# Queue Length Estimation at Traffic Signals: Connected Vehicles with Range
Measurement Sensors

Gurcan Comert<EMAIL_ADDRESS>Mecit Cetin<EMAIL_ADDRESS>Department of Computer Sc., Phys., and Eng., Benedict College, 1600 Harden
St., Columbia, SC 29204 USA  Department of Civil and Env. Engineering, Old
Dominion University, 135 Kaufman Hall, Norfolk, VA 23529 USA

###### Abstract

Today vehicles are becoming a rich source of data as they are equipped with
localization or tracking and with wireless communications technologies. With
the increasing interest in automated- or self- driving technologies, vehicles
are also being equipped with range measuring sensors (e.g., LIDAR, stereo
cameras, and ultrasonic) to detect other vehicles and objects in the
surrounding environment. It is possible to envision that such vehicles could
share their data with the transportation infrastructure elements (e.g., a
traffic signal controller) to enable different mobility and safety
applications. Data from these connected vehicles could then be used to
estimate the system state in real-time. This paper develops queue length
estimators from connected vehicles equipped with range measurement sensors.
Simple plug-and-play models are developed for queue length estimations without
needing ground truth queue lengths by extending the previous formulations. The
proposed method is simple to implement and can be adopted to cyclic queues at
traffic signals with known phase lengths. The derived models are evaluated
with data from microscopic traffic simulations. From numerical experiments,
the QLE model with range sensors improves the errors as much as $25\%$ in
variance-to-mean ratio and $5\%$ in coefficient of variation at low $\leq
20\%$ market penetration rates.

###### keywords:

Applied probability, queue length estimation, connected and autonomous
vehicles, range measurement sensor, sensor fusion.

##  1 Introduction

Tracking individual vehicles traversing the transportation network in real
time has gained significant momentum lately due to recent developments in
Connected Vehicles (CVs) technologies. Real-time data from CV enable observing
complex system dynamics and support new applications to improve system
efficiency and safety. However, not all vehicles may be equipped with CV
technologies nor all drivers are willing to share their locations.
Consequently, predicting the overall system behavior based on data from a
fraction of vehicles has been an area of active research for more than three
decades now. Queue Length (QL) is a fundamental performance criteria for any
system with multiple users demanding service from a server as for real-time
control of signalized intersections or port terminal gates. It is critical for
applications such as emergency management and reliability of the network.
Models based on performance measures can be input for traffic control,
facility design, and construction planning purposes. Specifically, the real-
time (adaptive) control demands high accuracy from such models. Today,
vehicles are becoming a rich data and computational resource as they are
equipped with localization or tracking, wireless communication technologies,
and microprocessors. Furthermore, with the increasing interest on automated-
or self- driving technologies, vehicles are also being equipped with range
measuring sensors (e.g., LIDAR, stereo cameras, and ultrasonic) to detect
other vehicles and objects in the surrounding environment. It is possible to
envision that such vehicles could share their data with the transportation
infrastructure elements (e.g., a signal controller) to enable various traffic
mobility and safety applications. Data from these CVs could then be used to
estimate the system state in real-time. Consequently, this paper is focused on
estimation of QL from such CVs that may represent partial vehicles in the
traffic or other systems (Nam et al. (2009)). Connected and autonomous
vehicles (CAVs) or autonomous vehicles can also be treated similarly.

###  1.1 Background

Queue length estimation (QLE) from connected vehicle at traffic signals is a
well-investigated problem. In basic form, there is no feedback from
environment, thus, true queue length values are not known. This makes critical
difference with the prediction formulations with series of true queue lengths
are available where researchers used filtering, fit models, and deep learning
methods (Tiaprasert et al. (2015); Yin et al. (2018); Gao et al. (2020);
Comert et al. (2019)).

In the literature, researchers obtained CV based queues via reconstructing
trajectories (Argote et al. (2011); Cetin (2012); Cheng et al. (2012); Xu et
al. (2017); Rompis et al. (2018); Zhao et al. (2019b)). Stochastic
formulations are desirable as increase in number of related paper show.
Researchers developed important models using similar approach. Within the
scope and space, for more information on general queue length estimation,
readers are referred to recent survey studies on QLE methods from CAVs (Zhao
et al. (2019b); Guo et al. (2019)) and queue estimation bibliography by
(Asanjarani et al. (2017)). Hence here, we focus on related previous works on
stochastic QLEs from CVs are discussed.

Stochastic models of QLEs from CVs mainly concerns with stationary (behavior
after sufficiently long time interval) or non-stationary (short-time
intervals, by cycle, and red phase). Such classification is impacted by
treatment of arrival rate, distribution, and market penetration rate (MPR)
(probe percentage). For instance, cycle-to-cycle estimation of QLs including
green arrivals was modeled in (Tan et al. (2019)) and real-time applications
with high frequency connected vehicle data was presented by(Mei et al.
(2019)).

In queueing, detection the stopped vehicles accurately is crucial as it will
be impacting the back of queue. This problem was addressed in (Liu et al.
(2019)). Reported error levels in similar QLE studies may exceed $10\%$ of
true queue lengths (Argote et al. (2011)) which would require $80\%$ CV
penetration rate. In order to improve the accuracy, data fusion from different
traffic surveillance technologies are also considered such as loops (Li et al.
(2013); Comert (2013a)), license plate recognition (Tan et al. (2020)), and
inclusion of radio frequency identification Wu and Yang (2013) which acts as
filtered QLE would give higher errors. Queue lengths for constant arrival rate
at the end of red phase act like autoregressive (2) time series with large
coefficients with opposite signs or moving average (1) with positive high
coefficient. Therefore, filtering for a true QLs would over-smooth the series.
With back of queue estimation and convex optimization, similar error levels at
$p=20\%$ was observed (Yang and Menendez (2018)). Range sensor applications in
traffic in some capabilities are discussed in (Mandal et al. (2020)). Authors,
placed ultrasonic sensors capable Arduinos on lane markings and reported
distance that vehicles could be detected for queue lengths. Although we seek
higher penetration levels, efficient estimators performing at low MPRs would
be desirable as we will need to address efficient data management. Question of
how much data we need is important not to overload systems or security
reasons. Partially supporting these, in an interesting report, researchers
found in a simulation environment, over 200 vehicles at $100\%$ penetration
level signal control from CVs becomes difficult (University of Arizona (2016))
due to computational load.

In this study, simple plug-and-play expressions are developed for QLE from CVs
in the case of connected vehicles are also equipped with a range sensor by
extending the authors’ previous formulations in (Comert and Cetin (2009,
2011); Comert (2013b, 2016)). Derived formulate in this paper incorporate the
measured range from each CV as an additional source of real-time data
supplementing the typical CV information (e.g., location and time). In a
standing queue, the range will inform whether there is a vehicle behind a CV.
Steady-state behavior of estimated QLs and their errors are formulated. These
are then compared against and validated with results from point queue as well
as microscopic simulations.

###  1.2 Contributions of this study

Our modular expectation-based approach enables application of filtering
algorithms to deal with measurement or low MPR related errors. Note also that
the estimation problem is interesting where only assume partially observed
system. Thus, for any similar problem such as autonomous vehicles, our
approach can be adopted. One of limitations for connected vehicles with low
MPRs, as we discussed in (Comert (2016)), is distinguishing whether the last
connected vehicle is the last vehicle in the queue. This information would be
available from a range sensor (ultrasonic , LIDAR) Cetin et al. (2017) showed
the accuracy levels in LIDAR-based estimations. Moreover, presented formulae
do not depend on any primary parameter such as arrival rate or MPR. So, method
can be used for time dependent arrival rates or $p$. Estimation of these
primary parameters are becoming interest also (Comert (2016); Zhao et al.
(2019a); Van Phu and Farhi (2018)).

Impact of additional sensors in terms of estimation errors for different
arrival rates $\lambda$ and MPR (probe percentage) $p$ are presented. This
research gives insights about the benefit of additional data from range
measuring sensors that can be common with the upcoming autonomous vehicles.
The results can be incorporated for more accurate queue lengths and delay
calculations in new control heuristics (Comert et al. (2009)).

##  2 Problem Statement and Notation

Fig. 1 shows a view of an approach lane right before turning green, vehicles
with darker colors represent CVs and shaded ones depict the detected vehicles
via range sensors. Location of the connected vehicle information in queue $L$
are inferred as number of vehicles. Number of CVs ($M$) is 3, location of the
last CV ($L$) is $9$, and queue time tag of the last probe $T$ is $35$ seconds
(s). Moreover, last CV is able to detect the vehicle behind it, so,
$L^{\prime}$=($9+1$) and queuing time of the following vehicle detected by
sensor on the last CV is $T^{\prime}=38$ s.

Basic problem is defined as deriving estimators for the total queue length $N$
given the information from CVs and other detected vehicles by CVs’ range
measuring sensors such as LIDARs (i.e., CV locations and time tags). Range
measuring sensors provide the critical information of whether the last CV
actually is the last vehicle in the queue (_i.e._ , there is at least one more
vehicle joined the queue after it). The queue length ($N$) is 11 which is
estimated in real-time at the end of red phase $R$ using the location and the
time of CVs ($L^{\prime}=10$, $T^{\prime}=38$, $M=3$, and $R$). Estimators are
developed for known $R$, but they can be used for actuated or adaptive signal
phase lengths at the time instance of interest.

Fig. 1: Approach lane right before the green phase

###  2.1 Limitations of the study

In this study, we aim to develop analytical input-output models. For
simplifications we assume following.

  1. 1.

Approach lane has no capacity limit and there is no loss time due to
acceleration. These assumptions are for adopted overflow queue models (Haight
(1959); Ohno (1978); Medhi (1991); Heidemann (1994)). Usefulness of the models
in the proposed estimators are shown via comparing with errors from
microscopic traffic simulations (i.e., VISSIM) which provide more realistic
arrivals and queuing at the signal.

  2. 2.

Poisson arrivals to the approach lane are assumed which can describe isolated
intersections, low demand scenarios of more complex intersections, and enable
simple closed-form derivations.

  3. 3.

Lane level accuracy for CV locations is assumed. Position and time tags are
accurately received. Connected vehicle architectures especially at
intersections will satisfy this by map information and timing passed by
roadside units. When location is provided as latitude and longitude, using
average vehicle length or passenger car equivalencies, spot in number of
vehicles can be estimated. Basic safety messages also include vehicle type,
using vehicle lengths by type locations can also be estimated.

  4. 4.

Uniform vehicle type is assumed. Mixed traffic could be incorporated to the
models by arrival rates in Poisson arrivals (Walraevens et al. (2017)). Impact
of different vehicle types would impact intersection capacity and location
information.

  5. 5.

Connected vehicles are able to sense accurately if there is a vehicle behind
with the help of range sensors.

Estimators presented are unbiased under these assumptions. They would hold for
estimating real-time queues at isolated signals, signals with low to moderate
volume-to-capacity ratios, Poisson, or wherever approximately random arrivals
can be assumed (Newell (1982); Fiems and De Turck (2019)). Such QLEs are going
to be unbiased. In fact, any approach queue length can be estimated given
formulae. Amount of deviations from random arrivals will be reflected into the
estimation errors as bias.

##  3 Analysis for the Case without Overflow Queue

First, it is assumed that vehicles that have arrived in one given cycle clear
the intersection in that cycle (the overflow queue ($Q$) is zero). At each
cycle, the total number of vehicles in the queue $N$ is estimated given $L$,
$T$, and primary parameters arrival rate $\lambda$, $p$, and $R$. Throughout
the examples, signal related parameters are chosen to be effective red
duration of $45$ $s$, green time of $43.2$ $s$, and discharge rate $1.8$
seconds per vehicle ($spv$). Other selected parameter values are
$\lambda=\\{0.111,0.133,0.163,0.190,0.218,0.239,0.267\\}$ $vps$ which
correspond to volume-to-capacity ratios of
$\rho=\\{0.41,0.49,0.60,0.70,0.80,0.88,0.98\\}$ and $0.001\leq p\leq 1.00$
(Comert (2016)).

As shown in Eq. 1, estimator of the queue length $N$ can be written as
summation of expected values of the number of vehicles known until last CV
$N_{1}$ and inferred $N_{2}$ after this CV for given conditions: (i) whether
the last CV is also the last vehicle $I(L=l_{v})=1$ and (ii) the last CV is
not the last vehicle $I(L\neq l_{v})=1$. Here, $I()$ is the indicator
function.

$\displaystyle
E(N|{L=l,T=t})=\begin{cases}[E(N_{1}|{L=l,T=t})+E(N_{2}|{L=l,T=t})]I(L=l_{v})+\\\
[E(N_{1}|{L=l,T^{\prime}=t^{\prime}})+E(N_{2}|{L=l,T^{\prime}=t^{\prime}})]I(L\neq
l_{v})\end{cases}$ (1)

In Eq. (1), for condition (i) $I(L=l_{v})$, $N_{1}$ is known and $N_{1}=l$ and
$E(N_{2}|{L=l,T=t)}$ is number of arrivals after the last CV during
$(R-T)$=$\delta$ which is zero leading to estimator with no error $N=l$. For
condition (ii) $(L\neq l_{v})$, after the last CV, only non-CVs would arrive,
thus $N_{1}=l+1$ and $\delta=(R-T^{\prime})$. Arrival rate during $\delta$ is
$(1-p)\lambda$. Denoting $(1-p)\lambda$ by $\theta$, we can simply write
$N_{2}|{L=l,T=t^{\prime}}$$\sim$Poisson($\theta\delta$). Simplified estimator
is given in Eq. (2).

For the estimation error when first condition, queue length in Eq. (2) is
known ($N=l$), thus, the error is $zero$. For the second condition, first term
is known ($N_{1}=l+1$) and there is no error. The second term is
$\sim$Poisson($\theta\delta$) and the variance is $\theta\delta$. Thus, the
conditional variance of Eq. (1) for $L\geq 0,T\geq 0$ can simply be written as
Eq. 3. Practically, given the time information and if there is a vehicle
behind or not, we can calculate the estimation errors under the assumptions
listed in 2.

$\displaystyle E(N|{L=l,T=t})=\begin{cases}l,&L=l_{v}\\\
l+1+\theta\delta,&\delta=(R-t^{\prime}),L\neq l_{v}\end{cases}$ (2)
$V(N|{L=l,T=t})=\theta(R-t^{\prime})I(L\neq l_{v}),L\geq 0,\ T\geq 0$ (3)

Formula for the errors can be expressed as deviations from $D=N-E(N|L=l,T=t)$.
Thus, the variance of $D$ for $L\geq 0$ can be written as
$V(D)=E[V(N|L=l,T=t)]$ and given in Eq. (4). Suppose that $E(T)$ and
$E(T^{\prime})$ are known (without and with range sensor, respectively),
errors can be evaluated and simplified to Eq. 4.

