1 Introduction
The gravity is an essential attribute of any physical matter. Therefore, the study of gravity has great theoretical significance and extensive application value.
The theory of satellite is important in space science. In [
1‐
4], the authors systematically studied the theory of satellite and obtained some interesting results. In particular, in [
2], the authors defined the centered 2-surround system, established several geometric inequalities for the centered 2-surround system under the proper hypotheses, and illustrated the background of the centered 2-surround system in space science.
It is well known that the Moon is a satellite of the Earth. In space science, we are concerned with the gravity of the Moon since the gravity may be disastrous causing tsunami and tidal wave, etc.
In this paper, we first define the mean central distance
\(\bar{r}_{P}\) of a centered 2-surround system \(S^{(2)} \{ P,\varGamma ,l \}\). Next, we study the boundary curve of the l-central regions and the properties of the asymptotic system and establish several identities and inequalities involving the centered 2-surround system. Next, we prove an isoperimetric inequality in the centered 2-surround system. Finally, we demonstrate the application of our results in space science and obtain the best lower bounds of the mean
λ-gravity norm
\(\overline {\Vert {\mathbf{F}}_{\lambda} ( \varGamma ,P )\Vert }\).
A large number of algebraic, functional analysis, differential equation, convex geometry, physics, computer, and inequality theories are used in this paper. The proofs of our results are both interesting and difficult, as well as which are depend on our previous work. Some of our proof methods can also be found in the references of this paper, such as [
1‐
3].
2 Basic concepts and main results
We continue to use the notation of the references [
1‐
3].
We begin by recalling some of the basic concepts and preliminary results of [
1‐
3].
Let
\(\gamma:I\rightarrow\mathbb{R}^{2}\) be a continuous function, where
\(I\subset\mathbb{R}\) is an interval, and let the image
$$\varGamma \triangleq\gamma(I)= \bigl\{ \gamma(t)\in\mathbb{R}^{2}|\gamma (t)=x(t) \mathbf{i}+y(t) \mathbf{j}, t \in I \bigr\} $$
of
γ be a smooth curve [
5], that is, the derivatives
\(x'(t)\) and
\(y'(t)\) are continuous, and the derivative of the vector
\(\gamma(t)\) satisfies the condition
$$\gamma'(t)\triangleq x'(t) \mathbf{i}+y'(t) \mathbf{j} \ne \mathbf{0},\quad \forall t\in I, $$
where
$$\mathbf{0}=(0,0),\qquad \mathbf{i}=(1,0),\qquad \mathbf{j}=(0,1),\qquad \mathbb {R}\triangleq(-\infty,\infty), \qquad \mathbb{R}^{2}\triangleq \mathbb{R} \times \mathbb{R}. $$
Then the length
\(|\varGamma |\) of the curve
Γ exists:
$$|\varGamma |\triangleq \int_{I}\bigl\Vert \gamma'(t)\bigr\Vert \, \mathrm{d}t= \int_{I}\sqrt {\bigl[x'(t)\bigr]^{2}+ \bigl[y'(t)\bigr]^{2}} \,\mathrm{d}t>0, $$
and
\(|\varGamma |<\infty\) if
I is a bounded interval, where the norm
\(\| x\mathbf{i}+y\mathbf{j}\|\) of the vector
\(x\mathbf{i}+y\mathbf{j}\in \mathbb{R}^{2}\) is defined as
$$\|x\mathbf{i}+y\mathbf{j}\|\triangleq\sqrt{x^{2}+y^{2}}. $$
In this paper, we assume that
Γ is a smooth and convex Jordan closed curve in
\(\mathbb{R}^{2}\) [
1‐
3]. Then
$$\varGamma \triangleq\gamma(\mathbb{R})= \bigl\{ \gamma(t)\in\mathbb {R}^{2}|\gamma(t)=x(t) \mathbf{i}+y(t) \mathbf{j}, t \in\mathbb {R} \bigr\} \quad \text{and}\quad \gamma(t)\equiv\gamma\bigl(t+\vert \varGamma \vert \bigr), \quad \forall t\in \mathbb{R}, $$
that is,
\(\gamma(t)\) is a periodic function, where the parameter
t is the natural parameter, that is,
$$0< l\leqslant|\varGamma |\quad \Rightarrow\quad \bigl\vert \gamma\bigl([t,t+l] \bigr)\bigr\vert \triangleq \int _{t}^{t+l}\sqrt{\bigl[x'(t) \bigr]^{2}+\bigl[y'(t)\bigr]^{2}} \, \mathrm{d}t=l, \quad \forall t\in \mathbb{R}. $$
We denote by
\(D(\varGamma )\) the convex region enclosed by the Jordan closed curve
Γ, that is,
$$P_{1},P_{2}\in D(\varGamma )\quad \Rightarrow\quad \lambda P_{1}+(1-\lambda)P_{2}\in D(\varGamma ), \quad \forall \lambda \in[0,1] $$
and
$$\bigl\vert D(\varGamma )\bigr\vert \triangleq\operatorname{Area}D(\varGamma ) $$
denote the area of the region
\(D(\varGamma )\).
