首先第一章，先介紹Ostrowski不等式令 f: [a，b] → R 在 [a，b] 上是一個有界變分的函數。則下列不等式

 |∫_a^b▒〖f(x)  dx-(b-a)f(x)〗|≤[1/2 (b-a)+|x-(a+b)/2|] V_a^b (f)   

對於每一個 x 在 (a，b)上都成立，這裡的 V_a^b (f) 是 f 在 [a,b] 上的全變分。
第二章，我們介紹一些已建立有關於Ostrowski型的不等式。
第三章，我們要展示我們所建立的Ostrowski不等式。
第四章，我們要介紹一些特殊的加權的 Ostrowski不等式和一些特殊的改良的 Ostrowski 不等式。我們得到了幾個重要的不等式。像是不等式在有界變分函數之下加權的梯形積分及在有界變分函數之下 ‘‘加權的 Ostrowski’’ 不等式。
最後，我們要介紹特殊平均數應用在我們的主要結果上。

In this dissertation, it consists of five chapters.
In the first chapter, we introduce Ostrowski inequality for function of bounded variation. The inequality 
|∫_a^b▒〖f(x)  dx-(b-a)f(x)〗|≤[1/2 (b-a)+|x-(a+b)/2|] V_a^b (f)
holds for all x∈(a,b)  where f: [a,b] → R is a mapping of bounded variation on [a,b] and V_a^b (f) is the total variation of f on the interval [a,b].
In the second chapter, we introduce Some established Ostrowski's type inequalities.
In the third chapter, we present some refinements of Ostrowski inequalities.
In the forth chapter, we present some particular weighted ostrowski inequality and some particular integral of improved ostrowski Inequality. We get some important results. Some inequalities like the weighted trapezoid inequality for mappings of bounded variation and the ‘weighted Ostrowski inequality for mappings of bounded variation.
Finally, we discuss Some Particular integral inequality about my main results.

目錄
第一章  導論 ..............................................1
1.1 簡介...................................................1
1.2 Ostrowski 不等式.......................................2
第二章 一些已建立的 Ostrowski’s型不等式...................5
2.1由 Dragomir 所建立的結果................................5
2.2由Tseng, Hwang 和 Dragomir， Tseng 和 Hwang所建立的結果.9
第三章 一些加權的 Ostrowski 不等式的改良..................13
3.1在函數為有界變分下，加權之 Ostrowski’s 型不等式.......13
3.2在函數為有界變分下，改良之Ostrowski’s 型不等式........23
第四章 一些特殊的積分不等式...............................33
4.1一些特殊的加權的 Ostrowski不等式.......................33
4.2一些特殊的改良的 Ostrowski不等式.......................40
第五章 特殊平均數應用 ....................................44
5.1 一些特殊平均數........................................44
5.2一些特殊平均數的應用...................................46
參考文獻..................................................49

Contents
Chapter 1.  Introduction..................................52
1.1 Introduction .........................................52
1.2 Ostrowski Inequality..................................53
Chapter 2.Some established Ostrowski's type Inequalities..55
2.1 Inequality established by Drangomir...................55
2.2 Inequality established by Tseng, Hwang and Drangomir..57
Chapter 3. Some Refinements of Ostrowski Inequality.......62
3.1 Weighted Ostrowski Inequality for mappings of bounded variation.................................................62
3.2 Improved Ostrowski Inequality for mappings of bounded variation.................................................72
Chapter 4. Some Particular integral inequality............82
4.1 Some Particular Weighted Ostrowski Inequality.........82
4.2 Some Particular Improved Ostrowski Inequality.........88
Chapter 5. Applications to special means..................92
5.1 Some special means....................................92
5.2 Some applications to special means....................93
References................................................97

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