# Static boson stars in the Einstein-Friedberg-Lee-Sirlin theory and their
astrophysical images

Pedro L. Brito de Sá<EMAIL_ADDRESS>Programa de Pós-Graduação em
Física, Universidade Federal do Pará, 66075-110, Belém, Pará, Brazil.  Haroldo
C. D. Lima Jr<EMAIL_ADDRESS>Programa de Pós-Graduação em Física,
Universidade Federal do Pará, 66075-110, Belém, Pará, Brazil.  Departamento de
Física - CCET, Universidade Federal do Maranhão, Campus Universitário do
Bacanga, 65080-805, São Luís, Maranhão, Brazil.  Carlos A. R. Herdeiro
<EMAIL_ADDRESS>Departamento de Matemática da Universidade de Aveiro and Centre
for Research and Development in Mathematics and Applications (CIDMA), Campus
de Santiago, 3810-183 Aveiro, Portugal.  Luís C. B. Crispino<EMAIL_ADDRESS>Programa de Pós-Graduação em Física, Universidade Federal do Pará, 66075-110,
Belém, Pará, Brazil.

(June 2024)

###### Abstract

We investigate the static boson star solutions in the so-called Einstein-
Friedberg-Lee-Sirlin (E-FLS) theory, performing a complete analysis of the
solution space in this model. We study the phenomenological aspects of E-FLS
stars, for instance, by investigating the timelike and null geodesics with an
emphasis on the analysis of circular timelike orbits and light rings. In order
to study the astrophysical signatures of such stars, their images were
obtained considering them surrounded by a geometrically thin accretion disk.
Our results comprise two different models of accretion disks, namely the
optically thin and thick disk models. We present a selection of our findings
for the astrophysical images of E-FLS stars and discuss their relevance as a
possible black hole mimicker.

##  I Introduction

The last decades have represented a golden age concerning observational data
on the strong field regime for the theory of General Relativity (GR). The
detection of gravitational waves (GW) by the collaboration LIGO/VIRGO LIGO-
VIRGO2016  inaugurated the era of GW astronomy, i.e. the exploration of the
cosmos through a new type of radiation other than the electromagnetic one. Up
to date, several events have been observed through GWs, such as, the merger of
black hole-black hole and black hole-neutron star systems LIGO-VIRGO . In
2019, the Event Horizon Telescope (EHT) collaboration imaged, for the first
time ever, the structure close to the central regions of both the M87 and
Milky Way galaxies M87_1:2019 ; M87_2:2019 ; M87_3:2019 ; M87_4:2019 ;
M87_5:2019 ; M87_6:2019 ; sgra_1:2022 ; sgra_2:2022 ; sgra_3:2022 ;
sgra_4:2022 ; sgra_5:2022 ; sgra_6:2022 , using Very Long Baseline
Interferometry (VLBI) techniques. The images present a inner dark region which
is compatible to the shadow of a supermassive compact object.

In the near future, additional GW detectors will become operational, improving
the precision of GW astronomy. Moreover the next generation Event Horizon
Telescope (ng-EHT) is planed to increase the current baselines in order to
capture images with a higher level of details. The forthcoming data on the
gravitational and electromagnetic channels can be used to constrain models
that challenge the universality of the Kerr black hole (BH) hypothesis
Kerr_Hypothesis . The departure from the Kerr hypothesis may arise from either
(i) alternative black hole models, in particular, described by additional
parameters beyond mass and angular momentum within the Kerr metric, or (ii)
compact objects lacking a horizon yet exhibiting characteristics that closely
resemble those of BHs.

Among such horizonless BH mimickers, non-topological solitonic solutions known
as boson stars stand as robust viable candidates. Some boson stars are
linearly stable in regions of their parameter space LeePang ; Santos:2024vdm .
Additionally, they admit a robust formation mechanism dubbed as gravitational
cooling grav_cool1 ; grav_cool2 ; grav_cool3 . The pioneering work on boson
star solutions dates back to 1968 with the studies of Kaup Kaup , where he
obtained the boson star solutions in the context of GR in the presence of a
complex scalar field with a potential without self interaction terms, known
nowadays as mini-boson stars. After that, boson star solutions for different
sorts of potential were studied in the literature. Additionally, analogous
compact objects in GR composed of massive vector fields have been discovered
more recently and they are known as Proca stars Proca1 ; Proca2 .

In 1976, a pioneer work showed that solitons in flat spacetime can emerge as
solutions of a system composed of a complex scalar field coupled with a real
scalar field, known as Friedberg-Lee-Sirlin (FLS) model FLS . Remarkably, the
FLS model provide a simple example of a renormalizable theory. The masses of
the complex and real scalar fields are generated due to the coupling between
them. The FLS model minimally coupled to the Einstein gravity, dubbed as E-FLS
theory, has been studied in different works. In Ref. Rotating_FLS , the
rotating boson stars and the Kerr BHs with synchronized scalar hair were
studied in the context of FLS model. The non-rotating gauged boson stars and
gauged Q-balls were obtained in Ref. U1_FLS . Furthermore, a vector version of
the FLS model was considered in Ref Proca-Higgs , motivated by the need to
avoid hyperbolicity issues commonly encountered in the standard Proca models.

Although the rotating and gauged solutions in the E-FLS theory have been
studied, a detailed analysis of the space of solutions in the non-rotating and
ungauged case has not been presented in the literature, as far as we are
aware. Additionally, the study of the phenomenology of such static and
ungauged E-FLS stars is of uttermost importance and has not been done so far.
In this paper we provide a complete analysis of the solution space in the
static case, exploring the full range of parameter within the E-FLS model.
Moreover, we also study the timelike and lightlike geodesics around E-FLS
stars. Finally, we obtain the astrophysical images of such E-FLS stars when
surrounded by different accretion disk models, which are important in the
light of the recent EHT results.

This paper is organized as follows. In Sec. II, we present the FLS model
minimally coupled to gravity, the field equations and the appropriate boundary
conditions. In Sec. III, we discuss the numerical method and present a
selection of our numerical results for boson star solutions. In Sec. IV we
analyze the timelike and lightlike geodesics around such stars, obtaining, for
instance, the circular timelike orbits and the light rings. In Sec. V, we
apply the backwards ray-tracing method in order to obtain the astrophysical
images of these stars, when surrounded by an accretion disk. We present our
final remarks in Sec. VI. In the remainder of this paper, we adopt geometrical
units such that $c=\hbar=1$ and the metric signature is $+2$.

##  II The E-FLS theory

We explore solitonic solutions in (3+1)-dimensions with two scalar fields (a
complex field and a real field), minimally coupled to gravity, whose
interaction is governed by the FLS model FLS . The E-FLS theory studied here
is defined by the following action:

$\displaystyle S=\int
d^{4}x\sqrt{-g}\left(\frac{R}{2\kappa^{2}}-\mathcal{L}_{m}\right),\qquad\kappa^{2}\equiv
8\pi G,$ (1)

where $R$ is the Ricci scalar, $g$ is the metric determinant, and

$\displaystyle\mathcal{L}_{m}=\frac{1}{2}\nabla_{\mu}\psi\,\nabla^{\mu}\psi+\nabla_{\mu}\Phi\,\nabla^{\mu}\Phi^{*}+U(\psi,\Phi),$
(2) $\displaystyle
U(\psi,\Phi)=m^{2}\psi^{2}\left|\Phi\right|^{2}+\mu^{2}\left(\psi^{2}-v^{2}\right)^{2},$
(3)

being $\psi$ a self-interacting real scalar field and $\Phi$ a complex scalar
field. The real and complex scalar fields are coupled through the coupling
constants $m$ and $\mu$. The constant $v$ is the vacuum expectation value of
the real scalar field. The presence of a coupling term in the potential
$U(\psi,\Phi)$ generates the effective masses for the real and complex scalar
fields. By expanding $U(\psi,\Phi)$ around the minimum $\psi=v$ and $\Phi=0$,
we can compute the masses of the scalar fields to be given by
$m_{\psi}=\sqrt{8}\mu\,v$ and $m_{\phi}=m\,v$. We observe that as
$\mu\rightarrow\infty$, the real scalar field becomes infinitely massive and
decouples from the complex scalar field. In this limit, the real scalar field
tends to $\psi\rightarrow v$, and the E-FLS theory recovers the Einstein-
Klein-Gordon model Kaup . On the other hand, as $\mu\rightarrow 0$ the real
scalar field becomes massless. The E-FLS star solutions in this limit are the
ones that differ most from the mini-boson stars.

The E-FLS field equations can be obtained by varying the action (1) with
respect to the metric $g_{\mu\nu}$ and the scalar fields $\psi$ and $\Phi$.
The variation with respect to the metric leads to the Einstein’s equation:

$\displaystyle R_{\mu\nu}-\frac{1}{2}R\,g_{\mu\nu}=\kappa^{2}T_{\mu\nu},$ (4)

where the corresponding stress-energy tensor associated to the scalar fields
reads

$\displaystyle
T_{\mu\nu}=\nabla_{\mu}\Phi\nabla_{\nu}\Phi^{*}+\nabla_{\mu}\Phi^{*}\nabla_{\nu}\Phi+\nabla_{\mu}\psi\nabla_{\nu}\psi-\mathcal{L}_{m}\,g_{\mu\nu}.$
(5)

On the other hand, by varying the action (1) with respect to the real and
complex scalar fields, we obtain, respectively:

$\displaystyle\square\Phi=m^{2}\,\psi^{2}\Phi,$ (6)
$\displaystyle\square\psi=2\psi\left(m^{2}\left|\Phi\right|^{2}+2\mu^{2}\psi^{2}-2\,v^{2}\,\mu^{2}\right),$
(7)

where $\square\equiv\nabla^{\mu}\,\nabla_{\mu}$ is the d’Alembertian operator.
We notice that the parameters $v$ and $m$ on Eqs. (4)-(7) can be eliminated by
performing the following rescaling:

$\displaystyle
r=\frac{\tilde{r}}{mv},\quad\psi=v\tilde{\psi},\quad\phi=v\tilde{\phi},\quad\omega=mv\tilde{\omega},$
$\displaystyle\kappa=\frac{\tilde{\kappa}}{v},\quad\mu=m\tilde{\mu},$ (8)

and the rescaled field equations are equivalent to set $v=m=1$ in Eqs.
(4)-(7). In the remainder of this paper, we work with the rescaled field
equation, omitting the tildes for the sake of simplicity.

The E-FLS action (1) possesses a global invariance under a $U(1)$
transformation of the complex scalar field ($\Phi\rightarrow
e^{i\alpha}\Phi$), where $\alpha$ is a constant phase. Hence, by the Noether
theorem, a conserved current associated with such symmetry exists and can be
expressed as:

$\displaystyle
j_{\mu}=i\left(\Phi\,\nabla_{\mu}\Phi^{*}-\Phi^{*}\,\nabla_{\mu}\Phi\right),$
(9)

such that it has null divergence $\nabla_{\mu}j^{\mu}=0$. The integration of
the timelike component of the conserved current over a spacelike hypersurface
$\Sigma$ yields a conserved quantity known as the Noether charge, namely

$\displaystyle N\equiv\frac{1}{4\pi}\int_{\Sigma}j^{0}\sqrt{-g}d^{3}x,$ (10)

which is related to the number of scalar particles that constitute the system.

Let us now explore the solitonic solutions in the context of the E-FLS theory.
We consider spherically symmetric solutions to the field equations (4)-(7),
and adopt the following ansatz for the metric tensor in Schwarzschild-like
coordinates:

$\displaystyle
ds^{2}=-e^{\Gamma(r)}dt^{2}+e^{\Lambda(r)}dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta\,d\varphi^{2}\right),$
(11)

where $\Gamma(r)$ and $\Lambda(r)$ are two real metric functions that depend
solely on the radial coordinate. The corresponding ansatzes for the
spherically symmetric real and complex scalar fields are

$\displaystyle\psi=\psi(r),$ (12) $\displaystyle\Phi=\phi(r)e^{i\omega\,t},$
(13)

respectively, where $\omega>0$ is a real parameter associated with the
oscillation frequency of the complex scalar field. Inserting the ansatzes
(11)-(13) into the E-FLS field equations (4)-(7), we obtain the following set
of coupled ordinary differential equations (ODEs):

$\displaystyle\frac{e^{-\Lambda}\,\Lambda^{\prime}}{r}+\frac{\left(1-e^{-\Lambda}\right)}{r^{2}}=\kappa^{2}\left[\omega^{2}\,e^{-\Gamma}\phi^{2}\right.$
$\displaystyle\left.+e^{-\Lambda}\phi^{\prime
2}+\frac{e^{-\Lambda}\,\psi^{\prime
2}}{2}+\psi^{2}\phi^{2}+\mu^{2}\left(\psi^{2}-1\right)^{2}\right],$ (14)
$\displaystyle\frac{e^{-\Lambda}\,\Gamma^{\prime}}{r}-\frac{\left(1-e^{-\Lambda}\right)}{r^{2}}=\kappa^{2}\left[-\omega^{2}\,e^{-\Gamma}\phi^{2}\right.$
$\displaystyle\left.-e^{-\Lambda}\phi^{\prime
2}-\frac{e^{-\Lambda}\,\psi^{\prime
2}}{2}-\psi^{2}\phi^{2}+\mu^{2}\left(\psi^{2}-1\right)^{2}\right],$ (15)
$\displaystyle\phi^{\prime\prime}+\left(\frac{2}{r}+\frac{\Gamma^{\prime}-\Lambda^{\prime}}{2}\right)\phi^{\prime}+e^{\Lambda}\left(\omega^{2}e^{-\Gamma}-\psi^{2}\right)\phi=0,$
(16)
$\displaystyle\psi^{\prime\prime}+\left(\frac{2}{r}+\frac{\Gamma^{\prime}-\Lambda^{\prime}}{2}\right)\psi^{\prime}+4\mu^{2}e^{\Lambda}\psi^{3}$
$\displaystyle-2\,\left(2\,\mu^{2}+\phi^{2}\right)e^{\Lambda}\psi=0,$ (17)

where the primes denote differentiation with respect to the radial coordinate
$r$.