$\displaystyle V(D)=E[\theta\delta]=\theta[R-E(T^{\prime})]P(L\neq
l_{v})=\frac{p(1-e^{-p\lambda R(1+p)})}{(1+p)},\ t<R$ (4)

where, $P(L\neq l_{v})=E(1-e^{-\lambda(R-t)}),\ t<R$ and $P(L=l_{v})=1-P(L\neq
l_{v})$ can be interpreted as $L=l_{v}$ probability of last CV is the last
vehicles meaning no arrival during $\delta=(R-t)$ (i.e., $e^{-\lambda(R-t)}$).
Using distribution of $T$ (Comert (2013b)) $f(t)=\frac{\lambda
pe^{-p\lambda(R-t)}}{(1-e^{-\lambda pR})},\ t<R$ is input for scenario
probabilities $P(L\neq l_{v})$ as well as for the expected value of
$T^{\prime}$ respectively. They can then be incorporated to compute the
variance given in Eq. (4).

###  3.1 Derivation of the expectations

Revising the the probability distributions derived in Comert (2013b), queue
length estimators and the errors are derived. Corresponding $E(L)$ and
$E(L^{\prime})$ revised for range sensor incorporation which are given in Eq.
(5). The impact of sensor information is simply in the denominator adding one
non-CV $(1-p)$.

$\displaystyle E(L)=\lambda R-[(1-e^{-\lambda pR})/p],\ m>0$ $\displaystyle
E(L^{\prime})=\lambda R-[(1-e^{-\lambda pR})/(p+(1-p))],\ m>0$ (5)

Similarly, in Eq. (6), $E(T)$ and $E(T^{\prime})$ are provided. Impact is
simple again by adding the term $(1-p)\lambda$. The revised variance of $D$
($V(D)$) can be calculated using Eq. (6) in Eq. (4) and shown in Eq. (7). Note
that the QLE error formula without sensors is
$V(D)=E[\theta\delta]=\theta[R-E(T)]$=$(1-p)(1-e^{-p\lambda R})/p$.

$\displaystyle E(T)=R-[(1-e^{-\lambda pR})/\lambda p],\ m>0$ $\displaystyle
E(T^{\prime})=R-[(1-e^{-\lambda pR})/\lambda p+(1-p)/\lambda],\ m>0$ (6)

One of the main contribution of Comert (2013b) was to be able to compute
$E(L)$ and $E(T)$ from one another, thus, $E(T^{\prime})$ can be utilized to
calculate $E(L^{\prime})$ and shows the impact of range sensor in Fig. (2)
which are derived using Eqs. (6) and (5). This error (Eq. 7) can be calculated
using estimators for $p$ and $\lambda$ and signal timing information.

$\displaystyle V(D)=E[\theta\delta]=\theta[R-E(T^{\prime})]P(L\neq lv)$
$\displaystyle=(1-p)\lambda[R-(R-(1-e^{-p\lambda R})/\lambda
p+(1-p)/\lambda)]P(L\neq lv)$ $\displaystyle=[1-p][(1-e^{-p\lambda
R})/p-(1-p)]P(L\neq lv)$ (7)

where, $P(L\neq lv)=[1-p(1-e^{-(1+p)\lambda R})/(1+p)]$.

Fig. 2: Impact of range sensor on $\\{E(T),E(L)\\}$ shown as
$\\{E(T^{\prime}),E(L^{\prime})\\}$ for $\lambda=0.239$ $vps$

To verify the formulation above, the results are compared with simple point
queue simulations with Poisson arrivals coded in R. From Fig. 2-a, it can be
seen that $E(T^{\prime})$ is following the simulated values very closely. As
$p$ is approaching $100\%$, the $E(T^{\prime})$ is getting closer to the red
duration (e.g., $45$ seconds). In addition, Fig. 2-b shows the values of
$V(D)$s computed with the analytical models with and without range measuring
sensors. These are then compared against simulations with $100,000$
replications. As it is evident analytical results follow simulation results
very closely.

Fig. 3: Improvements in QLE errors when each CV also provides its following
vehicle’s position in the queue (various $\lambda$ levels)

Figs. 3 a-b are given to illustrate how much improvement is achieved when CVs
are also equipped with range measuring sensor, i.e., these graphs show the
additional reduction in QLE error when $N$ is estimated from CVs that also
provide information about their following vehicles (as compared to CVs that
provide only their own positions). In Fig. 3 a, this improvement is shown via
graphing percent differences of $V(D)/E(N)$s (i.e., variance-to-mean ratio or
dispersion index) with respect to probe percentages $p$. Figure depicts
maximum gain (e.g., close to $20\%$ at $\lambda=0.133$ $vps$) at about
$p=10\%$ and increasing as $\lambda$ decreases. Figure suggests that about
$20\%$ probe vehicles are need to experience the maximum impact ($\approx
10\%$). Lower $p$ values are still showing $8$ to $10\%$ improvements relative
to mean QLs. And, the impact is diminishing as $p$ increases where $P(L=lv)$
also get higher.

Fig. 4: Relative errors $\%\Delta CV=\sqrt{V(D)}/E(N)$ and
$\%\Delta\sqrt{V(D)}$

Figs. 4 a-b demonstrate the impact of range sensors in terms of $\%$
difference of $\sqrt{V(D)}/E(N)$s (i.e., coefficient of variations) and
difference of $\sqrt{V(D)}$s changing as functions of market penetration rate
$p$ at different $\lambda$ values. The maximum improvement (e.g., close to
$7\%$ for $\lambda=0.133$ $vps$) at about $p=30\%$ and increasing as $\lambda$
decreases. Similarly, estimation error differences are higher for lower
$\lambda$ values. They get very close to each other after about $30\%$ $p$
level. Difference of $\sqrt{V(D)}$ values are given in Fig. 4 b that are QLE
error differences in terms of number of vehicles. Again, maximum impact $0.35$
vehicle per cycle is observed at $p\approx 30\%$. This behavior would make
such estimator more suitable for intersections with low demand or rural
intersections. Desirably, this would also satisfy the assumptions for the
derivations.

##  4 Analysis for the Case with Overflow Queue

When the overflow queue from signal cycles is incorporated in the queue length
estimation, we can express queue length $N$ as the sum of the overflow queue
$Q$ and the new red duration arrivals $A$. Then, for any cycle, based on the
last CV information (e.g., CV in $Q$ or $A$) and whether it is the last
vehicle ($L=lv$) (i.e., detected by range sensor), we can model the estimator
with the help of five possible scenarios:

  1. 1.

The last CV is within new arrivals and it is the last vehicle, $I(l\in A\land
L=lv)$.

  2. 2.

The last CV is within new arrivals and it is not the last vehicle, $I(l\in
A\land L\neq lv)$.

  3. 3.

The last CV is in the overflow queue and it is the last vehicle, $I(l\in
Q\land L=lv)$.

  4. 4.

The last probe may be within the overflow queue and it is not the last
vehicle, $I(l\in Q\land L\neq lv)$.

  5. 5.

There is no CV in the queue, $I(l=0)$.

Considering these scenarios, we can write
$N|L,T$=$I(L=lv)[Q|L,T+A|L,T]+I(L\neq lv)[Q|L,T+A|L,T]$. Then, the estimator
for the total queue length given CV information can be written as Eq. (8).
When utilized, Fig. 5 was given for the average estimation errors $V(D)$s from
with and without range sensors. From the figure, similar to $Q=0$ case,
improvements with sensors are higher in between $p$=$0.05-30\%$. The estimator
with overflow queue also contains a revised queue joining time $\tau$ that
defined as the remaining last CV’s joining time from previous green duration.
These observations are desired and expected. Benefit at lower CV penetration
levels would be more cost saving. As $p$ gets higher, it is expected that more
CVs will be the last vehicle in the queue, thus, saving would be less from the
range sensor fusion.

$E(N|L=l,L=lv,T=t)=\begin{cases}I(l\in Q\land L=lv)[l+\theta R]+\\\ I(l\in
Q\land L\neq lv)[l+1+\theta(C-\tau^{\prime})+\theta R]+\\\ I(l\in A\land
L=lv)[l]+\\\ I(l\in A\land L\neq lv)[l+1+\theta(R-t^{\prime})]+\\\
I(l=0)[(1-p)(E(Q)+\theta R)].\end{cases}$ (8)

Fig. 5: Estimation errors from VISSIM with and without $Q$ results for
different $\lambda$ and $p$ values

Average overflow queue in Eq. (8), $E(Q)$=$\frac{\rho^{2}}{2(1-\rho)}$, is
adopted from (Akçelik (1980)) where $\rho=\lambda C/X$ with $X=24$ vehicles
per cycle in our examples, $\lambda$ arrival rate per second, and $C$ is cycle
time $90$ seconds. Alternatively, from (Viti (2006); Medhi (1991)) different
average overflow queue formula can be adopted
$E(Q)$=$1.5(\rho-\rho_{o})/(1-\rho)$,
$E(Q)$=$\frac{(2\rho-1)\rho^{4}}{2(1-\rho)}$, and
$E(Q)$=$[\frac{\rho}{2(1-\rho)}]e^{-[\frac{(1-\rho)\sqrt{R/2}-R(1-\rho)^{2}}{4}]}$
where $\rho_{o}$=$0.67+X/600$. The queue length on the approach lane is
assumed to be zero for all examples in the study. Using the scenario
probabilities in Eq. (9), we can obtain Eq. (8).

$E(N)=\begin{cases}P(l\in Q\land L\neq
lv)[E(L)+\theta(C-E(\tau^{\prime}))+\theta R]+\\\ P(l\in Q\land
L=lv)[E(L)+\theta R]+\\\ P(l\in A\land L\neq lv)[E(L)+1+\theta(R-\tau)]+\\\
P(l\in A\land L=lv)[E(L)]+\\\ P(l=0)[(1-p)(E(Q)+\theta R)]\end{cases}$ (9)

Average or steady-state estimation error, $V(D)$s, can be given as in Eq. (10)
with $V(Q)$ in Eq. (12) from (Medhi (1991)).

$V(D)=\begin{cases}P(l\in Q\land L\neq lv)[\theta(C-E(T^{\prime}))+\theta
R]+\\\ P(l\in Q\land L=lv)[\theta R]+\\\ P(l\in A\land L\neq
Lv)[(1-p)(1-e^{-(1+p)\lambda R})/(1+p)]+\\\ P(l=0)[(1-p)(V(Q)+\theta
R)]\end{cases}$ (10)

Probability distributions of $Q$ and $\tau$ are needed to calculate Eq. (10).
These distributions can be obtained numerically (Olszewski (1990); Viti and
Van Zuylen (2010)) or by solving the probability generating function (Gray et
al. (1992); Tarko and Rouphail (1994); Mung et al. (1996); Van Den Broek et
al. (2006)). But, no exact simple form for $V(D)$ is available. Thus, in
Section 5 we used approximations in the estimators and the errors and
evaluated them with microsimulation results. Figs. 6 a-b are derived using the
approximation formulae and microsimulations using Eq. (14). Details of the
approximate estimator is explained next by utilizing steady-state $V(Q)$s by
the formulae adopted from (Medhi (1991); Fu and Hellinga (2000)). The
approximate $V(D)$s are also given against VISSIM results in Figs. 5-10.
Relative errors are expressed in coefficient of variations (i.e.,
$100\sqrt{V(D)}/E(N)$ or $\%\Delta CV$). Figures also given as
$\%\Delta\sqrt{V(D)}$ from the two evaluations.

Fig. 6: Comparing estimation errors obtained from VISSIM and approximations

##  5 Approximate Inference for $E(N)$ and $V(D)$

Approximation formulas for the estimators can be derived simply quantifying
the impact of overflow queue in terms of $E(Q)$ and $V(Q)$ in Eq. (11) and Eq.
(12), respectively. Note that these formulas assume $M/G/1$ queue and they are
based on the Pollaczek-Khintchine formula (Haight (1959); Heidemann (1994);
Ohno (1978)) and adopted from (Medhi (1991)). Numerical values are determined
and discussed in the simulation set up (Comert (2016)), mean discharge rate
from the intersection is determined as $1.8$ seconds per vehicle with the
capacity for effective green of 43.20 seconds ($(43.20/1.80)$=$24$ vehicles
per cycle). Demand levels of $\\{600,700,800,900,985\\}$ vph that corresponds
to volume-to-capacity ratios of $\rho$=$\\{0.60,0.70,0.80,0.88,0.98\\}$ are
used in the numerical examples. VISSIM is run $1000$ cycles ($25$ hours) for
$3$ random seeds for 11 $p=\\{0.1\%,5\%,10\%...,90\%\\}$ CV percentages at
each demand level. Average of $3$ random seeds are reported for estimation
errors. Default queue definition of VISSIM in terms of minimum speed$\leq 10$
kilometers per hour is used. For detailed discussions, readers are referred to
(Akçelik (1980); Medhi (1991); Fu and Hellinga (2000); Viti (2006)).

Note that we are presenting the steady-state QLE errors incorporating overflow
queue moments, however, cycle-to-cycle formulas were also given $E(Q_{i})$ and
results are given as figures in Section 6. $E(Q_{i})$ and $V(Q_{i})$ are
adopted from (Akçelik (1980)) and (Viti (2006)), respectively.

$E(Q)=\frac{\rho^{2}}{2(1-\rho)}$ (11)
$V(Q)=[4(1-\rho)\rho^{3}+3\rho^{4}]/(12(1-\rho)^{2})$ (12)

Simply, queue length estimation approximations for expected value and errors
are obtained by incorporating Eqs. (11) and (12) into appropriate cases in
Eqs. (8) and (10). Resulting approximations with range sensors are then
expressed as Eqs. (13) and (14). Scenario probabilities can be simply
approximated by quadratic equations.

$E(N)=\begin{cases}P(l\in Q)[E(Q)+\theta R]+\\\ P(l\in A)[E(Q)+\lambda R]+\\\
I(l=0)[(1-p)(E(Q)+\theta R)]\end{cases}$ (13)

$V(D)=\begin{cases}P(l\in Q)[(1-p)(1-e^{-pE(Q)})/p+\theta R]+\\\ P(l\in
A)[(1-p)(1-e^{-p\lambda R})/p]+\\\ P(l=0)[(1-p)(V(Q)+\theta R)]\end{cases}$
(14)

where, approximate probabilities are given in Eq. 15.

