Research Article | Open Access

Chaobang Gao, Jiajin Wen, "Theories and Inequalities on the Satellite System", *International Scholarly Research Notices*, vol. 2011, Article ID 909261, 22 pages, 2011. https://doi.org/10.5402/2011/909261

# Theories and Inequalities on the Satellite System

**Academic Editor:**D. Han

#### Abstract

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 , we establish the inequality: . For the satellite system with a central , we establish the following inequality under the proper hypothesis: . As an application, we get the inequalities , for the satellite system of single point with a central . For these results, there are generalized backgrounds in the fields of differential geometry and space science.

#### 1. Introduction and the Main Results

In this paper, we will use the following symbols: and to denote the set of real numbers and the set of integers, respectively, , and [1, 2].

*Definition 1.1. *Let be a smooth curve in three-space . and are called the initial point and terminal point of , respectively. If , , , and , then is called a coincident point of ; smooth curve without any coincident point is called a simple curve; If a simple curve satisfies , then is said be a simple closed curve. Especially, we say is a Jordan closed curve if is a plane curve in . For a Jordan closed curve , denotes the closed region bounded by and its area is written as , denotes the length of , 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.*

*Definition 1.3 (see [4, 5]). *Assume that is a smooth space curve with end points and , is a fixed point in , and , where is a moving point, then the trail of the line segment is called a bounded cone surface, written as . We Also say is the generating line, is the vertex, and is the generating curve of the bounded cone surface. denotes the area of .

If is a given smooth curve in , then its length is written as ; denotes the smooth curve segment with end points and , and .

*Definition 1.4. *Let be a fixed smooth closed curve in , the -polygon be inscribed in , and, . If all vertices of move continuously along and keep invariable, where , we say that the set is the satellite system without any central in three-space .

For , we write the set of the vertices of as and define , for all [6].

*Definition 1.5. *For , if the following statements are valid:(i)is a fixed closed Jordan curve in ,(ii) is a fixed point in the interior of , and(iii) is always a Jordan closed curve and as all vertices of move continuously along ,then we say the set := is the satellite system with a central and is its central.

For , denotes the distance from to the line , . And we write .

*Remark 1.6. *The may be explained as follows: the point denotes the central of earth, denotes the trajectory on which satellites move, and the vertexes of are viewed as satellites moving on the same trajectory . In order to avoid hitting, they must move by same curve velocity, that is, is invariable and .

*Definition 1.7. *The *t*th power mean of the positive real numbers sets , written as , is defined by [7–15]:

*Definition 1.8 (see [1, 7, 10, 11, 14]). * Let be a smooth curve in . For Riemann integrable function , is called function means of on . For the function and the real number , if and are Riemann integrable on , then
is called the *t*th power means of on .

Now, we give our main results as follow.

Theorem 1.9. *Let be a satellite system without any central. For any , we have inequality
**
that is,
**
where is the generating line of the bounded cone surface . A sufficient condition of equalities in (1.3) and (1.4) is that is a circle and is always a regular polygon with sides.*

Theorem 1.10. *Let be a satellite system with a central. If or , , we have inequality, for any the real number ,
**
that is,
**
The equalities of (1.5) and (1.6) occur if and only if is a circle, is the central of the circle, and is a regular polygon with sides .*

In Section 5, we will give some applications of these results and theories.

#### 2. Preliminaries

Lemma 2.1. *For the quadrilateral in , writing , we obtain
**
with equality if and only if . If the quadrilateral is convex and in plane, equality holds if and only if the quadrilateral is inscribed in a circle.*

*Proof. *Write . By the area formula of triangle and cosine theorem, we get
Consider as the implicit function with respect to . Finding the derivatives of the two sides of (2.2) and (2.3), respectively, we have
Therefore, and is increasing with respect to . Since
we have
Hence is max if and only if . When is max, . Thus,
In other words, (2.1) holds. Equality holds if and only if . This completes the proof.

Lemma 2.2. *Let be a satellite system without any central. For any , we have inequality:
**
where is the generating line of the bounded cone surface . The second equality occurs if is a regular polygon with sides .*

*Proof. *By the definition and geometric meaning of curve integral, we obtain
Now we prove the inequality in (2.8).

If , (2.8) is known; if , by Lemma 2.1, (2.8) holds. In the following, we suppose that . First, fix the value of . Without loss of generality, we set . By the theory of differential geometry, we know a bounded cone surface is a developable surface, which implies that can be developed into a bounded cone surface in the plane, where is the developed graph of in the plane and it is a polygon with sides (may not be a closed Jordan curve) and satisfies
When is max, for any , the quadrilateral must be convex and the polygon with 5 sides must be also convex by the plane geometry. Now, we prove the four points and are on a common circle. Otherwise, there exists some such that are not on a common circle. Therefore, fix the point and modify to such that are on a common circle and
Hence, we obtain a new polygon with sides and write
It follows that
By Lemma 2.1, we have
This contradicts the greatest . Since three points confirm a unique circle and for any , we know are on a common circle if is greatest, and since
is inscribed in a circle and . Otherwise, there exists such that . When the perimeter of a circle is a fixed value, the area of regular polygon with sides is greatest in all N-polygons inscribed in the circle [16]. Therefore, is a regular polygon with sides if is greatest.

