should select
the proper depth of vehicle $i$. Since the depth of parent $k$ is $d_{k}$, the
depth of vehicle $i$ is $d_{k}+1$. As mentioned before, $d_{k}+1$ should
neither smaller than $d_{\text{div}}$, nor have intersection with set
$D_{\text{swap}}$. For traffic efficiency consideration, the depth should be
as small as possible, as shown in Line 7 in Algorithm 2.

In Example 1, $\mathcal{C}_{1}=\varnothing,\mathcal{D}_{1}=\\{0\\}$,
$d_{\text{div}}=0,D_{\text{swap}}=\varnothing$, so the parent node of vehicle
$1$ is selected as node $0$ and $d_{1}=1$. Node $2$ is arranged the same as
node $1$. $\mathcal{C}_{3}=\\{1,2\\},\mathcal{D}_{3}=\\{0\\}$,
$d_{\text{div}}=0,D_{\text{swap}}=\\{1\\}$, so the parent node of vehicle $3$
is selected as node $1$ and $d_{3}=2$.
$\mathcal{C}_{4}=\\{2,3\\},\mathcal{D}_{4}=\\{0\\}$,
$d_{\text{div}}=0,D_{\text{swap}}=\\{1,2\\}$, so the parent node of vehicle
$4$ is selected as node $3$ and $d_{4}=3$.
$\mathcal{C}_{5}=\\{1,4\\},\mathcal{D}_{5}=\\{0\\}$,
$d_{\text{div}}=0,D_{\text{swap}}=\\{1,3\\}$, so the parent node of vehicle
$5$ is selected as node $2$ and $d_{5}=2$.
$\mathcal{C}_{6}=\\{1,4\\},\mathcal{D}_{5}=\\{0,5\\}$,
$d_{\text{div}}=2,D_{\text{swap}}=\\{1,3\\}$, so the parent node of vehicle
$5$ is selected as node $0$ and $d_{5}=4$. Based on this method we can obtain
the optimized depth-first spanning tree $\mathcal{G}_{N+1}^{\prime}$ as shown
in Fig. 4(b).

(a) DFST Spanning Tree with $d_{\text{all}}=5$ (b) OPT-DFST Spanning Tree with
$d_{\text{all}}=4$ (c) MM Spanning Tree with $d_{\text{all}}=3$

Figure 4: Different spanning tree results comparison of three different
method.

The main improvement of OPT-DFST method is on Line 6, i.e., Algorithm 2. In
[15], the diverging conflict and crossing conflict are treated as one union
conflict set. In this way, the largest depth $d_{k}$ of the parent vertices
are found in Line 6. For instance, in our scenario case in Fig. 4(a), vehicle
$5$ is ranked $d_{5}=2$ in OPT-DFST method. But in DFST method, the parent
nodes of vehicle $5$ are nodes $\\{0,1,4\\}$, which cause the largest depth
$d_{k}$ of the parent vertices is $d_{k}=3$, so DFST method gets $d_{5}=4$ as
shown in Fig. 4(a). As a result, the total depth of $6$ vehicles to
$d_{\text{all}}=5$ while the OPT-DFST method optimized it into
$d_{\text{all}}=4$.

###### Remark 1

We come to a conclusion that the DFST method basically focus on the
feasibility of the passing order problem. The proposed OPT-DFST method further
considers the optimality, i.e., tries to find the lowest depth for each
vehicle, but still, the solution is found vehicle by vehicle, i.e., based on
FIFO principle. In other words, the solution is a local optimal solution in
arranging each vehicle. Since each vehicle need to check all its conflict
relationship of the parent nodes, the computational complexity is $O(N^{2})$.

###  III-B Maximum Matching Method

As mentioned before, we aim to minimize the overall depth $d_{\text{all}}$ of
the spanning tree, which corresponds to the evacuation time of all the
vehicles. If we re-consider the case scenario in Fig. 3(a), the right-turn
traffic flows have no conflict with any other traffic flows. Considering the
rest eight kinds of left-turn or straight traffic flows, a maximum number of
two vehicles can pass the intersection simultaneously. From this point of
view, another method to describe the conflict relationships of the vehicle is
to describe coexisting relationships of them.

######  Definition 2 (Coexisting Undirected Graph)

The coexisting undirected graph is defined as the complement graph of the
conflict directed graph $\mathcal{G}_{N+1}$ excluding node $0$. It denotes
$\overline{\mathcal{G}_{N}}=\\{\overline{\mathcal{V}_{N}},\overline{\mathcal{E}_{N}}\\}$,
where $\overline{\mathcal{V}_{N}}=\mathcal{V}_{N+1}-\\{0\\}$,
$\overline{\mathcal{E}_{N}}=\\{(i,j)|i,j\in\overline{\mathcal{V}_{N}}\text{
and }(i,j)\notin\mathcal{E}_{N+1}\\}$.

In the case scenario, the conflict directed graph (CDG) is drawn in Fig. 3(b)
and coexisting undirected graph (CUG) in Fig. 3(c). Since CDG edges
$\mathcal{E}_{N+1}$ means two vehicles have conflicts and CUG is the
complement graph of CDG, CUG edges $\overline{\mathcal{E}_{N}}$ means two
vehicles are conflict-free, i.e., can pass the intersection simultaneously.

Recall that we expect to minimize the overall depth $d_{\text{all}}$ of the
spanning tree. Since the total vehicle number, i.e., the node number
$\left|\overline{\mathcal{V}_{N}}\right|$ is a constant value, minimizing the
overall depth of the spanning tree equals to widening the average width of the
spanning tree. Thus, solving the optimal passing solution is equivalent to
find the maximum pairings of coexisting vehicles combination in CUG. Note that
this conclusion corresponds to common sense. Maximum pairings of vehicles mean
maximizing the usage of the intersection, which lowers the overall evacuation
time.

In graph theory, maximum matching (MM) is a well-developed theory that is
capable of solving this problem [16]. Firstly, we define matching in our case
formally.

######  Definition 3 (Matching)

A matching $\mathcal{M}$ in $\overline{\mathcal{G}_{N}}$ is a subset of its
edge $\overline{\mathcal{E}_{N}}$, such that no vertex in
$\overline{\mathcal{V}_{N}}$ is incident to more that one edge in
$\mathcal{M}$.

The matching $\mathcal{M}$ is the aforementioned pairing set. Since the edges
are selected from CUG $\overline{\mathcal{G}_{N}}$, the two vehicles are
conflict-free in passing the intersection simultaneously. Because no two edges
in $\mathcal{M}$ have the same vertex, each vehicle only appears once in the
matching. In Example 1, $\mathcal{M}=\\{(2,6),(3,5)\\}$ is a matching, but it
is still not a maximal matching because $(1,4)$ can be add into the matching.
$\mathcal{M}=\\{(1,4),(2,6),(3,5)\\}$ is a maximal matching, but still we
cannot claim that it is the maximum number of pairing in CUG. Our target is to
find the maximum number of pairing in CUG, i.e., the maximum number of
pairing, so we further introduce maximum matching.

######  Definition 4 (Maximum Matching)

A matching $\mathcal{M}$ is said to be maximum if for any other matching
$\mathcal{M}^{\prime}$,
$\left|\mathcal{M}\right|>\left|\mathcal{M}^{\prime}\right|$. The maximum
matching is denoted as $\mathcal{M}_{\max}$ in rest of the paper.

Maximum matching $\mathcal{M}_{\max}$ is the maximum sized matching in CUG
$\overline{\mathcal{G}_{N}}$, which corresponds to find the maximum number of
vehicle pairing. In the case scenario, the maximum matching
$\mathcal{M}_{\max}=\\{(1,4),(2,6),(3,5)\\}$ with size of $3$. The maximum
matching have been solved in time
$O(\left|\overline{\mathcal{E}_{N}}\right|\left|\overline{\mathcal{V}_{N}}\right|^{2})$
using Edmonds’ blossom algorithm [16].

After maximum matching, we can add the rest single vehicles in the end,
forming the optimized feasible spanning tree which is the same as shown in
Fig. 4(b). The process which completes the spanning tree is introduced as
follows, denoting as SPANNING process in Line 4. CUG
$\overline{\mathcal{G}_{N}}$ only contains the coexisting information of the
vehicles, while the vehicle’s passing order from the same lane should be
strictly according to their relative position. The MM method could generate
solutions that cannot be directly executed, e.g.,
$\mathcal{M}_{\max}=\\{(1,4),(3,6),(2,5)\\}$ with size of $3$. The spanning
tree forms as $(1,4)\rightarrow(3,6)\rightarrow(2,5)$. However, this solution
is not feasible since vehicle $5$ is in front of vehicle $6$. We proof that
this can be solved by exchange the unfeasible sequence as shown in Theorem 1.

###### Theorem 1

If the maximum matching $\mathcal{M}_{\max}$ contains $\\{(i,j),(n,m)\\}$ that
is not feasible, $\\{(i,m),(j,n)\\}$ must be a feasible solution.

###### Proof:

Without loss of generality, we assume in the maximum matching solution, $m$
and $j$ are in the same lane with $m$ in front of $j$, causing the
unfeasibility. According to the definition of CDG, there exist a
unidirectional edge $(m,j)$ in CDG. If $(i,m),(j,n)$ is still not a feasible
solution, the unfeasibility can only be caused by two possibilities. The first
case is that $(i,j)$ or $(n,m)$ is in the same lane which is contradicted to
the maximum matching solution. Second case is that $(m,j)$ are in the same
lane with $j$ in front of $m$, which means there exist a unidirectional edge
$(j,m)$ in CDG and it is not possible. As a result, $(i,m),(j,n)$ must be a
feasible solution. ∎

If the original MM solution is $\mathcal{M}_{\max}=\\{(1,4),(3,6),(2,5)\\}$,
we can exchange vehicle $5$ and $6$, obtaining the feasible solution
$\mathcal{M}_{\max}=\\{(1,4),(3,5),(2,6)\\}$.

###### Remark 2

We sum up the MM method in Algorithm 3. Recall that in the process of solving
the passing order, the vehicles’ arrival sequence is neglected on purpose.
Therefore, the FIFO principle is abandoned and the fundamental vehicle
conflict relationship is highlighted, which refers that MM method obtains the
global optimal solution as indicating by Definition 4. Since the maximum
matching in Line 3 takes
$O(\left|\overline{\mathcal{E}_{N}}\right|\left|\overline{\mathcal{V}_{N}}\right|^{2})$
time and the adjusting and spanning in Line 4 takes
$O(\overline{\mathcal{V}_{N}})$, the overall MM method takes
$O(\left|\overline{\mathcal{E}_{N}}\right|\left|\overline{\mathcal{V}_{N}}\right|^{2}+\overline{\mathcal{V}_{N}})=O(N^{4})$
time.

Algorithm 3 Maximum Matching Method

1:Coexisting Undirected Graph $\overline{\mathcal{G}_{N}}$

2:Spanning Tree $\mathcal{G}_{N+1}^{\prime}$

3:$\mathcal{M}_{\max}$ = MAXIMUM-MATCHING($\mathcal{G}_{N+1}^{\prime}$)

4:$\mathcal{G}_{N+1}^{\prime}$ = SPANNING($\mathcal{M}$)

5:return $\mathcal{G}_{N+1}^{\prime}$

###  III-C Virtual Platooning Topology

As mentioned before, all the three method, i.e., DFST, OPT-DFST and MM, lead
to a feasible spanning tree
$\mathcal{G}_{N+1}^{\prime}=\left\\{\mathcal{V}_{N+1},\mathcal{E}_{N+1}^{\prime}\right\\}$
as shown in Fig. 4(b), which forms the geometry topology of the virtual
platoon. Since each node $\mathcal{V}_{i}$ in $\mathcal{G}_{N+1}^{\prime}$
represents a vehicle $i$ and the edge $\mathcal{E}_{i}^{\prime}=(j,i)$ stands
for the virtual preceding vehicle $j$, the vehicle can always follow its
parent node vehicle $\mathcal{V}_{j}$. In other words, vehicle $i$ and its
parent vehicle $j$ intend to keep a constant desired car-following distance
$D_{des}$ and the same velocity $v_{des}$.

$\left\\{\begin{array}[]{l}\displaystyle\lim_{t\rightarrow\infty}\left\|v_{i}(t)-v_{j}(t)\right\|=0\\\
\displaystyle\lim_{t\rightarrow\infty}\left(p_{j}(t)-p_{i}(t)-D_{des}\right)=0\end{array},i\in\\{1,2,\cdots,N\\},\right.$
(1)

in which $v_{i}(t)$ and $p_{j}(t)$ denote the velocity and position of vehicle
$i$. Lemma 2 has proven that the vehicles in the same depth have no conflict
relationship. Therefore, if vehicle $i$ and vehicle $j$ are of the same depth,
it stands

$\displaystyle\lim_{t\rightarrow\infty}\left(p_{i}(t)-p_{j}(t)\right)=0.$ (2)

The state of vehicle $i$ is defined as $x_{i}=\left[p_{i};v_{i}\right]$ and
second-order model is employed to describe the vehicle dynamics. As for the
low-level vehicle control, a linear feedback PID controller is designed to
execute the virtual platooning [17].

##  IV SIMULATIONS

###  IV-A Simulation Environment

The traffic simulation is conducted in SUMO, where we use the default electric
vehicle parameters. The intersection scenario and the lane direction settings
are the same as shown in Fig. 1, and the control zone length $R_{c}=1000m$.
The vehicle arrival is assumed to be binomially distributed flow with
probability $p$.

###  IV-B Simulation under Different Vehicle Number

(a) Maximum depth of the spanning tree of three methods. (b) Evacuation time
comparison.

Figure 5: Simulation results of different vehicle input number.

Since obtaining the optimal passing order solution is the critical point of
this research, the first simulation is conducted with different vehicle
numbers $N$. The vehicle arrival probability is $p=0.3$, and the maximum
vehicles input is $N=84$. Generally speaking, there are maximum $12^{72}\cdot
12!$ possible passing order solutions in $12$ lanes scenario.

