††thanks: corresponding author’s Email<EMAIL_ADDRESS>
# Proposal of A Linked Mirror Configuration for Magnetic Confinement
Experiment

Zhichen Feng  Guodong Yu  Peiyou Jiang  GuoYong Fu  Institute for Fusion
Theory and Simulation and Department of physics, Zhejiang University, Hangzhou
310027, China

###### Abstract

A new linked mirror device for magnetic confinement experiment is proposed.
The new linked mirror device consists of two straight magnetic mirrors
connected by two half-torus. The structure of the configuration as a whole is
three dimensional because the two linear mirror sections are not parallel. The
angle between the two mirror sections generates rotational transform which
results in good magnetic confinement of toroidally processing passing
particles. In this way the usual loss cone of the traditional linear mirror
machines is eliminated. The single particle confinement is similar to that of
tokamaks with most of particles well confined. The calculated neoclassical
confinement is very good and is even better than that of an equivalent
tokamak. Potentially the proposed linked mirror configuration can allow MHD
stable high beta equilibria with good plasma confinement suitable for neutron
sources and magnetic fusion reactors.

##  I introduction

Magnetically controlled fusion energy research has been pursued since early
1950’s. Many types of magnetic confinement devices have been invented, such as
magnetic mirrorsFowler ; Post , stellaratorsSpitzer ; Wakatani  and tokamaks.
One of earliest and simplest magnetic confinement devices is magnetic mirror.
In early days the magnetic mirror devices suffered from two major defects: the
fast end loss and strong MHD interchange instabilityPost . In particular the
end loss of passing particles is very fast leading to poor plasma confinement.
Since then various schemes have been developed to mitigate the fast end loss,
such as tandem mirror approachFowler2  and the more modern approach of
GDTBurdakov ; Ivanov . These approaches have indeed improved the plasma
confinement and MHD stability considerably. However some residual end loss
remains and as a result plasma confinement is still not good enough for fusion
reactors.

In this paper a new linear-torus hybrid device of Linked Mirror Configuration
(LMC) has been proposed. The main idea of the new configuration was inspired
by the concepts of linear magnetic mirror and the figure-8 stellaratorSpitzer
. The hybrid configuration consists of two straight mirror sections connected
by two half-torus sections. The two linear mirror sections are not parallel
but with a finite angle between them. This angle results in three dimensional
(3D) toroidal structure of magnetic field leading to finite rotational
transform, just as in the early figure-8 stellarator. Correspondingly the
toroidally transiting passing particles are well confined and the end loss of
a linear mirror is eliminated. Indeed our simulation results show that the
single particle confinement is very good and the neoclassical transport is
very low, even lower than that of an equivalent tokamak.

The linked mirror approach has been considered previously. The earlier ELMO
Bumpy Torus (EBT) consists of many mirrors linked toroidallyEBT . Later in
1981 the DRAKON closed fusion trap was proposed by Glagolev et al.Glagolev ;
Kondakov , consists of two long linear mirror sections joined by two
curvilinear sections (CRELs). More recently, Mazzucato proposed a fusion
reactor scheme of two linear mirrors joined by two toroidal sections with
helical coilsMazzucato . Conceptually our linked mirror approach is similar to
those of the DRAKONGlagolev ; Kondakov  and Mazzucato’sMazzucato . The main
difference is that in our approach the rotational transform is generated by
the angle between the two non-parallel mirror sections rather than the two 3D
half-torus sections.

As will be shown later, our LMC posses following good properties: (1) there is
no end loss; (2) neoclassical confinement is very good and better than that of
equivalent tokamaks; (3)All the current coils have the simple geometry of
circular planar shape. Finally the device is potentially MHD stable order-of-
unity plasma beta. Because of these good properties, the proposed LMC is
potentially suitable for neutron sources and fusion reactors.

The paper is organized as follows. Section II introduces the basic structure
of the LMC. Section III describes the rotational transform properties. Section
IV analyses the quasi-omnigeneity and single particle confinement of LMC.
Section V presents the numerical results of neoclassical transport. Section VI
briefly discusses MHD stability. Section VII gives conclusions.

##  II Basic structures of the linked mirror configuration

(a) Top view of the linked mirror configuration

(b) Side view of the linked mirror configuration

Figure 1: Basic structure of the linked mirror configuration

The basic structure of LMC is shown in Fig. 1 with cartesian coordinates
(x,y,z). From the top view (Fig. 1(a)), the configuration shape looks like a
race track. The configuration consists of two straight mirror sections and two
half-torus sections. However the two straight sections are not parallel as
seen from the side view in Fig. 1(b). The angle between the two straight
sections is defined as $2\Theta$ with $\Theta$ being the angle between one of
the straight section and the horizontal plane (z=0 plane). We will show in
next section that this angle is responsible for generating rotational
transform of the whole configuration. The magnetic field is provided by simple
circular planar coils of equal radius and different currents. For the two
mirror sections, the coils forms a circular cylinder. On the other hand, for
the half-torus sections, the circular coils are placed in such a way that the
centers of the coils form a half circle connecting the two ends of the mirror
sections. Obviously, the connections between the straight and half-torus
sections can not be tangent, but the magnetic field lines and the magnetic
flux surfaces are always smooth, as will be shown later. Besides the angle
$\Theta$, other key configuration parameters include $L$ the length of the
straight mirror section, $d$ the half distance between the two mirror
sections, $r$ the radius of the coils, and $I_{i}$ the current in each coil.

Fig. 1 plots the coils in blue, one of the magnetic surfaces in orange, and a
magnetic field line on the flux surface. The parameters used are:
$\Theta=\pi/20$, $L=12m$, $d=3m$, $r=1m$, $I_{torus}/I_{tube}=5I_{0}:1I_{0}$,
which means the currents in the coils of the half-torus sections are 5 times
of that in the straight mirror tubes. It should be pointed out that the
currents in the four end coils in each mirror section are $5I_{0}$ too.

(a) Cross section of magnetic flux surfaces at $\phi=0$ plane

(b) Cross section of magnetic flux surfaces at $\phi=\pi/2$ plane

Figure 2: Cross section of magnetic flux surfaces of the linked mirror
configuration, where $\phi$ is the toroidal angle

Fig. 2 plots the cross-section of the magnetic flux surfaces of LMC obtained
from field line tracing for the $\phi=0$ plane (a) and the $\phi=\pi/2$ plane
(b) respectively. Here $\phi$ is the toroidal angle about the z-axis through
the center of the whole configuration. It shows that by this type of simple
connections, nice magnetic flux surfaces are formed with finite rotational
transform. The magnetic flux surfaces plotted in red indicate that the flux
surfaces go out of the cylindrical tube within coils, as shown in Fig. 3. They
are still closed magnetic flux surfaces and can potentially be used as
divertors.

Figure 3: The last closed magnetic flux surface with the coils

##  III Rotational transform properties

The LMC as just defined above is a special toroidal magnetic confinement
configuration with finite external rotational transform generated by the
coils. The mechanism for the rotational transform is the 3D magnetic field
lines induced by finite $\Theta$ just as in the early figure-8 stellarator. In
this section, we will show the dependence of the rotational transform on key
configuration parameters.

