在這篇論文裡，我們首先探討了S_{p}及pi_{p}的函數值隨著p值增加的變化趨勢，其中S_{p}為下列積分之反函數
x=\int^{S_{p}(x)}_{0}(1-|t|^{p})^{-\frac{1}{p}}dt, p>1
且pi_{p}=2\int^{1}_{0}(1-t^{p})^{-\frac{1}{p}}dt。
我們修正了Binding的錯誤，重新證明S_{p}(x)及pi_{p} 對p的單調性。
在本論文的第二部分，我們將S_{p}與pi_{p}的定義推廣至S_{p,q}函數與pi_{p,q}，探討其相關微分方程及對p與q的單調性，此處S_{p,q}為下列積分之反函數x=\int^{S_{p,q}(x)}_{0}(1-|t|^{q})^{-\frac{1}{p}}dt, p,q>1
且pi_{p,q}=2\int^{1}_{0}(1-t^{q})^{-\frac{1}{p}}dt。
我們發現S_{p,q}與S_{p}、sin(x)有類似的性質，並且證明了下列等價條件
(a)\forall x\in[-\frac{ i_{p,q}}{2},\frac{ i_{p,q}}{2}]

x=\int^{S_{p,q}(x)}_{0}(1-|t|^{q})^{-\frac{1}{p}}dt.\forall x \in \mathbb{R}
S_{p,q}(x+k i_{p,q})=(-1)^{k}S_{p,q}(x).

(b)S_{p,q}(0)=0,S'_{p,q}(0)=1
|S_{p,q}|^{q}+|S'_{p,q}|^{p}=1
(c)S_{p,q}(0)=0,S'_{p,q}(0)=1
(|S'_{p,q}|^{p-2}S'_{p,q})'+\frac{q(p-1)}{p}|S_{p,q}|^{q-2}S_{p,q}=0

最後，我們還證明了pi_{p,q}與S_{p,q}( i_{p,q}t),\forall t\in[0,1]的函數值隨著p,q增加而遞減。

In this thesis, we firstly discuss the monotonicity of S_{p}(x) and pi_{p} in p. Here S_{p} is the inverse function of the integral
x=\int^{S_{p}(x)}_{0}(1-|t|^{p})^{-\frac{1}{p}}dt, p>1
and pi_{p}=2\int^{1}_{0}(1-t^{p})^{-\frac{1}{p}}dt.
We revise the mistake that Binding made ,and prove the function S_{p} and pi_{p} decrease in p.
In the second part, we extend the definitions of S_{p} and pi_{p} to S_{p,q} and pi_{p,q}, and discuss the relative differential equations and the monotonicity in p and q. Here S_{p,q} is the inverse function of the integral x=\int^{S_{p,q}(x)}_{0}(1-|t|^{q})^{-\frac{1}{p}}dt ,p,q>1
and pi_{p,q}=2\int^{1}_{0}(1-t^{q})^{-\frac{1}{p}}dt. We find that S_{p,q}, S_{p} and sin(x) have similar property, and prove the following equivalent conditions
\begin{enumerate}

(a)\forall x\in[-\frac{ i_{p,q}}{2},\frac{ i_{p,q}}{2}]

x=\int^{S_{p,q}(x)}_{0}(1-|t|^{q})^{-\frac{1}{p}}dt.\forall x \in \mathbb{R}
S_{p,q}(x+k i_{p,q})=(-1)^{k}S_{p,q}(x).

(b)S_{p,q}(0)=0,S'_{p,q}(0)=1
|S_{p,q}|^{q}+|S'_{p,q}|^{p}=1
(c)S_{p,q}(0)=0,S'_{p,q}(0)=1
(|S'_{p,q}|^{p-2}S'_{p,q})'+\frac{q(p-1)}{p}|S_{p,q}|^{q-2}S_{p,q}=0

Finally, we also prove that pi_{p,q} and S_{p,q}( i_{p,q}t),\forall t\in[0,1] are decreasing in p,q.

1 Introduction......................... 1
2 pi_{p}和S_{p}的單調性..................7
2.1 pi_{p}單調性.........................8
2.2 S_{p}單調性......................... 9
3 Proof of Theorem 1.2 .................12
4 pi_{p,q}與 S_{p,q} 的性質..............16
4.1 pi_{p,q}的極值...................... 17
4.2 pi_{p,q}對p,q的單調性................20
4.3 S_{p}對p,q的單調性...................21

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