Fig. 5 demonstrates the performance of different algorithms. Each vehicle
input consists of five-time simulations, where input flow is randomly
distributed according to $p$. Standard deviations of the result are also
presented as error bar in Fig. 5. Firstly, the OPT-DFST method outperforms the
original DFST method. By separating the conflict types as explained in Section
III-A, OPT-DFST is capable of finding the local optimal passing order
solution, i.e., smaller depth of the spanning tree for each vehicle. Although
both of these two methods are based on the FIFO principle, the OPT-DFST method
obtains a better passing order solution considering the total evacuation time.
However, because of the FIFO principle, OPT-DFST only searches for the local
optimal solution as each new vehicle comes. In contrast, the MM method is
capable of searching the global optimal solution of all passing order
possibilities by discarding the arrival sequence of the existing vehicles.
Besides, the error bars show that the MM method has the best performance in
standard deviations, i.e., finding the global optimal solution especially when
the vehicle input number is rather large. As a result, in the situation of
$84$ input vehicles, OPT-DFST saves $10.2\%$ evacuation time than that of
DFST, and MM further saves $2.2\%$ of it.

##  V CONCLUSIONS

This paper presents a graph-based vehicle cooperation method to guarantee
safety and efficiency at unsignalized intersections. Base on graphic analysis,
conflict directed graphs and coexisting directed graphs are drawn to model the
conflict relationship of the vehicles. Based on the existing research, an
optimized depth-first spanning tree method is proposed, which proves to be the
local optimal passing order solution. The maximum matching method is further
developed to find the global optimal solution. The computational complexity of
both methods is provided and traffic simulation verified the effectiveness of
the proposed methods.

## References

  * [1] Q. Xu, M. Cai, K. Li, B. Xu, J. Wang, and X. Wu, “Coordinated formation control for intelligent and connected vehicles in multiple traffic scenarios,” _IET Intelligent Transport Systems_ , vol. 15, no. 1, pp. 159–173, 2021.
  * [2] R. Azimi, G. Bhatia, R. R. Rajkumar, and P. Mudalige, “Stip: Spatio-temporal intersection protocols for autonomous vehicles,” in _2014 ACM/IEEE international conference on cyber-physical systems (ICCPS)_. IEEE, 2014, pp. 1–12.
  * [3] C. C. Chan and K. Chau, “An overview of power electronics in electric vehicles,” _IEEE transactions on Industrial Electronics_ , vol. 44, no. 1, pp. 3–13, 1997.
  * [4] Y. Zheng, J. Wang, and K. Li, “Smoothing traffic flow via control of autonomous vehicles,” _IEEE Internet of Things Journal_ , vol. 7, no. 5, pp. 3882–3896, 2020.
  * [5] C. Chen, J. Wang, Q. Xu, J. Wang, and K. Li, “Mixed platoon control of automated and human-driven vehicles at a signalized intersection: dynamical analysis and optimal control,” _arXiv preprint arXiv:2010.16105_ , 2020.
  * [6] V. Milanés, J. Pérez, E. Onieva, and C. González, “Controller for urban intersections based on wireless communications and fuzzy logic,” _IEEE Transactions on Intelligent Transportation Systems_ , vol. 11, no. 1, pp. 243–248, 2010.
  * [7] Y. Zheng, L. Jin, Y. Jiang, F. Wang, X. Guan, S. Ji, and J. Xu, “Research on cooperative vehicle intersection control scheme without using traffic lights under the connected vehicles environment,” _Advances in Mechanical Engineering_ , vol. 9, no. 8, p. 1687814017719219, 2017.
  * [8] A. A. Malikopoulos, C. G. Cassandras, and Y. J. Zhang, “A decentralized energy-optimal control framework for connected automated vehicles at signal-free intersections,” _Automatica_ , vol. 93, pp. 244–256, 2018.
  * [9] L. Li and F.-Y. Wang, “Cooperative driving at blind crossings using intervehicle communication,” _IEEE Transactions on Vehicular technology_ , vol. 55, no. 6, pp. 1712–1724, 2006.
  * [10] H. Xu, Y. Zhang, L. Li, and W. Li, “Cooperative driving at unsignalized intersections using tree search,” _IEEE Transactions on Intelligent Transportation Systems_ , vol. 21, no. 11, pp. 4563–4571, 2019.
  * [11] F. Yan, M. Dridi, and A. El Moudni, “Autonomous vehicle sequencing algorithm at isolated intersections,” in _2009 12th International IEEE conference on intelligent transportation systems_. IEEE, 2009, pp. 1–6.
  * [12] Y.-T. Lin, H. Hsu, S.-C. Lin, C.-W. Lin, I. H.-R. Jiang, and C. Liu, “Graph-based modeling, scheduling, and verification for intersection management of intelligent vehicles,” _ACM Transactions on Embedded Computing Systems (TECS)_ , vol. 18, no. 5s, pp. 1–21, 2019.
  * [13] H. Ahn and A. Colombo, “Abstraction-based safety verification and control of cooperative vehicles at road intersections,” _IEEE Transactions on Automatic Control_ , 2019.
  * [14] D. Miculescu and S. Karaman, “Polling-systems-based autonomous vehicle coordination in traffic intersections with no traffic signals,” _IEEE Transactions on Automatic Control_ , vol. 65, no. 2, pp. 680–694, 2019.
  * [15] B. Xu, S. E. Li, Y. Bian, S. Li, X. J. Ban, J. Wang, and K. Li, “Distributed conflict-free cooperation for multiple connected vehicles at unsignalized intersections,” _Transportation Research Part C: Emerging Technologies_ , vol. 93, pp. 322–334, 2018.
  * [16] J. Edmonds, “Maximum matching and a polyhedron with 0, 1-vertices,” _Journal of research of the National Bureau of Standards B_ , vol. 69, no. 125-130, pp. 55–56, 1965.
  * [17] L. Peppard, “String stability of relative-motion pid vehicle control systems,” _IEEE Transactions on Automatic Control_ , vol. 19, no. 5, pp. 579–581, 1974.


††footnotetext:  File: main.tex, printed: 2024-8-28, 4.20

# On multidimensional Bohr radii for Banach spaces

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

In this paper, we study a more general version of multidimensional Bohr radii
for the holomorphic functions defined on unit ball of $\ell^{n}_{q}\,\,(1\leq
q\leq\infty)$ spaces with values in arbitrary complex Banach spaces. More
precisely, we study the multidimensional Bohr radii for bounded linear
operators between complex Banach spaces, primarily motivated by the work of A.
Defant, M. Maestre, and U. Schwarting [Adv. Math. 231 (2012), pp. 2837–2857].
We obtain the exact asymptotic estimates of multidimensional Bohr radius for
both finite and infinite dimensional Banach spaces. As an application, we find
the lower bound of arithmetic Bohr radius.

###### Key words and phrases:

Bohr radius, Power series, Holomorphic functions, Homogeneous polynomials,
Banach spaces

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

Primary 32A05, 32A10, 46B07; Secondary 46B09, 46G20

##  1\. Introduction

In the beginning of the 20th century while studying the well-known Riemann
$\zeta$-function, a Danish mathematician Harald Bohr made significant
developments to the general theory of Dirichlet series. A Dirichlet series is
an summation expression of the form

(1.1) $D(s):=\sum_{n\geq 1}\frac{a_{n}}{n^{s}},$

where $a_{n}\in\mathbb{C}$ and $s=\sigma+it$ is a complex variable. It is
well-known that the region of convergence, absolutely convergence, and uniform
convergence of a Dirichlet series define to be the half planes of the form
$\\{s\in\mathbb{C}:\mbox{Re}(s)>\sigma_{0}\\}$ in the complex plane
$\mathbb{C}$. Bohr was mainly interested in tackling the region of convergence
of this Dirichlet series. More precisely, Bohr was investigating the width of
the greatest strip for which a Dirichlet series converges uniformly but not
absolutely. This problem is known as absolute convergence problem of Bohr in
the theory of Dirichlet series. In order to tackle this problem, Bohr has
obtained a nice connection between a Dirichlet series and a power series of
infinite variables. Indeed, given a Dirichlet series of the form (1.1), Bohr
considered for each $n\in\mathbb{N}$ the prime decomposition of
$n=p^{\alpha_{1}}_{1}\cdots p^{\alpha_{r}}_{r}$ (where
$(p_{m})_{m\in\mathbb{N}}$ denotes the sequence of order prime numbers) and
defined $z=(p^{-s}_{1},\ldots,p^{-s}_{r})$. Therefore, Bohr has established
the following identity:

(1.2) $D(s)=\sum_{n\geq
1}a_{n}(p^{-s}_{1})^{\alpha_{1}}\cdots(p^{-s}_{r})^{\alpha_{r}}=\sum
a_{n}z^{\alpha_{1}}_{1}\cdots z^{\alpha_{r}}_{r}.$

The above correspondence (1.2) is known as the Bohr transform. It helps us to
translate the problem on Dirichlet series in terms of power series and thus,
one can apply complex analysis techniques to solve this problem. As a part of
these studies, Bohr once asked whether is it possible to compare the absolute
value of a power series in one complex variable with the sum of the absolute
values of its coefficients. In 1914, Bohr [12] proved the following remarkable
result which nowadays known as Bohr’s inequality:

######  Theorem A.

[12] If $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ is a holomorphic function in unit
disk such that $\sup_{z\in\mathbb{D}}|f(z)|<\infty$, then

(1.3)
$\displaystyle\sum_{n=0}^{\infty}|a_{n}|r^{n}\leq\sup_{z\in\mathbb{D}}|f(z)|$

for $|z|=r\leq 1/3$, and the constant $1/3$, referred to as the Bohr radius,
which is sharp.

In fact, Bohr himself obtained the radius as $1/6$. Later, this radius was
further improved to $1/3$ by M. Riesz, I. Schur, and N. Wiener [37, 38],
independently. This significant inequality was forgotten for long eighty years
until Dixon [27] used the inequality in his paper to disprove a long-standing
conjecture in Operator Algebras: If a Banach algebra satisfies the von Neumann
inequality then it is necessarily an operator algebra, i.e., which is
isometrically isomorphic to a closed subalgebra of $\mathcal{L}(H)$, the
algebra of all bounded operators for some Hilbert space $H$. After that,
several researchers like Aizenberg [1, 2, 3], Boas and Khavinson [10], Boas
[11], Bombieri and Bourgain [15], Galicer, Mansilla and Muro [30], Hamada,
Honda and Kohr [31], Paulsen and Singh [35], Popescu [36] have paid much
attention to this result to use it in different contexts of mathematics and
generalized it. In this paper, we mostly focus on the unit ball of Minkowski
space $\ell^{n}_{q}$, where $q\in[1,\infty]$. For $q\in[1,\infty)$, we denote

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

and

$B_{\ell^{n}_{\infty}}=\left\\{z\in\mathbb{C}^{n}:\left\lVert
z\right\rVert_{\infty}=\sup_{1\leq i\leq n}|z_{i}|<1\right\\}.$

Before proceeding for further discussions, let us first introduce the
following notion of multidimensional Bohr radius for holomorphic functions
defined on $B_{\ell^{n}_{q}},\,1\leq q\leq\infty$ with values in a complex
Banach space $X$, primarily motivated by the work of Defant et. al. [25].

######  Definition 1.1.

Let $1\leq q\leq\infty$, $\lambda\geq 1$, $n\in\mathbb{N}$, and let
$T:X\rightarrow Y$ be a bounded linear operator between complex Banach spaces
$X$ and $Y$ such that $\left\lVert T\right\rVert\leq\lambda^{1/p}$. For $1\leq
p<\infty$, the $\lambda_{p}$-Bohr radius of $T$, denoted by
$K^{p}(B_{\ell^{n}_{q}},T,\lambda)$, is the supremum of all $r\geq 0$ such
that for all holomorphic functions
$f(z)=\sum_{\alpha\in\mathbb{N}^{n}_{0}}x_{\alpha}z^{\alpha}$ on
$B_{\ell^{n}_{q}}$ we have

(1.4) $\sup_{z\in
rB_{\ell^{n}_{q}}}\sum_{\alpha\in\mathbb{N}^{n}_{0}}\left\lVert
T(x_{\alpha})z^{\alpha}\right\rVert^{p}_{Y}\leq\lambda\left\lVert\sum_{\alpha\in\mathbb{N}^{n}_{0}}x_{\alpha}z^{\alpha}\right\rVert^{p}_{X}.$

We write $K^{p}(B_{\ell^{n}_{q}},T)$ for $\lambda=1$. If $T$ is the identity
map on $X$ then we use the notation $K^{p}(B_{\ell^{n}_{q}},X,\lambda)$ and
$K^{p}(B_{\ell^{n}_{q}},X)$. For $X=\mathbb{C}$, we write
$K^{p}(B_{\ell^{n}_{q}},\lambda)$ for
$K^{p}(B_{\ell^{n}_{q}},\mathbb{C},\lambda)$ and $K(B_{\ell^{n}_{q}})$
whenever $p=1$ and $\lambda=1$.

We observe that if $f$ is an unbounded holomorphic function on
$B_{\ell^{n}_{q}}$ then the inequality (1.4) holds trivially. Therefore,
throughout our discussion, we assume the holomorphic functions to be bounded
on $B_{\ell^{n}_{q}}$. Clearly, with the above notation, Theorem A reduces to
$K(\mathbb{D})=1/3$.

