By means of the algebraic, analysis, convex geometry, computer, and inequality theories we establish the following isoperimetric inequality in the centered 2-surround system \(S^{(2)} \{P,\varGamma ,l \}\):
As an application of the inequality in space science, we obtain the best lower bounds of the mean λ-gravity norm \(\overline{\Vert {\mathbf{F}}_{\lambda} ( \varGamma ,P )\Vert }\) as follows:
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.
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.
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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 meanλ-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].
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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
$$\gamma'(t)\triangleq x'(t) \mathbf{i}+y'(t) \mathbf{j} \ne \mathbf{0},\quad \forall t\in I, $$
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
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.
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,
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
then we say that \(A_{1},A_{2},\ldots,A_{N}\) are Nsatellites 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]
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]
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
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:
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:
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
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.
Corollary 1
Let\(S^{(2)} \{P,\varGamma \}\)be a centered surround system. Then for all\(p\in(-\infty,-2]\), we have the following isoperimetric inequality:
According to Theorem 1, equality in (14) holds if and only if Γ is a circle and P is the center of the circle. Corollary 1 is proved. □
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.
3 Preliminaries
In order to prove Theorem 1, we need some preliminaries involving the centered 2-surround system.
3.1 Boundary curve of the l-central region
In the definition of the centered 2-surround system \(S^{(2)} \{ P,\varGamma ,l \}\), an important assumption is that the l-central region \(D(\varGamma ,l)\) is nonempty, that is, the boundary curve \(\partial D(\varGamma ,l)\) of the l-central region \(D(\varGamma ,l)\) is a Jordan closed curve. Unfortunately, the l-central region \(D(\varGamma ,l)\) may be empty. For example, let Γ be a regular triangle of side length 3, then \(D(\varGamma ,4)=\emptyset\), where ∅ denote the empty set, see [2].
there exists \(\varepsilon\in(0,|\varGamma |/2)\) such that, for any \(l\in(0,\varepsilon )\), we have \(D(\varGamma ,l)\ne\emptyset\).
On the other hand, in [2], the following statement is proved (see Lemmas 2.1 and 2.3 in [2]): Let Γ be a smooth and convex Jordan closed curve. Then \(D(\varGamma ,l)\neq\emptyset\) for all \(l\in (0,|\varGamma |/2)\) if and only if Γ is a central symmetric curve.
According to this result, we know that if Γ is an ellipse, which is a central symmetric curve, then the l-central region \(D(\varGamma ,l)\) is nonempty. In space science, the orbit of a satellite is an ellipse, and P in \(S^{(2)} \{P,\varGamma ,l \}\) is one of the focuses of the ellipse [4]. Therefore, the centered 2-surround system \(S^{(2)} \{P,\varGamma ,l \}\) is of great application value in the theory of satellite.
Based on the definition of the l-central region \(D(\varGamma ,l)\), we know that the boundary curve \(\partial D(\varGamma ,l)\) of the l-central region \(D(\varGamma ,l)\) is the envelope curve of the family of straight line \(AA_{+}\), that is, for any point \(x\mathbf {i}+y\mathbf{j}\in\partial D(\varGamma ,l)\), there exists a line \(AA_{+}\) such that \(AA_{+}\) is tangent to \(\partial D(\varGamma ,l)\) at the point \(x\mathbf{i}+y\mathbf{j}\). Hence, the point \(x\mathbf{i}+y\mathbf{j}\) must satisfy the equation
Eliminating the parameter \(t_{A}\) from (16) and (17), we can obtain the equation of the boundary curve \(\partial D(\varGamma ,l)\); see the following Propositions 1 and 2.
Consider the regular region \(\widehat{A_{j-1}A_{j}A_{j+1}}\). Let the rays \(A_{i}A_{i-1}\) and \(A_{i}A_{i+1} \) be tangent to \(\partial D(\varGamma _{N},l)\) at the points \(T_{i}\) and \(T_{i}'\), respectively, and let
If\(S^{(2)} \{P,\varGamma _{N_{n}},\frac{k_{n}}{N_{n}}|\varGamma _{N_{n}}| \}\)is an asymptotic system of\(S^{(2)} \{P,\varGamma ,l \}\), then there exists a sequence
Let the image\(\varGamma =\gamma([a,b])\)of a continuous function\(\gamma:[a,b]\rightarrow\mathbb{R}^{m}\)be a smooth curve, and let\(f:\varGamma \rightarrow\mathbb{R}\)be a Riemann-integrable function overΓ. Suppose thatΓis partitioned by means of\(N+1\)points
Consider the asymptotic system \(S^{(2)} \{ P,\varGamma _{N_{n}},\frac {k_{n}}{N_{n}}|\varGamma _{N_{n}}| \}\). By Lemmas 1 and 2 we have that \(P\in D(\varGamma _{N_{n}})\) if n is sufficiently large. By Lemmas 1, 2, and 3 and by the identity
has no real roots and \({\mathrm{d}\varphi_{*}}/{\mathrm{d} \theta }<0\). Hence, the function \(\varphi_{*}\) is decreasing. The solution of the inequality
where\(\eta=1.7571802619873076 \ldots\)is the unique real root of equation (13). Equality in (36) holds if and only if\(P'\)is the midpoint of the closed straight line segment\([AA_{+}]\).
Proof
The relevant calculations in the proof are dependent on the Mathematica software since these calculations are very complex.
The equation \({\mathrm{d}g^{*}}/{\mathrm{d} \theta}=0\) has no real roots in the interval \((0, {\pi}/{2} )\), and \({\mathrm{d}g^{*}}/{\mathrm{d} \theta}>0\). Hence,
Based on this proof, we know that equality in (36) holds if and only if \(P'\) is the midpoint of the closed straight line segment \([AA_{+}]\). This completes the proof of Lemma 7. □
Let\(E\subset\mathbb{R}^{m}\)be a bounded and closed region (or curve), and let the functions\(f:E\rightarrow\mathbb{R}\)and\(\phi: f(E)\rightarrow\mathbb{R}\)be integrable, where\(f(E)\)is an interval. If\(\phi: f(E)\rightarrow \mathbb{R}\)is a convex function, then we have the following Jensen inequality:
The graph of the function \({\mathrm{d}\psi}/{\mathrm{d}\theta}\) is depicted in Figure 6.
×
The function \({\mathrm{d}\psi}/{\mathrm{d}\theta}\) has unique extremum point \(\theta_{1}=1.2313051084629325\ldots\) in the interval \((0,{\pi }/{2} )\).
The increasing and decreasing intervals of the function \({\mathrm{d}\psi}/{\mathrm{d}\theta}\) are \((0,\theta _{1} ]\) and \([\theta_{1},{\pi}/{2} )\), respectively. Hence, the convex and concave intervals of the function ψ are \((0,\theta_{1}]\) and \([\theta_{1},{\pi}/{2})\), respectively. Since
Based on this proof, we know that the equality in (12) holds if and only if Γ is a circle and P is the center of the circle. This completes the proof of Theorem 1. □
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
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
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.
the λ-gravity function, λ-gravity norm, and meanλ-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
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:
Equality in (68) holds if and only if Γ is a circle and P is the center of the circle.
Acknowledgements
The authors would like to acknowledge the support from the National Natural Science Foundation of China (No. 11161024), and the Foundation of Scientific Research Project of Fujian Province Education Department (No. JK2013051).
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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.