$P(L=l)=\begin{cases}e^{-p\lambda R}-e^{-p(\lambda
R+(9.87p^{2}-4.62p+0.991)E(Q)},L\in Q\\\ 1-e^{-p\lambda R},L\in A\\\
e^{-p(\lambda R+(9.87p^{2}-4.62p+0.991)E(Q))},L=0\end{cases}$ (15)
$V(D)=\begin{cases}P(l\in Q)[(1-p)p(1-e^{-(1+p)E(Q)})/(1+p)-p\theta R]+\\\
P(l\in A)[(1-p)(1-e^{-p\lambda R})/p-(1-p)][1-p(1-e^{-(1+p)\lambda
R})/(1+p)]+\\\ P(l=0)[(1-p)(V(Q)+\theta R)]\end{cases}$ (16)

Performance of the approximations are presented in Figs. 6 a-d. It can be seen
that approximate errors are following VISSIM values closely for both with and
without range sensors at all $\rho$ and $p$ levels. Moreover, Fig. 7 presents
linear regression lines without intercepts for $V(D)$s from VISSIM simulations
and approximate models without and with range sensor. Regression lines are
given for estimated and simulation results. The lines show high correlations
between analytical approximations and VISSIM errors without and with range
sensor at $R^{2}$=$0.986$ and $R^{2}$=$0.985$, respectively.

Fig. 7: Comparing estimation errors obtained from VISSIM and approximations

Fig. 8-a shows Eq. (14) values along with VISSIM QLs for various $\rho$
levels. Developed approximation for QL is a simple formula that needs $\rho$,
$\lambda$, and signal timing. Fig. 8 b shows the evolution of errors with
respect to cycles for $p\leq 0.05$ and $\rho\geq 0.80$. These errors are
calculated from microsimulations. We can see that approximate errors are able
to match simulation values even at high $\rho\geq 0.80$ and low probe
proportion $p\leq 0.05$ values. When we consider the error differences with
respect the mean QLs, they are very close.

Fig. 8: Estimation errors with and without results for different $\lambda$
$vps$ and $ps$

##  6 Estimation with Range Sensor, Unknown $\lambda$, and $p$

Known arrival $\lambda$ and market penetration rate $p$ parameters in Eqs. (2)
and (8) can be relaxed by replacing them with estimators for these low level
parameters. Detailed derivations of such estimators were given in (Comert
(2016)). Best performing combinations were adopted from this earlier study.
Without Q, the QL with unknown arrival rate and probe proportion can simply be
estimated using Eq. (17). For $E(N|l,t,m,Q=0$, referred as $Estimator$ $1$
with combination of $\hat{p}_{1}=m/l$ and $\hat{\lambda}_{1}=l/R$ simply Eq.
(18) is derived. $Estimator$ $2$ uses $\hat{p}_{2}=\frac{mt}{(mt+(l-m)R)}$ and
$\hat{\lambda}_{2}=\frac{l-m}{t}+\frac{m}{R}$ which simplifies to Eq. (19).

$E(N|l,t,m,Q=0)=l+(1-\hat{p})\hat{\lambda}(R-t)$ (17)
$E(N|l,t,m,Q=0)=l+(l-m)(1-t/R)$ (18)
$E(N|l,t,m,Q=0)=l+(1-\frac{mt}{mt+(l-m)R})((l-m)/t+m/R)(R-t)$ (19)

Similarly, estimation of queues real-time and steady-state including Q are
also presented. In this case, QLs can be considered as sum of random variables
given in Eq. (20). Indicator function $I(l\in Q)$ represents the case if the
last CV is in the overflow queue, $I(l\in A)$ indicates the last CV in new red
arrivals, and $I(l=0)$ indicates that there is no CV in the queue. At any
given cycle, these terms represent disjoint events, hence, only one term will
be positive where others will be zeros.

$E(N_{i}|L=l,T=t,Q_{i}\geq 0)=\begin{cases}I(l\in
Q)[l+\hat{\theta}(C-t^{\prime})+\hat{\theta}R]+\\\ I(l\in
A)[l+\hat{\theta}\delta]+\\\
I(l=0)[(1-\hat{p})(E(Q_{i})+\hat{\theta}R)]\end{cases}$ (20)
$E(N_{i}|L=l,L=lv,T=t,Q_{i}\geq 0)=\begin{cases}I(l\in Q\land
L=lv)[l+\hat{\theta}R]+\\\ I(l\in Q\land L\neq
lv)[l+1+\hat{\theta}(C-\tau^{\prime})+\hat{\theta}R]+\\\ I(l\in A\land
L=lv)[l]+\\\ I(l\in A\land L\neq lv)[l+1+\hat{\theta}(R-t^{\prime})]+\\\
I(l=0)[(1-\hat{p})(E(Q_{i})+\hat{\theta}R)].\end{cases}$ (21)

Overflow queue
$E(Q_{i})=\frac{Xi(\hat{\rho}-1)}{4}\sqrt{(\hat{\rho}-1)^{2}+\frac{12(\hat{\rho}-\rho_{o})}{Xi}}$
is adopted from (Akçelik (1980)) where $\rho_{o}$=$0.67+X/600$, $X=24$
vehicles per cycle, $\hat{\rho}=\hat{\lambda}C/X$, and $i=1,2,3...$ denotes
the cycle index. For $E(Q_{i})=E(Q)(1-e^{-\beta i})$ from (Viti (2006)) can
also be used which gives very close results (Comert (2013b)) where
$E(Q)=\frac{3(\hat{\rho}-\rho_{o})}{2(1-\hat{\rho)}}$.

Fig. 9: Estimation errors $\%\Delta V(D)/E(N)$ with and without range sensor
results for different $\lambda$ $vps$ and $ps$

Average true and estimated QLs derived from VISSIM runs. Note that QLs depend
on VISSIM queue definition where default in queue=$I(speed\leq 10$ kilometers
per hour. From Fig. 10, at 600 vehicles per hour, $7.34$ vehicles per 45
seconds red duration is expected. Similarly others $8.57,9.79,10.78$, and
$12.01$ vehicles per red. However, overflow queue becomes more important when
$\rho>0.80$ on average $0.68$, $2.13$, and $20.68$ vehicles in length for
$\rho=0.80$, $\rho=0.88$, and $\rho=0.98$, respectively. Impact of overflow
queue also observed accuracy of the estimators start to decrease after
$\rho>0.88$ from $\pm$ $1.5$ to $\pm$ $3.45$ vehicles. Overall, $Estimator$
$2$ can provide accuracy between $12\%$ to $19\%$ of the true average QLs.

Fig. 10: Relative estimation errors $\%\Delta\sqrt{V(D)}$ and
$\%\sqrt{V(D)}/E(N)$ from VISSIM simulations

With overflow queue combination of parameter estimators were input where
results are shown in Fig. 8. As queue length estimators closely follow true
queue lengths by $p$=$30\%$, queue length estimators with $\lambda$ and $p$
estimators also gets very close to the behavior of estimator with known
parameters.

Fig. 11: Estimation errors $\%\Delta\sqrt{V(D)}/E(N)$ from VISSIM and
approximations

$\displaystyle E(N_{i}|l,t,m,Q_{i}=0)=l+(l-m/l)(l/R)(R-t)=l+(l-m)(1-t/R)$ (22)
$\displaystyle
E(N_{i}|l,t,m,Q_{i}=0)=l+(1-\frac{mt}{mt+(l-m)R})((l-m)/t+m/R)(R-t)=m+\frac{R(l-m)}{t}$
(23)

Figs. 9-11 summarize the QLE errors with the overflow queue. The errors are
expressed in $\%\Delta{V(D)}$ and $\%\Delta{CV}$ where,
$CV$=$\sqrt{V(D)}/E(N)$. Since estimators are able to capture overflow queue,
errors are declining to zero as connected vehicle MPR increases. Results from
the literature cited above also confirmed that $p\geq 30\%$ queue length can
be predicted within $\pm 10\%$ across all all $\rho levels$. Similarly at
$p=50\%$, queue length can be predicted within $\pm 2\%$ in $\%\Delta{CV}$ and
$\%\Delta{EN}$.

Fig. 12 confirms these results with good agreement with the true QLs. The
figures depict about $20$ cycles for illustration of performance of
estimators. Based on the numerical results, for $p\geq 30\%$ QLEs with unknown
estimators shows good agreement with true average QLs. Thus, any of the
$Estimator$ $1$ or $Estimator$ $2$ can be used in accurate queue length
estimation in signal control. Estimator with range sensor can also be seen
that more closely following the true queue values.

Fig. 12: Estimation results for $\lambda=0.239$ $vps$ at $p=\\{30\%,50\%\\}$

##  7 Conclusions

In this study, models for queue length estimation from connected vehicles with
range sensors at signalized intersections are derived. Fully analytical models
without overflow queue and approximations with the overflow queue are
presented. Contribution of the proposed models are simple input-output models
without requiring true queue length observations, as a step towards
generalizing estimation problem with partially observed entities in a cycling
queueing system, and leverage of simple variations to previously derived
models to express additional complexity.

When range sensor included, estimators are more accurate as high as $25\%$ in
estimation errors relative to average queue lengths for high volume-to-
capacity ratio $\rho$=$0.98$ at market penetration ratio of $p=5\%$. About
$10\%$ improvement is observed when volume-to-capacity ratio is $\leq 0.88$.
Moreover, approximately $0.25$ vehicle per cycle was improved. For example, at
constant arrival rate in $10$ hours, total estimation error of $100$ vehicles
(i.e., energy consumption, emissions, and delay) would be improved.

## Acknowledgments

This study is supported by the Center for Connected Multimodal Mobility
($C^{2}$$M^{2}$) (USDOT Tier 1 University Transportation Center) Grant
headquartered at Clemson University, Clemson, South Carolina, USA. The authors
would also like to acknowledge U.S. Department of Homeland Security (DHS)
Summer Research Team Program Follow-On, FY19 US Department of Education MSEIP
Grant Award P120A190061, and National Science Foundation (NSF, No. 1719501,
1436222, and 1400991) grants. Any opinions, findings, conclusions or
recommendations expressed in this study are those of the author(s) and do not
necessarily reflect the views of ($C^{2}$$M^{2}$), USDOT, DHS, U.S. Department
of Education, or NSF and the U.S. Government assumes no liability for the
contents or use thereof.