In the classical sense, Bombieri and Bourgain [14, 15] have studied the
constant $K(\mathbb{D},\lambda)$ whenever $\lambda\geq 1$. In fact, Bombieri
[14] has obtained the exact value

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

where $\lambda\in[1,\sqrt{2}]$, and Bombieri and Bourgain [15] have determined
the exact asymptotic behavior of $K(\mathbb{D},\lambda)$ as
$\sqrt{\lambda^{2}-1}/\lambda$ whenever $\lambda\rightarrow\infty$. In 2010,
Blasco [8] established that $K(\ell^{2}_{q})=0$ for $1\leq q\leq\infty$. Thus,
the study of Bohr radius problem becomes irrelevant for $\text{dim}(X)>1$.
Hence, it seems reasonable to modify the Bohr inequality to obtain more
interesting result. That is why the authors [25] have introduced $\lambda$ in
the definition of Bohr radius $K(B_{\ell^{n}_{\infty}},T,\lambda)$ of bounded
linear operator $T$. In [25], the authors have established that
$K(B_{\ell^{n}_{\infty}},T,\lambda)>0$ for every bounded linear operator
$T:X\rightarrow Y$ between complex Banach spaces $X$ and $Y$. Later, in this
direction, the work was extended by Kumar et. al. [32] and studied the Bohr
radius $K(B_{\ell^{n}_{q}},T,\lambda)$ for any bounded operator $T$,
$\lambda\geq 1$, and $1\leq q\leq\infty$.

In $1989$, Dineen and Timoney [29] extensively studied the constant
$K(B_{\ell^{n}_{\infty}})$ and later, their result has been quickly refined in
[10]. In $1997$, Boas and Khavinson [10] proved the following estimate for
each $n\in\mathbb{N}$ with $n\geq 2$,

$\frac{1}{3\sqrt{n}}\leq K(B_{\ell^{n}_{\infty}})\leq 2\sqrt{\frac{\log
n}{n}}.$

In view of results of Aizenberg, Boas, Dineen, Khavinson and Timoney (see
[1],[10], [11], [29]), estimating the multidimensional Bohr radii, for every
$1\leq q\leq\infty$ and all $n$ we have

$\frac{1}{C}\left(\frac{1}{n}\right)^{1-\frac{1}{\min\\{q,2\\}}}\leq
K(B_{\ell^{n}_{q}})\leq C\left(\frac{\log
n}{n}\right)^{1-\frac{1}{\min\\{q,2\\}}},$

where $C$ is a constant independent of $q$ and $n$. One of the main reasons
behind the occurrence of $\log$-term in the upper estimates of all known non-
trivial results on multidimensional Bohr radii is the probabilistic methods
used by the authors. In [11, p. 326] Boas conjectured that “This logarithmic
factor, an artifact of the proof, presumably should not really be present”. In
2006, Dafant and Frerick [18] disproved the conjecture by obtaining a
logarithmic lower bound which is almost correct asymptotic estimate for the
Bohr radius $K(B_{\ell^{n}_{q}})\,\,1\leq q\leq\infty$. Using the methods
developed in [21], Defant and Frerick [18] have proved that there is a
constant $C>0$ such that for every $1\leq q\leq\infty$ and $n$ we have

(1.5) $\frac{1}{C}\left(\frac{\log n/\log\log
n}{n}\right)^{1-\frac{1}{\min\\{q,2\\}}}\leq K(B_{\ell^{n}_{q}}).$

Later, (1.5) was improved by the same authors in [19]. The case for $q=\infty$
was settled by Defant and his co-authors [20] and they showed that
$K(B_{\ell^{n}_{\infty}})$ behaves asymptotically as $\sqrt{\log n/n}$ as a
consequence of hypercontractivity of Bohnenblust-Hille inequality. Further,
Bayart et. al. [5] have extended the techniques and obtained the following:

$\lim_{n\rightarrow\infty}\frac{K(B_{\ell^{n}_{\infty}})}{\sqrt{\frac{\log
n}{n}}}=1.$

The powered versions of the Bohr radii $K^{p}(\mathbb{D})$ and
$K^{p}(B_{\ell^{n}_{\infty}})$ for scalar-valued holomorphic functions were
first studied by Djakov and Ramanujan [28]. Later, Bénéteau et. al. [7]
explored the analogous radii in the context of Hardy spaces. Apart from the
scalar valued functions, Blasco [9] has investigated the radius
$K^{p}(\mathbb{D},X)$ for arbitrary complex Banach spaces. In fact, Blasco [9,
Theorem 1.10] has proved that the Bohr radius $K^{p}(\mathbb{D},X)>0$ if, and
only if, $X$ is $p$-uniformly $\mathbb{C}$-convex, where $2\leq p<\infty$.
Recently, Das [16, 17] has showed the precise asymptotic value of
$K^{p}(B_{\ell^{n}_{\infty}})$ and investigated $p$-Bohr radius for Hardy
space of $X$-valued holomorphic functions defined on complete Reinhardt
domain.

To explore more general aspects of multidimensional Bohr radii, we introduce
$\lambda$ in our Definition 1.1, in more general sense. Indeed, we shall prove
that for every $1\leq p<\infty$ and bounded linear operator $T$ with
$\left\lVert T\right\rVert<\lambda^{1/p}$, the multidimensional Bohr radius
$K^{p}(B_{\ell^{n}_{q}},T,\lambda)$ is nonzero. Further, our aim is to study
the correct asymptotic estimates for the Bohr radii
$K^{p}(B_{\ell^{n}_{q}},T,\lambda)$ for arbitrary Banach spaces. As a
consequence, we obtain lower bound of arithmetic Bohr radius for Banach
spaces. The organization of this paper is as follows. After the preliminaries,
Section 3 accumulates all the main results of this paper. In this section,
Proposition 3.1 shows the for every bounded operator $T$ with $\left\lVert
T\right\rVert<\lambda^{1/p}$ the Bohr radius
$K^{p}(B_{\ell^{n}_{q}},T,\lambda)$ is positive. Further, Theorem 3.1 and
Theorem 3.2 give the asymptotic behavior of
$K^{p}(B_{\ell^{n}_{q}},X,\lambda)$ for finite and infinite dimensional Banach
space $X$ respectively, in case of identity operator on $X$. Finally, in
Theorem 3.4, we obtain lower estimates for
$K^{p}(B_{\ell^{n}_{q}},T,\lambda)$. Section 4 keeps two important lemmas
which play vital role to prove the main results. Section 5 contains the proof
of main results.

##  2\. Preliminaries

We employ standard terminologies and notions from classical theory of Banach
spaces (see e.g. [33, 34]). We denote $\mathbb{D}$ to be the open unit disk in
complex plane $\mathbb{C}$. Throughout the paper, it has been assumed that
every Banach space $X$ is complex. $X^{*}$ will denote the topological dual of
Banach space $X$ and we write $B_{X}$ for the open unit ball in $X$. As usual,
we denote $\ell^{n}_{q}$ for the Banach space of all $n$-tuples
$z=(z_{1},\ldots,z_{n})\in\mathbb{C}^{n}$ endowed with the norm

$\left\lVert
z\right\rVert_{q}:=\left(\sum_{i=1}^{n}|z_{i}|^{q}\right)^{\frac{1}{q}}\,\,\,\mbox{for}\,\,1\leq
q<\infty$

and $\left\lVert z\right\rVert_{\infty}:=\max_{1\leq i\leq n}|z_{i}|$ for
$q=\infty$. For $1\leq q\leq\infty$, the conjugate exponent $q^{\prime}$ for
$q$ is defined by $1/q+1/q^{\prime}=1$. For $x\in X$, the absolute value of
$x$ is defined by $|x|=x\vee(-x)$. Recall that a Banach lattice is a Banach
space $X$ which is a vector lattice with $|x|\leq|y|$ implies $\left\lVert
x\right\rVert\leq\left\lVert y\right\rVert$ for all $x,y\in X$. A Banach
lattice is said to be $r$-concave, where $1\leq r<\infty$, if there is a
constant $M>0$ such that

(2.1) $\left(\sum_{j=1}^{n}\left\lVert
x_{j}\right\rVert^{r}\right)^{\frac{1}{r}}\leq
M\left\lVert\left(\sum_{j=1}^{n}|x_{j}|^{r}\right)^{\frac{1}{r}}\right\rVert$

for every arbitrarily chosen finite elements $x_{1},\ldots,x_{n}\in X$. The
best such constant satisfying (2.1) is as usual denoted by $M_{r}(X)$. For
$2\leq r\leq\infty$, a Banach space $X$ is said to have cotype $r$, if there
is a constant $C>0$ such that for every choice of finitely many vectors
$x_{1},\ldots,x_{n}\in X$, we have

(2.2) $\left(\sum_{j=1}^{n}\left\lVert
x_{j}\right\rVert^{r}\right)^{\frac{1}{r}}\leq
C\left(\int_{0}^{1}\left\lVert\sum_{j=1}^{n}r_{j}(t)x_{j}\right\rVert^{2}\,dt\right)^{\frac{1}{2}},$

where $r_{j}$ is the $j$th Rademacher function on $[0,1]$. Further, the best
such constant $C$ satisfying (2.2) is usually denoted by $C_{r}(X)$. We write

$\text{Cot}(X):=\inf\left\\{2\leq r\leq\infty:X\,\,\mbox{has
cotype}\,r\right\\}.$

For a Banach space with cotype $\infty$ i.e., $\text{Cot}(X)=\infty$, we
denote $\frac{1}{\text{Cot}(X)}=0$. It is to be noted that the notions of
concavity and cotype are closely related concepts in the context of Banach
lattices. In fact, a $r$-concave Banach lattice with $r\geq 2$ is of cotype
$r$. On the other way, each Banach lattice of cotype $2$ is $2$-concave, and a
Banach lattice of cotype $r>2$ is $s$-concave for all $s>r$.

Given $m,n\in\mathbb{N}$, we consider the following sets of indices:

$\displaystyle\mathcal{M}(m,n)$
$\displaystyle=\left\\{\mathbf{j}=(j_{1},\ldots,j_{m}):1\leq
j_{1},\ldots,j_{m}\leq n\right\\}$
$\displaystyle=\left\\{1,2,\ldots,n\right\\}^{m},$
$\mathcal{J}(m,n)=\left\\{\mathbf{j}\in\mathcal{M}(m,n):1\leq
j_{1}\leq\cdots\leq j_{m}\leq n\right\\}.$

Furthermore, we write another closely related multi-index set

$\Lambda(m,n)=\left\\{\alpha=(\alpha_{1},\ldots,\alpha_{n})\in\mathbb{N}^{n}_{0}:|\alpha|=m\right\\},$

where $|\alpha|=\alpha_{1}+\cdots+\alpha_{n}$. We may identify $\Lambda(m,n)$
with $\mathcal{J}(m,n)$ and vice-versa. For
$\alpha=(\alpha_{1},\ldots,\alpha_{n})\in\Lambda(m,n)$, the associated index
$\mathbf{j}_{\alpha}\in\mathcal{J}(m,n)$ is given by
$\mathbf{j}_{\alpha}=\left(1,\stackrel{{\scriptstyle\alpha_{1}}}{{\ldots}},1,2,\stackrel{{\scriptstyle\alpha_{2}}}{{\ldots}}2,\ldots,n,\stackrel{{\scriptstyle\alpha_{n}}}{{\ldots}},n\right)$.
Moreover, for $\mathbf{j}\in\mathcal{J}(m,n)$ the associated multi-index
$\mathbf{j}_{\alpha}\in\Lambda(m,n)$ is given by
$\mathbf{j}_{\alpha_{r}}=|\\{k:j_{k}=r\\}|$. This gives a one-to-one
correspondence between $\Lambda(m,n)$ and $\mathcal{J}(m,n)$. To identify
$\mathcal{M}(m,n)$, we define the following equivalence relation on
$\mathcal{M}(m,n)$: for $\mathbf{i},\mathbf{j}\in\mathcal{M}(m,n)$, we denote
$\mathbf{i}\thicksim\mathbf{j}$ if, and only if, there is a permutation
$\sigma\in S_{m}$ such that $\mathbf{j}=(i_{\sigma(1)},\ldots,i_{\sigma(m)})$.
We observe that

$\mathcal{M}(m,n)=\cup_{\mathbf{j}\in\mathcal{J}(m,n)}[\mathbf{j}].$

Moreover, it is worth noting that for each $\alpha\in\Lambda(m,n)$ we have

$\text{card}[\mathbf{j}_{\alpha}]=\frac{m!}{\alpha!},$

where $\alpha!=\alpha_{1}!\cdots\alpha_{n}!$. Further, we consider the
following notation from [6]: given a subset
$\mathcal{J}\subset\mathcal{J}(m,n)$, we denote

$\mathcal{J}^{*}=\left\\{\mathbf{j}\in\mathcal{J}(m-1,n):\,\mbox{there
is}\,\,k\geq 1,(\mathbf{j},k)\in\mathcal{J}\right\\}.$

For $1\leq q\leq\infty$, let $\mathcal{P}(^{m}\ell^{n}_{q},X)$ be the linear
space of all $m$-homogeneous polynomial $Q:\ell^{n}_{q}\rightarrow X$ of $n$
complex variables with values in $X$, equipped with uniform (or sup) norm

$\left\lVert Q\right\rVert_{\ell^{n}_{q}}:=\sup_{z\in
B_{\ell^{n}_{q}}}\left\lVert Q(z)\right\rVert_{X}.$

Using the above multi-index notations, a polynomial
$Q\in\mathcal{P}(^{m}\ell^{n}_{q},X)$ can be expressed in the form

$Q(z)=\sum_{\alpha\in\Lambda(m,n)}x_{\alpha}z^{\alpha}=\sum_{\mathbf{j}\in\mathcal{J}(m,n)}c_{\mathbf{j}}z_{\mathbf{j}},$

where $x_{\alpha},c_{\mathbf{j}}\in X$ and $x_{\alpha}=c_{\mathbf{j}}$ with
$\mathbf{j}=\left(1,\stackrel{{\scriptstyle\alpha_{1}}}{{\ldots}},1,2,\stackrel{{\scriptstyle\alpha_{2}}}{{\ldots}}2,\ldots,n,\stackrel{{\scriptstyle\alpha_{n}}}{{\ldots}},n\right)$.
If $f:B_{\ell^{n}_{q}}\rightarrow X$ is an $X$-valued holomorphic function
then we write $f(z)=\sum_{\alpha\in\mathbb{N}^{n}_{0}}x_{\alpha}(f)z^{\alpha}$
as the monomial series expansion of $f$, where
$x_{\alpha}(f)=(\partial^{\alpha}f(0))/\alpha!$ is the $\alpha$th coefficient
of the expansion.