Figure 4: Rotational transform versus $x$, where $x$ is the starting point of
each field line with $z=0$ at $\phi=0$ plane

Fig. 4 plots the rotational transform profile against radius variable $x$ for
the LMC plotted in Fig. 1, where $x$ is the major radius at $\phi=0$ plane. We
observe that the rotational transform profile is almost constant with little
magnetic shear.

(a) Rotational transform versus $\Theta$

(b) Rotational transform versus $L$

(c) Rotational transform versus $d$

Figure 5: The dependence of rotational transform on the main parameters

Fig. 5(a), 5(b) and 5(c) plot the dependence of the rotational transform on
$\Theta$, $L$ and $d$ respectively. For each plot, the following parameters
are held fixed unless it is varied: $\Theta=\pi/20$, $L=12m$, $d=3m$, $r=1m$,
$I_{tours}/I_{tube}=5I_{0}:1I_{0}$. Fig. 5(a) shows that the rotational
transform of the LMC increases almost linearly with $\Theta$ and can vary in a
large range, from 0 to 0.58. The last point in Fig. 5(a) refers to the case
where the two straight mirror tubes are perpendicular to each other.
Configurations with $\Theta>\pi/4$ have higher rotational transforms. However,
the flux surfaces would become too small relative to the coil size when the
angle is larger than $45^{\circ}$. Furthermore Fig. 5(b) shows that the
rotational transform increases with $L$, the length of the straight mirror
sections. It should be noted that in the limit of $L=0$, the configuration
becomes an axisymmetric torus with $\iota=0$ (since there is no plasma
current). Finally, Fig. 5(c) shows that the rotational transform decreases
when $d$ increases, but it tends to a fixed value when $d\rightarrow\infty$.

In summary, the rotational transform profile of this linked mirror
configuration is shearless, and can be varied in a vast range by simply
changing one of the geometry parameters. This property can be used to avoid
low order rational iota values in order to control MHD instabilities.

##  IV Quasi-omnigeneity and single particle confinement

In this section we consider the collisionless single particle confinement in
the LMC. We will show that the magnetic configuration is quasi-Omnigeneous
resulting in good neoclassical particle confinement. Single particle orbit
simulations are carried out using the drift-kinetic equation with the magnetic
field calculated from all coils. A numerical code with high accuracy has been
developed for this purpose. The details of the code will be reported
elsewhere.

Figure 6: Magnetic field strength along the field line starting at
$(9.15,0,0)$

Fig. 6 plots the magnetic field strength along one field line starting at
$(x,y,z)=(9.15,0,0)$ at the reference coil current $I_{0}=2\times 10^{5}$A.
The valleys are in the straight mirror sections, while the peaks are in the
half-torus sections. The minimum values of the field strength along one field
line are almost the same, and the maximum values also do not vary very much.
This indicates that the magnetic configuration is quasi-omnigeneousmatt .
There are ripples along the field line (see the expanded view in Fig. 6),
which comes from the effect of discrete coils. Omnigeneity is defined
byLaurence ; Helander :

$<\mathbf{v_{d}}\cdot\nabla\psi>_{b}=\int\mathbf{v_{d}}\cdot\nabla\psi dt=0$
(1)

for all trapped particles, where $\mathbf{v_{d}}$ is the guiding center drift
velocity. In another word, the bounce-averaged drift velocity is zero or very
small.

Figure 7: Poincare plots of mirror trapped particles’ orbits at $\phi=\pi/2$
plane, energy E=100eV

Fig. 7 shows the Poincare plot for the orbits (colored lines) of mirror-
trapped particles as well as flux surfaces (black lines) at the $\phi=\pi/2$
plane. We observe that these orbits are all closed. Furthermore, the poloidal
processing frequency $\omega_{d}$ is much smaller than the bounce frequency
$\omega_{b}$ ($\omega_{b}\approx 500\omega_{d}$). These results indicate that
$<\mathbf{v_{d}}\cdot\nabla\psi>_{b}$ is vary small, which means that the
configuration is quasi-omnigeneous.

There exist finite shifts between the trapped particles’ orbits and the
magnetic flux surfaces. This is due to the fact that the magnetic flux
surfaces are not cylindrically symmetric as the coils in the mirror sections
because the configuration is 3D. In particular we observe that the position of
magnetic axis is not at the center of coils as shown in Fig. 1 and Fig. 3.
This orbital shift can be reduced to zero by careful placement of coil
locations in the half-torus sections. Furthermore we observe that, just like
trapped particles, passing particles’ orbits also have finite radial shift
away from the flux surfaces as shown in Fig. 8, although the size of orbital
shifts is usually quite modest.

Figure 8: Poincare plots of passing particles’ orbits at $\phi=\pi/2$ plane,
energy E=100eV

We found that by placing different numbers of coils in the half-torus
sections, the flux surfaces in straight section would move translationally,
while the trapped particle’s orbits keep almost fixed. In particular, the
configuration with 16 coils in half-torus sections aligns the magnetic
surfaces and the poloidally processing orbits of trapped particles better than
other configurations(Fig. 9).

Figure 9: Reference flux surface and Poincare plot for particle starting at
same point (0,2.7,0), with energy E=100ev and pitch angle
($v_{||}=v\cos(0.3\pi)$)

A systematic study of single particle confinement has been done. Simulation
results show that there are four types of particle orbits in the LMC: passing
particles, mirror trapped particles, ripple trapped particles, and particles
trapped by the magnetic field variation with higher magnetic field strength
mainly in the half-torus sections(particles with $E/\mu\sim 1.36-1.45T$ in
Fig. 6). These ”trapped particles” are in fact passing particles processing
toroidally. However, their parallel velocity changes sign after several
toroidal transits. Passing particles and mirror trapped particles are found to
be confined well. Ripple trapped particles are also confined in the
configuration although they may leave the last closed surface. Fig. 10 plots
the orbits for the typical passing particles, mirror trapped particles and
ripple trapped particles.

Figure 10: Orbits for the typical passing particles, mirror trapped particles
and ripple trapped particles

The particles barely trapped by the magnetic field in the mirror sections may
not be confined for a long time, since they bounce between two points with
high fields near the half-torus sections in one period and the magnetic drift
due to the toroidal curvature is not averaged out. The fraction of these lost
particles is small since they are barely trapped.

