Theories and Inequalities on the Satellite System

We deﬁne the satellite system without any central, the satellite system with a central, and the satellite system of single point with a central. For the satellite system without any central S { Γ , Γ N } , we establish the inequality: (cid:2) Γ ds (cid:2) Γ N | PQ | ds Q ≤ (cid:3) N | Γ | 3 / 4 π 2 (cid:4) sin (cid:3) 2 π/N (cid:4) . For the satellite system with a central S { O, Γ , Γ N } , we establish the following inequality under the proper hypothesis: (cid:2) Γ M (cid:5) t (cid:6) N (cid:3) r (cid:4) ds ≤ (cid:3) | Γ | 2 / 2 π (cid:4) cos (cid:3) π/N (cid:4)(cid:3) ∀ t ∈ (cid:6) − ∞ , − 2 (cid:6)(cid:4) . As an application, we get the inequalities (cid:2) Γ ρ P ds ≥ (cid:2) Γ r P ds ≥ 2 | D (cid:3)Γ(cid:4) | , for the satellite system of single point with a central S { O, Γ ,P } . For these results, there are generalized backgrounds in the ﬁelds of di ﬀ erential geometry and space science.


Introduction and the Main Results
In this paper, we will use the following symbols: R and Z to denote the set of real numbers and the set of integers, respectively, a a 1 , a 2 , . . . , a n , and R n {a|a i ∈ R, 1 ≤ i ≤ n} 1, 2 .
be a smooth curve in three-space R 3 . A r a and B r b are called the initial point and terminal point of Γ, respectively. If a < t 1 < b, a ≤ t 2 ≤ b, t 1 / t 2 , and r t 1 r t 2 , then r t 1 is called a coincident point of Γ; smooth curve without any coincident point is called a simple curve; If a simple curve Γ satisfies A B, then Γ is said be a simple closed curve. Especially, we say Γ is a Jordan closed curve if Γ is a plane curve in R 3 . For a Jordan closed curve Γ, D Γ denotes the closed region bounded by Γ and its area is written as |D Γ |, |Γ| denotes the length of Γ, and we have the following Jordan curve theorem. Theorem 1.2 Jordan Curve 3 . An arbitrary Jordan closed curve must divide a plane into two parts, where one part is bounded, and the other is unbounded. The bounded part is called interior and another is outside of the Jordan closed curve.

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Definition 1.3 see 4, 5 . Assume that Γ AB is a smooth space curve with end points A and B, P is a fixed point in R 3 , and Q ∈ Γ AB , where Q is a moving point, then the trail of the line segment PQ is called a bounded cone surface, written as Ω P, Γ AB . We Also say PQ is the generating line, P is the vertex, and Γ AB is the generating curve of the bounded cone surface. |Ω P, Γ AB | denotes the area of Ω P, Γ AB .
If Γ is a given smooth curve in R 3 , then its length is written as |Γ|; Γ AB denotes the smooth curve segment with end points A and B, and Γ AB ⊂ Γ.
For S{Γ, Γ N }, we write the set of the vertices of Γ N as V Γ N : {A 1 , A 2 , . . . , A N } and define A i A j ⇔ i ≡ j mod N , for all i, j ∈ Z 6 . Remark 1.6. The S{O, Γ, Γ N } may be explained as follows: the point O denotes the central of earth, Γ denotes the trajectory on which satellites move, and the vertexes of Γ N are viewed as N satellites moving on the same trajectory Γ. In order to avoid hitting, they must move by same curve velocity, that is, l i : |Γ A i A i 1 | is invariable and 0 < l i < |Γ|/2, i 1, 2, . . . , N.
Definition 1.7. The tth power mean of the positive real numbers sets a a 1 , a 2 , . . . , a N N ≥ 2 , written as M t N a , is defined by 7-15 : In Section 5, we will give some applications of these results and theories.