A bounded linear operator $T:X\rightarrow Y$ between two Banach spaces $X$ and
$Y$ is called $(r,s)$-summing, $1\leq r,s<\infty$, if there is a constant
$\widetilde{C}>0$ such that for each choice of finitely many vectors
$x_{1},\ldots,x_{n}\in X$, we have

(2.3) $\left(\sum_{j=1}^{n}\left\lVert
T(x_{j})\right\rVert^{r}\right)^{\frac{1}{r}}\leq\widetilde{C}\sup_{x^{*}\in
B_{X^{*}}}\left(\sum_{j=1}^{n}|x^{*}(x_{j})|^{s}\right)^{\frac{1}{s}}.$

Also, the best such constant $\widetilde{C}$ satisfying (2.3) is as usual
denoted by $\pi_{r,s}(T)$. In particular, if $r=s$, we call $T$ is $r$-summing
and we write the best constant as $\pi_{r}(T)$.

##  3\. Main Results

As a crucial property, we ensure that for every bounded operator $T$ with
$\left\lVert T\right\rVert<\lambda^{1/p}$, the $\lambda_{p}$-Bohr radius of
$T$ is non-zero.

######  Proposition 3.1.

Let $1\leq p<\infty$ and $\lambda>1$. Suppose $T:X\rightarrow Y$ is a bounded
operator between complex Banach spaces $X$ and $Y$ such that $\left\lVert
T\right\rVert<\lambda^{1/p}$. Then there exists a constant $M>0$ such that for
all $1\leq q\leq\infty$ and $n\in\mathbb{N}$ we have

$K^{p}(B_{\ell^{n}_{q}},T,\lambda)\geq
M\frac{1}{n^{\frac{1}{p}-\frac{1}{pq}}},$

where

$M=\left\\{\begin{array}[]{ll}\max\left\\{\left(\frac{\lambda-\left\lVert
T\right\rVert^{p}}{2\lambda-\left\lVert
T\right\rVert^{p}}\right)^{1/p},\left(\frac{\lambda-\left\lVert
T\right\rVert^{p}}{(\lambda-\left\lVert T\right\rVert^{p}+1)\left\lVert
T\right\rVert}\right)^{1/p}\right\\},&\mbox{ if \,$\left\lVert
T\right\rVert\geq 1$},\\\\[14.22636pt]
\max\left\\{\left(\frac{\lambda-\left\lVert
T\right\rVert^{p}}{2\lambda-\left\lVert
T\right\rVert^{p}}\right)^{1/p},\left(\frac{\lambda-\left\lVert
T\right\rVert^{p}}{\lambda-\left\lVert
T\right\rVert^{p}+1}\right)^{1/p}\right\\},&\mbox{ if \,$0<\left\lVert
T\right\rVert<1$}.\end{array}\right.$

As a consequence, whenever we consider the identity operator on $X$,
Proposition 3.1 demonstrates the lower bound for the classical $p$-Bohr radius
of Banach spaces.

######  Corollary 3.2.

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

$K^{p}(B_{\ell^{n}_{q}},X,\lambda)\geq\left(\frac{\lambda-1}{\lambda}\right)^{\frac{1}{p}}\frac{1}{n^{\frac{1}{p}-\frac{1}{pq}}}.$

######  Remark 3.1.

It is worth noting that for $\lambda=1$ and $p=1$, we actually obtain the
trivial lower bound $K(B_{\ell^{n}_{q}},X)\geq 0$. Indeed, as we have
discussed before, Blasco [8, Theorem 1.2] has also observed that
$K(\ell^{2}_{q})=0$ for every $1\leq q\leq\infty$.

Our main aim of this paper is to investigate precise asymptotic behavior of
$\lambda_{p}$-Bohr radius $K^{p}(B_{\ell^{n}_{q}},T,\lambda)$, where $T$ is a
bounded operator between two complex Banach spaces. Motivated by the work of
Defant et. al. [25], one of our contributions in this paper is the following
theorem, which provides the correct asymptotic estimates for
$\lambda_{p}$-Bohr radius of identity operator on Banach spaces, where $1\leq
p<\infty$. Let us first concentrate on finite dimensional Banach spaces.

######  Theorem 3.1.

Let $X$ be a finite dimensional complex Banach space and $\lambda>1$, $1\leq
p<\infty$. Then there are constants $B$ and $C$ such that for each $1\leq
q\leq\infty$ and $n\in\mathbb{N}$ we have

$K^{p}(B_{\ell^{n}_{q}},X,\lambda)\leq B\lambda^{2/p}n^{1-1/p}\left(\frac{\log
n}{n}\right)^{1-\frac{1}{\min\\{q,2\\}}}$

and

$K^{p}(B_{\ell^{n}_{q}},X,\lambda)\geq\left\\{\begin{array}[]{ll}C\left(\frac{\lambda-\left\lVert
T\right\rVert^{p}}{2\lambda-\left\lVert
T\right\rVert^{p}}\right)^{1/p}\left(\frac{\log
n}{n}\right)^{1-\frac{1}{r}},&\mbox{for \, $q\leq r$}\\\\[22.76219pt]
C\left(\frac{\lambda-\left\lVert T\right\rVert^{p}}{2\lambda-\left\lVert
T\right\rVert^{p}}\right)^{1/p}n^{\frac{1}{q}-\frac{1}{r}}\left(\frac{\log
n}{n}\right)^{1-\frac{1}{r}},&\mbox{for \, $r<q$}\end{array}\right.$

where $r=pq^{\prime}/(pq^{\prime}-1)$. Here $B$ is a universal constant and
$C$ depends only on $X$.

Now, we want to study the Bohr radius $K^{p}(B_{\ell^{n}_{q}},X,\lambda)$ for
the infinite dimensional Banach spaces. In this cases, we observe that the log
term is missing from the estimates of $K^{p}(B_{\ell^{n}_{q}},X,\lambda)$. In
fact, the asymptotic decay is very much influenced by geometry of the Banach
space $X$ i.e., by its optimal cotype $\text{Cot}(X)$.

######  Theorem 3.2.

Let $1\leq p<\infty$ and $1\leq q\leq\infty$. If $X$ is an infinite
dimensional complex Banach space of cotype $r$ with $2\leq r\leq\infty$, then
we have

$K^{p}(B_{\ell^{n}_{q}},X,\lambda)\leq\left\\{\begin{array}[]{ll}\frac{\lambda^{1/p}}{n^{\frac{1}{p}-\frac{1}{q}}},&\mbox{for
\, $q\leq\text{Cot}(X)=r$}\\\\[14.22636pt]
\frac{\lambda^{1/p}}{n^{\frac{1}{p}\left(1-\frac{1}{Cot(X)}\right)-\frac{1}{q}\left(1-\frac{1}{p}\right)}},&\mbox{for
\, $q>\text{Cot}(X)=r$}\end{array}\right.$

and

$K^{p}(B_{\ell^{n}_{q}},X,\lambda)\geq\left\\{\begin{array}[]{ll}\left(\frac{\lambda-1}{\lambda}\right)^{1/p}\frac{1}{e\,C_{r}(X)},&\mbox{for
\, $q\leq s$}\\\\[14.22636pt]
\left(\frac{\lambda-1}{\lambda}\right)^{1/p}\frac{1}{e\,C_{r}(X)n^{\frac{1}{s}-\frac{1}{q}}},&\mbox{for
\, $q>s$},\end{array}\right.$

where $s=r/(r-1)$.

Defant et. al. [25, Theorem 4.1] have used the hypercontractivity of the
Bohnenblust-Hille inequality to establish the asymptotic behavior of of the
Bohr radii $K^{p}(B_{\ell^{n}_{\infty}},X,\lambda)$ for $p=1$. Moreover, as a
technical tool, Defant et. al. [25, Theorem 5.3] have shown Bohnenblust-Hille
type inequality for $(s,1)$-summing operator to estimate the multidimensional
Bohr radii for operators: let $Y$ be a $r$-concave Banach lattice, with $2\leq
r<\infty$, and $T:X\rightarrow Y$ an $(s,1)$-summing operator with $1\leq
s\leq r$. Then there is a constant $C>0$ such that for every $m$-homogeneous
polynomial $Q:\ell^{n}_{\infty}\rightarrow X$,
$Q(z)=\sum_{\alpha\in\Lambda(m,n)}x_{\alpha}z^{\alpha}$

$\left(\sum_{\alpha\in\Lambda(m,n)}\left\lVert
T(x_{\alpha})\right\rVert^{\rho}_{Y}\right)^{\frac{1}{\rho}}\leq
C^{m}\left\lVert Q\right\rVert_{B_{\ell^{n}_{\infty}}},$

where

$\rho:=\frac{rsm}{r+(m-1)s}.$

Recently, by considering the diagonal operator
$D_{z}:\ell^{n}_{\infty}\rightarrow\ell^{n}_{q}$, Kumar and Manna [32] have
obtained an extended version of the above result for $m$-homogeneous
polynomials $Q\in\mathcal{P}(^{m}\ell^{n}_{q},X),\,\,1\leq q<\infty$.

######  Theorem 3.3.

[32] Let $2\leq r<\infty$. Suppose $Y$ is a $r$-concave Banach lattice, and
the operator $T:X\rightarrow Y$ is an $(s,1)$-summing operator with $1\leq
s\leq r$. Then there is a constant $\widetilde{C}>0$ such that the following
holds

$\left(\sum_{\alpha\in\Lambda(m,n)}\left\lVert
T(x_{\alpha})z^{\alpha}\right\rVert^{\rho}_{Y}\right)^{\frac{1}{\rho}}\leq\widetilde{C}^{m}\left\lVert
Q\right\rVert_{B_{\ell^{n}_{q}}}$

for every polynomial
$Q(z)=\sum_{\alpha\in\Lambda(m,n)}x_{\alpha}z^{\alpha}\in\mathcal{P}(^{m}\ell^{n}_{q}),\,\,1\leq
q<\infty$.

We are now ready to state our next main result. As a consequence of the above
technical discussions, Theorem 3.4 provides the lower estimates of the Bohr
radius $K^{p}(B_{\ell^{n}_{q}},T,\lambda)$ for any bounded operator $T$.

######  Theorem 3.4.

Let $1\leq p<\infty$ and $T:X\rightarrow Y$ be a bounded operator between
Banach spaces $X$ and $Y$.

  1. (1)

Assume that $X$ or $Y$ is of cotype $r$ with $2\leq r\leq\infty$. Then there
is a constant $D>0$ such that for every $\left\lVert
T\right\rVert<\lambda^{1/p}$, $1\leq q\leq\infty$, and $n$

$K^{p}(B_{\ell^{n}_{q}},T,\lambda)\geq\left\\{\begin{array}[]{ll}D\,\left(\frac{\lambda-\left\lVert
T\right\rVert^{p}}{\lambda}\right)^{\frac{1}{p}},&\mbox{for \, $q\leq
s$}\\\\[14.22636pt] D\,\left(\frac{\lambda-\left\lVert
T\right\rVert^{p}}{\lambda}\right)^{\frac{1}{p}}\left(\frac{1}{n}\right)^{\frac{1}{s}-\frac{1}{q}},&\mbox{for
\, $q>s$},\end{array}\right.$

where $s=r/(r-1)$.

  2. (2)

Assume that $Y$ is a $r$-concave Banach lattice with $2\leq r<\infty$ and
there is a $1\leq s<r$ such that the operator $T$ is $(s,1)$-summing. Then
there is a constant $D>0$ such that

$K^{p}(B_{\ell^{n}_{q}},T,\lambda)\geq D\,\left(\frac{\lambda-\left\lVert
T\right\rVert^{p}}{2\lambda-\left\lVert
T\right\rVert^{p}}\right)^{\frac{1}{p}}\left(\frac{\log
n}{n}\right)^{1-\frac{1}{r}}$

for every $\left\lVert T\right\rVert<\lambda^{1/p}$, $1\leq q\leq\infty$, and
$n\in\mathbb{N}$.

We now move our focus for a while on arithmetic Bohr radius of Banach spaces,
which is another interesting multidimensional variant of Bohr radius. In fact,
it has very close relationship with classical multidimensional Bohr radii. In
order to study the upper inclusion of domains of convergence of monomial
expansions, as a technical tool, Defant et. al. [24] introduced the arithmetic
Bohr radius for scalar-valued holomorphic functions. Later, it was studied
independently by the same authors in [23]. Recently, the study of arithmetic
Bohr radius has been extended for vector-valued holomorphic functions (see
[4]). Indeed, the authors [4] have introduced the following notion of
arithmetic Bohr radius:

######  Definition 3.1.

[4] Let $\mathcal{F}(B_{\ell^{n}_{q}},X)$ be set of all holomorphic functions
on $B_{\ell^{n}_{q}}$ into a complex Banach space $X$. For each $1\leq
p<\infty$ and $\lambda\geq 1$, the $\lambda_{p}$-arithmetic Bohr radius of
$B_{\ell^{n}_{q}}$ with respect to $\mathcal{F}(B_{\ell^{n}_{q}},X)$ is
defined as

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

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 $A_{p}(B_{\ell^{n}_{q}},X,\lambda)$ for
$A_{p}(H_{\infty}(B_{\ell^{n}_{q}},X),\lambda)$ and
$A_{p}(B_{\ell^{n}_{q}},X)$ for $A_{p}(B_{\ell^{n}_{q}},X,1)$.

In view of [4, Theorem 2.2] and Theorem 3.1, as an application, we obtain the
following lower estimate for the arithmetic Bohr radius
$A_{p}(B_{\ell^{n}_{q}},X,\lambda)$ in case of $X$ is finite dimensional.

