# ORIENT: A Priority-Aware Energy-Efficient Approach for Latency-Sensitive
Applications in 6G

Masoud Shokrnezhad1, and Tarik Taleb1, 2
1Oulu University, Oulu, Finland; {masoud.shokrnezhad<EMAIL_ADDRESS>
2Ruhr University Bochum, Bochum, Germany<EMAIL_ADDRESS>
###### Abstract

Anticipation for 6G’s arrival comes with growing concerns about increased
energy consumption in computing and networking. The expected surge in
connected devices and resource-demanding applications presents unprecedented
challenges for energy resources. While sustainable resource allocation
strategies have been discussed in the past, these efforts have primarily
focused on single-domain orchestration or ignored the unique requirements
posed by 6G. To address this gap, we investigate the joint problem of service
instance placement and assignment, path selection, and request prioritization,
dubbed PIRA. The objective function is to maximize the system’s overall profit
as a function of the number of concurrently supported requests while
simultaneously minimizing energy consumption over an extended period of time.
In addition, end-to-end latency requirements and resource capacity constraints
are considered for computing and networking resources, where queuing theory is
utilized to estimate the Age of Information (AoI) for requests. After
formulating the problem in a non-linear fashion, we prove its NP-hardness and
propose a method, denoted ORIENT. This method is based on the Double Dueling
Deep Q-Learning (D3QL) mechanism and leverages Graph Neural Networks (GNNs)
for state encoding. Extensive numerical simulations demonstrate that ORIENT
yields near-optimal solutions for varying system sizes and request counts.

###### Index Terms:

6G, Resource Allocation, Energy Consumption, Service Placement and Assignment,
Path Selection, Prioritization, E2E Latency, Age of Information (AoI),
Reinforcement Learning, Q-Learning, and Graph Neural Networks (GNNs).

##  I Introduction

The advent of the 6th generation of telecommunication systems (6G) signifies a
pivotal era marked by unparalleled connectivity and technological
advancements. With ultra-low End-to-End (E2E) latency (less than $1$
milisecond), exceeding $1$ terabit per second peak data rates, and ultra-high
reliability surpassing $99.99999\%$ [1], 6G promises to revolutionize
industries such as holographic telepresence utilizing extended reality [2],
dynamic metaverse empowered by semantic communications [3], and quantum
networking [4]. However, achieving these capabilities raises substantial
energy consumption concerns for both computing and networking resources.
Presently, these resources consume around $200$ terawatt-hours of electricity
annually, approximately $1\%$ of the global total [5]. Many quality-sensitive
applications may require uploading up to $50\%$ of data to computing
facilities for processing [6], adding even more strain to computing and
networking resources. Moreover, the projected surge in 6G-connected devices
and global data exacerbates the energy consumption challenge, underscoring the
need for sustainable solutions.

In order to realize a 6G-enabled future, it may be necessary to create novel
resource orchestration mechanisms to address impending energy challenges. The
subject has been extensively studied in the literature. Xuan et al. [7]
addressed the Service Function Chaining (SFC) problem with the objective of
minimizing energy consumption by proposing an algorithm based on multi-agent
Reinforcement Learning (RL) and a self-adaptive division strategy. Solozabal
et al. [8] investigated the same problem and proposed a single-agent solution.
Other authors have also examined the SFC problem. By proposing a sampling-
based Markov approximation method, Pham et al. [9] solved the problem in an
effort to minimize operational and traffic energy consumption. Santos et al.
[10] developed two policy-aware RL algorithms based on actor-critic and
proximal policy optimization to maximize availability while minimizing energy
consumption. Reducing energy consumption was considered in the Service
Function (SF) placement problem as well. Sasan et al. [11] presented a
heuristic algorithm to tackle the joint problem of network slicing, path
selection, and SF placement, with the objective of maximizing user acceptance
while minimizing energy consumption. Farhoudi [12] and He et al. [13]
investigated a comparable problem and proposed RL-based solutions, taking into
account the dynamic nature of service requests and overall cost considerations
(including operation, deployment, and transmission), respectively.

While effective in specific contexts, the mentioned methods may not be
suitable for 6G systems. These approaches prioritize energy efficiency over
maximizing device support, whereas achieving an E2E efficient solution
requires holistic management of computing and networking resources,
considering the stringent Quality of Service (QoS) demands of 6G. Furthermore,
certain studies overlook or oversimplify critical network parameters like
latency, which contradicts the intricate and dynamic requirements of 6G
systems. This paper addresses this gap by investigating the joint problem of
allocating computing and networking resources (service instance placement and
assignment, path selection, and request prioritization), termed PIRA. The
objective is to optimize the system’s overall profit (as a function of
supported concurrent requests) while minimizing energy consumption over time,
accounting for E2E latency and resource capacity constraints. The $M/M/1$
queuing model is employed to accurately evaluate request latency on compute
nodes and network devices. To solve this problem, we propose ORIENT, an
approach leveraging Double Dueling Deep Q-Learning (D3QL) reinforced by Graph
Neural Networks (GNNs). This hybrid method effectively encodes the system
state and facilitates the identification of near-optimal solutions.

The remainder of this paper is organized as follows. Section II introduces the
system model. PIRA is defined and formulated in Section III, and ORIENT is
presented in Section IV. Finally, numerical results are illustrated in Section
V, followed by concluding remarks in Section VI.

##  II System Model

As shown in Fig. 1, the following is an explanation of the two main components
of the system: resources and requests.

Figure 1: The system model, including network devices and distributed compute
nodes facilitating holographic telepresence services for end users.

###  II-A Resources

The 6G system examined in this paper is an integrated infrastructure of
computing and networking resources comprised of $\mathcal{N}$ network devices
and $\mathcal{V}$ compute nodes (radio resources are excluded) [14].
$\mathbb{N}=\\{n|0\leq n\leq\mathcal{N}\\}$ is the set of network devices, and
$\mathbb{V}=\\{v|0\leq v\leq\mathcal{V}\\}$ denotes the set of compute nodes.
Compute nodes are connected through network devices via $\mathcal{P}$ paths,
the set of which is denoted by $\mathbb{P}=\\{p|0\leq p\leq\mathcal{P}\\}$,
and the immediate network device of compute node $v$ is indicated by $n_{v}$.
Each path $p$ contains a set of links $\mathbb{L}_{p}\subset\mathbb{L}$, where
$\mathbb{L}=\\{l:(n,n^{\prime})|n,n^{\prime}\in\mathbb{N}\\}$ is the set of
all network links, and $\mathcal{L}$ is its size. Network devices and compute
nodes are priority-aware, i.e., $\mathbb{K}=\\{k|0\leq k\leq\mathcal{K}\\}$ is
regarded as the set of permissible priority levels (where lower levels
indicate higher priorities), and the resources in both domains are virtually
partitioned, isolated, and guaranteed for each priority level $k$. Note that
higher priorities receive a larger share of available resources than lower
priorities.

To evaluate the performance of allocated resources, we will employ the $M/M/1$
queuing model for each priority level on network devices and compute nodes
assuming that this theory’s stability requirements are met and that all queues
are independent. The service rate allocated to priority level $k$ on network
device $n$ is $\widehat{\mathcal{B}_{n,k}}$, and those packets leaving this
queue will be forwarded through their corresponding link, let’s call it $l$,
allocating $\widehat{\mathcal{B}_{l,k}}$ bandwidth. Note that the overall
capacity of this network device and link is constrained by
$\widehat{\mathcal{B}_{n}}$ and
$\widehat{\mathcal{B}_{l}{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>respectively. Similarly, the requests of priority level $k$ will be served on
compute node $v$ leveraging a queue with dedicated service rate
$\widehat{\mathcal{C}_{v,k}}$, and this node is equipped with computing
resources limited to a predefined capacity threshold, dubbed
$\widehat{\mathcal{C}_{v}}$. In addition, the energy consumption for
transmitting bandwidth units over network device $n$ is
$\widetilde{\mathcal{E}_{n}}$, and compute node $v$ consumes
$\widetilde{\mathcal{E}_{v}}$ energy per capacity unit and
$\overline{\mathcal{E}_{v}}$ energy when its state changes (from the idle mode
to the operation mode or vice versa).

###  II-B Requests

This paper investigates the system for $\mathcal{T}$ time slots while a set of
$\mathcal{R}_{t}$ requests, denoted by $\mathbb{R}_{t}=\\{r|0\leq
r\leq\mathcal{R}_{t}\\}$, arrives at time slot $t\in\mathbb{T}=\\{t|1\leq
t\leq\mathcal{T}\\}$. The set of all requests is
$\mathbb{R}=\\{\mathbb{R}_{t}|1\leq t\leq\mathcal{T}\\}$, and $\mathcal{R}$
represents the number of all requests. Each request $r$ enters the system
through an edge network device, denoted by $n_{r}$ and referred to as its
Point of Arrival (PoA), and orders service $s_{r}$ from the set of obtainable
service instances, that is $\mathbb{I}=\\{i|0\leq i\leq\mathcal{I}\\}$, where
instance $i$ provides service $s_{i}$. In order to successfully fulfill each
request, one instance of its target service must be replicated on one of the
compute nodes in order to receive the request, process it, and return it to
its entry point so that it can be delivered to the end user.
$\widehat{\mathcal{C}_{i{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>represents the maximum capacity of instance $i$. To fulfill each user’s
request, its QoS requirements must be met, including the minimum service
capacity and network bandwidth, as well as the maximum tolerable E2E latency,
denoted by
$\widecheck{\mathcal{C}_{r{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>$\widecheck{\mathcal{B}_{r}}$, and $\widecheck{\mathcal{D}_{r}}$,
respectively. Besides, the maximum permissible packet size for request $r$ is
$\widehat{\mathcal{H}_{r}}$. If request $r$ is successfully completed, the
system will achieve a profit, that is $\gamma_{r}$. Note that the arrival rate
for each queue is determined by a Poisson process and is assumed to be the sum
of
$\widecheck{\mathcal{C}_{r{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>(for compute queues) and $\widecheck{\mathcal{B}_{r}}$ (for network queues)
for all requests assigned to that queue, respectively.

##  III Problem Definition

This section discusses the joint problem of instance placement and assignment,
request prioritization, and path selection for integrated compute-network
infrastructures to maximize the overall profit of the system while minimizing
its energy consumption. In this section, the constraints and objective
function are formulated, followed by the problem statement as a Mixed-Integer
Non-Linear Programming (MINLP) formulation and its complexity analysis.

###  III-A Instance Orchestration Constraints

Constraints C1-C6 assign requests to instances and place them on compute nodes
while maintaining the capacity constraints of instances and compute nodes.
Considering that $\ddot{\mathcal{I}}^{t}_{r,i}$ is a binary variable whose
value is $1$ if request $r$ is assigned to instance $i$ at time slot $t$, C1
ensures that each request $r$ is assigned to no more than one instance of its
service for each time slot $t$. C2 defines a new binary variable,
$\dot{\mathcal{I}}^{t}_{i}$, which indicates that whether instance $i$ is
activated at time slot $t$. If
$\sum_{\mathbb{R}_{t}}\ddot{\mathcal{I}}^{t}_{r,i}$ is equal to or greater
than $1$ (i.e., at least one request is assigned to instance $i$),
$(\sum_{\mathbb{R}_{t}}\ddot{\mathcal{I}}^{t}_{r,i})/\mathcal{R}_{t}$ will be
a small number (between $0$ and $1$) and
$\sum_{\mathbb{R}_{t}}\ddot{\mathcal{I}}^{t}_{r,i}$ will be a large number, so
$\dot{\mathcal{I}}^{t}_{i}$ will be set to $1$. Otherwise, both sides of the
equation will equal $0$, causing $\dot{\mathcal{I}}^{t}_{i}$ to also equal
$0$. C3 ensures that each activated instance is assigned to exactly one
compute node, where $\ddot{\mathcal{G}}^{t}_{i,v}$ is a binary variable
indicating the compute node of instance $i$ at time slot $t$. Similar to C2,
C4 defines $\dot{\mathcal{G}}^{t}_{v}$ as a binary variable to determine
whether compute node $v$ should be activated at time slot $t$. Then, it must
be assured that assigned requests do not exceed the capacity limitations of
instances and compute nodes (C5 and C6).

$\displaystyle\sum\nolimits_{\mathbb{I}|s_{i}=s_{r}}\ddot{\mathcal{I}}^{t}_{r,i}\leq
1\quad\forall
t,r\in\mathbb{T},\mathbb{R}_{t}{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}ttttttttttttttttttttttttttttttttttt}$
(C1)
$\displaystyle\frac{1}{\mathcal{R}_{t}}\cdot\sum\nolimits_{\mathbb{R}_{t}}\ddot{\mathcal{I}}^{t}_{r,i}\leq\dot{\mathcal{I}}^{t}_{i}\leq\sum\nolimits_{\mathbb{R}_{t}}\ddot{\mathcal{I}}^{t}_{r,i}\quad\forall
t,i\in\mathbb{T},\mathbb{I}$ (C2)
$\displaystyle\sum\nolimits_{\mathbb{V}}\ddot{\mathcal{G}}^{t}_{i,v}=\dot{\mathcal{I}}^{t}_{i}\quad\forall
t,i\in\mathbb{T},\mathbb{I}$ (C3)
$\displaystyle\frac{1}{\mathcal{I}}\cdot\sum\nolimits_{\mathbb{I}}\ddot{\mathcal{G}}^{t}_{i,v}\leq\dot{\mathcal{G}}^{t}_{v}\leq\sum\nolimits_{\mathbb{I}}\ddot{\mathcal{G}}^{t}_{i,v}\quad\forall
t,v\in\mathbb{T},\mathbb{V}$ (C4)
$\displaystyle\sum\nolimits_{\mathbb{R}_{t}}\widecheck{\mathcal{C}_{r}}\cdot\ddot{\mathcal{I}}^{t}_{r,i}\leq\widehat{\mathcal{C}_{i{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>t,i\in\mathbb{T},\mathbb{I}$ (C5)
$\displaystyle\sum\nolimits_{\mathbb{I}}\widehat{\mathcal{C}_{i{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.}}}\cdot\ddot{\mathcal{G}}^{t}_{i,v}\leq\widehat{\mathcal{C}_{v}}\quad\forall
v,t\in\mathbb{V},\mathbb{T}$ (C6)

###  III-B Path Selection Constraints

Constraints C7-C9 ensure that an E2E path is selected for each request
considering the capacity constraints of network links and the traffic pattern,
where packets of each request enter the network through its PoA and, after
visiting its assigned instance, are returned to the same PoA to be handed off
to the corresponding end user. C7 determines the allocated path for each
request $r$, ensuring that it originates and terminates at $n_{r}$ and
traverses the network device directly connected to the compute node hosting
the instance assigned to the request. In this constraint, $\ddot{f}^{t}_{r,p}$
is a binary variable that represents the assigned path of request $r$ at time
slot $t$. Finally, C8 and C9 maintain the maximum capacity of network links
and devices.

