# Bifurcation and chaos in one dimensional chains of small particles

Fabiano C. Simas1,2, K. Z. Nobrega3, and D. Bazeia4<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>1Programa de Pós-Graduação em Física,
Universidade Federal do Maranhão (UFMA), Campus Universitário do Bacanga,
65085-580, São Luís, Maranhão, Brazil
2Centro de Ciências Agrárias e Ambientais (CCAA), Universidade Federal do
Maranhão (UFMA), 65000-000 Chapadinha, MA, Brazil
3Departamento de Engenharia Teleinformática, Universidade Federal do Ceará
(UFC), 60455-640, Fortaleza, Ceará, Brazil
4Departamento de Física, Universidade Federal da Paraíba (UFPB), 58051-970
João Pessoa, PB, Brazil

###### Abstract

This work deals with bifurcation and the chaotic behavior in one dimensional
chains of small particles. We consider two distinct possibilities, one where
the particles are modeled by a fourth-order potential which was already
studied. We revisit this system and bring novelties concerning the presence of
the three-armed star behavior and the study of the corresponding Lyapunov
exponent. The other system is new, and there the particles are more involved
and modeled by an eighth-order potential. We investigate this new system
within the same approach, emphasizing the behavior of the orbits, the
bifurcation profile and the corresponding Lyapunov exponent.

##  I  Introduction

The main interest of the present investigation is to study one dimensional
chain of small particles, with the particles having internal structure and so
making the system very attractive from the point of view of physical
applications of practical interest. In the literature, there are several
interesting studies related to the present investigation, and here we recall
the following possibilities: if the small particles are magnetic, the related
chain can be used to model magnetic quantum cellular automata, as explored in
science  with the magnetic particles used to implement logic operations and
propagate quantum information at room temperature. The one dimensional chain
of small magnetic particles can also be used to describe the presence of
fractal structures due to the nonlinear interactions among the particles, as
studied in Ref. magnetic . Moreover, the one dimensional chain can be used to
investigate the metal-insulator transition observed in doped polyacetylene
bak1  and also, the behavior of the nonlinear interactions leading to
bifurcation and chaos, as described in Ref. bak2 ; bak3 .

Figure 1: The one dimensional chain of small particles.

There are many other distinct motivations to study chaotic systems, some
interesting possibilities have been discussed in bak3 ; book ; Lya  and in
references therein, including applications to Physics, Chemistry, Biology and
Engineering. In this work, we shall treat the system of small particles as
depicted in Fig. 1. The blue springs are used to construct the one dimensional
chain and they respond connecting the next neighbour in the usual manner. Its
contribution to the Hamiltonian adds in the form of kinetic energy and has the
standard form

$T=\frac{1}{2}\sum_{n}(z_{n}-z_{n-1})^{2}.$ (1)

In the above expression, $z_{n}$ identifies the $n$-th site of the chain and
we have omitted a real parameter, in general used to weigh the kinetic
contribution of the system. Here we use dimensionless quantities, so we will
add the parameter $a$ to the potential energy, to do the task of weighing the
potential contributions relative to the kinetic energy of the system. In the
one dimensional chain, however, the red closed curves depicted in Fig. 1
describe the small particles, and they then respond for the presence of
nonlinear interactions in the model under consideration. The contribution to
the one dimensional chain adds in the form of the potential energy. In order
to specify the system, we have to identify its contribution to the chain, and
an interesting standard nonlinear situation is in general described by a
potential having the fourth-order power in the coordinate. It can be written
as

$V=\frac{1}{4}a\sum_{n}(1-z_{n}^{2})^{2}.$ (2)

It is very important in physics and engenders the basics of the Ginzburg-
Landau theory for a second-order phase transition Landau , the Higgs mechanism
that introduces spontaneous symmetry breaking that responds for the generation
of mass for the elementary particles field , the Kerr effect in nonlinear
optics which allows for the presence of bright and dark solitons in fibers
optics ; APR  or the Landau-Ginzburg-Devonshire theory to describe negative
capacitance in ferroelectric materials ferro ; applied . This potential is
also present in Refs. magnetic ; bak1 ; bak2 ; bak3 . It is also known as the
double-well or $\phi^{4}$ potential, and has been used to describe other
issues, in particular, movability properties of localized excitations PRL96
and thermal conduction thermal  in discrete one dimensional lattices, to quote
two interesting possibilities.

In the present work we want to investigate the one dimensional chain depicted
in Fig. 1, but we shall suppose the small particles have another internal
structure, different from the above fourth-order potential. In fact, inspired
by the work on kinks in scalar field theory with polynomial potential baz1 ,
here we shall study the eighth-order potential described by

$V=\frac{8}{9}\;a\;\sum_{n}\left(\frac{1}{4}-z_{n}^{2}\right)^{2}(1-z_{n}^{2})^{2}.$
(3)

Differently from the fourth-order potential (2) which engenders two distinct
minima at each site, the potential (3) has four minima, so it is more complex
and describes a chain with small particles of different nature. Evidently, in
this new chain the particles are somehow more complex then the previous ones
and this will lead to distinct results, which we want to describe in this
work. We implement the investigation organizing the study as follows: In Sec.
II we describe the main results, firstly reviewing the system described by the
potential (2). However, we add new information, in particular, the presence of
the three-armed star and the behavior of the Lyapunov exponent. We then go on
and study the new system with the potential (3), carefully investigating
orbits, bifurcations, starfish, three-armed star, the banana behavior and the
Lyapunov exponent. We conclude the work in Sec. III, adding some comments on
the present investigation and describing new possibilities for future work.

##  II Results

The models to be considered in this work are described the by Hamiltonian

$H=\frac{1}{2}\biggl{[}\sum_{n}\left(\frac{1}{2}(z_{n}-z_{n-1})^{2}+V(z_{n})\right)\biggr{]}.$
(4)

As we have already informed, we are considering dimensionless quantities, so
$z_{n}$ stands for the dimensionless position identifying the $n$-th small
particle in the one dimensional chain, and $V(z_{n})$ represents the
corresponding potential. A sum in $n$ is considered to add all the particles.
The equations of motion that follow from the above system can be obtained in
the usual way. Here we follow the standard route considered in bak2  and we
get the set of equations

$\displaystyle x_{n+1}$ $\displaystyle=$ $\displaystyle-
y_{n}+2x_{n}+\frac{dV(x_{n})}{dx_{n}},$ $\displaystyle y_{n+1}$
$\displaystyle=$ $\displaystyle x_{n},$ (5)

where we have changed $z_{n}\to x_{n}$ and $z_{n-1}\to y_{n}$. Each model is
now identified by specific potential, and below we consider the fourth-order
and the eighth-order models separately.

###  II.1 The fourth-order model

The fourth-order potential is specified by

$V(x_{n})=\frac{1}{4}a(1-x_{n}^{2})^{2},$ (6)

where $a$ is a real and dimensionless parameter that controls the potential
energy of the system, in comparison with the kinetic energy. This potential is
non-negative, has a local maximum at zero and two minima at
${\bar{x}}_{n}^{\pm}=\pm 1$. It shows that the small particles in the chain
have internal structure and this allows for the presence of nonlinear
excitation in the chain, which we study below. The derivative of the potential
to be used in Eq. (5) has the form

$\frac{dV(x_{n})}{dx_{n}}=-a\,x_{n}+a\,x_{n}^{3}.$ (7)

Evidently, the driving parameter is $a$, and it measures the importance of the
potential behavior compared to the kinetic contribution to the chain. In bak1
, infinite series of bifurcations were discovered. Using the mapping (5), the
authors discovered an elliptical structure centered at the fixed point at the
origin for $a={8}/{9}$. The same outcome can be found in the Ref. bak2 . We
have also investigated the same system, and in Fig. 2 we display the orbits
for the fourth-order theory calculated for $a=0.8$ and $a=1.5$. This result
demonstrates that for small values of $a$, there is just one fixed point at
the origin.

Figure 2: (a) Orbits of the fourth-order model calculated for $a=0.8$ and (b)
$a=1.5$.

The stability of fixed point can be studied by the tangent space orbits. We
consider small fluctuations $x_{n}=\hat{x}_{n}+\delta x_{n}$ around
$\hat{x}_{n}$; we implement this to get

$\displaystyle\left(\begin{array}[]{c}\delta x_{n+1}\\\ \delta
x_{n}\end{array}\right)=M_{n}\left(\begin{array}[]{c}\delta x_{n}\\\ \delta
x_{n-1}\end{array}\right),$ (12)

where

$\displaystyle M_{n}=\left(\begin{array}[]{cc}3a\hat{x}_{n}-a+2&-1\\\
1&0\end{array}\right).$ (15)

The eigenvalues of $M_{n}$ are determined by the trace and determinant, in the
form

$\frac{trM_{n}}{2}\pm\sqrt{\frac{|trM_{n}|^{2}}{4}-\det M_{n}}.$ (16)

We realize that $\det M_{n}=1$, then the eigenvalues are solely determined by
the trace bou1 ; gree1 . This allows us to determine whether the eigenvalues
are real or not. The system will oscillate with non-real eigenvalues,
consequently, the orbits will be stable under the following condition
$|trM_{n}|<2$.

As a result of investigation of stability, it has been shown that the fixed
point at the origin becomes unstable for $a>4$. Therefore, in bak1  for values
below the threshold $a=4$, bifurcations are impossible to observe. However,
when one increases $a$ above $a=4$, the system bifurcates into $2$-cycle
orbits. This is shown in Fig. 3, which displays the results for $a=3.95$ and
$a=4.1$: in Fig. 3a, we observe the features of elliptic orbits around
$(0,0)$; however, in Fig. 3b one shows that the original fixed point presents
features of hyperbolic orbit. As a result, the fixed point around the origin
gives rise to two new fixed points which are now encircled by orbits. This
indicates that the original fixed point has become unstable, and a bifurcation
has appeared, as it is illustrated in Fig. 3.

Figure 3: (a) Orbits of the fourth-order model calculated for $a=3.95$ and (b)
$a=4.1$.

With the increase of the value of parameter $a$, another fascinating result
emerges. The 2-cycle orbits become slightly unstable at $a=5$. However, no
bifurcations are envisaged bak2 . The Fig. 4 show the result of computer
iterations for $a=4.995$. In the Fig. 4a, the entire pattern of behavior is
visible. In Fig. 4b, we observe a zoom in on the positive component of $x$.
The Fig. 4 depicts the starfish behavior identified before in Ref. bak2 . The
regular orbit eventually deforms as $a$ becomes larger ($a>5$), resulting in
an irregular starfish.

Figure 4: (a) The starfish of the fourth-order model calculated for $a=4.995$
and (b) the zoom from figure (a) to $x>0$ and $y<0$.

The starfish is no longer visible for $a=5.078$ because its arms disappear, as
seen in Fig. 5. The orbits that identified the starfish slowly disappear as we
increase the parameter $a$. As illustrated in Fig. 6, however, the increasing
of $a$ resulted in the identification of a novel behavior for the fourth-order
model. For $a=5.405$, we see the creation of three arms, each with its own
island. Fig. 6a depicts the whole pattern, whereas Fig. 6b displays a zoom to
the positive component of $x$. This was not noticed in bak2 , the presence of
the three-armed star depicted in Fig. 6, in contrast to the starfish in Fig.
4. It is important to observe that there is no bifurcation since the fixed
points, like in the starfish, are surrounded by orbits.

Figure 5: Orbits of the fourth-order model calculated for $a=5.078$.

Figure 6: (a) Orbits of the fourth-order model calculated for $a=5.405$ and
(b) the zoom from figure (a) to $x>0$ and $y<0$.

We notice that as the parameter $a$ is increased, the starfish disappears,
becoming the three-armed star which also changes to become the banana
structure found before in Ref. bak2 . The orbits calculated for $a=5.98$ is
shown in Fig. 7a. There is no bifurcation at this point, and the fixed point
is surrounded by curves. However, when $a>6$, the initial fixed point becomes
hyperbolic and, consequently, unstable. The banana becomes a banana split at
$a=6.02$ in Fig. 7b, and it further splits in the form depicted in Fig. 7c for
$a=6.05$.

Figure 7: The banana behavior of the fourth-order model calculated for (a)
$a=5.98$, (b) $a=6.02$ and (c) $a=6.05$.