######  Theorem 3.5.

Let $1\leq p<\infty$, $\lambda\geq 1$, and $1\leq q\leq\infty$. Suppose $X$ is
a finite dimensional Banach space. Then there is a constant $C>0$ such that
for every $n>1$ we have

$A_{p}(B_{\ell^{n}_{q}},X,\lambda)\geq\left\\{\begin{array}[]{ll}C\left(\frac{\lambda-1}{2\lambda-1}\right)^{1/p}n^{-\frac{1}{q}}\left(\frac{\log
n}{n}\right)^{1-\frac{1}{r}},&\mbox{for \, $q\leq r$}\\\\[14.22636pt]
C\left(\frac{\lambda-1}{2\lambda-1}\right)^{1/p}n^{-\frac{1}{r}}\left(\frac{\log
n}{n}\right)^{1-\frac{1}{r}},&\mbox{for \, $r<q$},\end{array}\right.$

where $r=pq^{\prime}/(pq^{\prime}-1)$, and the constant $C$ depends only on
$X$.

##  4\. Key Lemmas

As demonstrated by the scalar case, studying the $m$-homogeneous polynomial
first is typically a beneficial approach for obtaining nontrivial estimates
for Bohr radii. The following definition is the $m$-homogeneous analogue of
Definition 1.1 for $m\in\mathbb{N}$.

######  Definition 4.1.

Let $1\leq p<\infty$, $1\leq q\leq\infty$, $\lambda\geq 1$,
$m,n\in\mathbb{N}$, and $T:X\rightarrow Y$ be a bounded operator in complex
Banach spaces such that $\left\lVert T\right\rVert\leq\lambda^{1/p}$. Then
$K^{p}_{m}(B_{\ell^{n}_{q}},X,\lambda)$ is defined to be the supremum of all
$r\geq 0$ such that for every $m$-homogeneous polynomials
$Q\in\mathcal{P}(^{m}\ell^{n}_{q},X)$,
$Q(z)=\sum_{\alpha\in\Lambda(m,n)}x_{\alpha}z^{\alpha}$, we have

$\sup_{z\in rB_{\ell^{n}_{q}}}\sum_{\alpha\in\Lambda(m,n)}\left\lVert
T(x_{\alpha})z^{\alpha}\right\rVert^{p}_{Y}\leq\lambda\left\lVert\sum_{\alpha\in\Lambda(m,n)}x_{\alpha}z^{\alpha}\right\rVert^{p}_{X}.$

It is worth mentioning that if $T$ is a null operator, then the above supremum
is $\infty$. Thus, by assuming that $T\neq 0$ throughout the paper, we are
avoiding this trivial scenario. The constant
$K^{p}_{m}(B_{\ell^{n}_{q}},X,\lambda)$ in the Definition 4.1 can be rephrased
as

(4.1) $\displaystyle K^{p}_{m}(B_{\ell^{n}_{q}},X,\lambda):=\sup\biggl{\\{}$
$\displaystyle r\geq 0:\sup_{z\in
B_{\ell^{n}_{q}}}\sum_{\alpha\in\Lambda(m,n)}\left\lVert
T(x_{\alpha})z^{\alpha}\right\rVert^{p}_{Y}\leq\frac{\lambda}{r^{mp}}\sup_{z\in
B_{\ell^{n}_{q}}}\left\lVert Q(z)\right\rVert^{p}_{X}$ $\displaystyle\mbox{for
all}\,\,Q\in\mathcal{P}(^{m}\ell^{n}_{q},X)\biggr{\\}}.$

Further, it is worth noting that

(4.2)
$K^{p}_{m}(B_{\ell^{n}_{q}},T,\lambda)=\sqrt[mp]{\lambda}K^{p}_{m}(B_{\ell^{n}_{q}},T)$

and

(4.3)
$K^{p}(B_{\ell^{n}_{q}},T,\lambda)\geq\max\left\\{K^{p}\left(B_{\ell^{n}_{q}},X,\lambda/\left\lVert
T\right\rVert^{p}\right),K^{p}\left(B_{\ell^{n}_{q}},Y,\lambda/\left\lVert
T\right\rVert^{p}\right)\right\\}.$

The following lemma provides a nice connection between Definitions 1.1 and
4.1, which will be one of the most important technical tools to prove our main
results in this paper.

######  Lemma 4.4.

Let $1\leq p<\infty$ and $T:X\rightarrow Y$ be a bounded operator between
complex Banach spaces $X$ and $Y$ with $\left\lVert
T\right\rVert<\lambda^{1/p}$. Then we have

  1. (a)

$\left(\frac{\lambda-\left\lVert T\right\rVert^{p}}{2\lambda-\left\lVert
T\right\rVert^{p}}\right)^{1/p}\inf_{m\in\mathbb{N}}K^{p}_{m}(B_{\ell^{n}_{q}},T,\lambda)\leq
K^{p}(B_{\ell^{n}_{q}},T,\lambda)\leq\inf_{m\in\mathbb{N}}K^{p}_{m}(B_{\ell^{n}_{q}},T,\lambda)$

  2. (b)

$\left(\frac{\lambda-\left\lVert T\right\rVert^{p}}{\lambda-\left\lVert
T\right\rVert^{p}+1}\right)^{1/p}\inf_{m\in\mathbb{N}}K^{p}_{m}(B_{\ell^{n}_{q}},T)\leq
K^{p}(B_{\ell^{n}_{q}},T,\lambda)\leq\lambda^{1/p}\inf_{m\in\mathbb{N}}K^{p}_{m}(B_{\ell^{n}_{q}},T).$

Proof. The right hand side inequality of (a) is clear from the fact that
$K^{p}(B_{\ell^{n}_{q}},T,\lambda)\leq K^{p}_{m}(B_{\ell^{n}_{q}},T,\lambda)$
for all $m\geq 1$. So, we now proceed for the left hand side inequality of
(a). Let $f(z)=\sum_{\alpha\in\mathbb{N}^{n}_{0}}x_{\alpha}z^{\alpha}$ be a
holomorphic function on $B_{\ell^{n}_{q}}$ with values in the complex Banach
space $X$. Further, suppose $v\in\mathbb{C}^{n}$ such that $\left\lVert
v\right\rVert\leq\inf_{m\in\mathbb{N}}\\{K^{p}_{m}(B_{\ell^{n}_{q}},T,\lambda)\\}.$
It is worth noting that for $1\leq p<\infty$

$0<\left(\frac{\lambda-\left\lVert T\right\rVert^{p}}{2\lambda-\left\lVert
T\right\rVert^{p}}\right)^{1/p}<1.$

Since $v\in K^{p}_{m}(B_{\ell^{n}_{q}},T,\lambda)\overline{B_{\ell^{n}_{q}}}$
for all $m\in\mathbb{N}$ and using the Cauchy-Riemann inequalities we have
that

$\displaystyle\sum_{\alpha\in\mathbb{N}^{n}_{0}}\left\lVert
T(x_{\alpha})\left(\left(\frac{\lambda-\left\lVert
T\right\rVert^{p}}{2\lambda-\left\lVert
T\right\rVert^{p}}\right)^{1/p}v\right)^{\alpha}\right\rVert^{p}_{Y}$
$\displaystyle=\left\lVert
T(x_{0})\right\rVert^{p}+\sum_{m=1}^{\infty}\left(\frac{\lambda-\left\lVert
T\right\rVert^{p}}{2\lambda-\left\lVert
T\right\rVert^{p}}\right)^{m}\sum_{\alpha\in\Lambda(m,n)}\left\lVert
T(x_{\alpha})\right\rVert^{p}|v|^{p\alpha}$ $\displaystyle\leq\left\lVert
T\right\rVert^{p}\left\lVert
x_{0}\right\rVert^{p}+\sum_{m=1}^{\infty}\left(\frac{\lambda-\left\lVert
T\right\rVert^{p}}{2\lambda-\left\lVert
T\right\rVert^{p}}\right)^{m}\lambda\left\lVert\sum_{\alpha\in\Lambda(m,n)}x_{\alpha}z^{\alpha}\right\rVert^{p}_{B_{\ell^{n}_{q}}}$
$\displaystyle\leq\left(\left\lVert
T\right\rVert^{p}+\lambda\sum_{m=1}^{\infty}\left(\frac{\lambda-\left\lVert
T\right\rVert^{p}}{2\lambda-\left\lVert
T\right\rVert^{p}}\right)^{m}\right)\left\lVert
f\right\rVert^{p}_{B_{\ell^{n}_{q}}}=\lambda\left\lVert
f\right\rVert^{p}_{B_{\ell^{n}_{q}}}.$

We shall prove (b) in the similar way. Since
$K^{p}_{m}(B_{\ell^{n}_{q}},T,\lambda)=\lambda^{1/pm}K^{p}_{m}(B_{\ell^{n}_{q}},T)$,
we obtain

$K^{p}(B_{\ell^{n}_{q}},T,\lambda)\leq\inf_{m\in\mathbb{N}}\left\\{\lambda^{\frac{1}{pm}}K^{p}_{m}(B_{\ell^{n}_{q}},T)\right\\}\leq\lambda^{1/p}\inf_{m\in\mathbb{N}}\\{K^{p}_{m}(B_{\ell^{n}_{q}},T)\\}.$

In this case, we suppose again that $v\in\mathbb{C}^{n}$ as above but here we
proceed by considering $\left\lVert
v\right\rVert\leq\inf_{m\in\mathbb{N}}\\{K^{p}_{m}(B_{\ell^{n}_{q}},T)\\}$.
Thus, for $v\in K^{p}_{m}(B_{\ell^{n}_{q}},T)\overline{B_{\ell^{n}_{q}}}$, we
obtain

$\displaystyle\sum_{\alpha\in\mathbb{N}^{n}_{0}}\left\lVert
T(x_{\alpha})\left(\left(\frac{\lambda-\left\lVert
T\right\rVert^{p}}{\lambda-\left\lVert
T\right\rVert^{p}+1}\right)^{1/p}v\right)^{\alpha}\right\rVert^{p}_{Y}$
$\displaystyle=\left\lVert
T(x_{0})\right\rVert^{p}+\sum_{m=1}^{\infty}\left(\frac{\lambda-\left\lVert
T\right\rVert^{p}}{\lambda-\left\lVert
T\right\rVert^{p}+1}\right)^{m}\sum_{\alpha\in\Lambda(m,n)}\left\lVert
T(x_{\alpha})\right\rVert^{p}|v|^{p\alpha}$ $\displaystyle\leq\left\lVert
T\right\rVert^{p}\left\lVert
x_{0}\right\rVert^{p}+\sum_{m=1}^{\infty}\left(\frac{\lambda-\left\lVert
T\right\rVert^{p}}{\lambda-\left\lVert
T\right\rVert^{p}+1}\right)^{m}\left\lVert\sum_{\alpha\in\Lambda(m,n)}x_{\alpha}z^{\alpha}\right\rVert^{p}_{B_{\ell^{n}_{q}}}$
$\displaystyle\leq\left(\left\lVert
T\right\rVert^{p}+\sum_{m=1}^{\infty}\left(\frac{\lambda-\left\lVert
T\right\rVert^{p}}{\lambda-\left\lVert
T\right\rVert^{p}+1}\right)^{m}\right)\left\lVert
f\right\rVert^{p}_{B_{\ell^{n}_{q}}}=\lambda\left\lVert
f\right\rVert^{p}_{B_{\ell^{n}_{q}}}.$

This completes the proof. ∎

The following lemma of Bayart et. al. [6, Lemma 3.5] is a key tool to prove
Theorem 3.1. For a $m$-homogeneous polynomial
$Q\in\mathcal{P}(^{m}\ell^{n}_{q}),\,\,(1\leq q\leq\infty)$, it guarantees the
$\ell_{q^{\prime}}$-summability of certain coefficients of $Q$ in terms of its
uniform norm on $\ell^{n}_{q}$.