$\displaystyle\sum\nolimits_{\mathbb{P}|n_{r}\&n_{v}\in
p}\ddot{f}^{t}_{r,p}=\ddot{\mathcal{I}}^{t}_{r,i}\cdot\ddot{\mathcal{G}}^{t}_{i,v}\quad\forall
t,r,i,v,\in\mathbb{T},\mathbb{R},\mathbb{I},\mathbb{V}$ (C7)
$\displaystyle\sum\nolimits_{\mathbb{R}_{t},\mathbb{P}|l\in\mathbb{L}_{p}}\widecheck{\mathcal{B}_{r}}\cdot\ddot{f}^{t}_{r,p}\leq\widehat{\mathcal{B}_{l}{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>t,l\in\mathbb{T},\mathbb{L}{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}tttttttttttttttttttttttttt}$
(C8)
$\displaystyle\sum\nolimits_{\mathbb{R}_{t},\mathbb{P}|n\in\mathbb{L}_{p}}\widecheck{\mathcal{B}_{r}}\cdot\ddot{f}^{t}_{r,p}\leq\widehat{\mathcal{B}_{n}{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>t,n\in\mathbb{T},\mathbb{N}$ (C9)

###  III-C Request Prioritization Constraints

$\displaystyle\sum\nolimits_{\mathbb{K}}\ddot{\varrho}^{t}_{r,k}=\sum\nolimits_{\mathbb{I}}\ddot{\mathcal{I}}^{t}_{r,i}\quad\forall
t,r\in\mathbb{T},\mathbb{R}_{t}{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}ttttttttttttttttttttttttttttttt}$
(C10)
$\displaystyle\sum\nolimits_{\mathbb{R}_{t},\mathbb{I}}\widecheck{\mathcal{C}_{r}}\cdot\ddot{\varrho}^{t}_{r,k}\cdot\ddot{\mathcal{I}}^{t}_{r,i}\cdot\ddot{\mathcal{G}}^{t}_{i,v}<\widehat{\mathcal{C}_{v,k}}\quad\forall
t,k,v\in\mathbb{T},\mathbb{K},\mathbb{V}$ (C11)
$\displaystyle\sum\nolimits_{\mathbb{R}_{t},\mathbb{P}|l\in\mathbb{L}_{p}}\widecheck{\mathcal{B}_{r}}\cdot\ddot{\varrho}^{t}_{r,k}\cdot\ddot{f}^{t}_{r,p}<\widehat{\mathcal{B}_{l,k}{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>t,k,l\in\mathbb{T},\mathbb{K},\mathbb{L}$ (C12)
$\displaystyle\sum\nolimits_{\mathbb{R}_{t},\mathbb{P}|n\in\mathbb{L}_{p}}\widecheck{\mathcal{B}_{r}}\cdot\ddot{\varrho}^{t}_{r,k}\cdot\ddot{f}^{t}_{r,p}<\widehat{\mathcal{B}_{n,k}{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>t,k,n\in\mathbb{T},\mathbb{K},\mathbb{N}$ (C13)

To maintain integrity, it’s crucial to prevent any overuse of resources
allocated to each priority level. Given that $\ddot{\varrho}^{t}_{r,k}$ is the
priority of request $r$ at time slot $t$, C10 promises that the request’s
priority is determined if an instance is assigned to serve it. Then, C11 to
C13 satisfy the capacity constraints of priority queues on compute nodes and
network resources.

###  III-D Latency Constraints

Each packet has to wait for three sources of latency through its request’s
assigned E2E path in the system: 1) the service latency experienced at the
network devices included in the path, 2) the transmission latency over the
network links of the path, and 3) the service latency at the assigned compute
node. Since the average latency of a packet in a $M/M/1$ queue is equal to
$1/(\mu-\lambda)$ when the arrival rate is $\lambda$ and the service rate is
$\mu$, the average latency experienced by the packets of request $r$ at
network device $n$ allocated to priority level $k$ during time slot $t$ can be
calculated as C14. In this constraint, the numerator will be $0$ for network
devices and priority levels that have not been allocated to request $r$,
causing $\ddot{\mathcal{D}}^{t}_{r,n,k}$ to equal $0$. Otherwise, the
numerator will be $1$, and the latency will be determined following the
adopted queuing theorem with the arrival rate of the queue set to the overall
bandwidth of requests assigned to priority level $k$ and traversing network
device $n$. C15 follows the same logic to calculate the average latency of
request $r$ allocated to priority level $k$ at time slot $t$ on compute node
$v$. C16 calculates the transmission latency of request $r$ over link $l$ at
time slot $t$, considering its priority level and maximum packet size, if the
link is part of the path assigned to the request. Otherwise, the latency will
be $0$. Finally, C17 determines the Age of Information (AoI), followed by C18,
which ensures the maximum acceptable latency requirement of requests.

$\displaystyle\ddot{\mathcal{D}}^{t}_{r,n,k}=\frac{\sum\nolimits_{\mathbb{P}|n\in\mathbb{L}_{p}}\ddot{\varrho}^{t}_{r,k}\cdot\ddot{f}^{t}_{r,p}}{\widehat{\mathcal{B}_{n,k}}-\sum\nolimits_{\mathbb{R}_{t},\mathbb{P}|n\in\mathbb{L}_{p}}\widecheck{\mathcal{B}_{r^{\prime}}}\cdot\ddot{\varrho}^{t}_{r^{\prime},k}\cdot\ddot{f}^{t}_{r^{\prime},p}}\quad\begin{aligned}
&\forall t,r,k,n\in\\\ &\mathbb{T},\mathbb{R}_{t},\mathbb{K},\mathbb{N}\\\
\end{aligned}$ (C14)
$\displaystyle\ddot{\mathcal{D}}^{t}_{r,v,k}=\frac{\sum\nolimits_{\mathbb{I}}\ddot{\varrho}^{t}_{r,k}\cdot\ddot{\mathcal{I}}^{t}_{r,i}\cdot\ddot{\mathcal{G}}^{t}_{i,v}}{\widehat{C_{v,k}}-\sum\nolimits_{\mathbb{R}_{t},\mathbb{I}}\widecheck{\mathcal{C}_{r^{\prime}}}\cdot\ddot{\mathcal{I}}^{t}_{r^{\prime},i}\cdot\ddot{\mathcal{G}}^{t}_{i,v}}\quad\begin{aligned}
&\forall t,r,k,v\in\\\ &\mathbb{T},\mathbb{R}_{t},\mathbb{K},\mathbb{V}\\\
\end{aligned}$ (C15)
$\displaystyle\ddot{\mathcal{D}}^{t}_{r,l,k}=\frac{\widehat{\mathcal{H}_{r}}}{\widehat{\mathcal{B}_{l,k}}}\cdot\sum\nolimits_{\mathbb{P}|l\in\mathbb{L}_{p}}\ddot{\varrho}^{t}_{r,k}\cdot\ddot{f}^{t}_{r,p}\quad\forall
t,r,l,k\in\mathbb{T},\mathbb{R}_{t},\mathbb{L},\mathbb{K}{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}tttt}$
(C16)
$\displaystyle\ddot{\mathcal{D}}^{t}_{r}=\sum\nolimits_{\mathbb{N},\mathbb{K},\mathbb{V},\mathbb{L}}(\ddot{\mathcal{D}}^{t}_{r,n,k}+\ddot{\mathcal{D}}^{t}_{r,v,k}+\ddot{\mathcal{D}}^{t}_{r,l,k})\quad\forall
t,r\in\mathbb{T},\mathbb{R}_{t}$ (C17)
$\displaystyle\ddot{\mathcal{D}}^{t}_{r}\leq\widecheck{\mathcal{D}_{r}}\quad\forall
t,r\in\mathbb{T},\mathbb{R}_{t}$ (C18)

###  III-E Objective Function

The objective is to maximize the overall profit while minimizing the energy
consumption of resources, that is:

$\displaystyle\sum\nolimits_{\mathbb{T},\mathbb{R}_{t}}(\gamma_{r}\cdot\sum\nolimits_{\mathbb{I}}\ddot{\mathcal{I}}^{t}_{r,i})-\alpha\cdot(\sum\nolimits_{\mathbb{N}}\ddot{\mathcal{E}}_{n}+\sum\nolimits_{\mathbb{V}}\ddot{\mathcal{E}}_{v}),{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}ttttttttttttt}{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}ttt}$
(OF)

where $\sum_{\mathbb{I}}\ddot{\mathcal{I}}^{t}_{r,i}$ is $1$ if request $r$ is
supported at time slot $t$, $\alpha$ is a small positive number, and
$\ddot{\mathcal{E}}_{v}$ and $\ddot{\mathcal{E}}_{n}$ represent, respectively,
the total energy consumption of compute node $v$ and network device $n$. Note
that $\alpha$ must be set such that the total profit exceeds the total amount
of energy consumed. Otherwise, supporting requests would result in a negative
objective function value, and the only optimal solution would be to support no
requests, making OF equal to $0$. To determine $\ddot{\mathcal{E}}_{n}$, where
the only source of energy consumption is transmitting requests’ data, the
following equations are employed:

$\displaystyle\ddot{\mathcal{E}}_{n}=\widetilde{\mathcal{E}_{n}}\cdot\sum\nolimits_{\mathbb{T},\mathbb{R}_{t},\mathbb{P}|n\in\mathbb{L}_{p}}\widecheck{\mathcal{B}_{r}}\cdot\ddot{f}^{t}_{r,p}{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}ttttttttttttttttttttttttttttttttttt}$
(1)

To calculate $\ddot{\mathcal{E}}_{v}$, it should be noted that the energy
consumed on each compute node has two primary sources: 1) the energy consumed
to service each unit of requests’ data, and 2) the energy consumed during
booting up or shutting down the compute node. Consequently,
$\ddot{\mathcal{E}}_{v}$ for each $v\in\mathbb{V}$ is:

$\displaystyle\widetilde{\mathcal{E}_{v}}\cdot\sum\nolimits_{\mathbb{T},\mathbb{R}_{t},\mathbb{I}}\widecheck{\mathcal{C}_{r}}\cdot\ddot{\mathcal{I}}^{t}_{r,i}\cdot\ddot{\mathcal{G}}^{t}_{i,v}+\overline{\mathcal{E}_{v}}\cdot\sum\nolimits_{\mathbb{T}|0\leq
t<\mathcal{T}}\dot{\mathcal{G}}^{t}_{v}\oplus\dot{\mathcal{G}}^{t+1}_{v}{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}ttttttttt}$
(2)

In this equation,
$\ddot{\mathcal{I}}^{t}_{r,i}\cdot\ddot{\mathcal{G}}^{t}_{i,v}$ equals $1$ if
compute node $v$ is selected as the host for request $r$; therefore, the first
part of the equation calculates the total energy consumption of compute node
$v$ to service its assigned requests. Assuming $\dot{\mathcal{G}}^{0}_{v}$
equals $0$, $\dot{\mathcal{G}}^{t}_{v}\oplus\dot{\mathcal{G}}^{t+1}_{v}$ is
equal to $1$ in the second part if and only if $\dot{\mathcal{G}}^{t}_{v}$ and
$\dot{\mathcal{G}}^{t+1}_{v}$ differ, representing a boot-up or shutdown for
compute node $v$. Then, the second part demonstrates the total energy
consumption of state transitions in compute note $v$ within $\mathbb{T}$.

###  III-F Problem

Considering the constraints and the objective function, the Problem of
Integrated Resource Allocation (PIRA) is:

$\footnotesize\text{PIRA: }\max\text{ OF}\text{ s.t. }\text{C1 -
C18.}{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}tttttttttttttttttttttttttttttttttttttttttt}$
(3)

The optimal solution involves assigning requests with stringent latency
requirements to high-priority queues and keeping those queues as empty as
possible to minimize latency. Resource selection should aim to minimize
activated resources and consider future requests to reduce energy consumption
through fewer start-ups and shutdowns, improving energy efficiency.

###  III-G Complexity Analysis

The problem defined in (3) is an extended version of the Multi-Dimensional
Knapsack (MDK) problem. Assume the problem is relaxed and reformulated
specifically for time slot $t$ as the problem of maximizing profit and
minimizing energy consumption while the only decision is to assign requests to
instances concerning only their capacity constraints. Since the MDK problem is
NP-hard and this relaxed version is an MDK problem with at least
$\mathcal{R}_{t}$ items and $\mathcal{I}$ knapsacks, PIRA is at least as
difficult as the MDK problem and is also NP-hard.

##  IV ORIENT

This section proposes an RL-based priORIty-aware Energy-efficieNt laTency-
sensitive resource allocation approach (ORIENT) to find near-optimal solutions
for PIRA. Subsequently, the learning mechanism is elaborated upon, followed by
an explanation of the agent’s design, and concluding with a description of the
algorithm.

###  IV-A Learning Mechanism

Given the continuous operation of the system defined in this paper and the
recurring necessity for consistent resource allocation decisions at each time
slot of PIRA, the adoption of RL presents itself as a viable means to enhance
decision-making proficiency to solve it. Within the framework of RL, an agent
undergoes a process of learning by means of trial and error at each step
(here, time slot), with the primary aim of optimizing a specific decision-
making problem. The system’s designer defines a reward function in alignment
with the objectives of the problem. By learning and following the optimal
strategy derived from this reward function, the agent aims to maximize
cumulative discounted rewards, regardless of the initial state. Among various
RL-based algorithms, Q-Learning stands out as widely acknowledged.