The final feature of the fourth-order model that we consider is the connection
that exists between bifurcation and chaos. We observe an elliptic orbit around
the fixed point for smaller values of $a$. As this value increases, two new
orbits are formed and the original fixed point becomes unstable. Then, the
process of forming new orbits repeats itself as this parameter increases,
until the system becomes chaotic.

To inspect the behavior of the Lyapunov exponent we use Refs. book ; Lya . We
first illustrate the subject investigating the famous logistic map, which is
described by $x_{i+1}=rx_{i}(1-x_{i})$, where $r$ is a real parameter know as
the growth rate. In this case, we can calculate the bifurcation diagram and
Lyapunov exponent standardly. The results are displayed in Fig. 8, and
reproduces the corresponding results described in Ref. book . These are well-
known results, and we notice that for $r<r_{\infty}=3.569$ the Lyapunov
exponent is non-positive. Also, for $r$ small, in the region before reaching
the chaotic behavior, one identifies the values $r=3$, $r=3.449$ and $r=3.544$
where the Lyapunov exponent approaches zero; these values are also seen in the
bifurcation diagram, and there they identify the first, second and third
bifurcation, respectively. This behavior also appears in the one dimensional
chain with fourth-order potential. To see how it appear, let us now turn
attention to the presence of bifurcation and chaos in the fourth-order model.
We investigate both the bifurcation diagram and the Lyapunov exponent and
display the results in Fig. 9. There one sees that the Lyapunov exponent is
non-positive for $a$ smaller than $a_{\infty}=6.605$. Also, it approaches zero
at $a=4$, $a=6$ and $a=6.472$, before the system engenders chaotic behavior.
As expected, these values also identify the presence of the first, second and
third bifurcation in the bifurcation diagram in Fig. 9a.

In addition, there are minima in the figure, as seen at $a=2$ and $a=5$. The
minimum at $a=2$ corresponds to the boundary region, where from it, the
elliptical orbit rotates $90^{\circ}$. In order to view this, contrast Fig. 2a
for $a=0.8$ ($a<2$) with Fig. 3a for $a=3.95$ ($a>2$). Finally, for large
values of $a$, the Lyapunov exponent becomes positive, indicating the
appearance of chaos.

Figure 8: (a) Bifurcation diagram of the logistic map and (b) corresponding
behavior of Lyapunov exponent.

Although some of the above investigation are revisions of previous results, as
described in Ref. bak2 , for instance, the appearance of the three-armed star
and the study of the Lyapunov exponent are novel, and confirm the chaotic
behavior of the system for larger and larger values of $a$. We have found
another work, Ref. cama , where the authors implement similar investigations,
using another model, with the same double-well profile, but with the
interactions being controlled by hyperbolic functions. However, the
investigation does not study the Lyapunov exponent.

In order to better understand the importance of the non-linear interaction in
the model and also, to check the robustness of the numerical investigation
that we developed above, we have considered a similar study, with the
inclusion of another real and positive parameter, $b$, to control the weight
of the interaction; that is, we have changed the fourth-order potential in a
way such that the derivative in Eq. (7) changes to

$\frac{dV(x_{n})}{dx_{n}}=-a\,x_{n}+a\,b\,x_{n}^{3},$ (17)

where $b$ can be considered small or large, compared to $b=1$ which we have
just considered. We have developed several calculations, for $b$ smaller and
larger than unity, and we have identified the same qualitative behavior. To
illustrate this, in Fig. 10 we display both the bifurcation and the Lyapunov
exponent, for $b=0.5$, for instance. We have also checked that as we diminish
$b$, the fixed points separate from each other; this is visible when we
compare the bifurcation diagrams in Figs. 9a and 10a. Moreover, in the limit
$b\to 0$ the system trivializes, as expected. In Fig. 10, we also notice the
correct correspondence between the first, second and third bifurcations and
the three first zeroes of the Lyapunov exponent, as expected.

Figure 9: (a) Bifurcation diagram of the fourth-order model and (b)
corresponding behavior of Lyapunov exponent.

Figure 10: (a) Bifurcation diagram of the fourth-order model and (b)
corresponding behavior of Lyapunov exponent, for $b=0.5$.

###  II.2 The eighth-order model

Let us now investigate the eighth-order potential shown in Eq. (3). It is
inspired by Ref. baz1 , where it was shown that in the continuum, the
relativistic real scalar field model is capable of supporting kinklike
solutions in the distinct topological sectors present in the model. Since it
has a richer structure, when compared to the previous fourth-order model, it
is then worth examining its discrete version under similar procedure. The
potential in Eq. (3) has three local maxima, one at the origin and two other,
symmetric, at $\pm 0.79$. Also, it has four global minima, two at $\pm 0.5$
and two at $\pm 1$. The height of the maximum at the origin is $h_{0}=a/18$,
and the other two are at $\pm 0.79$; they are $h_{\pm}$, and are $3.16$ times
shorter. Moreover, the derivative of the potential to be used in Eq. (5) has
the form

$\frac{dV(x_{n})}{dx_{n}}=\frac{2}{9}\,a\,x_{n}\,(32\,x^{6}_{n}-60\,x^{4}_{n}+33\,x^{2}_{n}-5).$
(18)

This potential describes a much more involved situation, with the small
particles having a more complex internal structure. In the same way as the
fourth-order model, we can note the fixed point at the origin is stable for
small values of $a$. For example, in the Fig. 11a, we observe orbits for
$a=0.5$, which engenders the elliptical structure centered on the fixed point
$(0,0)$, but we also see the emergence of new elliptical structures related to
the other two maxima of the potential. The same appears in Fig. 11b, for
$a=1.83$, but now the orbits are different.

Figure 11: (a) Orbits of the eighth-order model calculated for $a=0.5$ and (b)
$a=1.83$.

Another intriguing phenomenon is seen with the increase in $a$. We can see in
Fig. 11b that the new orbits migrate away from the core of the fixed points.
The orbits begin to rotate and change their elliptical form. This behavior is
directly related to the first minimum point found in the Lyapunov exponent,
which will be addressed later. It is worth noting that the orbits around
$(0,0)$ and $\pm(0.79,0.79)$ behave similarly, distorting their initial
elliptical shape. Following the Refs. bak1 ; bou1 ; gree1 , we can analyze the
stability of the fixed points through the tangent space orbits, as done before
for the fourth-order potential. In comparison to the previous model, the
eighth-order potential is more involved. For this reason, let us now focus on
the behavior of the model around the maximum at $(0.79,0.79)$. It bifurcates
at $a=3.2$, so in Fig. 12 we depict the orbits for $a=3.1$ and $a=3.3$, which
illustrate the corresponding behavior. We increase the value of $a$, searching
for other possibilities. We have also found the starfish, the three armed star
and the banana behavior. They are displayed in Fig. 13 and 14, and show
behavior similar to the case of the fourth-order potential, but this happens
here in the interval $a\in[3.42,3.70]$ three times narrower than it appear in
the previous model.

We now focus on the fixed point at the origin. We implemented numerical
investigation and noted that it becomes unstable for $a>3.6$. In the Fig. 15,
we can see the orbits for $a=3.5$ and $a=3.7$, respectively, and they clearly
show the corresponding bifurcation. When comparing Fig. 15 to Fig. 11, we can
also see that the orbits have different inclination.

Figure 12: (a) Orbits of the eighth-order model calculated for $a=3.1$ and (b)
$a=3.3$.

Figure 13: (a) Orbits of the eighth-order model calculated for $a=3.422$ and
for (b) $a=3.524$, showing the starfish and the three-armed star,
respectively.

Figure 14:  The banana behavior of the eighth-order model calculated for (a)
$a=3.672$, (b) $a=3.682$ and (c) $a=3.69$.

Figure 15: (a) Orbits of the eighth-order model calculated for $a=3.5$ and (b)
$a=3.7$.

We increase the value of $a$ and in Fig. 16a we display the starfish behavior
for $a=4.56$. This is similar to the fourth-order model, so we increase even
more $a$ until we find in Fig. 16b the three-armed star, and then in Fig. 17
we depict the banana behavior for several values of $a$. These results show
that for $a$ in the interval from $a=4.5$ until $a=5.7$, the eighth-order
model behaves as the previous model, with the fourth-order potential, showing
the starfish, the three-armed star and the banana behavior as well.

Figure 16: (a) Orbits of the eighth-order model calculated for $a=4.56$ and
(b) $a=4.993$, showing the starfish and the three-armed star, respectively.

Figure 17: The banana behavior of the eighth-order model calculated for (a)
$a=5.6$, (b) $a=5.66$ and (c) $a=5.7$.

We have added more investigation, and in Fig. 18a one depicts the bifurcation
behavior of the model as the value of the parameter $a$ increases. The diagram
shows the path of new orbit formation until the dynamic of the system becomes
chaotic. As we can see, the bifurcation process continues for larger and
larger values of $a$, and leads to chaos. In order to reinforce this behavior,
in Fig. 18b, we calculate the Lyapunov exponent for the eighth-order model, in
order to identify the chaotic behavior. We did this following the behavior
associated to the maximum at the origin, in a way similar to the case
described in Fig. 9b for the fourth-order potential. For $a<a_{\infty}=6.333$,
we identify the presence of the values $a=3.598$, $a=5.637$ and $a=6.181$
where the Lyapunov exponent approaches zero, also showing the first, second
and third bifurcation, in direct connection with the corresponding bifurcation
diagram. Furthermore, for large values of $a$, the Lyapunov exponent becomes
positive, indicating that the system has reached the chaotic behavior. It is
also worth noting that the exponent has minima at $a=1.8$ and $a=4.56$, for
instance. The first minimum relates to the limit region in which elliptical
orbits begin to rotate at $90^{\circ}$, as compared to smaller values of $a$.
The second minimum refers to a region in which the 2-cycles become marginally
unstable, but no bifurcation is observed. The usual starfish behavior can be
found in this instability zone, as we can see in Fig. 16 for $a=4.56$.

In order to help us compare the results on bifurcation and the Lyapunov
exponent in the fourth- and eighth-order models, we have depicted both Figs.
9a and 18a with the same number of points, and Figs. 9b and 18b with curves
with the same width. As one can see, the fourth-order model is governed by a
double well potential, and the eighth-order model by a much more involved
potential. The two models are different from each other, and develop distinct
bifurcation behavior.

Figure 18: (a) Bifurcation diagram of the eighth-order model and (b)
corresponding behavior of Lyapunov exponent.

##  III Conclusion

In this work we have revisited the investigations described in Refs. bak1 ;
bak2 , concerning an one dimensional chain of small particles, which interact
via the fourth-order potential displayed in Eq. (2). As shown in bak2 , the
non-linearity introduced by the particles may induce the system to exhibit
bifurcation and chaos. Moreover, in the more recent investigation in magnetic
, one also sees the appearance of fractal structures in the one dimensional
chain of small magnetic particles.

The interesting behavior has attracted our attention, and here we considered
the study of the Lyapunov exponent, to see how it fits with the orbits and the
associated bifurcation pattern. The new results presented in Fig. 9 for the
fourth-order potential unveiled interesting relation between the bifurcation
diagram and the Lyapunov exponent, motivating us to go further and study other
possibilities. We have, in particular, found in between the starfish and the
banana behavior, the new three-armed star that appears in Fig. 6b, which was
not present in previous investigations. We have also considered the fourth-
order model with the inclusion of a new parameter, $b$, to control the fourth-
order interaction, and another model, with another potential. This is the
eighth-order potential displayed in Eq. (3), which has a more complex
structure and may be used to describe one dimensional chain of small particles
engendering other kind of non-linearity. The results reported in this work, in
particular the ones for bifurcation and the Lyapunov exponent displayed in
Fig. 18 show that modifications in the non-linearity related to the small
particles in the chain changes the behavior of the system. The two systems are
different, and we can see this from the two Figs. 9 and 18, so they suggest
that we further investigate other possibilities.

An interesting specific possibility of investigation could follow Ref.
magnetic , to see how the fractal structure appears in this new situation,
with the model being described by the eighth-order potential considered in the
present work. Another possibility could be the case where the distinct
interactions are controlled by distinct real parameters, which weight their
importance in the model, to see how the non-linearities contribute to induce
or generate the chaotic behavior in the chain. We can also suppose that in the
one dimensional chain of small particles, the particles interact with the
nearest-neighbour and also with the next-nearest-neighbour. The presence of
the next-to-nearest interaction will certainly change the kinematics of the
model and may perhaps bring new information on the behavior of the system. We
can also suppose that the small particles are composed, described by two (or
more) distinct degrees of freedom, to see how they can contribute to the
chaotic behavior. This is motivated by investigations developed before, on the
presence of localized structures in relativistic models described by coupled
real scalar fields; see, e.g., Ref. Baz ; 2fields1 ; Juan ; 2fields2  and
references therein for more information on this issue. In particular, in
2fields2  the study focuses on bifurcation and pattern changing. Evidently,
the presence of composed particles increases the possibility of applications
of practical importance.