In order to obtain boson star solutions in the E-FLS theory, we must impose
appropriate boundary conditions for Eqs. (14)-(17), as we outline now. We
study configurations that are both regular at the origin $(r=0)$ and
asymptotically flat far away from the center ($r=\infty$). The condition of
regularity of the E-FLS field equations at the origin implies that

$\displaystyle\Lambda^{\prime}(r=0)=0,\quad\phi^{\prime}(r=0)=0,\quad\psi^{\prime}(r=0)=0.$
(18)

Moreover, the asymptotically flatness condition leads to the following
boundary conditions:

$\displaystyle\Gamma(r=\infty)=0,\quad\phi(r=\infty)=0,\quad\psi(r=\infty)=1.$
(19)

For a given value of the scalar field at the origin $\phi(0)\equiv\phi_{0}$,
the system of equations (14)-(17), subjected to the boundary conditions
(18)-(19), defines an eigenvalue problem for the frequency $\omega$. For a
given $\phi_{0}$, there exists an infinite countable set of eigenfrequencies
$\\{\omega_{n}\\}$ that solves the eigenvalue problem described above. The
integer $n$ labels the number of nodes of the complex scalar field. The
excited states, characterized by $n\geq 1$, are generically unstable LeePang .
Therefore, we study only the solutions with $n=0$. In the next section, we
describe the numerical method used to compute the nodeless solutions and
present a selection of our numerical results.

##  III Results

###  III.1 The numerical method

In order to solve numerically the two-point boundary value problem described
in Sec. II, we used the FORTRAN package entitled COLSYS (Collocation for
Systems) originally proposed in Ref. colsys . The COLSYS package employs a
spline collocation method at Gaussian points for solving boundary value
problems associated with ODEs. The numerical solutions can be obtained by
providing an initial guess which is close to the actual solution of the
problem. Starting with large values of $\mu$, we use the uncharged mini-boson
star solution as our initial guess, since the E-FLS model recovers the EKG
model for $\mu\rightarrow\infty$. The mini-boson star initial guess was
constructed using a standard shooting method Cpp . Once we obtain the solution
in the large $\mu$ limit, we gradually decrease the value of this parameter
until $\mu=0$. Since the field equations are invariant under the
transformation $\mu\rightarrow-\mu$, it is not necessary to compute numerical
solutions for negative values of $\mu$. Thus, we were able to obtain a full
scan of the solution space within the E-FLS theory. In this paper, we employed
our numerical method using a grid of $2000$ points and a relative accuracy of
the order $10^{-11}$.

###  III.2 E-FLS stars

Let us now present a selection of our numerical results for boson star
solution in the E-FLS theory, termed as E-FLS stars in the remainder of this
work, for simplicity. We computed the numerical solutions for several values
of the parameters $\phi_{0}$ and $\mu$, in order to fully describe the
solution space in this theory. As an example, we show in Fig. 1, the numerical
solutions for the metric functions $e^{\Gamma(r)}$ and $e^{\Lambda(r)}$, the
complex scalar field $\phi(r)$ and the real scalar field $\psi(r)$ in terms of
the radial coordinate. In Fig. 1, we fixed the value $\mu=0.2$, and varied the
value of complex scalar field at the origin. For small values of $\phi_{0}$,
the solutions approach the Newtonian regime, where the star solutions are less
compact, and $\omega\approx 1$. As we increase the value of $\phi_{0}$, the
stars become more compact, and relativistic effects tend to dominate the
solution, for instance, light rings can emerge in pairs Cunha2017  (see Sec.
IV.2). We notice that the E-FLS stars exist in a range of frequency
$\omega_{\min}\leq\omega\leq\omega_{\rm max}$, where the upper bound is the
Newtonian limit $\omega_{\rm max}=1$ and the lower bound $\omega_{\min}$
depends on the specific value of the FLS parameter $\mu$.

Figure 1:  The numerical solutions for E-FLS stars as a function of the radial
coordinate. In the top left panel, we display the metric function
$e^{\Gamma(r)}$, while in the top right panel we show the metric function
$e^{\Lambda(r)}$. In the bottom left and right panels, we show the complex
scalar field and real scalar field, respectively. In this figure, we have
chosen ${\mu}=0.2$ and displayed three distinct solutions for E-FLS stars with
${\phi}_{0}=(0.25,0.5,0.75)$.

For a spherically symmetric spacetime, as it is in our case, we can define the
mass function $\mathcal{M}(r)$, as given by

$\mathcal{M}(r)=\frac{r}{2}\left(1-e^{-\Lambda(r)}\right).$ (20)

We compute the ADM mass of the E-FLS stars by taking the limit
$M\equiv\mathcal{M}(r\rightarrow\infty)$. In Fig. 2, we show the ADM mass for
E-FLS stars as a function of the oscillation frequency. In the left panel of
Fig. 2, we fix the value ${\mu}=0.2$ and show the value of $\phi_{0}$ as a
color map, with the blue (red) color representing the lowest (highest) value.
We notice that for small values of $\phi_{0}$, the Newtonian limit is
approached, being the oscillation frequency close to 1. As we increase the
value of the complex scalar field at the origin, the E-FLS star becomes more
massive, reaching a maximum value at ${\omega}\approx 0.83$. In the right
panel of Fig. 2, we show the ADM mass as a function of the oscillation
frequency for different values of $\mu$. It can be noticed that the maximum
mass depends on the FLS parameter $\mu$. There are essentially two distinct
cases: (i) For $\mu\neq 0$, the maximum mass decreases as we decrease the
value of $\mu$, being the mini-boson star ($\mu\rightarrow\infty$) case the
highest maximum mass. (ii) For $\mu=0$, we notice that the maximum mass
increases in comparison to low values of $\mu\neq 0$. Moreover, for the case
$\mu=0$, the minimum frequency $\omega_{\min}$ is smaller, in comparison to
the case $\mu\neq 0$. In order to explain the distinct behavior for $\mu=0$,
we recall that the mass of the real scalar field, arising due to the
interaction with the complex scalar field, is given by
$m_{\psi}=\sqrt{8}\mu\,v$. Hence, for $\mu=0$, the real scalar field becomes
massless. In Refs. Rubakov2011 ; Loiko2018 , it was proved that for $\mu\neq
0$ the real scalar field approaches the asymptotic value as $\psi\sim
1-e^{-\sqrt{\mu^{2}-\omega^{2}}r}$, while for $\mu=0$ the real massless scalar
field decays asymptotically as

$\displaystyle\psi\sim 1-\frac{C}{r}+\mathcal{O}(r^{-2}),$ (21)

which is akin to the Coulomb asymptotic decay. Therefore, the interaction is
long-ranged in the vanishing $\mu$ limit, what explains the distinct behavior
in this limit.

In general, the maximum mass solution in a mass-frequency diagram marks the
onset of the instability regime LeePang . The range of solutions, spanning
from the Newtonian limit to the maximum mass solution, constitutes the stable
branch. A remarkable characteristic of the E-FLS stars is that the family of
solutions with $\mu=0$ exhibits a significantly larger stable branch, in
comparison to the $\mu\neq 0$ case. Hence, the E-FLS stars are more prone to
be stable, as we vary the oscillation frequency, in comparison to the mini-
boson stars.

(a)  (b)

Figure 2: Left panel: The total mass $M$ for E-FLS stars, with ${\mu}=0.2$, as
a function of the normalized frequency ${\omega}$. Each point in this panel
represents an E-FLS star solution with a given $\phi_{0}$. The values of
$\phi_{0}$ are depicted using a color map, with the blue color indicating the
lowest value and the red color indicating the highest value. Right panel: The
total mass $M$ for E-FLS stars, with different values of $\mu$, as a function
of the normalized frequency ${\omega}$.  Figure 3: The Noether charge for
E-FLS stars as a function of the normalized frequency ${\omega}$, for distinct
values of the FLS parameter $\mu$.

We also computed the Noether charge, given in Eq. (10) for the family of E-FLS
star solutions. In Fig. 3, we present the Noether charge as a function of the
oscillation frequency, for different values of the FLS parameter $\mu$. As we
decrease the FLS parameter, the maximum value for the Noether charge also
decreases if $\mu\neq 0$. For $\mu=0$, the maximum value for the Noether
charge increases in comparison to the case $\mu\neq 0$, similarly to the
behavior observed for the ADM mass. The relation between the ADM mass and the
Noether charge can provide insights into the stability properties of the E-FLS
stars. The binding energy of the star is defined as

$\displaystyle E_{b}=M-m\,N,$ (22)

therefore boson star solutions with negative binding energy ($M<m\,N$) are
gravitationally bound, while solutions with positive binding energy ($M>m\,N$)
are gravitationally unbound. In Fig. 4, we show the comparison between the
total ADM mass and number of particles as a function of ${\omega}$ and for
different values of the FLS parameter. The region with gravitationally bound
configuration is centered around the maximum values of the Noether charge and
the ADM mass. For high values of ${\phi}_{0}$, the Noether charge becomes
smaller than the total ADM mass, hence the solutions become gravitationally
unbound. Decreasing the FLS parameter $\mu$ leads to an increasing of the
region associated with gravitationally bound configurations. In particular,
the case of $\mu=0$ exhibits the largest region with gravitationally bound
configurations, as can be seen in panel (a) of Fig. 4.

Besides the ADM mass and the Noether charge, we may also compute an effective
radius for the E-FLS stars. In contrast to perfect-fluid stars in GR, boson
stars do not posses a well-defined radius since the scalar field is spread up
to the spatial infinity. However, for large values of $r$, the scalar field
decays exponentially, allowing the definition of an effective radius
$R_{eff}$. We note that the definition for $R_{eff}$ is not unique. In the
present work, we define the effective radius to be the surface containing
$99\%$ of the ADM mass, i.e., $\mathcal{M}(R_{eff})=0.99M$. In the left panel
of Fig. 5, we show the effective radius $R_{eff}$ for E-FLS stars as a
function of the oscillation frequency ${\omega}$. We observe a slight change
in the effective radius for $\mu>0$. In contrast, a noticeable change in the
effective radius is evident for E-FLS stars with $\mu=0$. In particular, the
effective radius for $\mu=0$ is considerably larger compared to the other
cases, which is in accordance to the fact that for $\mu=0$ the interaction
becomes long ranged. In the bottom panel of Fig. 5, we plot the total ADM mass
as a function of the effective radius for different values of $\mu$. The
noticeable result is observed once more for $\mu=0$, where we note, for
instance, that the maximum mass occurs at a larger effective radius compared
to the other cases.

(a) $\mu=0$ (b) $\mu=0.2$

(c) $\mu=0.4$ (d) $\mu\rightarrow\infty$

Figure 4: The total mass $M$ and number of particles $N$ behavior related to
the complex scalar field at the origin $\phi_{0}$. We exhibit results for four
values of $\mu$, which demonstrate that not only the maximum total mass $M$
decreases for lower values of $\mu$, but also the total number of particles
$N$. The set of solutions in which $M<N$ are stable and the others are
unstable.

(a)

(b)

Figure 5: Top panel: The effective radius $R_{eff}$ as a function of the
oscillation frequency for distinct values of $\mu$. Bottom panel: The total
mass $M$ of the E-FLS stars in terms of the effective radius $R_{eff}$, for
distinct values of the FLS parameter $\mu$. The impact of decreasing the FLS
parameter $\mu$ is evident for the $\mu=0$ case.

These results for the ADM mass and effective radius suggest that E-FLS stars
with $\mu=0$ are less compact than the cases with $\mu>0$. In order to
investigate the compactness of the E-FLS stars, we can define the concept of
inverse compactness, as given by:

$\text{Compactness}^{-1}=\frac{R_{eff}}{2M_{eff}},$ (23)

where $M_{eff}$ is the effective mass of the star that represents $99\%$ of
its total ADM mass, i.e. $M_{eff}\equiv\mathcal{M}(R_{eff})$. For extremely
compact objects, such as black holes, we have $\text{Compactness}^{-1}\sim 1$,
while less compact objects have higher inverse compactness. In Fig. 6, we show
the inverse compactness as a function of the oscillation frequency, for
different values of $\mu$. For $\mu\neq 0$, we notice a slight variation of
the inverse compactness. However, for $\mu=0$ we notice that inverse
compactness increases considerably. This result confirms the expectation that
E-FLS stars with $\mu=0$ are less compact.

Figure 6: The inverse compactness as a function of the oscillation frequency,
for distinct values of the FLS parameter $\mu$. We notice that the impact of
decreasing the FLS parameter $\mu$ is evident for the inverse compactness in
the $\mu=0$ case.