(a) Number of left particles versus t for 10000 test particles with energy E =
1keV

(b) Distribution of test particles and loss particles with energy E = 1keV in
$v_{\parallel}/v$ and $\phi$ space

(c) Distribution of test particles and loss particles with energy E = 1keV in
$\mu B_{max}/E$ and $\phi$ space

Figure 11: Study of 10000 test particles energy E = 1keV in the linked mirror
configuration

Fig. 11 plots the simulation results of 10000 test particles with energy E =
1keV in the LMC given in Fig. 1. The particles are initially distributed
uniformly in real space on the middle magnetic flux surface going through the
point (9.1,0,0). The velocity distribution is isotropic. The particle boundary
is chosen to be at $r_{p}=0.99r$, where $r$ is the coil radius. Fig. 11(a)
shows that more than $95\%$ particles are still confined after a long duration
(about 0.17s). There is a thin envelope formed by loss particles in phase
space as shown in Fig. 11(b). These lost particles are barely trapped
particles near the trap-passing boundary in the mirror sections. All the
passing particles are well confined including those ”trapped” by the global
magnetic well at high fields. There is another type of particles:
$v_{//}/v\sim 0$, which are confined by ripple wells in both mirror sections
and the half-torus sections. They are well confined in the configuration as
shown in Fig. 10. We observe in Fig. 11(c) that all the lost particles have
similar $\mu B_{max}/E$ values ($\approx 1$). This indicates that the lost
particles in the LMC are all near the passing-trap boundary in phase space. As
a result the loss cone is very narrow. Furthermore, unlike first orbit loss in
tokamaks, these particles typically bounce back and forth for tens to hundreds
times before leaving the configuration as shown for one such orbit in Fig. 12.

Figure 12: Orbit of the typical loss particles

(a) Fraction of confined particles versus time for particles with different
energies

(b) Fraction of confined particles versus particle energy after 0.1668s

Figure 13: Particle confinement scaling study

Fig. 13(b) shows that the fraction of confined particles tends to a constant
after a long time ( 0.1668s ) for particle energy larger than 1600eV. It is a
sufficiently long time period to calculate the fraction because the number of
confined particles becomes almost a constant at that time, as shown in Fig.
13(a).

##  V Neoclassical transport

We have pointed out that the trapped particle’s bounce orbit width is very
small as compared with the device size and the neoclassical transport level is
expected to be very low. Here we investigate the neoclassical transport
numerically. The particle collisions are taken into account using the Monte
Carlo pitch angle scattering modelBoozer . We consider a linked mirror
configuration with 54 coils as shown in Fig. 14. The following key parameters
are used: $\Theta=\pi/20$, $L=4.8m$, $d=1.2m$, $r=0.4m$,
$I_{red}/I_{blue}=160kA:80kA$. The corresponding mirror ratio is about
$R=B_{max}/B_{min}=0.87T/0.22T\sim 4$. It should be pointed out that the
plasma minor radius ($a_{0}=0.2m$) is about $50\%$ of the size given in Fig.
1. More coils are used here because they help to align the magnetic axis with
the coil center in the straight mirror sections. As a result, the precession
orbits of mirror-trapped particles(Fig. 7) almost coincide with magnetic flux
surfaces.

Figure 14: Overview of the linked mirror with 54 coils

Fig. 15 plots the numerical results of neoclassical transport studies. At the
beginning of particle simulations, 4000 protons are loaded uniformly on the
middle flux surface. The velocity distribution is a Maxwellian with
temperature $T=160eV$. The plasma density is chosen to be $6.4\times
10^{17}m^{-3}$. Fig. 15 shows the fraction of confined ions as a function of
time (a) and the real time loss rate (b) for the new configuration (red line)
and an equivalent tokamak (black line). Here the real time loss rate is
estimated by $N^{-1}dN/dt$. It can be regarded as the inverse of particle
confinement time if the loss rate converges to a steady state value in time.
The results clearly show that the particle confinement of the considered
configuration is substantially better than that of an equivalent tokamak. We
find that the particle confinement time for the configuration is about 0.3s,
which is quite large considering the parameters used.

(a) Fraction of confined particles versus time for T=160eV protons

(b) Real time loss rate for T=160eV protons

Figure 15: Neoclassical transport study for linked mirror configuration and
tokamak. The collision rate is proportional to $E^{-3/2}$, and the same for
protons with same energy in both configuration.

##  VI MHD stability

So far we have only considered the property of vacuum magnetic field with
focus on particle confinement. Concerning MHD stability, our configuration at
zero or small plasma beta has a small magnetic hill on order of $1\%$, i.e.,
the strength of flux surface-averaged magnetic field decreases about $1\%$
from magnetic axis to the last closed flux surface. As a consequence the
interchange mode is expected to be unstable, at least for small plasma beta
assuming pressure is constant on magnetic surfaces. There are a number of ways
to stabilize this interchange modeRyutov  including stabilization by energetic
sloshing ionsHinton  and Finite ion Larmor Radius (FLR) effectsRosenbluth .
First, the instability can be mitigated or even stabilized by choosing a value
of rotational transform such that the low order rational resonances are
avoided since the rotational transform profile is basically uniform. Second, a
simple estimate has shown that the interchange mode can be stabilized by
energetic sloshing ions with a peak pitch angle distribution (i.e., via
neutral beam injection) if energetic ion pressure is comparable to background
plasma pressure. Finally the FLR stabilizing is expected to be significant
since the poloidal mode numbers of the interchange modes are large (note that
the value of rotational transform is small). Therefore our linked mirror
configuration as designed is potentially MHD stable at high beta on order of
unity. A more definitive conclusion requires calculation of finite beta
equilibria and a more precise determination of MHD stability. This will be
done in a future work.

##  VII Conclusions

In conclusion, a linked mirror configuration is proposed. The new device
consists of two linear mirror sections with ends connected by half-torus
sections. The magnetic field is generated by simple circular planar coils. The
passing particles are well confined due to rotational transform generated by a
finite angle between the two linear mirror sections. Therefore the usual end
loss of linear mirror devices is completely eliminated. Most particles are
well confined including the mirror trapped particles and the ripple-trapped
particles. Monte Carlo simulations show that the neoclassical confinement is
very good and is even better than that of an equivalent tokamak. The proposed
linked mirror configuration is potentially suitable for neutron sources and
fusion reactors.

###### Acknowledgements.

This work is supported by Zhejiang University’s startup funding for one of the
authors (Prof. Guoyong Fu).

## References

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# Contraction property of differential operator on Fock space

David Kalaj  University of Montenegro, Faculty of Natural Sciences and
Mathematics, Cetinjski put b.b. 81000 Podgorica, Montenegro<EMAIL_ADDRESS>
###### Abstract.

In the recent paper, [8] Nicola and Tilli proved the Faber-Krahn inequality,
which for $p=2$, states the following. If $f\in\mathcal{F}_{\alpha}^{2}$ is an
entire function from the corresponding Fock space, then

$\frac{1}{\pi}\int_{\Omega}|f(z)|^{2}e^{-\pi|z|^{2}}dxdy\leq(1-e^{-|\Omega|})\|f\|^{2}_{2,\pi}.$

Here $\Omega$ is a domain in the complex plane and $|\Omega|$ is its Lebesgue
measure. This inequality is sharp and equality can be attained. We prove the
following sharp inequality

$\int_{\Omega}\frac{|f^{(n)}(z)|^{2}e^{-\pi|z|^{2}}}{\pi^{n}n!L_{n}(-\pi|z|^{2})}dxdy\leq(1-e^{-(n+1)|\Omega|})\|f\|^{2}_{2,\pi},$

where $L_{n}$ is Laguerre polynomial, and $n\in\\{0,1,2,3,4\\}$. For $n=0$ it
coincides with the result of Nicola and Tilli.