There are several other problems of current interest, in particular, the ones
concerning changing the one dimensional chain of small particles and consider
a two dimensional lattice where several species evolve under cyclic and non
hierarchical rules similar to the rules of the rock, paper and scissors game.
Studies concerning the chaotic behavior of this kink of system were
implemented before in Refs. srep ; EPL , for instance, based on the Hamming
distance concept, but now we want to investigate the Lyapunov exponent. There
are several investigations dealing with this issue in predator-pray systems in
a two dimensional lattice, and we think the works AA ; BB  will certainly help
us implement other possibilities.

We can also suppose that the one dimensional chain of small particles is
composed of atomic elements. This may bring quantum effects and change
radically the models investigated in this work. However, it is of current
interest and here we recall, for instance, the case of quantum particles on a
chain of harmonically coupled quartic double-well potential prb  and also, the
systems composed of two distinct family of atomic bosonic and fermionic
particles, the one dimensional optical lattices of 7Li – 6Li and 23Na – 6Li
investigated before in Refs. A ; B . We are now considering some of the above
possibilities, and hope to report on them in the near future.

##  IV Acknowledgements

This work is supported by funds provided by Conselho Nacional de
Desenvolvimento Científico e Tecnológico, CNPq, Grant no. 303469/2019-6, and
by Paraiba State Research Foundation, FAPESQ-PB, Grant no. 0015/2019. F.C.S.
thanks FAPEMA, Fundação de Amparo à Pesquisa e ao Desenvolvimento Científico e
Tecnológico do Maranhão, Universal Grant 00920/19.

*****

The authors declare that there are no known conflicts of interest associated
with this publication and there has been no financial support for this work
that could have influenced its outcome.

*****

DB suggested the investigation. DB, KZN and FCS discussed the analytical
calculations and the organization of the work. KZN and FCS implemented the
numerical calculations. DB and FCS selected the main results. DB wrote the
paper and KZN and FCS agreed with the final version.

## References

  * (1) R. P. Cowburn and M. E. Welland, Room Temperature Magnetic Quantum Cellular Automata. Science 287, 1466 (2000).
  * (2) K. E. Kürten and F. V. Kusmartsev, Fractal structures in systems made of small magnetic particles. Phys Rev. B 72, 014433 (2005).
  * (3) P. Bak and V. L. Pokrovsky, Theory of Metal-Insulator Transition in Peierls Systems with Nearly Half-Filled Bands. Phys. Rev. Lett. 47, 958 (1981).
  * (4) P. Bak and M. Hogh Jensen, Bifurcations and chaos in the $\phi^{4}$ theory on a lattice. J. Phys. A: Math. Gen. 15 1893-1907 (1982).
  * (5) P. Bak Commensurate phases, incommensurate phases and the devil’s staircase. Rep. Prog. Phys. 45, 587 (1982).
  * (6) S.H. Strogatz, Nonlinear dynamics and chaos. (Perseus Books Publishing, 1994).
  * (7) J.M.T. Thompson and H.B. Stewart, Nonlinear Dynamics and Chaos (Wiley, 2002).
  * (8) L.D. Landau and E.M. Lifshitz, Statistical Physics (Elsevier, 2013).
  * (9) C. Itzykson and J.B. Zuber, Quantum Field Theory ( Dover, 2006).
  * (10) G. Agrawal, Nonlinear Fiber Optics (Elsevier, 2019).
  * (11) Y. Song at al., Recent progress of study on optical solitons in fiber lasers. Applied Phys. Rev. 6, 021313 (2019).
  * (12) M. Hoffmann et al., Unveiling the double-well energy landscape in a ferroelectric layer. Nature 565, 464 (2019).
  * (13) A. K. Saha and S.K. Gupta, Negative capacitance effects in ferroelectric heterostructures: A theoretical perspective. J. Applied Phys. 129, 080901 (2021).
  * (14) D. Chen, S. Aubry and G.P. Tsironis, Breather mobility in discrete phi-4 nonlinear lattices. Phys. Rev. Lett. 77, 4776 (1996).
  * (15) S. Lepri, R. Livi and A. Politi, Thermal conduction in classical low-dimensional lattices. Phys. Rep. 377, 1 (2003).
  * (16) D. Bazeia, M. A. González León, L. Losano and J. Mateos Guilarte, Deformed defects for scalar fields with polynomial interactions. Phys. Rev. D 73, 105008 (2006).
  * (17) T. Bountis and R.H.G. Helleman, On the Stability of Periodic Orbits of Two-Dimensional Mappings. J. Math. Phys., 22, 1867 (1981).
  * (18) J.M. Greene, A method for determining a stochastic transition. J. Math. Phys. 20, 1183 (1979).
  * (19) E. Epie Njumbe, A.M. Dikandé, and T.C. Kofané, Integrable discrete static Hamiltonian with a parametric double-well potential. Phys. Lett. A 372, 6890 (2008).
  * (20) D. Bazeia, M.J. dos Santos and R.F. Ribeiro, Solitons in model of coupled scalar fields. Phys. Lett. A 208, 84 (1995).
  * (21) D. Bazeia and F.A. Brito, Bags, junctions and networks of BPS and non-BPS defects. Phys. Rev. D 61, 105019 (2000).
  * (22) A. Alonzo Izquierdo, M.A. Gonzalez Leon and J.M. Guilarte, The kink variety in systems of two coupled scalar fields in two space-time dimensions. Phys. Rev. D 65, 085012 (2002).
  * (23) P.P. Avelino, D. Bazeia, R. Menezes, and J.C.R.E. Oliveira, Bifurcation and pattern changing with two real scalar fields, Phys. Rev. D 79, 085007 (2009).
  * (24) D. Bazeia et al. A novel procedure for the identification of chaos in complex biological systems. Scientific Reports 7, 44900 (2017).
  * (25) D. Bazeia, J. Menezes, B.F. de Oliveira and J. G. G. S. Ramos, Hamming distance and mobility behavior in generalized rock-paper-scissors models. EPL 119, 58003 (2017).
  * (26) Z. Jing and J. Yang, Bifurcation and chaos in discrete-time predator-pray system. Chaos, Solitons and Fractals, 27, 259 (2006).
  * (27) T. Huang and H. Zhang, Bifurcation, chaos and pattern formation in a space- and time-discrete predator-pray system. Chaos, Solitons and Fractals 91, 92 (2016).
  * (28) X. Wang, D.K. Campbell and J.E. Gubernatis, Quatum double-well chain: ground state phases and applications to hydrogen bonded materials. Phys. Rev. B 49, 15485 (1994).
  * (29) A.G. Truscott, et al. Observation of Fermi pressure in a gas of trapped atoms. Science 291, 2570 (2001).
  * (30) Z. Hadzibabic, et al. Fifty-fold improvement in the number of quantum degenerate fermionic atoms. Phys. Rev. Lett. 91, 160401 (2003).


# Convexity of quotients of theta functions

Atul Dixit, Arindam Roy and Alexandru Zaharescu  Department of Mathematics,
University of Illinois, 1409 West Green Street, Urbana, IL 61801, USA
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

For fixed $u$ and $v$ such that $0\leq u<v<1/2$, the monotonicity of the
quotients of Jacobi theta functions, namely, $\theta_{j}(u|i\pi
t)/\theta_{j}(v|i\pi t)$, $j=1,2,3,4$, on $0<t<\infty$ has been established in
the previous works of A.Yu. Solynin, K. Schiefermayr, and Solynin and the
first author. In the present paper, we show that the quotients
$\theta_{2}(u|i\pi t)/\theta_{2}(v|i\pi t)$ and $\theta_{3}(u|i\pi
t)/\theta_{3}(v|i\pi t)$ are convex on $0<t<\infty$.

2010 Mathematics Subject Classification. Primary 11F27, 33E05.
Keywords and phrases. Jacobi theta function, Weierstrass elliptic function,
Monotonicity, Heat equation.
The third author is supported by NSF grant number DMS - 0901621.

##  1\. Introduction

Let $q=e^{\pi i\tau}$ with Im $\tau>0$. The Jacobi theta functions are defined
by [8, p. 355, Section 13.19]

$\displaystyle\theta_{1}(z|\tau)$
$\displaystyle=2\sum_{n=0}^{\infty}(-1)^{n}q^{(n+\frac{1}{2})^{2}}\sin(2n+1)\pi
z,$ $\displaystyle\theta_{2}(z|\tau)$
$\displaystyle=2\sum_{n=0}^{\infty}q^{(n+\frac{1}{2})^{2}}\cos(2n+1)\pi z,$
$\displaystyle\theta_{3}(z|\tau)$
$\displaystyle=1+2\sum_{n=1}^{\infty}q^{n^{2}}\cos 2n\pi z,$
$\displaystyle\theta_{4}(z|\tau)$
$\displaystyle=1+2\sum_{n=1}^{\infty}(-1)^{n}q^{n^{2}}\cos 2n\pi z.$

We denote $\theta_{i}(z|\tau)$ by $\theta_{i}(z)$, $i=1,2,3$ and $4$, when the
dependence on $z$ is to be emphasized and that on $\tau$ is to be suppressed.
Moreover when $z=0$, we denote the above theta functions by $\theta_{i}$,
i.e., $\theta_{i}:=\theta_{i}(0|\tau),i=1,2,3$ and $4$, where it is easy to
see that $\theta_{1}=0$.

For $u,v\in\mathbb{C}$ and $\tau=i\pi t$ with Re $t>0$, define $S_{j}(u,v;t)$,
$j=1,2,3$ and $4$, to be the following quotient of theta functions:

$S_{j}:=S_{j}(u,v;t):=\frac{\theta_{j}(u/2|i\pi t)}{\theta_{j}(v/2|i\pi t)}.$
(1.1)

Monotonicity of these quotients has attracted a lot of attention in recent
years. Monotonicity of $S_{2}(u,v;t)$ on $0<t<\infty$ arose naturally in the
work of A.Yu. Solynin [14] where it is related to the steady-state
distribution of heat. In particular, Solynin used it to prove a special case
of a generalization of a conjecture due to A.A. Gonchar [4, Problem 7.45]
posed by A. Baernstein II [1]. (For complete history and progress on Gonchar’s
conjecture, the reader should consult [3, 7]). However, the proof for
$S_{2}(u,v;t)$ in [14] contained a small error. This was rectified by A.Yu.
Solynin and the first author in [7], where they also proved monotonicity of
$S_{1}(u,v;t),S_{3}(u,v;t)$ and $S_{4}(u,v;t)$. However, it turns out that K.
Schiefermayr [13, Theorem 1] obtained the same results as those in [7] on
monotonicity of $S_{3}(u,v;t)$ and $S_{4}(u,v;t)$ two years before the
appearance of [7], though the proofs in [7] and [13] use entirely different
ideas. These results on monotonicity of $S_{j}(u,v;t),j=1,2,3,4$, are stated
in [7] as follows.

For fixed $u$ and $v$ such that $0\leq u<v<1$, the functions $S_{1}(u,v;t)$
and $S_{4}(u,v;t)$ are positive and strictly increasing on $0<t<\infty$, while
the functions $S_{2}(u,v;t)$ and $S_{3}(u,v;t)$ are positive and strictly
decreasing on $0<t<\infty$.

At the end of the paper [7], based on numerical calculations, it was
conjectured that $S_{j}(u,v;t)$, $j=1,2,3,4$, are completely monotonic on
$0<t<\infty$. A function $f$ is said to be completely monotonic on
$[0,\infty)$ if $f\in C[0,\infty)$, $f\in C^{\infty}(0,\infty)$ and
$(-1)^{k}f^{(k)}(t)\geq 0$ for any $k$ non-negative and $t>0$. Several
functions related to gamma function, digamma function, polygamma function and
modified Bessel function etc. have been shown to be completely monotonic. See
[5, 9, 11]. For a survey on properties of completely monotonic functions, see
[12]. The above-mentioned conjecture can be precisely formulated (and
corrected) as follows.