In Q-Learning, every state-action pair is associated with a numeric value
referred to as the Q-value, where the agent selects the action with the
maximum Q-value at each step. In Deep Q-Learning (DQL), a Deep Neural Network
(DNN) serves as the approximator for these Q-values. In this arrangement, the
state and action are presented as inputs, and the DNN-based $Q$-function
encompassing all feasible actions, denoted by
$Q(s,.;\boldsymbol{\mathcal{W}})$, is generated as the output and
systematically updated over time according to the following equation:

$\footnotesize\boldsymbol{\mathcal{W}}^{t+1}=\boldsymbol{\mathcal{W}}^{t}+\sigma[Y^{t}-Q(\boldsymbol{S}^{t},a^{t};\boldsymbol{\mathcal{W}}^{t})]\nabla_{\boldsymbol{\mathcal{W}}^{t}}\cdot
Q(\boldsymbol{S}^{t},a^{t};\boldsymbol{\mathcal{W}}^{t}){\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}tttttttt}$
(4)

In this equation, $\boldsymbol{\mathcal{W}}$ is the set of DNN weights,
$\sigma$ is a scalar step size, $\boldsymbol{S}^{t}$ and $a^{t}$ are the
agent’s state and action at time slot $t$, and $Y^{t}$ (also known as the
target) shows the maximum value expected to be achieved by following $a^{t}$
at $\boldsymbol{S}^{t}$. $Y^{t}$ is the only variable that must be estimated
in this equation, and in Double DQL (DDQL), where the selection and evaluation
processes are decoupled, it can be expressed as follows:

$\footnotesize
Y^{t}=r^{t+1}+\gamma\;\widehat{Q}(\boldsymbol{S}^{t+1},a^{\prime},\boldsymbol{\mathcal{W}}^{t-}),{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}tttttttttttttttttttttttttttttttttttt}$
(5)

where $r^{t+1}$ is the earned reward at time slot $t+1$, $\gamma\in[0,1]$ is a
discount factor that balances the importance of immediate and future rewards,
$a^{\prime}=\text{argmax}_{a\in\boldsymbol{\mathcal{A}}}Q(\boldsymbol{S}^{t+1},a,\boldsymbol{\mathcal{W}}^{t})$,
and $\boldsymbol{\mathcal{A}}$ is the set of actions. In this equation,
$\boldsymbol{\mathcal{W}}$ represents the set of weights for the main $Q$ and
is updated in each step, whereas $\boldsymbol{\mathcal{W}}^{-}$ is for the
target $\widehat{Q}$ and is replaced with the weights of the main network
every $t$ steps. In other words, $\widehat{Q}$ remains a periodic copy of $Q$.

Furthermore, we augment DDQL by incorporating the dueling concept introduced
by Wang et al. [15]. Unlike conventional DDQL, which directly approximates
Q-values using DNNs, this method initially computes separate estimators for
state values ($\psi$) and action advantages ($\varphi$). Q-values are then
derived from these estimators, as illustrated below:

$\footnotesize
Q(\boldsymbol{S}^{t},a^{t},\boldsymbol{\mathcal{W}}^{t})=\psi(\boldsymbol{S}^{t},\boldsymbol{\mathcal{W}}^{t})\Bigg{(}\varphi(\boldsymbol{S}^{t},a^{t},\boldsymbol{\mathcal{W}}^{t})-\frac{\Phi}{\left|\boldsymbol{\mathcal{A}}\right|}\Bigg{)}{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}ttttttttttttt}$
(6)

where
$\Phi=\sum_{\boldsymbol{\mathcal{A}}}\varphi(\boldsymbol{S}^{t},a^{\prime},\boldsymbol{\mathcal{W}}^{t})$.
The primary benefit is the ability to generalize learning across actions
without modifying the learning algorithm, which improves policy evaluation in
the presence of numerous actions with similar state values. As a result of
combining the Dueling technique and DDQL, we can expect that the resultant
D3QL agent will outperform its predecessors. To bolster the effectiveness and
resilience of D3QL, observed transitions are archived in a memory bank known
as the experience memory. The learning process entails randomly selecting
transitions from this repository [16].

###  IV-B Agent Customization

The first step toward exploiting D3QL to solve PIRA is to define the agent’s
action space, state space, and reward.

#### Action Space

We define the action space as set
$\boldsymbol{\mathcal{A}}=\\{a:(i,v,p,k)|i,v,p,k\in\mathbb{I},\mathbb{V},\mathbb{P},\mathbb{K}\\}$.
During each time slot and for every request, a specific action must be
executed to finalize the resource allocation pertaining to that request.

#### State Space

For encoding the system’s state, an architecture involving aggregation GNN
layers is employed, constructing an aggregation sequence across all compute
nodes, iteratively facilitating information exchange with neighboring nodes.
Therefore, at time slot $t$ when request $r$ is on the verge of receiving
service, the system state is denoted as
$\boldsymbol{S}^{t}(r)=\\{\boldsymbol{S}^{t}_{\mathbb{V}}(r),\boldsymbol{S}^{t}_{\mathbb{P}}(r)\\}$
and can be formally defined as:

$\displaystyle\boldsymbol{S}^{t}_{\mathbb{V}}(r)=\Bigg{\\{}\Big{[}[\widehat{\mathcal{C}^{t}_{v,k}}-\widecheck{\mathcal{C}_{r{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.}}}]_{\mathbb{K}},[\ddot{\mathcal{D}}^{t}_{r,v,k}-\widecheck{\mathcal{D}_{r{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.}}}]_{\mathbb{K}},\widetilde{\mathcal{E}_{v}},\overline{\mathcal{E}_{v}}\cdot(1-\dot{\mathcal{G}}^{t}_{v})\Big{]}_{\mathbb{V}}\Bigg{\\}},$
(7)
$\displaystyle\boldsymbol{S}^{t}_{\mathbb{P}}(r)=\Bigg{\\{}\Big{[}\big{[}\wedge_{\mathbb{L}_{p}}-\widecheck{\mathcal{B}_{r{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.}}}\big{]}_{\mathbb{K}},[\ddot{\mathcal{D}}^{t}_{r}-\ddot{\mathcal{D}}^{t}_{r,v,k}-\widecheck{\mathcal{D}_{r{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.}}}]_{\mathbb{K}},\widetilde{\mathcal{E}_{n}}\Big{]}_{\mathbb{P}}\Bigg{\\}},{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}ttttttttt}$
(8)

where
$\wedge_{\mathbb{L}_{p}}=\min\nolimits_{\mathbb{L}|n,l\in\mathbb{L}_{p}}\\{\widehat{\mathcal{B}^{t}_{n,k}},\widehat{\mathcal{B}^{t}_{l,k}}\\}$.
$\boldsymbol{S}^{t}_{\mathbb{V}}(r)$ and $\boldsymbol{S}^{t}_{\mathbb{P}}(r)$
represent the embeddings of compute nodes and network paths, respectively,
which function as inputs for the GNN layers. These embeddings encompass the
remaining resource capacity and the anticipated latency when request $r$ is
allocated to them, as well as their associated energy consumption.

#### Reward

Since the agent is designated to maximize OF, the reward should be engineered
to reinforce the support of high-profit requests while selecting resources
with low energy consumption. This goal is satisfied in (9), that is:

$\footnotesize r^{t+1}=\left\\{\begin{array}[]{ll}0&\mbox{otherwise}\\\
\dfrac{\mathcal{M}}{\text{OF}\big{(}\boldsymbol{S}^{t}(r),a^{t}\big{)}-\min_{\boldsymbol{\mathcal{A}}}\text{OF}\big{(}\boldsymbol{S}^{t}(r),a^{\prime}\big{)}}&r\text{
is
met}{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>(9)

where
$\mathcal{M}=\max_{\boldsymbol{\mathcal{A}}}\text{OF}\big{(}\boldsymbol{S}^{t}(r),a^{\prime}\big{)}-\min_{\boldsymbol{\mathcal{A}}}\text{OF}\big{(}\boldsymbol{S}^{t}(r),a^{\prime}\big{)}$,
$\max\text{/}\min_{\boldsymbol{\mathcal{A}}}\text{OF}\big{(}\boldsymbol{S}^{t}(r),a^{\prime}\big{)}$
is the maximum/minimum profit that can be achieved by allocating the available
resources at time slot $t$ to request $r$ without considering any constraints
or requirements, and $\text{OF}\big{(}\boldsymbol{S}^{t}(r),a^{t}\big{)}$ is
the profit of the allocation provided by the agent. If the action fails to
meet the requirements of the request, it results in a reward of $0$.
Conversely, actions that yield greater profits correspond to higher rewards.

TABLE I: Simulation Parameters. Parameter | Value
---|---
number of priority levels | $4$
resource capacity bounds |  $\sim\mathcal{U}\\{250,300\\}$ mbps
Instance capacity bound |  $20$ mbps
energy consumptions per capacity unit | $\sim\mathcal{U}\\{10,20\\}$
energy consumptions per state transition | $\sim\mathcal{U}\\{100,200\\}$
capacity requirement per request |  $\sim\mathcal{U}\\{4,8\\}$ mbps
bandwidth requirement per request |  $\sim\mathcal{U}\\{2,10\\}$ mbps
latency requirement per request |  $\sim\mathcal{U}\\{1,3\\}$ ms
packet size per request | $1$
profit per request ($\gamma_{r}$) | $\mathcal{U}\\{5,15\\}$

Input: $\mathcal{T}$, $\epsilon^{\prime}$, and $\widetilde{\epsilon}$

1 $\boldsymbol{\Omega}\leftarrow\emptyset$,
$\boldsymbol{\mathcal{W}}\leftarrow\mathbf{0}$,
$\boldsymbol{\mathcal{W}^{-}}\leftarrow\mathbf{0}$, $\epsilon\leftarrow 1$,
$memory\leftarrow\\{\\}$

2 for  _each $t$ in $[0:\mathcal{T}]$_ do

3 if  _new request $r$ is arrived_ then

4 calculate
$\boldsymbol{S}^{t}(r)=\\{\boldsymbol{S}^{t}_{\mathbb{V}}(r),\boldsymbol{S}^{t}_{\mathbb{P}}(r)\\}$

5 $\zeta\leftarrow$ generate a random number from $[0:1]$

6 if  _$\zeta >\epsilon$_ then

7 $a^{t}=(i,v,p,k)\leftarrow$
argmax${}_{\boldsymbol{\mathcal{A}}}Q(\boldsymbol{S}^{t},a^{\prime},\boldsymbol{\mathcal{W}}^{t})$

8

9 else

10 select a random $a^{t}=(i,v,p,k)$ from $\boldsymbol{\mathcal{A}}$

11 calculate $r^{t+1}$

12 if  _$r^{t+1} >0$_ then

13 Establish request $r$ connection based on $a^{t}$

14

15 $memory\leftarrow memory\cup\\{(\boldsymbol{S}^{t},a^{t},r^{t+1})\\}$

16 choose a batch of samples from $memory$

17 train the agent

18 if  _$\epsilon >\widetilde{\epsilon}$_ then

19 $\epsilon\leftarrow\epsilon-\epsilon^{\prime}$

20 $\boldsymbol{\Omega}\leftarrow\boldsymbol{\Omega}\cup\\{(t,r,a^{t})\\}$

return $\boldsymbol{\Omega}$

Algorithm 1 ORIENT

###  IV-C ORIENT’s Algorithm

ORIENT is detailed in Algorithm 1. In this algorithm, $\epsilon^{\prime}$ and
$\widetilde{\epsilon}$ are small positive integers that control the
$\epsilon$-greedy mechanism. During each time slot $t$, the agent receives
notifications of new request arrivals ($r$), and it computes the state based
on request $r$ requirements and the current system state. The action is then
chosen using an $\epsilon$-greedy policy, which follows the evaluation
function of the corresponding agent with probability $(1-\epsilon)$ and
selects a random action with probability $\epsilon$. Subsequently, the reward
is calculated, and if it exceeds $0$, it indicates that $a^{t}$ is feasible
and meets request $r$’s QoS requirements, enabling its connection based on
$a^{t}$ allocations. Finally, the experience memory is updated, samples are
drawn from the memory bank, and the agent undergoes training. during the
training process, $\epsilon$ decreases from $1$ to $\widetilde{\epsilon}$. The
algorithm yields $\boldsymbol{\Omega}$ as the history of allocations.

##  V Performance Evaluation

Figure 2: The mean energy consumption of supported requests and the total
profit vs. the system size (A & B) and the number of all requests (C & D).

In this section, we present numerical results based on the system model
parameters listed in Table I. Other parameters can be chosen arbitrarily so
long as the logic outlined throughout the paper remains valid. To evaluate the
efficiency of ORIENT, we conduct a comparative analysis with OPT, D3QL, and
RND. OPT represents the optimal solution for PIRA, obtained through the use of
CPLEX 12.10. D3QL, on the other hand, bears similarities to the method
outlined in Algorithm 1, but employs exclusively simple linear layers in its
DNN, without utilizing any GNNs. This approach forms the foundation for
several related studies, including A-DDPG [13] and MDRL-SaDS [7], both of
which are RL methods designed to enhance the utility of individual requests by
considering factors such as resource cost, required bandwidth, and E2E path
latency. Lastly, RND represents a random allocation strategy, where resources
are allocated to active requests without considering any constraints.

The results are depicted in Fig. 2, where subfigures A and B represent the
average energy consumption per request and total profit across various system
sizes, with a constant of $300$ active requests. Here, incrementing the system
size entails the creation of a new system graph, incorporating $\mathcal{N}+1$
network devices and $\mathcal{V}+1$ compute nodes. Particularly, from $10$ to
$13$, resources with a significant energy consumption are introduced into the
graph. Between $14$ and $17$, resources with a moderate energy consumption are
added to the graph, and the remaining resources included from $18$ to $21$ are
characterized by low energy consumption. Subfigures C and D display similar
quality metrics, but for different numbers of requests, while keeping the
system size fixed at $12$ with equal numbers of resources ($4$) from each
level of energy consumption.