###### Key words and phrases:

Fock space, reproducing kernel, isoperimetric inequality

###### Contents

  1. 1 Introduction
  2. 2 Proof of main results
    1. 2.1 Proof Theorem 1.6 for $n=1$.
    2. 2.2 The proof of Theorem 1.6 for $n\geqslant 2$

##  1\. Introduction

Let $\mathbb{C}$ be the complex plane. Denote by $dA(z)(=dxdy)$ the Lebesgue
measure on the complex plane. Throughout the paper, we consider the Gaussian-
probability measure

$d\mu_{\alpha}(z)=\frac{\alpha}{\pi}e^{-\alpha|z|^{2}}dA(z),$

where $\alpha$ is a positive parameter. Also let
$dA_{\alpha}(z)=\frac{\alpha}{\pi}dA(z)$.

For $1\leq p<\infty,$ let $L^{p}(\mathbb{C},d{\mu}_{\alpha})$ denote the space
of all Lebesgue measurable functions $f$ on $\mathbb{C}$ such that

$\|f\|_{p,\alpha}^{p}=\frac{p\alpha}{2\pi}\int_{\mathbb{C}}|f(z)|^{p}e^{-\frac{p\alpha|z|^{2}}{2}}dA(z)<\infty.$

The Segal–Bargmann space also known as the Fock space, denoted by
$\mathcal{F}_{\alpha}^{2},$ consists of all entire functions $f$ in
$L^{2}(\mathbb{C},d\mu_{\alpha}).$ For any $p\geq 1$, $F_{\alpha}^{p}$ is a
closed subspace of $L^{p}(\mathbb{C},d\mu_{\alpha}).$ Therefore
$F_{\alpha}^{p}$ is a Banach space.

We refer to the book [9] for a good setting of the Fock space on the plane and
the papers [2], [3] and [5] for a higher-dimensional setting.

Since $L^{2}(\mathbb{C},d\mu_{\alpha})$ is a Hilbert space with the inner
product

$\left<f,g\right>_{\alpha}=\int_{\mathbb{C}}f(z)\overline{g(z)}d\mu_{\alpha}(z),$

the Fock space $\mathcal{F}_{\alpha}^{2}$ as its closed subspace determines a
natural orthogonal projection
$P_{\alpha}:L^{2}(\mathbb{C},d\mu_{\alpha})\rightarrow\mathcal{F}_{\alpha}^{2}.$

It can be shown (see [9]) that $P_{\alpha}$ is an integral operator induced by
the reproducing kernel

$K_{\alpha}(z,w)=e^{\alpha z\bar{w}}.$

More precisely,

$P_{\alpha}f(z)=\int_{\mathbb{C}}K_{\alpha}(z,w)f(w)d\mu_{\alpha}(w),f\in
L^{2}(\mathbb{C},d\mu_{\alpha}),$

and in particular

(1.1)
$f(z)=\int_{\mathbb{C}}K_{\alpha}(z,w)f(w)d\mu_{\alpha}(w),f\in\mathcal{F}_{\alpha}^{2}.$

Laguerre polynomial is defined by

$L_{n}(x)=\sum_{k=0}^{n}\binom{n}{k}\frac{(-1)^{k}}{k!}x^{k}.$

For $n=0$, $L_{0}(x)=1$.

Le $F={{}_{1}}F_{1}$ be the Kummer function defined by

$F[a,b,z]=\sum_{k=0}^{\infty}\frac{(a)_{k}}{(b)_{k}k!}z^{k},$

where $(c)_{k}=c(c+1)\cdot\dots\cdot(c+k-1)$ is denoted the shifter factoriel.
The connection between Kummer function and Laguerre polynomial is given by

$F(1+n,1,r)=e^{r}L_{n}(-r).$

For $\alpha>0$ and $n\geqslant 0$ an integer, we define the Hilbert space of
entire functions
$\mathcal{F}^{2}_{n,\alpha}=L^{2}(\mathbb{C},d\mu_{n,\alpha})$, where

$d\mu_{n,\alpha}(z)=\frac{1}{n!\alpha^{n}L_{n}(-\alpha|z|^{2})}d\mu_{\alpha}(z).$

In this paper we consider the differential operator
$\mathcal{D}_{n}[f](z)=f^{(n)}(z),$ and show that it maps
$\mathcal{F}^{2}_{\alpha}$ into $\mathcal{F}^{2}_{n,\alpha}$. Moreover, we
show that it is a contraction satisfying the local contraction property.

We start with a result of Haslinger, author, and Vujadinovic in [6], which can
be stated as the contraction property of differential operator from
$\mathcal{F}^{2}_{\alpha}$ into $\mathcal{F}^{\infty}_{n,\alpha}$, where the
last space is defined in a standard fashion. Note that the space
$F_{n,\alpha}^{\infty}$ is defined to be the space of all entire functions $f$
such that

$\|f\|_{\infty,n,\alpha}:=\alpha^{-n/2}(n!)^{-1/2}\sup\\{|f(z)|{e^{-\frac{\alpha|z|^{2}}{2}}}{L_{n}^{-1/2}(-\alpha|z|^{2})}:z\in\mathbb{C}\\}<\infty.$

######  Proposition 1.1.

Let $f\in\mathcal{F}_{\alpha}^{2}.$ Then we have the sharp point-wise
inequality

$|f^{(n)}(z)|\leq
e^{\alpha|z|^{2}/2}\sqrt{\alpha^{n}n!L_{n}(-\alpha|z|^{2})}\|f-T_{n}(f)\|_{2,\alpha},$

where $n\in\mathbf{N}_{0}$ and
$T_{n}(f)(w)=\sum_{k=0}^{n-1}\frac{f^{k}(0)}{k!}w^{k}$, and $L_{n}$ is
Laguerre polynomial. The extremal functions are of the form

$f(w)=\sum_{k=0}^{n-1}a_{k}w^{k}+\alpha^{n}e^{\alpha\bar{z}w}w^{n}.$

In particular, for $n=1$ we get the following best estimates

(1.2)
$\begin{split}|f^{\prime}(z)|\leq\sqrt{\alpha\left(1+\alpha|z|^{2}\right)}e^{\alpha|z|^{2}/2}\|f\|_{2,\alpha},\
\ \ z\in\mathbb{C}.\end{split}$

In other words we have the sharp inequality

$\|f^{(n)}\|_{\infty,n,\alpha}\leq\|f\|_{2,\alpha}.$

Proposition 1.1 for $n=0$ coincides with a corresponding result [9, Theorem
2.7] or [4, Theorem 3.4.2], and that result was crucial for proving the
following important result in [8], by Nicola and Tilli

######  Theorem 1.2.

For $f\in\mathcal{F}_{\alpha}^{2}$, and a domain $\Omega$ with a finite
measure, we have

$\int_{\Omega}{|f(z)|^{2}e^{-\pi|z|^{2}}}dxdy\leq(1-e^{-|\Omega|})\|f\|^{2}_{2,\alpha}.$

This result is sharp and equality is attained for certain balls and certain
specific functions related to extremal functions in Proposition 1.1.