######  Conjecture 1.1.

Let $S_{j}(u,v;t)$ be defined in (1.1). For fixed $u$ and $v$ such that $0\leq
u<v<1$, the functions $\frac{\partial}{\partial
t}S_{1}(u,v;t),S_{2}(u,v;t),S_{3}(u,v;t)$ and $\frac{\partial}{\partial
t}S_{4}(u,v;t)$ are completely monotonic on $0<t<\infty$.

If this conjecture is indeed true, by a theorem of S.N. Bernstein and D.
Widder [6, p. 95, Theorem 1] there exist non-decreasing bounded functions
$\gamma_{j}$ such that $S_{j}(u,v;t)=\int_{0}^{\infty}e^{-st}d\gamma_{j}(s)$
for $j=2,3,$ and $\frac{\partial}{\partial
t}S_{j}(u,v;t)=\int_{0}^{\infty}e^{-st}d\gamma_{j}(s)$ for $j=1,4$.

In the present paper, we study convexity of $S_{2}(u,v;t)$ and $S_{3}(u,v;t)$
as functions of $t$. Figures 1 and 2 seem to indicate that these quotients are
convex on $0<t<\infty$, which is consistent with the above conjecture.

|
---|---

Our main result given below shows that this is indeed true.

######  Theorem 1.2.

For fixed $u$ and $v$ such that $0\leq u<v<1$, the functions $S_{2}$ and
$S_{3}$ are strictly convex on $0<t<\infty$. In other words, $\frac{\partial
S_{2}}{\partial t}$ and $\frac{\partial S_{3}}{\partial t}$ are negative and
strictly increasing on $0<t<\infty$.

##  2\. Preliminary results

In this section, we collect main ingredients all of which are subsequently
required in the proofs of our results. We then prove certain lemmas also to be
used in the later sections. Then in Section $3$, we prove Theorem 1.2 for
$\frac{\partial S_{2}}{\partial t}$. Finally, Section $4$ is devoted to the
proof of Theorem 1.2 for $\frac{\partial S_{3}}{\partial t}$.

We first start with some important properties of Weierstrass elliptic
function. For $z\in\mathbb{C}$, let $\wp(z)$ denote the Weierstrass elliptic
function with periods $1$ and $\tau$. It is known [8, p. 376] that $\wp(z)$
maps the period parallelogram $R$ (rectangle in our case) with vertices $0$,
$\omega=1/2$, $\omega+\omega^{\prime}=1/2+\tau/2$ and $\omega^{\prime}=\tau/2$
conformally and one-to-one onto the lower half plane
$\\{\omega:\text{Im}\hskip 1.42262pt\omega<0\\}$. Moreover, $\wp(z)$ is real
and decreases from $\infty$ to $-\infty$ as $z$ describes the boundary of $R$
in the counterclockwise direction starting from $0$. It is known that $\wp(z)$
and $\wp^{\prime}(z)$ are respectively even and odd functions of $z$.

Let $g_{2}$ and $g_{3}$ denote the invariants of $\wp(z)$. The following
differential equations for $\wp$ are well-known and can be found in [8, p.
332]:

$\displaystyle{\wp^{\prime}}^{2}(z)$
$\displaystyle=4\wp^{3}(z)-g_{2}\wp(z)-g_{3},$
$\displaystyle\wp^{\prime\prime}(z)$
$\displaystyle=6\wp^{2}(z)-\frac{g_{2}}{2},$
$\displaystyle\wp^{\prime\prime\prime}(z)$
$\displaystyle=12\wp(z)\wp^{\prime}(z).$ (2.1)

The first equation in (2) can also be represented in the form [8, p. 331]

${\wp^{\prime}}^{2}(z)=4\left(\wp(z)-e_{1}\right)\left(\wp(z)-e_{2}\right)\left(\wp(z)-e_{3}\right),$
(2.2)

where $e_{1},e_{2}$ and $e_{3}$ are values of the $\wp(z)$ at
$z=1/2,(\tau+1)/2$ and $\tau/2$ respectively [8, p. 330]. As can be easily
seen from (2.2), $\wp^{\prime}(z)$ vanishes at these values of $z$. It is
known that $e_{3}<e_{2}<e_{1}$, that $e_{3}<0$ and that $e_{1}>0$. Again, from
[8, p. 332], we find that

$\displaystyle e_{1}$ $\displaystyle=-e_{2}-e_{3}$ $\displaystyle g_{2}$
$\displaystyle=-4(e_{1}e_{2}+e_{2}e_{3}+e_{3}e_{1})$ $\displaystyle g_{3}$
$\displaystyle=4e_{1}e_{2}e_{3}.$ (2.3)

Further, the quantities $e_{1},e_{2}$ and $e_{3}$ are related to theta
functions by [8, p. 361]

$\displaystyle(e_{1}-e_{3})^{1/2}$ $\displaystyle=\pi\theta_{3}^{2},$
$\displaystyle(e_{1}-e_{2})^{1/2}$ $\displaystyle=\pi\theta_{4}^{2}.$ (2.4)

An important quantity which arises while expressing $\wp(z)$ in terms of theta
functions is the following multiple of weight $2$ Eisenstein series [2, p. 87,
Equation 4.1.7] given by

$c_{0}=c_{0}(q)=-\frac{\pi^{2}}{3}\left(1-24\sum_{n=1}^{\infty}\frac{nq^{n}}{1-q^{n}}\right).$
(2.5)

See [7]. Using [7, Equation 4.4], we have

$e_{3}<c_{0}<e_{2}<e_{1}.$ (2.6)

We note that $\theta_{2}(x|i\pi t)$ and $\theta_{3}(x|i\pi t)$ are related to
$\theta_{1}(x|i\pi t)$ by following simple relations:

$\displaystyle\theta_{2}(x|i\pi t)$ $\displaystyle=\theta_{1}(1/2-x|i\pi t),$
$\displaystyle\theta_{3}(x|i\pi t)$ $\displaystyle=iq^{-1/4}e^{-i\pi
x}\theta_{1}(x|i\pi t).$ (2.7)

Observe that from [7, Equation (2.9)], we have on $0<x<1/2$,

$2\frac{\theta_{1}^{\prime}(x)}{\theta_{1}(x)}+\frac{\wp^{\prime}(x)}{\wp(x)-c_{0}}>0,$

which when combined with (2) implies that on $0<x<1/2$,

$2\frac{\theta_{2}^{\prime}(x)}{\theta_{2}(x)}+\frac{\wp^{\prime}(x-1/2)}{\wp(x-1/2)-c_{0}}<0.$
(2.8)

Finally, we use the fact that each of the theta functions $\theta_{j}(x/2|i\pi
t)$, $j=1,2,3$ and $4$, satisfies the heat equation [8, Section 13.19]

$\frac{\partial\theta}{\partial t}=\frac{\partial^{2}\theta}{\partial x^{2}}.$
(2.9)

We now prove an inequality which will be instrumental in our proof of
monotonicity of $S_{2}$ on $0<t<\infty$.

######  Lemma 2.1.

Let $0<q<1$. Let $e_{1},g_{2},g_{3}\mbox{ and }c_{0}$ be defined as above.
Then the following inequality holds:

$e_{1}^{2}(g_{2}-12c_{0}^{2})+e_{1}(6g_{3}+4g_{2}c_{0})+\left(\frac{g_{2}^{2}}{4}+g_{2}c_{0}^{2}+6g_{3}c_{0}\right)<0.$
(2.10)

  * Proof.

Let $T(q)$ denote the left-hand side of (2.10). We view $T(q)$ as a quadratic
function in $c_{0}$ rather than that in $e_{1}$, i.e.,

$T(q)=(g_{2}-12e_{1}^{2})c_{0}^{2}+(6g_{3}+4g_{2}e_{1})c_{0}+\left(\frac{g_{2}^{2}}{4}+g_{2}e_{1}^{2}+6g_{3}e_{1}\right).$
(2.11)

Employing (2) in (2.11), we see that

$\displaystyle T(q)$
$\displaystyle=-4(2e_{2}^{2}+5e_{2}e_{3}+2e_{3}^{2})c_{0}^{2}-8(2e_{2}^{3}+7e_{2}^{2}e_{3}+7e_{2}e_{3}^{3}+2e_{3}^{3})c_{0}$
$\displaystyle\quad+(8e_{2}^{4}+44e_{2}^{3}e_{3}+76e_{2}^{2}e_{3}^{2}+44e_{2}e_{3}^{3}+8e_{3}^{4})$
$\displaystyle=-4(2e_{2}+e_{3})(e_{2}+2e_{3})(c_{0}^{2}+2(e_{2}+e_{3})c_{0}-(e_{2}^{2}+3e_{2}e_{3}+e_{3}^{2})).$
(2.12)

The quadratic in $c_{0}$ in the last expression in (Proof.) has discriminant

$4(e_{2}+e_{3})^{2}+4(e_{2}^{2}+3e_{2}e_{3}+e_{3}^{2})=4(2e_{2}+e_{3})(e_{2}+2e_{3})=4(e_{1}-e_{2})(e_{1}-e_{3}),$

where we utilized (2) in the last equality. Hence,

$\displaystyle T(q)=$
$\displaystyle-4(e_{1}-e_{2})(e_{1}-e_{3})\left(c_{0}-\left(-(e_{2}+e_{3})+\pi^{2}\theta_{3}^{2}\theta_{4}^{2}\right)\right)\left(c_{0}-\left(-(e_{2}+e_{3})-\pi^{2}\theta_{3}^{2}\theta_{4}^{2}\right)\right)$
$\displaystyle\quad=-4(e_{1}-e_{2})(e_{1}-e_{3})(c_{0}-e_{1}-\pi^{2}\theta_{3}^{2}\theta_{4}^{2})(c_{0}-e_{1}+\pi^{2}\theta_{3}^{2}\theta_{4}^{2}),$
(2.13)

where we invoked (2) in the first equality and (2) in the second. Using (2.6)
and (Proof.), it suffices to show that
$e_{1}-c_{0}>\pi^{2}\theta_{3}^{2}\theta_{4}^{2}$. To that end, observe that
using [2, p. 15, Equation (1.3.32)], we have

$\theta_{3}\theta_{4}=\theta_{4}^{2}(0|2\tau).$ (2.14)

Also, from [10, Equation 4],

$\theta_{4}^{4}=1+8\sum_{n=1}^{\infty}\frac{(-1)^{n}q^{n}}{(1+q^{n})^{2}}.$
(2.15)

Using (2.14) and (2.15), we deduce that

$\pi^{2}\theta_{3}^{2}\theta_{4}^{2}=\pi^{2}+8\pi^{2}\sum_{n=1}^{\infty}\frac{(-1)^{n}q^{2n}}{(1+q^{2n})^{2}}.$
(2.16)

But from [7, Equation 4.1],

$e_{1}-c_{0}=\pi^{2}+8\pi^{2}\sum_{n=1}^{\infty}\frac{q^{2n}}{(1+q^{2n})^{2}}.$
(2.17)

Thus (2.16) and (2.17) along with the fact that $0<q<1$ imply the inequality
$e_{1}-c_{0}>\pi\theta_{3}^{2}\theta_{4}^{2}$. This proves (2.10). ∎

######  Lemma 2.2.

Let $0<q<1$. Let $e_{2},g_{2},g_{3}\mbox{ and }c_{0}$ be defined as above.
Then the following inequality holds:

$e_{2}^{2}(g_{2}-12c_{0}^{2})+e_{2}(6g_{3}+4g_{2}c_{0})+\left(\frac{g_{2}^{2}}{4}+g_{2}c_{0}^{2}+6g_{3}c_{0}\right)>0.$
(2.18)

  * Proof.