######  Lemma 4.5.

[6] Let $1\leq t\leq\infty$ and $P$ be an $m$-homogeneous polynomial in $n$
variables. Then for any $\mathbf{j}\in\mathcal{J}(m-1,n)$,

$\left(\sum_{k=j_{m-1}}^{n}|c_{(\mathbf{j},k)}(P)|^{t^{\prime}}\right)^{1/t^{\prime}}\leq
me^{1+\frac{m-1}{t}}|\mathbf{j}|^{1/t}\left\lVert
P\right\rVert_{\mathcal{P}(^{m}\ell^{n}_{t})}.$

##  5\. Proof of the main results

Proof of Proposition 3.1. Let
$Q(z)=\sum_{\alpha\in\Lambda(m,n)}x_{\alpha}z^{\alpha}\in\mathcal{P}(^{m}l^{n}_{q},X)$
be an $m$-homogeneous polynomial. By using Hölder’s inequality and for all
$z\in\mathbb{C}^{n}$, we have

$\displaystyle\sum_{\alpha\in\Lambda(m,n)}\left\lVert
T(x_{\alpha})z^{\alpha}\right\rVert^{p}$
$\displaystyle\leq\max_{\alpha\in\Lambda(m,n)}\left\lVert
T(x_{\alpha})\right\rVert^{p}\sum_{\alpha\in\Lambda(m,n)}|z^{\alpha}|^{p}$
$\displaystyle\leq\left\lVert
T\right\rVert^{p}\max_{\alpha\in\Lambda(m,n)}\left\lVert
x_{\alpha}\right\rVert^{p}\left(|z_{1}|^{p}+\cdots+|z_{n}|^{p}\right)^{m}$
$\displaystyle\leq\lambda\max_{\alpha\in\Lambda(m,n)}\left\lVert
x_{\alpha}\right\rVert^{p}\left(|z_{1}|^{pq}+\cdots+|z_{n}|^{pq}\right)^{\frac{m}{q}}n^{(1-\frac{1}{q})m}$
$\displaystyle\leq\lambda\left\lVert
Q\right\rVert^{p}_{\ell^{n}_{q}}\left\lVert
z\right\rVert^{mp}_{pq}n^{{mp}\left(\frac{1}{p}-\frac{1}{pq}\right)}$
$\displaystyle\leq\lambda\left\lVert
Q\right\rVert^{p}_{\ell^{n}_{q}}\left\lVert
z\right\rVert^{mp}_{q}n^{{mp}\left(\frac{1}{p}-\frac{1}{pq}\right)}.$

Therefore, for all $z\in\left(1/n^{(1/p-1/pq)}\right)B_{\ell^{n}_{q}}$, we
obtain

$\sum_{\alpha\in\Lambda(m,n)}\left\lVert
T(x_{\alpha})z^{\alpha}\right\rVert^{p}\leq\lambda\left\lVert
Q\right\rVert^{p}.$

Thus, for every $m$-homogeneous polynomial
$Q\in\mathcal{P}(^{m}\ell^{n}_{q})$, we obtain

$\sup\left\\{\sum_{\alpha\in\Lambda(m,n)}\left\lVert
T(x_{\alpha})z^{\alpha}\right\rVert^{p}\,|\,z\in\frac{1}{n^{\frac{1}{p}-\frac{1}{pq}}}B_{\ell^{n}_{q}}\right\\}\leq\lambda\left\lVert
Q\right\rVert^{p},$

which yields

$K^{p}_{m}(B_{\ell^{n}_{q}},T,\lambda)\geq\frac{1}{n^{\frac{1}{p}-\frac{1}{pq}}}.$

In view of Lemma 4.4(a), for each $1\leq p<\infty$, we have

$K^{p}(B_{\ell^{n}_{q}},T,\lambda)\geq\frac{1}{n^{\frac{1}{p}-\frac{1}{pq}}}\left(\frac{\lambda-\left\lVert
T\right\rVert^{p}}{2\lambda-\left\lVert T\right\rVert^{p}}\right)^{1/p}.$

By going similar lines of arguments as above for every $m$-homogeneous
polynomial $Q(z)=\sum_{|\alpha|=m}x_{\alpha}z^{\alpha}$, we obtain that

$\sup_{z\in\left(\frac{1}{\sqrt[m]{\left\lVert
T\right\rVert}n^{\frac{1}{p}-\frac{1}{pq}}}\right)B_{\ell^{n}_{q}}}\sum_{\alpha\in\Lambda(m,n)}\left\lVert
T(x_{\alpha})z^{\alpha}\right\rVert^{p}=\frac{1}{\left\lVert
T\right\rVert^{p}}\sup_{z\in\left(\frac{1}{n^{\frac{1}{p}-\frac{1}{pq}}}\right)B_{\ell^{n}_{q}}}\sum_{\alpha\in\Lambda(m,n)}\left\lVert
T(x_{\alpha})z^{\alpha}\right\rVert^{p}\leq\left\lVert
Q\right\rVert^{p}_{\mathcal{P}(^{m}\ell^{n}_{q})}.$

Hence, for every $m\in\mathbb{N}$, we have

$K^{p}_{m}(B_{\ell^{n}_{q}},T)\geq\frac{1}{\sqrt[m]{\left\lVert
T\right\rVert}n^{\frac{1}{p}-\frac{1}{pq}}}.$

Now, by the virtue of Lemma 4.4(b), we have

$K^{p}(B_{\ell^{n}_{q}},T,\lambda)\geq\left(\frac{\lambda-\left\lVert
T\right\rVert^{p}}{\lambda-\left\lVert
T\right\rVert^{p}+1}\right)^{1/p}\frac{1}{\left\lVert T\right\rVert
n^{\frac{1}{p}-\frac{1}{pq}}}\quad\mbox{if}\,\,\left\lVert T\right\rVert\geq
1$

and

$K^{p}(B_{\ell^{n}_{q}},T,\lambda)\geq\left(\frac{\lambda-\left\lVert
T\right\rVert^{p}}{\lambda-\left\lVert
T\right\rVert^{p}+1}\right)^{1/p}\frac{1}{n^{\frac{1}{p}-\frac{1}{pq}}}\quad\mbox{if}\,\,0<\left\lVert
T\right\rVert<1.$

This completes the proof. ∎

Proof of Theorem 3.1. We first check that the upper bound of
$K^{p}(B_{\ell^{n}_{q}},X,\lambda)$ is true. In view of [4, Lemma 2.3], we
have

(5.1) $K^{p}(B_{\ell^{n}_{q}},X,\lambda)\leq
n^{1/q}A_{p}(B_{\ell^{n}_{q}},X,\lambda),\,\,\lambda\geq 1$

for any $1\leq p<\infty$ and $n\in\mathbb{N}$, where
$A_{p}(B_{\ell^{n}_{q}},X,\lambda)$ is the $\lambda_{p}$-arithmetic Bohr
radius for the unit ball of $\ell^{n}_{q}$ spaces. In the same paper [4,
Theorem 2.2], the authors have established that there exists a constant $B>0$
such that

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

Using (5.1) and (5.2) and a simple computation yields

$K^{p}(B_{\ell^{n}_{q}},X,\lambda)\leq B\lambda^{\frac{2}{p\log
n}}\,\,\frac{(\log
n)^{1-\frac{1}{\min\\{q,2\\}}}}{n^{\frac{1}{p}-\frac{1}{\min\\{q,2\\}}}}=B\,\lambda^{\frac{2}{p\log
n}}\,n^{1-\frac{1}{p}}\left(\frac{\log
n}{n}\right)^{1-\frac{1}{\min\\{q,2\\}}}.$

Now we shall prove the lower bound for $K^{p}(B_{\ell^{n}_{q}},X,\lambda)$. By
the virtue of Lemma 4.4, it is sufficient to show the existence of a constant
$C(X)>0$ such that

$\inf_{m\in\mathbb{N}}K^{p}_{m}(B_{\ell^{n}_{q}},X,\lambda)\geq\left\\{\begin{array}[]{ll}C\,\left(\frac{\log
n}{n}\right)^{1-\frac{1}{r}},&\mbox{for \, $q\leq r$}\\\\[17.07164pt]
C\,n^{\frac{1}{q}-\frac{1}{r}}\left(\frac{\log
n}{n}\right)^{1-\frac{1}{r}},&\mbox{for \, $r<q$}\end{array}\right.$

for every $1\leq p<\infty$. Let us fix $Q\in\mathcal{P}(^{m}\ell^{n}_{q},X)$
and $u\in\ell^{n}_{q}$. Then, by Lemma 4.5, it is worth noting that

$\displaystyle\left(\sum_{k:\,\,(\mathbf{j},k)\in\mathcal{J}(m,n)}|c_{(\mathbf{j},k)}(P)|^{t^{\prime}}\right)^{1/t^{\prime}}$
$\displaystyle=\left(\sum_{k=j_{m-1}}^{n}|c_{(\mathbf{j},k)}(P)|^{t^{\prime}}\right)^{1/t^{\prime}}$
(5.3) $\displaystyle\leq me^{1+\frac{m-1}{t}}|\mathbf{j}|^{1/t}\left\lVert
P\right\rVert_{\mathcal{P}(^{m}\ell^{n}_{t})}$

holds for any $m$-homogeneous polynomial
$P\in\mathcal{P}(^{m}\ell^{n}_{t}),\,1\leq t\leq\infty$. Since $X$ is a finite
dimensional Banach space, the identity map $\mbox{Id}_{X}$ on $X$ is
absolutely summing i.e., $1$-summing and hence the operator $\mbox{Id}_{X}$ is
$s$-summing for every $s\geq 1$ with
$\pi_{s}(\mbox{Id}_{X})\leq\pi_{1}(\mbox{Id}_{X})$ (see [26, Theorem 10.4]).
In other words, there exists a constant $d>0$ such that for every $s\geq 1$
and for every finitely many $x_{1},\ldots,x_{k}\in X$ ,

$\left(\sum_{i=1}^{k}\left\lVert x_{i}\right\rVert^{s}\right)^{1/s}\leq
d\sup_{x^{*}\in
B_{X^{*}}}\left(\sum_{i=1}^{k}|x^{*}(x_{i})|^{s}\right)^{1/s}.$

Hence, if $Q(z)=\sum_{|\alpha|=m}x_{\alpha}(Q)z^{\alpha}$, then using the
above inequality (5), Hölder’s inequality (two times), and the multinomial
formula, for every $1\leq p<\infty$ we obtain

$\displaystyle\sum_{\mathbf{j}\in\mathcal{J}(m,n)}\left\lVert
x_{\mathbf{j}}(Q)\right\rVert^{p}|u_{\mathbf{j}}|^{p}=\sum_{\mathbf{j}\in\mathcal{J}(m,n)^{*}}\left(\sum_{k:\,\,(\mathbf{j},k)\in\mathcal{J}(m,n)}\left\lVert
x_{\mathbf{j}}(Q)\right\rVert^{p}|u_{\mathbf{j}}|^{p}|u_{k}|^{p}\right)$
$\displaystyle\leq\sum_{\mathbf{j}\in\mathcal{J}(m,n)^{*}}|u_{\mathbf{j}}|^{p}\left(\sum_{k:\,\,(\mathbf{j},k)\in\mathcal{J}(m,n)}\left\lVert
x_{\mathbf{j}}(Q)\right\rVert^{pq^{\prime}}\right)^{1/q^{\prime}}\left(\sum_{k}|u_{k}|^{pq}\right)^{1/q}$
$\displaystyle=\sum_{\mathbf{j}\in\mathcal{J}(m,n)^{*}}|u_{\mathbf{j}}|^{p}\left(\left(\sum_{k:\,\,(\mathbf{j},k)\in\mathcal{J}(m,n)}\left\lVert
x_{\mathbf{j}}(Q)\right\rVert^{pq^{\prime}}\right)^{1/pq^{\prime}}\left\lVert
u\right\rVert_{pq}\right)^{p}$
$\displaystyle\leq\pi^{p}_{1}(\mbox{Id}_{X})\sum_{\mathbf{j}\in\mathcal{J}(m,n)^{*}}|u_{\mathbf{j}}|^{p}\left(\sup_{x^{*}\in
B_{X^{*}}}\left(\sum_{k:\,\,(\mathbf{j},k)\in\mathcal{J}(m,n)}|x^{*}(x_{(\mathbf{j},k)}(Q))|^{pq^{\prime}}\right)^{1/pq^{\prime}}\right)^{p}\left\lVert
u\right\rVert^{p}_{pq}$
$\displaystyle\leq\pi^{p}_{1}(\mbox{Id}_{X})\left(me^{1+\frac{m-1}{r}}\right)^{p}\left\lVert
u\right\rVert^{p}_{pq}\sup_{x^{*}\in
B_{X^{*}}}\sup_{z\in\ell^{n}_{q}}\bigg{|}x^{*}\left(\sum_{\mathbf{j}\in\mathcal{J}(m,n)}x_{\alpha}(Q)z^{\alpha}\right)\bigg{|}^{p}\sum_{\mathbf{j}\in\mathcal{J}(m,n)^{*}}|\mathbf{j}|^{p/r}|u_{\mathbf{j}}|^{p}$
$\displaystyle\leq\pi^{p}_{1}(\mbox{Id}_{X})\left(me^{1+\frac{m-1}{r}}\right)^{p}\left\lVert
u\right\rVert^{p}_{q}\left\lVert
Q\right\rVert^{p}_{\mathcal{P}(^{m}\ell^{n}_{q})}\left(\sum_{\mathbf{j}\in\mathcal{J}(m,n)^{*}}|\mathbf{j}|^{1/r}|u_{\mathbf{j}}|\right)^{p}$
$\displaystyle\leq\pi^{p}_{1}(\mbox{Id}_{X})\left(me^{1+\frac{m-1}{r}}\right)^{p}\left\lVert
u\right\rVert^{p}_{q}\left\lVert
Q\right\rVert^{p}_{\mathcal{P}(^{m}\ell^{n}_{q})}\left(\left(\sum_{\mathbf{j}\in\mathcal{J}(m,n)^{*}}|\mathbf{j}||u_{\mathbf{j}}|^{r}\right)^{1/r}\left(\sum_{\mathbf{j}\in\mathcal{J}(m,n)^{*}}1\right)^{1-1/r}\right)^{p}.$

It is worth noting that

$\left(\sum_{\mathbf{j}\in\mathcal{J}(m,n)^{*}}|\mathbf{j}||u_{\mathbf{j}}|^{r}\right)^{1/r}=\left\lVert
u\right\rVert^{m-1}_{r}.$

Hence, we obtain

$\sum_{\mathbf{j}\in\mathcal{J}(m,n)}\left\lVert
x_{\mathbf{j}}(Q)\right\rVert^{p}|u_{\mathbf{j}}|^{p}\leq\pi^{p}_{1}(\mbox{Id}_{X})\left(me^{1+\frac{m-1}{r}}\right)^{p}\left\lVert
u\right\rVert^{p(m-1)}_{r}\left\lVert u\right\rVert^{p}_{q}\left\lVert
Q\right\rVert^{p}_{\mathcal{P}(^{m}\ell^{n}_{q})}\left(\sum_{\mathbf{j}\in\mathcal{J}(m,n)^{*}}1\right)^{p\left(1-\frac{1}{r}\right)}.$

Using the facts that

$\mbox{card}\,\\{\mathbf{j}=(j_{1},\ldots,j_{m-1}):1\leq j_{1}\leq\cdots\leq
j_{m-1}\leq n\\}=\binom{n+m-2}{n-1}$

and

$\binom{n+m-2}{n-1}\leq e^{m}\left(1+\frac{n}{m}\right)^{m-1}\,\,\,\mbox{for
all}\,\,n,m\in\mathbb{N},$

finally we have the following inequality

$\sum_{\mathbf{j}\in\mathcal{J}(m,n)}\left\lVert
x_{\mathbf{j}}(Q)\right\rVert^{p}|u_{\mathbf{j}}|^{p}\leq\pi^{p}_{1}(\mbox{Id}_{X})\left(me^{m+\frac{1}{r^{\prime}}}\right)^{p}\left(1+\frac{n}{m}\right)^{\frac{p(m-1)}{r^{\prime}}}\left\lVert
u\right\rVert^{p(m-1)}_{r}\left\lVert u\right\rVert^{p}_{q}\left\lVert
Q\right\rVert^{p}_{\mathcal{P}(^{m}\ell^{n}_{q})}.$