Theorem 1.2 is further extended in [8] for $p\geqslant 2$, and proved local
Lieb’s uncertainty inequality for the STFT (short-time Fourier transforms see
[8, Theorem 5.2]), by using Lieb’s uncertainty inequality for the STFT.
Theorem 1.2 proves a conjecture of Abreu and Speckbacher [1, Conjecture 1]
(for $p=2$). We extend Theorem 1.2 for higher derivatives. We believe that our
results can be also applied to STFT, especially because of the Laguerre
connection [2, Chapter 1]. By using the method developed in [8], Kulikov in
[7] proved some conjectures for Bergman and Hardy space of holomorphic
mappings. We also use the method developed in [8].

In this paper, we consider the following conjecture

######  Conjecture 1.3.

For $f\in\mathcal{F}_{\alpha}^{2}$, and a domain $\Omega$ with a finite
measure, we have

$\int_{\Omega}{|f^{(n)}(z)|^{2}}d\mu_{n,\alpha}(z)\leq(1-e^{-(n+1)|\Omega|})\|f\|^{2}_{2,\alpha},$

where $n\geqslant 0$.

We reduce Conjecture 1.3 to the following conjecture

######  Conjecture 1.4.

For $f\in\mathcal{F}_{\alpha}^{2}$, we have

$\|f^{(n)}\|^{2}_{2,n,\alpha}:=\int_{\mathbb{C}}{|f^{(n)}(z)|^{2}}d\mu_{n,\alpha}(z)\leq\|f\|^{2}_{2,\alpha}.$

Further Conjecture 1.4 is reduced to the following

######  Conjecture 1.5.

Let $k\geqslant n$ and

$a_{k}=\frac{\Gamma[1+k]}{n!\Gamma[1+k-n]^{2}}\int_{0}^{\infty}\frac{e^{-r}r^{k-n}}{L_{n}(-r)}dr,$

where $L_{n}$ is Laguerre polynomial, and $n\geqslant 0$.

Then $a_{k}<1$.

Then we prove Conjecture 1.3 for $n\leq 4$. In other words, we prove

######  Theorem 1.6.

For $f\in\mathcal{F}_{\alpha}^{2}$, and a domain $\Omega\subset\mathbb{C}$
with a finite Lebesgue measure $|\Omega|$, we have

$\frac{\alpha}{\pi}\int_{\Omega}\frac{|f^{(n)}(z)|^{2}e^{-\alpha|z|^{2}}}{\alpha^{n}n!L_{n}(-\alpha|z|^{2})}dxdy\leq(1-e^{-(n+1)|\Omega|})\|f\|^{2}_{2,\alpha},$

where $L_{n}$ is Laguerre polynomial, and $0\leq n\leq 4$. The inequality is
sharp but is never attained. In particular for $n=1$, we have

$\int_{\Omega}\frac{|f^{\prime}(z)|^{2}e^{-\alpha|z|^{2}}}{\alpha(1+\alpha|z|^{2})}dA_{\alpha}(z)\leq(1-e^{-2|\Omega|})\|f\|^{2}_{2,\alpha}.$

In other words the differential operator
$\mathcal{D}_{n}:\mathcal{F}_{\alpha}^{2}\to\mathcal{F}^{2}_{n,\alpha}$, is a
contraction and satisfies local contraction property on sub-domains of complex
plane.

######  Remark 1.7.

Note that for $n=0$ the previous theorem is exactly the main result of Nicola
and Tilli in [8]. Note also that $f\in\mathcal{F}_{\alpha}^{2}\not\Rightarrow
f^{\prime}\in\mathcal{F}_{\alpha}^{2}$ (see a counterexample in [6]), so
Laguerre polynomials are inevitable.

######  Corollary 1.8.

Under conditions of Theorem 1.6 we have

$\int_{\Omega}\frac{|f^{\prime}(z)|^{2}e^{-\alpha|z|^{2}}}{\alpha(1+\alpha|z|^{2})}dA_{\alpha}(z)\leq(1-e^{-2|\Omega|})d^{2}_{\alpha}(f,\mathcal{C}),$

where $\mathcal{C}$ is the space of all constants $d_{\alpha}$ is the induced
metric in the Fock space $\mathcal{F}_{\alpha}^{2}$.

##  2\. Proof of main results

We need the following corollary of Proposition 1.1.

######  Corollary 2.1.

If $f\in F_{\alpha}^{2}$, then

$\lim_{|z|\to\infty}\frac{|f^{(n)}(z)|}{\sqrt{\alpha^{n}n!L_{n}(-\alpha|z|^{2})}}e^{-\alpha|z|^{2}/2}=0.$

and in particular, for $n=1$, we have

$\lim_{|z|\to\infty}\frac{|f^{\prime}(z)|}{\sqrt{\alpha(1+\alpha|z|^{2})}}e^{-\alpha|z|^{2}/2}=0.$

###### Proof.

Let us prove for $n=1$ and note that the same proof works for every $n$. Since
the space of polynomials is dense in the Fock space, for given $\epsilon>0$,
there is a polynomial $P$ so that $\|f-P\|_{2,\alpha}\leq\epsilon$. Then we
have

$\|f\|_{2,\alpha}\leq\|f-P\|_{2,\alpha}+\|P\|_{2,\alpha}.$

From (1.2), we get

$|f^{\prime}(z)-P^{\prime}(z)|\leq\sqrt{\alpha\left(1+\alpha|z|^{2}\right)}e^{\alpha|z|^{2}/2}\|f-P\|_{2,\alpha}\leq\sqrt{\alpha\left(1+\alpha|z|^{2}\right)}e^{\alpha|z|^{2}/2}\epsilon.$

On the other hand

$\lim_{|z|\to\infty}\frac{|P^{\prime}(z)|}{\sqrt{\alpha(1+\alpha|z|^{2})}}e^{-\alpha|z|^{2}/2}=0.$

So

$\limsup_{|z|\to\infty}\frac{|f^{\prime}(z)|}{\sqrt{\alpha(1+\alpha|z|^{2})}}e^{-\alpha|z|^{2}/2}\leq\epsilon.$

Thus

$\limsup_{|z|\to\infty}\frac{|f^{\prime}(z)|}{\sqrt{\alpha(1+\alpha|z|^{2})}}e^{-\alpha|z|^{2}/2}=0$

as it is stated. ∎

######  Lemma 2.2.

Let

$u(z)=\frac{|f^{(n)}(z)|^{2}e^{-\alpha|z|^{2}}}{\alpha^{n}n!L_{n}(-\alpha|z|^{2})}$

and $A_{t}=\\{z:u(z)>t\\}$. If $|\Omega|=s=\mu(t)$, then

$\int_{\Omega}u(z)dxdy\leq\int_{A_{t}}u(z)dxdy.$

###### Proof of Lemma 2.2.

As in [8], let $\Omega_{1}=\Omega\cap A_{t}$. Then
$\Omega=\Omega_{1}\cup(\Omega\setminus\Omega_{1})$ and
$A_{t}=\Omega_{1}\cap(A_{1}\setminus\Omega_{1})$. We also have

$|(\Omega\setminus\Omega_{1})|=|\Omega|-|\Omega_{1}|=|A_{t}|-|\Omega_{1}|=|A_{t}\setminus\Omega_{1}|.$

If $z\in(\Omega\setminus\Omega_{1})$, then $u(z)\leq t$ and if $w\in
A_{t}\setminus\Omega_{1}$ then $u(w)>t$.