##  IV Geodesic Analysis

###  IV.1 Timelike geodesics in E-FLS stars

In order to investigate the phenomenological aspects of E-FLS stars, we study
the motion of timelike and null geodesics around these compact objects, which
is important for the description of accretion disks. Let us first consider
timelike geodesics, describing the motion of massive particles around the
E-FLS stars, given by the following Lagrangian:

$\displaystyle\mathcal{L}_{t}=g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=-e^{\Gamma}\,\dot{t}^{2}+e^{\Lambda}\,\dot{r}^{2}+r^{2}\dot{\varphi}^{2}=-1,$
(24)

where we considered $\theta=\pi/2$, without loss of generality, since the
spacetime is spherically symmetric. The overdots denote derivatives with
respect to the proper time along the timelike geodesics. We note that the
Lagrangian $\mathcal{L}_{t}$ is independent of $t$ and $\varphi$, hence there
are two conserved quantities associated to such symmetries, namely

$\displaystyle E=e^{\Gamma}\,\dot{t},$ (25) $\displaystyle
L=r^{2}\dot{\varphi},$ (26)

such that $E$ and $L$, respectively, represent the energy and angular momentum
of the particle. Substituting Eqs. (25)-(26) into the Lagrangian (24), we
obtain an equation describing the radial motion, namely:

$\displaystyle
e^{\left(\Gamma+\Lambda\right)}\,\dot{r}^{2}=E^{2}-e^{\Gamma}\left(1+\frac{L^{2}}{r^{2}}\right).$
(27)

The set of circular timelike orbits around E-FLS stars is obtained from the
radial equation by imposing $\left.\dot{r}\right|_{r=r_{c}}=0$ and
$\left.\ddot{r}\right|_{r=r_{c}}=0$. The energy and angular momentum of such
orbits at a given radius $r_{c}$ are given, respectively, by:

$\displaystyle
E_{c}^{2}=\left.\frac{2\,e^{\Gamma}}{2-r\,\Gamma^{\prime}}\right|_{r=r_{c}},\qquad
L_{c}^{2}=\left.\frac{r^{3}\,\Gamma^{\prime}}{2-r\,\Gamma^{\prime}}\right|_{r=r_{c}}.$
(28)

Moreover, using Eqs. (25)-(26) and (28) we can calculate the orbital frequency
of circular geodesics at a radius $r_{c}$, as follows:

$\displaystyle\Omega(r_{c})=\left.\frac{d\phi}{dt}\right|_{r=r_{c}}=\left.\left[\frac{e^{\Gamma}\,\Gamma^{\prime}}{2\,r}\right]^{\frac{1}{2}}\right|_{r=r_{c}}.$
(29)

In Fig. 7, we show the orbital frequency for circular geodesics in E-FLS stars
as a function of the radius $r_{c}$, corresponding to the configurations
presented in Table 1. We notice that the orbital frequency for circular orbits
located at a large radius remains unaltered for the different configurations.
On the other hand, the orbital frequency close to the origin present distinct
behavior for different E-FLS solutions. For instance, the configurations FLS2
and FLS4 present a much larger orbital frequency close to the origin, in
comparison to the configurations FLS1 and FLS3. Additionally, the
configurations FLS2 and FLS4 present a maximum value for the orbital
frequency, which does not occur for the configurations FLS1 and FLS3.

The properties of timelike circular geodesics studied in the present section
are important for the description of accretion disks around E-FLS stars, and
consequently for the astrophysical images of such compact objects, as we
discuss in details in Sec. V.

Figure 7: The orbital frequency for timelike circular geodesics as a function
of the radius $r_{c}$, for the different E-FLS configurations described in
Table 1. We notice that more compact configurations, for instance FLS2 and
FLS4, admit a local maximum for the orbital frequency, as shown in the inset.

###  IV.2 Null-like geodesics in E-FLS stars

The null geodesic trajectories represent the path followed by light in a
curved spacetime. The study of null geodesics around E-FLS stars is important
due to the connection with the recent images of compact objects at the centers
of the M87 and Milky Way galaxies, obtained by the EHT collaboration
M87_1:2019 ; M87_2:2019 ; M87_3:2019 ; M87_4:2019 ; M87_5:2019 ; M87_6:2019 ;
sgra_1:2022 ; sgra_2:2022 ; sgra_3:2022 ; sgra_4:2022 ; sgra_5:2022 ;
sgra_6:2022 . The Lagrangian for null geodesics around E-FLS stars is given by

$\displaystyle
2\,\mathcal{L}_{ph}=g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=-e^{\Gamma}\,\dot{t}^{2}+e^{\Lambda}\,\dot{r}^{2}+r^{2}\dot{\varphi}^{2}=0,$
(30)

where we considered (again) $\theta=\pi/2$ due to spherical symmetry.
Similarly to the timelike geodesic analysis, we may compute the conserved
quantities along null geodesics and consequently obtain the radial equation of
motion for photons to be given by

$\displaystyle\dot{r}^{2}+V(r,E,L)=0,$ (31)

being $V(r,E,L)$ the effective potential for null geodesics, namely

$\displaystyle
V(r,L,E)=e^{-\Lambda}\left(\frac{L^{2}}{r^{2}}-\frac{E^{2}}{e^{\Gamma}}\right).$
(32)

Several properties of the null geodesic trajectories in a E-FLS star can be
extracted from the radial equation (31). In particular, closed circular photon
orbits at the equatorial plane, commonly dubbed as light rings (LRs), are
described by $\left.\dot{r}\right|_{r=r_{ph}}=0$ and
$\left.\ddot{r}\right|_{r=r_{ph}}=0$. The LRs can also be described as
critical points of the effective potential:

$\displaystyle V(r_{ph},L,E)=0,\quad V^{\prime}(r_{ph},L,E)=0.$ (33)

We observe that $V(r,L,E)$ has the disadvantage of depending on the constants
of motion $E$ and $L$. For this reason, we choose to work with a new effective
potential $\mathcal{H}(r)$, which depends solely on the radial coordinate and
is defined by:

$\displaystyle\mathcal{H}(r)\equiv\frac{e^{\frac{\Gamma}{2}}}{r},$ (34)
$\displaystyle
V(r,L,E)=\frac{L^{2}}{e^{\Gamma+\Lambda}}\left(\mathcal{H}(r)+\frac{1}{b}\right)\left(\mathcal{H}(r)-\frac{1}{b}\right),$
(35)

where $b\equiv L/E$ is the impact parameter for photons. From Eqs. (33) and
(34), we can obtain the conditions for the existence of a LR in terms of the
new effective potential $\mathcal{H}(r)$, namely:

$\displaystyle\mathcal{H}(r)=\frac{1}{b},\quad\mathcal{H}^{\prime}(r)=0,$ (36)

where we notice that LRs are critical points of the new effective potential as
well.

In Fig. 8, we show different examples for the effective potential
$\mathcal{H}(r)$ as a function of the radial coordinate, computed for the
distinct configurations presented in Table 1. We observe that for the
configurations FLS1 and FLS3, there are no light rings at all. This result is
expected, as these configurations are close to the Newtonian regime, where
E-FLS stars are not sufficiently compact. On the other hand, LRs emerge for
the configurations FLS2 and FLS4. We note that they always appear as pairs of
unstable/stable LRs, in accordance with the theorems for the conservation of
the total topological charge for horizonless compact objects Cunha2017 ;
Cunha2020 . Denoting the radial coordinate of the unstable LR as $r_{+}$ and
the radial coordinate of the stable LR as $r_{-}$, we have that $r_{+}>r_{-}$
for any E-FLS star admitting LRs.

In Fig. 9, we show the null geodesic trajectories around the different E-FLS
stars configurations presented in Table 1. The null geodesics are restricted
to the equatorial plane and have distinct impact parameters. We notice that
for the configurations FLS1 and FLS3 the null geodesics are slightly
deflected, while for the configurations FLS2 and FLS4 they can be scattered
with arbitrarily large angles. These results are in accordance with the fact
that the configurations FLS1 and FLS3 are not compact enough, in contrast to
the configurations FLS2 and FLS4 which posses light rings.

Table 1: Four distinct E-FLS star solutions and their properties. Configuration | $\mu$ | $\omega$ | $M$ | Number of LRs | $r_{+}$
---|---|---|---|---|---
FLS1 |  0.2 | $0.8873$ | 0.5523 | 0 | –
FLS2 |  0.2 | $0.8280$ | 0.3715 | 2 | $0.0794$
FLS3 | 0 | 0.8323 | 0.5043 | 0 | –
FLS4 | 0 | 0.7828 | 0.3855 | 2 | $0.0896$
Figure 8: The effective potential $\mathcal{H}(r)$ for null geodesics around
E-FLS stars, as a function of the radial coordinate, and for the different
configurations presented in Table 1. We notice that, depending on the
particular configuration, the star can have two LRs or no LR at all.

(a) FLS1

(b) FLS2

(c) FLS3

(d) FLS4

Figure 9: The trajectories described by null geodesics with different impact
parameters along the equatorial plane of the E-FLS stars. The color scheme
represents the different impact parameters for null geodesics.

The existence of such LRs for the configurations FLS2 and FLS4 is a purely
relativistic effect and can impact significantly to the visual appearance of
the E-FLS stars. In order to investigate the images of these configurations,
we discuss in the next section the implementation of the backwards ray-tracing
technique for the E-FLS stars surrounded by different accretion disk models.

##  V Astrophysical Images of E-FLS stars

We found that some E-FLS star configurations admit LRs for given values of the
FLS parameter and of the frequency. The presence of LRs can impact
significantly the gravitational lensing, and consequently the images of E-FLS
stars, as seen by a distant observer. In order to simulate such images, we
apply the so-called backwards ray-tracing technique to the E-FLS stars
Bohn2015 ; Cunha2018 ; Cunha20202 ; HCDL2021 ; Proca_Star_Shadow ;
Ivo_Proca_Shadow ; Caio_shadows . The backwards ray-tracing consists in
evolving the light rays from the observer’s position, backwards in time, until
some event is met, for instance, when the light ray intersects the accretion
disk or escape to infinity. The implementation of the backwards ray-tracing
method involves solving the first order equations of motion for null geodesics

$\displaystyle\dot{t}=\frac{E}{e^{\Gamma}},$ (37)
$\displaystyle\dot{\phi}=\frac{L}{r^{2}},$ (38)

coupled to the second order equations of motion for the radial and polar
coordinates:

$\displaystyle\ddot{r}+\Gamma^{r}_{\ \mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=0,$
(39) $\displaystyle\ddot{\theta}+\Gamma^{\theta}_{\
\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=0,$ (40)

where $\Gamma^{\alpha}_{\ \mu\nu}$ denotes the components of the Christoffel
symbol associated to the E-FLS star geometry. We note that for the backwards
ray-tracing technique, the motion is not assumed to be restricted to the
equatorial plane, hence we need to solve also the equation of motion for the
polar coordinate (40).

Since we are interested in simulating the images of E-FLS stars surrounded by
an emitting accretion disk, we need to solve the so-called radiative transfer
equation, coupled to the geodesic equations. Assuming that the beam of photons
are unpolarized, the radiative transfer equation is given by Lindquist .

$\displaystyle\frac{d}{d\lambda}\left(\frac{I_{\nu}}{\nu^{3}}\right)=\frac{j_{\nu}}{\nu^{2}}-\nu\,\alpha_{\nu}\,\left(\frac{I_{\nu}}{\nu^{3}}\right),$
(41)

where $I_{\nu},j_{\nu}$ and $\alpha_{\nu}$ are the specific intensity,
emission coefficient and absorption coefficient, respectively. These
quantities are measured by an observer comoving with the accretion disk. They
are related to the invariant quantities, i.e. observer independent objects, as
follows:

$\displaystyle\mathcal{I}=\frac{I_{\nu}}{\nu^{3}},\quad\eta=\frac{j_{\nu}}{\nu^{2}},\quad\chi=\nu\,\alpha_{\nu},$
(42)

being $\nu$ the frequency of emission.

In order to obtain the initial conditions for the set of Eqs. (37)-(38) and
(39)-(40), we apply a procedure akin to the one detailed in Refs. Bohn2015 ;
Cunha2018 ; Cunha20202 ; HCDL2021 , which we summarize as follows:

  * •

(i) Since we are propagating the light rays starting at the observer’s
position, the first set of initial conditions is given by $t=0,\ r=r_{obs},\
\theta=\theta_{obs},\ \varphi=0;$

  * •

(ii) A second set of initial conditions can be obtained by computing the
initial direction of the light ray, as measured by a static local observer. We
introduce a local tetrad associated to the static observer and compute the
projection of the 4-momentum of the light ray into the tetrad basis;

  * •

(iii) Each initial direction, as seen by the local observer, defines a
geodesic to be numerically evolved;

  * •

(iv) Finally, by evolving the geodesic equation coupled to the radiative
transfer equation, for a given set of initial directions parametrized by two
angles $(\alpha,\beta)$, we produce a map: ($\alpha$, $\beta$) $\mapsto
I^{obs}(x,y)$, where $(x,y)$ are two coordinates on the image plane, which is
perpendicular to the observer’s line of sight, and $I^{obs}(x,y)$ is the
corresponding observed intensity, which represents the astrophysical image of
the E-FLS stars.