Let $U(q)$ denote the left-hand side of (2.18). From (2) and (2.6),

$\displaystyle U(q)$
$\displaystyle=(g_{2}-12e_{2}^{2})c_{0}^{2}+(6g_{3}+4g_{2}e_{2})c_{0}+\left(\frac{g_{2}^{2}}{4}+g_{2}e_{2}^{2}+6g_{3}e_{2}\right)$
$\displaystyle=-4(e_{2}-e_{3})(2e_{2}+e_{3})(c_{0}^{2}-2e_{2}c_{0}-(e_{2}^{2}-e_{2}e_{3}-e_{3}^{2}))$
$\displaystyle=4(e_{1}-e_{2})(e_{2}-e_{3})((c_{0}-e_{2})^{2}+(e_{1}-e_{2})(e_{2}-e_{3}))$
$\displaystyle>0.$

∎

##  3\. Proof of monotonicity of $\displaystyle\frac{\partial S_{2}}{\partial
t}$

From [7, Theorem 1], since $S_{2}(u,v;t)$ is decreasing on $0<t<\infty$, we
see at once that $\frac{\partial S_{2}}{\partial t}<0$. Let $L_{2}:=\log
S_{2}(u,v;t)$. Observe that

$\frac{\partial S_{2}}{\partial t}=S_{2}\frac{\partial L_{2}}{\partial t}.$
(3.1)

In order to show that $\frac{\partial S_{2}}{\partial t}$ is increasing on
$0<t<\infty$, it suffices to show that $\frac{\partial^{2}S_{2}}{\partial
t^{2}}>0$. Now from (3.1),

$\frac{\partial^{2}S_{2}}{\partial t^{2}}=\frac{\partial}{\partial
t}\left(S_{2}\frac{\partial L}{\partial
t}\right)=S_{2}\left(\frac{\partial^{2}L_{2}}{\partial
t^{2}}+\left(\frac{\partial L_{2}}{\partial t}\right)^{2}\right).$

We claim that $\frac{\partial^{2}L_{2}}{\partial t^{2}}>0$ whence we will be
done. Using (2.9) twice, we see that

$\frac{\partial^{2}}{\partial t^{2}}\theta_{2}(x/2|i\pi
t)=\frac{\partial}{\partial t}\left(\frac{\partial^{2}}{\partial
x^{2}}\theta_{2}(x/2|i\pi t)\right)=\frac{\partial^{2}}{\partial
x^{2}}\left(\frac{\partial}{\partial t}\theta_{2}(x/2|i\pi
t)\right)=\frac{\partial^{4}}{\partial x^{4}}\theta_{2}(x/2|i\pi t).$

Hence,

$\displaystyle\frac{\partial^{2}L_{2}}{\partial t^{2}}$
$\displaystyle=\frac{\partial}{\partial t}\left(\frac{\frac{\partial}{\partial
t}\theta_{2}(u/2|i\pi t)}{{\theta_{2}(u/2|i\pi
t)}}-\frac{\frac{\partial}{\partial t}\theta_{2}(v/2|i\pi
t)}{{\theta_{2}(v/2|i\pi t)}}\right)$
$\displaystyle=\frac{\theta_{2}^{(4)}(u/2|i\pi t)}{\theta_{2}(u/2|i\pi
t)}-\frac{\theta_{2}^{(4)}(v/2|i\pi t)}{\theta_{2}(v/2|i\pi t)}$
$\displaystyle\quad-\bigg{(}\bigg{(}\frac{\theta_{2}^{\prime\prime}(u/2|i\pi
t)}{\theta_{2}(u/2|i\pi
t)}\bigg{)}^{2}-\bigg{(}\frac{\theta_{2}^{\prime\prime}(v/2|i\pi
t)}{\theta_{2}(v/2|i\pi t)}\bigg{)}^{2}\bigg{)}.$

Thus it suffices to show that the function $\theta_{2}^{(4)}(x|i\pi
t)/\theta_{2}(x|i\pi t)-\left(\theta_{2}^{\prime\prime}(x|i\pi
t)/\theta_{2}(x|i\pi t)\right)^{2}$ decreases on $0<x<1/2$. From now on, we
fix $t$ where $0<t<\infty$ and henceforth suppress the dependence of
$\theta_{2}(x/2|i\pi t)$ on $t$. From (2) and the relation [7, Equation (2.6)]

$\left(\frac{\theta_{1}^{\prime}(x)}{\theta_{1}(x)}\right)^{\prime}=-\left(\wp(x)-c_{0}\right),$
(3.2)

we find that

$\left(\frac{\theta_{2}^{\prime}(x)}{\theta_{2}(x)}\right)^{\prime}=-\left(\wp\left(x-1/2\right)-c_{0}\right),$
(3.3)

since $\wp(x)$ is an even function of $x$. Then by a repeated application of
quotient rule for derivatives and (3.3), it is easy to see that the following
are true:

$\displaystyle\frac{\theta_{2}^{\prime\prime}(x)}{\theta_{2}(x)}$
$\displaystyle=\left(\frac{\theta_{2}^{\prime}(x)}{\theta_{2}(x)}\right)^{2}-\left(\wp\left(x-1/2\right)-c_{0}\right),$
$\displaystyle\frac{\theta_{2}^{\prime\prime\prime}(x)}{\theta_{2}(x)}$
$\displaystyle=\left(\frac{\theta_{2}^{\prime}(x)}{\theta_{2}(x)}\right)^{3}-3\frac{\theta_{2}^{\prime}(x)}{\theta_{2}(x)}\left(\wp\left(x-1/2\right)-c_{0}\right)-\wp^{\prime}\left(x-1/2\right),$
$\displaystyle\frac{\theta_{2}^{(4)}(x)}{\theta_{2}(x)}$
$\displaystyle=\left(\frac{\theta_{2}^{\prime}(x)}{\theta_{2}(x)}\right)^{4}-6\left(\frac{\theta_{2}^{\prime}(x)}{\theta_{2}(x)}\right)^{2}\left(\wp\left(x-1/2\right)-c_{0}\right)-4\frac{\theta_{2}^{\prime}(x)}{\theta_{2}(x)}\wp^{\prime}\left(x-1/2\right)$
$\displaystyle\quad+3\left(\wp\left(x-1/2\right)-c_{0}\right)^{2}-\wp^{\prime\prime}\left(x-1/2\right),$

from which it easily follows that

$\displaystyle\frac{\theta_{2}^{(4)}(x)}{\theta_{2}(x)}-\left(\frac{\theta_{2}^{\prime\prime}(x)}{\theta_{2}(x)}\right)^{2}$
$\displaystyle=-4\left(\frac{\theta_{2}^{\prime}(x)}{\theta_{2}(x)}\right)^{2}\left(\wp\left(x-1/2\right)-c_{0}\right)+2\left(\wp\left(x-1/2\right)-c_{0}\right)^{2}$
$\displaystyle\quad-4\frac{\theta_{2}^{\prime}(x)}{\theta_{2}(x)}\wp^{\prime}\left(x-1/2\right)-\wp^{\prime\prime}\left(x-1/2\right).$

Again using (3.3), we find that

$\displaystyle\frac{d}{dx}\left(\frac{\theta_{2}^{(4)}(x)}{\theta_{2}(x)}-\left(\frac{\theta_{2}^{\prime\prime}(x)}{\theta_{2}(x)}\right)^{2}\right)$
$\displaystyle=8\frac{\theta_{2}^{\prime}(x)}{\theta_{2}(x)}\left(\wp\left(x-1/2\right)-c_{0}\right)^{2}-4\left(\frac{\theta_{2}^{\prime}(x)}{\theta_{2}(x)}\right)^{2}\wp^{\prime}\left(x-1/2\right)$
$\displaystyle\quad+8\left(\wp\left(x-1/2\right)-c_{0}\right)\wp^{\prime}\left(x-1/2\right)-4\frac{\theta_{2}^{\prime}(x)}{\theta_{2}(x)}\wp^{\prime\prime}\left(x-1/2\right)$
$\displaystyle\quad-\wp^{\prime\prime\prime}\left(x-1/2\right).$

From the monotonicity of $\wp$ along the boundary of the rectangular lattice
as mentioned in Section 2, in the case at hand, we have in particular that
$\wp(x)$ is strictly decreasing on $0<x<1/2$. Hence $\wp(1/2-x)$ is strictly
increasing on $0<x<1/2$. Since $\wp(1/2-x)=\wp(x-1/2)$, this implies that
$\wp^{\prime}(x-1/2)>0$ on $0<x<1/2$. Define the function $F_{2}(x)$ as

$\displaystyle F_{2}(x)$
$\displaystyle:=\frac{1}{\wp^{\prime}(x-1/2)}\frac{d}{dx}\left(\frac{\theta_{2}^{(4)}(x)}{\theta_{2}(x)}-\left(\frac{\theta_{2}^{\prime\prime}(x)}{\theta_{2}(x)}\right)^{2}\right)$
$\displaystyle=8\frac{\theta_{2}^{\prime}(x)}{\theta_{2}(x)}\frac{\left(\wp\left(x-1/2\right)-c_{0}\right)^{2}}{\wp^{\prime}\left(x-1/2\right)}-4\left(\frac{\theta_{2}^{\prime}(x)}{\theta_{2}(x)}\right)^{2}+8\left(\wp\left(x-1/2\right)-c_{0}\right)$
$\displaystyle\quad-4\frac{\theta_{2}^{\prime}(x)}{\theta_{2}(x)}\frac{\wp^{\prime\prime}\left(x-1/2\right)}{\wp^{\prime}\left(x-1/2\right)}-\frac{\wp^{\prime\prime\prime}\left(x-1/2\right)}{\wp^{\prime}\left(x-1/2\right)}.$
(3.4)

It suffices to prove that $F_{2}(x)<0$. We prove this by showing that
$F_{2}(1/2)=0$ and $F_{2}^{\prime}(x)>0$, since then, the mean value theorem
implies that for any $x\in(0,1/2)$,
$F_{2}(x)-F_{2}(1/2)=F_{2}^{\prime}(c)(x-1/2)$ for some $c\in(x,1/2)$. We
begin by showing $F_{2}(1/2)=0$. We require the following series expansions in
order to establish this. First, from [8, p. 358, Section 13.19],

$\displaystyle\frac{\theta_{2}^{\prime}(z)}{\theta_{2}(z)}$
$\displaystyle=-\pi\tan\pi
z+4\pi\sum_{n=1}^{\infty}(-1)^{n}\frac{q^{2n}}{1-q^{2n}}\sin 2n\pi z$
$\displaystyle=\left(\frac{1}{z-1/2}-\frac{\pi^{2}}{3}(z-1/2)-\cdots\right)+4\pi\sum_{n=1}^{\infty}(-1)^{n}\frac{q^{2n}}{1-q^{2n}}\sin
2n\pi z.$ (3.5)

Further, the Laurent series expansions of $\wp(z-1/2)$ and
$\wp^{\prime}(z-1/2)$ around $z=1/2$ are as follows [8, p. 330, Section
13.12].

$\displaystyle\wp(z-1/2)$
$\displaystyle=\frac{1}{(z-1/2)^{2}}+\frac{g_{2}(z-1/2)^{2}}{2^{2}.5}+\frac{g_{3}(z-1/2)^{4}}{2^{2}.7}+\frac{g_{2}^{2}(z-1/2)^{6}}{2^{4}.3.5^{2}}+...,$
$\displaystyle\wp^{\prime}(z-1/2)$
$\displaystyle=\frac{-2}{(z-1/2)^{3}}+\frac{g_{2}(z-1/2)}{10}+\frac{g_{3}(z-1/2)^{3}}{7}+\frac{g_{2}^{2}(z-1/2)^{5}}{2^{3}.5^{2}}+....$
(3.6)

Using (3), (3), the third differential equation in (2) and simplifying, we
find that $F_{2}(1/2)=0$. Differentiating both sides of (3) with respect to
$x$, using (2), (3.3) and simplifying, we get

$\displaystyle\frac{F_{2}^{{}^{\prime}}(x)}{4}$
$\displaystyle=\frac{\theta_{2}^{\prime}(x)}{\theta_{2}(x)}\cdot\frac{\wp^{2}(x-1/2)\left(g_{2}-12c_{0}^{2}\right)+\wp\left(x-1/2\right)\left(6g_{3}+4g_{2}c_{0}\right)+\left(6g_{3}c_{0}+g_{2}c_{0}^{2}+\frac{g_{2}^{2}}{4}\right)}{{\wp^{\prime}}^{2}(x-1/2)}$
$\displaystyle\quad+\frac{\wp\left(x-1/2\right)\left(g_{2}/2-6c_{0}^{2}\right)+g_{3}+2c_{0}^{3}+g_{2}c_{0}/2}{\wp^{\prime}(x-1/2)}.$
(3.7)