For $q\leq r$, we have $\left\lVert u\right\rVert_{r}\leq\left\lVert
u\right\rVert_{q}$. Then, we obtain

(5.4) $\sum_{\mathbf{j}\in\mathcal{J}(m,n)}\left\lVert
x_{\mathbf{j}}(Q)\right\rVert^{p}|u_{\mathbf{j}}|^{p}\leq\pi^{p}_{1}(\mbox{Id}_{X})\left(me^{m+\frac{1}{r^{\prime}}}\right)^{p}\left(1+\frac{n}{m}\right)^{\frac{p(m-1)}{r^{\prime}}}\left\lVert
u\right\rVert^{mp}_{q}\left\lVert
Q\right\rVert^{p}_{\mathcal{P}(^{m}\ell^{n}_{q})}$

and hence, in view of (4.1) and (5.4), it follows that

(5.5)
$K^{p}_{m}(B_{\ell^{n}_{q}},X,\lambda)\geq\sqrt[mp]{\lambda}\left(\pi_{1}(\mbox{Id}_{X})\,me^{m+\frac{1}{r^{\prime}}}\right)^{-1/m}\left(1+\frac{n}{m}\right)^{-\frac{m-1}{mr^{\prime}}}.$

On the other hand, we have

$\left\lVert u\right\rVert_{r}\leq n^{\frac{1}{r}-\frac{1}{q}}\left\lVert
u\right\rVert_{q}\,\,\,\mbox{for}\,\,r<q.$

Thus, we obtain

$\sum_{\mathbf{j}\in\mathcal{J}(m,n)}\left\lVert
x_{\mathbf{j}}(Q)\right\rVert^{p}|u_{\mathbf{j}}|^{p}\leq\pi^{p}_{1}(\mbox{Id}_{X})\left(me^{m+\frac{1}{r^{\prime}}}\right)^{p}\left(1+\frac{n}{m}\right)^{\frac{p(m-1)}{r^{\prime}}}\left(n^{\frac{1}{r}-\frac{1}{q}}\right)^{p(m-1)}\left\lVert
u\right\rVert^{mp}_{q}\left\lVert
Q\right\rVert^{p}_{\mathcal{P}(^{m}\ell^{n}_{q})}$

and hence (4.1) gives

(5.6)
$K^{p}_{m}(B_{\ell^{n}_{q}},X,\lambda)\geq\sqrt[mp]{\lambda}\left(\pi_{1}(\mbox{Id}_{X})\,me^{m+\frac{1}{r^{\prime}}}\right)^{-1/m}\left(n^{\frac{1}{r}-\frac{1}{q}}\right)^{-\frac{m-1}{m}}\left(1+\frac{n}{m}\right)^{-\frac{m-1}{mr^{\prime}}}.$

From (5.5) and (5.6), we observe that there exists a constant $E(X)$,
depending on $X$, such that

$K^{p}_{m}(B_{\ell^{n}_{q}},X,\lambda)\geq\left\\{\begin{array}[]{ll}E(X)\sqrt[mp]{\lambda}\left(1+\frac{n}{m}\right)^{-\frac{m-1}{mr^{\prime}}},&\mbox{for
\, $q\leq r$}\\\\[14.22636pt]
E(X)\sqrt[mp]{\lambda}\left(n^{\frac{1}{r}-\frac{1}{q}}\right)^{-\frac{m-1}{m}}\left(1+\frac{n}{m}\right)^{-\frac{m-1}{mr^{\prime}}},&\mbox{for
\, $r<q$}\end{array}\right.$

for all $1\leq p<\infty$ and $m\in\mathbb{N}$. The concluding part of this
proof lies on the minimizing the right side of this inequality in $m$. We
shall use the following important estimate

$\left(1+\frac{n}{m}\right)^{-\frac{m-1}{m}}\geq
2^{-\frac{m-1}{m}}\min\left\\{1,\left(\frac{n}{m}\right)^{-\frac{m-1}{m}}\right\\}\geq\frac{1}{2}\min\left\\{1,\frac{mn^{1/m}}{nm^{1/m}}\right\\}.$

Now, we observe that the function $h:(0,\infty)\rightarrow\mathbb{R}$ defined
by $h(x)=xn^{1/x}$ attains a minimum value $e\log n$ at $x=\log n$. This
implies that

$\left(1+\frac{n}{m}\right)^{-\frac{m-1}{m}}\geq\frac{1}{2}\left(\frac{\log
n}{n}\right).$

Therefore, there exists a constant $C(X)$ such that

$\inf_{m\in\mathbb{N}}K^{p}_{m}(B_{\ell^{n}_{q}},X,\lambda)\geq\left\\{\begin{array}[]{ll}C(X)\,\left(\frac{\log
n}{n}\right)^{\frac{1}{r^{\prime}}},&\mbox{for \, $q\leq r$}\\\\[14.22636pt]
C(X)\,n^{\frac{1}{q}-\frac{1}{r}}\left(\frac{\log
n}{n}\right)^{\frac{1}{r^{\prime}}},&\mbox{for \, $r<q$}.\end{array}\right.$

This completes the proof. ∎

Proof of Theorem 3.2. At first, we establish the claim of the upper bound for
$K^{p}(B_{\ell^{n}_{q}},X,\lambda)$. To prove this, we need an important
result by Maurey and Pisier [26, Theorem 14.5]: For every infinite dimensional
Banach space $X$,

$\inf\left\\{2\leq r\leq\infty:X\,\,\mbox{has
cotype}\,\,r\right\\}=\sup\left\\{2\leq r\leq\infty:X\,\,\mbox{finitely
factors}\,\,\ell_{r}\hookrightarrow\ell_{\infty}\right\\}.$

We say that $X$ finitely factors $\ell_{r}\hookrightarrow\ell_{\infty}$ if for
every $\epsilon>0$ and $n\in\mathbb{N}$, there exist $x_{1},\ldots,x_{n}\in X$
such that

(5.7) $\frac{1}{1+\epsilon}\left\lVert
z\right\rVert_{\infty}\leq\left\lVert\sum_{j=1}^{n}z_{j}x_{j}\right\rVert\leq\left\lVert
z\right\rVert_{r}$

for all $z=(z_{1},\ldots,z_{n})\in\mathbb{C}^{n}$. In particular, by
considering $z=e_{j}=(0,\ldots,0,1,0,\ldots,0)$, (5.7) reduces to

$\frac{1}{1+\epsilon}=\frac{1}{1+\epsilon}\left\lVert
e_{j}\right\rVert_{\infty}\leq\left\lVert x_{j}\right\rVert,\,\,\,1\leq j\leq
n.$

Hence, for a given $\epsilon>0$, there exist $x_{1},\ldots,x_{n}\in X$ such
that

(5.8) $\frac{n}{(1+\epsilon)^{p}}\leq\sum_{j=1}^{n}\left\lVert
x_{j}\right\rVert^{p},\,\,1\leq p<\infty.$

In view of (5.7) and the definition of $K^{p}_{1}(B_{\ell^{n}_{q}},X)$ for
$1$-homogeneous polynomial in $X$, for any $1\leq p<\infty$ we obtain

$\displaystyle\sum_{j=1}^{n}\left\lVert
x_{j}\right\rVert^{p}\frac{1}{n^{p/q}}$
$\displaystyle\leq\frac{1}{\left(K^{p}_{1}(B_{\ell^{n}_{q}},X)\right)^{p}}\sup_{z\in
B_{\ell^{n}_{q}}}\left\lVert\sum_{j=1}^{n}z_{j}x_{j}\right\rVert^{p}$ (5.9)
$\displaystyle\leq\frac{1}{\left(K^{p}_{1}(B_{\ell^{n}_{q}},X)\right)^{p}}\sup_{z\in
B_{\ell^{n}_{q}}}\left\lVert z\right\rVert^{p}_{\text{Cot}(X)}.$

Now, we note that if $q\leq\text{Cot}(X)$ then $\sup_{z\in
B_{\ell^{n}_{q}}}\left\lVert z\right\rVert_{\text{Cot}(X)}\leq 1$. Then, using
(5.8) and (5), we have that

$\left(K^{p}_{1}(B_{\ell^{n}_{q}},X)\right)^{p}\leq\frac{(1+\epsilon)^{p}}{n^{1-\frac{p}{q}}}$

for every $1\leq p<\infty$ and for all $\epsilon>0$. Thus, we have

$K^{p}_{1}(B_{\ell^{n}_{q}},X)\leq\frac{1}{n^{\frac{1}{p}-\frac{1}{q}}},\,\,1\leq
p<\infty.$

On the other hand, if $q>\text{Cot}(X)=r$ then we have

$\left\lVert z\right\rVert_{\text{Cot}(X)}\leq
n^{\frac{1}{\text{Cot}(X)}-\frac{1}{q}}\left\lVert
z\right\rVert_{q}.\quad(\mbox{ We assume the
convention}\,\,n^{\frac{1}{\infty}}=1)$

In view of (5.8) and (5), for any $1\leq p<\infty$ it is easy to see that

$\left(K^{p}_{1}(B_{\ell^{n}_{q}},X)\right)^{p}\leq\frac{(1+\epsilon)^{p}}{n^{1-\frac{p}{q}-\frac{1}{\text{Cot}(X)}+\frac{1}{q}}}$

holds for all $\epsilon>0$. Consequently, it follows that

$K^{p}_{1}(B_{\ell^{n}_{q}},X)\leq\frac{1}{n^{\frac{1}{p}\left(1-\frac{1}{\text{Cot}(X)}\right)-\frac{1}{q}\left(1-\frac{1}{p}\right)}}.$

Now, the conclusion follows immediately from the following observation:

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

By the virtue of Corollary 3.2, the lower bound for
$K^{p}(B_{\ell^{n}_{q}},X,\lambda)$ has already been discussed for complex
Banach spaces with no finite cotype. Suppose that $X$ has cotype $r$. To prove
the lower bound of $K^{p}(B_{\ell^{n}_{q}},X,\lambda)$ in this case, we use
the following technical result by Bombal [13, Theorem 3.2]: Given an
$m$-homogeneous polynomial $Q:\mathbb{C}^{n}\rightarrow X$,
$Q(z)=\sum_{|\alpha|=m}x_{\alpha}z^{\alpha}$, we denote by
$H:\mathbb{C}^{n}\times\cdots\times\mathbb{C}^{n}\rightarrow X$ the unique
symmetric $m$-linear mapping associated to $Q$. Then we have

$\left(\sum_{i\in\mathcal{M}(m,n)}\left\lVert
H(e_{i_{1}},\dots,e_{i_{m}})\right\rVert^{r}\right)^{\frac{1}{r}}\leq
C_{r}(X)^{m}\left\lVert H\right\rVert_{B_{\ell^{n}_{q}}}.$

Let us consider $s>1$ be such that $1/r+1/s=1$. Then, by applying Hölder’s
inequality, for all $z\in\mathbb{C}^{n}$, we obtain

$\displaystyle\sum_{\alpha\in\Lambda(m,n)}\left\lVert
x_{\alpha}z^{\alpha}\right\rVert^{p}$
$\displaystyle=\sum_{\mathbf{j}\in\mathcal{J}(m,n)}\text{card}[\mathbf{j}]^{p}\left\lVert
H(e_{j_{1}},\ldots,e_{j_{m}})\right\rVert^{p}|z_{j_{1}}\cdots z_{j_{m}}|^{p}$
$\displaystyle=\sum_{i\in\mathcal{M}(m,n)}\left\lVert
H(e_{i_{1}},\ldots,e_{i_{m}})\right\rVert^{p}|z_{i_{1}}\cdots z_{i_{m}}|^{p}$
$\displaystyle\leq\left(\sum_{i\in\mathcal{M}(m,n)}\left\lVert
H(e_{i_{1}},\ldots,e_{i_{m}})\right\rVert|z_{i_{1}}\cdots
z_{i_{m}}|\right)^{p}$
$\displaystyle\leq\left[\left(\sum_{i\in\mathcal{M}(m,n)}\left\lVert
H(e_{i_{1}},\ldots,e_{i_{m}})\right\rVert^{r}\right)^{\frac{1}{r}}\left(\sum_{i\in\mathcal{M}(m,n)}|z_{i_{1}}\cdots
z_{i_{m}}|^{s}\right)^{\frac{1}{s}}\right]^{p}$ $\displaystyle\leq
C_{r}(X)^{mp}\left\lVert
H\right\rVert^{p}_{B_{\ell^{n}_{q}}}\left(\sum_{i\in\mathcal{M}(m,n)}|z_{i_{1}}\cdots
z_{i_{m}}|^{s}\right)^{\frac{p}{s}}$ $\displaystyle=C_{r}(X)^{mp}\left\lVert
H\right\rVert^{p}_{B_{\ell^{n}_{q}}}\left(|z_{1}|^{s}+\cdots+|z_{n}|^{s}\right)^{\frac{mp}{s}}.$

Now, for $q\leq s$, we have $\left\lVert z\right\rVert_{s}\leq\left\lVert
z\right\rVert_{q}$, whereas

$\left\lVert z\right\rVert_{s}\leq n^{\frac{1}{s}-\frac{1}{q}}\left\lVert
z\right\rVert_{q}\,\,\,\mbox{for}\,\,s<q.$