As the solutions for E-FLS stars are numerical rather than analytical, we have
also incorporated an interpolation scheme in our ray-tracing code. The chosen
interpolation method is the cubic spline, providing a ${C}^{2}$ interpolation
function.

We have chosen the four distinct configurations presented in Table 1 for the
backward ray-tracing simulation. In the next subsections, we explore the
astrophysical images for these configurations surrounded by two different
models for the accretion disk.

(a) FLS1

(b) FLS2

(c) FLS3

(d) FLS4

Figure 10: The gravitational lensing for the distinct E-FLS star
configurations, presented in Table 1, surrounded by an optically thick
accretion disk. In this figure we have positioned the observer at
$r_{obs}=20\,M$, $\theta_{\textrm{obs}}=80^{\circ}$, i.e. the observer is
slightly displaced from the equatorial plane.

(a) FLS1

(b) FLS2

(c) FLS3

(d) FLS4

Figure 11: The gravitational lensing for the distinct E-FLS star
configurations, presented in Table 1, surrounded by an optically thick
accretion disk. In this figure the observer is placed at $r_{obs}=20\,M$,
$\theta_{\textrm{obs}}=5^{\circ}$, i.e. the observer is close to a face-on
observation of the accretion disk.

(a) FLS1

(b) FLS2

(c) FLS3

(d) FLS4

Figure 12: The gravitational lensing for the distinct E-FLS star
configurations, presented in Table 1, surrounded by an optically thin
accretion disk. Similarly to Fig 10, we have positioned the observer at
$r_{obs}=20\,M$, $\theta_{\textrm{obs}}=80^{\circ}$.

(a) FLS1

(b) FLS2

(c) FLS3

(d) FLS4

Figure 13: The gravitational lensing for the distinct E-FLS star
configurations, presented in Table 1, surrounded by an optically thin
accretion disk. Similarly to Fig 11, we have positioned the observer at
$r_{obs}=20\,M$, $\theta_{\textrm{obs}}=5^{\circ}$.

###  V.1 Astrophysical Images of E-FLS stars surrounded by an optically thick
disk

Let us first consider a geometrically thin and opaque accretion disk (also
known as optically thick) surrounding the E-FLS star. The accretion disk is
located along the equatorial plane. Due to the opacity, we evolve the light
rays from the observer’s position until they escape to spatial infinity or
intersect the accretion disk. Moreover, we consider that the disk is
constituted by massive particles following circular geodesic motion, as
described in Sec IV.1. The energy, angular momentum and orbital frequency for
a particle at radius $r_{c}$ are given by Eqs. (28) and (29), respectively.

For a geometrically thin and optically thick accretion disk, the solution for
the radiative transfer equation (41) is quite straightforward. We consider
that the accretion disk’s emission is monochromatic with frequency $\nu_{em}$:

$\displaystyle I_{\nu}\propto\delta(\nu-\nu_{em})\,\epsilon(r),$ (43)

as measured by an observer comoving with the accretion disk. Moreover based on
Refs. Proca_Star_Shadow ; Ivo_Proca_Shadow , we assume that $\epsilon(r)$
behaves as follows:

$\displaystyle\epsilon(r)\equiv\frac{1+\tanh[50(r-6M)]}{2}\left(\frac{6\,M}{r}\right)^{3}.$
(44)

This expression serves as an ad-hoc emission profile, yet effectively captures
the essential features of more intricate emission profiles. The specific
intensity as measured by the observer can be obtained from the invariant
intensity (42), namely

$\displaystyle I^{obs}_{\nu^{\prime}}=\frac{\nu^{\prime 3}}{\nu^{3}}I_{\nu},$
(45)

where $\nu^{\prime}$ is the observed frequency.

In Fig. 10, we show the astrophysical images for the different E-FLS
configurations (presented in Table 1) surrounded by a geometrically thin and
optically thick accretion disk. In Fig. 10, we considered the observer at a
radius $r_{obs}=20\,M$ and the polar angle $\theta_{\textrm{obs}}=80^{\circ}$,
hence the observer is slightly displaced from the equatorial plane. We notice
that the gravitational lensing effect is not strong for configurations FLS1
and FLS3, as can be seen in panels (a) and (c) of Fig. 10, where the image of
the accretion disk is slightly distorted. These results are in accordance with
the fact that configurations FLS1 and FLS3 are not compact enough to possess
light rings. On the other hand, we notice a strong gravitational lensing
effects for the configurations FLS2 and FLS4, as shown in panels (b) and (d)
of Fig. 10. In particular, the gravitational lensing effect in this
configuration is so pronounced that we can observe the back part of the
accretion disk being gravitationally lensed, appearing both above and below
the center of the final image. These results are in accordance with the fact
that configurations FLS2 and FLS4 are more compact and they even allow the
presence of a light ring.

Furthermore, as depicted in Fig. 10, we note that the right side of the image
appears brighter than the left side. This discrepancy arises from the rotation
of the accretion disk, inducing a Doppler effect for the observer. The Doppler
effect and the invariance of $I_{\nu}/\nu^{3}$ cause the difference in the
observed brightness on each side of the accretion disk Misner_book .

We also obtained the astrophysical images of E-FLS stars as viewed by an
observer positioned at an observational angle $\theta_{obs}=5^{\circ}$, which
corresponds to a face-on perspective of the disk. The motivation for this
choice of angle is based on the recent EHT results that disfavor observations
at a high observational angles M87_1:2019 ; M87_2:2019 ; M87_3:2019 ;
M87_4:2019 ; M87_5:2019 ; M87_6:2019 ; sgra_1:2022 ; sgra_2:2022 ; sgra_3:2022
; sgra_4:2022 ; sgra_5:2022 ; sgra_6:2022 . For instance, in the case of Sgr
A*, observational angles $\theta_{obs}>50^{\circ}$ are disfavored by the
comparison between the theoretical model and the observations sgra_1:2022 . In
Fig. 11, we show the gravitational lensing for E-FLS stars, as seen by an
observer at $r_{obs}=20\,M$ and $\theta_{obs}=5^{\circ}$. We notice that the
images for the different configurations are more similar, in comparison to the
images for an observer at the equatorial plane. The major difference relies in
the brightness difference for each configuration. The less compact
configurations (FLS1 and FLS3) appear brighter than the more compact ones
(FLS2 and FLS4). These results arises from the stronger gravitational redshift
associated with the more compact configuration, in conjunction with the
formula for the invariant intensity, as provided in Eq. (45).

###  V.2 Astrophysical Images of E-FLS stars surrounded by an optically thin
disk

Let us now consider an optically thin accretion disk surrounding the E-FLS
star. An optically thin disk is transparent to radiation and the light rays
can cross it several times before being scattered. Each time that a given
light ray intersects the accretion disk, it acquires more intensity. We
consider the same emission profile as given in Eqs. (43)-(44). We also keep
the position of the observer at $r_{obs}=20\,M$ and vary the polar angle of
observation.

In Fig. 12, we show the gravitational lensing for the different E-FLS
configurations in Table 1, for a fixed polar observation angle
$\theta_{obs}=80^{\circ}$. We notice that for less compact configurations
(FLS1 and FLS3), the images for the thin disk are akin to the images with
optically thick disk. However, for more compact configurations that have a
light ring (FLS2 and FLS4), the images can be quite different when compared to
the optically thick case. The major difference arises due to the thin
signatures on the images, dubbed as photon ring Universal_Int . These thin
signatures are associated to the existence of light rings. The light rays that
have an impact parameter close to the critical one [see Eq. (36)] complete
several turns around the center before being scattered, intersecting the disk
many times and consequently increasing the intensity.

In Fig. 13, we present the gravitational lensing for the E-FLS stars in Table
1 for an observer at $r_{obs}=20\,M$ and $\theta_{obs}=5^{\circ}$, i.e. an
almost face-on observation. We notice that the configurations FLS2 and FLS4
present a photon ring close to the center of the image, while the
configurations FLS1 and FLS3 do not. As discussed previously, such thin
signature close to the center of the image arises due to the light rays that
cross the accretion disk several times for certain values of the impact
parameter.

We notice that the presence of thin photon rings on the image leave a
characteristic fingerprint on the interferometric signature on very long
baselines Universal_Int . The presence (or the absence) of such photon rings
can, in principle, be used as a test for distinguishing different
configurations with the future advent of ng-EHT.

##  VI Conclusions

We have studied the boson star solutions in the context of the E-FLS theory,
which we dubbed as E-FLS stars. The matter sector of the Lagrangian in the
E-FLS theory is constituted by two interacting scalar fields, one being real
and the other complex. By assuming a spherically symmetric ansatz for the
background metric and the scalar fields, we computed numerical solutions for
the E-FLS field equations. Within our numerical scheme, we were able to obtain
a full scan of the solution space in the E-FLS theory. We explored the E-FLS
star solutions ranging from the limit in which $\mu$ approaches infinity,
where we recovered mini-boson star solutions, to the $\mu=0$ limit, where the
effects of the FLS coupling became pronounced. In particular, we found
interesting results for the case $\mu=0$, in which the real scalar field
becomes massless.

We computed some properties of the E-FLS stars, such as, the ADM mass, the
total Noether charge, the effective radius and the compactness. An intriguing
feature of E-FLS stars is that as we decrease the value of $\mu$, the range of
frequencies for which E-FLS stars exist increases. Moreover, we also computed
the binding energy and showed that the case $\mu=0$ exhibits the largest
region with bound configurations and, consequently, the largest region with
linearly stable configurations. Concerning the compactness of the E-FLS stars,
we showed that the configurations with $\mu=0$ are less compact, in comparison
to the cases with $\mu>0$.

Motivated by the recent EHT image results, we studied the geodesics around the
E-FLS stars and the astrophysical images of such compact objects, when
surrounded by an accretion disk. We analyzed the timelike geodesics,
describing the motion of massive particles around the E-FLS stars. For
instance, we computed the angular velocity for timelike circular geodesics.
Our findings revealed that more compact configurations exhibit a maximum value
for the angular velocity, whereas less compact configurations do not exhibit
such a maximum. Concerning the null geodesic motion, we computed the light
rings in E-FLS star spacetimes. We obtained that the E-FLS stars close to the
newtonian regime do not have light rings. However, as the configurations
become more compact, pairs of stable and unstable light rings emerge. This
result is a consequence of the conservation of the total topological charge
for horizonless exotic compact objects.

We computed the astrophysical images of E-FLS stars using the backwards ray-
tracing technique. We assumed that the E-FLS stars were surrounded by two
different models of disk, namely, an optically thick (opaque) and an optically
thin (transparent). We found that the configuration close to the Newtonian
regime do not exhibit a prominent gravitational lensing, neither a prominent
gravitational redshift, since the images are brighter. On the other hand, the
configurations that are more compact, and posses a light ring, exhibit strong
gravitational lensing and gravitational redshift effects in the astrophysical
images. For the optically thin accretion model, we noticed that a thin
signature is present in the images due to the light rays that orbits several
times around the center before being scattered. The images of such compact
configuration can, in principle, mimic the visual appearance of a
Schwarzschild black hole.

###### Acknowledgements.

The authors would like to acknowledge Fundação Amazônia de Amparo a Estudos e
Pesquisas (FAPESPA), Conselho Nacional de Desenvolvimento Científico e
Tecnológico (CNPq) and Coordenação de Aperfeiçoamento de Pessoal de Nível
Superior (CAPES) – Finance Code 001, from Brazil, for partial financial
support. H. L. J. thanks Etevaldo Costa and Pedro Cunha for useful
discussions. This work is supported by the Center for Research and Development
in Mathematics and Applications (CIDMA) through the Portuguese Foundation for
Science and Technology Fundação para a Ciência e a Tecnologia),
UIDB/04106/2020, UIDP/04106/2020, https://doi.org/10.54499/UIDB/04106/2020 and
https://doi.org/10.54499/UIDP/04106/2020. The authors acknowledge support from
the projects http://doi.org/10.54499/PTDC/FISAST/3041/2020,
http://doi.org/10.54499/CERN/FIS-PAR/0024/2021 and
https://doi.org/10.54499/2022.04560.PTDC. This work has further been supported
by the European Horizon Europe staff exchange (SE) programme HORIZON-
MSCA2021-SE-01 Grant No. NewFunFiCO101086251.

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# Non-Archimedean second main theorem sharing small functions

Ta Thi Hoai An  Institute of Mathematics, Vietnam Academy of Science and
Technology
18 Hoang Quoc Viet Road, Cau Giay District
10307 Hanoi, Vietnam  and: Institute of Mathematics and Applied Sciences
(TIMAS)
Thang Long University
Hanoi, Vietnam<EMAIL_ADDRESS>and  Nguyen Viet Phuong  Thai Nguyen
University of Economics and Business Administration, Vietnam
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we establish a new second main theorem for meromorphic
functions on a non-Archimedean field and small functions with counting
functions truncated to level $1.$ As an application, we show that two
meromorphic functions on a non-Archimedean field must coincide to each other
if they share $q\,(q\geq 5)$ distinct small functions ignoring multiplicities.
In particular, if two non-Archimedean meromorphic functions share $5$ small
functions ignoring multiplicities, they must be identical. Thus, our work
improves the results in [1].

Key words: Non-Archimedean field, meromorphic functions, Nevanlinna theory,
small functions, uniqueness.