Now we show that $F_{2}^{\prime}(x)>0$. Let

$\displaystyle A_{1}(x)$
$\displaystyle:=\wp(x-1/2)\left(g_{2}/2-6c_{0}^{2}\right)+g_{3}+2c_{0}^{3}+g_{2}c_{0}/2,$
$\displaystyle A_{2}(x)$
$\displaystyle:=\wp^{2}(x-1/2)\left(g_{2}-12c_{0}^{2}\right)+\wp\left(x-1/2\right)\left(6g_{3}+4g_{2}c_{0}\right)+\left(6g_{3}c_{0}+g_{2}c_{0}^{2}+g_{2}^{2}/4\right).$
(3.8)

By Remark 1 in [7], we have

$e_{1}<\frac{-(2g_{3}+4c_{0}^{3}+g_{2}c_{0})}{g_{2}-12c_{0}^{2}}.$ (3.9)

This along with the fact that $\wp(x-1/2)$ is strictly increasing on $0<x<1/2$
from $e_{1}$ to $\infty$ implies that $A_{1}$ has a unique zero, say $a_{1}$
in $(0,1/2)$. Now Lemma 2 from [7] implies that $g_{2}-12c_{0}^{2}>0$. This
along with the fact that $\wp\left(x-1/2\right)\rightarrow\infty$ as
$x\rightarrow{\frac{1}{2}}^{-}$ implies that $A_{2}(x)\to\infty$ as
$x\rightarrow{\frac{1}{2}}^{-}$. Using the fact that
$\wp(1/2)=\wp(-1/2)=e_{1}$ and Lemma 2.1, we have $A_{2}(0)<0$. Since $A_{2}$
is quadratic in $\wp(x-1/2)$ and $\wp(x-1/2)$ is strictly increasing on
$0<x<1/2$, there exists a unique value $a_{2}$ of $x$ in $(0,1/2)$ such that
$A_{2}(a_{2})=0$. Let $P:=\wp(a_{2}-1/2)$. Note that $a_{2}$ is not a double
root of $A_{2}$. Next, $P$ has two possibilities, say,

$P=P_{1}:=\frac{-6g_{3}-4g_{2}c_{0}-\sqrt{\Delta}}{2(g_{2}-12c_{0}^{2})}\mbox{
or }P=P_{2}:=\frac{-6g_{3}-4g_{2}c_{0}+\sqrt{\Delta}}{2(g_{2}-12c_{0}^{2})},$

where

$\Delta:=(6g_{3}+4g_{2}c_{0})^{2}-4(g_{2}-12c_{0}^{2})(6g_{3}c_{0}+g_{2}c_{0}^{2}+g_{2}^{2}/4)>0,$
(3.10)

the last inequality coming from the above discussion. We now claim that
$P=P_{2}$. Now

$P_{2}>\frac{-6g_{3}-4g_{2}c_{0}}{2(g_{2}-12c_{0}^{2})}$ (3.11)

and

$\frac{-6g_{3}-4g_{2}c_{0}}{2(g_{2}-12c_{0}^{2})}+\frac{2g_{3}+4c_{0}^{3}+g_{2}c_{0}}{g_{2}-12c_{0}^{2}}=\frac{-g_{3}-g_{2}c_{0}/2-2c_{0}^{3}}{(g_{2}-12c_{0}^{2})}+\frac{6c_{0}^{3}-g_{2}c_{0}/2}{g_{2}-12c_{0}^{2}}>\frac{e_{1}-c_{0}}{2}>0,$
(3.12)

where we utilized (3.9) in the penultimate step and (2.6) in the ultimate
step. Therefore, by (3.9), (3.11) and (3.12),

$e_{1}<\frac{-(2g_{3}+4c_{0}^{3}+g_{2}c_{0})}{g_{2}-12c_{0}^{2}}<P_{2}.$
(3.13)

This shows that $\wp(x-1/2)$ attains the value $P_{2}$ for a unique $x$ in the
interval $(0,1/2)$. This combined with the facts that $P_{1}<P_{2}$ and
$A_{2}$ has a unique root in $0<x<1/2$ implies that $P=P_{2}$.

Remark 1. The above discussion implies that $P_{1}<e_{1}<P_{2}$. As the real
period of $\wp$ is $1$, this tells us that there is no real number $x$ such
that $\wp(x-1/2)=P_{1}$.

Using $P=P_{2}$ and (3.13), it is clear that $0<a_{1}<a_{2}<1/2$. Figure 3
shows the graphs of $10A_{1}(x)$111The graph of $A_{1}(x)$ is scaled by the
factor of $10$ for better view without changing the fact $0<a_{1}<a_{2}<1/2$.
and $A_{2}(x)$.

Define

$\displaystyle G_{2}(x)$
$\displaystyle:=\frac{F_{2}^{\prime}(x){\wp^{\prime}}^{2}(x-1/2)}{4A_{2}(x)}$
$\displaystyle=\frac{\theta_{2}^{\prime}(x)}{\theta_{2}(x)}+\frac{\wp^{\prime}(x-1/2)\left(\wp\left(x-1/2\right)+\frac{2g_{3}+4c_{0}^{3}+g_{2}c_{0}}{g_{2}-12c_{0}^{2}}\right)}{2\left(\wp^{2}(x-1/2)+\wp(x-1/2)\frac{6g_{3}+4g_{2}c_{0}}{g_{2}-12c_{0}^{2}}+\frac{6g_{3}c_{0}+g_{2}c_{0}^{2}+g_{2}^{2}/4}{g_{2}-12c_{0}^{2}}\right)}.$
(3.14)

Next, we differentiate extreme sides of (3) with respect to $x$ and use (3.3)
so that $\theta_{2}^{\prime}(x)/\theta_{2}(x)$ is eliminated from the right-
hand side of (3) and we have everything in terms of $\wp$ and $\wp^{\prime}$.
This along with the second differential equation in (2) gives

$\displaystyle G_{2}^{\prime}(x)$
$\displaystyle=-(\wp(x-1/2)-c_{0})+\frac{(6\wp^{2}(x-1/2)-\frac{g_{2}}{2})\left(\wp(x-1/2)+\frac{2g_{3}+4c_{0}^{3}+g_{2}c_{0}}{g_{2}-12c_{0}^{2}}\right)}{2\left(\wp^{2}(x-1/2)+\wp(x-1/2)\frac{6g_{3}+4g_{2}c_{0}}{g_{2}-12c_{0}^{2}}+\frac{6g_{3}c_{0}+g_{2}c_{0}^{2}+g_{2}^{2}/4}{g_{2}-12c_{0}^{2}}\right)}$
$\displaystyle\quad+\frac{{\wp^{\prime}}^{2}(x-1/2)}{2\left(\wp^{2}(x-1/2)+\wp(x-1/2)\frac{6g_{3}+4g_{2}c_{0}}{g_{2}-12c_{0}^{2}}+\frac{6g_{3}c_{0}+g_{2}c_{0}^{2}+g_{2}^{2}/4}{g_{2}-12c_{0}^{2}}\right)}$
$\displaystyle\quad-\frac{{\wp^{\prime}}^{2}(x-1/2)\left(\wp(x-1/2)+\frac{2g_{3}+4c_{0}^{3}+g_{2}c_{0}}{g_{2}-12c_{0}^{2}}\right)\left(2\wp(x-1/2)+\frac{6g_{3}+4g_{2}c_{0}}{g_{2}-12c_{0}^{2}}\right)}{2\left(\wp^{2}(x-1/2)+\wp(x-1/2)\frac{6g_{3}+4g_{2}c_{0}}{g_{2}-12c_{0}^{2}}+\frac{6g_{3}c_{0}+g_{2}c_{0}^{2}+g_{2}^{2}/4}{g_{2}-12c_{0}^{2}}\right)^{2}}.$
(3.15)

Simplifying the first three terms of (3), we obtain

$\displaystyle G_{2}^{\prime}(x)$
$\displaystyle=\frac{{\wp^{\prime}}^{2}(x-1/2)}{\left(\wp^{2}(x-1/2)+\wp(x-1/2)\frac{6g_{3}+4g_{2}c_{0}}{g_{2}-12c_{0}^{2}}+\frac{6g_{3}c_{0}+g_{2}c_{0}^{2}+g_{2}^{2}/4}{g_{2}-12c_{0}^{2}}\right)}$
$\displaystyle\quad-\frac{{\wp^{\prime}}^{2}(x-1/2)\left(\wp(x-1/2)+\frac{2g_{3}+4c_{0}^{3}+g_{2}c_{0}}{g_{2}-12c_{0}^{2}}\right)\left(2\wp(x-1/2)+\frac{6g_{3}+4g_{2}c_{0}}{g_{2}-12c_{0}^{2}}\right)}{2\left(\wp^{2}(x-1/2)+\wp(x-1/2)\frac{6g_{3}+4g_{2}c_{0}}{g_{2}-12c_{0}^{2}}+\frac{6g_{3}c_{0}+g_{2}c_{0}^{2}+g_{2}^{2}/4}{g_{2}-12c_{0}^{2}}\right)^{2}}.$
(3.16)

Consider three cases: $0<x<a_{1}$, $a_{1}\leq x\leq a_{2}$ and $a_{2}<x<1/2$.

Case 1: $0<x<a_{1}$.

Then, $A_{1}(x)<0$ and $A_{2}(x)<0$. We show that $G_{2}(x)<0$. Note that from
(2.2), (3), (3.9) and Lemma 2.1, it readily follows that $G_{2}(0)=0$. Since
$A_{1}(x)<0$, $A_{2}(x)<0$ and $g_{2}-12c_{0}^{2}>0$, we have

$\displaystyle\wp\left(x-1/2\right)+\frac{2g_{3}+4c_{0}^{3}+g_{2}c_{0}}{g_{2}-12c_{0}^{2}}<0,$
(3.17)
$\displaystyle\wp^{2}(x-1/2)+\wp(x-1/2)\frac{6g_{3}+4g_{2}c_{0}}{g_{2}-12c_{0}^{2}}+\frac{6g_{3}c_{0}+g_{2}c_{0}^{2}+g_{2}^{2}/4}{g_{2}-12c_{0}^{2}}<0.$
(3.18)

From (3.17) and (3.12), we see that

$\displaystyle 2\wp(x-1/2)+\frac{6g_{3}+4g_{2}c_{0}}{g_{2}-12c_{0}^{2}}<0.$
(3.19)

Therefore, (3.17), (3.18) and (3.19) imply that $G_{2}^{\prime}(x)<0$. By the
mean value theorem, for any $x\in(0,a_{1})$, $G_{2}(x)=xG_{2}^{\prime}(d)$ for
some $d\in(0,x)$. Hence $G_{2}(x)<0$. Thus $F_{2}^{\prime}(x)>0$ in
$0<x<a_{1}$.

Case 2: $a_{1}\leq x\leq a_{2}$.

Note that $A_{1}(a_{1})=0$, $A_{2}(a_{1})<0$, $A_{1}(a_{2})>0$ and
$A_{2}(a_{2})=0$. Also, $A_{1}(x)>0$ and $A_{2}(x)<0$ when $a_{1}<x<a_{2}$.

Since $\wp(x-1/2)$ is strictly increasing on $0<x<1/2$, we have
$\wp^{\prime}(x-1/2)>0$ and $\wp(x-1/2)-c_{0}>e_{1}-c_{0}>0$, where we invoked
(2.6) in the last step. This along with (2.8) shows that
$\theta_{2}^{\prime}(x)/\theta_{2}(x)<0$ on $0<x<1/2$. Using all of the above
facts and (3), we observe that $F_{2}^{\prime}(x)>0$ on $a_{1}\leq x\leq
a_{2}$.