## References

  * Akçelik (1980) Akçelik, R., 1980. Time-dependent expressions for delay, stop rate and queue length at traffic signals. Australian Road Research Board Melbourne, Australia.
  * Argote et al. (2011) Argote, J., Christofa, E., Xuan, Y., Skabardonis, A., 2011. Estimation of measures of effectiveness based on connected vehicle data. In: 2011 14th International IEEE Conference on Intelligent Transportation Systems (ITSC). IEEE, pp. 1767–1772.
  * Asanjarani et al. (2017) Asanjarani, A., Nazarathy, Y., Pollett, P. K., 2017. Parameter and state estimation in queues and related stochastic models: A bibliography. arXiv preprint arXiv:1701.08338.
  * Cetin (2012) Cetin, M., 2012. Estimating queue dynamics at signalized intersections from probe vehicle data: Methodology based on kinematic wave model. Transportation research record 2315 (1), 164–172.
  * Cetin et al. (2017) Cetin, M., Sazara, C., Nezafat, R. V., et al., 2017. Exploring the use of lidar data from autonomous cars for estimating traffic flow parameters and vehicle trajectories.
  * Cheng et al. (2012) Cheng, Y., Qin, X., Jin, J., Ran, B., 2012. An exploratory shockwave approach to estimating queue length using probe trajectories. Journal of intelligent transportation systems 16 (1), 12–23.
  * Comert (2013a) Comert, G., 2013a. Effect of stop line detection in queue length estimation at traffic signals from probe vehicles data. European Journal of Operational Research 226 (1), 67–76.
  * Comert (2013b) Comert, G., 2013b. Simple analytical models for estimating the queue lengths from probe vehicles at traffic signals. Transportation Research Part B: Methodological 55, 59–74.
  * Comert (2016) Comert, G., 2016. Queue length estimation from probe vehicles at isolated intersections: Estimators for primary parameters. European Journal of Operational Research 252 (2), 502–521.
  * Comert and Cetin (2009) Comert, G., Cetin, M., 2009. Queue length estimation from probe vehicle location and the impacts of sample size. European Journal of Operational Research 197 (1), 196–202.
  * Comert and Cetin (2011) Comert, G., Cetin, M., 2011. Analytical evaluation of the error in queue length estimation at traffic signals from probe vehicle data. IEEE Transactions on Intelligent Transportation Systems 12 (2), 563–573.
  * Comert et al. (2009) Comert, G., Cetin, M., Nichols, A. P., 2009. Incorporating queue length measurements into actuated signal control: evaluation of efficiency benefits at an intersection. Tech. rep.
  * Comert et al. (2019) Comert, G., Khan, Z., Rahman, M., Chowdhury, M., 2019. Grey models for short-term queue length predictions for adaptive traffic signal control. arXiv preprint arXiv:1912.12676.
  * Fiems and De Turck (2019) Fiems, D., De Turck, K., 2019. Taylor-series approximations for queues with arrival correlation. Applied Mathematical Modelling 69, 113–126.
  * Fu and Hellinga (2000) Fu, L., Hellinga, B., 2000. Delay variability at signalized intersections. Transportation Research Record 1710, 215–221.
  * Gao et al. (2020) Gao, K., Huang, S., Han, F., Li, S., Wu, W., Du, R., 2020. An integrated algorithm for intersection queue length estimation based on iot in a mixed traffic scenario. Applied Sciences 10 (6), 2078.
  * Gray et al. (1992) Gray, W. J., Wang, P., Scott, M., 1992. An m/g/1-type queuing model with service times depending on queue length. Applied mathematical modelling 16 (12), 652–658.
  * Guo et al. (2019) Guo, Q., Li, L., Ban, X. J., 2019. Urban traffic signal control with connected and automated vehicles: A survey. Transportation research part C: emerging technologies.
  * Haight (1959) Haight, F. A., 1959. Overflow at a traffic flow. Biometrika 46 (3-4), 420–424.
  * Heidemann (1994) Heidemann, D., 1994. Queue length and delay distributions at signalized intersections. Transportation Research Part B 28 (5), 624–635.
  * Li et al. (2013) Li, J.-Q., Zhou, K., Shladover, S. E., Skabardonis, A., 2013. Estimating queue length under connected vehicle technology: Using probe vehicle, loop detector, and fused data. Transportation research record 2356 (1), 17–22.
  * Liu et al. (2019) Liu, H., Liang, W., Rai, L., Teng, K., Wang, S., 2019. A real-time queue length estimation method based on probe vehicles in cv environment. IEEE Access 7, 20825–20839.
  * Mandal et al. (2020) Mandal, A., Sadhukhan, P., Gaji, F., Sharma, P., 2020. Measuring real-time road traffic queue length: A reliable approach using ultrasonic sensor. In: Proceedings of the 2nd International Conference on Communication, Devices and Computing. Springer, pp. 391–398.
  * Medhi (1991) Medhi, J., 1991. Stochastic models in queuing theory. Academic Press, Inc. Harcourt Brace Jovanovich Publishers.
  * Mei et al. (2019) Mei, Y., Gu, W., Chung, E. C., Li, F., Tang, K., 2019. A bayesian approach for estimating vehicle queue lengths at signalized intersections using probe vehicle data. Transportation Research Part C: Emerging Technologies 109, 233–249.
  * Mung et al. (1996) Mung, K. S. G., Poon, A. C. K., Lam, W. H. K., 1996. Distributions of queue lengths at fixed time traffic signals. Transportation Research Part B:Methodological 30 (6), 421–439.
  * Nam et al. (2009) Nam, S. Y., Kim, S., Sung, D. K., 2009. Estimation of available bandwidth for an m/g/1 queueing system. Applied mathematical modelling 33 (8), 3299–3308.
  * Newell (1982) Newell, G. F., 1982. Applications of queueing theory, 2nd Edition.
  * Ohno (1978) Ohno, K., 1978. Computational algorithm for a fixed cycle traffic signal and new approximate expressions for average delay. Transportation Science 12 (1), 29–47.
  * Olszewski (1990) Olszewski, P., 1990. Modeling of queue probability distribution at traffic signals. In: M.Koshi (Ed.), Transportation and Traffic Theory. Elsevier, Amsterdam.
  * Rompis et al. (2018) Rompis, S. Y., Cetin, M., Habtemichael, F., 2018. Probe vehicle lane identification for queue length estimation at intersections. Journal of Intelligent Transportation Systems 22 (1), 10–25.
  * Tan et al. (2020) Tan, C., Liu, L., Wu, H., Cao, Y., Tang, K., 2020. Fuzing license plate recognition data and vehicle trajectory data for lane-based queue length estimation at signalized intersections. Journal of Intelligent Transportation Systems, 1–18.
  * Tan et al. (2019) Tan, C., Yao, J., Tang, K., Sun, J., 2019. Cycle-based queue length estimation for signalized intersections using sparse vehicle trajectory data. IEEE Transactions on Intelligent Transportation Systems.
  * Tarko and Rouphail (1994) Tarko, A., Rouphail, N., 1994. Distribution-free model for estimating random queues in signalized networks. Transportation Research Record 1457, 192–197.
  * Tiaprasert et al. (2015) Tiaprasert, K., Zhang, Y., Wang, X. B., Zeng, X., 2015. Queue length estimation using connected vehicle technology for adaptive signal control. IEEE Transactions on Intelligent Transportation Systems 16, 2129–2140.
  * University of Arizona (2016) University of Arizona, University of California PATH Program, S. N. I. E., 2016\. Multi-modal intelligent traffic signal system—phase ii: System development, deployment and field test.
  * Van Den Broek et al. (2006) Van Den Broek, M., Van Leeuwaarden, J., Adan, I., Boxma, O., 2006. Bounds and approximations for the fixed-cycle traffic-light queue. Transportation science 40 (4), 484–496.
  * Van Phu and Farhi (2018) Van Phu, C. N., Farhi, N., 2018. Estimation of urban traffic state with probe vehicles. arXiv preprint arXiv:1811.05394.
  * Viti (2006) Viti, F., 2006. The dynamics and the uncertainty of delays at signals. Ph.D. thesis, Delft University of Technology, Delft, the Netherlands.
  * Viti and Van Zuylen (2010) Viti, F., Van Zuylen, H., 2010. Probabilistic models for queues at fixed control signals. Transportation Research Part B 44 (1), 120–135.
  * Walraevens et al. (2017) Walraevens, J., Bruneel, H., Fiems, D., Wittevrongel, S., 2017. Delay analysis of multiclass queues with correlated train arrivals and a hybrid priority/fifo scheduling discipline. Applied Mathematical Modelling 45, 823–839.
  * Wu and Yang (2013) Wu, A., Yang, X., 2013. Real-time queue length estimation of signalized intersections based on rfid data. Procedia-Social and Behavioral Sciences 96, 1477–1484.
  * Xu et al. (2017) Xu, H., Ding, J., Zhang, Y., Hu, J., 2017. Queue length estimation at isolated intersections based on intelligent vehicle infrastructure cooperation systems. In: 2017 IEEE Intelligent Vehicles Symposium (IV). IEEE, pp. 655–660.
  * Yang and Menendez (2018) Yang, K., Menendez, M., 2018. Queue estimation in a connected vehicle environment: A convex approach. IEEE Transactions on Intelligent Transportation Systems 20 (7), 2480–2496.
  * Yin et al. (2018) Yin, J., Sun, J., Tang, K., 2018. A kalman filter-based queue length estimation method with low-penetration mobile sensor data at signalized intersections. Transportation Research Record 2672 (45), 253–264.
  * Zhao et al. (2019a) Zhao, Y., Zheng, J., Wong, W., Wang, X., Meng, Y., Liu, H. X., 2019a. Estimation of queue lengths, probe vehicle penetration rates, and traffic volumes at signalized intersections using probe vehicle trajectories. Transportation Research Record 2673 (11), 660–670.
  * Zhao et al. (2019b) Zhao, Y., Zheng, J., Wong, W., Wang, X., Meng, Y., Liu, H. X., 2019b. Various methods for queue length and traffic volume estimation using probe vehicle trajectories. Transportation Research Part C: Emerging Technologies 107, 70–91.


# Decomposition of Approximately Monotone and Convex Sequences

Angshuman R. Goswami  Department of Mathematics, University of Pannonia,
Veszprém, Egyetem u. 10, H-8200<EMAIL_ADDRESS>
###### Abstract.

In this paper, we primarily deal with approximately monotone and convex
sequences. We start by showing that any sequence can be expressed as the
difference between two nondecreasing sequences. One of these two monotone
sequences act as the majorant of the original sequence, while the other
possesses non-negativity. Another result establishes that an approximately
monotone(increasing) sequence can be closely approximated by a non-decreasing
sequence. A similar assertion can be made for approximately convex sequence. A
sequence $\big{<}u_{n}\big{>}_{n=0}^{\infty}$ is said to be approximately
convex (or $\varepsilon$-convex) if the following inequality holds under the
mentioned assumptions

$u_{i}-u_{i-1}\leq u_{j}-u_{j-1}+\varepsilon\quad\quad\mbox{ where}\quad\quad
i,j\in\mathbb{N}\quad\mbox{with}\quad i<j.$

We proved that an approximately convex sequence can be written as the
algebraic difference of two specific types of sequences. The initial sequence
contains sequential convexity property, while the other sequence possesses the
Lipschitz property. Moreover, we introduce an operator $\mathscr{T}$, that
termed as a twisting operator. In a compact interval $I(\subseteq\mathbb{R})$;
we characterized the convex function with this newly introduced operator.
Besides various sequence decomposition results and study related to
$\mathscr{T}$-operator on convex function; a characterization regarding non-
negative sequential convexity, a fractional inequality, implication of
$\mathscr{T}$ operator on various types of functions, relationship between a
convex function and a convex sequence are also included.

Motivation, previous research in this direction, various applications and
linkage with some other branches of mathematics are discussed in the
introduction section.

###### Key words and phrases:

Approximately monotone & Convex sequences, Decomposition and approximation of
sequences

###### 2000 Mathematics Subject Classification:

Primary: . 26A51; Secondary: 39B62

## Introduction

Functional equation and inequalities is one of the few branches of
Mathematics; which is very much interdisciplinary. One of the popular research
area in this field is generalizing the concept of convexity and then
establishing various characterizations of it. There are hundreds of books and
scientific articles from the last century that contributes in this direction.
[14, 13, 16, 17, 18, 19, 20, 21] and the reference therein plays an important
role in the development of convex function theory and also reflects immense
research possibility in this subject. However the introduction to the concept
of sequential convexity is relatively new. The first mention of the term
convex sequence is found in [1]. The importance of this terminology can be
understood with the simple fact that arithmetic, Fibonacci and many other
familiar sequences are only some specific types of convex sequences.
Therefore, investigation of various properties of such sequences can be
carried out under the umbrella of sequential convexity. Since the discovery of
it, several advancements have been made and most of these are closely
analogous to the ordinary convex functions. For instance; studies related to
higher-order convex sequence, the establishment of a discrete version of
Hermite-Hadamard type inequalities, various implications and applications of
sequential convexity in the theory of trigonometric functions etc. can be
found in the papers [2, 3, 4, 5, 6, 7, 8]. Due to this reason, convex
sequences are often labeled as discrete versions of convex functions.

A function $f:I(\subseteq\mathbb{R})\to\mathbb{R}$ is said to be convex if for
any $x,y\in I$ and for all $t\in]0,1[$; it satisfies the following functional
inequality

$\displaystyle f(tx+(1-t)y)\leq tf(x)+(1-t)f(y).$

Besides this standard definition of convex function; we can define functional
convexity in many other ways. The one obvious way is replacing $u:=tx+(1-t)y$
in the above inequality. After simplification, we can obtain the following
equivalent form that also represents convexity of $f$

(1) $\displaystyle\dfrac{f(u)-f(x)}{u-x}\leq\dfrac{f(y)-f(u)}{y-u}\quad
x,u,y\in I\quad\mbox{with}\quad x<u<y.$

Both of these definitions can be used to represent/determine the convex
function $f$. However, throughout this paper, we are going to extensively use
(1) to establish various results.

The researchers have been working in the field of convex analysis for
centuries. The first breakthrough in the field of approximate convexity was
found in the paper [16]. It is shown that for any $\delta>0$, a function
$f:I\to\mathbb{R}$, satisfying the inequality

$\displaystyle f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)+\delta\quad\mbox{for any}\quad
x,y\in I\quad\mbox{and for all}\quad t\in[0,1]$

can be expressed as the algebraic summation of two functions. One of these
possesses convexity; while the supremum norm of the other function is less
than $\delta/2.$ In the last few decades, there have been many decomposition,
approximation, and characterization of various versions of approximate and
higher-order convex functions. These can be found in papers like [15, 16, 17,
18, 19, 20] etc. One such generalization is $\varepsilon$-convexity which was
introduced by Páles in his paper [14]. Let $\varepsilon>0$ to be fixed. A
function $f:I\to\mathbb{R}$ is said to be $\varepsilon$-convex if for any
$x,y\in I$ and $t\in[0,1]$; the following inequality holds true

$\displaystyle f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)+\varepsilon t(1-t)|y-x|.$

Later in this paper, we are going to discuss more about this special type of
approximate convexity.

These findings open up new possibilities for studying the known definition of
sequential convexity in an approximate sense and finding a linkage of it with
approximate convex functions. So far we have not found any article that
closely deals with this direction. Besides, there are many scopes to provide
new characterization to convex functions. In this paper, our findings are
mostly related to these two topics.

In the first fold of the paper, we primarily deal with sequences. We begin
with showing that any sequence $\big{<}u_{n}\big{>}_{n=0}^{\infty}$ can be
decomposed as the sum of two sequences $\big{<}v_{n}\big{>}_{n=0}^{\infty}$
and $\big{<}w_{n}\big{>}_{n=0}^{\infty}$, where
$\big{<}v_{n}\big{>}_{n=0}^{\infty}$ is a non-decreasing majorant of
$\big{<}u_{n}\big{>}_{n=0}^{\infty}$, while
$\big{<}w_{n}\big{>}_{n=0}^{\infty}$ is a non-negative and
monotone(increasing) sequence.

Then we define the notion of an approximately convex sequence about which we
have already mentioned in the abstract. We prove that $\varepsilon$-convex
sequence can be closely approximated by an ordinary convex sequence. More
precisely a $\varepsilon$-convex sequence $\big{<}u_{n}\big{>}_{n=0}^{\infty}$
can be decomposed as the algebraic summation of a convex sequence
$\big{<}v_{n}\big{>}_{n=0}^{\infty}$ and a controlled wavering sequence
$\big{<}w_{n}\big{>}_{n=0}^{\infty}$ which rate of variation is bounded above
by ${\varepsilon}/{2}.$ A similar type of assertion can also be made for
approximately monotone sequences. We showed that a Cauchy sequence can be
tightly approximated by a non-decreasing sequence in $\mathbb{R}$. Moreover,
we also establish that if we draw a graph by joining the points of a convex
sequence with free hands; we will ultimately end up with a convex function.
Mathematically, for any convex sequence $\big{<}u_{n}\big{>}_{n=0}^{\infty}$;
there exists a continuous function $f:\mathbb{R}_{+}\to\mathbb{R}$; that
satisfies $u_{n}=f(n)$ for all $n\in\mathbb{N}\cup\\{0\\}$. This more
concretely justifies that the convex sequence is nothing but a discrete
version of the convex function. This simplified several results of the well-
cited paper [4]. We will discuss it in detail later in the concerned section.
Finally, we showed that; for any arbitrarily chosen convex sequence
$\big{<}u_{n}\big{>}_{n=0}^{\infty}$ from the domain of a non-decreasing
convex function $f:I\to\mathbb{R}$; the corresponding functional sequence
$\big{<}f(u_{n})\big{>}_{n=0}^{\infty}$ is convex as well. Also, at the end of
the first section, we mentioned some arising open problems regarding higher-
order and approximate convex sequences.

In the second part of the paper, we introduce a new operator which is termed
as a twisting operator ($\mathscr{T}$). The implication of the twisting
operator $\mathscr{T}$ on any function $f:[a,b]\to\mathbb{R}$; results a new
function $\mathscr{T}_{f}$; as shown below

$\mathscr{T}(f(x)):=\mathscr{T}_{f}(x)=f(a+b-x)\quad\mbox{for all}\quad
x\in[a,b].$

We studied changes in functional properties of monotone and periodic functions
upon application of the operator $\mathscr{T}$. Besides, characterizing convex
functions in terms of this newly introduced operator, we provide results to
some functional equations too.