2010 Mathematics Subject Classification 30D35

##  1\.  Introduction and main results

Let $\mathbf{K}$ be an algebraically closed field of arbitrary characteristic,
complete with respect to a non-Archimedean absolute value $|.|.$ For any
nonconstant non-Archimedean meromorphic function $f,$ we denote by $S(r,f)$
any quantity satisfying $S(r,f)=o(T(r,f))$ for $r\rightarrow\infty.$ A non-
Archimedean meromorphic function $a$ is called a small function with respect
to $f$ if $T(r,a)=S(r,f).$ Let $k$ is a positive integer or $\infty,$ we
denote by $\overline{E}(a,k,f)$ the set of distinct zeros of $f-a$ with
multiplicities at most $k,$ where a zero of $f-\infty$ means a pole of $f.$ In
the case that $k=\infty,$ we denote $\overline{E}(a,\infty,f)$ simply by
$\overline{E}(a,f).$ We say that $f$ and $g$ share a function $a$ ignoring
multiplicities if $\overline{E}(a,f)=\overline{E}(a,g).$

In 1926, as an application of the celebrated Nevanlinna’s value distribution
theory of meromorphic functions, R. Nevanlinna [4] himself proved that for two
distinct nonconstant meromorphic functions $f$ and $g$ on the complex plane
$\mathbb{C},$ they cannot have the same inverse images for five distinct
values. This result is known as Nevalinna’s five values theorem and is
considered as the first result on the uniqueness problem of meromorphic
functions sharing values. Then, some authors have generalized the result of
Nevanlinna to the case where the five distinct values are replaced by five
small functions, for example Yuhua and Jianyong [9], Yao [7], Yi [8], Thai and
Tan [5]. The proofs of the above results are based straightforwardly on the
Cartan’s auxialiary functions and the sharp second main theorem for
meromorphic functions and small functions on $\mathbb{C}$ with counting
functions truncated to level $1$ of Yamanoi [6].

Nevanlinna theory in complex analysis is so beautiful that one would naturally
be interested in determining how such a theory would look in the $p$-adic
analysis. In 1997, Hu and Yang [2] proved that for two distinct nonconstant
$p$-adic meromorphic functions $f$ and $g$ on $\mathbb{C}_{p}$ they cannot
have the same inverse images for four distinct values. This result can be
regarded as the $p$-adic analogue of corresponding uniqueness theorem in value
distribution theory of meromorphic functions of one complex variable. More
general, this result can be proved for any algebraically closed field
$\mathbf{K}$ complete with respect to a non-Archimedean absolute value (cf.
[3]). Corresponding to such a uniqueness theorem of non-Archimedean
meromorphic functions which have the same inverse images for distinct values,
one expects a generalized result in which in this case the distinct values
replaced by the small functions. However, as we known, there is no second main
theorem similar to the theorem of Yamanoi in non-Archimedean analysis.
Recently, A. Escassut and C. C. Yang [1] given a second fundamental theorem
for non-Archimedean meromorphic functions and small functions with reduced
counting functions which makes a tool other than theorem of Yamanoi in complex
analysis to derive a uniqueness theorem for non-Archimedean meromorphic
functions sharing $7$ small functions ignoring multiplicities. However, their
results are not sharp.

Our aim in this paper is to contruct a better second main theorem for small
functions to study the uniqueness problem for meromorphic functions sharing
small functions. First, we give a second main theorem for small functions,
which improves Theorem 2 in [1] by increasing the coefficent $\frac{q}{3}$ in
front of the characteristic function to $\frac{2q}{5}.$ Our first result is as
follows.

######  Theorem 1.

Let $f$ be a nonconstant meromorphic function on $\mathbf{K}.$ Let
$a_{1},\dots,a_{q}\,\,(q\geq 5)$ be $q$ distinct small functions with respect
to $f.$ We have

$\frac{2q}{5}T(r,f)\leq\sum_{i=1}^{q}\overline{N}\big{(}r,\frac{1}{f-a_{i}}\big{)}+S(r,f).$

As an application of Theorem 1, we get a uniquenes theorem for the meromorphic
functions sharing a few small functions as follows.

######  Theorem 2.

Let $f$ and $g$ be two nonconstant meromorphic functions on $\mathbf{K}.$ Let
$a_{1},\dots,a_{q}\quad(q\geq 5)$ be $q$ distinct small functions with respect
to $f$ and $g.$ Let $k_{1},\dots,k_{q}$ be $q$ positive integers or $+\infty$
with

$\sum_{j=1}^{q}\frac{1}{k_{j}+1}<\frac{2q(q-4)}{5(q+4)}.$

If

$\overline{E}(a_{j},k_{j},f)=\overline{E}(a_{j},k_{j},g)\quad(j=1,\dots,q),$

then $f\equiv g.$

In the case $k_{1}=\dots=k_{q}=k,$ we can get the result with slightly smaller
multiples as follows.

######  Theorem 3.

Let $f$ and $g$ be two nonconstant meromorphic functions on $\mathbf{K}.$ Let
$a_{1},\dots,a_{q}\quad(q\geq 5)$ be $q$ distinct small functions with respect
to $f$ and $g.$ Let $k$ be a positive integer or $+\infty$ with
$k>\frac{3(q+4)}{2(q-4)}.$ If

$\overline{E}(a_{j},k,f)=\overline{E}(a_{j},k,g)\quad(j=1,\dots,q),$

then $f\equiv g.$

By Theorem 3, we obtain the following corollary, which is a uniqueness theorem
for non-Archimedean meromorphic functions sharing $5$ small functions ignoring
multiplicities.

######  Corollary 1.

Let $f$ and $g$ be two nonconstant meromorphic functions on $\mathbf{K}.$ Let
$a_{1},\dots,a_{5}$ be $5$ distinct small functions with respect to $f$ and
$g.$ If $f$ and $g$ share $a_{j}$ ignoring multiplicities $(j=1,\dots,5,)$
then $f\equiv g.$

Note that this Corollary 1 improves a result of A. Escassut and C. C. Yang [1,
Theorem 3], where the number of small functions is reduced to $5.$

##  2\. Preliminary on Nevanlinna Theory for non-Archimedean meromorphic
functions

We recall the following definitions and results (cf. [3]). Let $\mathbf{K}$ be
an algebraically closed field of arbitrary characteristic, complete with
respect to a non-Archimedean absolute value $|.|.$ If

$h(z)=\sum_{j=0}^{\infty}a_{n}z^{n}$

is an entire function on $\mathbf{K},$ then for each real number $r\geq 0,$ we
define

$\displaystyle|h|_{r}=\sup_{j}|a_{j}|r^{j}$
$\displaystyle=\sup\\{|h(z)|:z\in\mathbf{K}\text{ with }|z|\leq r\\}$
$\displaystyle=\sup\\{|h(z)|:z\in\mathbf{K}\text{ with }|z|=r\\}.$

Let $n\big{(}r,\frac{1}{h}\big{)}$ denote the number of zeros of $h$ in
$\\{z||z|<r\\},$ counting multiplicity. Define the valence function of $h$ by

$N\big{(}r,\frac{1}{h}\big{)}=\int_{0}^{r}\frac{n\big{(}r,\frac{1}{h}\big{)}-n\big{(}0,\frac{1}{h}\big{)}}{t}dt+n\big{(}0,\frac{1}{h}\big{)}\log
r,$

where $n\big{(}0,\frac{1}{h}\big{)}$ is order of zero of $h$ at $z=0.$

A non-Archimedean meromorphic function $f$ on $\mathbf{K}$ is the quotient of
two non-Archimedean entire functions $\frac{h}{g}$ such that $h,g$ do not have
common zeros on $\mathbf{K}.$ Therefore, we can uniquely extend $|.|_{r}$ to
the non-Archimedean meromorphic function $f=\frac{h}{g}$ by defining

$|f|_{r}=\frac{|h|_{r}}{|g|_{r}}.$

Take $a\in K$ and the counting function $n\big{(}r,\frac{1}{f-a}\big{)}$ of
$f$ for $a$ by

$n\big{(}r,\frac{1}{f-a}\big{)}=\begin{cases}n(r,f)=n\big{(}r,\frac{1}{g}\big{)}&:\,\,a=\infty\\\
n\big{(}r,\frac{1}{h-ag}\big{)}&:\,\,a\neq\infty,\end{cases}$

and define the valence function $N\big{(}r,\frac{1}{f-a}\big{)}$ of $f$ for
$a$ by

$N\big{(}r,\frac{1}{f-a}\big{)}=\begin{cases}N(r,f)=N\big{(}r,\frac{1}{g}\big{)}&:\,\,a=\infty\\\
N\big{(}r,\frac{1}{h-ag}\big{)}&:\,\,a\neq\infty.\end{cases}$

Similarly, we can define $\overline{N}(r,f)$ and
$\overline{N}\big{(}r,\frac{1}{f-a}\big{)}.$

Let $f$ be a nonconstant non-Archimedean meromorphic function, $a$ be a small
function with respect to $f,$ and $k$ be a positive integer. We denote by
$\overline{N}_{k)}\big{(}r,\frac{1}{f-a}\big{)}$ the counting function of
zeros of $f-a$ with multiplicities at most $k,$ by
$\overline{N}_{(k+1)}\big{(}r,\frac{1}{f-a}\big{)}$ the counting function of
zeros of $f-a$ with multiplicities at least $k+1,$ where each multiple zero in
these counting functions counted only once.

We define the compensation function by

$m(r,f)=\log^{+}|f|_{r}=\max\\{0,\log|f|_{r}\\},$

and the characteristic function

$T(r,f)=m(r,f)+N(r,f).$

The logarithmic derivative lemma can be stated as follows (see [3]).

######  Lemma 1 (Logarithmic Derivative Lemma).

Let $f$ be a non-constant meromorphic function on $\mathbf{K}.$ Then for any
integer $k>0,$ we have

$m\big{(}r,\frac{f^{(k)}}{f}\big{)}=O(1)$

as $r\rightarrow\infty.$

We state the first and second fundamental theorem in Nevanlinna theory (see
e.g. [3]):

######  Theorem 4 (The First Main Theorem).

Let $f(z)$ be a no-Archimedean meromorphic function and $c\in\mathbf{K}.$ Then

$T(r,\frac{1}{f-c})=T(r,f)+O(1).$

######  Theorem 5 (Second fundamental theorem).

Let $a_{1},\cdots,a_{q}$ be a set of distinct numbers of $\mathbf{K}.$ Let $f$
be a non-constant meromorphic function on $\mathbf{K}.$ Then, the inequality

$(q-2)T(r,f)\leq\sum_{j=1}^{q}\overline{N}\big{(}r,\frac{1}{f-a_{j}}\big{)}-\log
r+O(1).$

##  3\. Proof of Theorem 1

We first consider the following lemma.

######  Lemma 2.

Let $f$ be a nonconstant meromorphic function on $\mathbf{K}.$ Let
$a_{1},\dots,a_{5}$ be distinct small functions with respect to $f.$ We have

$2T(r,f)\leq\sum_{i=1}^{5}\overline{N}\big{(}r,\frac{1}{f-a_{i}}\big{)}+S(r,f).$

###### Proof.

By the transformation

$F=\frac{f-a_{2}}{f-a_{1}}\cdot\frac{a_{3}-a_{1}}{a_{3}-a_{2}},$

we just need to prove the theorem in the case that
$a_{1}=\infty,a_{2}=0,a_{3}=1,$ $a_{4},a_{5}\not\equiv
0,1,\infty,a_{4}\not\equiv a_{5}.$ If one of $a_{4}$ and $a_{5}$ is constant,
then we need to prove nothing according to the second main theorem for
constants. Thus, we may assume that both $a_{4}$ and $a_{5}$ are nonconstant
small functions of $f.$ Set

$H=\begin{vmatrix}ff^{\prime}&f^{\prime}&f(f-1)\\\
a_{4}a^{\prime}_{4}&a^{\prime}_{4}&a_{4}(a_{4}-1)\\\
a_{5}a^{\prime}_{5}&a^{\prime}_{5}&a_{5}(a_{5}-1)\end{vmatrix}$ (3.1)

By a simple computation, we get

$\displaystyle H=$ $\displaystyle
f(f-1)a_{4}(a_{4}-1)a_{5}(a_{5}-1)\Big{[}\big{(}\frac{a^{\prime}_{4}}{a_{4}}-\frac{a^{\prime}_{5}}{a_{5}}\big{)}\big{(}\frac{f^{\prime}}{f-1}-\frac{a^{\prime}_{5}}{a_{5}-1}\big{)}$
$\displaystyle-\big{(}\frac{a^{\prime}_{4}}{a_{4}-1}-\frac{a^{\prime}_{5}}{a_{5}-1}\big{)}\big{(}\frac{f^{\prime}}{f}-\frac{a^{\prime}_{5}}{a_{5}}\big{)}\Big{]}.$
(3.2)

We claim that $H\not\equiv 0.$ Indeed, on the contrary, assume that $H\equiv
0.$ Since $f$ is not constant and $a_{4},a_{5}\not\equiv 0,1,$ it follows from
(3.1) that

$\Big{(}\frac{a^{\prime}_{4}}{a_{4}}-\frac{a^{\prime}_{5}}{a_{5}}\Big{)}\frac{f^{\prime}}{f-1}-\Big{(}\frac{a^{\prime}_{4}}{a_{4}-1}-\frac{a^{\prime}_{5}}{a_{5}-1}\Big{)}\frac{f^{\prime}}{f}\equiv\Big{(}\frac{a^{\prime}_{4}}{a_{4}}-\frac{a^{\prime}_{5}}{a_{5}}\Big{)}\frac{a^{\prime}_{5}}{a_{5}-1}-\Big{(}\frac{a^{\prime}_{4}}{a_{4}-1}-\frac{a^{\prime}_{5}}{a_{5}-1}\Big{)}\frac{a^{\prime}_{5}}{a_{5}}.$
(3.3)

We now distinguish four cases

Case 1. $\frac{a^{\prime}_{4}}{a_{4}}\equiv\frac{a^{\prime}_{5}}{a_{5}}.$ It
follows from (3.3) that
$\frac{a^{\prime}_{4}}{a_{4}-1}\equiv\frac{a^{\prime}_{5}}{a_{5}-1}$ or
$\frac{f^{\prime}}{f}\equiv\frac{a^{\prime}_{5}}{a_{5}}.$ If
$\frac{a^{\prime}_{4}}{a_{4}-1}\equiv\frac{a^{\prime}_{5}}{a_{5}-1}$ then
$a_{4}$ and $a_{5}$ are constants, which contradicts our assumption. This
means $\frac{f^{\prime}}{f}\equiv\frac{a^{\prime}_{5}}{a_{5}}.$ Hence, we get
$f=ca_{5},$ where $c$ is a constant. This is a contradiction.