Within the subfigures, it is evident that OPT serves as an upper performance
bound, while RND serves as the lower bound. Furthermore, when all resources
exhibit high energy consumption rates or are fully occupied (with
$\mathcal{N}\leq 13$ in A and $\mathcal{R}\geq 200$ in C), RND shows a similar
energy consumption pattern to D3QL-based techniques, but its support rate is
limited due to the absence of intelligence and feasibility checks. In
contrast, ORIENT excels in both scenarios. As demonstrated in A and B, it
achieves near-optimal results by prioritizing high-capacity resources with
minimal energy consumption, especially when multiple choices are available for
each request ($\mathcal{N}\geq 13$). Similarly, regardless of whether all
requests can be supported (C and D, with $\mathcal{R}\leq 200$), near-optimal
solutions are consistently attained. However, D3QL exhibits less efficiency
and stability compared to ORIENT, mainly due to its inferior state decoding
capability.

##  VI Conclusion

In this paper, we examined the joint problem of service instance placement and
assignment, path selection, and request prioritization, dubbed PIRA, with the
objective of maximizing the overall profit of the system (as a function of the
number of supported concurrent requests) while minimizing the overall energy
consumption over a continuous period of time, taking into account E2E latency
and resource capacity constraints. This problem was formulated as a MINLP
problem, its complexity was analyzed, and it was demonstrated that it is an
NP-hard problem. Subsequently, a technique named ORIENT was introduced to
address the problem in a near-optimal manner, utilizing a GNN-empowered D3QL
strategy. The effectiveness of the suggested technique was validated through
numerical results. As potential future work, our intention is to tackle the
problem in the context of dynamic environments characterized by
temporal/spatial fluctuations in requests and resources.

## Acknowledgment

It is partially supported by the European Union’s Horizon 2020 Research and
Innovation Program through the aerOS project under Grant No. 101069732; the
Business Finland 6Bridge 6Core project under Grant No. 8410/31/2022; the
European Union’s HE research and innovation program HORIZON-JUSNS-2022 under
the 6GSandbox project (Grant No. 101096328); and the Research Council of
Finland 6G Flagship Programme under Grant No. 346208. This research was also
conducted at ICTFICIAL Oy. The paper reflects only the authors’ views, and the
commission bears no responsibility for any utilization of the information
contained herein.

## References

  * [1] M. Latva-aho, “Key Drivers and Research Challenges for 6G Ubiquitous Wireless Intelligence,” 2019, publisher: University of Oulu.
  * [2] H. Yu _et al._ , “Toward 6G-Based Metaverse: Supporting Highly-Dynamic Deterministic Multi-User Extended Reality Services,” _IEEE Network_ , vol. 37, no. 4, pp. 30–38, 2023.
  * [3] H. Mazandarani _et al._ , “A Semantic-Aware Multiple Access Scheme for Distributed, Dynamic 6G-Based Applications,” _arXiv preprint arXiv:2401.06308_ , 2024.
  * [4] J. Prados-Garzon _et al._ , “Deterministic 6GB-Assisted Quantum Networks with Slicing Support: A New 6GB Use Case,” _IEEE Network_ , 2023.
  * [5] Z. Yang _et al._ , “Increasing the Energy Efficiency of a Data Center Based on Machine Learning,” _Journal of Industrial Ecology_ , vol. 26, no. 1, pp. 323–335, 2022.
  * [6] “Cisco Annual Internet Report (2018–2023) White Paper.”
  * [7] H. Xuan _et al._ , “Multi-Agent Deep Reinforcement Learning Algorithm with Self-adaption Division Strategy for VNF-SC Deployment in SDN/NFV-Enabled Networks,” _Applied Soft Computing_ , vol. 138, p. 110189, May 2023.
  * [8] R. Solozabal _et al._ , “Virtual Network Function Placement Optimization With Deep Reinforcement Learning,” _IEEE Journal on Selected Areas in Communications_ , vol. 38, no. 2, pp. 292–303, Feb. 2020\.
  * [9] C. Pham _et al._ , “Traffic-Aware and Energy-Efficient VNF Placement for Service Chaining: Joint Sampling and Matching Approach,” _IEEE Trans. on Services Computing_ , vol. 13, no. 1, pp. 172–185, Jan. 2020.
  * [10] G. L. Santos _et al._ , “Availability-Aware and Energy-Aware Dynamic SFC Placement using Reinforcement Learning,” _The Journal of Supercomputing_ , vol. 77, no. 11, Nov. 2021.
  * [11] Z. Sasan _et al._ , “Joint Network Slicing, Routing, and In-Network Computing for Energy-Efficient 6G,” _arXiv preprint arXiv:2401.06306_ , 2024\.
  * [12] M. Farhoudi _et al._ , “Qos-Aware Service Prediction and Orchestration in Cloud-Network Integrated Beyond 5G,” _arXiv preprint arXiv:2309.10185_ , 2023.
  * [13] N. He _et al._ , “Leveraging Deep Reinforcement Learning With Attention Mechanism for Virtual Network Function Placement and Routing,” _IEEE Trans. on Parallel and Distributed Systems_ , vol. 34, no. 4, Apr. 2023.
  * [14] M. Shokrnezhad _et al._ , “Double Deep Q-Learning-based Path Selection and Service Placement for Latency-Sensitive Beyond 5G Applications,” _IEEE Transactions on Mobile Computing_ , pp. 1–14, 2023.
  * [15] Z. Wang _et al._ , “Dueling Network Architectures for Deep Reinforcement Learning,” in _Proceedings of The 33rd International Conference on Machine Learning_ , vol. 48, Jun. 2016, pp. 1995–2003.
  * [16] V. Mnih _et al._ , “Human-level Control through Deep Reinforcement Learning,” _Nature_ , vol. 518, no. 7540, pp. 529–533, Feb. 2015.


# Means refinements via convexity

M. Sababheh  Department of Basic Sciences, Princess Sumaya University For
Technology, Al Jubaiha, Amman 11941, Jordan<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

The main goal of this article is to find the exact difference between a convex
function and its secant, as a limit of positive quantities. This idea will be
expressed as a convex inequality that leads to refinements and reversals of
well established inequalities treating different means. The significance of
these inequalities is to write one inequality that brings together and refine
almost all known inequalities treating the arithmetic, geometric, harmonic and
Heinz means, for numbers and operators.

###### Key words and phrases:

convex functions, means inequalities, norm inequalities.

###### 2010 Mathematics Subject Classification:

15A39, 15B48, 26D15, 26B25, 47A30, 47A63.

##  1\. introduction

Convex functions and their inequalities have played a major role in the study
of various topics in Mathematics; including applied Mathematics, Mathematical
Analysis and Mathematical Physics. Means and their comparison is indeed an
important application of convexity.
Recall that a function $f:\mathbb{I}\to\mathbb{R}$, defined on a real interval
$\mathbb{I}$, is said to be convex if $f(\alpha x_{1}+\beta x_{2})\leq\alpha
f(x_{1})+\beta f(x_{2})$, when $x_{1},x_{2}\in\mathbb{I}$ and
$\alpha,\beta\geq 0$ satisfying $\alpha+\beta=1.$ On the other hand,
$f:\mathbb{I}\to\mathbb{R}^{+}$ is said to be log-convex if $g(x)=\log f(x)$
is convex, or equivalently if $f(\alpha x_{1}+\beta x_{2})\leq
f^{\alpha}(x_{1})f^{\beta}(x_{2})$ for the above parameters.

Speaking of means, the comparison between the weighted arithmetic, geometric
and harmonic means is an immediate consequence of convexity or log-convexity
of the functions $x\nabla_{t}y=(1-t)x+ty,x\\#_{t}y=x^{1-t}y^{t}$ and
$x!_{t}y=((1-t)x^{-1}+ty^{-1})^{-1},x,y>0,$ defined for $0\leq t\leq 1$.
Adopting these notations, we drop $t$ when $t=\frac{1}{2}.$
Convexity of the function $f(t)=x\\#_{t}y$ implies the well known Young’s
inequality $x\\#_{t}\leq x\nabla_{t}y.$ On the other hand, convexity of the
function $g(t)=x!_{t}y$ implies the arithmetic-harmonic mean inequality
$x!_{t}y\leq x\nabla_{t}y$, while log-convexity of $g$ implies the geometric-
harmonic mean inequality $x!_{t}y\leq x\\#_{t}y.$

These inequalities, though very simple, have some significant applications.
For example, the above Young’s inequality implies the celebrated Holder’s
inequality $\|fg\|_{1}\leq\|f\|_{p}\|g\|_{q}$ for $f\in L^{p}(X)$ and $g\in
L^{q}(X)$, for the conjugate exponents $p,q$, where $X$ is some measure space.

Among the most interesting applications of the above mean inequalities is the
possible comparison between operators acting on a finite dimensional Hilbert
space $H$. In the sequel, $\mathbb{M}_{n}$ will denote the space of operators
acting on an $n-$deimentional Hilbert space $H$, $\mathbb{M}_{n}^{+}$ will
denotes the cone of semi positive operators in $\mathbb{M}_{n}$ while
$\mathbb{M}_{n}^{++}$ will denotes the cone of strictly positive operators in
$\mathbb{M}_{n}.$ Then the above numerical inequalities have their operator
versions such as $A\\#_{t}B\leq A\\#_{t}B$, where
$A,B\in\mathbb{M}_{n}^{++},A\nabla_{t}B=(1-t)A+tB$ and
$A\\#_{t}B=A^{\frac{1}{2}}\left(A^{-\frac{1}{2}}BA^{-\frac{1}{2}}\right)^{t}A^{\frac{1}{2}}.$
In this context, we say that $A\leq B$ for two self-adjoint operators $A$ and
$B$ if $B-A\in\mathbb{M}_{n}^{+}.$

Obtaining the operator versions from the corresponding numerical versions can
be done in different approaches, among which is the application of the
following lemma [2].

######  Lemma 1.1.

Let $X\in\mathcal{M}_{n}$ be self-adjoint and let $f$ and $g$ be continuous
real valued functions such that $f(t)\geq g(t)$ for all $t\in{\text{Sp}}(X),$
the spectrum of $X$. Then $f(X)\geq g(X).$

Recent studies of the topic have investigated possible refinements of the
above inequalities, where adding a positive term to the left side becomes
possible. This idea has been treated in [3, 5, 6, 7, 9, sabjmaa, 10, 11, 12],
where not only refinements have been investigated, but reversed versions and
much more have been discussed.

Keeping our paper concise, we will not go through the exact results done in
the above references now, however we will comment later how the results in
this paper generalize almost all results in these references, regarding the
refinements and the reverses of the above mean inequalities.

The main goal of this article is to avoid dealing with the specific means, and
to treat a general convexity argument that leads to these refinements. In
particular, we prove that for certain positive quantities
$A_{j}(\nu)\Delta_{j}f(\nu;a,b),$ we have

$\displaystyle f\left((1-\nu)a+\nu
b\right)+\sum_{j=1}^{N}A_{j}(\nu)\Delta_{j}f(\nu;a,b)\leq(1-\nu)f(a)+\nu
f(b),N\in\mathbb{N},$

for the convex function $f:[a,b]\to\mathbb{R}$. This provides $N$ refining
terms of the inequality $f\left((1-\nu)a+\nu b\right)\leq(1-\nu)f(a)+\nu
f(b)$, which follows from convexity of $f$. Furthermore, we prove a reversed
version and we prove that as $N\to\infty$ the above inequality becomes an
equality. As a natural consequence, we obtain some refinements and reverses
for log-convex functions.

As we will see, the above inequality and its consequences happen to be
generalizations that imply almost all inequalities in the references [3, 5, 6,
9, sabjmaa, 10, 11, 12]. This is our main motivation behind this work; to find
a formula that implies and generalizes all other formulae and hence, to
enhance our understanding of these inequalities.

We remark that the proof of the first main result in this work is inspired by
our recent work in [sabjmaa].

##  2\. main results

For the rest of the paper, the following notations will be adopted. For
$0\leq\nu\leq 1$ and $j\in\mathbb{N}$, let

$\left\\{\begin{array}[]{cc}k_{j}(\nu)=[2^{j-1}\nu],r_{j}(\nu)=[2^{j}\nu]\;{\text{and}}\\\
A_{j}(\nu)=(-1)^{r_{j}(\nu)}2^{j-1}\nu+(-1)^{r_{j}(\nu)+1}\left[\frac{r_{j}(\nu)+1}{2}\right]\end{array}\right..$
(2.1)

Moreover, if $f:[a,b]\to\mathbb{R}$ is any function, define

$\displaystyle\Delta_{j}f(\nu;a,b)$ $\displaystyle=$ $\displaystyle
f\left(\left(1-\frac{k_{j}(\nu)}{2^{j-1}}\right)a+\frac{k_{j}(\nu)}{2^{j-1}}b\right)+f\left(\left(1-\frac{k_{j}(\nu)+1}{2^{j-1}}\right)a+\frac{k_{j}(\nu)+1}{2^{j-1}}b\right)$
(2.2) $\displaystyle-$ $\displaystyle
2f\left(\left(1-\frac{2k_{j}(\nu)+1}{2^{j}}\right)a+\frac{2k_{j}(\nu)+1}{2^{j}}b\right),0\leq\nu\leq
1.$

###  2.1. Convex functions

We discuss first the inequalities that govern convex functions, then we apply
these inequalities to log-convex functions.