The description of notions can be found at the very beginning of each of the
sections.

##  1\. On monotone and convex sequences

Throughout this section the symbols $\mathbb{N}$,$\,\mathbb{R}$ and
$\mathbb{R}_{+}$ will denote the sets of natural, real and non-negative real
numbers respectively. A sequence $\big{<}u_{n}\big{>}_{n=0}^{\infty}$ is said
to be convex if it satisfies the following discrete functional inequality

(2) $\displaystyle 2u_{n}\leq u_{n-1}+u_{n+1}\quad\mbox{for all}\quad
n\in\mathbb{N}.$

In other words, $\big{<}u_{n}\big{>}_{n=0}^{\infty}$ is said to be
sequentially convex, if the sequence
$\big{<}u_{n}-u_{n-1}\big{>}_{n=1}^{\infty}$ possesses monotonicity.

######  Proposition 1.1.

Let $\big{<}u_{n}\big{>}_{n=0}^{\infty}$ be a non-negative sequence such that
$u_{n}\leq\sqrt{u_{n-1}u_{n+1}}$ holds for all $n\in\mathbb{N}$. Then the
sequence is convex.

###### Proof.

Since $\big{<}u_{n}\big{>}_{n=0}^{\infty}$ is non-negative; we can conclude
$\Big{(}\sqrt{u_{n+1}}-\sqrt{u_{n}}\Big{)}^{2}\geq 0.$ By expanding this
expression and utilizing the condition on
$\big{<}u_{n}\big{>}_{n=0}^{\infty}$, we obtain the following inequality

$\displaystyle 2u_{n}\leq 2\sqrt{u_{n-1}u_{n+1}}\leq u_{n-1}+u_{n+1}.$

This yields that $\big{<}u_{n}\big{>}_{n=0}^{\infty}$ is convex and completes
the proof. ∎

Besides, the above proposition also shows that geometric sequences possess
convexity. The next proposition gives a characterization to convex sequence
under some minimal assumptions.

######  Proposition 1.2.

Let $\big{<}u_{n}\big{>}_{n=0}^{\infty}$ be a decreasing, convex sequence with
positive terms if and only if the reciprocal sequence
$\big{<}v_{n}\big{>}_{n=0}^{\infty}$ defined by $v_{n}=\dfrac{1}{u_{n}}$ is
positive, non-decreasing and concave.

###### Proof.

The non-negativity and monotonicity (increasingness)of the sequence
$\big{<}v_{n}\big{>}_{n=0}^{\infty}$ is self-explanatory. The convexity of the
sequence $\big{<}u_{n}\big{>}_{n=0}^{\infty}$ yields (2). Utilizing convexity,
decreasingness and non-negativity of the sequence
$\big{<}u_{n}\big{>}_{n=0}^{\infty}$, we can obtain the following inequality

$\displaystyle(u_{n}-u_{n-1})u_{n+1}\leq(u_{n+1}-u_{n})u_{n-1}\quad\mbox{for
all}\quad n\in\mathbb{N}.$

Dividing both sides of the above inequality by $(u_{n-1}u_{n}u_{n+1})$, and
rearranging the terms we get

$\displaystyle\dfrac{1}{u_{n-1}}+\dfrac{1}{u_{n+1}}\leq\dfrac{2}{u_{n}}.$

This shows that the sequence $\big{<}v_{n}\big{>}_{n=0}^{\infty}$ is concave.
The reverse implication can be obtained in a similar way. ∎

The next proposition shows that any sequence can be represented as the
difference of two non-deceasing sequences.

######  Proposition 1.3.

Any sequence $\big{<}u_{i}\big{>}_{i=0}^{\infty}$ can be expressed as the
difference of two non-decreasing sequences
$\big{<}v_{i}\big{>}_{i=0}^{\infty}$ and $\big{<}w_{i}\big{>}_{i=0}^{\infty}$
of $\big{<}u_{i}\big{>}_{i=0}^{\infty}$. Where the sequence
$\big{<}v_{i}\big{>}_{i=0}^{\infty}$ also act as a majorant of
$\big{<}u_{i}\big{>}_{i=0}^{\infty}$ and the second sequence
$\big{<}w_{i}\big{>}_{i=0}^{\infty}$ is non-negative.

###### Proof.

Let $\big{<}u_{i}\big{>}_{i=0}^{\infty}$ be any arbitrary sequence. We can
define two sequences $\big{<}v_{i}\big{>}_{i=0}^{\infty}$ and
$\big{<}w_{i}\big{>}_{i=0}^{\infty}$ as follows

$\displaystyle v_{i}:=\begin{cases}u_{0},\quad\mbox{if}\quad i=0\\\
u_{0}+\overset{n}{\underset{i=1}{\sum}}|u{{}_{i}}-u_{i-1}|\quad
n\in\mathbb{N}\end{cases}\quad\quad\quad\mbox{and}\quad\quad\quad
w_{i}=v_{i}-u_{i}.$

We can express $u_{i}=v_{i}-w_{i}$ for all $i\in\mathbb{N}\cup\\{0\\}$. By
definition, It can be easily observed that

$\displaystyle v_{i}\geq u_{i}\quad\mbox{and}\quad w_{i}\geq 0\quad\mbox{for
all}\quad i\in\mathbb{N}\cup\\{0\\}.$

Therefore, only remains to show the monotonicity of these two newly defined
sequences.

It can be easily observed that $u_{0}\leq u_{1}$. Let $n\in\mathbb{N}$ to be
arbitrary. Then we can compute

$\displaystyle
v_{n+1}=u_{0}+\sum_{i=1}^{n+1}|u_{i}-u_{i-1}|=u_{0}+\sum_{i=1}^{n}|u_{i}-u_{i-1}|+|u_{n+1}-u_{n}|\geq
v_{n}.$

This establishes the monotonicity of the sequence
$\big{<}v_{i}\big{>}_{i=0}^{\infty}$. We can see that

(3) $\displaystyle
w_{1}-w_{0}=(v_{1}-u_{1})-(v_{0}-u_{0})=v_{1}-u_{1}=u_{0}+|u_{1}-u_{0}|-u_{1}\geq
0.$

Also, for any $n\in\mathbb{N}$ we have the inequality

$\displaystyle
v_{n+1}=u_{0}+\sum_{i=1}^{n+1}|u_{i}-u_{i-1}|=u_{0}+\sum_{i=1}^{n}|u_{i}-u_{i-1}|+|u_{n+1}-u_{n}|\geq
v_{n}+u_{n+1}-u_{n}.$

It shows $w_{n+1}\geq w_{n}$ for all $n\in\mathbb{N}$. This together with (3)
proves the monotonicity of the sequence $\big{<}w_{i}\big{>}_{i=0}^{\infty}$
and completes the proof. ∎

A sequence $\big{<}u_{i}\big{>}_{i=0}^{\infty}$ is said to be
$\varepsilon$-monotone, if for any
$u_{m}\in\big{<}u_{i}\big{>}_{i=0}^{\infty}$; the inequality

(4) $\displaystyle u_{m}\leq u_{n}+\varepsilon\quad\mbox{for
all}\quad{m<n}\quad(n\in\mathbb{N}).$

holds. In the next theorem, we are going to show that a $\varepsilon$-monotone
sequence can be approximated by an ordinary monotone(increasing) sequence. The
theorem extensively uses the decomposition result of $\varepsilon$-monotone
function; which was introduced by Páles in his paper [88]

######  Theorem 1.4.

Let $\big{<}u_{n}\big{>}_{n=0}^{\infty}$ be a $\varepsilon$-monotone sequence.
Then there exists a continuous, almost everywhere differentiable
$\varepsilon$-monotone function $f:\mathbb{R}_{+}\to\mathbb{R}$ satisfying
$f(n)=u_{n}$ for all $n\in\mathbb{N}\cup\\{0\\}.$ Additionally, there exists a
monotone sequence $\big{<}v_{n}\big{>}_{n=0}^{\infty}$, such that

(5) $\displaystyle|u_{n}-v_{n}|\leq\dfrac{\varepsilon}{2}\quad\mbox{holds for
all}\quad n\in\mathbb{N}\cup\\{0\\}.$

###### Proof.

To prove this theorem, We construct a function $f:\mathbb{R}_{+}\to\mathbb{R}$
as

$\displaystyle
f(x):=tu_{n-1}+(1-t)u_{n}\quad\mbox{;}\,\,x\in[{n-1},n]\,\,\mbox{satisfying}\,\,x=t(n-1)+(1-t)n\quad(t\in[0,1]).$

From above, the continuity of the function $f$ is clearly visible. Besides it
is evident that the constructed function is differentiable everywhere except
the points at $\mathbb{N}\cup\\{0\\}$. Substituting $t=0$; at the definition
of $f$, we obtain $f(n)=u_{n}$ for all $\mathbb{N}\cup\\{0\\}$. Now to show
$f$ is $\varepsilon$-monotone; let $x,y\in\mathbb{R}_{+}$ with $x<y$. It leads
to the following

$\displaystyle x\in[{n_{1}-1},{n}]\quad\mbox{and}\quad
y\in[{n_{2}-1},{n_{2}}],\quad\mbox{where}\quad n_{1}\leq
n_{2}\quad\quad(n_{1},n_{2}\in\mathbb{N}).$

Based on it $f(x)$ and $f(y)$ will satisfy the following two inequalities

$\displaystyle\min\Big{\\{}u_{n_{1}-1},u_{n_{2}}\Big{\\}}\leq
f(x)\leq\max\Big{\\{}u_{n_{1}-1},u_{n_{1}}\Big{\\}}\quad\mbox{and}\quad\min\Big{\\{}u_{n_{2}-1},u_{n_{2}}\Big{\\}}\leq
f(y)\leq\max\Big{\\{}u_{n_{2}-1},u_{n_{2}}\Big{\\}}$

By, using $\varepsilon$-monotonicity; from the two inequalities above, we can
compute

$\displaystyle
f(x)-f(y)\leq\max\Big{\\{}u_{n_{1}-1},u_{n_{1}}\Big{\\}}-\min\Big{\\{}u_{n_{2}-1},u_{n_{2}}\Big{\\}}\leq\varepsilon.$

This yields that $f$ is $\varepsilon$-monotone and completes the first part of
the theorem.

To prove the second part, we will utilize one of the results of
$\varepsilon$-monotonicity that appeared in the paper [14]. The statement of
the theorem can also be formulated as follows

If a function $f:\mathbb{R}_{+}\to\mathbb{R}$ is $\varepsilon$-monotone then
there exists a monotone(increasing) function $g:\mathbb{R}_{+}\to\mathbb{R}$
that satisfies the inequality

$\displaystyle|f(x)-g(x)|\leq\dfrac{\varepsilon}{2}\quad\mbox{for all}\quad
x\in I.$

Now choosing the elements of the sequence $\big{<}v_{n}\big{>}_{n=0}^{\infty}$
as $v_{n}=g(n)$ for all $n\in\mathbb{N}\cup\\{0\\}$ and using the above
mentioned result, we can obtain the inequality (5). Besides, monotonicity of
the function $g:\mathbb{R}_{+}\to\mathbb{R}$ ensures that the sequence
$\big{<}v_{n}\big{>}_{n=0}^{\infty}$ is non-decreasing. This completes the
proof and establishes the result. ∎

In the upcoming result, we will show the relationship of the Cauchy sequence
with $\varepsilon$-monotonicity. A sequence
$\big{<}u_{n}\big{>}_{n=0}^{\infty}$ is said to be Cauchy sequence if for any
given $\delta\geq 0$ there exists $r\in\mathbb{N}$; that satisfies the
following inequality under the mentioned conditions

(6) $\displaystyle|u_{m}-u_{n}|\leq\delta\quad\mbox{for all}\quad m,n\geq
r\quad(m,n,r\in\mathbb{N}).$

######  Proposition 1.5.

Every Cauchy sequence is approximately monotone.

###### Proof.

To prove the proposition, we assume $\big{<}u_{n}\big{>}_{n=0}^{\infty}$ be a
Cauchy sequence such that for a fixed $\delta\geq 0$, it satisfies (6). We
consider

(7) $\displaystyle\gamma=\max\Big{\\{}\max|u_{p}-u_{q}|,\delta\Big{\\}}\quad
p,q\leq r\quad(p,q,r\in\mathbb{N}).$

We take $\varepsilon=\gamma+\delta$ and will show that
$\big{<}u_{i}\big{>}_{n=0}^{\infty}$ is $\varepsilon$-monotone. For that we
have to consider three cases. At first, we assume
$m,n\in\mathbb{N}\cup\\{0\\}$ with $m<n\leq r$. Using (7), we can compute the
following

$\displaystyle u_{m}-u_{n}\leq\max_{p,q\leq
r}|u_{p}-u_{q}|\leq\gamma<\varepsilon.$

For $r\leq m<n,$ utilizing (6) we can proceed as follows

$\displaystyle u_{m}-u_{n}\leq|u_{m}-u_{n}|\leq\delta\leq\varepsilon.$

Finally for $m<r<n$, utilizing triangular inequality, we obtain the following

$\displaystyle
u_{m}-u_{n}\leq|u_{p}-u_{r}|+|u_{r}-u_{n}|\leq\gamma+\delta=\varepsilon.$

The analogous inequalities under these three situations establish that
$\big{<}u_{i}\big{>}_{n=0}^{\infty}$ possesses $\varepsilon$-monotonicity. ∎

The above result leaves us with the question, how effectively we can
approximate a Cauchy sequence with a non-decreasing sequence. The next
proposition shows the close relationship of a Cauchy sequence with a usual
monotonic sequence.