Thus

$\begin{split}\int_{\Omega}u(z)dxdy&=\int_{\Omega_{1}}u(z)dxdy+\int_{\Omega\setminus\Omega_{1}}u(z)dxdy\\\
&\leq\int_{\Omega_{1}}u(z)dxdy+s\int_{\Omega\setminus\Omega_{1}}dxdy\\\
&=\int_{\Omega_{1}}u(z)dxdy+s\int_{A_{t}\setminus\Omega_{1}}dxdy\\\
&\leq\int_{\Omega_{1}}u(z)dxdy+\int_{A_{t}\setminus\Omega_{1}}u(z)dxdy\\\
&=\int_{A_{t}}u(z)dxdy.\end{split}$

∎

###  2.1. Proof Theorem 1.6 for $n=1$.

We need the following general global estimate.

######  Lemma 2.3.

Assume that $\alpha>0$ and that $f\in F_{\alpha}^{2}$. Then

$\frac{1}{\pi}\int_{\mathbb{C}}\frac{|f^{\prime}(z)|^{2}e^{-\alpha|z|^{2}}}{1+\alpha|z|^{2}}dxdy\leq\|f\|_{2,\alpha}^{2}.$

###### Proof.

Since $f$ is an entire holomorphic functions, it has a Taylor representation:

$f(w)=\sum_{k=0}^{\infty}a_{k}w^{k}.$

Then

$f^{\prime}(w)=\sum_{k=1}^{\infty}ka_{k}w^{k-1}.$

Thus

$\|f\|_{2,\alpha}^{2}=\sum_{k=0}^{\infty}|a_{k}|^{2}\frac{k!}{\alpha^{k}}.$

Moreover

$\begin{split}\frac{\alpha}{\pi}\int_{\mathbb{C}}\frac{|f^{\prime}(z)|^{2}e^{-\alpha|z|^{2}}}{\alpha(1+\alpha|z|^{2})}dxdy&=2\alpha\sum_{k=1}^{\infty}k^{2}|a_{k}|^{2}\int_{0}^{\infty}\frac{r^{2(k-1)+1}e^{-\alpha
r^{2}}}{\alpha(1+\alpha r^{2})}dr\\\
&=\sum_{k=1}^{\infty}k^{2}|a_{k}|^{2}eE_{k}\frac{\Gamma(k)}{\alpha^{k}}\\\
&=\sum_{k=0}^{\infty}g_{k}|a_{k}|^{2}\frac{k!}{\alpha^{k}},\end{split}$

where

$E_{k}=E_{k}(1)=\int_{1}^{\infty}\frac{e^{-t}}{t^{k}}dt,$

and $g_{k}=keE_{k}.$ We need the following
Claim. _For every $k\geqslant 1$ we have $g_{k}<1$. Moreover_

(2.1) $\lim_{k\to\infty}g(k)=1.$

By using partial integration we get

$g_{k+1}=\frac{k+1}{k^{2}}(k-g_{k}).$

Further $1/2\leq g_{1}=0.596347<1$. Now we prove by induction that

$\frac{k}{k+1}<g_{k}<1$

for $k\geqslant 2$. In order to do so let

$h(y):=\frac{k+1}{k^{2}}(k-y).$

Then it is clear $h(\frac{k}{k+1})=1$ and

$h(1)=\frac{k^{2}-1}{k^{2}}\geqslant\frac{k+1}{k+2}.$

Thus the claim follows.

Since $ekE_{k}<1$, we obtain that

$\frac{1}{\pi}\int_{\mathbb{C}}\frac{|f^{\prime}(z)|^{2}e^{-\alpha|z|^{2}}}{1+\alpha|z|^{2}}dxdy\leq\|f\|_{2,\alpha}^{2},$

as claimed.

∎

###### Proof of Theorem 1.6 for $n=1$..

Let

$u(z)=\frac{|f^{\prime}(z)|^{2}e^{-\alpha|z|^{2}}}{\alpha(1+\alpha|z|^{2})}$

and define $\mu(t)=|A_{t}|.$ Then by Corollary 2.1, $A_{t}=\\{z:u(z)>t\\}$ is
compactly supported in $\mathbb{C}$ and has a smooth boundary $\partial
A_{t}=\\{z:u(z)=t\\}$ for almost every $t\in(0,\max u)$. Also
$\lim_{|z|\to\infty}u(z)=0$ uniformly provided that $f\in
L^{2}(\mathbf{C},d\mu_{\alpha})$, where

$d\mu_{\alpha}=\frac{\alpha}{\pi}e^{-\alpha|z|^{2}}dxdy.$

Let, $s=\mu(t)$ and define

$I(s)=\frac{\alpha}{\pi}\int_{A_{t}}u(z)dxdy.$

Then as in [8, Lemma 3.4] we can prove that $I$ is smooth and

$I^{\prime}(s)=\mu^{-1}(s).$

In view of Lemma 2.2, we need to prove that

$I(s)\leq(1-e^{-2|A_{t}|})\|f\|_{2,\alpha}.$

Assume as we may that $\|f\|_{2,\alpha}=1$. Further we have $|A_{t}|=s$.
Therefore we need to show that

$I(s)\leq(1-e^{-2s}).$

Then by using the Cauchy inequality to the length of $\partial A_{t}$ we
obtain

$|\partial A_{t}|^{2}\leq\int_{\partial A_{t}}|\nabla
u|^{-1}ds\cdot\int_{\partial A_{t}}|\nabla u|ds.$

Further by [8, Lemma 3.2] we have

$\int_{\partial A_{t}}|\nabla u|^{-1}ds=-\mu^{\prime}(t).$

Let $\nu$ be the out-pointing unit normal at the smooth curve $\partial
A_{t}$. Note that this curve is smooth for almost every $t>0$. This can be
shown by using a similar approach as in [8]. Now

$|\nabla u|=-\left<\nabla u,\nu\right>=-t\left<\nabla\log u,\nu\right>,$

for $z\in\partial A_{t}=\\{z:u(z)=t\\}$. Thus

$\int_{\partial A_{t}}|\nabla u|ds=-t\int_{\partial A_{t}}\left<\nabla\log
u,\nu\right>ds.$

By Green theorem we have

$\int_{\partial A_{t}}\left<\nabla\log u,\nu\right>ds=\int_{A_{t}}\Delta\log
udA(z)$

Since

(2.2) $\Delta\log
u=-4\alpha\left(1+\frac{1}{\left(1+\alpha|z|^{2}\right)^{2}}\right),$

because $f^{\prime}(z)$ is nonvanishing holomorphic function on $A_{t}$, we
obtain