######  Lemma 2.1.

If $f:[a,b]\to\mathbb{R}$ is convex, then $\Delta_{j}f(\nu;a,b)\geq 0$ for
$j\in\mathbb{N}$ and $0\leq\nu\leq 1.$

###### Proof.

Letting
$x_{j}(\nu)=\left(1-\frac{k_{j}(\nu)}{2^{j-1}}\right)a+\frac{k_{j}(\nu)}{2^{j-1}}b,y_{j}(\nu)=\left(1-\frac{k_{j}(\nu)+1}{2^{j-1}}\right)a+\frac{k_{j}(\nu)+1}{2^{j-1}}b$
and
$z_{j}(\nu)=\left(1-\frac{2k_{j}(\nu)+1}{2^{j}}\right)a+\frac{2k_{j}(\nu)+1}{2^{j}}b,$
it is easy that $z_{j}(\nu)=\frac{x_{j}(\nu)+y_{j}(\nu)}{2}.$ The
$\Delta_{j}f(\nu;a,b)=f(x_{j}(\nu))+f(y_{j}(\nu))-2f(z_{j}(\nu))\geq 0,$ by
convexity of $f$. ∎

######  Remark 2.2.

When $f:[a,b]\to\mathbb{R}$, we adopt the convention that $f(x)=0$ for
$x\not\in[a,b].$ This convention will be needed, for example, in the next
lemma, when $N=1$ and $\nu=1.$

######  Lemma 2.3.

Let $f:[0,1]\to\mathbb{R}$ be a function and let $N\in\mathbb{N}$. Then

$\displaystyle(1-\nu)f(0)$ $\displaystyle+$ $\displaystyle\nu
f(1)-\sum_{j=1}^{N}A_{j}(\nu)\Delta_{j}f(\nu;0,1)$ $\displaystyle=$
$\displaystyle\left([2^{N}\nu]+1-2^{N}\nu\right)f\left(\frac{[2^{N}\nu]}{2^{N}}\right)+\left(2^{N}\nu-[2^{N}\nu]\right)f\left(\frac{[2^{N}\nu]+1}{2^{N}}\right).$

###### Proof.

We proceed by induction on $N$.
When $N=1$ and $0\leq\nu<\frac{1}{2},$ $r_{1}(\nu)=0$ and $k_{1}(\nu)=0$.
Hence $A_{1}(\nu)=\nu$ and
$\Delta_{1}f(\nu;a,b)=f(a)+f(b)-2f\left(\frac{a+b}{2}\right).$ Then direct
computations show the result.
Now if $\frac{1}{2}\leq\nu<1$, then $r_{1}(\nu)=1$ and $k_{1}(\nu)=0$, hence
$A_{1}(\nu)=1-\nu$ and
$\Delta_{1}f(\nu;a,b)=f(a)+f(b)-2f\left(\frac{a+b}{2}\right).$ Again, direct
computations show the result.
When $\nu=1,$ the result follows immediately.
Now assume that (LABEL:exact_difference) is true for some $N\in\mathbb{N}$. We
assert its truth for $N+1.$ Notice that, using the inductive step,

$\displaystyle(1-\nu)f(0)$ $\displaystyle+$ $\displaystyle\nu
f(1)-\sum_{j=1}^{N+1}A_{j}(\nu)\Delta_{j}f(\nu;0,1)$ (2.4) $\displaystyle=$
$\displaystyle(1-\nu)f(0)+\nu
f(1)-\sum_{j=1}^{N}A_{j}(\nu)\Delta_{j}f(\nu;0,1)-A_{N+1}(\nu)\Delta_{N+1}f(\nu;0,1)$
$\displaystyle=$
$\displaystyle\left([2^{N}\nu]+1-2^{N}\nu\right)f\left(\frac{[2^{N}\nu]}{2^{N}}\right)+\left(2^{N}\nu-[2^{N}\nu]\right)f\left(\frac{[2^{N}\nu]+1}{2^{N}}\right)$
$\displaystyle-$
$\displaystyle\left((-1)^{[2^{N+1}\nu]}2^{N}\nu+(-1)^{[2^{N+1}\nu]+1}\left[\frac{[2^{N+1}\nu]+1}{2}\right]\right)\times$
$\displaystyle\times$
$\displaystyle\left(f\left(\frac{[2^{N}\nu]}{2^{N}}\right)+f\left(\frac{[2^{N}\nu]+1}{2^{N}}\right)-2f\left(\frac{2[2^{N}\nu]+1}{2^{N+1}}\right)\right).$

Now we treat two cases.
Case I If $[2^{N+1}\nu]$ is odd, then we easily see that
$[2^{N}\nu]=\frac{[2^{N+1}\nu]-1}{2}.$ Therefore,

$f\left(\frac{[2^{N}\nu]+1}{2^{N}}\right)=f\left(\frac{[2^{N+1}\nu]+1}{2^{N+1}}\right)\;{\text{and}}\;f\left(\frac{2[2^{N}\nu]+1}{2^{N+1}}\right)=f\left(\frac{[2^{N+1}\nu]}{2^{N+1}}\right).$

Substituting these values in (2.4) and simplifying imply

$\displaystyle(1-\nu)f(0)+\nu
f(1)-\sum_{j=1}^{N+1}A_{j}(\nu)\Delta_{j}f(\nu;0,1)$ $\displaystyle=$
$\displaystyle\left(2^{N+1}\nu-[2^{N+1}\nu]\right)f\left(\frac{[2^{N+1}\nu]+1}{2^{N+1}}\right)+\left([2^{N+1}\nu]+1-2^{N+1}\nu\right)f\left(\frac{[2^{N+1}\nu]}{2^{N+1}}\right),$

which completes the proof, when $[2^{N+1}\nu]$ is odd.
Case II If $[2^{N+1}\nu]$ is even, then $2[2^{N}\nu]=[2^{N+1}\nu]$ and

$f\left(\frac{[2^{N}\nu]}{2^{N}}\right)=f\left(\frac{[2^{N+1}\nu]}{2^{N+1}}\right)\;{\text{and}}\;f\left(\frac{2[2^{N}\nu]+1}{2^{N+1}}\right)=f\left(\frac{[2^{N+1}\nu]+1}{2^{N+1}}\right).$

Substituting these values in (2.4) and simplifying imply

$\displaystyle(1-\nu)f(0)+\nu
f(1)-\sum_{j=1}^{N+1}A_{j}(\nu)\Delta_{j}f(\nu;0,1)$ $\displaystyle=$
$\displaystyle\left(2^{N+1}\nu-[2^{N+1}\nu]\right)f\left(\frac{[2^{N+1}\nu]+1}{2^{N+1}}\right)+\left([2^{N+1}\nu]+1-2^{N+1}\nu\right)f\left(\frac{[2^{N+1}\nu]}{2^{N+1}}\right).$

This completes the proof. ∎

######  Corollary 2.4.

Let $f:[0,1]\to\mathbb{R}$ be convex and let $N\in\mathbb{N}$. Then

$f(\nu)+\sum_{j=1}^{N}A_{j}(\nu)\Delta_{j}f(\nu;0,1)\leq(1-\nu)f(0)+\nu f(1).$
(2.5)

###### Proof.

From Lemma 2.3 and convexity of $f$, we have

$\displaystyle(1-\nu)f(0)$ $\displaystyle+$ $\displaystyle\nu
f(1)-\sum_{j=1}^{N}A_{j}(\nu)\Delta_{j}f(\nu;0,1)$ $\displaystyle=$
$\displaystyle\left([2^{N}\nu]+1-2^{N}\nu\right)f\left(\frac{[2^{N}\nu]}{2^{N}}\right)+\left(2^{N}\nu-[2^{N}\nu]\right)f\left(\frac{[2^{N}\nu]+1}{2^{N}}\right)$
$\displaystyle\geq$ $\displaystyle
f\left(\left([2^{N}\nu]+1-2^{N}\nu\right)\frac{[2^{N}\nu]}{2^{N}}+\left(2^{N}\nu-[2^{N}\nu]\right)\frac{[2^{N}\nu]+1}{2^{N}}\right)$
$\displaystyle=$ $\displaystyle f(\nu).$

This completes the proof. ∎

Now our first main result in its general form can be stated as follows.

######  Theorem 2.5.

Let $f:[a,b]\to\mathbb{R}$ be convex. Then for each $N\in\mathbb{N}$ and
$0\leq\nu\leq 1,$ we have

$\displaystyle f\left((1-\nu)a+\nu
b\right)+\sum_{j=1}^{N}A_{j}(\nu)\Delta_{j}f(\nu;a,b)\leq(1-\nu)f(a)+\nu
f(b).$ (2.6)

###### Proof.

For the given $f$, define $g:[0,1]\to\mathbb{R}$ by $g(x)=f((1-x)a+xb).$ Then
$g$ is convex on $[0,1].$ Applying Corollary 2.4 on the function $g$ implies
the result. ∎

######  Remark 2.6.

We remark that a negative version of the above theorem has been recently shown
in [8]. Namely, it was proved

$\displaystyle(1+\nu)f(a)-\nu f(b)$
$\displaystyle+\sum_{j=1}^{N}2^{j}\nu\left[\frac{f(a)+f\left(\frac{(2^{j-1}-1)a+b}{2^{j-1}}\right)}{2}-f\left(\frac{(2^{j}-1)a+b}{2^{j}}\right)\right]$
$\displaystyle\leq f\left((1+\nu)a-\nu b\right),\nu\geq 0,a<b,$ (2.7)

for the convex function $f:\mathbb{R}\to\mathbb{R}.$ However, the method of
proof is considerably easier than the above proofs and the applications are
different.

Our next step is to prove a reversed version of (2.6).

######  Theorem 2.7.

Let $f:[a,b]\to\mathbb{R}$ be convex and let $N\in\mathbb{N}$. Then for
$0\leq\nu\leq\frac{1}{2},$ we have

$\displaystyle f\left((1-\nu)a+\nu b\right)$ $\displaystyle+$
$\displaystyle(1-A_{1}(\nu))\Delta_{1}f(\nu;a,b)$ $\displaystyle\geq$
$\displaystyle(1-\nu)f(a)+\nu
f(b)+\sum_{j=1}^{N}A_{j}(1-2\nu)\Delta_{j}f\left(1-2\nu;\frac{a+b}{2},b\right).$

On the other hand, if $\frac{1}{2}\leq\nu\leq 1,$ we have

$\displaystyle f\left((1-\nu)a+\nu b\right)$ $\displaystyle+$
$\displaystyle(1-A_{1}(\nu))\Delta_{1}f(\nu;a,b)$ $\displaystyle\geq$
$\displaystyle(1-\nu)f(a)+\nu
f(b)+\sum_{j=1}^{N}A_{j}(2-2\nu)\Delta_{j}f\left(2-2\nu;a,\frac{a+b}{2}\right).$

###### Proof.

For $0\leq\nu\leq\frac{1}{2},$ we have

$\displaystyle f\left((1-\nu)a+\nu
b\right)+(1-A_{1}(\nu))\Delta_{1}f(\nu;a,b)-\left((1-\nu)f(a)+\nu f(b)\right)$
$\displaystyle=$ $\displaystyle 2\nu
f\left(\frac{a+b}{2}\right)+(1-2\nu)f(b)+f\left((1-\nu)a+\nu
b\right)-2f\left(\frac{a+b}{2}\right)$ $\displaystyle\geq$
$\displaystyle\sum_{j=1}^{N}A_{j}(1-2\nu)\Delta_{j}f\left(1-2\nu;\frac{a+b}{2},b\right)+f\left(2\nu\frac{a+b}{2}+(1-2\nu)b\right)$
$\displaystyle+f\left((1-\nu)a+\nu b\right)-2f\left(\frac{a+b}{2}\right)$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{N}A_{j}(1-2\nu)\Delta_{j}f\left(1-2\nu;\frac{a+b}{2},b\right)+f\left(\nu
a+(1-\nu)b\right)$ $\displaystyle+f\left((1-\nu)a+\nu
b\right)-2f\left(\frac{a+b}{2}\right)$ $\displaystyle\geq$
$\displaystyle\sum_{j=1}^{N}A_{j}(1-2\nu)\Delta_{j}f\left(1-2\nu;\frac{a+b}{2},b\right),$

where the last line follows from convexity of $f$, where one has

$\displaystyle f\left(\nu a+(1-\nu)b\right)+f\left((1-\nu)a+\nu b\right)$
$\displaystyle\geq 2f\left(\frac{\nu a+(1-\nu)b+(1-\nu)a+\nu
b}{2}\right)=2f\left(\frac{a+b}{2}\right).$

This completes the proof for $0\leq\nu\leq\frac{1}{2}.$ Similar computations
imply the desired inequality for $\frac{1}{2}\leq\nu\leq 1.$ ∎

In fact, the above reversed version turns out to be equivalent to convexity.

######  Proposition 2.8.

Let $f:\mathbb{I}\to\mathbb{R}$ be a function defined on the interval
$\mathbb{I}$. Assume that for all $a<b$ in $\mathbb{I}$ and all $0\leq\nu\leq
1$, we have

$f\left((1-\nu)a+\nu
b\right)+(1-A_{1}(\nu))\Delta_{1}f(\nu;a,b)\geq(1-\nu)f(a)+\nu f(b),$ (2.8)

then $f$ is convex on $\mathbb{I}.$

###### Proof.

Observe that when $0\leq\nu\leq\frac{1}{2},$ (2.8) is equivalent to

$f\left((1-\nu)a+\nu
b\right)+(1-\nu)\left(f(a)+f(b)-2f\left(\frac{a+b}{2}\right)\right)\geq(1-\nu)f(a)+\nu
f(b),$

or

$f\left(\frac{a+b}{2}\right)\leq\frac{1}{2-2\nu}f\left((1-\nu)a+\nu
b\right)+\frac{1-2\nu}{2-2\nu}f(b).$ (2.9)

On the other hand, if $\frac{1}{2}\leq\nu\leq 1,$ (2.8) is equivalent to

$f\left(\frac{a+b}{2}\right)\leq\frac{2\nu-1}{2\nu}f(a)+\frac{1}{2\nu}f\left((1-\nu)a+\nu
b\right).$ (2.10)

Let $x_{1}<x_{2}\in\mathbb{I}$ and let $0<\lambda<1.$ We assert that
$f((1-\lambda)x_{1}+\lambda x_{2})\leq(1-\lambda)f(x_{1})+\lambda f(x_{2}).$
If $0<\lambda\leq\frac{1}{2},$ let

$\nu=\frac{1-2\lambda}{2(1-\lambda)},a=(2-2\lambda)x_{1}+(2\lambda-1)x_{2}\;{\text{and}}\;b=x_{2}.$

Then one can easily check that when $0<\lambda\leq\frac{1}{2},$ we have
$0<\nu\leq\frac{1}{2}$ and $a<b$. With these choices, we have

$\frac{a+b}{2}=(1-\lambda)x_{1}+\lambda x_{2}\;{\text{and}}\;(1-\nu)a+\nu
b=x_{1}.$

Substituting these quantities in (2.9) implies $f((1-\lambda)x_{1}+\lambda
x_{2})\leq(1-\lambda)f(x_{1})+\lambda f(x_{2}).$ This proves the desired
inequality for $0<\lambda\leq\frac{1}{2}.$
Now if $\frac{1}{2}\leq\lambda<1,$ let

$\nu=\frac{1}{2\lambda},a=x_{1}\;{\text{and}}\;b=(1-2\lambda)x_{1}+2\lambda
x_{2}.$

With these choices, we have $\frac{1}{2}<\nu\leq 1$ and $a<b$. Now
substituting these quantities in (2.10) implies the desired inequality for
$\frac{1}{2}\leq\lambda<1.$ This completes the proof. ∎

As for the geometric meaning of these refinements, it turns out we are dealing
with the interpolation of the function $f$ over the dyadic partition.