2.7
In other words, 2.1 holds. Equality holds if and only if α β π ⇔ BAD BCD π. This completes the proof. Lemma 2.2. Let S{Γ, Γ N } be a satellite system without any central. For any P ∈ V Γ N , we have inequality: where PQ is the generating line of the bounded cone surface Ω P, Γ N . The second equality occurs if Γ N is a regular polygon with sides N.
Proof. By the definition and geometric meaning of curve integral, we obtain Now we prove the inequality in 2.8 . If N 3, 2.8 is known; if N 4, by Lemma 2.1, 2.8 holds. In the following, we suppose that N ≥ 5. First, fix the value of |Γ N |. Without loss of generality, we set P A 1 . By the theory of differential geometry, we know a bounded cone surface is a developable surface, which implies that Ω P, Γ N can be developed into a bounded cone surface Ω A * 1 , Γ * N in 6 ISRN Mathematical Analysis the plane, where Γ * N : A * 1 A * 2 · · · A * N A * 1 P A 1 A * 1 is the developed graph of Γ N in the plane and it is a polygon with sides N may not be a closed Jordan curve and satisfies

2.10
When must be convex and the polygon with 5 sides A * 1 must be also convex by the plane geometry. Now, we prove the four points are not on a common circle. Therefore, fix the point Hence, we obtain a new polygon with N sides and write It follows that

2.13
By Lemma 2.1, we have This contradicts the greatest |Ω A * 1 , Γ * N |. Since three points confirm a unique circle and for When the perimeter of a circle is a fixed value, the area of regular polygon with N sides is greatest in all N-polygons inscribed in the circle 16 . Therefore, Γ * N is a regular polygon with N sides if |Ω A * 1 , Γ * N | is greatest. Now let Γ * N be a regular polygon with N sides. By the plane geometry, we know Hence, the inequality holds in 2.8 . Equality occurs if Γ * N is a regular N-polygon. This completes the proof.
a, b denotes the inner product of vectors a and b, especially, a 2 a · a |a| 2 . Thus, 2.17 is expressed as and a : a b c d 0, which implies that

2.20
it follows that we get the following inequality:

2.25
It follows from Lemma 2.3 and 2.25 that the inequality 2.27 is equivalent to

2.29
Inequality 2.23 is proved. From this proof and Lemma 2.3, we know that a sufficient condition of equality is that Γ N is a regular polygon with N sides in R 3 .

2.31
A sufficient condition of equality is that Γ N is a regular N-polygon in R 3 .
Proof. We prove it by mathematical induction with respect to j. i When j 1, let C k 1, Let Γ N be a regular N-polygon. By Remark 2.6, we know L k L k 1 L k−1 L 1 R 2 0 > 0; it follows from Lemma 2.4 that the equality of 2.32 holds, thus, C k 1,1 C k−1,1 C 1,1 1.
ii Assume that Lemma 2.7 holds for j n ≥ 1. Now we want to prove that Lemma 2.7 holds for j n 1. By the hypothesis k, n 1 : k ≥ 2, k n 1 ≤ N − 1, we know that k, n : k ≥ 2, k n ≤ N − 2 ≤ N − 1. Thus, from the induction hypothesis, there exist constants C k n,n , C k−1,n , C 1,n such that C k n,n C k−1,n C 1,n 1, 2.33 L k ≤ C k n,n L k n C k−1,n L k−1 C 1,n L 1 .

2.39
Let Γ N be a regular N-polygon. From Remark 2.6, we know L k L k n 1 L k−1 L 1 R 2 0 > 0. It follows from Lemma 2.4 and induction hypothesis that the equality of 2.38 holds. Thus, C * * k n 1,n 1 C * * k−1,n 1 C * * 1,n 1 1, that is, Lemma 2.7 holds for j n 1. This completes the proof.