$\begin{split}\int_{\partial A_{t}}\left<\nabla\log
u,\nu\right>ds&=4t\alpha\int_{A_{t}}\left(1+\frac{1}{\left(1+\alpha|z|^{2}\right)^{2}}\right)dA(z)\\\
&=4t\pi\int_{A_{t}}\left(1+\frac{1}{\left(1+\alpha|z|^{2}\right)^{2}}\right)dA_{\alpha}(z)\\\
&\leq 8t\pi|A_{t}|.\end{split}$

Now we use the isoperimetric inequality for the plane:

$4\pi|A_{t}|\leq|\partial A_{t}|^{2},$

and obtain

$4\pi\mu(t)\leq(-\mu^{\prime}(t))(8t\pi(\mu(t))).$

The last inequality implies the inequality:

$1+2\frac{q(s)}{q^{\prime}(s)}\leq 0,$

where $q(s)=\mu^{-1}(s)$. Because $q^{\prime}(s)<0$, we obtain
$q^{\prime}(s)+2q(s)\geqslant 0$. Thus the function

$Q(s)=q(s)e^{2s}$

is increasing, because

$Q^{\prime}(s)=e^{2s}(2q(s)+q^{\prime}(s))\geqslant 0.$

Therefore

$G(\sigma)=I(-1/2\log\sigma)$

is a convex function. Namely

$G^{\prime}(\sigma)=-\frac{\mu^{-1}(-1/2\log\sigma)}{2\sigma}=-e^{2s}q(s)=-Q(s),$

where $s=e^{-2\sigma}$. So

$G^{\prime\prime}(\sigma)=2Q^{\prime}(s)e^{-2\sigma}>0.$

Moreover $G$ satisfies the conditions $G(0)\leq 1$ and $G(1)=0$. The first
condition is satisfied because of Lemma 2.3. Namely

$G(0)=I(+\infty)=\int_{A_{0}}u(z)dA_{\alpha}\leq 1.$

On the other hand

$G(1)=I(0)=\int_{A_{\infty}}u(z)dA_{\alpha}=0.$

By using the convexity of $G$ we have $G(\sigma)\leq(1-\sigma)$ and this
implies the inequality

(2.3) $I(s)\leq(1-e^{-2s}).$

To prove the sharpness, let $f(z)=z^{m}$. Then in view of (2.1) we have

$\lim_{R\to\infty,m\to\infty}J(R,m)=1,$

where

$J(R,m)=\frac{\int_{|z|<R}\frac{|f^{\prime}(z)|^{2}e^{-\alpha|z|^{2}}}{\alpha(1+\alpha|z|^{2})}dxdy}{(1-e^{-2\pi
R^{2}})\|f\|^{2}_{2,\alpha}}.$

∎

###  2.2. The proof of Theorem 1.6 for $n\geqslant 2$

The proof is just an imitation of the case $n=1$, so we omit some details. We
begin by the following lemma

######  Lemma 2.4.

Let $L(x,y)=\log F[1+n,1,\alpha(x^{2}+y^{2})]$, where $F$ is Kummer function.
Then

$\Delta L(x,y)\leq 4\alpha(1+n).$

###### Proof.

First of all, we have $\Delta L(x,y)=g(t)$ where $t=\left(x^{2}+y^{2}\right)$
and

$\begin{split}&\frac{g(t)}{2(1+n)\alpha}F\left[1+n,1,\alpha
t\right]^{2}=-2(1+n)\alpha tF\left[2+n,2,\alpha t\right]^{2}\\\
&+F\left[1+n,1,\alpha t\right]\left(2F\left[2+n,2,\alpha t\right]+(2+n)\alpha
tF\left[3+n,3,\alpha t\right]\right).\end{split}$

We need to prove that $g(t)\leq g(0)=4\alpha(1+n)$. It is equivalent with the
following

$k(t)=g(t)F\left[1+n,1,\alpha t\right]^{2}-4\alpha(1+n)F\left[1+n,1,\alpha
t\right]^{2}\leq 0.$

Then after some straightforward computations we get

$k(t)=-4\alpha^{2}ntF[1+n,2,\alpha t](F[1+n,1,\alpha t]+nF[1+n,2,\alpha t])$

which is certainly a non-positive function. This finishes the proof of the
lemma. ∎

Now define

$J(s)=\frac{\alpha}{\pi}\int_{A_{t}}u_{n}(z)dxdy,$

where

$u_{n}=\frac{|f^{(n)}(z)|^{2}e^{-\alpha|z|^{2}}}{\alpha^{n}n!L_{n}(-\alpha|z|^{2})}.$

Now instead of (2.2), we use $-t\Delta u\leq 4t\alpha(1+n)$, and instead of
(2.3) we have

$J_{n}(s)\leq(1-e^{-ms}),$

provided that Conjecture 1.5 is correct. This finishes the proof of Theorem
1.6 for the case $n\geqslant 2$, up to Lemma 2.5 below. The sharpness part can
be proved in a similar way as in the case $n=1$, but using, in this case, the
equation (2.4) below.

######  Lemma 2.5.

For $f\in\mathcal{F}_{\alpha}^{2}$, we have the following sharp inequality

$I_{n}(f):=\frac{\alpha}{\pi}\int_{\mathbb{C}}\frac{|f^{(n)}(z)|^{2}e^{-\alpha|z|^{2}}}{\alpha^{n}n!L_{n}(-\alpha|z|^{2})}dxdy\leq\|f\|^{2}_{2,\alpha},$

provided that Conjecture 1.5 is true.

###### Proof.

Since

$f^{(n)}(w)=\sum_{k=n}^{\infty}\frac{\Gamma(k+1)}{\Gamma(k-n+1)}a_{k}w^{k-n},$

and because the polynomials $w^{n}$ and $w^{m}$ are orthogonal the Fock space
for $n\neq m$, it is clear that it is enough to prove the inequality for such
functions. So we need to show that

$I_{n}(w^{k})\leq\|w^{k}\|^{2}_{2,\alpha}.$

Let $k\geqslant n$ and

$a_{k}=\frac{I_{n}(w^{k})}{\|w^{k}\|^{2}_{2,\alpha}}.$

Then after straightforward computation we get

$I_{n}(w^{k})=\frac{a^{-k}\Gamma^{2}(1+k)}{n!\Gamma^{2}(1+k-n)}\int_{0}^{\infty}\frac{e^{-s}s^{k-n}}{L_{n}(-s)}ds$

and

$\|w^{k}\|^{2}_{2,\alpha}=\alpha^{-k}k!.$

Thus

$a_{k}=\frac{\Gamma[1+k]}{n!\Gamma[1+k-n]^{2}}\int_{0}^{\infty}\frac{e^{-r}r^{k-n}}{L_{n}(-r)}dr$

We need to show that

  1. (1)

$\lim_{k\to\infty}a_{k}=1$.