######  Proposition 2.9.

Let $f:[0,1]\to\mathbb{R}$ be any function, and let $N\in\mathbb{N}$. Then, if
$\nu_{i}=\frac{i}{2^{N}}$ for some $i=0,1,\cdots,2^{N},$ we have

$f(\nu_{i})+\sum_{j=1}^{N}A_{j}(\nu_{i})\Delta_{j}f(\nu_{i};0,1)=(1-\nu_{i})f(0)+\nu_{i}f(1).$
(2.11)

###### Proof.

Observe that when $\nu_{i}=\frac{i}{2^{N}},$ we have
$[2^{N}\nu_{i}]=2^{N}\nu_{i}=i.$ From Lemma LABEL:exact_difference, we have

$\displaystyle(1-\nu_{i})f(0)$ $\displaystyle+$
$\displaystyle\nu_{i}f(1)-\sum_{j=1}^{N}A_{j}(\nu_{i})\Delta_{j}f(\nu_{i};0,1)$
$\displaystyle=$
$\displaystyle\left([2^{N}\nu_{i}]+1-2^{N}\nu_{i}\right)f\left(\frac{[2^{N}\nu_{i}]}{2^{N}}\right)+\left(2^{N}\nu_{i}-[2^{N}\nu_{i}]\right)f\left(\frac{[2^{N}\nu_{i}]+1}{2^{N}}\right)$
$\displaystyle=$ $\displaystyle f\left(\frac{i}{2^{N}}\right)=f(\nu_{i}).$

This completes the proof. ∎

######  Proposition 2.10.

Let $f:[0,1]\to\mathbb{R}$ be a given function. If $f$ is continuous, then

$f(\nu)+\lim_{N\to\infty}\sum_{j=1}^{N}A_{j}(\nu)\Delta_{j}f(\nu;0,1)=(1-\nu)f(0)+\nu
f(1),$ (2.12)

uniformly in $\nu\in[0,1].$

###### Proof.

Let $N\in\mathbb{N}$ and define the function

$g_{N}(\nu)=\sum_{j=1}^{N}A_{j}(\nu)\Delta_{j}f(\nu;0,1).$

From Proposition 2.9, we have
$g(\nu_{i})=(1-\nu_{i})f(0)+\nu_{i}f(1)-f(\nu_{i}),$ when
$\nu_{i}=\frac{i}{2^{N}}$ for some $i=1,\cdots,2^{N}.$ Noting the definitions
of $A_{j}$ and $\Delta_{j}f$, one can easily see that $g_{N}$ is linear on
each dyadic interval
$I_{i}:=\left[\frac{i}{2^{N}},\frac{i+1}{2^{N}}\right],i=0,\cdots,2^{N}-1.$
Now since $g_{N}$ is linear on $I_{i}$ and $g_{N}$ coincides with the
continuous function $h(\nu):=(1-\nu)f(0)+\nu f(1)-f(\nu),$ it follows that
$g_{N}$ is the linear interpolation of $h$ at the dyadic partition of $[0,1].$
Since $f$ is continuous, it follows that $g_{N}\to h$ uniformly, completing
the proof. ∎

###  2.2. Log-Convex function

The proof of the following result follows from Theorem 2.5 on replacing $f$ by
$\log f.$

######  Corollary 2.11.

Let $f:[a,b]\to(0,\infty)$ be log-convex. Then for $0\leq\nu\leq 1$ and
$N\in\mathbb{N}$, we have

$f\left((1-\nu)a+\nu
b\right)\prod_{j=1}^{N}\left(\frac{f(x_{j}(\nu))f(y_{j}(\nu))}{f^{2}(z_{j}(\nu))}\right)^{A_{j}(\nu)}\leq
f^{1-\nu}(a)f^{\nu}(b),$ (2.13)

where $x_{j}(\nu),y_{j}(\nu)$ and $z_{j}(\nu)$ are as in the proof of Lemma
2.1.

On the other hand, applying Theorem 2.7 implies the following.

######  Corollary 2.12.

Let $f:[a,b]\to(0,\infty)$ be log-convex. Then for $0\leq\nu\leq\frac{1}{2}$
and $N\in\mathbb{N}$, we have

$f\left((1-\nu)a+\nu
b\right)\left(\frac{f(a)f(b)}{f^{2}(\frac{a+b}{2})}\right)^{1-A_{1}(\nu)}\geq
f^{1-\nu}(a)f^{\nu}(b)\prod_{j=1}^{N}\left(\frac{f(t_{j}(\nu))f(u_{j}(\nu))}{f^{2}(w_{j}(\nu))}\right)^{A_{j}(1-2\nu)},$
(2.14)

where $t_{j}(\nu),u_{j}(\nu)$ and $w_{j}(\nu)$ are obtained from the above
$x_{j}(\nu),y_{j}(\nu)$ and $z_{j}(\nu)$ on replacing $(\nu,a,b)$ by
$\left(1-2\nu,\frac{a+b}{2},b\right).$
On the other hand, if $\frac{1}{2}\leq\nu\leq 1,$ we have

$f\left((1-\nu)a+\nu
b\right)\left(\frac{f(a)f(b)}{f^{2}(\frac{a+b}{2})}\right)^{1-A_{1}(\nu)}\geq
f^{1-\nu}(a)f^{\nu}(b)\prod_{j=1}^{N}\left(\frac{f(t_{j}(\nu))f(u_{j}(\nu))}{f^{2}(w_{j}(\nu))}\right)^{A_{j}(2-2\nu)},$
(2.15)

where $t_{j}(\nu),u_{j}(\nu)$ and $w_{j}(\nu)$ are obtained from the above
$x_{j}(\nu),y_{j}(\nu)$ and $z_{j}(\nu)$ on replacing $(\nu,a,b)$ by
$\left(2-2\nu,a,\frac{a+b}{2}\right).$

The following is a squared additive version for log-convex functions. This
inequality will help prove some squared versions of certain means.

######  Theorem 2.13.

Let $f:[a,b]\to[0,\infty)$ be log-convex. Then for $0\leq\nu\leq 1$ and $N\geq
2$, we have

$\displaystyle f^{2}((1-\nu)a+\nu b)$ $\displaystyle+$ $\displaystyle
A_{1}^{2}(\nu)\Delta_{1}f^{2}(\nu;a,b)+\sum_{j=2}^{N}A_{j}(\nu)\Delta_{j}f^{2}(\nu;a,b)$
$\displaystyle\leq$ $\displaystyle\left((1-\nu)f(a)+\nu f(b)\right)^{2}.$

###### Proof.

We prove the result for $0\leq\nu\leq\frac{1}{2}.$ Since $f$ is log-convex, it
follows that $g=f^{2}$ is log-convex too, and hence is convex. Therefore,
Theorem 2.5 implies

$g((1-\nu)a+\nu
b)+\sum_{j=1}^{N}A_{j}(\nu)\Delta_{j}g(\nu;a,b)\leq(1-\nu)g(a)+\nu g(b),$

which implies, for $0\leq\nu\leq\frac{1}{2},$

$\displaystyle f^{2}((1-\nu)a+\nu b)$ $\displaystyle+$
$\displaystyle\nu^{2}\Delta_{1}f^{2}(\nu;a,b)+\sum_{j=2}^{N}A_{j}(\nu)\Delta_{j}f^{2}(\nu;a,b)$
(2.16) $\displaystyle\leq$ $\displaystyle\left((1-\nu)f(a)+\nu
f(b)\right)^{2}+H(\nu;a,b),$

where

$\displaystyle H(\nu;a,b)$ $\displaystyle=$ $\displaystyle(1-\nu)f^{2}(a)+\nu
f^{2}(b)+\nu^{2}\Delta_{1}f^{2}(\nu;a,b)-\nu\Delta_{1}f^{2}(\nu;a,b)$
$\displaystyle-\left((1-\nu)f(a)+\nu f(b)\right)^{2}$ $\displaystyle=$
$\displaystyle
2\nu(1-\nu)\left(f^{2}\left(\frac{a+b}{2}\right)-f(a)f(b)\right)$
$\displaystyle\leq$ $\displaystyle 0,$

where the last inequality follows from log-convexity of $f$. Since
$H(\nu;a,b)\leq 0$, it follows from (2.16) that

$\displaystyle f^{2}((1-\nu)a+\nu b)$ $\displaystyle+$
$\displaystyle\nu^{2}\Delta_{1}f^{2}(\nu;a,b)+\sum_{j=2}^{N}A_{j}(\nu)\Delta_{j}f^{2}(\nu;a,b)$
$\displaystyle\leq$ $\displaystyle\left((1-\nu)f(a)+\nu f(b)\right)^{2}.$

Similar computations imply the result for $\frac{1}{2}\leq\nu\leq 1.$ ∎

Then reversed squared versions maybe obtained in a similar way from Theorem
2.7 as follows.

######  Theorem 2.14.

Let $f:[a,b]\to[0,\infty)$ be log-convex and $N\in\mathbb{N}$. If
$0\leq\nu\leq\frac{1}{2},$ we have

$\displaystyle f^{2}((1-\nu)a+\nu b)$ $\displaystyle+$
$\displaystyle(1-\nu)^{2}\Delta_{1}f^{2}(\nu;a,b)+2\nu(1-\nu)\left(f(a)f(b)-f^{2}\left(\frac{a+b}{2}\right)\right)$
$\displaystyle\geq$ $\displaystyle\left((1-\nu)f(a)+\nu f(b)\right)^{2}$
$\displaystyle+\sum_{j=1}^{N}A_{j}(1-2\nu)\Delta_{j}f^{2}\left(1-2\nu;\frac{a+b}{2},b\right).$

If $\frac{1}{2}\leq\nu\leq 1,$ we have

$\displaystyle f^{2}((1-\nu)a+\nu b)$ $\displaystyle+$
$\displaystyle\nu^{2}\Delta_{1}f^{2}(\nu;a,b)+2\nu(1-\nu)\left(f(a)f(b)-f^{2}\left(\frac{a+b}{2}\right)\right)$
$\displaystyle\geq$ $\displaystyle\left((1-\nu)f(a)+\nu f(b)\right)^{2}$
$\displaystyle+\sum_{j=1}^{N}A_{j}(2-2\nu)\Delta_{j}f^{2}\left(2-2\nu;a,\frac{a+b}{2}\right).$

##  3\. Application

###  3.1. Refinements of means inequalities

In this section we present some interesting applications of the above
inequalities. The first result is the following refinement of Young’s
inequality.

######  Corollary 3.1.

Let $x,y>0,N\in\mathbb{N}$ and $0\leq\nu\leq 1.$ Then

$x\\#_{\nu}y+\sum_{j=1}^{N}A_{j}(\nu)\left(\sqrt[2^{j}]{y^{k_{j}(\nu)}x^{2^{j-1}-k_{j}(\nu)}}-\sqrt[2^{j}]{y^{k_{j}(\nu)+1}x^{2^{j-1}-k_{j}(\nu)-1}}\right)^{2}\leq
x\nabla_{\nu}y.$ (3.1)

###### Proof.

This follows from Theorem 2.5, on letting $f(t)=x^{1-t}y^{t},a=0,b=1.$ Then
$f$ is convex. Moreover, direct computations show that

$\Delta_{j}f(\nu;0,1)=\left(\sqrt[2^{j}]{y^{k_{j}(\nu)}x^{2^{j-1}-k_{j}(\nu)}}-\sqrt[2^{j}]{y^{k_{j}(\nu)+1}x^{2^{j-1}-k_{j}(\nu)-1}}\right)^{2}.$

∎

The above theorem has been recently proved in [sabjmaa] as a refinement of
Young’s inequality. This inequality refines the corresponding refinements
appearing in [5] and [11], where the inequality was proved only for $N=1,2.$

On the other hand, letting $f(t)=x!_{t}y$, the weighted harmonic mean, we
obtain the following refinement of the arithmetic-harmonic mean inequality.

######  Corollary 3.2.

Let $x,y>0,N\in\mathbb{N}$ and $0\leq\nu\leq 1.$ Then

$x!_{\nu}y+\sum_{j=1}^{N}A_{j}(\nu)\left(x!_{\alpha_{j}(\nu)}y+x!_{\beta_{j}(\nu)}y-2x!_{\gamma_{j}(\nu)}y\right)\leq
x\nabla_{\nu}y,$ (3.2)

where
$\alpha_{j}(\nu)=\frac{[2^{j-1}\nu]}{2^{j-1}},\beta_{j}(\nu)=\frac{[2^{j-1}\nu]+1}{2^{j-1}}$
and $\gamma_{j}(\nu)=\frac{\alpha_{j}(\nu)+\beta_{j}(\nu)}{2}.$

This inequality is a significant refinement of the corresponding inequality in
[12], where the inequality was proved only for $N=1.$

Now noting log-convexity of the function $t\mapsto x!_{t}y$ on $[0,1],$ and
applying Corollary 2.11, we get the following multiplicative refinement of the
geometric-harmonic mean inequality.

######  Corollary 3.3.

Let $x,y>0,N\in\mathbb{N}$ and $0\leq\nu\leq 1$, we have

$x!_{\nu}y\prod_{j=1}^{N}\left(\frac{(x!_{\alpha_{j}(\nu)}y)(x!_{\beta_{j}(\nu)}y)}{(x!_{\gamma_{j}(\nu)}y)^{2}}\right)^{A_{j}(\nu)}\leq
x\\#_{\nu}y,$

where
$\alpha_{j}(\nu)=\frac{[2^{j-1}\nu]}{2^{j-1}},\beta_{j}(\nu)=\frac{[2^{j-1}\nu]+1}{2^{j-1}}$
and $\gamma_{j}(\nu)=\frac{\alpha_{j}(\nu)+\beta_{j}(\nu)}{2}.$

When $N=1$, Corollary 3.3, reduces to

$(x!_{\nu}y)\left(\frac{x\nabla y}{x\\#y}\right)^{2\nu}\leq
x\\#_{\nu}y,0\leq\nu\leq\frac{1}{2}$

and

$(x!_{\nu}y)\left(\frac{x\nabla y}{x\\#y}\right)^{2(1-\nu)}\leq
x\\#_{\nu}y,\frac{1}{2}\leq\nu\leq 1.$

The constant $\left(\frac{x\nabla y}{x\\#y}\right)^{2}$ appearing in these
inequalities is called the Kantorovich constant, and has appeared in recent
refinements of these mean inequalities. One can see [6] as a recent reference
treating some inequalities using this constant.