2.40
A sufficient condition of equality is that Γ N is a regular polygon with N sides in R 3 .
Proof. Setting k j N − 1 in 2.31 , we get

2.42
It follows from 2.41 and 2.42 that there exist constants C k−1 , C 1 : C k−1 C 1 1 such that Using 2.43 repeatedly, we get

2.45
Set Γ N is a regular polygon with N sides in R 3 . By Remark 2.6, we know L k L 1 R 2 0 > 0; from Lemma 2.7, we have the equality of 2.45 holds, which implies that C 1. A sufficient condition of equality of 2.40 is that Γ N is a regular N-polygon in R 3 . This completes the proof.  N − 1 N ≥ 3 points A A 0 , A 1 , . . . , It follows from the definition of the curve integral Thus,

2.52
It follows that In view of 2.50 , we get

2.54
This completes the proof of Lemma 2.9.

2.55
A sufficient condition of equality is that Γ is a circle in R 3 .
Proof. By the theory of real number, there exists an increasing sequence of positive integers {k n } such that lim n → ∞ k n 10 n l |Γ| .

2.56
Write k : k n , N : 10 n . Consider a partition of Γ by means of N N ≥ 10 points A 1 , A 2 , . . . , First, we give the following fact: if the point A moves to some point A i along Γ, the point B moves to A * i along Γ, in other words,

2.60
By Lemma 2.9, we get

2.61
It follows from inequality 2.40 that

2.62
It follows from 2.61 and 2.62 that

2.63
Namely, 2.55 holds. From Lemma 2.8, a sufficient condition of equality of 2.55 is that Γ is a circle in R 3 . This completes the proof.

Lemma 2.11
Cauchy inequality 1, page 6 . Let Γ be a smooth closed curve in R 3 . If the function f : Γ → R and g : Γ → R are smooth, we have the inequality as follows:

2.65
If O ∈ Q i Q i 1 , then
Lemma 2.13. For S{O, Γ, Γ N }, we have the following inequality: with equality if and only if Γ N is a regular N-polygon in R 3 and O is its central.
Proof. By Definition 1.5 and Lemma 2.12, for any

2.73
The

Proof of Theorem 1.9
By 2.8 , 2.64 , 2.55 , and 2.72 in order, we get

3.2
Hence, 1.3 and 1.4 are proved. By this proof, a sufficient condition of equalities is that Γ is a circle and Γ N is always a regular polygon with N sides. This completes the proof of Theorem 1.9.

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Remark 3.1. We may extend the concepts of the bounded cone surface and the satellite system without any central to R m m ≥ 2 . And Theorem 1.9 also holds for the satellite system without any central in R m .

Proof of Theorem 1.10
Since power mean M t N r is increasing with respect to t and by 2.68 , we have the following inequalities, for t ≤ −2,

4.2
Therefore, 1.5 and 1.6 are proved. From this proof, the equalities of 1.5 and 1.6 occur if and only if Γ is a circle, O is the central of the circle, and Γ N is a regular polygon with sides N. This completes the proof of Theorem 1.10.

Applications
In Theorem 1.9, if both Γ and Γ N are simple closed curves in R 3 , and Ω Γ N denotes the minimal surface 4, 5, 19-21 bounded by Γ N , then for any P ∈ V Γ N , |Ω Γ N | ≤ |Ω P, Γ N |, where |Ω Γ N | denotes the area of Ω Γ N . Thus, we get the theorem as follows. where Ω Γ N is a minimal surface bounded by Γ N and |Ω Γ N | denotes its area. A sufficient condition of equality in 5.1 is that Γ is a circle and Γ N is always a regular N-polygon.  Proof. Insert the inscribed polygon Γ N : A 1 A 2 · · · A N A 1 N ≥ 3 such that then the point O is in the inner of Γ N when N is sufficiently large. In the following, we assume that O is in the inner of Γ N . According to Definition 1.8, we only need to prove 5.3 . Since for any P ∈ Γ, ρ P ≥ r P , the first inequality of 5 |Γ| Γ r P ds 1 2 Γ r P ds.

5.9
The second inequality of 5.3 is proved. By this proof, the first equalities of 5. Definition 5.6. Theorem 5.5 may be explained as follows: the point O denotes the central of earth; P is a satellite and Γ denotes the moving trajectory of satellites; M r P , Γ is the function mean of r P on Γ and M ρ P , Γ is the function mean of ρ P on Γ.
Definition 5.7. For the S{O, Γ, P 1 , P 2 }, which denotes the satellite system of two points with a central, in other words, there are only two satellites on same trajectory, we will discuss in another paper.