Case 3: $a_{2}<x<1/2$. Since $A_{1}(x)>0$, $A_{2}(x)>0$ and
$g_{2}-12c_{0}^{2}>0$, we have

$\displaystyle\wp\left(x-1/2\right)+\frac{2g_{3}+4c_{0}^{3}+g_{2}c_{0}}{g_{2}-12c_{0}^{2}}>0,$
(3.20)
$\displaystyle\wp^{2}(x-1/2)+\wp(x-1/2)\frac{6g_{3}+4g_{2}c_{0}}{g_{2}-12c_{0}^{2}}+\frac{6g_{3}c_{0}+g_{2}c_{0}^{2}+g_{2}^{2}/4}{g_{2}-12c_{0}^{2}}>0.$
(3.21)

From (3), as $x\to{\frac{1}{2}}^{-}$,

$\displaystyle
G_{2}(x)=\frac{\theta_{2}^{\prime}(x)}{\theta_{2}(x)}+\frac{\wp^{\prime}(x-1/2)}{2\wp(x-1/2)}\left(1+O\left(\frac{1}{\wp(x-1/2)}\right)\right).$

Using (3) and (3), it is easy to check that $G_{2}(1/2)=0$. Next we show that
$G_{2}^{\prime}(x)<0$. From (3),

$\displaystyle
G_{2}^{\prime}(x)=\frac{{\wp^{\prime}}^{2}(x-1/2)(1-Q(x))}{\left(\wp^{2}(x-1/2)+\wp(x-1/2)\frac{6g_{3}+4g_{2}c_{0}}{g_{2}-12c_{0}^{2}}+\frac{6g_{3}c_{0}+g_{2}c_{0}^{2}+g_{2}^{2}/4}{g_{2}-12c_{0}^{2}}\right)},$

where

$Q(x):=\frac{\left(\wp(x-1/2)+\frac{2g_{3}+4c_{0}^{3}+g_{2}c_{0}}{g_{2}-12c_{0}^{2}}\right)\left(2\wp(x-1/2)+\frac{6g_{3}+4g_{2}c_{0}}{g_{2}-12c_{0}^{2}}\right)}{2\left(\wp^{2}(x-1/2)+\wp(x-1/2)\frac{6g_{3}+4g_{2}c_{0}}{g_{2}-12c_{0}^{2}}+\frac{6g_{3}c_{0}+g_{2}c_{0}^{2}+g_{2}^{2}/4}{g_{2}-12c_{0}^{2}}\right)}.$
(3.22)

We claim that $Q(x)>1$. Note that the denominator of $Q(x)$ can be simplified
as follows:

$\displaystyle\quad 2\bigg{(}\wp^{2}(x$
$\displaystyle-1/2)+\wp(x-1/2)\frac{6g_{3}+4g_{2}c_{0}}{g_{2}-12c_{0}^{2}}+\frac{6g_{3}c_{0}+g_{2}c_{0}^{2}+g_{2}^{2}/4}{g_{2}-12c_{0}^{2}}\bigg{)}$
$\displaystyle=\left(2\wp(x-1/2)+\frac{6g_{3}+4g_{2}c_{0}}{g_{2}-12c_{0}^{2}}\right)\left(\wp(x-1/2)+\frac{6g_{3}+4g_{2}c_{0}}{2(g_{2}-12c_{0}^{2})}\right)$
$\displaystyle\quad+\left(2\frac{6g_{3}c_{0}+g_{2}c_{0}^{2}+g_{2}^{2}/4}{g_{2}-12c_{0}^{2}}-\frac{(6g_{3}+4g_{2}c_{0})^{2}}{2(g_{2}-12c_{0}^{2})^{2}}\right).$
(3.23)

Now

$\displaystyle 2\wp(x-1/2)+\frac{6g_{3}+4g_{2}c_{0}}{g_{2}-12c_{0}^{2}}$
$\displaystyle>2\wp(a_{2}-1/2)+\frac{6g_{3}+4g_{2}c_{0}}{g_{2}-12c_{0}^{2}}$
$\displaystyle=2P+\frac{6g_{3}+4g_{2}c_{0}}{g_{2}-12c_{0}^{2}}$
$\displaystyle=\frac{\sqrt{\Delta}}{(g_{2}-12c_{0}^{2})}$ $\displaystyle>0.$
(3.24)

From (3.12), we have

$\wp(x-1/2)+\frac{2g_{3}+4c_{0}^{3}+g_{2}c_{0}}{g_{2}-12c_{0}^{2}}>\wp(x-1/2)+\frac{6g_{3}+4g_{2}c_{0}}{2(g_{2}-12c_{0}^{2})}.$
(3.25)

By (3.10), the last term on the right-hand side of (3) is negative. Hence,
(3), (3), (3.25) and (3.21) imply that $Q(x)>1$. Therefore
$G_{2}^{\prime}(x)<0$. By the mean value theorem, for any $x\in(a_{2},1/2)$,
$G_{2}(x)-G_{2}(1/2)=G_{2}^{\prime}(b)(x-1/2)$ for some $b\in(x,1/2)$. Hence
$G_{2}(x)>0$. Since $A_{2}(x)>0$, this implies that
$F_{2}^{{}^{\prime}}(x)>0$.

From the above three cases, we conclude that $F_{2}^{{}^{\prime}}(x)>0$ in
$0<x<1/2$. Since $F_{2}(1/2)=0$, by another application of the mean value
theorem, we conclude that $F_{2}(x)<0$ in $0<x<1/2$. This completes the proof.

Figure 4 shows the graph of $G_{2}(x)$ on $0<x<1/2$.

##  4\. Proof of monotonicity of $\displaystyle\frac{\partial S_{3}}{\partial
t}$

The method for proving monotonicity of $\frac{\partial S_{3}}{\partial t}$ is
similar to that of $\frac{\partial S_{2}}{\partial t}$ and so we will be
brief. From [7, Theorem 1], since $S_{3}(u,v;t)$ is decreasing on
$0<t<\infty$, we see at once that $\frac{\partial S_{3}}{\partial t}<0$. Let
$L_{3}:=\log S_{3}(u,v;t)$. Observe that

$\frac{\partial S_{3}}{\partial t}=S_{3}\frac{\partial L_{3}}{\partial t}.$

It suffices to show that $\frac{\partial^{2}S_{3}}{\partial t^{2}}>0$. Now,

$\frac{\partial^{2}S_{3}}{\partial t^{2}}=\frac{\partial}{\partial
t}\left(S_{3}\frac{\partial L}{\partial
t}\right)=S_{3}\left(\frac{\partial^{2}L_{3}}{\partial
t^{2}}+\left(\frac{\partial L_{3}}{\partial t}\right)^{2}\right).$

We show that $\frac{\partial^{2}L_{3}}{\partial t^{2}}>0$. Observe that using
(2.9) twice, we have $\frac{\partial^{2}}{\partial t^{2}}\theta_{3}(x/2|i\pi
t)=\frac{\partial^{4}}{\partial x^{4}}\theta_{3}(x/2|i\pi t)$. It suffices to
show that the function $\theta_{3}^{(4)}(x|i\pi t)/\theta_{3}(x|i\pi
t)-\left(\theta_{3}^{\prime\prime}(x|i\pi t)/\theta_{3}(x|i\pi t)\right)^{2}$
decreases on $0<x<1/2$. Fix $t$ where $0<t<\infty$. Using (2) and (3.2), we
find that

$\left(\frac{\theta_{3}^{\prime}(x)}{\theta_{3}(x)}\right)^{\prime}=-\left(\wp\left(x+\frac{\tau-1}{2}\right)-c_{0}\right).$
(4.1)

Observe that

$\displaystyle\frac{\theta_{3}^{(4)}(x)}{\theta_{3}(x)}-\left(\frac{\theta_{3}^{\prime\prime}(x)}{\theta_{3}(x)}\right)^{2}$
$\displaystyle=-4\left(\frac{\theta_{3}^{\prime}(x)}{\theta_{3}(x)}\right)^{2}\left(\wp\left(x+\frac{\tau-1}{2}\right)-c_{0}\right)+2\left(\wp\left(x+\frac{\tau-1}{2}\right)-c_{0}\right)^{2}$
$\displaystyle\quad-4\frac{\theta_{3}^{\prime}(x)}{\theta_{3}(x)}\wp^{\prime}\left(x+\frac{\tau-1}{2}\right)-\wp^{\prime\prime}\left(x+\frac{\tau-1}{2}\right).$

Using (4.1), we find that

$\displaystyle\frac{d}{dx}\left(\frac{\theta_{3}^{(4)}(x)}{\theta_{3}(x)}-\left(\frac{\theta_{3}^{\prime\prime}(x)}{\theta_{3}(x)}\right)^{2}\right)$
$\displaystyle=8\frac{\theta_{3}^{\prime}(x)}{\theta_{3}(x)}\left(\wp\left(x+\frac{\tau-1}{2}\right)-c_{0}\right)^{2}-4\left(\frac{\theta_{3}^{\prime}(x)}{\theta_{3}(x)}\right)^{2}\wp^{\prime}\left(x+\frac{\tau-1}{2}\right)$
$\displaystyle\quad+8\left(\wp\left(x+\frac{\tau-1}{2}\right)-c_{0}\right)\wp^{\prime}\left(x+\frac{\tau-1}{2}\right)$
$\displaystyle\quad-4\frac{\theta_{3}^{\prime}(x)}{\theta_{3}(x)}\wp^{\prime\prime}\left(x+\frac{\tau-1}{2}\right)-\wp^{\prime\prime\prime}\left(x+\frac{\tau-1}{2}\right).$

Since $\wp\left(x+\frac{\tau-1}{2}\right)$ decreases on $0<x<1/2$, we have
$\wp^{\prime}\left(x+\frac{\tau-1}{2}\right)<0$. Define a function $F_{3}(x)$
as

$\displaystyle F_{3}(x)$
$\displaystyle:=\frac{1}{\wp^{\prime}(x+\frac{\tau-1}{2})}\frac{d}{dx}\left(\frac{\theta_{3}^{(4)}(x)}{\theta_{3}(x)}-\left(\frac{\theta_{3}^{\prime\prime}(x)}{\theta_{3}(x)}\right)^{2}\right)$
$\displaystyle=8\frac{\theta_{3}^{\prime}(x)}{\theta_{3}(x)}\frac{\left(\wp\left(x+\frac{\tau-1}{2}\right)-c_{0}\right)^{2}}{\wp^{\prime}\left(x+\frac{\tau-1}{2}\right)}-4\left(\frac{\theta_{3}^{\prime}(x)}{\theta_{3}(x)}\right)^{2}+8\left(\wp\left(x+\frac{\tau-1}{2}\right)-c_{0}\right)$
$\displaystyle\quad-4\frac{\theta_{3}^{\prime}(x)}{\theta_{3}(x)}\frac{\wp^{\prime\prime}\left(x+\frac{\tau-1}{2}\right)}{\wp^{\prime}\left(x+\frac{\tau-1}{2}\right)}-\frac{\wp^{\prime\prime\prime}\left(x+\frac{\tau-1}{2}\right)}{\wp^{\prime}\left(x+\frac{\tau-1}{2}\right)}.$
(4.2)

It suffices to prove that $F_{3}(x)>0$. We prove this by showing that
$F_{3}^{\prime}(x)<0$ and $F_{3}(1/2)>0$, because then by the mean value
theorem, for any $x\in(0,1/2)$, we have
$F_{3}(x)-F_{3}(1/2)=F_{3}^{\prime}(e)(x-1/2)$ for some $e\in(x,1/2)$ whence
$F_{3}(x)>0$. We first show that $F_{3}(1/2)>0$. Using the thirs differential
equation in (2), we have

$\displaystyle F_{3}(1/2)$
$\displaystyle=8(e_{3}-c_{0})^{2}\lim_{x\to{\frac{1}{2}}^{-}}\frac{\theta_{3}^{\prime}(x)/\theta_{3}(x)}{\wp^{\prime}\left(x+\frac{\tau-1}{2}\right)}-4\lim_{x\to{\frac{1}{2}}^{-}}\left(\frac{\theta_{3}^{\prime}(x)}{\theta_{3}(x)}\right)^{2}+8(e_{3}-c_{0})$
$\displaystyle\quad-4\wp^{\prime\prime}(\tau/2)\lim_{x\to{\frac{1}{2}}^{-}}\frac{\theta_{3}^{\prime}(x)/\theta_{3}(x)}{\wp^{\prime}\left(x+\frac{\tau-1}{2}\right)}-12e_{3}.$
(4.3)

Now [8, p. 358, Section 13.19]

$\displaystyle\frac{\theta_{3}^{\prime}(z)}{\theta_{3}(z)}=4\pi\sum_{n=1}^{\infty}(-1)^{n}\frac{q^{n}}{1-q^{2n}}\sin
2n\pi z$ (4.4)

implies that $\theta_{3}^{\prime}(x)/\theta_{3}(x)$ vanishes at $x=1/2$. Note
that $\wp^{\prime}\left(x+\frac{\tau-1}{2}\right)=0$ at $x=1/2$ too. Hence,
using L’Hopital’s rule in (4), then (4.1), the second differential equation in
(2) and simplifying, we see that

$F_{3}(1/2)=\frac{16(e_{3}-c_{0})^{3}}{g_{2}-12e_{3}^{2}}-12c_{0}.$

Now using (2) and (2.6), note that

$\displaystyle g_{2}-12e_{3}^{2}$
$\displaystyle=-4(e_{1}e_{2}+e_{2}e_{3}+e_{3}e_{1})-12e_{3}^{2}$
$\displaystyle=4(e_{3}-e_{1})(e_{2}-e_{3})$ $\displaystyle<0.$

Thus, we need to show that $16(e_{3}-c_{0})^{3}-12c_{0}(g_{2}-12e_{3}^{2})<0$
or equivalently, $(e_{3}-c_{0})^{3}<3c_{0}(e_{3}-e_{1})(e_{2}-e_{3})$.
Consider two cases.