We remark here that, for the Jordan closed curve, we have the following
Jordan theorem [
3]: An arbitrary Jordan closed curve must divide a plane into two regions, and one of the regions is bounded and the another is unbounded. The bounded region is called the
interior and the another is called the
outside of the Jordan closed curve.
In this paper, we also assume that
$$ A_{-}=\gamma (t_{A}-l ),\qquad A=\gamma (t_{A} ),\qquad A_{+}=\gamma (t_{A}+l ),\quad t_{A}\in \mathbb{R}. $$
(1)
If
l is a fixed real number such that
\(0< l<|\varGamma |/2\), then we say that the plane point set
$$D(\varGamma ,l)\triangleq\bigcap_{A\in \varGamma }\widehat{A_{-}AA_{+}} \subset D(\varGamma )\subset\mathbb{R}^{2} $$
is an
l-
central region of the curve
Γ, where the
angular region
$$\widehat{A_{-}AA_{+}}\triangleq \bigl\{ (1-\lambda)\gamma(t_{A})+ \lambda \gamma(t) |0< \lambda< \infty, t_{A}+l< t< t_{A}-l+| \varGamma | \bigr\} . $$
Let the
l-central region
\(D(\varGamma ,l)\) be nonempty, and let
\(P\in D(\varGamma ,l)\) be a fixed point. We say that the set
$$S^{(2)} \{P,\varGamma ,l \}\triangleq \{P,\varGamma ,l \} $$
is a
centered 2-
surround system or
centered 2-
satellite system,
P is a
center and
\(A,A_{+}\in \varGamma \) are two
satellites of the system [
1‐
3].
For the centered 2-surround system
\(S^{(2)} \{P,\varGamma ,l \} \), we may think of the point
P as the center of the Earth,
Γ as the orbit of two satellites
A,
\(A_{+}\). In order to avoid hitting, the satellites
A,
\(A_{+}\) must move by the same curve velocity, that is,
$$l\triangleq\bigl|\gamma \bigl( [{t_{A},t_{A}+l} ] \bigr)\bigr|\in \biggl(0, \frac{|\varGamma |}{2} \biggr) $$
is invariable. This is the significance of the centered 2-surround system
\(S^{(2)} \{ P,\varGamma ,l \}\) in the theory of satellites.