######  Corollary 1.6.

Let $\big{<}u_{n}\big{>}_{n=0}^{\infty}$ be a Cauchy sequence in $\mathbb{R}$.
Then there exists a monotone(increasing) sequence
$\big{<}v_{n}\big{>}_{n=0}^{\infty}$ satisfying
$\underset{n\to\infty}{\lim}u_{n}=\underset{n\to\infty}{\lim}v_{n}$

###### Proof.

Since $\big{<}u_{n}\big{>}_{n=0}^{\infty}$ is a Cauchy sequence in
$\mathbb{R}$; it converges to a finite value. Without losing the generality,
we assume that there exists $l\in\mathbb{R}$ such that
$\underset{n\to\infty}{\lim}u_{n}=l$. Now we define a sequence
$\big{<}u_{n}\big{>}_{n=0}^{\infty}$ as follows

$\displaystyle v_{n}=\inf_{n\leq
k}v_{k}\qquad\qquad(n\in\mathbb{N}\cup\\{0\\})$

By definition; it is evident that $v_{n}\leq u_{n}$ for all
$n\in\mathbb{N}\cup\\{0\\}$. Also for arbitrary $n_{1}$ and
$n_{2}\in\mathbb{N}\cup\\{0\\}$ with $n_{1}\leq n_{2}$; we can compute the
inequality below

$\displaystyle v_{n_{1}}=\inf_{n_{1}\leq k}v_{k}\leq\inf_{n_{2}\leq
k}v_{k}=v_{n_{2}}.$

This yields the monotonicity of the sequence
$\big{<}v_{n}\big{>}_{n=0}^{\infty}$. Now, we only need to show that
$\underset{n\to\infty}{\lim}v_{n}=l$.

Since $\big{<}u_{n}\big{>}_{n=0}^{\infty}$ converges to $l$; it also indicates
that $l$ is both limit superior and limit inferior for this sequence. By using
the definition of limit inferior, we obtain the following

$\displaystyle\lim_{n\to\infty}v_{n}=\lim_{n\to\infty}\Big{(}\inf_{n\leq
k}u_{k}\Big{)}=l\qquad\qquad(n\in\mathbb{N}\cup\\{0\\}).$

This completes the proof. ∎

Before proceeding to the next theorem, we must establish the following lemma.

######  Lemma 1.7.

Let $n\in\mathbb{N}$ be arbitrary. Then for any
$a_{1},\cdots,a_{n}\in\mathbb{R}$ and $b_{1},\cdots,b_{n}\in]0,\infty[$; the
following inequality holds

(8)
$\displaystyle\min\bigg{\\{}\dfrac{a_{1}}{b_{1}},\cdots,\dfrac{a_{n}}{b_{n}}\bigg{\\}}\leq\dfrac{a_{1}+\cdots+a_{n}}{b_{1}+\cdots+b_{n}}\leq\max\bigg{\\{}\dfrac{a_{1}}{b_{1}},\cdots,\dfrac{a_{n}}{b_{n}}\bigg{\\}}.$

###### Proof.

We will prove the theorem using the principle of mathematical induction. For
$n=1$, there is nothing to prove. For $n=2$; without loss of generality; for
$a_{1},a_{2}\in\mathbb{R}$ and $b_{1},b_{2}\in]0,\infty[$, we assume that
$\dfrac{a_{1}}{b_{1}}\leq\dfrac{a_{2}}{b_{2}}$. We can also express this in
terms of two inequalities as follows

$\displaystyle
a_{1}(b_{1}+b_{2})\leq(a_{1}+a_{2})b_{1}\quad\mbox{and}\quad(a_{1}+a_{2})b_{2}\leq(b_{1}+b_{2})a_{2}.$

The two inequalities above together yield the following

(9)
$\displaystyle\dfrac{a_{1}}{b_{1}}\leq\dfrac{a_{1}+a_{2}}{b_{1}+b_{2}}\leq\dfrac{a_{2}}{b_{2}}$

and validates the result for $n=2.$ Now we assume that the statement is true
for any fixed $n\in\mathbb{N}$. That is (8) holds true. Now, to establish the
statement, we need to validate it for $n+1\in\mathbb{N}$.

We assume $a_{1},\cdots,a_{n},a_{n+1}\in\mathbb{R}$ and
$b_{1},\cdots,b_{n},b_{n+1}\in]0,\infty[$. First utilizing (9) and then (8),
we can compute the following inequality

$\displaystyle\dfrac{a_{1}+\cdots+a_{n}+a_{n+1}}{b_{1}+\cdots+b_{n}+b_{n+1}}\leq\max\bigg{\\{}\dfrac{a_{1}+\cdots+a_{n}}{b_{1}+\cdots+b_{n}},\dfrac{a_{n+1}}{a_{n+1}}\bigg{\\}}\leq\max\bigg{\\{}\dfrac{a_{1}}{b_{1}},\cdots,\dfrac{a_{n}}{b_{n}},\dfrac{a_{n+1}}{b_{n+1}}\bigg{\\}}.$

In a similar way, we can show the inequality below

$\displaystyle\min\bigg{\\{}\dfrac{a_{1}}{b_{1}},\cdots,\dfrac{a_{n}}{b_{n}},\dfrac{a_{n+1}}{b_{n+1}}\bigg{\\}}\leq\min\bigg{\\{}\dfrac{a_{1}+\cdots+a_{n}}{b_{1}+\cdots+b_{n}},\dfrac{a_{n+1}}{a_{n+1}}\bigg{\\}}\leq\dfrac{a_{1}+\cdots+a_{n}+a_{n+1}}{b_{1}+\cdots+b_{n}+b_{n+1}}.$

Combining the above two inequalities, we will have our desired result. This
completes the proof of the lemma. ∎

The origin $\Gamma$-function is directly linked to the study of factorial
sequences. It states that a continuous function
$\Gamma:\mathbb{R}_{+}\to\mathbb{R}_{+}$ is obtainable from the sequence
$\big{<}n!\big{>}_{n=0}^{\infty}$ that possesses all the factorial properties.
Similarly, the upcoming theorem shows that any discrete graph of a convex
sequence can be extendable to a convex function. In other words, for every
convex sequence $\big{<}u_{n}\big{>}_{n=0}^{\infty}$ there exists atleast one
continuous convex function $f:\mathbb{R}_{+}\to\mathbb{R}$ that precisely
approximates the sequential points.

######  Theorem 1.8.

Let $\big{<}u_{n}\big{>}_{n=0}^{\infty}$ be a convex sequence. There there
exists a continuous almost everywhere differentiable convex function
$f:\mathbb{R}_{+}\to\mathbb{R}$ such that $f(n)=u_{n}$ for all
$n\in\mathbb{N}\cup\\{0\\}.$
Conversely, if $f$ is convex in $\mathbb{R}_{+}$; then the sequence
$<f(n)>_{n=0}^{\infty}$ is convex in $\mathbb{N}\cup\\{0\\}$.

###### Proof.

To prove the statement, we define a function $f:\mathbb{R}_{+}\to\mathbb{R}$
as follows

(10) $\displaystyle f(x):=tu_{n}+(1-t)u_{n+1}\quad\mbox{where}\quad
x:=tn+(1-t)(n+1)\quad(t\in[0,1]\,\mbox{ and }\,n\in\mathbb{N}\cup\\{0\\}).$

From the construction, it is clearly visible that $f$ is continuous in
$\mathbb{R}$. Besides it is also evident that $f$ is differentiable everywhere
except at the points $N\cup\\{0\\}$. Substituting $x=n$ in (10), we obtain
$f(n)=u_{n}$ for all $N\cup\\{0\\}$. We are only left to show the convexity of
the function $f$. Before proceeding, we need to go through some tangential
properties of the function $f$.

Overall, the function $f$ can be stated as the joining of line segments that
are defined consecutively in between the points $n-1$ and $n$ for all
$n\in\mathbb{N}$. For simplicity, we term these as functional line segments of
$f$. Based on it, for any $x,y\in\mathbb{R}_{+}$ with $x<y$; we can conclude
the statement as mentioned below

(11) Slope of the functional line segment(segments) of $f$ that contains
$(x,f(x))$ $\displaystyle\leq\mbox{{Slope of the line joining the points
$(x,f(x))$ and $(y,f(y))$}}$ $\displaystyle\leq\mbox{{Slope of the functional
line segment(segments) of $f$ that contains $(y,f(y))$}.}$

If both $(x,f(x))$ and $(y,f(y))$ lies in the same functional line segment,
the statement is obvious.

Next we consider the case when $x\in[n-1,n[$ and $y\in]n,n+1]$,
($n\in\mathbb{N}$) that is $x$ and $y$; lies in two consecutive intervals.
Utilizing the sequential convexity property of
$\big{<}u_{n}\big{>}_{n=0}^{\infty}$ (or $\big{<}f(n)\big{>}_{n=0}^{\infty}$)
and basic geometry of slopes in straight lines, we can compute the inequality
below

$\displaystyle
f(n)-f(n-1)=\dfrac{f(n)-f(x)}{n-x}\leq\dfrac{f(y)-f(n)}{y-n}=f(n+1)-f(n).$

The expression $\dfrac{f(y)-f(x)}{y-x}$ can also be written as
$\dfrac{f(y)-f(n)+f(n)-f(x)}{(y-n)+(n-x)}$.

Together with this, Using (8) of Lemma 1.7; the above inequality can be
extended as

(12) $\displaystyle f(n)-f(n-1)\leq\dfrac{f(y)-f(x)}{y-x}\leq f(n+1)-f(n).$

This yields the validity of statement (LABEL:E200) for this particular case.
Finally we assume $x\in[n_{1},n_{1}+1]$ and $y\in[n_{2},n_{2}+1]$ where
$n_{1},n_{2}\in\mathbb{N}\cup\\{0\\}$ such that $n_{2}-n_{1}>1$. From (8) of
Lemma 1.7 by utilizing (12); we can obtain the inequality below

$\displaystyle\dfrac{f(y)-f(x)}{y-x}$
$\displaystyle=\dfrac{\Big{(}f(y)-f(n_{2})\Big{)}+\Big{(}f(n_{2})-f(n_{2}-1)\Big{)}+\cdots+\Big{(}f(n_{1}+1)-f(x)\Big{)}}{\Big{(}y-n_{2}\Big{)}+\Big{(}n_{2}-(n_{2}-1)\Big{)}+\cdots+\Big{(}(n_{1}+1)-x\Big{)}}$
$\displaystyle\leq\max\bigg{\\{}\dfrac{f(y)-f(n_{2})}{y-n_{2}},\dfrac{f(n_{2})-f(n_{2}-1)}{n_{2}-(n_{2}-1)},\cdots,\dfrac{f(n_{1})-f(x)}{n_{1}-x}\bigg{\\}}$
$\displaystyle=\dfrac{f(y)-f(n_{2})}{y-n_{2}}=f(n_{2}+1)-f(n_{2}).$

In a similar way, we can compute the following inequality as well

$\displaystyle f(n_{1}+1)-f(n_{1})$
$\displaystyle=\dfrac{f(n_{1}+1)-f(x)}{(n_{1}+1)-x}$
$\displaystyle=\min\bigg{\\{}\dfrac{f(n_{1}+1)-f(x)}{(n_{1}+1)-x},\dfrac{f(n_{1}+2)-f(n_{1}+1)}{(n_{1}+2)-(n_{1}+1)},\cdots,\dfrac{f(y)-f(n_{2})}{y-n_{2}}\bigg{\\}}$
$\displaystyle\leq\dfrac{\Big{(}f(n_{1}+1)-f(x)\Big{)}+\Big{(}f(n_{1}+2)-f(n_{1}+1)\Big{)}+\cdots+\Big{(}f(y)-f(n_{2})\Big{)}}{\Big{(}(n_{1}+1)-x\Big{)}+\Big{(}(n_{1}+2)-(n_{1}+1)\Big{)}+\cdots+\Big{(}y-n_{2}\Big{)}}$
$\displaystyle=\dfrac{f(y)-f(x)}{y-x}.$

The above two inequalities validate the statement (LABEL:E200).

Now we can proceed to establish the convexity of $f$. Without loss of
generality, we assume $x,u,y\in\mathbb{R}_{+}$ with $x<u<y$ and $u\in[n,n+1]$
for a fixed $n\in\mathbb{N}\cup\\{0\\}$. Then first using the last part of
inequality in (LABEL:E200) and then applying the initial part of the same
inequality, we obtain the following

$\displaystyle\dfrac{f(u)-f(x)}{u-x}$ $\displaystyle\leq{f(n+1)-f(n)}$ and
$\displaystyle f(n+1)-f(n)$ $\displaystyle\leq\dfrac{f(y)-f(u)}{y-u}.$

Combining the above inequalities, we arrive at (1). This shows that the
function $f$ is convex and completes the first part of the proof.

To show the converse part, we assume three consecutive arbitrary $n-1,n$ and
$n+1$ from $\mathbb{N}$. Then the convexity of $f$ yields the following
inequality

$\displaystyle 2f(n)=2f\bigg{(}\dfrac{(n-1)+(n+1)}{2}\bigg{)}\leq
f(n-1)+f(n+1).$

This establishes the sequential convexity of
$\big{<}f_{n}\big{>}_{n=0}^{\infty}$ and completes the proof. ∎

One of the important papers of convex sequence is [4], which was written by
Debnath and Wu. In Lemma 7 of that paper, without establishing the convexity
of $f$, they showed that for a non-decreasing convex function $g$ on
$\mathbb{R}$, the composite function $g\circ f:\mathbb{R}_{+}\to\mathbb{R}$
also possesses convexity. On the other hand, if the function $g$ is concave
and decreasing then $g\circ f$ is concave on $\mathbb{R}_{+}$. Later this
lemma was used to prove one of the main theorems and other subsequent results.
From Theorem 1.8 of our paper, it is clear that the function $f$ is convex.
And from the well known results of composite functions, now it turns out that
the lemma 7 is obvious. Some of the other results of [4] can also be shortened
in the same manner.

We are now defining three approximate sequential terminologies. We will see
how closely, they are interconnected with one another. A sequence
$\big{<}u_{n}\big{>}_{n=0}^{\infty}$ is said to be $\varepsilon$-convex if for
any $i,j\in\mathbb{N}\cup\\{0\\}$ with $i<j$; it satisfies the following
inequality

(13) $\displaystyle u_{i}-u_{i-1}\leq u_{j}+u_{j-1}+\varepsilon.$

It is clearly evident that a convex sequence naturally possesses
$\varepsilon$-convexity. Substituting $j=i+1$ in (13), we can also observe
that sequential $\varepsilon$-convexity leads to the discrete inequality
$2u_{i}\leq u_{i-1}+u_{i+1}+\varepsilon$ for all $i\in\mathbb{N}$.