Now let be a regular polygon with sides. By the plane geometry, we know
Hence, the inequality holds in (2.8). Equality occurs if is a regular -polygon. This completes the proof.

Lemma 2.3. *For the quadrilateral in , we have
**
with equality if and only if .*

*Proof. *Write , denotes the inner product of vectors and , especially, . Thus, (2.17) is expressed as
Since
and , which implies that
it follows that
(2.18) is proved. Equality of (2.18) holds if and only if . This completes the proof.

Lemma 2.4. *Let be a polygon with sides in . Setting
**
we get the following inequality:
**
for . A sufficient condition of equality is that is a plane regular -polygon in .*

*Proof. *Consider the quadrilateral . From
we obtain
It follows from Lemma 2.3 and (2.25) that
which implies that
Since
the inequality (2.27) is equivalent to
Inequality (2.23) is proved. From this proof and Lemma 2.3, we know that a sufficient condition of equality is that is a regular polygon with sides in .

*Remark 2.5. *A sufficient condition of equality of (2.23) is that is a regular polygon with sides in . This condition is not necessary. For example, when , the equality holds in (2.23) if and only if is a parallelogram in .

*Remark 2.6. *If is a regular -polygon, defined by Lemma 2.4 is equal to , where denotes the circumradius of . Namely,

Lemma 2.7. *Suppose is defined by Lemma 2.4, for any positive integer , there exist constants which is only related to such that
**
A sufficient condition of equality is that is a regular -polygon in .*

*Proof. *We prove it by mathematical induction with respect to .

(i) When , let from Lemma 2.4, we have
Let be a regular -polygon. By Remark 2.6, we know ; it follows from Lemma 2.4 that the equality of (2.32) holds, thus, .

(ii) Assume that Lemma 2.7 holds for . Now we want to prove that Lemma 2.7 holds for . By the hypothesis , we know that . Thus, from the induction hypothesis, there exist constants such that
A sufficient condition of equality of (2.34) is that is a regular polygon with sides. Since , by induction hypothesis, substitute by in (2.33) and (2.34), in other words, there exist constants such that
Substituting (2.36) into (2.32), we get the following inequality:
Notice . Solving inequality (2.37) with respect to , we obtain
where
Let be a regular -polygon. From Remark 2.6, we know . It follows from Lemma 2.4 and induction hypothesis that the equality of (2.38) holds. Thus, , that is, Lemma 2.7 holds for . This completes the proof.

Lemma 2.8. *Suppose is defined by Lemma 2.4, we have the inequality as follows:
**
A sufficient condition of equality is that is a regular polygon with sides in .*

*Proof. *Setting in (2.31), we get
Since ,
It follows from (2.41) and (2.42) that there exist constants such that
where , for any . Using (2.43) repeatedly, we get
therefore,
Set is a regular polygon with sides in . By Remark 2.6, we know ; from Lemma 2.7, we have the equality of (2.45) holds, which implies that . A sufficient condition of equality of (2.40) is that is a regular -polygon in . This completes the proof.

Lemma 2.9. *Let be a smooth curve with the end points and in . If the function is Riemann integrable on , considering a partition of by means of points such that , where is a broken line, we have
*

*Proof. *Since is a smooth curve with the end points and in ,
exists and . And is Riemann integrable on , it follows that is bounded on , that is, there exists a constant such that for any . For any ,
It follows from the definition of the curve integral
that
Since , for any , there exists such that , for , therefore
Thus,
It follows that
In view of (2.50), we get
This completes the proof of Lemma 2.9.

Lemma 2.10. *Let be a smooth closed curve in . If the points move continuously on and keep invariable, where , we obtain
**
A sufficient condition of equality is that is a circle in .*

*Proof. *By the theory of real number, there exists an increasing sequence of positive integers such that
Write . Consider a partition of by means of points such that , where is a polygon with sides.

First, we give the following fact: if the point moves to some point along , the point moves to along , in other words, , then
In fact, from , we only need to prove
Since
(2.58) holds, it follows that (2.57) holds.

From (2.57), we know for any , there exists , when , such that
By Lemma 2.9, we get
It follows from inequality (2.40) that
It follows from (2.61) and (2.62) that
Namely, (2.55) holds. From Lemma 2.8, a sufficient condition of equality of (2.55) is that is a circle in . This completes the proof.

Lemma 2.11 (Cauchy inequality [1, page 6]). * Let be a smooth closed curve in . If the function and are smooth, we have the inequality as follows:
*

Lemma 2.12. *Assume is a closed Jordan curve in and , then .*

*Proof. *Let , denote the straight line through the points and , and let denote the oriented line segment with the initial point and terminal point .

For any , if , then ; if , then
If , then
If , without loss of generality, we set , then
In all, for any , hence, . This completes the proof.

Lemma 2.13. *For , we have the following inequality:
*