We remark here that, in [
1,
3], the authors extended the centered 2-surround system
\(S^{(2)} \{P,\varGamma ,l \}\) and defined the centered
N-surround system, that is, if
\(S^{(2)} \{ P,\varGamma ,l_{j} \}\) is a centered 2-surround system, where
\(j=1,2,\ldots,N\),
\(N\geqslant3\), then we say that the set
$$S^{(2)} \{P,\varGamma ,\mathbf{l} \}\triangleq \{P,\varGamma ,\mathbf{l} \} $$
is a
centered
N-
surround system and
P is a
center of the system, where
$$\mathbf{l}\triangleq (l_{1},l_{2},\ldots,l_{N} )\in\mathbb {R}^{N}, \qquad 0< l_{j}< \frac{|\varGamma |}{2}, \quad \forall j\mbox{: }1\leqslant j \leqslant N, \qquad \sum _{j=1}^{N}l_{j}=|\varGamma |, $$
and if
$$A_{j}\triangleq\gamma \Biggl(t_{A}+\sum _{k=1}^{j}l_{k} \Biggr),\quad j=1,2, \ldots,N, $$
then we say that
\(A_{1},A_{2},\ldots,A_{N}\) are
N
satellites of the system.
We remark here that, where the
\((2)\) in
\(S^{(2)} \{P,\varGamma ,\mathbf {l} \}\) means that
\(P\in\mathbb{R}^{2}\) and
\(\varGamma \subset\mathbb {R}^{2}\). If
\(P\in\mathbb{R}^{m}\) and
\(\varGamma \subset\mathbb {R}^{m}\),
\(m\geqslant3\) [
6], then we can define [
1,
3]
$$S^{(m)} \{P,\varGamma ,\mathbf{l} \}\triangleq \{P,\varGamma ,\mathbf{l} \} $$
as a
centered
N-
surround system and
$$S^{(m)} \{\varGamma ,\mathbf{l} \}\triangleq \{\varGamma ,\mathbf {l} \} $$
as a
N-
surround system without any central.
For centered 2-surround system
\(S^{(2)} \{P,\varGamma ,l \}\), let
$$P'\triangleq\operatorname{Projection}_{AA_{+}}P $$
denote the projection of the point
P in the line
\(AA_{+}\), that is,
$$PP'\perp AA_{+} \quad \text{and}\quad PP' \cap AA_{+}=P'. $$
In the centered 2-surround system
\(S^{(2)} \{P,\varGamma ,l \} \), we say that the distance
$$r_{P}\triangleq\operatorname{Distance}(P,AA_{+})=\bigl\Vert P'-P\bigr\Vert $$
from the point
P to line
\(AA_{+}\) is a
central distance of the system, the distances
$$r_{A}\triangleq\operatorname{Distance}(A,PA_{+}) \quad \text{and} \quad r_{A_{+}}\triangleq \operatorname{Distance}(A_{+},PA) $$
are the
Brocard distances of the system [
2], and the positive real number
$$\bar{r}_{P}\triangleq\frac{1}{\|A_{+}-A\|} \int_{M\in[AA_{+}]}\| M-P\| $$
is the
mean central distance of the system, which is the mean of the distance between the point
P and the point
M in the straight line segment
$$[AA_{+}]\triangleq \bigl\{ (1-\lambda)A+\lambda A_{+}|0 \leqslant\lambda \leqslant1 \bigr\} . $$
We remark here that if
\(l=0\), then
\(\bar{r}_{P}=\|A-P\|\). This is another geometrical meaning of
\(\bar{r}_{P}\), which has applications in space science; see Section
5.
According to the definitions of the central distance and the
l-central region, we know that
\(r_{P}\) is a
support function of the curve
\(\partial D(\varGamma ,l)\), which is the boundary curve of the
l-central region
\(D(\varGamma ,l)\), and we have that [
7]
$$ \bigl\vert D(\varGamma ,l)\bigr\vert =\frac{1}{2} \oint_{\partial D(\varGamma ,l)}r_{P}, $$
(2)
where
$$\bigl\vert D(\varGamma ,l)\bigr\vert \triangleq\operatorname{Area}D(\varGamma ,l) $$
is the area of the
l-central region
\(D(\varGamma ,l)\).