Previously, we discussed approximately monotone sequences. A sequence
$\big{<}v_{n}\big{>}_{n=0}^{\infty}$ is termed as $\varepsilon$-Hölder if and
only if both $\big{<}v_{n}\big{>}_{n=0}^{\infty}$ and
$\big{<}-v_{n}\big{>}_{n=0}^{\infty}$ are $\varepsilon$-monotone. That is the
following inequality holds true

$\displaystyle|v_{n}-v_{m}|\leq\varepsilon\quad\mbox{for all}\quad
m,n\in\mathbb{N}\cup\\{0\\}.$

On the other hand, a sequence $\big{<}w_{i}\big{>}_{i=0}^{\infty}$ is said to
be a Lipschitz sequence if there exists a $l\geq 0$ such that it satisfies the
following inequality alongside mentioned conditions

$\displaystyle|w_{n}-w_{m}|\leq l|n-m|\quad\mbox{for all}\quad
n,m\in\mathbb{N}\cup\\{0\\}.$

The infimum of all such $l$ that satisfies the above inequality is termed as
Lipschitz modulo of the sequence $\big{<}w_{i}\big{>}_{i=0}^{\infty}.$ These
definitions lead us to the following proposition.

######  Proposition 1.9.

Let $\big{<}u_{n}\big{>}_{n=0}^{\infty}$ and
$\big{<}v_{n}\big{>}_{n=0}^{\infty}$ are respectively convex and
${\varepsilon}/{2}$-Hölder sequences. Then the algebraic sum of these two
sequences forms a $\varepsilon$-convex sequence.

###### Proof.

Let consider the sequence
$\big{<}w_{n}\big{>}_{n=0}^{\infty}=\big{<}u_{n}+v_{n}\big{>}_{n=0}^{\infty}$
and $i,j\in\mathbb{N}$ such that $i<j$. The mentioned sequences give the
following three inequalities

$\displaystyle u_{i}-u_{i-1}\leq u_{j}-u_{j-1}\qquad
v_{i}-v_{i-1}\leq\varepsilon/2\qquad\mbox{and}\qquad-\varepsilon/2\leq
v_{j}-v_{j-1}.$

Summing up these inequalities side by side, we arrive at the following

$\displaystyle w_{i}-w_{i-1}\leq w_{j}-w_{j-1}+\varepsilon.$

This yields that $\big{<}w_{i}\big{>}_{i=0}^{\infty}$ is $\varepsilon$-convex
and completes the proof. ∎

The theorem below shows that a $\varepsilon$-convex sequence can be decomposed
as a usual convex sequence and a Lipschitz sequence. To establish the result
we will need the notion of $\varepsilon$-convex function that we have already
discussed in the introduction section. As shown in (1) of the introduction;
following the same substitution; an equivalent definition of functional
$\varepsilon$-convexity can be formulated as follows. For a given
$\varepsilon>0$; a function $f:I\to\mathbb{R}$ is termed as
$\varepsilon$-convex if for any $x,u,y\in I$ with $x<u<y$; we have the
following functional inequality

(14)
$\displaystyle\dfrac{f(u)-f(x)}{u-x}-\dfrac{\varepsilon}{2}\leq\dfrac{f(y)-f(u)}{y-u}+\dfrac{\varepsilon}{2}$

Now we can proceed to the theorem which can also be seen as the Hyers-Ulam
stability result in the discrete version of approximate convexity.

######  Theorem 1.10.

Let $\big{<}u_{n}\big{>}_{n=0}^{\infty}$ be a $\varepsilon$-convex sequence.
Then there exists a $\varepsilon$-convex function
$f:\mathbb{R}_{+}\to\mathbb{R}$ such that it satisfies $f(n)=u_{n}$ for all
$n\in\mathbb{N}\cup\\{0\\}$. Moreover, there exists a convex sequence
$\big{<}v_{n}\big{>}_{n=0}^{\infty}$ such that
$\big{<}u_{n}-v_{n}\big{>}_{n=0}^{\infty}$ results in a Lipschitz sequence
with the Lipschitz modulo bounded above by $\varepsilon$.

###### Proof.

To prove the statement, first we construct a function
$f:\mathbb{R}_{+}\to\mathbb{R}$ as defined in (10) of Theorem 1.8. Clearly $f$
interpolates $\big{<}u_{n}\big{>}_{n=0}^{\infty}$ in all points of
$\mathbb{N}\cup\\{0\\}$. We will now show that $f$ is a $\varepsilon$-convex
function. To establish the result, we need to show two important inequalities.
Let $x,y\in\mathbb{R}_{+}$ with $x<y$ such that $x\in[n_{1},n_{1}+1]$ and
$y\in[n_{2},n_{2}+1]$. Then

(15) $\displaystyle\dfrac{f(y)-f(x)}{y-x}-\dfrac{\varepsilon}{2}$
$\displaystyle\leq\max_{n_{1}<n<n_{2}}\big{\\{}f(n+1)-f(n)-\varepsilon/2\big{\\}}\quad\quad\quad(n\in\mathbb{N})$
and $\displaystyle\min_{n_{1}<n<n_{2}}\big{\\{}f(n+1)-$ $\displaystyle
f(n)+\varepsilon/2\big{\\}}\leq\dfrac{f(y)-f(x)}{y-x}+\dfrac{\varepsilon}{2}\quad\quad\quad(n\in\mathbb{N}).$

For readability purposes, we will only prove the first inequality. The
establishment of the second inequality will be analogous. With our assumptions
regarding $x$ and $y$, we can rewrite the expression
$\dfrac{f(y)-f(x)}{y-x}-\dfrac{\varepsilon}{2}$ as follows

$\displaystyle\dfrac{\Big{(}f(y)-f(n_{2})-(y-n_{2})\varepsilon/2\Big{)}+\cdots+\Big{(}f(n_{1}+1)-f(x)-(n_{1}+1-x)\varepsilon/2\Big{)}}{\big{(}y-n_{2}\big{)}+\cdots+\big{(}n_{1}+1-x\big{)}.}$

Using (8) of Lemma 1.7, we can conclude the following

$\displaystyle\dfrac{f(y)-f(x)}{y-x}-\dfrac{\varepsilon}{2}$
$\displaystyle=\dfrac{f(y)-f(x)-(y-x)\varepsilon/2}{y-x}$
$\displaystyle\leq\max\bigg{\\{}\dfrac{f(y)-f(n_{2})-(y-n_{2})\varepsilon/2}{y-n_{2}},\cdots$
$\displaystyle\quad\quad\cdots\cdots\dfrac{f(n_{1}+1)-f(x)-(n_{1}+1-x)\varepsilon/2}{n_{1}+1-x}\bigg{\\}}$
$\displaystyle=\max\Big{\\{}f(n_{1}+1)-f(n_{1})-\varepsilon/2,\cdots\cdots,f(n_{2}+1)-f(n_{2})-\varepsilon/2\Big{\\}}.$

This establishes the first part of (15). Now to complete the proof we assume
that $x,u,y\in\mathbb{R}_{+}$ with $x<u<y$. Then according to the results
obtained above; we will have distinct $i$ and $j\in\mathbb{N}\cup\\{0\\}$ with
$i\leq u\leq j$ satisfying the following two inequalities

(16) $\displaystyle\dfrac{f(u)-f(x)}{u-x}-\dfrac{\varepsilon}{2}$
$\displaystyle\leq f(i+1)-f(i)-\dfrac{\varepsilon}{2}$ and $\displaystyle
f(j+1)-f(j)+\dfrac{\varepsilon}{2}$
$\displaystyle\leq\dfrac{f(y)-f(u)}{y-u}+\dfrac{\varepsilon}{2}.$

But $\varepsilon$-convexity of the sequence
$\Big{<}f(n)\Big{>}_{n=0}^{\infty}$ yields

$\displaystyle f(i+1)-f(i)-\dfrac{\varepsilon}{2}\leq
f(j+1)-f(j)+\dfrac{\varepsilon}{2}.$

This together with (16), establishes (14).

Since $x,u$, and $y$ arbitrary with $x<u<y$, according to the definition of
$\varepsilon$-convex function, $f$ possesses $\varepsilon$-convexity. This
completes the first part of the theorem.

To prove the second part, we will utilize one of the decomposition results of
functional $\varepsilon$-convexity that was formulated by Pales in his paper
[14]. In view of our paper, It can be formulated as follows

"If a function $f:\mathbb{R}_{+}\to\mathbb{R}$ is $\varepsilon$-convex then it
can be expressed as the algebraic summation of an ordinary convex function
$g:\mathbb{R}_{+}\to\mathbb{R}$ and a Lipschitz function
$h:\mathbb{R}_{+}\to\mathbb{R}$ such that Lipschitz modulo $lip(h)$ is bounded
above by $\varepsilon.$ That is

$\underset{x,y\,\in\,\mathbb{R}_{+}}{\sup}\left|\dfrac{f(y)-f(x)}{y-x}\right|\leq\varepsilon.$

" Now we choose the elements of the sequence
$\big{<}v_{n}\big{>}_{n=0}^{\infty}$ as $v_{n}=g(n)$ for all
$n\in\mathbb{N}\cup\\{0\\}$. From the converse part of Theorem 1.11, we can
conclude that $\big{<}v_{i}\big{>}_{n=0}^{\infty}$ is a convex sequence. Using
the stated result above, we can obtain the inequality (5) as well. This proves
all the assertions in the theorem and completes the proof. ∎

######  Theorem 1.11.

Let $f:\mathbb{R}_{+}\to\mathbb{R}$ be a non-decreasing convex function. Then
for any strictly non-decreasing convex sequence
$\big{<}u_{n}\big{>}_{n=0}^{\infty}\in\mathbb{R}_{+}$; the corresponding
functional sequence $\big{<}f(u_{n})\big{>}_{n=0}^{\infty}$ is convex.

###### Proof.

Since $f$ is convex; then for any non-decreasing convex sequence
$\big{<}u_{n}\big{>}_{n=0}^{\infty}\in\mathbb{R}_{+}$; it will satisfy the
following inequality

$\displaystyle\dfrac{f(u_{n})-f(u_{n-1})}{u_{n}-u_{n-1}}\leq\dfrac{f(u_{n+1})-f(u_{n})}{u_{n+1}-u_{n}}\qquad\mbox{for
all}\quad n\in\mathbb{N}.$

Again using the monotonicity convexity property of the convex sequence
$\big{<}u_{n}\big{>}_{n=0}^{\infty}$, we obtain

$\displaystyle u_{n}-u_{n-1}\leq u_{n+1}-u_{n}\qquad\mbox{for all}\quad
n\in\mathbb{N}.$

Multiplying the above two inequalities side by side and utilizing the non-
negativity of all the involved terms, we arrive at the following

$\displaystyle f(u_{n})-f(u_{n-1})\leq f(u_{n+1})-f(u_{n})\qquad\mbox{for
all}\quad n\in\mathbb{N}.$

This yields the monotonicity of the sequence
$\big{<}f(u_{n})-f(u_{n-1}\big{>}_{n=0}^{\infty}$; in other words establishes
that $\big{<}f(u_{n})\big{>}_{n=0}^{\infty}$ is convex. ∎

Higher-order convexity plays an important role in the development of function
theory. To define the notion of higher-order convexity, we first need to
become familiar with the concept of divided difference (see the papers [17,
18, 19, 20] and the referred articles in those). In the inductive way, the
first and the $n^{th}$ ordered divided difference of the function
$f:I(\subseteq\mathbb{R})\to\mathbb{R}$ can be represented as follows

$\displaystyle=\dfrac{f(u_{i})-f(u_{i+1})}{u_{i}-u_{i+1}}\qquad\quad{i\in\\{0,1,\cdots,n\\}}\qquad\quad
u_{i}\in I$ $\displaystyle[u_{0},\cdots,u_{n};f]$
$\displaystyle=\dfrac{[u_{0}\cdots,u_{n-1};f]-[u_{1}\cdots,u_{n};f]}{u_{n}-u_{0}}\qquad\qquad
u_{0},\cdots,u_{n}\in I.$

A function $f:I\to\mathbb{R}$ can be termed as $n$-convex if the following
assertion satisfies

$[u_{0},\cdots,u_{n};f]\geq 0\qquad\qquad u_{0},\cdots,u_{n}\in I.$

From the definition, It can be easily observed that the $1$-convex denotes the
ordinary monotonicity(increasingness) of the function $f$. While the
$2^{nd}$-order convexity is our well-known ordinary functional convexity.

A sequence $\big{<}u_{i}\big{>}_{i=0}^{\infty}$ is said to be $n^{th}$-order
convex, if it satisfies the below mentioned non-negativity

$\displaystyle\sum_{r=0}^{n}(-1)^{r}\binom{n}{r}u_{n-r}\geq 0\qquad\qquad
0\leq r\leq n\quad(r,n\in\mathbb{N}).$

Studies related to higher-ordered sequential convexity can be found in the
papers [I]t is evident that in sequence the first order convexity denotes the
usual monotonicity. And one can also see that the $2^{nd}$-order convexity is
our usual sequential convexity.

It is clearly visible that $n^{th}$ order sequential convexity is just the
discrete version of $n^{th}$-order convex function. Now the very basic
question that we can formulate is, "For a given $n^{th}$-order convex
sequence, is it possible to define a continuous convex function
$f:\mathbb{R}_{+}\to\mathbb{R}$ of the same order that passes through the
sequence? Another important open question is the possibility of approximating
a $\varepsilon$\- $n^{th}$ order convex sequence by an ideal $n^{th}$ order
convex sequence.

In the next section, we are going to study an operator and its various
implications in monotone, periodic and convex functions.

##  2\. The Twisting Operator

Throughout this section, $I$ will denote the non-empty and non-singleton
interval $[a,b]$. For a function $f:I\to\mathbb{R}$; the operator
$\mathscr{T}$ will be termed as twisting operator and defined as bellow

$\mathscr{T}({f}(x)):=f(a+b-x)\quad\mbox{for all}\quad x\in I.$

For the upcoming results $\mathscr{T}_{f}$ will denote the newly generated
function after implication of the twisting operator $\mathscr{T}$ on $f$.

It can be easily observed that under the respective functional operations, the
set consists of the identity operator $\mathscr{I}$; defined as
$\mathscr{I}(x):=x$ (for all $x\in I$) and the twisting $\mathscr{T}$ together
forms a group. More specifically $\\{\mathscr{I},\mathscr{T}\\}$ is an
commutative group. One can also observe the following

(17)
$\displaystyle\mathscr{T}^{2}(f)=\mathscr{T}(\mathscr{T}(f))=T_{\mathscr{T}_{f}}=\mathscr{I}\quad\mbox{that
is}\quad\mathscr{T}^{2}(f)(x)=x\quad\quad(x\in I).$

The first proposition below shows the effect of the twisting operator in
monotone functions.