Case 1: $c_{0}\leq 0$.

By (2.6), the left-hand side is less than zero but the right-hand side is
greater than or equal to zero. This proves the required inequality.

Case 2: $c_{0}>0$.

Using (2),

$\displaystyle 3c_{0}(e_{3}-e_{1})(e_{2}-e_{3})-(e_{3}-c_{0})^{3}$
$\displaystyle=(e_{1}+e_{2}+c_{0})^{3}-3c_{0}(2e_{1}+e_{2})(e_{1}+2e_{2})$
$\displaystyle=\frac{1}{27}\left(((2e_{1}+e_{2})+(e_{1}+2e_{2})+3c_{0})^{3}-27\cdot
3c_{0}(2e_{1}+e_{2})(e_{1}+2e_{2})\right).$

The last expression is clearly positive by the Arithmetic mean-Geometric mean
inequality and since $2e_{1}+e_{2}$, $e_{1}+2e_{2}$ are positive by (2.6) and
since $3c_{0}$ is positive. From the above two cases, we conclude that
$F_{3}(1/2)>0$. Our next task is to show that $F_{3}^{\prime}(x)<0$. From (4),
we have

$\displaystyle\frac{F_{3}^{{}^{\prime}}(x)}{4}=\frac{\theta_{3}^{\prime}(x)}{\theta_{3}(x)}\frac{A_{2}(x+\frac{\tau}{2})}{{\wp^{\prime}}^{2}\left(x+\frac{\tau-1}{2}\right)}+\frac{A_{1}(x+\frac{\tau}{2})}{\wp^{\prime}\left(x+\frac{\tau-1}{2}\right)},$

where $A_{1}(x)$ and $A_{2}(x)$ are defined in (3). Now

$A_{2}^{\prime}\left(x+\frac{\tau}{2}\right)=\wp^{\prime}\left(x+\frac{\tau-1}{2}\right)\left(2\left(g_{2}-12c_{0}^{2}\right)\wp\left(x+\frac{\tau-1}{2}\right)+\left(6g_{3}+4g_{2}c_{0}\right)\right).$

From (2.6), (3.9) and the facts that
$e_{3}<\wp\left(x+\frac{\tau-1}{2}\right)<e_{2}$ and
$\wp^{\prime}\left(x+\frac{\tau-1}{2}\right)<0$ on $0<x<1/2$, we find that
$A_{2}^{\prime}(x+\frac{\tau}{2})>0$. Also by Lemma 2.2,
$A_{2}(\frac{\tau}{2})>0$. By the mean value theorem, for any $x\in(0,1/2)$,
we have
$A_{2}(x+\frac{\tau}{2})=A_{2}(\frac{\tau}{2})+xA_{2}^{\prime}(k+\frac{\tau}{2})>0$
for some $k\in(0,x)$.

Figure 5 shows the graphs of $A_{1}(\frac{\tau}{2})$ and
$A_{2}(\frac{\tau}{2})$ on $0<x<1/2$. Now define $G_{3}$ by

$\displaystyle G_{3}(x)$
$\displaystyle:=\frac{F_{3}^{\prime}(x){\wp^{\prime}}^{2}(x+\frac{\tau-1}{2})}{4A_{2}(x+\frac{\tau}{2})}$
$\displaystyle=\frac{\theta_{3}^{\prime}(x)}{\theta_{3}(x)}+\frac{\wp^{\prime}(x+\frac{\tau-1}{2})\left(\wp\left(x+\frac{\tau-1}{2}\right)+\frac{2g_{3}+4c_{0}^{3}+g_{2}c_{0}}{g_{2}-12c_{0}^{2}}\right)}{2\left(\wp^{2}(x+\frac{\tau-1}{2})+\wp(x+\frac{\tau-1}{2})\frac{6g_{3}+4g_{2}c_{0}}{g_{2}-12c_{0}^{2}}+\frac{6g_{3}c_{0}+g_{2}c_{0}^{2}+g_{2}^{2}/4}{g_{2}-12c_{0}^{2}}\right)}.$
(4.5)

From the above discussion, it suffices to show that $G_{3}(x)<0$. Now, from
(4.4) and the fact that
$\wp^{\prime}\left(\frac{\tau-1}{2}\right)=0=\wp^{\prime}\left(\frac{\tau}{2}\right)$,
it is easy to see that $G_{3}(0)=0=G_{3}(1/2)$. This implies that
$G_{3}^{\prime}(x)$ has at least one zero in $0<x<1/2$. Differentiating both
sides of (4) with respect to $x$ and simplifying, we observe that

$\displaystyle
G_{3}^{\prime}(x)=\frac{{\wp^{\prime}}^{2}(x+\frac{\tau-1}{2})(1-Q(x+\frac{\tau}{2}))}{\left(\wp^{2}(x+\frac{\tau-1}{2})+\wp(x+\frac{\tau-1}{2})\frac{6g_{3}+4g_{2}c_{0}}{g_{2}-12c_{0}^{2}}+\frac{6g_{3}c_{0}+g_{2}c_{0}^{2}+g_{2}^{2}/4}{g_{2}-12c_{0}^{2}}\right)},$

where $Q(x)$ is defined in (3.22). Now

$\displaystyle 1-Q\left(x+\frac{\tau}{2}\right)$
$\displaystyle=1-\frac{\left(\wp(x+\frac{\tau-1}{2})+\frac{2g_{3}+4c_{0}^{3}+g_{2}c_{0}}{g_{2}-12c_{0}^{2}}\right)\left(2\wp(x+\frac{\tau-1}{2})+\frac{6g_{3}+4g_{2}c_{0}}{g_{2}-12c_{0}^{2}}\right)}{2\left(\wp^{2}(x+\frac{\tau-1}{2})+\wp(x+\frac{\tau-1}{2})\frac{6g_{3}+4g_{2}c_{0}}{g_{2}-12c_{0}^{2}}+\frac{6g_{3}c_{0}+g_{2}c_{0}^{2}+g_{2}^{2}/4}{g_{2}-12c_{0}^{2}}\right)}$
$\displaystyle=\frac{2\wp(x+\frac{\tau-1}{2})\frac{g_{3}+g_{2}c_{0}-4c_{0}^{3}}{g_{2}-12c_{0}^{2}}+C}{2\left(\wp^{2}(x+\frac{\tau-1}{2})+\wp(x+\frac{\tau-1}{2})\frac{6g_{3}+4g_{2}c_{0}}{g_{2}-12c_{0}^{2}}+\frac{6g_{3}c_{0}+g_{2}c_{0}^{2}+g_{2}^{2}/4}{g_{2}-12c_{0}^{2}}\right)},$
(4.6)

where

$C:=\frac{2(6g_{3}c_{0}+g_{2}c_{0}^{2}+g_{2}^{2}/4)}{g_{2}-12c_{0}^{2}}-\frac{(6g_{3}+4g_{2}c_{0})(2g_{3}+4c_{0}^{3}+g_{2}c_{0})}{(g_{2}-12c_{0}^{2})^{2}}.$

The numerator in the last expression of (4) has atmost one zero since it is
linear in $\wp(x+\frac{\tau-1}{2})$ and $\wp(x+\frac{\tau-1}{2})$ is monotone.
Hence, $G_{3}^{\prime}(x)$ has exactly one zero, say $x_{0}$, in $0<x<1/2$.
Thus we will be done if we can show that $G_{3}(x)<0$ at some point in the
interval $0<x<1/2$. In fact, we show that $G_{3}(x)<0$ on $(0,x_{0})$.

For any $x$ in $(0,x_{0})$, we have
$\wp\left(x+\frac{\tau-1}{2}\right)>\wp\left(x_{0}+\frac{\tau-1}{2}\right)$.
Also,

$\displaystyle\frac{g_{3}+g_{2}c_{0}-4c_{0}^{3}}{g_{2}-12c_{0}^{2}}=\frac{g_{3}+g_{2}c_{0}/2+2c_{0}^{3}}{g_{2}-12c_{0}^{2}}+\frac{c_{0}(g_{2}/2-6c_{0}^{2})}{g_{2}-12c_{0}^{2}}<\frac{-(e_{1}-c_{0})}{2}<0,$

where last two inequalities follows from (3.9) and (2.6). Therefore

${2\wp\left(x+\frac{\tau-1}{2}\right)\frac{g_{3}+g_{2}c_{0}-4c_{0}^{3}}{g_{2}-12c_{0}^{2}}+C}<{2\wp\left(x_{0}+\frac{\tau-1}{2}\right)\frac{g_{3}+g_{2}c_{0}-4c_{0}^{3}}{g_{2}-12c_{0}^{2}}+C}=0,$

where the last equality comes from the fact that $G_{3}^{\prime}(x_{0})=0$.
Hence, $G_{3}^{\prime}(x)<0$ for $0<x<x_{0}$. Then it is clear by the mean
value theorem that for any $x\in(0,x_{0})$,
$G_{3}(x)=xG_{3}^{\prime}(x_{1})<0$ for some $x_{1}\in(0,x)$. So finally
$G_{3}(x)<0$ for $0<x<1/2$. This completes the proof. Figure 6 shows the graph
of $G_{3}(x)$ on $0<x<1/2$.

## References

  * [1] A. Baernstein II, _On the harmonic measure of slit domains_ , Complex Var. Theory Appl. 9 (1987), 131–142.
  * [2] B.C. Berndt, _Number theory in the spirit of Ramanujan_ , American Mathematical Society, Providence, RI, 2006.
  * [3] D. Betsakos, _Geometric theorems and problems for harmonic measure_ , Rocky Mountain J. Math. 31 (2001), no.3, 773–795.
  * [4] D.M. Campbell, J.G. Clunie and W.K. Hayman, _Research problem in complex analysis_ , in _Aspects of Contemporary Complex Analysis_ , D.A. Brannan, J.Clunie, Eds., Academic Press, London-New York, 1980.
  * [5] C.-P. Chen, Complete monotonicity and logarithmically complete monotonicity properties for the gamma and psi functions, J. Math. Anal. Appl. 336 (2007), No. 2, 812–822.
  * [6] W. Cheney and W. Light, _A course in Approximation Theory_ , Graduate Text in Mathematics, 101, American Mathematical Society, Providence, RI, 2009.
  * [7] A. Dixit and A.Yu. Solynin, _Monotonicity of quotients of theta functions related to an extremal problem on harmonic measure_ , J. Math. Anal. Appl. 336, No. 2 (2007), 1042–1053.
  * [8] A. Erldelyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, _Higher Transcendental Functions_ , Vol. 2, McGraw-Hill, New York, 1955.
  * [9] B.-N. Guo, S. Guo and F. Qi, _Complete monotonicity of some functions involving polygamma functions_ , J. Comput. Appl. Math., 233 (2010), No. 9, 2149–2160.
  * [10] M.D. Hirschhorn, _Partial fractions and four classical theorems of number theory_ , Amer. Math. Monthly 107, No. 3 (2007), 260–264.
  * [11] M.E.H. Ismail, _Integral representations and complete monotonicity of various quotients of Bessel functions_ , Can. J. Math. 29, No. 6 (1977), 1198–1207.
  * [12] K.S. Miller and S.G. Samko, _Completely monotonic functions_ , Integr. Transf. and Spec. Funct. 12 (2001), No. 4, 389–402.
  * [13] K. Schiefermayr, _Some new properties of Jacobi’s theta functions_ , J. Comput. Appl. Math. 178 (2005), 419–424.
  * [14] A.Yu. Solynin, _Harmonic measure of radial segments and symmetrization_ (Russian) Mat. Sb. 189 (1998), No. 11, 121–138; translation in Sb. Math. 189 (1998), no. 11-12, 1701- 1718.