Let
\(f:\varGamma \rightarrow(0,\infty)\) be a continuous function defined on the curve
Γ. Then the functional
$$M^{[p]}_{\varGamma }(f)\triangleq \left \{ \textstyle\begin{array}{l@{\quad}l} ( \frac{1}{|\varGamma |}\int_{\varGamma }f^{p} )^{1/p}, & p\in\mathbb{R}, p\neq0, \\ \exp ( \frac{1}{|\varGamma |}\int_{\varGamma }\ln f ),& p=0, \end{array}\displaystyle \right . $$
is called the
p-
power mean of the function
f, where
$$M_{\varGamma }(f)\triangleq M^{[1]}_{\varGamma }(f)= \frac{1}{|\varGamma |} \int _{\varGamma }f $$
is the mean of the function
f.
We remark here that
\(M^{[p]}_{\varGamma }(f)\) is increasing with respect to
p [
8‐
12], that is,
$$ p< q\quad \Rightarrow \quad M^{[p]}_{\varGamma }(f) \leqslant M^{[q]}_{\varGamma }(f), $$
(3)
where equality in (
3) holds if and only if
f is a constant function.
As pointed out in [
13], the theory of inequalities plays an important role in all the fields of mathematics. The concept of mean is the most prominent in the theory, and the
p-power mean is the crucial one. The references [
8‐
13] studied the sharp bounds of the
p-power mean.
In the convex geometry, a well-known isoperimetric inequality can be expressed as follows: If
Γ is a smooth Jordan closed curve, then we have
$$ \bigl\vert {D(\varGamma )}\bigr\vert \leqslant\frac{|\varGamma |^{2}}{4\pi}. $$
(4)
Equality in (
4) holds if and only if
Γ is a circle.
In the convex geometry, a large number of isoperimetric inequalities similar to (
4) was obtained [
14‐
16]. Recently, we obtained some new isoperimetric inequalities in the surround system [
1‐
3].
In [
1], the authors obtained the following results. For the centered 2-surround system
\(S^{(2)} \{P,\varGamma ,l \}\), we have the following isoperimetric inequalities:
$$ \biggl(\frac{1}{|\varGamma |} \oint_{\varGamma }r_{P}^{p} \biggr)^{1/p} \leqslant\frac {|\varGamma |}{2\pi} \cos\frac{l\pi}{|\varGamma |}, \quad \forall l\mbox{: }0< l \leqslant|\varGamma |/3, \forall p\leqslant-2 $$
(5)
and
$$ \biggl(\frac{1}{|\varGamma |} \oint_{\varGamma }r_{P}^{q} \biggr)^{1/q} \geqslant {\inf}_{\frac{l\pi}{|\varGamma |}\leqslant t< \pi} \biggl\{ \biggl(\frac {2|D(\varGamma )|}{|\varGamma |}t - \frac{l}{2} \biggr)\csc t+\frac{l}{2}\frac{\cos t}{t} \biggr\} , \quad \forall q\geqslant2. $$
(6)
Equalities in (
5) and (
6) hold if and only if
Γ is a circle and
P is the center of the circle.
In [
2], the authors obtained the following results. For the centered 2-surround system
\(S^{(2)} \{P,\varGamma ,l \}\), we have the following isoperimetric inequalities:
$$\begin{aligned}& \exp \biggl( \frac{1}{|\varGamma |} \oint_{\varGamma }\ln r_{A} \biggr) + \biggl( \frac{1}{|\varGamma |}\oint_{\varGamma }r_{A}^{\frac{2}{3}} \biggr) ^{\frac{3}{2}}\leqslant \frac {|\varGamma |}{\pi}\sin\frac{2l\pi}{|\varGamma |}, \end{aligned}$$
(7)
$$\begin{aligned}& \biggl[ \frac{1}{|\varGamma |} \oint_{\varGamma } \biggl( \frac{r_{A}+r_{A_{+}}}{2} \biggr) ^{\frac {2}{3}} \biggr] ^{\frac{3}{2}}\leqslant \frac{|\varGamma |}{2\pi}\sin\frac {2l\pi}{|\varGamma |}, \end{aligned}$$
(8)
$$\begin{aligned}& \biggl( \frac{1}{|\varGamma |} \oint_{\varGamma }r_{P}^{-2} \biggr) ^{-\frac{1}{2}}\leqslant \frac{|\varGamma |}{2\pi }\cos\frac{l\pi}{|\varGamma |}, \end{aligned}$$
(9)
and
$$ \biggl( \frac{1}{|\varGamma |} \oint_{\varGamma }r_{P}^{-p} \biggr) ^{-\frac{1}{p}}\leqslant \biggl( \frac{1}{|\varGamma |}\oint_{\varGamma }\bigl\Vert P'-A\bigr\Vert ^{q} \biggr) ^{\frac{1}{q}}\cot\frac{l\pi }{|\varGamma |}. $$
(10)
In (
10), where
\(p>1\),
\(p^{-1}+q^{-1}=1\) and
$$0< \angle A_{-}AA_{+}\leqslant\pi-\arctan \biggl( 2\sin \frac{2l\pi }{|\varGamma |} \biggr) ,\quad \forall A\in \varGamma . $$
Equalities in (
7)-(
10) hold if and only if
Γ is a circle and
P is the center of the circle.