As for the squared version, applying Theorem 2.13 to the log-convex functions
$t\mapsto x\\#_{t}y$ and $t\mapsto x!_{t}y$ implies the following. The first
inequality refines the corresponding results in [3] and [11], while the other
inequality is new.

######  Corollary 3.4.

Let $x,y>0,0\leq\nu\leq 1,N\geq 2$ and

$\alpha_{j}(\nu)=\frac{k_{j}(\nu)}{2^{j-1}},\beta_{j}(\nu)=\frac{k_{j}(\nu)+1}{2^{j-1}}\;{\text{and}}\;\gamma_{j}(\nu)=\frac{\alpha_{j}(\nu)+\beta_{j}(\nu)}{2}.$

Then

$\displaystyle\left(x\\#_{\nu}y\right)^{2}$ $\displaystyle+$ $\displaystyle
A_{1}^{2}(\nu)(x-y)^{2}+\sum_{j=2}^{N}A_{j}(\nu)\left(x^{1-\alpha_{j}(\nu)}y^{\alpha_{j}(\nu)}-x^{1-\beta_{j}(\nu)}y^{\beta_{j}(\nu)}\right)^{2}$
$\displaystyle\leq$ $\displaystyle(x\nabla_{\nu}y)^{2},$

and

$\displaystyle\left(x!_{\nu}y\right)^{2}$ $\displaystyle+$ $\displaystyle
2A_{1}^{2}(\nu)(x^{2}\nabla
y^{2}-(x!y)^{2})+\sum_{j=2}^{N}A_{j}(\nu)\left((x!_{\alpha_{j}(\nu)}y)^{2}+(x!_{\beta_{j}(\nu)}y)^{2}-2(x!_{\gamma_{j}(\nu)}y)^{2}\right)$
$\displaystyle\leq$ $\displaystyle(x\nabla_{\nu}y)^{2}.$

###  3.2. Reversed Version

Applying Theorem 2.7 to the function $f(t)=x\\#_{t}y$ implies the following
reversed version of Young’s inequality.

######  Corollary 3.5.

For $x,y>0$, let $f(t)=x\\#_{t}y$ and let $N\in\mathbb{N}$. If
$0\leq\nu\leq\frac{1}{2},$ we have

$\displaystyle x\\#_{\nu}y+(1-\nu)(\sqrt{x}-\sqrt{y})^{2}\geq
x\nabla_{\nu}y+\sum_{j=1}^{N}A_{j}(1-2\nu)\Delta_{j}f\left(1-2\nu;\frac{1}{2},1\right).$

On the other hand, if $\frac{1}{2}\leq\nu\leq 1,$ we have

$\displaystyle x\\#_{\nu}y+\nu(\sqrt{x}-\sqrt{y})^{2}\geq
x\nabla_{\nu}y+\sum_{j=1}^{N}A_{j}(2-2\nu)\Delta_{j}f\left(2-2\nu;0,\frac{1}{2}\right).$

These inequalities refine those in [5] and [11]. Then an arithmetic-harmonic
reversed version maybe obtained by applying Theorem 2.7 to the function
$f(t)=x!_{t}y$ as follows.

######  Corollary 3.6.

For $x,y>0$, let $f(t)=x\\#_{t}y$ and let $N\in\mathbb{N}$. If
$0\leq\nu\leq\frac{1}{2},$ we have

$\displaystyle x!_{\nu}y+(1-\nu)\left(x+y-2x!y\right)\geq
x\nabla_{\nu}y+\sum_{j=1}^{N}A_{j}(1-2\nu)\Delta_{j}f\left(1-2\nu;\frac{1}{2},1\right).$

On the other hand, if $\frac{1}{2}\leq\nu\leq 1,$ we have

$\displaystyle x!_{\nu}y+\nu(x+y-2x!y)\geq
x\nabla_{\nu}y+\sum_{j=1}^{N}A_{j}(2-2\nu)\Delta_{j}f\left(2-2\nu;0,\frac{1}{2}\right).$

These inequalities refine those in [6].

Similarly, noting log-convexity of the function $f(t)=x!_{t}y$, we may apply
Corollary 2.12 to obtain reversed multiplicative version of the harmonic-
geometric mean inequality. We leave the application to the reader.

Following the same guideline, we may obtain reversed squared versions by
applying Theorem 2.14 to the functions $t\mapsto x\\#_{t}y$ and $x\mapsto
x!_{t}y.$ Observe that when $f(t)=x\\#_{t}y$ we have
$f(a)f(b)-f^{2}\left(\frac{a+b}{2}\right)=0.$ Therefore, applying Theorem 2.14
implies the following inequalities, which refine the corresponding
inequalities in [3] and [11].

######  Corollary 3.7.

Let $x,y>0$ and $N\geq 1.$ If $0\leq\nu\leq 1,$ we have

$\displaystyle\left(x\\#_{\nu}y\right)^{2}+(1-\nu)^{2}(x-y)^{2}\geq(x\nabla_{\nu}y)^{2}+\sum_{j=1}^{N}A_{j}(1-2\nu)\Delta_{j}f^{2}\left(1-2\nu;\frac{1}{2},1\right).$

If $\frac{1}{2}\leq\nu\leq 1,$ we have

$\displaystyle\left(x\\#_{\nu}y\right)^{2}+\nu^{2}(x-y)^{2}\geq(x\nabla_{\nu}y)^{2}+\sum_{j=1}^{N}A_{j}(2-2\nu)\Delta_{j}f^{2}\left(1-2\nu;0,\frac{1}{2}\right).$

Now letting $g(t)=x!_{t}y$ we obtain the following new inequalities for the
arithmetic-harmonic means.

######  Corollary 3.8.

Let $x,y>0$ and $N\geq 1.$ If $0\leq\nu\leq 1,$ we have

$\displaystyle\left(x!_{\nu}y\right)^{2}$ $\displaystyle+$ $\displaystyle
2(1-\nu)^{2}(x^{2}\nabla
y^{2}-(x!y)^{2})+2\nu(1-\nu)\left(xy-\left(\frac{2xy}{x+y}\right)^{2}\right)$
$\displaystyle\geq$
$\displaystyle(x\nabla_{\nu}y)^{2}+\sum_{j=1}^{N}A_{j}(1-2\nu)\Delta_{j}g^{2}\left(1-2\nu;\frac{1}{2},1\right).$

If $\frac{1}{2}\leq\nu\leq 1,$ we have

$\displaystyle\left(x!_{\nu}y\right)^{2}$ $\displaystyle+$ $\displaystyle
2\nu^{2}(x^{2}\nabla
y^{2}-(x!y)^{2})+2\nu(1-\nu)\left(xy-\left(\frac{2xy}{x+y}\right)^{2}\right)$
$\displaystyle\geq$
$\displaystyle(x\nabla_{\nu}y)^{2}+\sum_{j=1}^{N}A_{j}(2-2\nu)\Delta_{j}g^{2}\left(1-2\nu;0,\frac{1}{2}\right).$

###  3.3. Some $L^{p}$ inequalities

Let $(X,\mathcal{M},\mu)$ be a measure space, and let $0<p<q<r.$ Then
$L^{p}\cap L^{r}\subset L^{q}$ and

$\|f\|_{q}\leq\|f\|_{p}^{\nu}\|f\|_{r}^{1-\nu},\;{\text{where}}\;f\in
L^{p}\cap L^{r}\;{\text{and}}\;\nu=\frac{q^{-1}-r^{-1}}{p^{-1}-r^{-1}}.$

This inequality can be modified using Corollary 2.11 and a reversed version
can be obtained using Corollary 2.12.

######  Proposition 3.9.

Let $(X,\mathcal{M},\mu)$ be a measure space, $0<p<q<r$ and $\nu$ be as above.
If $f\in L^{p}\cap L^{r}$ and $N\in\mathbb{N}$, then we have

$\|f\|_{q}\prod_{j=1}^{N}\left(\frac{\|f\|_{x_{j}^{-1}(\nu)}\|f\|_{y_{j}^{-1}(\nu)}}{\|f\|^{2}_{z_{j}^{-1}(\nu)}}\right)^{A_{j}(\nu)}\leq\|f\|_{p}^{\nu}\|f\|_{r}^{1-\nu}.$

###### Proof.

It is easy to check that the function $h(t)=\|f\|_{1/t}$ is log-convex on
$[r^{-1},p^{-1}]$. Then direct application of Corollary 2.11 implies the
result. ∎

In particular, when $N=1$, the above proposition implies

$\|f\|_{q}\leq\left\\{\begin{array}[]{cc}\|f\|_{r}^{1-2\nu}\|f\|^{2\nu}_{\frac{2pr}{p+r}},&0\leq\nu\leq\frac{1}{2}\\\
\|f\|_{p}^{2\nu-1}\|f\|^{2-2\nu}_{\frac{2pr}{p+r}},&\frac{1}{2}\leq\nu\leq
1\end{array}\right\\}\leq\|f\|_{p}^{\nu}\|f\|_{r}^{1-\nu}.$

The condition $0\leq\nu\leq\frac{1}{2}$ can be interpreted as
$\frac{2pr}{p+r}\leq q\leq r$, while $\frac{1}{2}\leq\nu\leq 1$ means $p\leq
q\leq\frac{2pr}{p+r}.$

Moreover, a reversed version maybe obtained using Corollary 2.12.

######  Proposition 3.10.

Let $(X,\mathcal{M},\mu)$ be a measure space, $0<p<q<r$ and
$\nu=\frac{q^{-1}-r^{-1}}{p^{-1}-r^{-1}}$. If $\frac{2pr}{p+r}\leq q\leq r$,
$f\in L^{p}\cap L^{r}$ and $N\in\mathbb{N}$, then

$\|f\|_{q}\geq\|f\|^{2-2\nu}_{\frac{2pr}{p+r}}\|f\|_{p}^{2\nu-1}\prod_{j=1}^{N}\left(\frac{\|f\|_{t_{j}^{-1}(\nu)}\|f\|_{u_{j}^{-1}(\nu)}}{\|f\|^{2}_{w_{j}^{-1}(\nu)}}\right)^{A_{j}(1-2\nu)}\geq\|f\|^{2-2\nu}_{\frac{2pr}{p+r}}\|f\|_{p}^{2\nu-1},$

where $t_{j},u_{j}$ and $z_{j}$ are obtained from $x_{j},y_{j}$ and $z_{j}$ by
replacing $(\nu,a,b)$ with $\left(1-2\nu,\frac{p+r}{2pr},p^{-1}\right).$ On
the other hand, if $p\leq q\leq\frac{2pr}{p+r},$ then

$\|f\|_{q}\geq\|f\|^{2\nu}_{\frac{2pr}{p+r}}\|f\|_{r}^{1-2\nu}\prod_{j=1}^{N}\left(\frac{\|f\|_{t_{j}^{-1}(\nu)}\|f\|_{u_{j}^{-1}(\nu)}}{\|f\|^{2}_{w_{j}^{-1}(\nu)}}\right)^{A_{j}(1-2\nu)}\geq\|f\|^{2\nu}_{\frac{2pr}{p+r}}\|f\|_{r}^{1-2\nu},$

where $t_{j},u_{j}$ and $z_{j}$ are obtained from $x_{j},y_{j}$ and $z_{j}$ by
replacing $(\nu,a,b)$ with $\left(2-2\nu,r^{-1},\frac{p+r}{2pr}\right).$

Propositions 3.9 and 3.10 have been obtained using log-convexity of the
function $h(t)=\|f\|_{t^{-1}}.$ In fact, noting log-convexity of the function
$h(t)=\|f\|_{t}^{t}$, we obtain the same results! This is due to the
equivalence of log-convexity of the functions $t\mapsto\|f\|_{t^{-1}}$ and
that of $\|f\|_{t}^{t}.$ We refer the reader to [9] where these relations
between the different log-convex function criteria have been discussed.

The celebrated three lines lemma of Hadamard states the following.

######  Lemma 3.11.

Let $\mathbb{D}=\\{z\in\mathbb{C}:0\leq\Re z\leq 1\\}$ and let
$\varphi:\mathbb{D}\to\mathbb{C}$ be continuous on $\mathbb{D}$ and analytic
in the interior of $\mathbb{D}$. Then the function $f:[0,1]\to\mathbb{R}$
defined by $f(x)=\sup_{y}|\varphi(x+iy)|$ is log-convex.

This lemma is an extremely useful tool in the theory of complex functions. In
particular, this lemma becomes handy in proving different interpolation
versions of bounded linear operators between $L^{p}$ spaces.

Log-convexity implied by Lemma 3.11 allows us to apply our refined and
reversed versions for log-convex functions. In the following proposition, we
present one term refinement and reverse.

######  Proposition 3.12.

Let $\mathbb{D}=\\{z\in\mathbb{C}:0\leq\Re z\leq 1\\}$ and let
$\varphi:\mathbb{D}\to\mathbb{C}$ be continuous on $\mathbb{D}$ and analytic
in the interior of $\mathbb{D}$. Then the function $f:[0,1]\to\mathbb{R}$
defined by $f(x)=\sup_{y}|\varphi(x+iy)|$ satisfies the following

$f(x)\leq\left\\{\begin{array}[]{cc}f^{1-2x}(0)f^{2x}\left(\frac{1}{2}\right),&0\leq
x\leq\frac{1}{2}\\\
f^{2x-1}(1)f^{2-2x}\left(\frac{1}{2}\right),&\frac{1}{2}\leq x\leq
1\end{array}\right.$

and

$f(x)\geq\left\\{\begin{array}[]{cc}f^{2x-1}(1)f^{2-2x}\left(\frac{1}{2}\right),&0\leq
x\leq\frac{1}{2}\\\ f^{1-2x}(0)f^{2x}\left(\frac{1}{2}\right),&\frac{1}{2}\leq
x\leq 1\end{array}\right..$

###  3.4. Operator versions

The following theorem provides a refinement of the well known Heinz inequality
and its reverse. The proof follows immediately noting convexity of the Heinz
means, see [1].