Hence, we obtain

$\sum_{\alpha\in\Lambda(m,n)}\left\lVert
x_{\alpha}z^{\alpha}\right\rVert^{p}\leq\left\\{\begin{array}[]{ll}C_{r}(X)^{mp}\left\lVert
H\right\rVert^{p}_{B_{\ell^{n}_{q}}}\left\lVert
z\right\rVert^{mp}_{q},&\mbox{for \, $q\leq s$}\\\\[14.22636pt]
C_{r}(X)^{mp}\left(n^{\frac{1}{s}-\frac{1}{q}}\right)^{mp}\left\lVert
H\right\rVert^{p}_{B_{\ell^{n}_{q}}}\left\lVert
z\right\rVert^{mp}_{q},&\mbox{for \, $q>s$}.\end{array}\right.$

Therefore, by the virtue of polarization formula for multilinear forms (see
e.g. [22, Section 2.6]), for all $z\in B_{\ell^{n}_{q}}$ we obtain

$\sum_{\alpha\in\Lambda(m,n)}\left\lVert
x_{\alpha}z^{\alpha}\right\rVert^{p}\leq\left\\{\begin{array}[]{ll}C_{r}(X)^{mp}\left(\frac{m^{m}}{m!}\right)^{p}\left\lVert
Q\right\rVert^{p}_{B_{\ell^{n}_{q}}},&\mbox{for \, $q\leq s$}\\\\[14.22636pt]
C_{r}(X)^{mp}\left(n^{\frac{1}{s}-\frac{1}{q}}\right)^{mp}\left(\frac{m^{m}}{m!}\right)^{p}\left\lVert
Q\right\rVert^{p}_{B_{\ell^{n}_{q}}},&\mbox{for \, $q>s$}\end{array}\right.$

and whence,

$\sum_{\alpha\in\Lambda(m,n)}\left\lVert
x_{\alpha}z^{\alpha}\right\rVert^{p}\leq\left\\{\begin{array}[]{ll}C_{r}(X)^{mp}e^{mp}\left\lVert
Q\right\rVert^{p}_{B_{\ell^{n}_{q}}},&\mbox{for \, $q\leq s$}\\\\[14.22636pt]
C_{r}(X)^{mp}\left(n^{\frac{1}{s}-\frac{1}{q}}\right)^{mp}e^{mp}\left\lVert
Q\right\rVert^{p}_{B_{\ell^{n}_{q}}},&\mbox{for \, $q>s$}.\end{array}\right.$

In view of Definition 4.1, for every $m\in\mathbb{N}$ and $1\leq p<\infty$ we
have

$K^{p}_{m}(B_{\ell^{n}_{q}},X)\leq\left\\{\begin{array}[]{ll}\frac{1}{e\,C_{r}(X)},&\mbox{for
\, $q\leq s$}\\\\[14.22636pt]
\frac{1}{e\,C_{r}(X)n^{\frac{1}{s}-\frac{1}{q}}},&\mbox{for \,
$q>s$}.\end{array}\right.$

Finally, by applying Lemma 4.4(b), for every $\lambda>1$, $1\leq p<\infty$,
and $1\leq q\leq\infty$, we obtain

$K^{p}(B_{\ell^{n}_{q}},X,\lambda)\leq\left\\{\begin{array}[]{ll}\left(\frac{\lambda-1}{\lambda}\right)^{1/p}\frac{1}{e\,C_{r}(X)},&\mbox{for
\, $q\leq s$}\\\\[14.22636pt]
\left(\frac{\lambda-1}{\lambda}\right)^{1/p}\frac{1}{e\,C_{r}(X)n^{\frac{1}{s}-\frac{1}{q}}},&\mbox{for
\, $q>s$}.\end{array}\right.$

This completes the proof. ∎

Proof of Theorem 3.4. We note that the proof of (1) is just a consequence of
(4.3) and Theorem 3.2. With the help of these observations, we obtain that

$\displaystyle K^{p}(B_{\ell^{n}_{q}},T,\lambda)$
$\displaystyle\geq\max\left\\{K^{p}(B_{\ell^{n}_{q}},X,\lambda/\left\lVert
T\right\rVert^{p}),K^{p}(B_{\ell^{n}_{q}},Y,\lambda/\left\lVert
T\right\rVert^{p})\right\\}$
$\displaystyle\geq\begin{cases}\left(\frac{\lambda-\left\lVert
T\right\rVert^{p}}{\lambda}\right)^{\frac{1}{p}}\frac{1}{e\,\min\\{C_{r}(X),C_{r}(Y)\\}}&\text{if
}q\leq s\\\\[14.22636pt] \left(\frac{\lambda-\left\lVert
T\right\rVert^{p}}{\lambda}\right)^{\frac{1}{p}}\frac{1}{e\,\min\\{C_{r}(X),C_{r}(Y)\\}}\left(\frac{1}{n}\right)^{\frac{1}{s}-\frac{1}{q}}&\text{if
}q>s,\end{cases}$

where $s=r/(r-1)$. Hence, the result follows.

Next, for the proof of (2), we partly follow the techniques of the proof of
Theorem 3.1 and hypercontractivity of vector-valued Bohnenblust-Hille type
inequality. Applying Theorem 3.3 and Hölder’s inequality, there is a constant
$\widetilde{C}>0$ such that for every $m$-homogeneous polynomial
$Q(z)=\sum_{\alpha\in\Lambda(m,n)}x_{\alpha}z^{\alpha}\in\mathcal{P}(^{m}\mathbb{C}^{n},X)$,
we have

$\displaystyle\sum_{\alpha\in\Lambda(m,n)}\left\lVert
T(x_{\alpha})z^{\alpha}\right\rVert^{p}$
$\displaystyle\leq\left[\left(\sum_{\alpha\in\Lambda(m,n)}1\right)^{\frac{(r-1)ms-r+s}{rms}}\times\left(\sum_{\alpha\in\Lambda(m,n)}\left\lVert
T(x_{\alpha})z^{\alpha}\right\rVert^{\frac{rsm}{r+(m-1)s}}\right)^{\frac{r+(m-1)s}{rsm}}\right]^{p}$
$\displaystyle\leq\left(\sum_{\alpha\in\Lambda(m,n)}1\right)^{\frac{((r-1)ms-r+s)p}{rms}}\widetilde{C}^{mp}\left\lVert
Q\right\rVert^{p}_{\ell^{n}_{q}}.$

We observe that

(5.10) $\sum_{\alpha\in\Lambda(m,n)}1=\binom{n+m-1}{m}\leq
e^{m}\left(1+\frac{n}{m}\right)^{m}.$

By the virtue of (4.1) and (4.2) and using the fact (5.10), there is a
constant $E>0$ such that

(5.11) $K^{p}_{m}(B_{\ell^{n}_{q}},T,\lambda)\geq
E\,\lambda^{\frac{1}{mp}}\left(1+\frac{n}{m}\right)^{-\frac{(r-1)ms-r+s}{rms}}$

for every $m$, $1\leq p<\infty$, and $\left\lVert
T\right\rVert<\lambda^{1/p}$. By minimizing the right side of (5.11), as
demonstrated in the proof of Theorem 3.1, we observe that there is a constant
$D>0$ such that for $1\leq p<\infty$ and $\left\lVert
T\right\rVert<\lambda^{1/p}$

$\inf_{m\in\mathbb{N}}\left\\{K^{p}_{m}(B_{\ell^{n}_{q}},T,\lambda)\geq
C\,\left(\frac{\log n}{n}\right)^{1-\frac{1}{r}}\right\\}.$

Finally, using Lemma 4.4(a), we obtain desired conclusion. This completes the
proof. ∎

Acknowledgment: The research of the second named author is supported by DST-
INSPIRE Fellowship (IF 190721), Department of Science and Technology, New
Delhi, India.

## References

  * [1] L. Aizenberg, Multidimensional analogues of Bohr’s theorem on power series, Proc. Amer. Math. Soc. 128 (2000), 1147–1155.
  * [2] L. Aizenberg, A. Aytuna, and P. Djakov, An abstract approach to Bohr’s phenomenon, Proc. Amer. Math. Soc. 128 (2000), 2611–2619.
  * [3] 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.
  * [4] V. Allu, H. Halder, and S. Pal, Multidimensional Bohr radii for vector-valued holomorphic functions, see https://arxiv.org/abs/2308.07825 (2023).
  * [5] F. Bayart, D. Pellegrino, and J. B. Seoane-Sep$\rm\acute{U}$lveda, The Bohr radius of the $n$-dimensional polydisk is equivalent to $\sqrt{(log\,n)/n}$, Adv. Math. 264 (2014), 726–746.
  * [6] F. Bayart, A. Defant, and S. Schlüters, Monomial convergence for holomorphic functions on $\ell_{r}$, J. Anal. Math. 138 (2019), 107–134.
  * [7] C. Bénéteau, A. Dahlner and D. Khavinson, Remarks on the Bohr phenomenon, Comput. Methods Funct. Theory 4(1) (2004), 1–19.
  * [8] O. Blasco, The Bohr radius of a Banach space, In Vector measures, integration and related topics, 5964, Oper. Theory Adv. Appl., 201, Birkhäuser Verlag, Basel, 2010.
  * [9] O. Blasco, The $p$-Bohr radius of a Banach space, Collect. Math. 68 (2017), 87–100.
  * [10] H. P. Boas and D. Khavinson, Bohr’s power series theorem in several variables, Proc. Amer. Math. Soc. 125 (1997), 2975–2979.
  * [11] H. P. Boas, Majorant Series, J. Korean Math. Soc. 37 (2000), 321–337.
  * [12] H. Bohr, A theorem concerning power series, Proc. Lond. Math. Soc. s2-13 (1914), 1–5.
  * [13] F. Bombal, D. Pérez-García and I. Villanueva, Multilinear extensions of Grothendieck’s theorem, Q. J. Math 55 (2004), 441–450.
  * [14] E. Bombieri, Sopra un teorema di H. Bohr e G. Ricci sulle funzioni maggioranti delle serie di potenze, Boll. Un. Mat. Ital. 17 (1962), 276–282.
  * [15] E. Bombieri and J. Bourgain, A remark on Bohr’s inequality, Internat. Math. Res. Notices 80 (2004), 4307–4330.
  * [16] N. Das, Estimates for generalized Bohr radii in one and higher dimensions, Canad. Math. Bull. 66 (2023), 682–699.
  * [17] N. Das, The $p$-Bohr radius for vector-valued holomorphic and pluriharmonic functions, Forum Math. 36 (2024), 765–782.
  * [18] A. Defant and L. Frerick, A logarithmic lower bound for multi-dimenional Bohr radii, Israel J. Math. 152 (2006), 17–28.
  * [19] A. Defant and L. Frerick, The Bohr radius of the unit ball of $\ell^{n}_{p}$, J. Reine Angew. Math. 660 (2011), 131–147.
  * [20] A. Defant, L. Frerick, J. Ortega-Cerd${\rm\grave{A}}$, M. Ounaïes, and K. Seip, The Bohnenblust-Hille inequality for homogeneous polynomils in hypercontractive, Ann. of Math. 174 (2011), 512–517.
  * [21] A. Defant, D. García, and M. Maestre, Bohr power series theorem and local Banach space theory, J. Reine Angew. Math. 557 (2003), 173–197.
  * [22] A. Defant, D. García, M. Maestre, and P. Sevilla-Peris, Dirichlet Series and Holomorphic Functions in High Dimensions, New Mathematical Monographs: 37, Cambridge University Press, Cambridge, (2019).
  * [23] A. Defant, M. Maestre, and C. Prengel, The arithmetic Bohr radius, Q. J. Math. 59 (2008), 189–205.
  * [24] A. Defant, M. Maestre, and C. Prengel, Domains of convergence for monomial expansions of holomorphic functions in infinitely many variables, J. Reine Angew. Math. 634 (2009), 13–49.
  * [25] A. Defant, M. Maestre and U. Schwarting, Bohr radii of vector valued holomorphic functions, Adv. Math. 231 (2012), 2837–2857.
  * [26] J. Diestel, H. Jarchow, and A. Tonge, Absolutely Summing Operators, in: Cambridge Studies in Advance Mathematics 43, Cambridge University Press, Cambridge, (1995).
  * [27] P. G. Dixon, Banach algebras satisfying the non-unital von Neumann inequality, Bull. Lond. Math. Soc. 27 (4) (1995), 359–362.
  * [28] P. B. Djakov and M. S. Ramanujan, A remark on Bohr’s theorem and its generalizations, J. Anal. 8 (2000), 65–77.
  * [29] S. Dineen and R. M. Timoney, Absolute bases, tensor products and a theorem of Bohr, Studia Math. 94 (1989), 227–234.
  * [30] D. Galicer, M. Mansilla and S. Muro, Mixed Bohr radius in several variables, Trans. Amer. Math. Soc. 373 (2020), 777–796.
  * [31] H. Hamada, T. Honda and G. Kohr, Bohr’s theorem for holomorphic mappings with values in homogeneous balls, Israel J. Math. 173 (2009), 177–187.
  * [32] S. Kumar and R. Manna, Multi-dimensional Bohr radii of Banach space valued holomorphic functions, see https://arxiv.org/pdf/2303.17416 (2023).
  * [33] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I: Sequence Spaces, in: Ergebnisse der Mathematik und ihrer Grenzgebiete, 92, Springer-Verlag, Berlin-Heidelberg-New York, (1977).
  * [34] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II: Function Spaces, in: Ergebnisse der Mathematik und ihrer Grenzgebiete, 97, Springer-Verlag, Berlin-Heidelberg-New York, (1979).
  * [35] V. I. Paulsen, G. Popescu and D. Singh, On Bohr’s inequality, Proc. Lond. Math. Soc. s3-85 (2002), 493–512.
  * [36] G. Popescu, Bohr inequalities for free holomorphic functions on polyballs, Adv. Math. 347 (2019), 1002–1053.
  * [37] S. Sidon, Uber einen satz von Hernn Bohr, Math. Zeit. 26 (1927), 731–732.
  * [38] M. Tomic, Sur un theoreme de H. Bohr, Math. Scand. 11 (1962), 103–106.