In [
3], the authors established a isoperimetric inequality in the
N-surround system without any central
\(S^{(3)}\{\varGamma ,\mathbf{l}\}\):
$$ \frac{1}{|\varGamma |} \oint_{\varGamma }\operatorname{Area} \bigl(\varOmega [P, \varGamma _{N} ] \bigr)\leqslant\frac {N|\varGamma |^{2}}{8\pi^{2}}\sin\frac{2\pi}{N}, $$
(11)
where the
N-polygon
\(\varGamma _{N}\) is inscribed in
Γ [
17] and
P is a vertex of
\(\varGamma _{N}\), and the
\(\varOmega [P,\varGamma _{N} ]\) is a cone surface its vertex is
P and alignment is
\(\varGamma _{N}\).
Convexity and concavity are essential attributes of any real-variable function, their research and applications are important topics in mathematics and, in particular, the convex analysis [
18].
In [
19], the authors generalized the traditional covariance and variance of random variables, defined the
ϕ-covariance,
ϕ-variance,
ϕ-Jensen variance,
ϕ-Jensen covariance, integral variance, and
γ-order variance, and studied the relationships among these variances. They also studied the monotonicity of the interval function
\(\operatorname{JVar}_{\phi}\varphi ( {{X _{ [ {a,b} ] }}} )\) and proved an interesting quasi-log-concavity conjecture. They also demonstrated the applications of these results in higher education. Based on the monotonicity of the interval function
\({\operatorname{Var}^{ [ \gamma ] }}{X _{ [ {{a},{b}} ] }}\), they show that the hierarchical teaching model is normally better than the traditional teaching model under the hypothesis that
$${X _{I}}\subset X \sim{N_{k}} ( {\mu,\sigma} ),\quad k>1. $$
In this paper, we study the best upper bounds of the
p-power mean
$$M^{[p]}_{\varGamma }(\bar{r}_{P})\triangleq \biggl( \frac{1}{|\varGamma |} \oint_{\varGamma }\bar{r}_{P}^{p} \biggr)^{1/p} $$
and establish a new isoperimetric inequality in the centered 2-surround system
\(S^{(2)} \{P,\varGamma ,l \}\) as follows.
Let
\(S^{(2)} \{P,\varGamma ,l \}\) be a centered 2-surround system. We say that the set
$$S^{(2)} \{P,\varGamma \}\triangleq\lim_{l\rightarrow 0}{S^{(2)} \{P,\varGamma ,l \}}= \{P,\varGamma ,0 \} $$
is a
centered surround system and
P is the
center of the system.
For the centered surround system \(S^{(2)} \{P,\varGamma \}\), we may think of the point P as the center of the Earth and Γ as the orbit of a satellite A (such as the Moon or an artificial Earth satellite). This is the significance of the centered surround system \(S^{(2)} \{P,\varGamma \}\) in the theory of satellite.
From (
15) in Section
3 we know that the centered surround system
\(S^{(2)} \{P,\varGamma \}\) exists for any smooth and convex Jordan closed curve
Γ.