  2. (2)

$a_{k}<1$ and

We will prove the first assertion, while the second is proved for few small
integers. This is the content of the following lemma. ∎

######  Lemma 2.6.

Let $p\geqslant n$ and

$A(p)=\frac{\Gamma[1+p]}{n!\Gamma[1+p-n]^{2}}\int_{0}^{\infty}\frac{e^{-r}r^{p-n}}{L_{n}(-r)}dr,$

where $L_{n}(-r)$ is the Laguerre polynomial. Then

(2.4) $\lim_{p\to\infty}A(p)=1,$

and if $n\leq 4$, then $A(p)<1$.

###### Proof.

First of all we have

$L_{n}(-r)=\sum_{m=0}^{n}\frac{\Gamma(1+n)}{\Gamma(1+m)^{2}\Gamma(1+n-m)}r^{m}.$

Fix $k$ and let $p=k+n$. It follows from the definition of $A(h+k)$,
$h\in\\{0,\dots,n\\}$ that

(2.5)
$\sum_{h=0}^{n}\frac{A(h+k)\Gamma(1+h+k-n)^{2}}{\Gamma(1+h)^{2}\Gamma(1+h+k)\Gamma(1+n-h)}=\frac{\Gamma(1+k-n)}{(n!)^{2}}.$

In order to prove $\lim_{k\to\infty}A(k+n)=1$, observe that, in view of (2.5),
for every $n$ we have

(2.6) $A(k+n)=C(k,n)-\sum_{h=0}^{n-1}C_{h}(k,n)A(k+h),$

where

$C(k,n)=\frac{(k-n)!(k+n)!}{(k!)^{2}}$

and

$C_{h}(k,n)=\frac{((h+k-n)!)^{2}(n!)^{2}(k+n)!}{(h!)^{2}(k!)^{2}(h+k)!(-h+n)!}.$

Then we easily conclude that

$A(k+n)<C(k,n)\leq\frac{(2n)!}{(n!)^{2}}.$

So $A(k+n)$ is bounded for fixed $n$. The same hold for $A(k+h)$ for $h\leq
n$. Namely for $k\geqslant 2n$,

$A(k+h)\leq\frac{(2h)!}{(h!)^{2}}\leq\frac{(2n)!}{(n!)^{2}}.$

Further we have

$\lim_{k\to\infty}C(k,n)=1$

and

$\lim_{k\to\infty}C_{h}(k,n)=0.$

This and (2.6) implies that

$\lim_{k\to\infty}A(k+n)=1.$

Let us now prove $A(k)<1$ for first few integers $n$. For $n=2$ we have

$A(2+k)=\frac{(2+k)\left(-k+k^{3}-2(1+k)A(k)-4(-1+k)^{2}A(1+k)\right)}{(-1+k)^{2}k^{2}}.$

Let

$B(x,y)=\frac{(2+k)\left(-k+k^{3}-2(1+k)x-4(-1+k)^{2}y\right)}{(-1+k)^{2}k^{2}}.$

Further $A(2)=0.427393$, $A(3)=0.662691$, $A(4)=0.784876$, $A(5)=0.852841$,
$A(6)=0.893707$, $A(7)=0.919951$. So we get $A(j)<1$ for $j\leq 7$ and
$a(j)>j/(j+1)$ for $4\leq j\leq 7$. Now we use mathematical induction to prove
that $\frac{m}{m+1}\leq A(m)<1$ for $m>7$. Let $m=k>7$ and assume that
$\frac{m}{m+1}\leq A(m)<1$ and $\frac{m+1}{m+2}\leq A(m+1)<1$. Then
$A(m+2)\geqslant B(1,1)$, and $B(1,1)\geqslant\frac{m+2}{m+2+1}$ iff

$(-1+m)^{2}m^{2}(2+m)(3+m)(-18+m(9+(-8+m)m))\geqslant 0,$

and the last inequality is true for every $m\geqslant 8$. Moreover

$\begin{split}A(m+2)&\leq B(m/(m+1),(m+1)/(m+2))\\\
&=\frac{(2+m)\left(-3m+m^{3}-\frac{4(-1+m)^{2}(1+m)}{2+m}\right)}{(-1+m)^{2}m^{2}}<1,\end{split}$

because the last inequality is equivalent with

$(-1+m)^{2}m^{2}(2+m)>0.$

For $n=3$, we have

$\begin{split}A(3+k)&=\frac{(1+k)(2+k)(3+k)}{(-2+k)(-1+k)k}-\frac{(1+k)(2+k)(3+k)}{(-2+k)(-1+k)k}\\\
&\times\bigg{(}\frac{6A(k)}{(-2+k)(-1+k)k}\\\
&+\frac{18(-2+k)A(1+k)}{(-1+k)k(1+k)}+\frac{9(-2+k)(-1+k)A(2+k)}{k(1+k)(2+k)}\bigg{)}.\end{split}$

By proceeding similarly as above, we can prove that $\frac{k}{k+1}\leq A(k)<1$
for $k\geqslant 14$, and that $A(k)<1$ for $k\in\\{1,\dots,13\\}$.

For $n=4$, we have

(2.7) $\begin{split}A(4+k)&=X-\bigg{(}\frac{24XA(k)}{(-3+k)(-2+k)(-1+k)k}\\\
&+\frac{96X(-3+k)A(1+k)}{(-2+k)(-1+k)k(1+k)}+\frac{72X(-3+k)(-2+k)A(2+k)}{(-1+k)k(1+k)(2+k)}\\\
&+\frac{16X(-3+k)(-2+k)(-1+k)A(3+k)}{k(1+k)(2+k)(3+k)}\bigg{)},\end{split}$

where

$X=\frac{(1+k)(2+k)(3+k)(4+k)}{(-3+k)(-2+k)(-1+k)k}.$

Then, by using Mathematica software we obtain $A(j)<j/(j+1)$, $j=1,\dots,16$
but $A(j)>j/(j+1)$ for $j=17,\dots,27$. By induction, and previous formula we
obtain that $j/(j+1)<A(j)<1$, for $j>27$. Namely, LHS of (2.7), for
$A(j)\equiv 1$ is equal to
$\frac{(4+k)(-13680+k(11316+k(-5464+k(1921+k(-520+k(106+(-16+k)k))))))}{(-3+k)^{2}(-2+k)^{2}(-1+k)^{2}k^{2}}$
and the last expression is bigger than $(k+4)/(k+5)$ if and only if
$k\geqslant 28$. Further the LHS of (2.7), for $A(j)=j/(j+1)$ is equal to
$\frac{-32832+k(14544+k(-2916+k(-180+k(433+k(-216+k(58+(-12+k)k))))))}{(-3+k)^{2}(-2+k)^{2}(-1+k)^{2}k^{2}}$
and the last expression is smaller than $1$ for every $k$.

∎

######  Remark 2.7.

Similar to the proof of the Lemma 2.5, one can proceed to prove the statement
for some constants greater than 4, but this still does not solve the general
problem. We believe that the recurrent formula (2.6) is suitable for proving
the general case, and we also believe that (2.6) implies $j/(j+1)<A(j)<1$ for
$j>n^{2}$. This and some good estimations of $A(j)$ for $j\leq n^{2}$ would
conclude the proof of the general case, but we fail to do it.

### Acknowledgments

I would like to thank Dr. P. Melentijević for helpful comments regarding this
paper.

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