Case 2. $\frac{a^{\prime}_{4}}{a_{4}-1}\equiv\frac{a^{\prime}_{5}}{a_{5}-1}.$
By an argument similar to Case 1, we also get a contradiction.

Case 3.
$\frac{a^{\prime}_{4}}{a_{4}}-\frac{a^{\prime}_{5}}{a_{5}}\equiv\frac{a^{\prime}_{4}}{a_{4}-1}-\frac{a^{\prime}_{5}}{a_{5}-1}\not\equiv
0.$ It follows from (3.3) that

$\frac{f^{\prime}}{f-1}-\frac{f^{\prime}}{f}\equiv\frac{a^{\prime}_{5}}{a_{5}-1}-\frac{a^{\prime}_{5}}{a_{5}},$

which implies

$\frac{f-1}{f}\equiv C\frac{a_{5}-1}{a_{5}},$

where $C$ is a constant. Thus, we obtain

$\frac{1}{f}\equiv 1-C\frac{a_{5}-1}{a_{5}}.$

It follows that

$T(r,f)=T\big{(}r,\frac{1}{f}\big{)}+O(1)=S(r,f).$

This is a contradiction.

Case 4. $\frac{a^{\prime}_{4}}{a_{4}}\not\equiv\frac{a^{\prime}_{5}}{a_{5}},$
$\frac{a^{\prime}_{4}}{a_{4}-1}\not\equiv\frac{a^{\prime}_{5}}{a_{5}-1}$ and
$\frac{a^{\prime}_{4}}{a_{4}}-\frac{a^{\prime}_{5}}{a_{5}}\not\equiv\frac{a^{\prime}_{4}}{a_{4}-1}-\frac{a^{\prime}_{5}}{a_{5}-1}.$
Then, it follows from (3.3)that the zeros of $f-1$ can only occur at the zeros
or $1-$points or the poles of $a_{j}\quad(j=4,5)$ or the zeros of
$\frac{a^{\prime}_{4}}{a_{4}}-\frac{a^{\prime}_{5}}{a_{5}}.$ Similar, the
zeros of $f$ can only occur at the zeros or $1-$points or the poles of
$a_{j}\quad(j=4,5)$ or the zeros of
$\frac{a^{\prime}_{4}}{a_{4}-1}-\frac{a^{\prime}_{5}}{a_{5}-1}.$ Furthermore,
from (3.3), we also can be seen that the poles of $f$ can only occur at the
zeros or $1-$points or the poles of $a_{j}\quad(j=4,5)$ or the zeros of
$\frac{a^{\prime}_{4}}{a_{4}}-\frac{a^{\prime}_{5}}{a_{5}}-\frac{a^{\prime}_{4}}{a_{4}-1}+\frac{a^{\prime}_{5}}{a_{5}-1}.$
Therefore, we get

$\overline{N}(r,f)+\overline{N}\big{(}r,\frac{1}{f}\big{)}+\overline{N}\big{(}r,\frac{1}{f-1}\big{)}=S(r,f).$
(3.4)

By (3.4) and applying the Second Main Theorem for $f$ and $0,1,\infty,$ we
have

$T(r,f)\leq\overline{N}(r,f)+\overline{N}\big{(}r,\frac{1}{f}\big{)}+\overline{N}\big{(}r,\frac{1}{f-1}\big{)}-\log
r+O(1)=S(r,f).$

This is a contradiction again. Thus, we must have $H\not\equiv 0.$

Given a real number $0<r<\infty.$ Let

$\delta(r)=\mathop{\mathrm{min}}\\{1,|a_{4}|_{r},|a_{5}|_{r},|a_{4}-1|_{r},|a_{5}-1|_{r},|a_{4}-a_{5}|_{r}\\}.$

Then, we have

$\displaystyle\log^{+}\frac{1}{\delta(r)}$
$\displaystyle\leq\log^{+}\max\\{1,\frac{1}{|a_{4}|_{r}},\frac{1}{|a_{5}|_{r}},\frac{1}{|a_{4}-1|_{r}},\frac{1}{|a_{5}-1|_{r}},\frac{1}{|a_{4}-a_{5}|_{r}}\\}$
$\displaystyle\leq\log^{+}\Big{(}1+\frac{1}{|a_{4}|_{r}}+\frac{1}{|a_{5}|_{r}}+\frac{1}{|a_{4}-1|_{r}}+\frac{1}{|a_{5}-1|_{r}}+\frac{1}{|a_{4}-a_{5}|_{r}}\Big{)}$
$\displaystyle\leq\log^{+}\frac{1}{|a_{4}|_{r}}+\log^{+}\frac{1}{|a_{5}|_{r}}+\log^{+}\frac{1}{|a_{4}-1|_{r}}+\log^{+}\frac{1}{|a_{5}-1|_{r}}$
$\displaystyle\quad+\log^{+}\frac{1}{|a_{4}-a_{5}|_{r}}+\log 6$
$\displaystyle=m\big{(}r,\frac{1}{a_{4}}\big{)}+m\big{(}r,\frac{1}{a_{5}}\big{)}+m\big{(}r,\frac{1}{a_{4}-1}\big{)}+m\big{(}r,\frac{1}{a_{5}-1}\big{)}$
$\displaystyle\quad+m\big{(}r,\frac{1}{a_{4}-a_{5}}\big{)}+\log 6.$

Hence, we get

$\log^{+}\frac{1}{\delta(r)}=S(r,f).$ (3.5)

We first consider the case that

$|f-a_{j}|_{r}>\frac{1}{2}\delta(r),$

for all $2\leq j\leq 5.$ In this case,

$\displaystyle
m\big{(}r,\frac{1}{f}\big{)}+m\big{(}r,\frac{1}{f-1}\big{)}+m\big{(}r,\frac{1}{f-a_{4}}\big{)}+m\big{(}r,\frac{1}{f-a_{5}}\big{)}$
$\displaystyle<5\log^{+}\frac{1}{\delta(r)}+O(1)$ $\displaystyle=S(r,f).$
(3.6)

Now let $i,\,2\leq i\leq 5,$ be the index among $\\{2,3,4,5\\}$ such that

$|f-a_{i}|_{r}\leq\frac{1}{2}\delta(r).$

Then for any $j\neq i,\,2\leq j\leq 5,$ we have

$\delta(r)\leq|a_{i}-a_{j}|_{r}\leq|f-a_{i}|_{r}+|f-a_{j}|_{r}\leq\frac{1}{2}\delta(r)+|f-a_{j}|_{r},$

so

$|f-a_{j}|_{r}\geq\frac{1}{2}\delta(r).$

Therefore, for $j\neq i,$ we have

$\displaystyle\sum_{\begin{subarray}{c}j=2\\\ j\neq
i\end{subarray}}^{5}m\big{(}r,\frac{1}{f-a_{j}}\big{)}=\sum_{\begin{subarray}{c}j=2\\\
j\neq i\end{subarray}}^{5}\log^{+}\frac{1}{|f-a_{j}|_{r}}\leq
3\log^{+}\frac{1}{\delta(r)}.$

Combining (3.5) and the above inequality, we get

$\sum_{\begin{subarray}{c}j=2\\\ j\neq
i\end{subarray}}^{5}m\big{(}r,\frac{1}{f-a_{j}}\big{)}=S(r,f).$ (3.7)

On the other hand, for $2\leq i\leq 5,$ we can write

$\displaystyle
ff^{\prime}=(f-a_{i})(f^{\prime}-a^{\prime}_{i})+a^{\prime}_{i}(f-a_{i})+a_{i}(f^{\prime}-a^{\prime}_{i})+a_{i}a^{\prime}_{i},$
$\displaystyle f^{\prime}=(f^{\prime}-a^{\prime}_{i})+a^{\prime}_{i},$
$\displaystyle
f(f-1)=f^{2}-f=(f-a_{i})^{2}+(2a_{i}-1)(f-a_{i})+a_{i}^{2}-a_{i}.$

By substituting the above equalities into (3.1) and using the determinant’s
properties, we get

$H=\begin{vmatrix}g_{i}&f^{\prime}-a^{\prime}_{i}&h_{i}\\\
a_{4}a^{\prime}_{4}&a^{\prime}_{4}&a_{4}(a_{4}-1)\\\
a_{5}a^{\prime}_{5}&a^{\prime}_{5}&a_{5}(a_{5}-1)\end{vmatrix},$ (3.8)

where

$g_{i}=(f-a_{i})(f^{\prime}-a^{\prime}_{i})+a^{\prime}_{i}(f-a_{i})+a_{i}(f^{\prime}-a^{\prime}_{i}),$
$h_{i}=(f-a_{i})^{2}+(2a_{i}-1)(f-a_{i})$

for $2\leq i\leq 5,$ note that $a_{2}=0,a_{3}=1.$ By the definition of
$\delta(r),$ we have $\delta(r)\leq 1+|a_{i}|_{r}.$ Hence,

$\log^{+}\delta(r)\leq\log^{+}(1+|a_{i}|_{r})\leq\log^{+}|a_{i}|_{r}+\log
2=m(r,a_{i})+\log 2=S(r,f).$

Thus, it follows from (3.8) and the Logarithmic Derivative Lemma that

$\displaystyle\log^{+}\Big{|}\frac{H}{f-a_{i}}\Big{|}_{r}$
$\displaystyle\leq\log^{+}\Big{|}\frac{f^{\prime}-a^{\prime}_{i}}{f-a_{i}}\Big{|}_{r}+\log^{+}|f-a_{i}|_{r}$
$\displaystyle\quad+O(\log^{+}|a_{i}|_{r}+\log^{+}|a^{\prime}_{i}|_{r}+\log^{+}|a_{4}|_{r}+\log^{+}|a^{\prime}_{4}|_{r}$
$\displaystyle\quad+\log^{+}|a_{5}|_{r}+\log^{+}|a^{\prime}_{5}|_{r})$
$\displaystyle\leq
m\big{(}\frac{f^{\prime}-a^{\prime}_{i}}{f-a_{i}}\big{)}+\log^{+}\delta(r)+S(r,f)$
$\displaystyle=S(r,f).$

Hence, we get

$\displaystyle m\big{(}r,\frac{1}{f-a_{i}}\big{)}$
$\displaystyle=\log^{+}\frac{1}{|f-a_{i}|_{r}}\leq\log^{+}\Big{|}\frac{H}{f-a_{i}}\Big{|}_{r}+\log^{+}\Big{|}\frac{1}{H}\Big{|}_{r}$
$\displaystyle\leq m\big{(}r,\frac{1}{H}\big{)}+S(r,f).$ (3.9)

It follows from (3), (3.8) and (3) that in any case, we have

$\displaystyle
m\big{(}r,\frac{1}{f}\big{)}+m\big{(}r,\frac{1}{f-1}\big{)}+m\big{(}r,\frac{1}{f-a_{4}}\big{)}+m\big{(}r,\frac{1}{f-a_{5}}\big{)}\leq
m\big{(}r,\frac{1}{H}\big{)}+S(r,f).$ (3.10)

Hence, by the First Main Theorem, we get

$\displaystyle 4T(r,f)$ $\displaystyle\leq
N\big{(}r,\frac{1}{f}\big{)}+N\big{(}r,\frac{1}{f-1}\big{)}+N\big{(}r,\frac{1}{f-a_{4}}\big{)}+N\big{(}r,\frac{1}{f-a_{5}}\big{)}$
$\displaystyle\quad+T(r,H)-N\big{(}r,\frac{1}{H}\big{)}+S(r,f).$ (3.11)

On the other hand, suppose that $z_{0}$ be a zero of $f-a_{i},$ $(2\leq i\leq
5)$ of order $s>1$ which is not a pole of $a_{4}$ or $a_{5}.$ Then, it follows
from (3.8) that $z_{0}$ is also a zero of $H$ of order at least $s-1.$ Hence,
from (3) and the above observations, we get