Theorem
1 implies the following interesting corollary.
In Section
5, we will demonstrate the applications of Corollary
1 in space science and establish an isoperimetric inequality involving the
λ-gravity of the Moon to the Earth.
5 Applications in space science
Corollary
1 is of great significance in space science.
Let
\(S^{(2)} \{P, \varGamma \}\) be a centered surround system. We may regard
P as the Earth (or an atomic nucleus,
etc.) with mass
M,
A as the Moon (or an electron of the atom,
etc.) with mass
m, which is a satellite of the Earth, and
Γ as the orbit of the Moon. According to the law of universal gravitation, the gravity of the Moon
A to the Earth
P is
$$ \mathbf{F}(A,P)=\frac{GmM(A-P)}{{{{\Vert {{A}-{P}}\Vert }^{3}}}}, $$
(58)
and the norm
\(\Vert \mathbf{F}(A,P)\Vert \) of the gravity
\(\mathbf {F}(A,P)\) between the Moon
A and the Earth
P is
$$ \bigl\Vert \mathbf{F}(A,P)\bigr\Vert =\frac{GmM}{{{{\Vert {{A}-{P}}\Vert }^{2}}}}, $$
(59)
where
G is the gravitational constant of the solar system. Without loss of generality, we may assume that
\(GmM=1\).
When the Moon
A traverses one cycle along its orbit
Γ, the mean of the norm
\(\Vert {\mathbf{F}} ( \varGamma ,P )\Vert \) of the gravity
\(\mathbf{F}(A,P)\) between the Moon
A and the Earth
P is
$$ \overline{\bigl\Vert {\mathbf{F}} ( \varGamma ,P )\bigr\Vert }\triangleq M_{\varGamma }\bigl({\bigl\Vert {\mathbf{F}} ( \varGamma ,P )\bigr\Vert }\bigr) =\frac {1}{|\varGamma |} \oint_{\varGamma }\frac{1}{\|A-P\|^{2}}. $$
(60)
In [
10], the authors defined the
λ-
gravity as follows:
$$ \mathbf{F}_{\lambda} (A,P )=\frac{GmM (A-P )}{{{{ \Vert {A-P}\Vert }^{\lambda+1}}}}= \frac{A-P}{{{{\Vert A-P\Vert }^{\lambda+1}}}}, \quad \lambda>0, $$
(61)
where
$$\mathbf{F}_{2} (A,P )=\mathbf{F} (A,P ). $$
In the solar system, the gravity of the physical matter X to another physical matter P is \(\mathbf{F} (A,P )\), whereas for another galaxy in the universe, the gravity may be \(\mathbf {F}_{\lambda} (A,P )\), where \(\lambda\in(0, 2)\cup(2, +\infty)\). For example, in the black hole of the universe, we conjecture that the gravity is \(\mathbf{F}_{\lambda} (A,P )\) with \(\lambda \in(0, 2)\), P can be regarded as an atomic nucleus of an atom, A can be regarded as an electron of the atom, and Γ can be regarded as the orbit of the electron.
We define as
$$\begin{aligned}& \mathbf{F}_{\lambda}(A,P)\triangleq\frac{A-P}{{{{\Vert {{A}-{P}}\Vert }^{\lambda+1}}}}, \end{aligned}$$
(62)
$$\begin{aligned}& \bigl\Vert \mathbf{F}_{\lambda}(A,P)\bigr\Vert \triangleq\frac{1}{{{{\Vert {{A}-{P}}\Vert }^{\lambda}}}}, \end{aligned}$$
(63)
and
$$ \overline{\bigl\Vert {\mathbf{F}}_{\lambda} ( \varGamma ,P ) \bigr\Vert }\triangleq\frac{1}{|\varGamma |} \oint_{\varGamma }\frac{1}{\|A-P\|^{\lambda}} $$
(64)
the
λ-
gravity function,
λ-
gravity norm, and
mean
λ-
gravity norm between the Moon
A and the Earth
P, respectively, where
\(\lambda\in(0,\infty)\).