######  Proposition 2.1.

Let $f:I\to\mathbb{R}$ be a non-decreasing function. Then $\mathscr{T}_{f}$
will generate a decreasing function

###### Proof.

To prove the statement, we assume $x,y\in I$ with $x<y$. This implies the
inequality $a+b-y\leq a+b-x$. Now using the monotonicity(increasingness) of
$f$, we can obtain

$\displaystyle\mathscr{T}_{f}(y)=f(a+b-y)\leq f(a+b-x)=\mathscr{T}_{f}(x).$

The above inequality establishes decreasingness of $\mathscr{T}_{f}$ and
completes the proof. ∎

A function $f:I\to\mathbb{R}$ is said to be periodic with a period $L$, where
$L<b-a$; if for any $x\in I$, there exists some $n\in\mathbb{Z}$ such that the
inequality $f(x)=f(x+nL)$ holds.

The stated proposition below shows the impact twisting operator on periodic
function.

######  Proposition 2.2.

Let $0<L<\ell(I)$, where $\ell(I)$ denotes the length of the interval $I$. If
$f:I\to\mathbb{R}$ is a periodic function with the period $L$, then
$\mathscr{T}_{f}$ also results in a periodic function.

###### Proof.

To prove this statement, we assume $x\in I$ be arbitrary. Then due to our
assumption regarding $\ell(I)$; there must exists some $n\in\mathbb{Z}$ such
that $x+nL\in I.$ This implies for all those selective $n$, the elements
$a+b-(x+nL)^{\prime}s$ and $a+b-x\in I$. Using the $L$-periodicity of $f$, we
can compute the following

$\displaystyle\mathscr{T}_{f}(x)=f(a+b-x)=f(a+b-(x+nL))=\mathscr{T}_{f}(x+nL).$

Since $x$ is arbitrary, we can conclude that $\mathscr{T}_{f}$ is periodic.
This completes the proof. ∎

Next, we will study two functional equations and the underlying results in
them. Before proceeding to these propositions, we must mention that
$I^{\prime}$ will denote the compact interval
$\bigg{[}-\dfrac{b-a}{2},\dfrac{b-a}{2}\bigg{]}.$ This first proposition shows
that if both $f$ and $\mathscr{T}_{f}$ represent the same function, then it is
almost equivalent to the evenness property of the function.

######  Proposition 2.3.

Let $f:I\to\mathbb{R}$ satisfies the functional equation $f(x)=f(a+b-x)$. Then
the function $g:I^{\prime}\to\mathbb{R}$ defined by
$g(\theta):=f\bigg{(}\dfrac{a+b}{2}+\theta\bigg{)}$ is even.

###### Proof.

To prove the proposition, we assume $\theta\in I^{\prime}$ to be arbitrary.
This implies both $\bigg{(}\dfrac{a+b}{2}-\theta\bigg{)}$ and
$\left(\dfrac{a+b}{2}+\theta\right)\in I$. By using the substitution,
$x=\dfrac{a+b}{2}-\theta$ in the functional equation $f(x)=f(a+b-x)$, we
obtain the following

$f\bigg{(}\dfrac{a+b}{2}-\theta\bigg{)}=f\bigg{(}\dfrac{a+b}{2}+\theta\bigg{)}.$

This shows that $g(-\theta)=g(\theta)$ and validates the statement. ∎

The proposition below deals with the scenario when $f+T_{f}$=0.

######  Proposition.

Let $f:I\to\mathbb{R}$ satisfies the functional equation $f(x)+f(a+b-x)=0$.
Then the function $g:I^{\prime}\to\mathbb{R}$ defined by
$g(\theta):=f\bigg{(}\dfrac{a+b}{2}+\theta\bigg{)}$ is even.

###### Proof.

The proof of the proposition is analogous to the previous result. Hence
omitted. ∎

In the upcoming theorem, we provide a characterization of the convex function
in terms of the twisting operator $\mathscr{T}$. we can also see that the
twisting operator results in a convex function when it acts on a convex
function.

######  Theorem 2.4.

Let $f:I\to\mathbb{R}$. Then the following conditions are equivalent to each
other:

  1. (i)

$f$ is convex;

  2. (ii)

For all $x,u,y\in I$ with $x<u<y$,

$\displaystyle\dfrac{(\mathscr{T}_{f})(u)-(\mathscr{T}_{f})(x)}{u-x}\leq\dfrac{(\mathscr{T}_{f})(y)-(\mathscr{T}_{f})(u)}{y-u};$

  3. (iii)

There exists a function $\varphi:I\to\mathbb{R}$ such that for all $x,u\in I$,

(18)
$\displaystyle(\mathscr{T}_{f})(u)+(x-u)\varphi(u)\leq(\mathscr{T}_{f})(x);$

  4. (iv)

For all $n\in\mathbb{N}$, $x_{1},\dots,x_{n}\in I,$ $t_{1},\dots,t_{n}\geq 0$
with $t_{1}+\dots+t_{n}=1$,

$\displaystyle(\mathscr{T}_{f})(t_{1}x_{1}+\dots+t_{n}x_{n})\leq
t_{1}\big{(}\mathscr{T}_{f})(x_{1})+\cdots+t_{n}(\mathscr{T}_{f})(x_{n}).$

  5. (v)

$\mathscr{T}_{f}$ is convex

###### Proof.

(i)$\Rightarrow$(ii): Assume that $f$ is $\Phi$ convex and let $x<u<y$ to be
arbitrary elements of $I$. This implies $a+b-y<a+b-u<a+b-x$. We choose
$t\in[0,1]$ such that $a+b-u=t(a+b-y)+(1-t)(a+b-x)$. In other words, we select
$\mathscr{T}=\dfrac{u-x}{y-x}$; and hence (2) can be rewritten as follows

$\displaystyle f(a+b-u)\leq\dfrac{u-x}{y-x}f(a+b-y)+\dfrac{y-u}{y-x}f(a+b-x).$

Therefore,

$\displaystyle(y-u+u-x)(\mathscr{T}_{f})(u)\leq$
$\displaystyle(u-x)(\mathscr{T}_{f})(y)+(y-u)(\mathscr{T}_{f})(x)$

Rearranging the above inequality we obtain,

$\displaystyle\dfrac{(\mathscr{T}_{f})(u)-(\mathscr{T}_{f})(x)}{u-x}$
$\displaystyle\leq\frac{(\mathscr{T}_{f})(y)-(\mathscr{T}_{f})(u)}{y-u}.$

(ii)$\Rightarrow$(iii): Assume that (ii) holds and define the function
$\varphi$ on $I$ by

$\displaystyle\varphi(u):=\underset{x<u}{\mathrm{\sup}}\bigg{(}\dfrac{(\mathscr{T}_{f})(u)-(\mathscr{T}_{f})(x)}{u-x}\bigg{)}\qquad(u\in
I).$

In view of condition (ii), for all $x<u<y$ in $I$, we have

(19)
$\displaystyle\dfrac{(\mathscr{T}_{f})(u)-(\mathscr{T}_{f})(x)}{u-x}\leq\varphi(u)\leq\dfrac{(\mathscr{T}_{f})(y)-(\mathscr{T}_{f})(u)}{y-u}.$

From the left hand side inequality in (19), we get

$\displaystyle(\mathscr{T}_{f})(u)+(x-u)\varphi(u)\leq(\mathscr{T}_{f})(x)\qquad(x\in
I,\,x<u).$

Similarly, from the right hand side inequality in (19) (replacing $y$ by $x$,
it follows that

$\displaystyle(\mathscr{T}_{f})(u)+(x-u)\varphi(u)\leq(\mathscr{T}_{f})(x)\qquad(x\in
I,\,u<x).$

Now, combining the two inequalities above, the condition (iii) follows (also
in the case $x=u$).

(iii)$\Rightarrow$(iv): To deduce (iv) from (iii), let $x_{1},\dots,x_{n}\in
I,$ and $t_{1},\dots,t_{n}\geq 0$ with $t_{1}+\dots+t_{n}=1$ and
$u:=t_{1}x_{1}+\dots+t_{n}x_{n}$. Substituting $x$ by $x_{i}$ in the
inequality of condition (iii) we will obtain $n$ inequalities. Then
multiplying this inequality by $t_{i}$, and finally adding up the inequalities
side by side, we get

$\displaystyle\sum_{i=1}^{n}t_{i}\big{(}(\mathscr{T}_{f})(u)+(x_{i}-u)\varphi(u)\big{)}\leq$
$\displaystyle\sum_{i=1}^{n}t_{i}\big{(}(\mathscr{T}_{f})(x_{i}).$

Using that $\sum_{i=1}^{n}t_{i}(x_{i}-u)=0$, the above inequality simplifies
to the inequality of condition (iv).

(iv)$\Rightarrow$(v): To deduce the $\Phi$-convexity of $f$ from condition
(iv)

let $x,y\in I$ and $t\in[0,1]$. Taking $n=2$, $x_{1}:=x$, $x_{2}:=y$,
$t_{1}:=t$ and $t_{2}:=1-t$ in condition (iv), it is immediate to see that the
inequality reduces to

$\displaystyle(\mathscr{T}_{f})(tx+(1-t)y)\leq
t(\mathscr{T}_{f})(x)+(1-t)(\mathscr{T}_{f})(y).$

This yields the convexity of the function $\mathscr{T}_{f}$.

(v)$\Rightarrow$(vi): The following the same steps from $\it(i)$ to $\it(v)$;
and using (17) we can finally obtain $\mathscr{T}_{f}$, that is $f$ is convex.

This establishes equivalency among all the implications and completes the
proof. ∎

Motivated by the condition (iii) of Theorem LABEL:T1, we say that
$\varphi:I\to\mathbb{R}$ is a _$\Phi$ -slope function for $\mathscr{T}_{f}$_
if it satisfies inequality (18) for all $x,u\in I$. The next proposition
establishes the monotonic property of $\phi$.

######  Proposition 2.5.

The slope function $\varphi$ defined in (18) is nondecreasing.

###### Proof.

To prove the statement, we assume that $x,u\in I$ with $x<u$. Then
interchanging the roles of $x$ and $u$ in the (18)(third assertion of the
above theorem), we get the following

$\displaystyle(\mathscr{T}_{f})(x)+(u-x)\varphi(x)\leq(\mathscr{T}_{f})(u).$

Now adding up the above inequality with (18) side by side, we arrive at

$\displaystyle(u-x)(\varphi(x)-\varphi(u))\leq 0.$

This implies $\Phi(u)\leq\Phi(x)$ and due to the arbitrariness of $u$ and $x$;
the monotonicity is established. ∎

## References

  * [1] Mitrinovic, Dragoslav S., and Petar M. Vasic.  Analytic inequalities.  Vol. 1. Berlin: Springer-verlag, 1970.
  * [2] Sofonea, Daniel Florin, Ioan Ţincu, and Ana Maria Acu.  Convex sequences of higher order.  Filomat 32, no. 13 (2018): 4655-4663.
  * [3] Molnár, Gábor Marcell, and Zsolt Páles.  On convex and concave sequences and their applications..  Mathematical Inequalities & Applications 25, no. 1 (2022): 727-750.
  * [4] Wu, Shanhe, and Lokenath Debnath.  Inequalities for convex sequences and their applications.  Computers & Mathematics with Applications 54, no. 4 (2007): 525-534.
  * [5] Gaubert, Stéphane, Peter Butkovic, and Raymond Cuninghame-Green.  Minimal (max,+) realization of convex sequences.  SIAM Journal on Control and Optimization 36, no. 1 (1998): 137-147.
  * [6] Latreuch, Zinelaâbidine, and Benharrat Belaıdi.  New inequalities for convex sequences with applications.  Int. J. Open Problems Comput. Math 5, no. 3 (2012): 15-27.
  * [7] Pečarić, Josip E.  On some inequalities for convex sequences.  Publications de l’Institut Mathématique. Nouvelle Série 33 (1983): 173-178.
  * [8] Essén, Matts.  A theorem on convex sequences.  Analysis 2, no. 1-4 (1982): 231-252.
  * [9] Jiménez-Garrido, Javier, Javier Sanz, and Gerhard Schindl.  Log-convex sequences and nonzero proximate orders.  Journal of Mathematical Analysis and Applications 448, no. 2 (2017): 1572-1599.
  * [10] Hyers, Donald H., George Isac, and Themistocles M. Rassias.  Approximately convex functions.  In Stability of Functional Equations in Several Variables, pp. 166-179. Boston, MA: Birkhäuser Boston, 1952.
  * [11] Milovanovic, I. Z., and E. I. Milovanovic.  Some properties of Lk p-convex sequences.  Bull. Int. Math. Virtual Inst 5, no. 1 (2015): 33-36.
  * [12] Mercer, A. McD.  Polynomials and convex sequence inequalities.  J. Inequal. Pure Appl. Math 6, no. 1 (2005).
  * [13] Kuczma, M. (1885)  An Introduction to the Theory of Functional Equations and Inequalities.  Polish Scientific Editors and Silesian University, Warszawa-Krakow-Katowice
  * [14] Páles, Z. (2003).  On approximately convex functions.  Proceedings of the American Mathematical Society, 131(1), 243-252.
  * [15] G. H. Hardy, J. E. Littlewood, and G. Pólya,  Inequalities,  Cambridge University Press, Cambridge, 1934, (first edition), 1952 (second edition). MR 13,727e
  * [16] D. H. Hyers and S. M. Ulam,  Approximately convex functions,  Proc. Amer. Math. Soc. 3 (1952), 821–828. MR 14:254b
  * [17] E. F. Beckenbach,  Generalized convex functions,  Bull. Amer. Math. Soc. 43 (1937), 363-371.
  * [18] Ger, Roman, and Kazimierz Nikodem.  Strongly convex functions of higher order.  Nonlinear Analysis: Theory, Methods & Applications 74, no. 2 (2011): 661-665.
  * [19] Ger, Roman.  Convex functions of higher orders in Euclidean spaces.  In Annales Polonici Mathematici, vol. 3, no. 25, pp. 293-302. 1972.
  * [20] Mishra, S. K., and Nidhi Sharma.  On strongly generalized convex functions of higher order.  Math. Inequal. Appl 22, no. 1 (2019): 111-121.
  * [21] Krasniqi, Xhevat Z.  On$\alpha$-convex sequences of higher order.  Journal of Numerical Analysis and Approximation Theory 45, no. 2 (2016): 177-182.