$\displaystyle 4T(r,f)$
$\displaystyle\leq\overline{N}\big{(}r,\frac{1}{f}\big{)}+\overline{N}\big{(}r,\frac{1}{f-1}\big{)}+\overline{N}\big{(}r,\frac{1}{f-a_{4}}\big{)}+\overline{N}\big{(}r,\frac{1}{f-a_{5}}\big{)}$
$\displaystyle\quad+T(r,H)+S(r,f).$ (3.12)

From (3), we have

$\displaystyle m(r,H)$ $\displaystyle\leq 2m(r,f)+S(r,f),$ $\displaystyle
N(r,H)$ $\displaystyle\leq 2N(r,f)+\overline{N}(r,f)+S(r,f).$

Hence, we get

$T(r,H)\leq 2T(r,f)+\overline{N}(r,f)+S(r,f).$ (3.13)

Combining (3) and (3.13), we obtain

$2T(r,f)\leq\overline{N}\big{(}r,\frac{1}{f}\big{)}+\overline{N}\big{(}r,\frac{1}{f-1}\big{)}+\overline{N}\big{(}r,\frac{1}{f-a_{4}}\big{)}+\overline{N}\big{(}r,\frac{1}{f-a_{5}}\big{)}+S(r,f).$

This completes the proof of Lemma 2. ∎

###### Proof of Theorem 1..

By Lemma 2, for every subset $\\{i_{1},\dots,i_{5}\\}$ of $\\{1,\dots,q\\}$
such that $1\leq i_{1}<\dots<i_{5}\leq q,$ we have

$2T(r,f)\leq\sum_{s=1}^{5}\overline{N}\big{(}r,\frac{1}{f-a_{i_{s}}}\big{)}+S(r,f).$
(3.14)

It is easily seen that the number of such inequalities is
$\mathrm{C}_{q}^{5}.$ Taking summing up of (3.14) over all subsets
$\\{i_{1},\dots,i_{5}\\}$ of $\\{1,\dots,q\\}$ as above, we get

$\displaystyle 2\mathrm{C}_{q}^{5}T(r,f)\leq$
$\displaystyle\sum_{\begin{subarray}{c}\\{i_{1},\dots,i_{5}\\}\subset\\{1,\dots,q\\}\\\
1\leq i_{1}<\dots<i_{5}\leq
q\end{subarray}}\Big{(}\overline{N}\big{(}r,\frac{1}{f-a_{i_{1}}}\big{)}+\overline{N}\big{(}r,\frac{1}{f-a_{i_{2}}}\big{)}+\overline{N}\big{(}r,\frac{1}{f-a_{i_{3}}}\big{)}$
$\displaystyle+\overline{N}\big{(}r,\frac{1}{f-a_{i_{4}}}\big{)}+\overline{N}\big{(}r,\frac{1}{f-a_{i_{5}}}\big{)}\Big{)}+S(r,f).$
(3.15)

In (3), for each index $i_{k},$ the number of terms
$\overline{N}\big{(}r,\frac{1}{f-a_{i_{k}}}\big{)}$ is $\mathrm{C}_{q-1}^{4}.$
Hence, from (3), we get

$2\mathrm{C}_{q}^{5}T(r,f)\leq\mathrm{C}_{q-1}^{4}\sum_{i=1}^{q}\overline{N}\big{(}r,\frac{1}{f-a_{i}}\big{)}+S(r,f).$

It follows that

$\frac{2q}{5}T(r,f)\leq\sum_{i=1}^{q}\overline{N}\big{(}r,\frac{1}{f-a_{i}}\big{)}+S(r,f).$

This completes the proof of Theorem 1. ∎

##  4\. Proof of Theorem 2

To prove Theorem 2, we need to prove the following lemma.

######  Lemma 3.

Let $f$ and $g$ be nonconstant meromorphic functions on $\mathbf{K}$ and
$a_{1},\dots,a_{q}$ be $q$ distinct small functions with respect to $f$ and
$g.$ Let $k_{1},\dots,k_{q}$ be $q$ positive integers or $+\infty.$ Suppose
that

$\overline{E}(a_{j},k_{j},f)=\overline{E}(a_{j},k_{j},g)\quad(j=1,\dots,q).$

If $f\not\equiv g,$ then for every subset $\\{i_{1},i_{2},i_{3},i_{4}\\}$ of
$\\{1,\dots,q\\},$ we have

$\displaystyle\sum_{j\in\\{1,\dots,q\\}\setminus\\{i_{1},\dots,i_{4}\\}}\overline{N}_{k_{j})}\big{(}r,\frac{1}{f-a_{j}}\big{)}\leq$
$\displaystyle\sum_{s=1}^{4}\Big{(}\overline{N}_{(k_{i_{s}}+1}\big{(}r,\frac{1}{f-a_{i_{s}}}\big{)}+\overline{N}_{(k_{i_{s}}+1}\big{(}r,\frac{1}{g-a_{i_{s}}}\big{)}\Big{)}$
$\displaystyle+S(r,f)+S(r,g).$

###### Proof.

Without losing generality, we just need to prove that

$\displaystyle\sum_{i=5}^{q}\overline{N}_{k_{i})}\big{(}r,\frac{1}{f-a_{i}}\big{)}\leq$
$\displaystyle\sum_{j=1}^{4}\Big{(}\overline{N}_{(k_{j}+1}\big{(}r,\frac{1}{f-a_{j}}\big{)}+\overline{N}_{(k_{j}+1}\big{(}r,\frac{1}{g-a_{j}}\big{)}\Big{)}$
$\displaystyle+S(r,f)+S(r,g).$ (4.1)

If
$\sum_{i=5}^{q}\overline{N}_{k_{i})}\big{(}r,\frac{1}{f-a_{i}}\big{)}=S(r,f)+S(r,g),$
then (4) obviously holds. Thus, in the following we may assume that

$\sum_{i=5}^{q}\overline{N}_{k_{i})}\big{(}r,\frac{1}{f-a_{i}}\big{)}\neq
S(r,f)+S(r,g).$ (4.2)

By using the transformation

$L(w)=\frac{w-a_{1}}{w-a_{2}}\cdot\frac{a_{3}-a_{2}}{a_{3}-a_{1}}$

and considering two functions $F=L(f),G=L(g)$ if necessary, we may assume that
$a_{1}=0,a_{2}=\infty,a_{3}=1$ and $a_{4},\dots,a_{q}$ are distinct small
functions with respect to $f$ and $g,$ $a_{i}\not\equiv
0,1,\infty\,\,(i=4,\dots,q).$

Set

$T:=\frac{f^{\prime}(a^{\prime}g-ag^{\prime})(f-g)}{f(f-1)g(g-a)}-\frac{g^{\prime}(a^{\prime}f-af^{\prime})(f-g)}{g(g-1)f(f-a)},$
(4.3)

where $a=a_{4}.$ Then we have

$T=\frac{(f-g)Q}{f(f-1)(f-a)g(g-1)(g-a)},$ (4.4)

where

$\displaystyle Q$
$\displaystyle=f^{\prime}(a^{\prime}g-ag^{\prime})(f-a)(g-1)-g^{\prime}(a^{\prime}f-af^{\prime})(g-a)(f-1)$
$\displaystyle=a^{\prime}ff^{\prime}g^{2}-a^{\prime}ff^{\prime}g-a(a-1)ff^{\prime}g^{\prime}-aa^{\prime}f^{\prime}g^{2}+aa^{\prime}f^{\prime}g-a^{\prime}f^{2}gg^{\prime}$
$\displaystyle\quad+a^{\prime}fgg^{\prime}+a(a-1)f^{\prime}gg^{\prime}+aa^{\prime}f^{2}g^{\prime}-aa^{\prime}fg^{\prime}.$
(4.5)

Suppose that $T\equiv 0.$ Then from (4.3) we have

$\frac{f^{\prime}(a^{\prime}g-ag^{\prime})(f-g)}{f(f-1)g(g-a)}\equiv\frac{g^{\prime}(a^{\prime}f-af^{\prime})(f-g)}{g(g-1)f(f-a)}.$
(4.6)

If $a$ is a constant then $f\equiv g,$ which contradicts our assumption. Thus,
$a$ is not a constant. It follows from (4.6) that

$\frac{(f-1)(g-a)}{(g-1)(f-a)}-1\equiv\frac{f^{\prime}(a^{\prime}g-ag^{\prime})}{g^{\prime}(a^{\prime}f-af^{\prime})}-1,$

which implies

$\frac{(f-g)(1-a)}{(g-1)(f-a)}\equiv\frac{a^{\prime}[(f^{\prime}-g^{\prime})g-(f-g)g^{\prime}]}{g^{\prime}(a^{\prime}f-af^{\prime})}.$

This yield that

$\frac{f^{\prime}-g^{\prime}}{f-g}\equiv\frac{(1-a)g^{\prime}(a^{\prime}f-af^{\prime})}{a^{\prime}g(g-1)(f-a)}+\frac{g^{\prime}}{g}.$
(4.7)

It follows from (4.3) that there exists a point $z_{0}$ such that $z_{0}$ is a
common zero of $f-a_{j}$ and $g-a_{j}$ that is not a zero or a pole of
$a,a^{\prime},a_{j},a_{j}-1,a_{j}-a$ $(5\leq j\leq q).$ Then, $z_{0}$ must be
a pole of the left hand side of (4.7), and not be pole of the right hand side
of (4.7). This is a contradiction. Thus $T\not\equiv 0.$

Suppose that $z_{1}$ is a common zero of $f-a_{j}$ and $g-a_{j}$ that is not a
zero or a pole of $a,a_{j},a_{j}-1,a_{j}-a$ $(5\leq j\leq q).$ Then, it is a
zero of $f-g$ and is not a pole of

$\frac{Q}{f(f-1)(f-a)g(g-1)(g-a)},$

and hence it is a zero of $T.$ Therefore, since
$\overline{E}(a_{j},k_{j},f)=\overline{E}(a_{j},k_{j},g)\quad(j=1,\dots,q),$
we have

$\displaystyle\sum_{i=5}^{q}\overline{N}_{k_{i})}\big{(}r,\frac{1}{g-a_{i}}\big{)}$
$\displaystyle=\sum_{i=5}^{q}\overline{N}_{k_{i})}\big{(}r,\frac{1}{f-a_{i}}\big{)}$
$\displaystyle\leq N\big{(}r,\frac{1}{T}\big{)}+S(r,f)+S(r,g)$
$\displaystyle\leq m(r,T)+N(r,T)+S(r,f)+S(r,g).$ (4.8)

We estimate $m(r,T).$ From (4.3) we get

$\displaystyle T$
$\displaystyle=\frac{f^{\prime}}{f-1}\frac{a^{\prime}g-ag^{\prime}}{g(g-a)}-\Big{(}\frac{f^{\prime}}{f-1}-\frac{f^{\prime}}{f}\Big{)}\frac{a^{\prime}g-ag^{\prime}}{g-a}$
$\displaystyle\quad+\frac{g^{\prime}}{g-1}\frac{a^{\prime}f-af^{\prime}}{f(f-a)}-\Big{(}\frac{g^{\prime}}{g-1}-\frac{g^{\prime}}{g}\Big{)}\frac{a^{\prime}f-af^{\prime}}{f-a}$
$\displaystyle=\frac{f^{\prime}}{f-1}\Big{(}\frac{g^{\prime}}{g}-\frac{g^{\prime}-a^{\prime}}{g-a}\Big{)}-\Big{(}\frac{f^{\prime}}{f-1}-\frac{f^{\prime}}{f}\Big{)}\Big{(}a^{\prime}-a\frac{g^{\prime}-a^{\prime}}{g-a}\Big{)}$
$\displaystyle\quad\frac{g^{\prime}}{g-1}\Big{(}\frac{f^{\prime}}{f}-\frac{f^{\prime}-a^{\prime}}{f-a}\Big{)}-\Big{(}\frac{g^{\prime}}{g-1}-\frac{g^{\prime}}{g}\Big{)}\Big{(}a^{\prime}-a\frac{f^{\prime}-a^{\prime}}{f-a}\Big{)}.$
(4.9)

Combining (4) and lemma of the logarithmic derivative, we obtain

$m(r,T)=S(r,f)+S(r,g).$ (4.10)

Next, we estimate the counting function $N(r,T).$ it follows from (4.3) that
the poles of $T$ can only occur at the zeros of $f,g,f-1,g-1,f-a,g-a$ and the
poles of $f,g,a.$ Let $A$ be the set of all zeros, $1-$points and poles of
$a.$ We consider all the following possibilities.