In [
21], the authors defined the planet system
\({\operatorname{PS}} \{ {P,m, {\mathrm{B}} ( {g,r} )} \}_{\mathbb{E}}^{n} \) in an Euclidean space
\(\mathbb{E}\) and the
λ-gravity function
$$\mathbf{F}_{\lambda}:\mathbb{E}^{n} \to\mathbb{E}, \qquad \mathbf{F}_{\lambda}( P )\triangleq\sum_{i = 1}^{n} {\frac{{m_{i} p_{i} }}{{\Vert {p_{i} } \Vert ^{\lambda + 1} }}} $$
in the planet system, and obtained some interesting results. For example, in the planet system
\({\operatorname{PS}} \{ {P,m, {\mathrm{B}} ( {g,r} )} \}_{\mathbb{E}}^{n} \), if
\(\lambda> \mu>2\) and
\(\|g\| \geqslant\sqrt{2}\), then we have the following inequality:
$$ \frac{\operatorname{Var}^{*}_{\lambda}(P)}{\operatorname{Var}^{*}_{\mu}(P)}\geqslant \frac{\mu}{\lambda} \biggl[\frac{\Vert {\mathbf{F}}_{2}(P)\Vert }{ \Vert {\mathbf{F}}_{0}(P)\Vert } \biggr]^{\lambda-\mu}, $$
(65)
where
$$ \operatorname{Var}^{*}_{\lambda}(P)\triangleq\frac{8}{\lambda(\lambda -2)} \biggl[ \biggl(\frac{\Vert {\mathbf{F}}_{\lambda}(P)\Vert }{ \Vert {\mathbf{F}}_{0}(P)\Vert } \biggr)^{2}- \biggl( \frac{\Vert {\mathbf {F}}_{2}(P)\Vert }{\Vert {\mathbf{F}}_{0}(P)\Vert } \biggr)^{\lambda} \biggr], $$
and the coefficient
\(\mu/\lambda\) in (
65) is the best constant.
In this section, we will establish a new isoperimetric inequality involving the λ-gravity.
Corollary
1 implies the following interesting result, which is significant in space science.
In [
10], the authors obtained the following interesting inequality:
$$ \sum_{j=1}^{m} \mu_{j} \overline{\bigl\Vert {\mathbf{F}}_{\alpha_{j}}(A,P)\bigr\Vert }^{\frac{\gamma }{\alpha_{j}}}\geqslant \overline{\bigl\Vert {\mathbf{F}}_{\lambda} ( \varGamma ,P )\bigr\Vert }^{\frac{\gamma}{\lambda}}, $$
(67)
where
$$\alpha, \mu\in(0,\infty)^{m},\qquad \sum _{j=1}^{m}\mu_{j}=1, \quad m\geqslant2, \qquad \gamma \in(0,\infty), \qquad 0< \lambda\leqslant \Biggl(\sum _{j=1}^{m} \frac{\mu_{j}}{\alpha _{j}} \Biggr)^{-1}. $$
According to Proposition
3 and (
67), we know that in the centered surround system
\(S^{(2)} \{P,\varGamma \}\), if
$$\alpha, \mu\in(0,\infty)^{m},\qquad \sum _{j=1}^{m}\mu_{j}=1,\quad m\geqslant2, \qquad \gamma \in(0,\infty),\qquad \Biggl(\sum_{j=1}^{m} \frac{\mu_{j}}{\alpha_{j}} \Biggr)^{-1}\geqslant2, $$
then we have the following isoperimetric inequality:
$$ \sum_{j=1}^{m} \mu_{j} \overline{\bigl\Vert {\mathbf{F}}_{\alpha_{j}}(A,P)\bigr\Vert }^{\frac{\gamma }{\alpha_{j}}}\geqslant \biggl(\frac{2\pi}{|\varGamma |} \biggr)^{\gamma}. $$
(68)
Equality in (
68) holds if and only if
Γ is a circle and
P is the center of the circle.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in this paper. All authors read and approved the final manuscript.