######  Theorem 3.13.

For $A,B\in\mathbb{M}_{n}^{+},X\in\mathbb{M}_{n},0\leq\nu\leq 1$ and any
unitarily invariant norm $\||\;\;\||$, let

$f(\nu)=\||A^{\nu}XB^{1-\nu}+A^{1-\nu}XB^{\nu}\||.$

Then we have the following refinement of Heinz inequality

$\displaystyle\||A^{\nu}XB^{1-\nu}+A^{1-\nu}XB^{\nu}\||+\sum_{j=1}^{N}A_{j}(\nu)\Delta_{j}f(\nu;0,1)\leq\||AX+XB\||.$

Moreover, if $0\leq\nu\leq\frac{1}{2},$ we have

$\displaystyle\||A^{\nu}XB^{1-\nu}+A^{1-\nu}XB^{\nu}\||$ $\displaystyle+$
$\displaystyle 2(1-\nu)\left(\||AX+XB\||-\||\sqrt{A}X\sqrt{B}\||\right)$
$\displaystyle\geq$
$\displaystyle\||AX+XB\||+\sum_{j=1}^{N}A_{j}(1-2\nu)\Delta_{j}f\left(1-2\nu;\frac{1}{2},1\right).$

On the other hand, if $\frac{1}{2}\leq\nu\leq 1,$ we have

$\displaystyle\||A^{\nu}XB^{1-\nu}+A^{1-\nu}XB^{\nu}\||$ $\displaystyle+$
$\displaystyle 2\nu\left(\||AX+XB\||-\||\sqrt{A}X\sqrt{B}\||\right)$
$\displaystyle\geq$
$\displaystyle\||AX+XB\||+\sum_{j=1}^{N}A_{j}(2-2\nu)\Delta_{j}f\left(2-2\nu;0,\frac{1}{2}\right).$

In [9], it is shown that for $A,B\in\mathbb{M}_{n}^{+}$ and
$X\in\mathbb{M}_{n},$ the functions

$t\to\||A^{t}XB^{1-t}\||,t\to\||A^{t}XB^{1-t}\||\;\||A^{1-t}X^{B}{t}\||,t\to{\rm{tr}}(A^{t}XB^{1-t}X^{*})$

are log-convex on $[0,1].$ Therefore, we may apply Corollaries 2.11 and 2.12
to obtain refinements and reversed versions for such functions.
For the $\|\;\;\|_{2}$ norm, we can prove log convexity of the Heinz means,
which allows us to obtain further refinements of the Heinz inequality by
applying Corollaries 2.11 and 2.12.

######  Proposition 3.14.

Let $A,B\in\mathbb{M}_{n}^{+}$ and $X\in\mathbb{M}_{n}$, and define
$f(\nu)=\|A^{\nu}XB^{1-\nu}+A^{1-\nu}XB^{\nu}\|_{2}.$ Then $f$ is log-convex
on $[0,1]$.

###### Proof.

Since $A,B\in\mathbb{M}_{n}^{+}$, there are diagonal matrices
$D_{1}:={\text{diag}}(\lambda_{i}),D_{2}:={\text{diag}}(\mu_{i})$ and
unitarily matrices $U,V$ such that $\lambda_{i},\mu_{i}\geq 0,$
$A=UD_{1}U^{*}$ and $B=VD_{2}V^{*}.$ Letting $Y=U^{*}XV,$ we have

$A^{\nu}XB^{1-\nu}+A^{1-\nu}XB^{\nu}=U(\lambda_{i}^{\nu}y_{ij}\mu_{j}^{1-\nu}+\lambda_{i}^{1-\nu}y_{ij}\mu_{j}^{\nu})V^{*}.$

Since $\|\;\;\|_{2}$ is a unitarily invariant norm, we have

$\displaystyle f^{2}(\nu)$ $\displaystyle=$
$\displaystyle\|U(\lambda_{i}^{\nu}y_{ij}\mu_{j}^{1-\nu}+\lambda_{i}^{1-\nu}y_{ij}\mu_{j}^{\nu})V^{*}\|_{2}^{2}$
$\displaystyle=$
$\displaystyle\sum_{i,j}\left(\lambda_{i}^{\nu}\mu_{j}^{1-\nu}+\lambda_{i}^{1-\nu}\mu_{j}^{\nu}\right)^{2}|y_{ij}|^{2}.$

Notice that each summand is log-convex, being the square of a log-convex
function. This implies that $f^{2}$ is log-convex. Consequently, $f$ is log-
convex. ∎

Letting $f(\nu)=\|A^{\nu}XB^{1-\nu}+A^{1-\nu}XB^{\nu}\|_{2}$ and applying
Theorem 2.13 imply the following squared version of Heinz inequality.

######  Corollary 3.15.

Let $A,B\in\mathbb{M}_{n}^{+},X\in\mathbb{M}_{n},0\leq\nu\leq 1$ and $N\geq
2.$ Then

$\displaystyle\|A^{\nu}XB^{1-\nu}+A^{1-\nu}XB^{\nu}\|_{2}^{2}+2A_{1}^{2}(\nu)\left(\|AX+XB\|_{2}^{2}-2\|A^{\frac{1}{2}}XB^{\frac{1}{2}}\|_{2}^{2}\right)$
$\displaystyle\hskip
8.5359pt+\sum_{j=2}^{N}A_{j}(\nu)\Delta_{j}f^{2}(\nu;0,1)\leq\|AX+XB\|_{2}^{2}.$

We leave the application of Corollaries 2.11 and 2.12 to the reader.

Further operator versions maybe obtained using Lemma 1.1. The following
operator versions refine the corresponding results in [5] and [11].

######  Proposition 3.16.

Let $A,B\in\mathbb{M}_{n}^{++}$ and $0\leq\nu\leq 1.$ Then for
$\alpha_{j}(\nu)=\frac{k_{j}(\nu)}{2^{j-1}},\beta_{j}(\nu)=\frac{k_{j}(\nu)+1}{2^{j-1}},\gamma_{j}(\nu)=\frac{\alpha_{j}(\nu)+\beta_{j}(\nu)}{2}$
and $N\in\mathbb{N}$, we have

$\displaystyle
A\\#_{\nu}B+\sum_{j=1}^{N}A_{j}(\nu)\left(A\\#_{\alpha_{j}(\nu)}+A\\#_{\beta_{j}(\nu)}-2A\\#_{\gamma_{j}(\nu)}B\right)\leq
A\nabla_{\nu}B.$

###### Proof.

In Corollary 3.1, let $x=1$, expand the summand and apply Lemma 1.1 with $y$
replaced by $X=A^{-\frac{1}{2}}BA^{-\frac{1}{2}}.$ Then the result follows
upon conjugating both sides with $A^{\frac{1}{2}}.$ ∎

In a similar way one may obtain reversed versions by applying Corollary 3.5.
This provides refinements of the reversed versions of [11]. The following is
an operator arithmetic-harmonic version, refining the corresponding results in
[12].

######  Proposition 3.17.

Let $A,B\in\mathbb{M}_{n}^{++}$ and $0\leq\nu\leq 1.$ Then for
$\alpha_{j}(\nu)=\frac{k_{j}(\nu)}{2^{j-1}},\beta_{j}(\nu)=\frac{k_{j}(\nu)+1}{2^{j-1}},\gamma_{j}(\nu)=\frac{\alpha_{j}(\nu)+\beta_{j}(\nu)}{2}$
and $N\in\mathbb{N}$, we have

$\displaystyle
A!_{\nu}B+\sum_{j=1}^{N}A_{j}(\nu)\left(A!_{\alpha_{j}(\nu)}+A!_{\beta_{j}(\nu)}-2A!_{\gamma_{j}(\nu)}B\right)\leq
A\nabla_{\nu}B.$

The proof follows immediately on applying Lemma 1.1 together with Corollary
3.2. On the other hand, applying Corollary 3.6 implies the following
refinement of the corresponding inequalities in [6].

######  Proposition 3.18.

Let $A,B\in\mathbb{M}_{n}^{++}$ and $N\in\mathbb{N}$. If $0\leq\nu\leq 1,$ we
have

$\displaystyle A!_{\nu}B$ $\displaystyle+$ $\displaystyle(1-\nu)(A+B-2A!B)$
$\displaystyle\geq$ $\displaystyle
A\nabla_{\nu}B+\sum_{j=1}^{N}A_{j}(1-2\nu)\left(A!_{\alpha_{j}(\nu)}B+A!_{\beta_{j}(\nu)}B-2A!_{\gamma_{j}(\nu)}B\right),$

where
$\alpha_{j}(\nu)=\frac{1}{2}\left(1-\frac{k_{j}(1-2\nu)}{2^{j-1}}\right)+\frac{k_{j}(1-2\nu)}{2^{j-1}},$
$\beta_{j}(\nu)=\frac{1}{2}\left(1-\frac{k_{j}(1-2\nu)+1}{2^{j-1}}\right)+\frac{k_{j}(1-2\nu)+1}{2^{j-1}}$
and $\gamma_{j}(\nu)=\frac{\alpha_{j}(\nu)+\beta_{j}(\nu)}{2}.$
On the other hand, if $\frac{1}{2}\leq\nu\leq 1,$ we have

$\displaystyle A!_{\nu}B$ $\displaystyle+$ $\displaystyle\nu(A+B-2A!B)$
$\displaystyle\geq$ $\displaystyle
A\nabla_{\nu}B+\sum_{j=1}^{N}A_{j}(2-2\nu)\left(A!_{\alpha_{j}(\nu)}B+A!_{\beta_{j}(\nu)}B-2A!_{\gamma_{j}(\nu)}B\right),$

where $\alpha_{j}(\nu)=\frac{k_{j}(2-2\nu)}{2^{j-1}},$
$\beta_{j}(\nu)=\frac{k_{j}(2-2\nu)+1}{2^{j-1}}$ and
$\gamma_{j}(\nu)=\frac{\alpha_{j}(\nu)+\beta_{j}(\nu)}{2}.$

The following is an interesting one-term multiplicative refinement of the
operator geometric-harmonic mean inequality.

######  Theorem 3.19.

Let $A,B\in\mathbb{M}_{n}^{++}$ and $0\leq\nu\leq 1.$ Then

$\displaystyle(A!_{\nu}B)\left(\frac{A^{-1}B+2I+B^{-1}A}{4}\right)^{r}\leq
A\\#_{\nu}B,$

where $r=\min\\{\nu,1-\nu\\}.$

###### Proof.

We prove the desired inequality for $0\leq\nu\leq\frac{1}{2}.$ In Corollary
3.3, let $N=1$ and $x=1$, to get
$(1!_{\nu}y)\left(\frac{1+y}{2\sqrt{y}}\right)^{2\nu}\leq 1\\#_{\nu}y,$ or

$\displaystyle\frac{1}{4^{\nu}}\left((1-\nu)+\nu
y^{-1}\right)^{-1}\left(y+2+y^{-1}\right)^{\nu}\leq y^{\nu}.$ (3.3)

Let $X=A^{-\frac{1}{2}}BA^{-\frac{1}{2}}$ and apply Lemma 1.1. The left hand
side of (3.3) becomes

$\displaystyle\frac{1}{4^{\nu}}\left((1-\nu)I+\nu
A^{\frac{1}{2}}B^{-1}A^{\frac{1}{2}}\right)^{-1}\left(A^{-\frac{1}{2}}BA^{-\frac{1}{2}}+2I+A^{\frac{1}{2}}B^{-1}A^{\frac{1}{2}}\right)^{\nu}$
(3.4) $\displaystyle=$
$\displaystyle\frac{1}{4^{\nu}}\left[A^{-\frac{1}{2}}(A!_{\nu}B)A^{-\frac{1}{2}}\right]\left[A^{\frac{1}{2}}\left(A^{-1}B+2I+B^{-1}A\right)^{\nu}A^{-\frac{1}{2}}\right]$
$\displaystyle=$ $\displaystyle
A^{-\frac{1}{2}}(A!_{\nu}B)\left(\frac{A^{-1}B+2I+B^{-1}A}{4}\right)^{\nu}A^{-\frac{1}{2}}.$

On the other hand, the right hand side of (3.3) is simply
$\left(A^{-\frac{1}{2}}BA^{-\frac{1}{2}}\right)^{\nu}.$ This together with
(3.4) imply the desired inequality, upon conjugating both sides with
$A^{\frac{1}{2}}.$ This completes the proof. ∎

## References

  * [1] R. Bhatia, Matrix analysis, Springer-Verlag, New York, 1997.
  * [2] T. Furuta, J. Micic Hot and J. Pecaric, Mond-Pecaric Method in Operator Inequalities, Element, Zagreb, 2005.
  * [3] O. Hirzallah and F. Kittaneh, Matrix Young inequalities for the Hilbert-Schmidt norm, Linear Algebra Appl. 308 (2000), 77–84.
  * [4] F. Kittaneh, On the convexity of the Heinz mean, Integr. Equ. Oper. Theory 68 (2010), 519–527.
  * [5] F. Kittaneh and Y. Manasrah, Improved Young and Heinz inequalities for matrices, J. Math. Anal. Appl. 36 (2010), 262–269.
  * [6] W. Liao and J. Wu, Reverse arithmetic-harmonic mean and mixed mean operator inequalities, J. Inequal. Appl., 2015:215.
  * [7] Y. Manasrah and F. Kittaneh, A generalization of two refined Young inequalities, Positivity 19 (2015), 757–768.
  * [8] M. Sababheh, Convex functions and means of matrices, Math. Ineq. Appl., accepted.
  * [9] M. Sababheh, Log and harmonically log-convex functions related to matrix norms, Operators and Matrices, in press.
  * [10] M. Sababheh, Integral inequalities of the Heinz means as convex functions, J. Math. Ineq., 10 (2) (2016), 313–325.
  * [11] J. Zhao and J. Wu, Operator inequalities involving improved Young and its reverse inequalities, J. Math. Anal. Appl. 421 (2015) 1779–1789.
  * [12] H. Zuo, G. Shi and M. Fujii, Refined Young inequality with Kantorovich constant, J. Math. Inequal. 5(4)(2011), 551-556 .