If $z\not\in A$ is a common pole of $f$ and $g$ of order $p_{1}$ and $q_{1},$
respectively. It follows from (4) that $z$ is a pole of $Q$ of order at most
$2p_{1}+2q_{1}+1.$ We have $z$ is a pole of $f-g$ of order
$\max\\{p_{1},q_{1}\\}.$ Hence, from (4.4) we see that $z$ is a pole of the
numerator of $T$ of order at most $2p_{1}+2q_{1}+1+\max\\{p_{1},q_{1}\\}$ and
it is a pole of the denominator of $T$ of order $3p_{1}+3q_{1}.$ Since

$2p_{1}+2q_{1}+1+\max\\{p_{1},q_{1}\\}-(3p_{1}+3q_{1})=1+max\\{p_{1},q_{1}\\}-(p_{1}+q_{1})\leq
0,$

it follows that $z$ is not a pole of $T.$

If $z\not\in A$ is a common zero of $f$ and $g$ of order $p_{2}$ and $q_{2},$
respectively. Then $z$ is a zero of $f-g$ of order
$\mathop{\mathrm{min}}\\{p_{2},q_{2}\\}.$ From (4) we have $z$ is a zero of
$Q$ of order at least $p_{2}+q_{2}-1.$ From (4.4) we have

$\mathop{\mathrm{min}}\\{p_{2},q_{2}\\}+p_{2}+q_{2}-1-(p_{2}+q_{2})=\mathop{\mathrm{min}}\\{p_{2},q_{2}\\}-1\geq
0.$

Hence, $z$ is not a pole of $T.$

If $z\not\in A$ is a common zero of $f-a_{i}$ and $g-a_{i}$ with $i=3,4,$ note
that $a_{3}=1,a_{4}=a.$ Then $z$ is a zero of $f-g.$ On the other hand, we
have $z$ is a simple pole of $\frac{f^{\prime}}{f-a_{i}}$ and
$\frac{g^{\prime}}{g-a_{i}}.$ Hence, it follows from (4.4) that $z$ is not a
pole of $T.$

If $z\not\in A$ is a pole of $f$ but is not a pole of $g$ and is not a zero of
$g,g-1$ and $g-a$ or $z\not\in A$ is a pole of $g$ but is not a pole of $f$
and is not a zero of $f,f-1$ and $f-a.$ It follows from (4.4) that $z$ is a
pole of $T$ of order at most $1.$

If $z\not\in A$ is a zero of $f$ but is not a pole of $g$ and is not a zero of
$g,g-1$ and $g-a$ or $z\not\in A$ is a zero of $g$ but is not a pole of $f$
and is not a zero of $f,f-1$ and $f-a.$ Then, it is easily derived from (4.4)
that $z$ is a pole of $T$ of order at most $1.$

If $z\not\in A$ is a zero of $f-a_{i}$ with $i=3,4$ but is not a pole of $g$
and is not a zero of $g,g-1$ and $g-a$ or $z\not\in A$ is a zero of $g-a_{i}$
with $i=3,4$ but is not a pole of $f$ and is not a zero of $f,f-1$ and $f-a.$
Then, from (4.4) we see that $z$ is a pole of $T$ of order at most $1.$

If $z\not\in A$ is a common zero of $f-a_{i}$ and $g-a_{j}$ with $1\leq
i,j\leq 4,\,\,i\neq j,$ here we regard $f-\infty=\frac{1}{f}$ and
$g-\infty=\frac{1}{g}.$ Then, it follows from (4.4) that $z$ is a pole of $T$
of at most $2.$

So, from the assumption and the above observations, we get

$N(r,T)\leq\sum_{j=1}^{4}\Big{(}\overline{N}_{(k_{j}+1}\big{(}r,\frac{1}{f-a_{j}}\big{)}+\overline{N}_{(k_{j}+1}\big{(}r,\frac{1}{g-a_{j}}\big{)}\Big{)}+S(r,f)+S(r,g).$
(4.11)

By combining (4), (4.10) and (4.11), we get (4). The proof of Lemma 3 is
completed. ∎

###### Proof of Theorem 2..

Suppose that $f\not\equiv g.$ By Lemma 3, for every subset
$\\{i_{1},\dots,i_{4}\\}$ of $\\{1,\dots,q\\},$ we have

$\displaystyle\sum_{j=1}^{q}\overline{N}_{k_{j})}\big{(}r,\frac{1}{f-a_{j}}\big{)}-\sum_{s=1}^{4}\overline{N}_{k_{i_{s}})}\big{(}r,\frac{1}{f-a_{i_{s}}}\big{)}$
$\displaystyle\leq\sum_{s=1}^{4}\Big{(}\overline{N}_{(k_{i_{s}}+1}\big{(}r,\frac{1}{f-a_{i_{s}}}\big{)}+\overline{N}_{(k_{i_{s}}+1}\big{(}r,\frac{1}{g-a_{i_{s}}}\big{)}\Big{)}+S(r,f)+S(r,g).$
(4.12)

Taking summing up of (4) over all subsets $\\{i_{1},\dots,i_{4}\\}$ of
$\\{1,\dots,q\\},$ we get

$\displaystyle\mathrm{C}_{q}^{4}\sum_{j=1}^{q}\overline{N}_{k_{j})}\big{(}r,\frac{1}{f-a_{j}}\big{)}-\sum_{\begin{subarray}{c}\\{i_{1},\dots,i_{4}\\}\subset\\{1,\dots,q\\}\\\
1\leq i_{1}<\dots<i_{4}\leq
q\end{subarray}}\sum_{s=1}^{4}\overline{N}_{k_{i_{s}})}\big{(}r,\frac{1}{f-a_{i_{s}}}\big{)}$
$\displaystyle\leq\sum_{\begin{subarray}{c}\\{i_{1},\dots,i_{4}\\}\subset\\{1,\dots,q\\}\\\
1\leq i_{1}<\dots<i_{4}\leq
q\end{subarray}}\sum_{s=1}^{4}\Big{(}\overline{N}_{(k_{i_{s}}+1}\big{(}r,\frac{1}{f-a_{i_{s}}}\big{)}+\overline{N}_{(k_{i_{s}}+1}\big{(}r,\frac{1}{g-a_{i_{s}}}\big{)}\Big{)}$
$\displaystyle\quad+S(r,f)+S(r,g).$

In the above inequality, for each index $i_{s},$ the number of terms
$\overline{N}\big{(}r,\frac{1}{f-a_{i_{s}}}\big{)}$ is $\mathrm{C}_{q-1}^{3}.$
Hence, it follows that

$\displaystyle(q-4)\sum_{j=1}^{q}\overline{N}_{k_{j})}\big{(}r,\frac{1}{f-a_{j}}\big{)}\leq$
$\displaystyle
4\sum_{j=1}^{q}\Big{(}\overline{N}_{(k_{j}+1}\big{(}r,\frac{1}{f-a_{j}}\big{)}+\overline{N}_{(k_{j}+1}\big{(}r,\frac{1}{g-a_{j}}\big{)}\Big{)}$
$\displaystyle+S(r,f)+S(r,g).$

By an argument similar, we have

$\displaystyle(q-4)\sum_{j=1}^{q}\overline{N}_{k_{j})}\big{(}r,\frac{1}{g-a_{j}}\big{)}\leq$
$\displaystyle
4\sum_{j=1}^{q}\Big{(}\overline{N}_{(k_{j}+1}\big{(}r,\frac{1}{f-a_{j}}\big{)}+\overline{N}_{(k_{j}+1}\big{(}r,\frac{1}{g-a_{j}}\big{)}\Big{)}$
$\displaystyle+S(r,f)+S(r,g).$

Hence, we get

$\displaystyle(q-4)\sum_{j=1}^{q}\Big{(}\overline{N}\big{(}r,\frac{1}{f-a_{j}}\big{)}+\overline{N}\big{(}r,\frac{1}{g-a_{j}}\big{)}\Big{)}$
$\displaystyle\quad\leq(q+4)\sum_{j=1}^{q}\Big{(}\overline{N}_{(k_{j}+1}\big{(}r,\frac{1}{f-a_{j}}\big{)}+\overline{N}_{(k_{j}+1}\big{(}r,\frac{1}{g-a_{j}}\big{)}\Big{)}+S(r,f)+S(r,g).$
(4.13)

By Theorem 1, we have

$\displaystyle\frac{2q}{5}(T(r,f)+T(r,g))$
$\displaystyle\leq\sum_{j=1}^{q}\Big{(}\overline{N}\big{(}r,\frac{1}{f-a_{j}}\big{)}+\overline{N}\big{(}r,\frac{1}{g-a_{j}}\big{)}\Big{)}$
$\displaystyle\quad+S(r,f)+S(r,g).$ (4.14)

Combining (4) and (4), we get

$\displaystyle\frac{2q(q-4)}{5}(T(r,f)+T(r,g))$
$\displaystyle\leq(q+4)\sum_{j=1}^{q}\Big{(}\overline{N}_{(k_{j}+1}\big{(}r,\frac{1}{f-a_{j}}\big{)}+\overline{N}_{(k_{j}+1}\big{(}r,\frac{1}{g-a_{j}}\big{)}\Big{)}$
$\displaystyle\quad+S(r,f)+S(r,g).$ (4.15)

On the other hand, we have

$\displaystyle\sum_{j=1}^{q}\Big{(}\overline{N}_{(k_{j}+1}\big{(}r,\frac{1}{f-a_{j}}\big{)}+\overline{N}_{(k_{j}+1}\big{(}r,\frac{1}{g-a_{j}}\big{)}\Big{)}$
$\displaystyle\leq\sum_{j=1}^{q}\frac{1}{k_{j}+1}\Big{(}N_{(k_{j}+1}\big{(}r,\frac{1}{f-a_{j}}\big{)}+N_{(k_{j}+1}\big{(}r,\frac{1}{g-a_{j}}\big{)}\Big{)}$
$\displaystyle\leq\sum_{j=1}^{q}\frac{1}{k_{j}+1}(T(r,f)+T(r,g)).$ (4.16)

The inequalities (4) and (4) imply

$\Big{(}\frac{2q(q-4)}{5(q+4)}-\sum_{j=1}^{q}\frac{1}{k_{j}+1}\Big{)}(T(r,f)+T(r,g))\leq
S(r,f)+S(r,g).$

Hence, when $\sum_{j=1}^{q}\frac{1}{k_{j}+1}<\frac{2q(q-4)}{5(q+4)},$ we have
a contradiction. Thus, $f\equiv g.$ The proof of Theorem 2 is completed. ∎

##  5\. Proof of Theorem 3

First, we prove the following lemma.

######  Lemma 4.

Let $f$ be a nonconstant meromorphic function on $\mathbf{K}.$ Let
$a_{1},\dots,a_{q}$ be $q$ distinct small functions with respect to $f.$ Let
$k$ be a positive integer or $+\infty.$ Then

$\sum_{i=1}^{q}\overline{N}_{(k+1}\big{(}r,\frac{1}{f-a_{i}}\big{)}\leq\frac{3q}{5k}T(r,f)+S(r,f).$

###### Proof.

For each $1\leq i\leq q,$ we have

$\displaystyle
k\overline{N}_{(k+1}\big{(}r,\frac{1}{f-a_{i}}\big{)}+\overline{N}\big{(}r,\frac{1}{f-a_{i}}\big{)}$
$\displaystyle=(k+1)\overline{N}_{(k+1}\big{(}r,\frac{1}{f-a_{i}}\big{)}+\overline{N}_{k)}\big{(}r,\frac{1}{f-a_{i}}\big{)}$
$\displaystyle\leq
N_{(k+1}\big{(}r,\frac{1}{f-a_{i}}\big{)}+N_{k)}\big{(}r,\frac{1}{f-a_{i}}\big{)}$
$\displaystyle=N\big{(}r,\frac{1}{f-a_{i}}\big{)}\leq T(r,f)+S(r,f).$

Hence, we get

$k\overline{N}_{(k+1}\big{(}r,\frac{1}{f-a_{i}}\big{)}\leq
T(r,f)-\overline{N}\big{(}r,\frac{1}{f-a_{i}}\big{)}+S(r,f).$

Combining this and Theorem 1, we obtain

$k\sum_{i=1}^{q}\overline{N}_{(k+1}\big{(}r,\frac{1}{f-a_{i}}\big{)}\leq\frac{3q}{5}T(r,f)+S(r,f).$

This completes the proof of Lemma. ∎

###### Proof of Theorem 3..

Suppose that $f\not\equiv g.$ By arguments similar to the inequality (4) in
the proof of Theorem 2, we get

$\displaystyle\frac{2q(q-4)}{5}(T(r,f)+T(r,g))\leq$
$\displaystyle(q+4)\sum_{j=1}^{q}\Big{(}\overline{N}_{(k+1}\big{(}r,\frac{1}{f-a_{j}}\big{)}+\overline{N}_{(k+1}\big{(}r,\frac{1}{g-a_{j}}\big{)}\Big{)}$
$\displaystyle+S(r,f)+S(r,g).$ (5.1)

By Lemma 4, we have

$\displaystyle\sum_{j=1}^{q}\Big{(}\overline{N}_{(k+1}\big{(}r,\frac{1}{f-a_{j}}\big{)}+\overline{N}_{(k+1}\big{(}r,\frac{1}{g-a_{j}}\big{)}\Big{)}\leq$
$\displaystyle\frac{3q}{5k}(T(r,f)+T(r,g))$ $\displaystyle+S(r,f)+S(r,g).$
(5.2)

Combining (5) and (5), we obtain

$\frac{q}{5}\Big{(}2(q-4)-\frac{3(q+4)}{k}\Big{)}(T(r,f)+T(r,g))\leq
S(r,f)+S(r,g).$

Thus, when $k>\frac{3(q+4)}{2(q-4)},$ we have a contradiction. Hence, $f\equiv
g.$ ∎

Acknowledgments. A part of this article was written while the first author was
visiting Vietnam Institute for Advanced Study in Mathematics (VIASM). She
would like to thank the institute for warm hospitality and partial support.

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